Analysis of many-body methods for quantum dotsfolk.uio.no/simenkva/simenkvaal_thesis_final.pdf ·...

Analysis of many-body methodsfor quantum dots

Simen Kvaal

Centre of Mathematics for Applicationsand

Department of PhysicsUniversity of Oslo

November 2008

Dissertation presented for the degree ofPhilosophiae Doctor (PhD)

Happy is he who gets to know the reasons for things.

— Virgil, Georgics book II

Preface

Writing a PhD thesis is no walk in the sun. But I don’t think it should be either, because youlearn a lot more that way, and at times when doing research is a struggle uphill, the reward ofattaining new understanding and connecting a few more dots is profoundly rewarding. Thefeeling of illumination is priceless. So, looking back, it has been an experience I wouldn’twant to be without. It would not be possible without the help of others, however.

First of all, I would like to thank my supervisor Morten Hjorth-Jensen for his friendship,support, enthusiasm and advice during the last years. Through our countless discussions inhis office or in picturesque Italian villages, I have learned a lot. He truly believes in me, andfor that I am very grateful.

I would also like to thank my secondary supervisor Ragnar Winther, CEO at CMA, aswell as senior adviser Helge Galdal at CMA, for continuous support, both financially andotherwise. At times when life has been less enjoyable, they have shown great kindness andunderstanding, cutting me some slack.

I have great colleagues and friends at CMA. I would especially like to thank formercolleagues Halvor Møll Nilsen, Per Christian Moan and Jan Brede Thomassen for their col-laboration, friendship, advice, and for the discussions we’ve had. Per Christian and Jan:Hopefully our Skyrme model project will be continued next year. Tore G. Halvorsen, partnerin crime at CMA (the crime being physics), has been responsible for at least one laugh eachday. I would also thank Dr. Giuseppe M. Coclite of University of Bari, Italy, who has beenvery helpful at the occasions he has visited CMA.

The constant encouragement of my many friends has been a comfort to me when I havebeen working at its hardest. To all of you, I’m really looking forward to going to the movieswithout thinking about electrons, having some brain capacity left for a board game, andhaving a beer at a friday quiz again.

When I was nine or ten years old, my father introduced me to BASIC. The shelves inthe basement were also filled with strange and wonderful books on physics, chemistry andprogramming. To this day, I return to the basement for bedtime reading when I am visiting.So thank you, dad, for spurring my imagination!

Mom, dad and my sisters Camilla and Mathilde have been with my all of my life, andwithout their love and support, I would not be who or where I am today.

Last, but not at all least, Hilde, my better half, love of my life, I am profoundly gratefulfor your love and support. It must have been quite lonely at times, me being holed up in mycave and barely dropping by for some hours of sleep. Yes, I am going to fix those clotheshooks now, if you make some chili.

November 2008 Simen KvaalOslo

Contents

I Introduction 1

1 Background and theory 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Second quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Properties of harmonic oscillator functions . . . . . . . . . . . . . . . . . . 111.5 Properties of the spectrum and eigenfunctions . . . . . . . . . . . . . . . . 141.6 Variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Many-body techniques 192.1 Configuration interaction method . . . . . . . . . . . . . . . . . . . . . . . 192.2 Effective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 Scaling and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 Hartree-Fock method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Coupled cluster method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Introduction to the papers 313.1 A reader’s guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Discussion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Bibliography 39

II Papers 41

1 Paper 1 43

2 Paper 2 57

3 Paper 3 75

4 Paper 4 85

Part I : Introduction

1 Background and theory

1.1 INTRODUCTION

On the surface, this thesis is about quantum dots. However, I admit that this is more or lessan excuse to study many-body techniques, which I consider to be the real topic of the thesis.Even though some of the quantum dot applications of many-body methods are novel, thequantum dot-interested reader may be disappointed, while I think the many-body techniques-interested on the other hand will find more of interest. The research has been motivated notprimarily by fascination of quantum dots, but rather the ab initio methods that are used tostudy them.

Ab initio many-body theory is, in many ways, a mature field, ranging from advanced per-turbation theoretical approaches to Monte Carlo techniques, via coupled cluster methods andfull configuration interaction. On the other hand, many-body theory is notoriously difficult,with many pitfalls and with, for the uninitiated, an obscure and esoteric way of speakingand formulating problems. In some cases, it is obvious that the theory still lacks a soundmathematical foundation, and that there are still many open yet very important questions topose as well as answer. In addition, there is the curse of dimensionality, which makes goodmethods so hard to invent.

For this reason, I have focused on two main many-body techniques, and tried to do a thor-ough study of their properties when applied to quantum dots: Full configuration-interactionmethod (FCI, or “exact diagonalization”) in a harmonic oscillator basis, and sub-cluster ef-fective interactions as employed in recent years’ no-core shell model calculations in nuclearphysics. The reason for choosing quantum dots as subject of application is, (a) that it is inter-esting in itself, and (b), that harmonically confined quantum dots can be viewed as simplifiednuclear systems, with lower spatial dimensionality and a well-defined, well-known interac-tion term. The knowledge gained should then have wide applicability in both quantum dotstudies and nuclear physics, as well as some quantum chemistry problems. The reason forchoosing the FCI method, was that other methods, such as coupled cluster (CC) and Hartree-Fock (HF) methods, are closely related to the FCI method. Insight can then be gained forthese methods as well.

Another reason for studying the FCI method is the prospect of accurately computingthe time-evolution of a system with a time-dependent Hamiltonian, such as a quantum dot


manipulated with electromagnetic fields. The period of time for which such a calculationwould be valid depends on the accuracy of the eigenvalues (or more precisely, their spacing).Except for the equation-of-motion coupled cluster method (see Section 2.5), FCI is the onlyab initio method that gives, in principle, all the eigenvalues and eigenfunctions of a many-body system.

The central part of this thesis deals with the analysis of the approximation properties ofharmonic oscillator eigenfunctions, i.e., many-dimensional Hermite functions. As these areusually used as basis functions in the FCI and CC calculations, and in many HF-calculationsas well, detailed knowledge of such properties is essential for understanding the numericalbehaviour of virtually every application, in both nuclear physics and quantum dot studies.Surprisingly, except for important work on ordinary one-dimensional Hermite functions byHille (1939), studies of hermite functions are hard to find in the literature. The analysis,carried out for the most part in Paper 2, results in unambiguous a priori error estimates forexpansions of functions in a harmonic oscillator basis, as well as error estimates for FCIcalculations.

Effective interactions are widely used in nuclear physics, and applying them to the sim-pler quantum dots could give great insight into the method, which is hard to obtain fromthe nuclear studies. Surprisingly, few studies of this kind have been done, even though itis fairly obvious that they are needed. As is shown in Papers 1 and 2, the quantum dotFCI calculations obtain a dramatic increase in convergence when using a two-body effectiveinteraction.

Paper 3 reformulates and simplifies parts of the standard effective Hamiltonian theory,and points out a simpler and more transparent procedure than the one used today for comput-ing the effective Hamiltonian used in no-core shell model calculations. I apply this procedurein all the other papers, and also give a thorough discussion of the two-body effective interac-tion in Paper 4.

This thesis is organized as follows. Part I continues with some preliminaries. In Section1.2 we give a brief introduction to quantum dots. Following that, we discuss the formalismof second-quantization needed for discussing many-body theory in Section 1.3. Because oftheir importance, a gentle introduction to the properties of harmonic oscillator basis functionsis given in Section 1.4; these are discussed in detail in Paper 2. Section 1.5 discusses someproperties of the quantum dot Hamiltonian compared to the molecular Hamiltonian, and wediscuss the variational principle in Section 1.6.

We are then ready to discuss many-body techniques in Chapter 2, starting with the FCImethod in Section 2.1 before discussing effective Hamiltonians and interactions in Section2.2. Section 2.3 discusses the scaling of the FCI method when a parallel implementation isconsidered, and Sections 2.4 and 2.5 briefly discuss the Hartree-Fock and coupled clustermethods, respectively.

Part II of the thesis contains the journal publications, referred to as Paper 1 through 4 inPart I.

1.2 Quantum dots 5

1.2 QUANTUM DOTS

During the last twenty years, the research activity on so-called quantum dots has grownrapidly.1 Put simply, quantum dots are nanometre-scale semiconductor devices confining afew electrons spatially. The quantum dots we consider here, so-called lateral quantum dots,are usually fabricated by etching techniques. Such methods are, despite their sophistication,essentially macroscopic. Still, the resulting structures are small enough to allow behaviourin the quantum mechanical regime of the confined electrons, such as shell structure (Taruchaet al. 1996) and entanglement (Weiss et al. 2006). Moreover, the confinement parametersmay be directly controlled via applied electromagnetic fields. For this reason, quantum dotsare often called “designer atoms”, and it is clear that these offer exciting possibilities to studyelementary quantum mechanics and the transition to classical mechanics, direct manipulationand imaging of quantum states (Fitzgerald 2003), and so on.

For a general introduction to the topic, see the popularized article by Reed (1993) andthe review by Reimann and Manninen (2002).

In Figure 1.1 a micrograph of a so-called double quantum dot (DQD) is shown. Thestarting point is a two-dimensional electron gas layer (2DEG), here a planar sheet of AlGaAssandwiched between insulating GaAs layers. Metal gate electrodes are then applied to thetop surface using lithographic etching techniques. Adjusting the voltage over some of thesegates will define a double-well potential that will squeeze and confine a few electrons in the2DEG, which otherwise were free to move in the plane. Other gates can again be used toload and remove single electrons from the DQD.

The DQD is actually a system of two lateral quantum dots coupled together. A modelfor this and similar quantum dots can be obtained in the effective mass approximation as asystem of N interacting electrons of effective mass m∗ ≈ 0.067me in a double-well potentialin d = 2 spatial dimensions. Such a double-well potential may be given by

V(~r) =12

m∗ω2 min‖~r−~a‖2,‖~r +~a‖2 ,

where ±~a are the locations of the minima of the well, and ω is a parameter defining thecharacteristic length unit a0 of the well by a0 =

√h/mω. Typical sizes are a0 ≈ 20 nm.

Whenever ‖~r‖ ‖~a‖, the potential is essentially a harmonic oscillator. The Hamiltonian forthis system reads

H =

[N

∑i=1− h2

2m∗∇2i +V(~ri)

]+

e2

4πε0ε∑i< j

1ri j, (1.1)

where e2/4πε0 ≈ 1.44 eVnm, and where ε ≈ 12 is the dielectric of the surrounding semicon-ductor bulk. The electrons carry a spin degree of freedom σ∈ ±1, and the total wavefunc-tion for the system must be anti-symmetric with respect to interchanges of any two particles’

1A search at ISI Web of Science reveals that the number of publications with topic “quantum dots” isapproximately n≈ 0.4(y−1985)3.1, where y is the year of publication.


Figure 1.1: Scanning electron micrograph of a double quantum dot defined by electrostatic gates(expanded area) applied to a sandwich of GaAs/AlGaAs/GaAs, surrounded by a loop antenna thatproduces a magnetic field. Also shown is a sample holder optimized for radio frequency measure-ments on the double quantum dot. Image courtesy of the Nanophysics Group at the University ofMünchen, Germany.

coordinates xi = (~ri,σ1) and x j = (~r j,σ j).Figure 1.2 illustrates the quantum dot model. The double-well potential is depicted along

with some (classical) electrons. The motion of classical electrons would be that of slidingon the surface (viewed from above), taking into account the Coulomb repulsion between theelectrons.

In this thesis, we shall almost exclusively discuss the so-called parabolic quantum dot, forwhich ~a = 0. In this case, V(~r) = m∗ω2‖~r‖2/2 which is a pure harmonic oscillator. Introduc-ing the energy unit hω and length unit a0, the Hamiltonian can be written on dimensionlessform, viz,

H =

[N

∑i=1−1

2∇

2i +

12‖~ri‖2

]+λ∑

i< j

1ri j, (1.2)

where the interaction strength parameter λ is given by

λ=e2

4πε0ε

1hωa0

=e2

2πε0ε

m∗ah2 ≈ 2.11.

This is a typical value in real quantum dots.One of the possible uses for quantum dots is as basic constituents of future quantum

computers. In one of the first proposed schemes, now a classic reference, Loss and DiVin-cenzo (1998) proposed to utilize the spins of coupled single-electron quantum dots (such as

1.3 Second quantization 7

~r1

~r2~r3

Figure 1.2: Illustration of quantum dot model. N = 3 electrons with coordinates~ri move in a potentialtrap V(~r) (gold wireframe) and interact via the Coulomb force; illustrated with squiggly lines. Theclassical motion of the electrons can be pictured by the rolling of small balls in a bowl, with additionalrepulsive forces

the DQD with N = 2 electrons) to encode qubits; the quantum-counterpart of a digital bit.However, one of the main obstacles turns out to be that of decoherence: Our model is overlysimplistic, as small irregularities and interactions with the environment (for example, due to anonzero temperature or the nuclei in the semiconductor bulk) breaks down the painstakinglyengineered state of the system after very short time (Cerletti et al. 2005).

Our model Hamiltonian (1.1) or (1.2) should therefore not be interpreted as the Hamilto-nian, but instead as a test case that must at least be tackled before we can predict the outcomeof more realistic cases. Moreover, such an idealized model is very well suited for analysis ofmany-body methods.

1.3 SECOND QUANTIZATION

The quantum dot model described in the preceding section is very well suited for many-bodytechniques like full configuration interaction, Hartree-Fock, and the coupled cluster method.Before we can describe these methods, we will need to rewrite the Hamiltonian (1.2) usingsecond quantization. This is a formalism which in an elegant way gets rid of difficulties suchas wave-function symmetry and dependence on the number of particles N. The formulationof the many-body techniques then becomes much simpler.

The uninitiated may find this exposition rather brief, and I refer to any good text on many-body quantum mechanics, such as the books by Raimes (1972) and Lindgren and Morrison(1982), for details. The book by Ballentine (1998) also gives an introduction to the topic ina more mathematical way, discussing advanced concepts such as rigged Hilbert spaces.

Our goal is to find a few of the lowest eigenvalues of the Hamiltonian (1.2), i.e., weseek real numbers Ek and wavefunctions Ψk = Ψk(x1, x2, · · · , xN) (where xi = (~ri,σi), with


σi ∈ +,− being the electron spin degree of freedom) such that

HΨk(x1, x2, · · · , xn) = EkΨk(x1, x2, · · · , xN).

The wavefunction Ψk is sought in the Hilbert space of anti-symmetric N-body functions, viz,

Ψk ∈HN := H1∧H1∧·· ·∧H1,

where the single-particle space H1 is the space of square integrable functions over Rd×±,i.e.,

H1 := L2(Rd×±)≈ L2(Rd)⊗C2,

that is, C2-valued L2-functions.Given a complete orthonormal set φα(x)∞

α=1 in H1, a corresponding set in HN is con-veniently given by Slater determinants. These are defined by

Φα1,··· ,αN (x1, · · · , xN) :=1√N!

∣∣∣∣∣∣∣φα1(x1) · · · φα1(xN)

......

φαN (x1) · · · φαN (xN)

∣∣∣∣∣∣∣ ,where

α1 < α2 < · · · < αN

is imposed to make the determinants unique. The indexing of φα(x) by α may be arbitrary,as long as we have an ordering.

It is common to introduce a coordinate-free notation for wavefunctions using the so-called ket-notation, viz, |Ψ〉, |Φα1,··· ,αN 〉, etc. One introduces a “basis” of kets |x1, x2, · · · , xN〉so that any |Ψ〉 ∈HN , may be expanded as

|Ψ〉=∫

dx1 · · ·dxN |x1, · · · , xN〉〈x1, · · · , xN |Ψ〉,

where

Ψ(x1, · · · , xN) = 〈x1, · · · , xN |Ψ〉,

i.e., the wavefunction’s value at (x1, · · · , xN) is the component of |Ψ〉 along the “basis vector”|x1, · · · , xN〉. The integral

∫dxi is defined as∫

dxi := ∑σi

∫ddr.

We will furthermore sometimes use ket-notation |Ψ〉 for Hilbert space elements, and some-

1.3 Second quantization 9

times plain function-notation Ψ(x1, · · · , xN), depending on the context.Let us introduce the so-called Fock space HFock, which embodies the idea of a Hilbert

space of functions containing arbitrary many particles. Simply by defining Slater determi-nants of N and N′(6= N) particles to be orthogonal to each other, we define

HFock :=∞⊕

N=0

HN .

The space H0 is defined as an arbitrary one-dimensional space, which is interpreted as con-taining zero particles. We pick a normalized |−〉 ∈ H0 and call it the vacuum; the single“Slater determinant” of zero particles. So-called creation and annihilation operators c†

α andcα (being Hermitian adjoints of each other) are defined for each orbital φα(x). They aredefined such that

|Φα1,··· ,αN 〉= c†α1

c†α2· · ·c†

αN|−〉. (1.3)

Thus, c†α : HN →HN+1 and cα : HN →HN−1. The important anti-commutator relations

cα,cβ= c†α,c

†β= 0 and c†

α,cβ= δα,β

follow from from Eqn. (1.3).As a simple example of the usefulness of cα and c†

α, consider the particle number opera-tor N defined by

N := ∑α

c†αcα,

where the caret notation distinguishes it from N. It is relatively easy to see that for any Slaterdeterminant

N|Φα1,··· ,αN 〉= N|Φα1,··· ,αN 〉,

that is, the Slater determinants are the eigenvectors of N with eigenvalue N. The eigenspaceis simply HN .

The Hamiltonian (1.2) can also be written in terms of c†α and cα as

H = ∑αβ

hαβc†αcβ+

12 ∑αβγδ

uαβγδc†αc†

βcδcγ, (1.4)

where the order of the indices in the last term should be noted. The matrix elements hαβ and

uαβγδ are defined by

hαβ :=∫

dx φα(x)(−1

2∇

2 +12‖~r‖2

)φβ(x),


and

uαβγδ := λ

∫dx1dx2 φα(x1)φβ(x2)

1r12

φγ(x1)φδ(x2).

The new Hamiltonian (1.4) is seen to be independent of N, and is given naturally as an oper-ator in the Fock space HFock. As such, it is seen as a particle number conserving operator –the subspace HN ⊂HFock is invariant under H, viz,

[H, N] = 0.

This is seen by noting that the terms of H create and destroy the same number of particles.Notice, that the second-quantized Hamiltonian is really defined as an infinite matrix withrespect to the Slater determinants.

We will take the single-particle orbitals φα(x) to be eigenfunctions of the single-particleHarmonic oscillator. In d = 2 spatial dimensions, the indices α are then mapped one-to-oneon the set (n,m,σ), where n ≥ 0 and m are integers. The single-particle functions take theform φα(x) = ϕn,m(~r)χσ, where χσ are the spin-dependent basis functions; a basis for C2.The functions ϕn,m(~r) are called Fock-Darwin orbitals, and are given in polar coordinates by

ϕ(n,m)(r, θ) := Neimθr|m|L|m|n (r2)e−r2/2, (1.5)

where N is a normalization constant, and where Lkn(x) is the generalized Laguerre polyno-

mial. The corresponding eigenvalues are εα = 2n + |m|+ 1. Notice, that we then obtainhαβ = εαδα,β.

The presentation of second-quantization given here is sketchy and omits some importantdetails, such as whether it is possible at all to write Eqn. (1.2) as a matrix with respect to thechosen Slater determinant basis. For example, the bare interaction in the nuclear shell modelis so strong that the matrix elements in the chosen basis are infinite, and one must do furtheranalysis. Moreover, H is not even defined on the whole of HN . As it stands, H is a formalsecond-order differentiation operator, which is unnecessary restricting. In abstract partialdifferential equation analysis it is common to introduce a weak formulation which makes Ha self-adjoint operator on a subspace HN , with a different norm. Typically, this would be thespace of weakly differentiable functions. We touch upon this in the next section.

In this thesis, we mostly follow the physicist notation and formalism, in accordance withvirtually all standard quantum mechanics texts. Ballentine (1998) gives a further accountof the mathematical foundation of quantum mechanics. See Weidmann (1980) for a discus-sion of operators defined in terms of infinite matrices, and Babuska and Osborn (1996) fora discussion of the eigenvalue problem of self-adjoint operators and its variational formula-tion and subsequent approximation by the Ritz-Galerkin method. This is the FCI method,and using the Fock-Darwin single particle orbitals, it can be shown to fit perfectly into theframework so that convergence estimates hold for quite general interaction potentials. (Seehowever Section 1.5.) With a little goodwill, this justifies the physicist’s common view of an

1.4 Properties of harmonic oscillator functions 11

operator as a matrix, even in the infinite dimensional case.

1.4 PROPERTIES OF HARMONIC OSCILLATOR FUNCTIONS

As stated in the Introduction, an important part of the thesis is the analysis of the harmonicoscillator (HO) eigenfunctions in Paper 2 (with some surface-scratching in Paper 1), whichgeneralize the standard Hermite functions. In this Section, I will give a general introductionto the ideas and concepts used to derive approximation properties. Hille (1939) gives a verydetailed analysis of expansions in standard Hermite functions, but it is not easy to generalizehis work to arbitrary dimensions, as the results are not proven or stated within the moremodern framework of weak differentiation.

The standard Hermite functions are defined by

φn(x) := [2nn!√π]−1/2Hn(x)e−x2/2,

where Hn(x), n = 0,1, · · · are the usual Hermite polynomials. These obey the recurrencerelation

Hn+1(x) = 2xHn(x)−2nHn−1(x), (1.6)

where we take H−1 ≡ 0 and where H0(x) = 1.Now, the functions φn(x) constitute a complete orthonormal sequence in L2(R), and they

are eigenfunctions of the one-dimensional HO Hamiltonian HHO defined by

HHO :=−12∂2

∂x2 +12

x2 = a†xax +1,

where

ax :=1√2

(x+

∂

∂x

), and a†

x :=1√2

(x− ∂

∂x

)are called ladder operators. The eigenvalue of φn is n + 1/2, as is easily verified. The mostimportant feature of a†

x and ax is that, for all n,

a†xφn(x) =

√n+1 φn+1(x) and axφn(x) =

√n φn−1(x).

This can be shown rather easily by using the recurrence formula (1.6). In Figure 1.3 a fewof the Hermite functions are shown, translated vertically according to their HO energy.

If we consider an arbitrary ψ∈ L2(R), it has a unique expansion in the Hermite functions,viz,

ψ(x) =∞

∑n=0

〈φn,ψ〉φn(x) =∞

∑n=0

cnφn(x).


−5 0 50123456789

101112

x

φn(

x)

Figure 1.3: The ten first Hermite functions φn(x) (blue) plotted together with the harmonic potentialx2/2 (green). Each φn(x) is shifted vertically by n+1/2 for easy identification. This shift is equal tothe eigenvalue of φn(x). Notice the penetration into the classically forbidden region by the Gaussiantail for each φn(x).

The coefficients satisfy

‖ψ‖2 =∞

∑n=0

|cn|2 < +∞, (1.7)

which automatically implies that

|cn| −→ 0 as n−→ ∞.

In fact, by the comparison test, |cn| must fall off faster than n−1/2, viz,

|cn|= o(n−1/2).

If we assume, that for a given ψ ∈ L2(R), also a†xψ ∈ L2(R), we obtain

‖a†xψ‖2 =

∞

∑n=0

(n+1)|cn|2. (1.8)

Comparing this with Eqn. (1.7), we see that the coefficients |cn| in this case must fall offfaster, i.e.,

|cn|= o(n−1).

Moreover, we can easily see that a†xψ ∈ L2(R) if and only if xψ ∈ L2(R) and ∂xψ ∈ L2(R).

The first is equivalent to ∂xψ ∈ L2(R), where the caret denotes the Fourier transform. Here,

1.4 Properties of harmonic oscillator functions 13

∂x is a shorthand for the partial derivative.The notion of derivative we use here is the so-called weak derivative, which generalizes

the classical one. The advantage of the weak derivative is that the space of k times weaklydifferentiable functions Hk(R) becomes a Hilbert space. This space is defined by

H1(R) :=ψ ∈ L2(R) : ∂ j

xψ ∈ L2(R) ∀ j≤ k

(1.9)

with inner product

〈ψ1,ψ2〉Hk :=k

∑j=0〈∂ j

xψ1,∂jxψ2〉. (1.10)

The spaces Hk(R) can be generalized to arbitrary domains Ω⊂Rn using weak partial deriva-tives of order ≤ k. The definition of the weak derivative is not entirely trivial, and we referto Evans (1998) for a general introduction. See also the discussion in Paper 2.

If ψ(x) is known a priori to decay exponentially fast as |x| →∞, we obtain the followingproposition:

Proposition 1 Let ψ∈ L2(R). Assume that, for all m> 0, xmψ∈ L2(R) (implied by exponen-tial decay). Then, (a†

x)kψ ∈ L2(R) if and only if ∂k

xψ ∈ L2(R), i.e., ψ ∈ Hk(R). Equivalently,

∞

∑n=0

nm|cn|2 < +∞. (1.11)

The latter implies, in particular, that

|cn|= o(n−(k+1)/2). (1.12)

In numerical applications, one usually does not observe an integral or half-integral expo-nent, as in Eqn. (1.12). Assume that ψ∈Hk(R) but ψ /∈Hk+1(R), i.e., ψ can be differentiatedat most k times. The upper bound on the decay rate is now |cn|= o(n−(k+2)/2). This explainsthat the actual behaviour observed is often

|cn|= o(n−(k+1+ε)/2), 0≤ ε < 1.

Proposition 1 has an analogue for Fourier series, which is perhaps more well-known. Letψ ∈ L2[0,2π] be given by a Fourier series, viz,

ψ(x) =∞

∑n=−∞

cneinx√

2π.


Assume now that ψ ∈ H1[0,2π]. Differentiating the function exp(inx)/√

2π, we obtain

∂xψ(x) =∞

∑n=−∞

(in)cneinx√

2π

with norm given by

‖∂xψ(x)‖2 =∞

∑n=−∞

n2|cn|2,

which should be compared with Eqn. (1.8). An interesting point here is that differentiationin the Fourier case gives a weight n2 in the sum, while the Hermite case only gives n1.Notice that the Fourier basis functions only approximate over a bounded interval [0,2π],while the Hermite functions must deal with the whole line R. Consequently, the width of theoscillations of einx/

√2π decreases faster than those of φn(x). Intuitively, einx/

√2π resolves

details in the function ψ(x) more efficiently.In Paper 2 Proposition 1 is generalized to n-dimensional Hermite functions, being defined

as tensor products of n one-dimensional Hermite functions. These constitute an orthonormalsequence for L2(Rn), and are eigenfunctions for the n-dimensional HO Hamiltonian, i.e.,

HHO =−12

∇2 +

12‖~r‖2 =

n

∑i=1

(−1

2∂2

∂r2i

+12

r2i

)=

n

∑i=1

(a†

i ai +1).

The n-dimensional results are completely analogous to the one-dimensional case, giving arapid decay of the expansion coefficients if and only if ψ is many times (weakly) differen-tiable and decays sufficiently fast as ‖~r‖→ ∞. Let me comment that the Slater determinantsare eigenfunctions of HHO with n = Nd, which automatically gives a starting point for theapproximation properties of these as well.

The relevance of Proposition 1, is that when using HO eigenfunctions as basis in amany-body calculation, the convergence rate with respect to the basis size (as representedby, for example, the truncation parameter R in Section 2.1) depends precisely on how fastthe coefficients of the exact eigenfunction (or more precisely, the spin component functions)Ψ(~r1, · · · ,~rN) fall off, which in turn is equivalent to the smoothness properties of the eigen-function.

1.5 PROPERTIES OF THE SPECTRUM AND EIGENFUNCTIONS

We briefly mention some of the results from the spectral theory of Schrödinger operators andthe analytic properties of electronic wavefunctions. In the light of the above discussion, thelatter is especially interesting. The material is taken mostly from Hoffmann-Ostenhof et al.

1.5 Properties of the spectrum and eigenfunctions 15

continuous spectrum

resonanceisolated eigenvalue

0

Figure 1.4: The spectrum of a molecular Hamiltonian in the Born-Oppenheimer approximation.There is a set of discrete, isolated eigenvalues Ek < 0, which may be infinitely many, and a continuousspectrum for E > 0. Moreover, there may be resonances, i.e., embedded discrete eigenvalues in thecontinuous spectrum

(1994) and Hislop and Sigal (1996).The spectral theory of unbounded operators (such as H) is complex. There is a large

amount of literature devoted to this, but most of the quantum mechanics-related literature isdevoted to molecular problems, where the Hamiltonian describes a system of atomic nucleiand electrons in d = 3 dimensions. The central result states that the molecular Hamilto-nian in the Born-Oppenheimer approximation (in which the positions of the nuclei are heldfixed) has a point spectrum σp ⊂ (−∞,0) (identical to the isolated eigenvalues and possi-bly empty) and a continuous spectrum σc = [0,+∞), consisting of approximate eigenvalues(whose “eigenvectors” are simply delta-function normalizable eigenfunctions, such as planewaves or the basis vectors |x1, · · · , xN〉 of Section 1.3) and possibly embedded eigenvalues,i.e., resonances. This is illustrated in Figure 1.4. These concepts are perhaps most easilyunderstood in the framework of rigged Hilbert space (Ballentine 1998).

When it comes to quantum dot Hamiltonians, the problem simplifies on one hand, but be-comes more complicated on the other. First, the presence of the confining harmonic potentialmakes the continuous spectrum empty: The spectrum consists entirely of ordinary eigenval-ues. Second, the two-dimensional Coulomb interaction actually violates the assumptionscommon in the literature (Hislop and Sigal 1996), namely that

1‖~r‖

∈ L2(Ω), ∀Ω ∈ R3,

where Ω is assumed to be a bounded subset. To see this, note that the only non-trivialdomains to check in both d = 2 and d = 3 are arbitrary domains containing ~r = 0. Considerthe unit sphere Ω = ~r : ‖~r‖ ≤ 1 for which we obtain the norm

∥∥‖~r‖−1∥∥2L2(Ω) =

∫Ω

(1‖~r‖

)2

d3r = 4π∫ 1

0

1r2 r2 dr = 4π <+∞.


In d = 2 dimensions, we obtain

∥∥‖~r‖−1∥∥2L2(Ω) =

∫Ω

1‖~r‖2 d2r = 2π

∫ 1

0

1r2 r dr = ∞.

Thus, the Coulomb interaction is, in a sense, too strong in two dimensions, and we shouldreally consider the problem further before going about and diagonalizing using the FCImethod, since the variational formulation of the eigenvalue problem is, strictly speaking,not valid without further analysis (Babuska and Osborn 1996).

On the other hand, as argued in Paper 2, the known exact solutions in the two-particlecase behave virtually identically in both d = 2 and d = 3. Moreover, Hoffmann-Ostenhofet al. (1994) have thoroughly analyzed the local analytic behaviour of many-electron wave-functions in d = 3 dimensions. The central result is that near a so-called coalesce pointξCP = (~r1, · · · ,~rN), where at least one ri j = 0, the wavefunction’s 2N spin components all canbe written on the form

Ψ(ξ+ ξCP) = ‖ξ‖kPk

(ξ

‖ξ‖

)(1+a‖ξ‖)+O(‖ξ‖k+1),

where Pk is a hyperspherical harmonic (a polynomial on the sphere S 3N−1) of degree k.Away from ξCP, Ψ(ξ) is real analytic. The number k varies for coalesce points where adifferent number of the ri j vanish. Thus, it is immediate, that Ψ ∈ Hmin(k)+1(R3N).

The actual behaviour of the FCI results fits very well with this for d = 2 dimensions aswell, indicating that the technicality with respect to the interaction should resolve. The FCIcalculations in Paper 2 clearly indicate that Ψ ∈ Kk+1(R2N) for some k ≥ 0, where k variesfor different eigenfunctions.

1.6 VARIATIONAL PRINCIPLE

We now discuss the variational principle, which is heavily relied upon by many many-bodytechniques in some way or another. The treatment by Babuska and Osborn (1996) is moredetailed, but for our purposes the present discussion is sufficient.

Let |Ψ〉 ∈HN . The Rayleigh quotient E[Ψ] is defined by

E[Ψ] :=〈Ψ|H|Ψ〉〈Ψ|Ψ〉

,

whenever this expression is finite. Physically, E[Ψ] is the total energy of the quantum systemin the state |Ψ〉.

Assume that the Hamiltonian is self-adjoint and bounded from below, and that its groundstate is |Ψ0〉 with eigenvalue E0. Any wavefunction |Ψ〉 will satisfy the variational principle

E[Ψ]≥ E[Ψ0],

1.6 Variational principle 17

with equality if and only if |Ψ〉= |Ψ0〉. Thus,

E0 = min|Φ〉∈HN

E[Φ].

Also the other eigenvalues and eigenfunctions can be characterized by similar principles(Babuska and Osborn 1996).

Estimates for the ground state can now be obtained by taking the minimum over subsetsof HN . For example, the FCI ground state |ΨFCI

0 〉 can be characterized as

|ΨFCI0 〉= argmin

|Φ〉∈PE[Φ],

i.e., minimization of the energy when the wavefunction is allowed to be in the model spaceP ⊂ HH only. The Hartree-Fock method to be described in Section 2.4 minimizes E[Φ]over the nonlinear manifold of Slater determinants, when also the single-particle functionsφα(x) are allowed to vary.

2 Many-body techniques

2.1 CONFIGURATION INTERACTION METHOD

The configuration interaction method is perhaps the conceptually simplest of the commonmany-body techniques based on second quantization. It is also the most accurate, in thesense that it converges to the exact solution, and that the other methods are approximations tothe FCI method. The main drawback of FCI is that the problem scales almost exponentiallywith the number of particles N, see Section 2.3. This is called the curse of dimensionalityand is, in fact, the main obstacle and motivation for new many-body methods.

Let us select a finite number of Slater determinants B := |Φi〉mi=1 with N particles.

These span a finite-dimensional space P ⊂HN called the model space. The FCI method isthen simply a Ritz-Galerkin approximation with respect to P (Babuska and Osborn 1996).This is equivalent to the eigenvalue problem of the matrix H whose matrix elements are

Hi, j = 〈Φi|H|Φ j〉, i, j≤ m.

Let us consider two widely used model spaces. Recall that the single-particle functionsφα(x) = ϕn,m(~r)χσ are eigenfunctions of the operator HHO given by

HHO =−12

∇2 +

12‖~r‖2,

which gives hαβ = h(n,m,σ)(n′,m′,τ) = εαδα,β, with ε(n,m,σ) = 2n+ |m|+1 in two-dimensional systems.

Let B be given by

B = B(R) :=

|Φα1,··· ,αN 〉 :

N

∑i=1

εαi −Nd2≤ R

.

Then B is a basis for what is commonly called the energy cut model space, as we includeall the Slater determinants with single-particle energy less than R + Nd/2. Notice that theenergy of a Slater determinant is on the form R′+ Nd/2 where R′ ≥ 0 is an integer.

This is the basis in which I did most of the numerical simulations in the papers. Another


00

4

4

8

8

12

12

16

16

α1

α2 P

P ′

Figure 2.1: Illustration of N = 2 model spaces when εn = n + 1/2. This is equivalent to a d = 1-dimensional harmonic oscillator. The Slater determinants included in each model space are on theform Φα1,α2 . The shaded areas indicate with Slater determinants are included in P , R = 16 (green)and P ′, R = 12 (blue), respectively.

basis B′ is even more common to encounter, defined by

B′ = B′(R) :=|Φα1,··· ,αN 〉 : max

iεαi−

d2≤ R

.

The resulting model space P ′ is much bigger than P , and is often called the direct productmodel space. Figure 2.1 illustrates P and P ′ in the case of N = 2 and εn = n +1/2, i.e., aone-dimensional system.

Which model space to choose is more or less a matter of taste. It is not obvious whichis the better choice with respect to accuracy of the FCI method. Even though dim(P ′)grows much faster with R than does dim(P), simple physical arguments indicate that P

will include “more physically relevant” basis functions on average than P ′. As far as Iknow, no results in this direction have been published.

Papers 2 and 3 discuss the FCI method, properties of the Slater determinant basis func-tions, convergence of the eigenvalues with respect to R, etc, in detail.

Before we move on to the Hartree-Fock method, let us mention the results in Paper 2on the error of the FCI method. For simplicity, we consider a one-dimensional one-particleproblem, and we ignore spin. Let the exact (unknown) ground state eigenfunction ψ(x) ∈Hk(R) (where k is perhaps known from some analysis) be given by

ψ(x) =∞

∑n=0

cnφn(x), |cn|= o(n−(k+1+ε)/2),

where φn(x) are the standard Hermite functions considered in Section 1.4, and where thedecay behaviour of |cn| is discussed in Section 1.4. The model spaces P and P ′ coincide

2.2 Effective Hamiltonians 21

in this case, and

B = B′ = φn(x)Rn=0

is the FCI basis. The main result (Babuska and Osborn 1996) on the Ritz-Galerkin methodstates that the error ∆E in a non-degenerate eigenvalue (i.e., with unit multiplicity) is boundedby

∆E ≤C infφ∈P

‖ψ−φ‖2HO

= C‖ψ−Pψ‖2HO = C

∞

∑n=R+1

(n+12)|cn|2, (2.1)

where P projects orthogonally onto P , and where where C is a constant. We have definedthe HO-norm by

‖ψ‖HO := 〈ψ,HHOψ〉1/2 =

[∞

∑n=0

(n+12)|〈φn,ψ〉|2

]1/2

and used the fact that we expand the numerical wavefunction in eigenfunctions for HHO =(−∂2

x + x2)/2.We can approximate the last sum in Eqn. (2.1) by an integral to obtain

∆E = O(R−(k+ε−1)), (2.2)

which is valid for k = 1 if ε > 0 and for k > 1. Otherwise, the integral approximation makesno sense. It turns out that Eqn. (2.2) is also valid in the general N-particle d-dimensionalcase, as described in Paper 2, assuming that the technicality of the Coulomb interaction intwo dimensions resolves. The error estimate is discussed in more detail further down, inSection 2.3.

2.2 EFFECTIVE HAMILTONIANS

The many-body problem of the atomic nucleus is perhaps the most difficult many-body prob-lem of modern physics: First, it has so many particles that a direct approach starting fromthe degrees of freedom from quantum chromodynamics is almost impossible, except whenvery few particles are present (Ishii, Aoki, and Hatsuda 2007). Second, there are too fewparticles to use statistical mechanics effectively. Furthermore, the fundamental interaction,i.e., the matrix elements uαβγδ are unknown, and one has to rely on phenomenological models.It is known that the interaction is very singular, with a so-called hard core potential growingrapidly towards infinity as ri j → 0; additionally, we have non-central tensor forces, violatingconservation of orbital angular momentum. Finally, there is no natural centre to the system


unlike the quantum dot model or the molecular system.These considerations have led nuclear physicists on a 50-year long search (Bloch 1958;

Bloch and Horowitz 1958) for ways to compute an effective Hamiltonian; a Hamiltonianthat will reproduce exact eigenvalues of the full problem within a finite-dimensional modelspace.

A detailed account of the nuclear effective interaction theory is not possible to give here(for several reasons, of course); see Ellis and Osnes (1977), Hjorth-Jensen et al. (1995) andDean et al. (2004). A lot of the theory also depends on perturbation theoretic approaches(Klein 1974; Shavitt and Redmon 1980; Lindgren and Morrison 1982; Schucan and Weiden-müller 1973). Instead, we will briefly discuss the most basic features of the standard theory.This is reformulated in Paper 3 in a geometric way, which relies heavily on the singularvalue decomposition (SVD), and leads to (in my opinion) a much simpler and transparentformalism.

An effective Hamiltonian Heff, which is not unique, is usually defined as follows: Assumethat HN is finite-dimensional, n = dim(HN), and that P is the projector onto a model spaceP ⊂HN with m = dim(P). Let G be such that

Qe−GHeGP = QHP = 0,

where Q is the orthogonal projector onto the complement of P , i.e., P + Q = 1. Then Heff

is defined by

Heff := PHP.

This is to be interpreted as an operator defined only in P .Since similarity transforms preserve eigenvalues (i.e., H has the same eigenvalues as H),

the m eigenvalues of Heff are identical to some of the n eigenvalues of H. If H has the spectraldecomposition

H =n

∑k=1

Ek|Ψk〉〈Ψk|,

we assume that the eigenvalues Ek are arranged so that E1 through Em are reproduced byHeff.

There are still many degrees of freedom in G to be determined. These fix the effectiveeigenvectors, i.e., the eigenvectors of Heff. Two choices for G are common: An operator onthe form G = ω= QωP defined by

ω(Q|ψk〉) = P|ψk〉, ∀k ≤ m

and the operator

G = artanh(ω−ω†).

2.2 Effective Hamiltonians 23

The first choice is called the Bloch-Brandow choice (Bloch 1958; Brandow 1967), and thesecond is the canonical Van Vleck choice (Van Vleck 1929). The first case leads to a non-Hermitian HBB

eff , while the second a Hermitian Hceff (where the superscript “c” stands for

Canonical, see Paper 3.)Finding the exact Heff is usually out of question. Indeed, it is equivalent to the original

diagonalization problem which we are trying to approximate. The usual way to tackle theproblem is to expand Heff in a perturbation series, with the interaction being the perturbation.To apply perturbation theory one must use the model space P , where the unperturbed modelspace spectrum is separated from the excluded space spectrum. It has been known for sometime, however, that this series is most likely to diverge in virtually all interesting cases in nu-clear physics due to the presence of so-called intruder states, see Schucan and Weidenmüller(1973) and also Schaefer (1974) for a discussion.

The no-core shell model approach (basically FCI) was introduced in the 90’s (Zhenget al. 1993), and today one can successfully handle, for example, the 12C nucleus, i.e., anN = 12 problem (Navrátil, Vary, and Barrett 2000). The usual approach here is to computethe Hermitian effective Hamiltonian for the two-body problem, which can be computed moreor less exactly, and to extract an effective interaction from this. Insertion into the full N-bodyproblem then gives an approximation to the full Hermitian Heff. This procedure is describedin detail in Paper 4. Attempts at creating a three-body effective interaction have also beenundertaken, using the same approach (Navrátil and Ormand 2003).

Let us give a general idea of the procedure. Formally, assume that H = H(2) for the two-body problem has been solved exactly, giving eigenvectors |Ψ(2)

k 〉 and eigenvalues E(2)k . By

computing the two-body effective Hamiltonian in second quantization we obtain

H(2)eff = P(2)

[∑α

εαc†αcα+

12 ∑αβγδ

uαβγδc†αc†

βcδcγ

]P(2),

where P(2) projects onto the two-body model space. The effective interaction matrix ele-ments uαβγδ are then inserted into H(N)

eff , which in general is an N-body operator, viz,

H(N)eff = P(N)

[∑α

εαc†αcα+

1N! ∑

α1···αN

∑β1···βN

vα1···αNβ1···βN

c†α1· · ·c†

αNcβN · · ·cα1

]P(N)

≈ P(N)

[∑α

εαc†αcα+

12 ∑αβγδ

uαβγδc†αc†

βcδcγ

]P(N).

This is easily seen to be equivalent to simply replacing the matrix elements uαβγδ in the originalHamiltonian (1.4). It is crucial that the all the orbitals that appear in the space P(2) alsoappear in P(N) in order for this prescription to be well-defined.

In Papers 1, 2 and 4 the application of the two-body effective interaction to quantum dotsis discussed. The motivation for this is two-fold: First, we may improve FCI calculationsthat already consume enormous amounts of CPU time worldwide. Second, the knowledge


thereby gained may give a general insight into the behaviour of such effective interactions inan easier way than if we were to study the nuclear Hamiltonian directly.

2.3 SCALING AND ACCURACY

The curse of dimensionality is simply the fact that dim(P) grows almost exponentially withN. In this section, we study the effects of this on the accuracy and scaling with respectto parallelization of the FCI calculations. Consider the model space P ′, in which L single-particle orbitals can be occupied by the N particles without restriction. We assume the worst-case scenario with respect to the symmetries of H, i.e., that there are no observables Ω suchthat [H,Ω] = 0, and we must include all Slater determinants in B′ in our FCI calculation.Thus,

dim(P ′) =(

LN

).

Using a HO basis in d spatial dimensions and excluding spin, we obtain L ≈ Rd/d!. Tosee this, consider the single-particle basis functions φα(~r) given by tensor products of dstandard Hermite functions φni(ri). (This gives no loss of generalization.) The energies areεα−d/2 = n1 + · · ·+nd, and the number L of φα(~r) with ∑

di=1 ni ≤ R approximately equals

the volume of a d-dimensional hyperpyramid, being 1/d!’th of the cube with sides R. Thus,

dim(P ′)≈ LN

N!≈ 1

d!N N!RNd. (2.3)

For the model space P the growth is similar. The spinless model space is characterized bythose Slater determinants for which

N

∑i=1

εαi −Nd2

=N

∑i=1

d

∑k=1

ni,k ≤ R.

This time, the hyperpyramid is in Nd dimensions, so that the number of possibilites is ap-proximately RNd/(Nd)!. We obtain an additional factor 1/N! due to the permutation sym-metry of the Slater determinants. Hence,

dim(P)≈ 1(Nd)!N!

RNd. (2.4)

Let us consider, for simplicity, the problem of storing a single vector on a supercomputercluster with ncpu nodes, each having the capability of storing Mcpu floating point numbers.In a real setting, we need to store several vectors and maybe the Hamiltonian matrix, whichhas at least ∼ dim(P) nonzero entries. Therefore, we set

ncpuMcpu = dim(P),

2.3 Scaling and accuracy 25

which is a conservative estimate on the requirements of the supercomputer. Using Eqns. (2.3)and (2.4), the largest model space parameter R that the supercomputer can handle becomes

R≈ (McpuncpuA)1/Nd,

where A = d!N N! for the model space P ′, and A = (Nd)!N! for P , respectively. Consider-ing the estimate (2.2) for the error ∆E in a non-degenerate eigenvalue, we assume

∆E ≈CR−α, α > 0.

The error in the FCI calculation on the supercomputer becomes, roughly,

∆E ≈C(McpuncpuA)−α/Nd

If we compare the performance with ncpu = 1, i.e., a single desktop computer, the erroris improved by a factor

n−α/Ndcpu ,

a very small number for interesting computations. For example, halving the error requiresn1/2

cpu = 2Nd/α nodes. In Paper 2 we see that α≈ 1.4 for the ground state of N = 5 particles ind = 2 dimensions, which requires about

n12cpu = 210/1.5 ≈ 100 nodes

to halve the error. Halving the error again would need ∼ 10,000 nodes. For N = 4 particles,α≈ 1.4, giving

n12cpu = 28/1.4 ≈ 50 nodes,

which is still very large for halving the error, requiring about 2,500 nodes to halve the erroragain.

These considerations lend support to the idea that running FCI codes in parallel is notthat useful. The combination of slow convergence with respect to R and the curse of dimen-sionality, leads to a very small benefit from parallelization. On the other hand, if one utilizesthe effective interaction, the convergence rate is improved to α ≈ 3.8 for N = 5. Then, wewould require only ncpu ≈ 6 nodes to halve the error.

For N = 4, α≈ 4.6, requiring ncpu ≈ 3 nodes. It also seems, that the constant C becomessmaller when using the effective interaction, improving the error even more.

In conclusion, using a two-body effective interaction – which will not make the schememore complicated – will improve the accuracy of parallel quantum dot calculations drasti-cally. Even a single desktop computer with effective interactions will rival parallel imple-mentations that use the bare Coulomb interaction.


2.4 HARTREE-FOCK METHOD

The exact eigenfunction Ψk(x1, x2, · · · , xN) ∈ HN of H is, on average, extremely compli-cated, as a function of N electron coordinates. Even if the basis functions used for HN areSlater determinants, which are fairly simple to visualize, linear combinations of such arenot. The function |Ψk〉 would be a Slater determinant only if the interactions were absent.Since in this case the electrons are independent, we say that their motion is uncorrelated,even though a degree of statistical correlation is present due to anti-symmetry. The Coulombinteraction introduces strong correlations, so that the probability of locating a particle at, say~r, depends strongly on the whereabouts of the others.

With these considerations at hand, it is natural to try and find a Hamiltonian describingelectrons that move independently without interacting. We therefore consider the interactionas a perturbation and seek a modified single-particle potential V(~r) such that the correspond-ing ground state Slater determinant |ΦHF〉 has a minimum variational energy E[ΦHF], i.e.,

|ΦHF〉 := argmin|Φ〉∈S

〈Φ|H|Φ〉〈Φ|Φ〉

, (2.5)

as discussed in Section 1.6. Here S consists of all N-electron functions that can be writtenas Slater determinants, with arbitrary orthonormal single-particle functions. The solution|ΦHF〉 to this problem is then the optimal independent particle (i.e., uncorrelated) pictureof the interacting system. This is the Hartree-Fock method, and it is called a mean-fieldmethod, since the interaction is replaced by an “average” particle-independent potential, ina very specific sense.

Let u(i, j) = u(xi, x j) = λ/ri j be the interaction between particles i and j. The Hamilto-nian (1.2), where we now replace the harmonic potential with a more general V(~r), can thenbe written

H =N

∑i=1

h(i)+∑i< j

u(i, j)

=

[N

∑i=1

h(i)+ f (i)

]+

[N

∑i< j

u(i, j)−N

∑i=1

f (i),

]

where

f (i) := V(~ri)−V(~ri)

is called the fluctuation potential, and is the modification of V(~ri) needed to obtain theHartree-Fock single-particle potential V . Equation (2.5) leads to a coupled non-linear setof equations called the Hartree-Fock equations. It has the form of a non-linear eigenvalue

2.4 Hartree-Fock method 27

problem for the eigenfunctions ψα(x) of h+ f with corresponding eigenvalues εα, and reads[h+

∫ρ(y,y)u(x,y) dy

]ψα(x)−

∫ρ(x,y)u(x,y)ψα(y) dy = εαψα(x), (2.6)

where

ρ(x,y) :=N

∑α=1

ψα(x)ψα(y).

The action of f = f (ψ1, · · · ,ψN) on an arbitrary φ ∈H1 depends on the unknown single-particle functions ψα, viz,

fφ(x) =[∫

ρ(y,y)u(x,y) dy]φ(x)−

∫ρ(x,y)u(x,y)φ(y) dy,

where the last term, called the exchange term, is seen to be non-local, in contrast to ordinarypotential function operators.

The Hartree-Fock solution is the Slater determinant constructed from the N first ψα(x),viz,

|ΦHF〉= b†1b†

2 · · ·b†N |−〉,

where b†α creates a particle in the single-particle orbital ψα(x). The Hartree-Fock energy

EHF = E[ΦHF] can be shown to be (Gross et al. 1991)

EHF = [ΦHF] =12

N

∑i=1

εi + 〈ψi|h|ψi〉.

To solve the Hartree-Fock equations (2.6), one usually uses fix-point iterations, startingwith the original single-particle functions φα(x). Thus, one iterates[

h+ f (ψ(k)1 , · · ·ψ(k)

N )]ψ

(k+1)α (x) = ε

(k+1)α ψα(x)(k+1),

with the initial condition ψ(0)α = φα. On each step, an infinite set of eigenfunctions are found

(in principle), and the lowest-lying are chosen for the next iteration.The Hartree-Fock orbitals ψα(x) are called canonical, because they constitute the best

possible non-correlated single-particle basis functions, in contrast to the original φα(x),which were arbitrary. It is common practice in for example quantum chemistry, to do FCIcalculations in the Slater determinant basis generated by the Hartree-Fock orbitals ψα(x).

For a detailed account of the properties of the Hartree-Fock method, such as convergenceissues, see Schneider (2006) and Lions (1987). For an application to parabolic quantum dots,see Waltersson and Lindroth (2007).


2.5 COUPLED CLUSTER METHOD

The coupled-cluster method (CC) is today probably the most powerful ab initio method toobtain ground state eigenvalues of many-body problems. It was introduced in the contextof nuclear physics by Coester and Kümmel around 1960. While it gained little attention innuclear physics communities, with only sporadic applications before the 1990’s, quantumchemists took on to the idea and developed the method further (Kümmel 1991).

Coupled cluster calculations have been very successful for molecular problems (Bartlettand Musiał 2007), and in recent years, nuclear physicists have been further developing andapplying the CC method to nuclei hitherto out of reach of FCI (Dean and Hjorth-Jensen2004). To name an example, Hagen et al. (2008) have computed the ground state energyof 48Ca (an N = 48 problem in d = 3 dimensions) to about ten percent accuracy. Keep-ing in mind that if the corresponding calculation should be done with FCI we would havedim(P)≈ 1071, it is clear that such CC calculations are a huge step forward, and that FCI ishard pressed to compete with the results, see for example the heated correspondence betweenDean et al. (2008) and Roth and Navrátil (2007, 2008). For a detailed analysis of the CCmethod, see the technical report by Schneider (2006). In this section we consider a simplifieddescription of the CC method, outlining the features more than giving calculational recipes.

Recently, the singles and doubles CC method has been used on quantum dots, see forexamlpe Heidari et al. (2007). We mention here also the so-called equation of motionCC method (Stanton and Bartlett 1993), which is able to extract other eigenfunctions andeigenvalues than the ground state. Henderson et al. (2003) have used it on parabolic quantumdots.

Consider an N-body problem, and let the model space P ′ be given, with truncationparameter R. In this section, we shall for simplicity assume that the Hamiltonian is definedby it’s projection onto P ′, i.e., that FCI is defined to be exact. In other words, we work in afinite-dimensional Hilbert space.

Let |Φ0〉 ∈ B′ be an arbitrary Slater determinant in the basis for P ′, say, the (unperturbed)ground state, viz,

|Φ0〉= c†1c†

1 · · ·c†N |−〉.

By reordering the orbitals φα(x), this form is always possible. Let us define an excitationoperator Tk by

Tk :=1

k!2

N

∑a1,··· ,ak=1

L

∑β1,··· ,βk=N+1

tβ1,··· ,βka1,··· ,ak

c†β1· · ·c†

βkcak · · ·ca1 , (2.7)

parametrized by excitation coefficients (usually called amplitudes) tβ1,··· ,βka1,··· ,ak , where the Latin

indices take values ai ≤ N only, and the Greek indices take values βi > N only. Each term,when applied to |Φ0〉, moves k particles “up” from the N first orbitals into the remainingL−N. Notice, that all |Φ〉 ∈ B′ can be written on the form Tk|Φ0〉, for suitable excitation

2.5 Coupled cluster method 29

amplitudes, see Eqn. (2.9) below.It can now be shown, that any |Ψ〉 ∈P ′ with 〈Φ0|Ψ〉= 1 can be written

|Ψ〉= eT |Φ0〉, where T =N

∑k=1

Tk.

In particular, assuming that we can normalize the exact ground state |Ψ0〉 (identical to theFCI ground state in the model space P ′) so that 〈Φ0|Ψ0〉= 1, we have

E0 = 〈Φ0|e−T HeT |Φ0〉= 〈Φ0|H|Φ0〉.

Using the Baker-Campbell-Hausdorff formula, we obtain the well-known expansion

H = H +[H,T ]+12!

[[H,T ],T ]+13!

[[[H,T ],T ],T ]+14!

[[[[H,T ],T ],T ],T ]+ · · ·

Using the fact that the Hamiltonian (1.4) contains only up to two-body terms, it is straight-forward to show that all the terms not listed in this expansion vanish identically. We obtainthe following equations:

E0 = 〈Φ0|H|Φ0〉= 〈Φ0|H(1+T2 +12

T 21 )|Φ0〉

0 = 〈Φ|e−T HeT |Φ0〉, ∀Φ ∈ B′,Φ 6= Φ0. (2.8)

It is interesting to note, that E0 only depends on T1 and T2, while |Ψ0〉 depends on all theTk. In Eqn (2.8), |Φ〉 can be written on the form

|Φ〉= c†β1

c†β2· · ·c†

βkcak· · ·ca2

ca1|Φ0〉. (2.9)

Inserting this into Eqn. (2.8), one obtains a set of nonlinear equations for the excitationamplitudes tβa, tβ1,β2

a1,a2 , etc, which can be solved by iterative techniques. We will not list theequations here, as they are quite complicated.

We have not yet introduced any approximation, and the number of unknowns is presentlydim(P ′), so that the problem is, in fact, no simpler than the FCI problem. We thereforetruncate the expansion T at, say, two-particle excitations, i.e.,

T ≈ T1 +T2,

and solve the resulting nonlinear equations. This is referred to as the “singles and doublescoupled cluster method” (CCSD). By truncating at T3, we obtain the “singles, doubles andtriplets” method (CCSDT), and so on, producing successively more complicated non-linearequations.

An important feature of the CC method is that it is not variational. The ground stateenergy computed is not a priori an upper bound on the ground state energy. On the otherhand, it scales much better with increasing model space size than does the FCI method.

3 Introduction to the papers

3.1 A READER’S GUIDE

There are four papers in this thesis. At the time of writing, Papers 1 and 3 have been pub-lished, while Papers 2 and 4 is awaiting editorial decisions. They are presented in chrono-logical order with respect to submission to the journals.

Paper 1

Paper 1, written in collaboration with Morten Hjorth-Jensen and Halvor Møll Nilsen, isa study of one-dimensional quantum dots, focusing on the two-particle case. It may beconsidered as a preliminary study of the general case, which is the topic for Paper 2. The one-dimensional two-particle parabolic quantum dot is reduced to a one-dimensional problem byapplying centre of mass-coordinates, in which only the relative coordinate problem is non-trivial, having Hamiltonian

Hrel =−12∂2

∂x2 +12

x2 +U(√

2|x|),

where U is the interaction potential. The effects of the trap alone was studied by replacing Uwith some other potential v(x), producing the one-body Hamiltonian. Arguments along theline of Section 1.4 are then used to analyze and discuss the numerical properties of the fullconfiguration interaction (FCI) method in this particular case.

Paper 2

After finishing Paper 1 (which admittedly bears marks of being my first publication), I triedto handle the general case: N electrons in d dimensions. In the end it sorted out, but notexactly as I thought, however. The keen reader may have noticed an error in Paper 1 con-cerning the decay of the coefficients |cn| in Sec. III C: xkψ(x) ∈ L2(R) and ∂k

xψ(x) ∈ L2(R)are not sufficient conditions for ψ(x) ∈ Hk(R) – the correct condition is x j∂

k− jx ψ(x) ∈ L2(R)

for all 0 ≤ j ≤ k. However, as shown in Paper 2 and indicated in Section 1.4, this has nobearing on the result when we consider exponentially decaying functions ψ(x).

Paper 2 develops a series of results concerning approximation properties of n-dimensional


Hermite functions, defined as tensor products of standard one-dimensional Hermite func-tions. These show that for an FCI calculation where the exact wavefunction is known to benon-smooth, the convergence will be at most algebraic, cf. Eqn. (2.2). This is demonstratedwith numerical simulations. It is also demonstrated numerically that the different eigenfunc-tions |Ψk〉 have differing smoothness properties, in accordance with the results mentioned inSection 1.5. By applying a two-electron effective interaction, the convergence with respectto basis size is seen to accelerate drastically. This motivates further analysis of the effectiveinteractions themselves.

Paper 3

Paper 3 deals with effective Hamiltonians at the stage before we extract an effective inter-action. It gives a description of the common effective Hamiltonians which I have termed“geometric” due to its extensive use of the characterization of pairs of linear subspaces byso-called canonical angles and vectors. The canonical angles generalizes angles betweenvectors to angles between general subspaces. The singular value decomposition is essentialhere. I also discuss the consequences of symmetries in the system (i.e., observables S suchthat [H,S ] = 0), and demonstrate that it is important to reflect such symmetries in the modelspace. That is, if angular momentum is conserved, viz, [H,Lz] = 0, then the model spaceshould be rotationally symmetric, viz, [P,Lz] = 0.

I continue to propose a new way to compute the effective interaction in the sub-clustercase, such as employed in all the other papers. The standard method is somewhat morecomplicated (Navrátil, Kamuntavicius, and Barrett 2000).

Paper 4

Paper 4 implements the effective two-body interaction and describes it in detail for theCoulomb case. The algorithm is also used in Papers 1 and 2, but I felt there was no room andthat it was inappropriate to spend much space on it there and then. Also, my own knowledgeof the method had increased since Paper 1 was written, so that a relatively succinct and cleardescription could be given.

The main contribution of Paper 4 is however an open source implementation of the FCImethod with effective two-body interactions called OPENFCI. The idea is that an openlyavailable code released under the Gnu GPL and documented in a scientific journal will spurmore activity among students and researchers new to the field. OPENFCI is also well-suitedas starting point for implementations of Hartree-Fock, coupled cluster methods and so on.

3.2 DISCUSSION AND OUTLOOK

Much of this work has been devoted to and motivated by the convergence properties of theFCI method with respect to the basis size parameter R. Let us summarize and discuss some

3.2 Discussion and outlook 33

of the findings.

Properties of the harmonic oscillator basis

The harmonic oscillator (HO) basis is on one side very well suited for many-body calcula-tions, since it easily allows for manipulation of operators using centre-of-mass and relativecoordinates. Furthermore, HO basis functions easily accommodate various common symme-tries, such as total angular momentum and reflection symmetries. On the other hand, it hasproblems resolving the non-smooth wavefunctions that invariably appear as eigenfunctionsof the quantum dot Hamiltonian. This has the important consequence that one will neverachieve better convergence than algebraic convergence, i.e., of R−α-type, when using FCI.

The FCI convergence estimates have obvious consequences for the coupled cluster (CC)class of methods, and the approximation estimates for the HO basis functions can prove tobe useful for analyzing this class of methods from first principles as well.

Other methods in other areas of research may benefit from the theory as well, such asapproximating general partial differential equations using pseudospectral methods (Boyd2001).

Properties of the effective two-body interaction

In Papers 1 and 2, it was demonstrated that the effective two-body interaction improvesthe convergence of the FCI calculations by up to several orders of magnitude, for examplefor systems containing N = 5 electrons. For N > 6, it is difficult to get good results usingFCI anyway, because of the extreme increase in basis size. However, what if the effectiveinteraction could be utilized in, say, a coupled cluster calculation? We can assume that theerror in a coupled cluster with singles and doubles calculation (cf. Section 2.5) is comparableto that of the FCI calculation. Can the error be reduced by the same order of magnitude inthe CC case as in the FCI case? To me, it seems very likely. Studies in this direction areobvious candidates for future research.

On the other hand, I suspect that the improvement the two-body effective interactionoffers would decrease with increasing N. Studies in this direction would be feasible if theeffective interaction-approach to coupled cluster calculations works. This would be highlyinteresting – both with respect to reaching higher N and understanding the effective interac-tions.

We have not studied the properties of the wavefunctions when using effective two-bodyinteractions. It is well-known that even if an effective Hamiltonian may reproduce nuclearspectra, the transition probabilities may be completely wrong. Thus, a study of the error inthe wavefunctions using different norms should be undertaken.

Consequences for other physical systems

Can we extrapolate the results obtained for quantum dots to other systems, such as the no-core nuclear shell model? How closely related are the different models? The no-core shell


model is formally equivalent to the parabolic quantum dot model, except for the compli-cations mentioned in Section 2.2 concerning a lack of natural centre of the system and theunknown interaction properties. The spectrum of the nucleus is not purely discrete, so somework must be done in order to apply what we have learned. It should be evident, however,that an interesting approach to nuclear properties may come from applying the approxima-tion properties of Hermite functions in the other direction, so to speak. By studying thenumerical eigenfunctions, one may extract information about the system. For example, ifa certain R−α-behaviour is found in the FCI coefficients, conclusions about the cusps andsimilar in the wavefunctions may be drawn.

Properties of many-body effective interactions

In the nuclear physics community, the need for more accurate effective interactions, i.e.,three-body interactions, is realized. Moreover, there are fundamental three-body forces thatneed to be accounted for. A study of such three-body effective interactions in the quantumdot model may give significant insight into the properties of such.

The reason why the effective two-body interaction can be computed unambiguously inthe Coulomb case, is that the two-body central force problem is classically integrable. (Thenuclear forces have non-central terms, so conclusions for nuclei cannot be drawn in thisway.) By transforming to the centre-of-mass frame and separating out angular momentum,we obtain a one-dimensional radial equation, which has no degeneracies, as discussed inPaper 4. In particular, all eigenvalues of the two-body quantum dot are complex analyticfunctions of the interaction strength λ. This rules out the existence of intruder states in thiscase.

If we move to the three-body problem, however, it is almost certainly classically chaotic,a fact that can be assessed by numerical simulations (Ott 1993). Is is well-known that thehelium atom, for example, exhibits classically chaotic motion. The attempt at creating athree-body effective interaction will then necessarily have to deal with this fact, since inclassically non-integrable systems there will be intruder states. Perturbation theory is there-fore likely to fail, and selecting the “proper” eigenpairs for an effective Hamiltonian is nottrivial.

Research along this direction will almost certainly prove very interesting.

Do we need the extra convergence?

This thesis has been about striving for greater accuracy in eigenvalues. But is it really nec-essary? Can we not be happy, so to speak, with what we’ve got? After all, the quantum dotmodel is only a model, and a rough one as well.

On the other hand, in future technologies, being able to successfully predict the outcomeof an experiment from first principles is crucial. Even if our models are inaccurate, futuremodels based on the same ideas may be much better. To be able to simulate a system underthe influence of a time-dependent perturbation, such as a laser, also requires highly accu-


rate eigenvalues in order to work, as the the time limit of accurate simulations is directlyproportional to the largest energy difference error.

Additionally, there is the desire to find out all there is to know about a given model.Good and well-understood numerical methods are then essential. The impact on the amassedknowledge on fundamental quantum systems when good numerical tools are available shouldnot be underestimated. Thus, we should strive for faster, more accurate, and economicalmethods.

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Part II : Papers

1 Effective interactions,large-scale diagonalization,and one-dimensionalquantum dots

This paper appeared in volume 76 of Physical Review B, 2007.

Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

Simen Kvaal,1,* Morten Hjorth-Jensen,2,1 and Halvor Møll Nilsen1

1Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway2Department of Physics, University of Oslo, N-0316 Oslo, Norway

Received 18 April 2007; published 17 August 2007

The widely used large-scale diagonalization method using harmonic oscillator basis functions an instance ofthe Rayleigh-Ritz method S. Gould, Variational Methods for Eigenvalue Problems: An Introduction to theMethods of Rayleigh, Ritz, Weinstein, and Aronszajn Dover, New York, 1995, also called a spectral method,configuration-interaction method, or “exact diagonalization” method is systematically analyzed using resultsfor the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensionalmodel of a quantum dot. The method is shown to converge slowly, and the nonsmooth character of theinteraction potential is identified as the main problem with the chosen basis, while, on the other hand, itsimportant advantages are pointed out. An effective interaction obtained by a similarity transformation isproposed for improving the convergence of the diagonalization scheme, and numerical experiments are per-formed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed.

DOI: 10.1103/PhysRevB.76.085421 PACS numbers: 73.21.La, 71.15.m, 31.15.Pf

I. INTRODUCTION

Large-scale diagonalization is widely used in many areasof physics, from quantum chemistry1 to nuclear physics.2 Itis also routinely used to obtain spectra of model quantumdots, see, for example, Refs. 3–19. The method is based on aprojection of the model Hamiltonian onto a finite-dimensional subspace of the many-body Hilbert space inquestion, hence the method is an instance of the Rayleigh-Ritz method.20 Usually, one takes the stance that the many-

body Hamiltonian is composed of two parts H0 and H1, treat-ing the latter as a perturbation of the former, whoseeigenfunctions are assumed to be a basis for the Hilbertspace. This leads to a matrix diagonalization problem, hence

the name of the method. As H1 often contains the interactionterms of the model, “perturbing” the electronic configuration

states of H0, the method is also called the configuration-interaction method. In the limit of an infinite basis, themethod is in principle exact, and for this reason, it is alsocalled “exact diagonalization.” Usually, however, this

method is far from exact, as H1 is rarely a small perturbationin a sense to be specified in Sec. III E, while limited com-puting resources yield a tight bound on the number of de-grees of freedom available per particle.

In this work, we provide mathematical convergence crite-ria for configuration-interaction calculations. More specifi-

cally, we address this problem in the case where H0 is aharmonic oscillator or HO for short, concentrating on asimple one-dimensional problem. A common model for aquantum dot is, indeed, a perturbed harmonic oscillator, andusing HO basis functions is also a common approach in otherfields of many-body physics and partial differential equationssettings in general, as it is also known as the Hermite spectralmethod.21 When we, in the following, refer to theconfiguration-interaction method, or CI for short, it is as-sumed that a HO basis is used.

Studying a one-dimensional problem may seem undulyrestrictive, but will, in fact, enable us to treat realistic multi-

dimensional problems as well due to the symmetries of theharmonic oscillator. Moreover, we choose a worst-case sce-nario, in which the interaction potential decays very slowly.We argue that the nature of the perturbation H1, i.e., thenonsmooth character of the Coulomb potential or the trappotential, hampers the convergence properties of the method.To circumvent this problem and improve the convergencerate, we construct an effective two-body interaction via asimilarity transformation. This approach, also using a HObasis, is routinely used in nuclear physics,22–24 where theinteractions are of a completely different nature.

The effective interaction is defined for a smaller spacethan the original Hilbert space, but it reproduces exactly thelowest-lying eigenvalues of the full Hamiltonian. This can beaccomplished by a technique introduced by Suzuki andOkamoto.25–28 Approaches based on this philosophy for de-riving effective interactions have been used with great suc-cess in the nuclear many-body problem.22–24 For light nuclei,it provides benchmark calculations of the same quality asGreen’s function Monte Carlo methods or other ab initiomethods. See, for example, Ref. 29 for an extensive compari-son of different methods for computing properties of thenucleus 4He. It was also used in a limited comparative studyof large-scale diagonalization techniques and stochasticvariational methods applied to quantum dots.30

We demonstrate that this approach to the CI method forquantum dots yields a considerable improvement to the con-vergence rate. This has important consequences for studies ofthe time development of quantum dots with two or moreelectrons, as reliable calculations of the eigenstates are cru-cial ingredients in studies of coherence. This is of particularimportance in connection with the construction of quantumgates based on quantum dots.31 Furthermore, the introductionof an effective interaction allows for studies of many-electron quantum dots via other many-body methods likeresummation schemes such as various coupled cluster theo-ries as well. As the effective interaction is defined onlywithin the model space, systematic and controlled conver-gence studies of these methods in terms of the size of thisspace are possible.

PHYSICAL REVIEW B 76, 085421 2007

1098-0121/2007/768/08542113 ©2007 The American Physical Society085421-1

The article is organized as follows: In Sec. II, the modelquantum dot Hamiltonian is discussed. In Sec. III, we dis-cuss the CI method and its numerical properties. Central tothis section are results concerning the convergence of Her-mite function series.32,33 We also demonstrate the resultswith some numerical experiments.

In Sec. IV, we discuss the similarity transformation tech-nique of Suzuki and Okamoto 25–28 and replace the Coulombterm in our CI calculations with this effective interaction. Wethen perform numerical experiments with the modifiedmethod and discuss the results.

We conclude the article with generalizations to more par-ticles in higher dimensions and possible important applica-tions of the modified method in Sec. V.

II. ONE-DIMENSIONAL QUANTUM DOTS

A widely used model for a quantum dot containing Ncharged fermions is a perturbed harmonic oscillator withHamiltonian

H = j=1

N −1

2 j

2 +1

2r j2 + vr j +

j=1

N

k=j+1

N

Urj − rk ,

1

where r j R2, j=1, . . . ,N are each particle’s spatial coordi-nate, r is a small modification of the HO potential r2 /2,and Ur is the Coulomb interaction, viz., Ur= /r, where is a constant. Modeling the quantum dot geometry by aperturbed harmonic oscillator is justified by self-consistentcalculations,34–36 and is the stance taken by many other au-thors using the large-scale diagonalization technique aswell.3–5,8–10,12,14–18 The potential modification r is chosento emulate different trap geometries, such as a double well ora flat-bottomed well. For instance, in Ref. 16, a nonsmoothperturbation in two dimensions of the form vx ,y=−Ax−By is used, creating four coupled wells. In this article, weuse a Gaussian perturbation in one dimension cf. Eq. 11.

Electronic structure calculations amount to finding eigen-pairs E ,, e.g., the ground state energy and wave function,such that

H = E, H and E R .

Here, even though the Hamiltonian only contains spatial co-ordinates, the eigenfunction is a function of both the spa-tial coordinates rkR2 and the spin degrees of freedom k −1/2 , +1/2, i.e.,

H = L2R2N C2.

The actual Hilbert space is the space of the antisymmetricfunctions, i.e., functions for which

xP1,xP2, . . . ,xPN = sgnPx1,x2,…,xN ,

for all permutations P of N symbols. Here, xk= rk ,k.For simplicity, we concentrate on one-dimensional quan-

tum dots. Even though this is not an accurate model for realquantum dots, it offers several conceptual and numerical ad-

vantages. Firstly, the symmetries of the harmonic oscillatormakes the numerical properties of the configuration-interaction method of this system very similar to a two oreven three-dimensional model, as the analysis extends al-most directly through tensor products. Secondly, we may in-vestigate many-body effects for moderate particle numbers Nwhile still allowing a sufficient number of HO basis func-tions for unambiguously addressing accuracy and conver-gence issues in numerical experiments.

In this article, we further focus on two-particle quantumdots. Incidentally, for the two-particle case, one can showthat the Hilbert space of antisymmetric functions is spannedby functions of the form

r1,1,r2,2 = r1,r21,2 ,

where the spin wave function can be taken as symmetric orantisymmetric with respect to particle exchange, leading toan antisymmetric or symmetric spatial wave function , re-

spectively. Inclusion of a magnetic field B poses no addi-tional complications,37 but for simplicity, we presently omitit. Thus, it is sufficient to consider the spatial problem andproduce properly symmetrized wave functions.

Due to the peculiarities of the bare Coulomb potential inone dimension,38,39 we choose a screened approximationUx1−x2 ; , given by

Ux;, =

x + ,

where is the strength of the interaction and 0 is ascreening parameter which can be interpreted as the width ofthe wave function orthogonal to the axis of motion. Thischoice is made since it is nonsmooth, like the bare Coulombpotential in two and three dimensions. The total Hamiltonianthen reads

H = −1

2 2

x12 +

2

x22 +

1

2x1

2 + x22 + vx1 + vx2

+ Ux1 − x2;, . 2

Observe that for U=0, i.e., =0, the Hamiltonian is sepa-

rable. The eigenfunctions of H disregarding proper symme-trization due to the Pauli principle become n1

x1n2x2,

where nx are the eigenfunctions of the trap Hamiltonian

Ht given by

Ht = −1

2

2

x2 +1

2x2 + vx . 3

Similarly, for a vanishing trap modification vx=0, theHamiltonian is separable in normalized center-of-mass co-ordinates given by

X =x1 + x2

2and x =

x1 − x2

2.

Indeed, any orthogonal coordinate change leaves the HOHamiltonian invariant see Sec. III, and hence

KVAAL, HJORTH-JENSEN, AND NILSEN PHYSICAL REVIEW B 76, 085421 2007

085421-2

H = −1

2 2

X2 +2

x2 +1

2X2 + x2 + vX + x/2

+ vX − x/2 + U2x;, .

The eigenfunctions become nXmx, where nX are theHermite functions, i.e., the eigenfunctions of the HO Hamil-tonian see Sec. III, and where mx are the eigenfunctionsof the interaction Hamiltonian, viz.,

Hi = −1

2

2

x2 +1

2x2 + U2x;, . 4

Odd even functions mx yield antisymmetric symmetricwave functions with respect to particle interchange.

III. CONFIGURATION-INTERACTION METHOD

A. Harmonic oscillator and model spaces

The configuration-interaction method is an instance of theRayleigh-Ritz method,20 employing eigenfunctions of theunperturbed HO Hamiltonian as basis for a finite-dimensional Hilbert space P, called the model space, ontowhich the Hamiltonian 1, or in our simplified case, theHamiltonian 2, is projected and then diagonalized. As men-tioned in Sec. I, this method is, in principle, exact, if thebasis is large enough.

We write the N-body Hamiltonian 1 as

H = H0 + H1,

with H0 being the HO Hamiltonian, viz.,

H0 = −1

2j=1

N

j2 +

1

2j=1

N

r j2,

and H1 being a perturbation of H0, viz.,

H1 = j=1

N

vr j + j=1

N

k=j+1

N

Urj − rk .

For a simple one-dimensional model of two particles, weobtain

H0 = hx1 + hx2 ,

where hx is the well-known one-dimensional harmonic os-cillator Hamiltonian, viz.,

hx = −1

2

2

x2 +1

2x2.

Clearly, H0 is just a two-dimensional HO Hamiltonian, if wedisregard symmetrization due to the Pauli principle. For theperturbation, we have

H1 = vx1 + vx2 +

x1 − x2 + .

In order to do a more general treatment, let us recall somebasic facts about the harmonic oscillator.

If we consider a single particle in D-dimensional space, itis clear that the D-dimensional harmonic oscillator Hamil-tonian is the sum of one-dimensional HO Hamiltonians foreach Euclidean coordinate, viz.,

hD = −1

22 +

1

2x2 =

k=1

D

hxk . 5

We indicate the variables on which the operators depend byparentheses if there is danger of confusion. Moreover, theHO Hamiltonian for N distinguishable particles in d dimen-

sions is simply hNd. The D-dimensional HO Hamiltonian ismanifestly separable, and the eigenfunctions are

nx = k=1

D

nkxk ,

with energies

n =D

2+

k=1

D

nk,

where n denotes the multi-index of quantum numbers nk. Theone-dimensional HO eigenfunctions are given by

nx = 2nn! 1/2−1/2Hnxe−x2/2,

where Hnx are the usual Hermite polynomials. These func-tions are the Hermite functions and are treated in furtherdetail in Sec. III C.

As for the discretization of the Hilbert space, we employa so-called energy-cut model space P, defined by the span ofall HO eigenfunctions with energies up to a given =Nmax+D /2, viz.,

P ª spnx0 k

nk Nmax ,

where we bear in mind that the D=Nd dimensions are dis-tributed among the N particles.

For the one-dimensional model with only one particle, themodel space reduces to

P1 = spnx0 n Nmax . 6

Thus, one particle is associated with one integer quantumnumber n, denoting the “shell number where the particle re-sides,” in typical terms. For two particles, we get

P2 = spn1x1n2

x20 n1 + n2 Nmax .

We illustrate this space in Fig. 1.Proper symmetrization must also be applied. However,

the Hamiltonian 1 commutes with particle permutations,meaning that the eigenfunctions will be symmetric or anti-symmetric, assuming that the eigenvalues are distinct. In thecase of degeneracy, we may simply produce antisymmetriceigenfunctions by taking linear combinations.

We mention that other model spaces can also be used;most common is perhaps the direct product model space,defined by N direct products of P1 rather than a cut in energyas above.

EFFECTIVE INTERACTIONS, LARGE-SCALE… PHYSICAL REVIEW B 76, 085421 2007

085421-3

B. Coordinate changes and the harmonic ascillator

It is obvious that any orthogonal coordinate change y

=Sx, where STS=1, commutes with hD. In particular, energyis conserved under the coordinate change. Therefore, theeigenfunctions of the transformed Hamiltonian will be a lin-ear combination of the original eigenfunctions of the sameenergy, viz.,

nSx = n

n,Tnnx ,

where the sum is over all n such that n=n. Here, T per-forms the coordinate change, viz.,

Tnx = nSx , 7

where T is unitary. Also note that energy conservation im-plies that P is invariant with respect to the coordinatechange, implying that the CI method is equivalent in the twocoordinate systems.

An important example is the center-of-mass transforma-tion introduced in Sec. II. This transformation is essentialwhen we want to compute the Hamiltonian matrix since theinteraction is given in terms of these coordinates.

Observe that in the case when the Hamiltonian is, in fact,separated by such a coordinate change, the formulation ofthe exact problem using HO basis is equivalent to two one-particle problems using HO basis in the new coordinates.

C. Approximation properties of the Hermite functions

In order to understand the accuracy of the CI method, weneed to study the approximation properties of the Hermitefunctions. Note that all the Hermite functions nx spanningL2R are smooth. Indeed, they are holomorphic in the entirecomplex plane. Any finite linear combination of these willyield another holomorphic function, so any nonsmooth func-tion will be badly approximated. This simple fact is sadlyneglected in the configuration-interaction literature, and wechoose to stress it here: Even though the Hermite functionsare simple to compute and deal with, arising in a natural wayfrom the consideration of a perturbation of the HO and obey-ing a wealth of beautiful relations, they are not very wellsuited for the computation of functions whose smoothness is

less than infinitely differentiable or whose decay behavior forlarge x is algebraic, i.e., fx=ox for some 0. Due tothe direct product nature of the N-body basis functions, it isclear that these considerations are general and not restrictedto the one-dimensional one-particle situation.

Consider an expansion x=n=0 cnnx in Hermite

functions of an arbitrary L2R. The coefficients aregiven by

cn = n, = −

xHnxe−x2/2dx .

Here, Hnx= 2nn! −1/2Hnx are the normalized Hermitepolynomials. If x is well approximated by the basis, thecoefficients cn will decay quickly with increasing n. Theleast rate of convergence is a direct consequence of

2 = n=0

cn2 ,

hence we must have cn=on−1/2. This is not a sufficientcondition, however. With further restrictions on the behaviorof x, the decay will be faster. This is analogous to thefaster decay of Fourier coefficients for smoother functions,40

although for Hermite functions smoothness is not the onlyparameter as we consider an infinite domain. In this case,another equally important feature is the decay of x as xgrows, which is intuitively clear given that all the Hermitefunctions decay as exp−x2 /2.

Let us prove this assertion. We give here a simple argu-ment due to Boyd,32 but we strengthen the result somewhat.

To this end, assume that x has k square integrable de-rivatives in the weak sense and that xmx is square inte-grable for m=0,1 , . . . ,k. Note that this is a sufficient condi-tion for

a†x =12

xx − x ,

and a†2x up to a†kx to be square integrable as well.Here, a† and its Hermitian conjugate a are the well-knownladder operators for the harmonic oscillator.41

Using integration by parts, the formula for cn becomes

cn = −

xHnxe−x2/2dx

= n + 1−1/2−

a†xHn+1xe−x2/2dx ,

or

cn = n + 1−1/2dn+1,

where dn are the Hermite expansion coefficients of a†xL2. Since dn2 by assumption, we obtain

n=0

ncn2 ,

implying

FIG. 1. Two-body model space defined by a cut in energy. Thetwo-body state has quantum numbers n1 and n2, the sum of whichdoes not exceed Nmax.


085421-4

cn = on−1 .

Repeating this argument k times, we obtain the estimate

cn = on−k+1/2 .

It is clear that if x is infinitely differentiable and if, inaddition, x decays faster than any power of x, such as, forexample, exponentially decaying functions or functions be-having like exp−x2, cn will decay faster than any power of1 /n, so-called infinite-order convercence or spectral conver-gence. Indeed, Hille33 gives results for the decay of the Her-mite coefficients for a wide class of functions. The mostimportant for our application being the following: If xdecays as exp−x2, with 0, and if 0 is the distancefrom the real axis to the nearest pole of x when consid-ered as a complex function, then

cn = On−1/4e−2n+1 , 8

a very rapid decay for even moderate .An extremely useful property32 of the Hermite functions

is the fact that they are uniformly bounded, viz.,

nx 0.816, ∀ x,n .

As a consequence, the pointwise error in a truncated series isalmost everywhere bounded by

x − n=0

Nmax

cnnx 0.816 n=Nmax+1

cn .

Thus, estimating the error in the expansion amounts to esti-mating the sum of the neglected coefficients. If cn=on,

x − n=0

Nmax

cnnx = oNmax+1 a.e.

For the error in the mean,

x − n=0

N

cnnx = ONmax+1/2 , 9

as is seen by approximating n=Nmax+1 cn2 by an integral.

In the above, “almost everywhere,” or “a.e.” for short,refers to the fact that we do not distinguish between squareintegrable functions that differ on a point set of Lebesguemeasure zero. Moreover, there is a subtle distinction betweenthe notations O(gn) and o(gn). For a given function f ,fn=o(gn) if limn→fn /gn=0, while fn=O(gn) ifwe have limn→fn /gn, a slightly weaker statement.

D. Application to the interaction potential

Let us apply the above results to the eigenproblem for aperturbed one-dimensional harmonic oscillator, i.e.,

x = x2 + 2fx − 2Ex , 10

which is also applicable when the two-particle Hamiltonian2 is separable, i.e., when U=0 or v=0.

It is now clear that under the assumption that fx is ktimes differentiable in the weak sense, and that fx

=ox2 as x→, the eigenfunctions will be k+2 timesweakly differentiable and decay as exp−x2 /2 for large x.Hence, the Hermite expansion coefficients of x will decayas on, =−k+3 /2.

If we further assume that fx is analytic in a strip ofwidth 0 around the real axis, the same will be true forx such that we can use Eq. 8 to estimate the coefficients.

A word of caution is, however, at its place. Although wehave argued that if a given function can be differentiated ktimes in the weak sense, then the coefficients decay ason, =−k+1 /2, it may happen that this decay “kicks in”too late to be observable in practical circumstances.

Consider, for example, the following function:

gx =e−x2/2

x + ,

which has exactly one almost everywhere continuous de-rivative and decays as exp−x2 /2. However, the derivativeis seen to have a jump discontinuity of magnitude 2/2 atx=0. From the theory, we expect on−1 decay of the coeffi-cients, but for small the first derivative is badly approxi-mated, so we expect to observe only on−1/2 decay for mod-erate n, due to the fact that the rate of decay of thecoefficients of gx are explicitly given in terms of the coef-ficients of a†gx.

In Fig. 2, the decay rates at different n and for various are displayed. The decay rate is computed by estimatingthe slope of the graph of ln cn versus ln n, a technique usedthroughout this article. Indeed, for small , we observe only−1/2 convergence in practical settings, where n is mod-erate, while larger gives −1 even for small n.

The above function was chosen due to its relation to theinteraction Hamiltonian 4. Indeed, its coefficients are givenby

cn = n,g = n,Ux;1,0 ,

i.e., proportional to the first row of the interaction matrix.Moreover, due to Eq. 10, the ground state of the interac-tion Hamiltonian has a second derivative with similar behav-ior near x=0 as gx. Thus, we expect to observe −3/2,rather than −2, for the available range of n in the large-scale diagonalization experiments.

We remark here that it is quite common to model quantumdot systems using nonsmooth potentials42 vr and even touse the CI method with HO basis functions on thesemodels.16,43,44 The present analysis clearly shows that the CImethod will behave badly with such potentials.

E. Numerical experiments

We wish to apply the above analysis by considering themodel Hamiltonian.2 We first consider the case where vx=0 or Ux=0, respectively, which reduces the two-particleproblem to one-dimensional problems through separation of

variables, i.e., the diagonalization of the trap Hamiltonian Ht

and the interaction Hamiltonian Hi in Eqs. 3 and 4. Thenwe turn to the complete nonseparable problem.


085421-5

For simplicity, we consider the trap x2 /2+vx with

vx = Ae−Cx − 2, A,C 0, R , 11

which gives rise to a double-well potential or a single-wellpotential, depending on the parameters, as depicted in Fig. 3.The perturbation is everywhere analytic and rapidly decay-ing. This indicates that the corresponding configuration-interaction energies and wave functions also should convergerapidly. In the numerical experiments below, we use A=4,C=2, and =0.75, creating the asymmetric double well inFig. 3.

For the interaction Hamiltonian Hi and its potential x2 /2+U2x ; ,, we arbitrarily choose =1 and =0.01, giv-ing a moderate jump discontinuity in the derivative.

As these problems are both one-dimensional, the modelspace reduces to P1 as given in Eq. 6. Each problem thenamounts to diagonalizing a matrix H with elements

Hn,m = n,Ht,im = n + 12n,m +

−

nxfxmxdx ,

0 n, m Nmax,

with fx=vx or fx=U2x ;1 ,0.01. We compute thematrix to the desired precision using Gauss-Legendrequadrature. In order to obtain reference eigenfunctions andeigenvalues, we use a constant reference potential method45

implemented in the MATSLISE package46 for MATLAB. Thisyields results accurate to about 14 significant digits.

In Fig. 4 left, the magnitude of the coefficients of theexact ground states alongside the ground state energy errorand wave function error right is graphed for each Hamil-tonian, using successively larger Nmax. The coefficients of theexact ground states decay according to expectations, as we

clearly have spectral convergence for the Ht ground state and

on−1.57 convergence for the Hi ground state.These aspects are clearly reflected in the CI calculations.

Both the Ht ground state energy and wave function converge

spectrally with increasing Nmax, while for Hi we clearly have

algebraic convergence. Note that for Ht, Nmax40 yields aground state energy accurate to 10−10, and that such preci-

sion would require Nmax1012 for Hi, which converges onlyalgebraically.

Intuitively, these results are easy to understand: For thetrap Hamiltonian, a modest value of Nmax produces almostexact results, since the exact ground state has extremelysmall components outside the model space. This is not pos-sible for the interaction Hamiltonian, whose exact groundstate is poorly approximated in the model space alone.

If we consider the complete Hamiltonian 2, we nowexpect the error to be dominated by the low-order conver-gence of the interaction Hamiltonian eigenproblem. Figure 4also shows the error in the ground state energy for the cor-responding two-particle calculation, and the error is, indeed,

seen to behave identically to the Hi ground state energy error.That the energy error curve is almost on top of the error in

the wave function for Hi is merely a coincidence.It is clear that the nonsmooth character of the potential U

destroys the convergence of the method. The eigenfunctionswill be nonsmooth, while the basis functions are all verysmooth. Of course, a nonsmooth potential vx would de-stroy the convergence as well.

In this sense, we speak of a “small perturbation H1” if theeigenvalues and eigenfunctions of the total Hamiltonian con-verge spectrally. Otherwise, the perturbation is so strong thatthe very smooth property of the eigenfunctions vanish. In ourcase, even for arbitrarily small interaction strengths , theeigenfunctions are nonsmooth, so that the interaction is neversmall in the sense defined here. On the other hand, the trapmodification vx represents a small perturbation of the har-monic oscillator if it is smooth and rapidly decaying. Thispoints to the basic deficiency of the choice of HO basis func-tions: They do not capture the properties of the eigenfunc-tions.

100

101

102

103

104

10−3

10−2

10−1

100

101

102

103

104

n

|c n|

Coefficients of f (x) for δ ∈ [0.01, 2]

δ = 0.01

δ = 2

ln10(n)

ln10(δ

)

Decay rate α for δ ∈ [0.01, 2]

−1.25

−1.2

−1.15−1.1

−1.05−1

−0.95−0.9−0.85−0.8−0.75−0.7

−0.65−0.6

−0.55

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6−2

−1.5

−1

−0.5

0

−1.2

−1.1

−1

−0.9

−0.8

−0.7

−0.6

FIG. 2. Top Coefficients cn of the function exp−x2 /2 / x+ for 0.01,2, n=0,2 , . . . ,5000. Bottom Estimated decayrate of cn, i.e., slope of the graphs in the left panel.


085421-6

We could overcome this problem by choosing a differentset of basis functions for the Hilbert space, and thus, a dif-ferent model space P altogether. However, the symmetries ofthe HO lets us treat the interaction potential with ease byexplicitly performing the center-of-mass transformation, asignificant advantage in many-body calculations. In our one-dimensional case, we could replace Ux1−x2 by a smoothpotential, after all, U is just an approximation somewhat ran-domly chosen. We would then obtain much better resultswith the CI method. However, we are not willing to trade thebare Coulomb interaction in two or even three dimensionsfor an approximation. After all, we know that the singularand long-range nature of the interaction is essential.

We, therefore, propose to use effective interaction theoryknown from many-body physics to improve the accuracy ofCI calculations for quantum dots. This replaces the matrix inthe HO basis of the interaction term with an approximation,giving exact eigenvalues in the case of no trap perturbationvx, regardless of the energy cut parameter Nmax. We cannothope to gain spectral convergence; the eigenfunctions arestill nonsmooth. However, we can increase the algebraic con-vergence considerably by modifying the interaction matrixfor the given model space. This is explained in detail in thenext section.

IV. EFFECTIVE HAMILTONIAN THEORY

A. Similarity transformation approach

The theories of effective interactions have been, and stillare, vital ingredients in many-body physics, from quantumchemistry to nuclear physics.1,2,47–50 In fields like nuclearphysics, due to the complicated nature of the nuclear inter-actions, no exact spatial potential exists for the interactions

between nucleons. Computation of the matrix elements ofthe many-body Hamiltonian then amounts to computing, forexample, successively complicated Feynman diagrams,48,49

motivating realistic yet tractable approximations such as ef-fective two-body interactions. These effective interactionsare, in turn, used as starting points for diagonalization calcu-lations in selected model spaces.2,22–24 Alternatively, they canbe used as starting point for the resummation of selectedmany-body correlations such as in coupled-cluster theories.1

In our case, it is the so-called curse of dimensionality thatmakes a direct approach unfeasible: The number of HOstates needed to generate accurate energies and wave func-tions grows exponentially with the number of particles in thesystem. Indeed, the dimension of P grows as Nmax

Nd / Nd!For the derivation of the effective interaction, we consider

the Hamiltonian 2 in center-of-mass coordinates, i.e.,

H = hX + hx + vX + x/2 + vX − x/2

+ U2x;, .

For vx0, the Hamiltonian is clearly not separable. Theidea is then to treat vxj as perturbations of a system sepa-rable in center-of-mass coordinates; after all, the trap poten-tial is assumed to be smooth. This new unperturbed Hamil-tonian reads

H = hX + hx + V ,

where V=U2x ; ,, or any other interaction in a more

general setting. We wish to replace the CI matrix of Hˆ with

a different matrix Heff , having the exact eigenvalues of , butnecessarily only approximate eigenvectors.

−4 −2 0 2 4−1

0

1

2

3

4

5

6

7

8

9

x

V(x

)

−4 −2 0 2 4−1

0

1

2

3

4

5

6

7

8

9

xV

(x)

Symmetric double−well

x2/2

Asymmetric double−wellAsymmetric single−well

x2/2

FIG. 3. Left Symmetric double-well potential created with the Gaussian perturbation A exp−Cx−2, with A=4, =0, and C=2.Right Asymmetric double-well potential created with the Gaussian perturbation with A=4, =0.75, and C=2, and single-well potentialusing C=0.25.


085421-7

The effective Hamiltonian Heff can be viewed as an op-erator acting in the model space while embodying informa-tion about the original interaction in the complete space H.We know that this otherwise neglected part of the Hilbert

space is very important if V is not small. Thus, the firstingredient is the splitting of the Hilbert space into the modelspace P= PH and the excluded space Q=QH= 1− PH.Here, P is the orthogonal projector onto the model space.

In the following, we let N be the dimension of the modelspace P. There should be no danger of confusion with thenumber of particles, N=2, as this is now fixed. Moreover, welet nn=1

N be an orthonormal basis for P, and nn=N+1 be

an orthonormal basis for Q.The second ingredient is a decoupling operator . It is an

operator defined by the properties

P = Q = 0,

which essentially means that is a mapping from the modelspace to the excluded space. Indeed,

= P + QP + Q = PP + PQ + QP + QQ = QP ,

which shows that the kernel of includes Q, while the rangeof excludes P, i.e., that acts only on states in P andyields only states in Q.

The effective Hamiltonian Heff= Phx+ hXP+ Veff,

where Veff is the effective interaction, is given by the simi-larity transformation28

Heff = Pe−zHezP ,

where z=artanh−†. The key point is that ez is a unitary

operator with ez−1=e−z, so that the N eigenvalues of Heff are

actually eigenvalues of H.In order to generate a well-defined effective Hamiltonian,

we must define =QP properly. The approach of Suzukiand Okamoto25–28 is simple: Select an orthonormal set of

vectors nn=1N . These can be some eigenvectors of H we

wish to include. Assume that Pnn=1N is a basis for the

model space, i.e., that for any nN, we can write

n = m=1

N

an,mPm

for some constants an,m. We then define by

Pn ª Qn, n = 1, . . . ,N .

Observe that defined in this way is an operator that recon-structs the excluded space components of n given its modelspace components, thereby, indeed, embodying informationabout the Hamiltonian acting on the excluded space.

Using the decoupling properties of , we quickly calcu-late

n = QPn = Qm=1

N

an,mm, n = 1, . . . ,N ,

and hence, for any nN, we have

n,n = m=1

N

an,mn,m ,

yielding all the nonzero matrix elements of .As for the vectors n, we do not know a priori the exact

eigenfunctions of H, of course. Hence, we cannot find Heffexactly. The usual way to find the eigenvalues is to solve amuch larger problem with NN and then assume that these

100

101

102

103

104

10−25

10−20

10−15

10−10

10−5

100

n

|c n|

Coefficients of ground states of Ht,i

Ground state of Hi

Ground state of Ht

100

101

102

103

10−15

10−10

10−5

100

Nmax

Err

or

Error in ground state energy and wave function

Error in E, for Ht

Error in ψ, for Ht

Error in E, for Hi

Error in ψ, for Hi

Error in E, full model

FIG. 4. Top Coefficients of the exact ground states of the

Hamiltonians Ht,i. For Hi, only even-numbered coefficients are non-zero and, thus, displayed. The almost straight line indicates approxi-mately on−1.57 decay of the coefficients around n=600, and

on−1.73 around n=5000 cf. Fig. 2. For the Ht ground state, weclearly have spectral convergence. Bottom The error in the groundstate energies and wave functions when using the CI method. For

Hi, we have on−1.24 decay for the energy error, and on−1.20 decay

for the wave function error, both evaluated at n=600. For Ht, weclearly have spectral convergence. A full two-particle CI calculationis superimposed, showing that the error in the interaction part of theHamiltonian 2 completely dominates. Here, the error in the energy

is oNmax−1.02 at n=70, while for Hi alone, we have oNmax

−1.01.


085421-8

eigenvalues are exact. The reason why this is possible at all

is that our Hamiltonian H is separable, and therefore, easierto solve. However, we have seen that this is a bad method:Indeed, one needs a matrix dimension of about 1010 to obtainabout ten significant digits. Therefore, we instead reuse theaforementioned constant reference potential method to ob-tain eigenfunctions and eigenvectors accurate to machineprecision.

Which eigenvectors of H do we wish to include? Intu-itively, the first choice would be the lowest N eigenvectors.However, simply ordering the eigenvalues “by value” is not

what we want here. Observe that H is block diagonal andthat the model space contains Nmax+1 blocks of sizes 1through Nmax+1. If we look at the exact eigenvalues, weknow that they have the structure

En,m = n + 1/2 + m,

where n is the block number and m are the eigenvalues of Hisee Eq. 4. However, it is easy to see that the large-scalediagonalization eigenvalues do not have this structure—weonly obtain this in the limit Nmax→. Therefore, we choosethe eigenvectors corresponding to the N eigenvalues En,m,n+mNmax, thereby achieving this structure in the eigenval-

ues of Heff .

In general, we wish to incorporate the symmetries of H

into the effective Hamiltonian Heff . In this case, it was theseparability and even eigenvalue spacing we wished to re-produce. In Sec. V, we treat the two-dimensional Coulombproblem similarly.

B. Numerical experiments with effective interactions

The eigenvectors of the Hamiltonian H differ from those

of the effective Hamiltonian Heff . In this section, we firstmake a qualitative comparison between the ground states ofeach Hamiltonian. We then turn to a numerical study of theerror in the CI method when using the effective interaction inthe model problem.

Recall that the ground state eigenvectors are of the form

X,x = 0Xx = 0Xn=0

cnnx .

For Heff , cn=0 for all nNmax, so that the excluded spacepart of the error coincides with the excluded space part of the

exact ground state. In Fig. 5, the coefficients cn for both H

and Heff are displayed. The pointwise error is also plotted,and the largest values are seen to be around x=0. This isexpected since U2x ; , and the exact ground state isnonsmooth there. Notice the slow spatial decay of the error,intuitively explained by the slow decay of the Coulomb in-teraction.

We now turn to a simulation of the full two-particleHamiltonian 2 and compare the decay of the ground stateenergy error with and without the effective interaction. Thus,we perform two simulations with Hamiltonians

H = H + vx1 + vx2 = hx1 + hx2 + vx1 + vx2 + TVT†

and

Heff = Heff + vx1 + vx2 = hx1 + hx2 + vx1 + vx2

+ TVeffT†,

respectively, where T is the center-of-mass transformationcf. Eq. 7.

We remark that the new Hamiltonian matrix has the samestructure as the original matrix. It is only the values of theinteraction matrix elements that are changed. Hence, the newscheme has the same complexity as the CI method if we

100

101

102

10−3

10−2

10−1

nco

effic

ient

s

102

102.2

10−3.4

10−3.3

10−3.2

n

coef

ficie

nts

Ground state of H’Ground state of H

eff

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

2.5x 10

−4

x

|ψ(x

)−ψ

eff(x

)|

Pointwise ground state error

FIG. 5. Top Plot of ground state coefficients of H and Heff.Bottom Pointwise error in relative coordinate x of effectiveHamiltonian ground state effx.


085421-9

disregard the computation of Veff, which is a one-time calcu-lation of low complexity.

The results are striking: In Fig. 6, we see that the groundstate error decays as ONmax

−2.57, compared to ONmax−0.95 for the

original CI method. For Nmax=40, the CI relative error isE /E02.610−3, while for the effective interaction ap-proach E /E01.010−5, a considerable gain.

The ground state energy E0 used for computing the errorswere computed using extrapolation of the results.

We comment that Nmax40 is the practical limit on asingle desktop computer for a two-dimensional two-particlesimulation. Adding more particles further restricts this limit,emphasizing the importance of the gain achieved in the rela-tive error.

In a more systematical treatment, we computed the errordecay coefficient for a range of trap potentials x2

+A exp−2x−2, where we vary A and to create single-and double-well potentials. In most cases, we could estimate successfully. For low values of , i.e., near-symmetricwells, the parameter estimation was difficult in the effectiveinteraction case due to very quick convergence of the energy.The CI calculations also converged quicker in this case. In-tuitively this is so because the two electrons are far apart inthis configuration.

The results indicate that at Nmax=60, we have

= − 0.96 ± 0.04 for H

and

= − 2.6 ± 0.2 for Heff

for the chosen model. Here, 0.61.8 and 2.9A4.7,and all the fits were successful. In Fig. 7, contour plots of the

obtained results are shown. For the range shown, resultswere unambiguous.

These numerical results clearly indicate that the effectiveinteraction approach will gain valuable numerical precisionover the original CI method in general; in fact, we havegained nearly 2 orders of magnitude in the decay rate of theerror.

V. DISCUSSION AND OUTLOOK

A. Generalizations

One-dimensional quantum dot models are of limited valuein themselves. However, as claimed in Sec. I, the analysisand experiments performed in this article are valid also inhigher-dimensional systems.

Consider two particles in two dimensions. Let hr be thetwo-dimensional harmonic oscillator Hamiltonian we omitthe superscript in Eq. 5 for brevity, and let the quantumdot Hamiltonian be given by

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

Nmax

rela

tive

erro

r

Ground state convergence

CI methodCI with eff. int.

−5 0 50

5

10

15

x

Potential v(x) +x2/2

α ≈−2.57

α ≈−1.02

FIG. 6. Ground state energy relative error for a two-particle simulation using the confinement potential Vx=x2 /2+4 exp−2x−0.752. For the CI method without effective interac-tions, we obtain −1.02, while the effective interactions give −2.57. The electron density is superimposed on the potential plot.

−1.0616−1.0459−1.0301

−1.0301

−1.0144

−1.0144

−0.99868

−0.99868

−0.98295

−0.98295

−0.96722

−0.96722

−0.95149

−0.95149

−0.95149

−0.93575

−0.93575

−0.93575

−0.92002

−0.92002

−0.92002

−0.90429

−0.90429

−0.90429

A

µ

CI energy error decay coefficient α

1 2 3 4 5 6 7 8

1

1.5

2

2.5

3

3.5

4

4.5

5

−1.06

−1.04

−1.02

−1

−0.98

−0.96

−0.94

−0.92

−3.0

894

−2.8664

−2.8664

−2.7549−2.7549 −2.7549

−2.7549

−2.6434−2.6434 −2.6434

−2.6434−2.6434−2.6434

−2.5319 −2.5319 −2.5319

−2.5319−2.5319−2.5319

−2.4204−2.4204 −2.4204

−2.4204−2.4204−2.4204

−2.7549−2.7549−2.7549

−2.6434−2.6434−2.6434−2.5319

A

µCI w/eff. int energy error decay coefficient α

1 2 3 4 5 6 7 8

1

1.5

2

2.5

3

3.5

4

4.5

5

−3.2

−3.1

−3

−2.9

−2.8

−2.7

−2.6

−2.5

FIG. 7. Estimates of for CI calculations with bottom andwithout top effective interactions.


085421-10

H = H + vr1 + vr2 ,

where

H = hr1 + hr2 +

r1 − r2.

The normalized center-of-mass and relative coordinates aredefined by

R =r1 + r2

2and r =

r1 − r2

2,

respectively, which gives

H = hR + hr +

2r.

The HO eigenfunctions in polar coordinates are given by9

n,mr, eimrmLnmr2e−r2/2,

and the corresponding eigenvalues are 2n+ m+1. Now, His further separable in polar coordinates, yielding a singleradial eigenvalue equation to solve, analogous to the single

one-dimensional eigenvalue equation of Hi in Eq. 4. Thisequation, in fact, has analytical solutions for a countable in-finite set of values, first derived in Ref. 51.

The eigenvalues of H have the structure

En,m,n,m = 2n + m + 1 + n,m,

where n ,m and n ,m are the center-of-mass and relativecoordinate quantum numbers, respectively. Again, the degen-eracy structure and even spacing of the eigenvalues are de-stroyed in the CI approach, and we wish to regain it with theeffective interaction. We then choose the eigenvectors corre-sponding to the quantum numbers

2n + m + 2n + m Nmax

to build our effective Hamiltonian Heff .Let us also mention that the exact eigenvectors n,m,n,m

are nonsmooth due to the 1/r singularity of the Coulombinteraction. The approximation properties of the Hermitefunctions are then directly applicable as before, when weexpand the eigenfunctions in HO basis functions. Hence, theconfiguration-interaction method will converge slowly alsoin the two-dimensional case. It is good reason to believe thateffective interaction experiments will yield similarly positiveresults with respect to convergence improvement.

Clearly, the above procedure is applicable to three-

dimensional problems as well. The operator H is separableand we obtain a single nontrivial radial equation, and thus,we may apply our effective Hamiltonian procedure. The ex-act eigenvalues will have the structure

En,l,m,n,l,m = 2n + l +3

2+ n,l,m,

on which we base the choice of the effective Hamiltonianeigenvectors as before.

The effective interaction approach to the configuration-interaction calculations is easily extended to a many-particleproblem, whose Hamiltonian is given by Eq. 1. The formof the Hamiltonian contains only interactions between pairs

of particles, and Veff as defined in Sec. IV can simply replacethese terms.

B. Outlook

A theoretical understanding of the behavior of many-bodysystems is a great challenge and provides fundamental in-sights into quantum mechanical studies, as well as offerspotential areas of applications. However, apart from someanalytically solvable problems, the typical absence of an ex-actly solvable contribution to the many-particle Hamiltonianmeans that we need reliable numerical many-body methods.These methods should allow for controlled expansions andprovide a calculational scheme which accounts for succes-sive many-body corrections in a systematic way. Typical ex-amples of popular many-body methods are coupled-clustermethods,1,52,53 various types of Monte Carlo methods,54–56

perturbative expansions,47,48 Green’s function methods,49,50

the density-matrix renormalization group,57,58 and large-scalediagonalization methods such as the CI method consideredhere.

In a forthcoming article, we will apply the similaritytransformed effective interaction theory to a two-dimensionalsystem and also extend the results to many-body situations.Application of other methods, such as coupled-cluster calcu-lations, is also an interesting approach and can give furtherrefinements on the convergence as well as insight into thebehavior of the numerical methods in general.

The study of this effective Hamiltonian is interesting froma many-body point of view: The effective two-body force isbuilt from a two-particle system. The effective two-body in-teraction derived from an N-body system, however, is notnecessarily the same. Intuitively, one can think of the formerapproach as neglecting interactions and scattering betweenthree or more two particles at a time. In nuclear physics, suchthree-body correlations are non-negligible and improve theconvergence in terms of the number of harmonic oscillatorshells.59 Our hope is that such interactions are much lessimportant for Coulomb systems.

Moreover, as mentioned in Sec. I, accurate determinationof eigenvalues is essential for simulations of quantum dots inthe time domain. Armed with the accuracy provided by theeffective interactions, we may commence interesting studiesof quantum dots interacting with their environment.

C. Conclusion

We have mathematically and numerically investigated theproperties of the configuration-interaction method, or “exactdiagonalization method,” by using results from the theory ofHermite series. The importance of the properties of the trapand interaction potentials is stressed: Nonsmooth potentialsseverely hampers the numerical properties of the method,while smooth potentials yield exact results with reasonablecomputing resources. On the other hand, the HO basis is


085421-11

very well suited due to the symmetries under orthogonal co-ordinate changes.

In our numerical experiments, we have demonstrated thatfor a simple one-dimensional quantum dot with a smoothtrap, the use of similarity transformed effective interactionscan significantly reduce the error in the configuration-interaction calculations due to the nonsmooth interaction,while not increasing the complexity of the algorithm. This

error reduction can be crucial for many-body simulations, forwhich the number of harmonic oscillator shells is verymodest.

ACKNOWLEDGMENT

This work was supported by the Norwegian ResearchCouncil.

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085421-12

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085421-13

2 Harmonic oscillatoreigenfunction expansions,quantum dots, and effectiveinteractions

This paper is submitted to Physical Review B.

Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions

Simen Kvaal1,∗

1Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway(Dated: November 1, 2008)

We present a mathematical analysis of the approximation properties of the harmonic oscillator eigenfunctionswidely used as basis for configuration-interaction calculations of harmonically confined quantum dots, as wellas other physical systems. We further givea priori error estimates for the diagonalization methods, that, forexample, can be used to estimate the computational parameters needed to give a desired accuracy. We show,that in general the method converges slowly for the quantum dot problem. In order to overcome this, wepropose to use an effective two-body interaction well-known from nuclear physics, and demonstrate a significantimprovement on the convergence by performing numerical simulations.

I. INTRODUCTION

The harmonic oscillator (HO) is ubiquitous in quantumphysics. It is the basis for extremely diverse many-body calcu-lations in nuclear physics,1 quantum chemistry,2 and quantumdot calculations in solid state physics.3 In these calculations,one typically seeks a few of the lowest eigenenergiesEk of thesystem HamiltonianH, and their corresponding eigenvectorsψk, viz,

Hψk = Ekψk, k = 1, · · · ,kmax.

In all cases, the many-body wave function is expanded in abasis of eigenfunctions of the HO, and then necessarily trun-cated to give an approximation. In fact, the so-called curse ofdimensionality implies that the number of degrees of freedomper particle available is severely limited. It is clear, that anunderstanding of the properties of such expansions, givingapriori error estimates on many-body calculations, is very im-portant. Unfortunately, this is a neglected topic in the physicsliterature. In this article, we give a thorough mathematicaland numerical analysis and give practical convergence esti-mates of HO basis expansions. It generalizes and refines thefindings of a recent study of one-dimensional systems.4

The HO eigenfunctions are popular for several reasons.Many quantum systems, such as the quantum dot model con-sidered here, are perturbed harmonic oscillatorsper se, so thatthe true eigenstates should be perturbations of the HO states.Moreover, the HO has many beautiful properties, such as sep-arability, invariance under orthogonal coordinate changes, andeasily computed eigenfunctions, so that computing matrix el-ements of relevant operators becomes relatively simple. TheHO eigenfunctions are defined on thewholeof Rd in whichthe particles live, so that truncation of the domain is unneces-sary. Indeed, this is one of the main problems with methodssuch as finite difference or finite element methods.5

The HO eigenfunctions for a many-body system are closelyrelated to the so-called Hermite functions, and theird-dimensional generalizations. We will derive approximationproperties of the Hermite functions in arbitrary dimensionsusing techniques based on the ladder operators for the HO, aswell as simple techniques from the abstract Sobolev space the-ory of partial differential equations.6 By approximation prop-erties, we mean estimates on the error‖ψ−Pψ‖, whereψ isany wave function andP projects onto a finite subspace of HO

eigenfunctions. Here,Pψ is in fact the best approximation inthe norm. The estimates will depend on analytic propertiesof ψ, i.e., if it is differentiable, and whether it falls of suffi-ciently fast at infinity. To our knowledge, these results are notpreviously published.

The results are crucial for obtaininga priori error boundsand rates of convergence for both eigenvalues and wave func-tions with respect to basis size of commonly used meth-ods such as the full configuration interaction method (CI),also known as “exact diagonalization”, and coupled clustermethods.7

We apply the obtained results to the CI method for a widely-used model quantum dot, namely that ofN interacting elec-trons trapped in a HO potential ind = 2 dimensions, and de-rive convergence estimates of the method as function of thebasis size. The main results are somewhat discouraging. Thecoefficients for typical eigenfunctions of the system are shownto decay very slowly, limiting the accuracy ofany practicalmethod using HO basis functions. We illustrate the results bystandard numerical calculations paralleling that of many otherpublished works, see for example Refs. 8–13.

Finally, we propose to use an effective two-body interac-tion to overcome, at least partially, the slow convergence rate.This is routinely used in nuclear physics1 (see for exampleRef. 14) where the interparticle forces are of a completely dif-ferent, and basically unknown, nature. For electronic systems,however, the interaction is well-known and easy to analyze,but effective interactions of the present kind have not beenapplied, at least to the author’s knowledge. We present re-sults forN = 3,4 and 5 electrons using the full configurationinteraction method with effective interactions, demonstratingthe usefulness of this approach. The modified method is seento have convergence rates of at least one order of magnitudehigher than the original CI method.

In fact, this remedy should be applicable to most systems ofinteracting electrons, independent of the confining trap poten-tial, and thereby quite universal. An important point here isthat the complexity of the CI calculations is not altered, as noextra non-zero matrix elements are introduced. All one needsis a relatively simple one-time calculation to produce the ef-fective interaction matrix elements.

The article is organized as follows. In Sec. II we discusstheN-body harmonic oscillator ind dimensions and its eigen-functions, in order to identify the relationship with Hermite

2

functions inn = Nd dimensions. We also briefly discuss theparabolic quantum dot model. In Sec. III, we give results forthe approximation properties of the Hermite functions inn di-mensions, and thus also of many-body HO eigenfunctions. InSec. IV, we briefly discuss the full configuration interactionmethod, using the results obtained in Sec. III to obtain conver-gence estimates of the method as function of the model spacesize. We also briefly discuss the effective interaction utilizedin the numerical simulations ford = and N ≤ 5, which arepresented in Sec. V. We conclude with a discussion of the re-sults, its consequences, and an outlook on further directionsof research in Sec. VI.

We have also included appendices with the most impor-tant facts and definitions from functional analysis and Sobolevspace theory that we will use, in addition to multi-indices forease of notation.

II. MANY-BODY STATES OF THE HARMONICOSCILLATOR

A. The Harmonic Oscillator

For completeness, we begin with a quick review of the har-monic oscillator inRd and establish some notation and units inwhich we work. A spinless particle of massm in an isotropicharmonic potential has Hamiltonian

HHO =− h2

2m∇2 +

12

mω2‖~r‖2,

where~r ∈Rd is the particle’s coordinates. By choosing properenergy and length units, i.e.,hω and

√h/mω, respectively, the

Hamiltonian becomes

HHO =−12

∇2 +12‖~r‖2.

This is the set of units in which we work.HHO can be written as a sum overd one-dimensional har-

monic oscillators, viz,

HHO =d

∑k=1

(−1

2∂2

∂r2k

+12

r2k

),

so that a complete specification of the HO eigenfunctions isgiven by

Φα1,α2,...,αd(~r) = φα1(r1)φα2(r2) · · ·φαd(rd), (1)

whereφαi (x), αi = 0,1, . . . are one-dimensional HO eigen-functions, also called Hermite functions. These are definedby

φn(x) = (2nn!π1/2)−1/2Hn(x)e−x2/2, n = 0,1, . . . , (2)

where the Hermite polynomialsHn(x) are given by

Hn(x) = (−1)nex2 ∂n

∂xn e−x2.

The Hermite polynomials also obey the recurrence formula

Hn+1(x) = 2xHn(x)−2nHn−1(x), (3)

with H0(x) = 1 andH1(x) = 2x. The Hermite polynomialHn(x) hasn zeroes, and the Gaussian factor will eventuallysubvert theHn(x) for large |x|. Thus, qualitatively, the Her-mite functions can be described as localized oscillations withn nodes with a Gaussian “tail” asx approaches±∞. One caneasily compute the quantum mechanical variance

(∆x)2 :=∫ ∞

−∞x2φn(x)2dx = n+

12,

showing that, loosely speaking, the width of the oscillatoryregion increases as(n+1/2)1/2.

The functionsΦα1,··· ,αd defined in Eqn. (1) are calledd-dimensional Hermite functions. In the sequel, we will writeα = (α1, · · · ,αd) ∈ Id for a tuple of non-negative integers,also called a multi-index, see Appendix A for definitions andnotation. Using multi-indices, we may write

Φα(~r) =(

2|α|α!πd/2)−1/2

Hα1(r1) · · ·Hαd(rd)e−‖~r‖2/2. (4)

The eigenvalue ofφn(x) is n+ 1/2, so that the eigenvalueof Φα(~r) is

εα =d2

+ |α|,

i.e., a zero-point energyd/2 plus a non-negative integer. Wedenote by|α| the shell numberof Φα, and the eigenspaceSr(Rd) corresponding to the eigenvalued/2+ r a shell. Wedefine theshell-truncated Hilbert spacePR(Rd)⊂ L2(Rd) as

PR(Rd) := spanΦα(~r) | |α| ≤ R=R⊕

r=0

Sr(Rd),

i.e., the subspace spanned by all Hermite functions with shellnumber less than or equal toR, or, equivalently, the direct sumof the shells up to and includingR.

Since the Hermite functions constitute an orthonormal basisfor L2(Rd) (a basic result from spectral theory15) PR(Rd)→L2(Rd), in the sense that for everyψ ∈ L2(Rd), limR→∞ ‖ψ−Pψ‖ = 0, whereP is the orthogonal projector onPR(Rd).Strictly speaking, we should use a symbol likePR or evenPR(Rd) for the projector. However,R andd will always beclear from the context, so we are deliberately sloppy to obtaina concise formulation. For the same reason, we will some-times simply writeP or PR for the spacePR(Rd).

An important fact is that sinceHHO is invariant underrigid spatial transformations (i.e., they conserve energy), sois each individual shell. That is, each shellSr(Rd), and thusPR(Rd), is independent of the spatial coordinates chosen.

For the cased = 1 each shellr is spanned by a single eigen-function, namelyφr(x). Ford= 2, each shellr has degeneracyr +1, with eigenfunctions

Φ(s,r−s)(~r) = φs(r1)φr−s(r2), 0≤ s≤ r.

3

The usual HO eigenfunctions used to construct many-bodywave functions are not the Hermite functionsΦα1,··· ,αd, how-ever, but rather those obtained by utilizing the spherical sym-metry of the HO. This gives a many-body basis diagonal inangular momentum, cf. Sec. II B. Ford = 2 we obtain theso-called Fock-Darwin orbitals given by

ΦFDn,m(r, θ) =

[2n!

(n+ |m|)!

]1/2 eimθ√

2πL|m|n (r2)e−r2/2. (5)

Here,n≥ 0 is the nodal quantum number, counting the nodesof the radial part, andm is the azimuthal quantum number.The eigenvalues are

εn,m = 2n+ |m|+1.

Thus,R= 2n+ |m| is the shell number. By construction, theFock-Darwin orbitals are eigenfunctions of the angular mo-mentum operatorLz =−i∂/∂θ with eigenvaluem. Of course,we may writeΦFD

n,m as a linear combination of the Hermitefunctions Φ(s,R−s), where 0≤ s≤ R = 2n+ |m|, and viceversa. The actual choice of form of eigenfunctions is imma-terial, as long as we may identify those belonging to a givenshell.

The spacePR=4(R2) is illustrated in Fig. 1 using both Her-mite functions and Fock-Darwin orbitals.

B. Many-body wave functions

We now discussN-body eigenfunctions of the HO ind di-mensions, including spin. This enables us to identify the ex-pansion of a many-body wave function with expansions inHermite functions forn = Nd dimensions.

Each particlek = 1, . . . ,N has both spatial degrees of free-dom~rk ∈ Rd and a spin coordinateτk ∈ ±1, correspondingto thez-projectionSz = ± h

2 of the electron spin. The config-uration space can thus be taken as two copiesX of Rd; onefor each spin value, i.e.,X = Rd×±1 andxk = (~rk, τk) ∈ Xare the particle coordinates. Due to the Pauli exclusion prin-ciple, the many-body Hilbert spaceH (X,N) now consists ofthe part ofL2(XN) anti-symmetric with respect to exchangesof any two particles’ coordinates, viz,

H (X,N) =N∧

k=1

L2(X) = PASL2(XN),

wherePAS is the orthogonal projector onto the totally anti-symmetric functions. An identity is

L2(XN)≡ L2(RNd)⊗ (C2)N,

i.e., eachψ∈ L2(XN) is equivalent to 2N component functionsψ(σ) ∈ L2(RNd), σ ∈ ±1N. Thus,

ψ(x1, · · · , xN) = ∑σ

ψ(σ)(~r1, · · · ,~rN)χσ(τ),

whereτ= (τ1, · · · , τN), and whereχσ(τ) = δσ,τ are basis func-tions for theN-spinor space(C2)N, being eigenfunctions for

Sz, viz,

Szχσ =h2

N

∑k=1

σk.

Now, consider theNd-dimensional HO as the sum ofNHOs ind dimensions, viz,

HHO(ξ) =N

∑k=1

HHO(~rk) =−12

∇2ξ +

12‖ξ‖2,

whereξ= (~r1, · · · ,~rN)∈RNd. One easily sees, that ifαk∈Id,runs over all multi-indices,k = 1, · · · ,N, then

β := (α1, · · · ,αN) ∈INd

is also an arbitrary multi-index. Moreover,

Φβ(ξ)≡Φα1(~r1) · · ·ΦαN(~rN)

covers all possible Hermite functions inNd dimensions.Since functions on the formψ(α,σ)(~r , τ) = Φα(~r)χ1

σ(τ)(whereχ1

σ are basis functions for the single-particle spinorspaceC2) constitute a basis forL2(X), we construct a basisfor L2(XN) by taking direct products, viz,

Ψα1,··· ,αN,σ1,··· ,σN(x1, · · · , xN) = Φβ(ξ)χσ(τ)

which identifies theσ-component as the Hermite functionΦβ(ξ).

A basis of Slater determinants forH (X,N) is now con-structed by anti-symmetrizing the basis forL2(XN), see forexample Ref. 16. To this end, we introduce a generalizedsingle-particle indexi = i(α,σ) ∈ N, so that an ordering onthe single-particle basis functionsΦαχ

1σ is apparent. We also

let ik := i(αk,σk). A Slater determinantΨi1,··· ,ik(x1, · · · , xN) isnow defined by

Ψi1,··· ,iN := A Ψi1,··· ,iN(x1, · · · , xN)

:= ∑p∈SN

sgn(p)pΨi1,··· ,iN(x1, · · · , xN).

where p ∈ SN is a permutation ofN symbols andA is theanti-symmetrization operator. The operator ˆp permutes thearguments of the function on which it acts. We obtain

pΨi1,··· ,iN(x1, · · · , xN) = pΦβ(ξ)χσ(τ)= Φβ′(ξ)χσ′(τ),

where

β′ = (αq(1), · · · ,αq(N)), σ′ = (σq(1), · · · ,σq(N)),

whereq is the inverse permutation ofp. Hence, ˆpΨi1,··· ,iN issimply a different Hermite functionwithin the same shell, i.e.,|β′|= |β|, with a different spin configuration.

It is now clear, that the Slater determinantΨi1,··· ,iN willhave spin components that are linear combinations of Hermitefunctions within a single shell|β|, i.e.,

Ψ(σ)i1,··· ,iN ∈S|β|(RNd).

4

n = 0

m = 0

n = 0

m = −1

n = 0

m = 1

n = 0

m = −2

n = 1

m = 0

n = 0

m = 2

n = 0

m = −3

n = 1

m = −1

n = 1

m = 1

n = 0

m = 3

n = 0

m = −4

n = 1

m = −2

n = 2

m = 0

n = 1

m = 2

n = 0

m = 4

α = (0, 0)

α = (0, 1) α = (1, 0)

α = (0, 2) α = (1, 1) α = (2, 0)

α = (0, 3) α = (1, 2) α = (2, 1) α = (3, 0)

α = (0, 4) α = (1, 3) α = (2, 2) α = (3, 1) α = (4, 0)

shell S3

Unitarily equivalent

FIG. 1: Illustration ofPR=4(R2): (Left) Fock-Darwin orbitals. (Right) Hermite functions. Basis functions with equal HO energy are shownat same line.

Moreover, if we define the shell-truncated Hilbert space

PR(XN) := PRL2(XN)

:= span

Ψi1,··· ,iN : |β|= ∑

k

|αk| ≤ R

= PR(RNd)⊗ (C2)N,

then it is easy to see that

PASPR(XN) = PASPRL2(XN)

= PRPASL2(XN) = PRH (X,N),

and that a basis forPASPR(XN) is given byΨi1,··· ,iN : i1 = i < · · · < iN, ∑

k

|αk| ≤ R

,

whereik = ik(αk,σk), i.e., the Slater determinants with totalsingle-particle shell number not greater thanR. Notice theordering oni j , which is necessary to select only linearly inde-pendent Slater determinants.

We stress, that,PASPR(XN) is independent of the actualone-body HO eigenfunctions used. In actual computations, itmay be more convenient to choose the Fock-Darwin orbitalsas basis, for example.

As should be clear now, studying approximation of Hermitefunctions in arbitrary dimensions automatically gives us thecorresponding many-body HO approximation theory, sincethe many-body eigenfunctions can be seen as 2N componentfunctions, and since the shell-truncated Hilbert space transfersto a many-body setting in a natural way, due to the separablenature of the many-dimensional harmonic oscillator.

C. Parabolic quantum dots

We considerN electrons confined in a harmonic oscilla-tor in d dimensions. This is a very common model for aquantum dot, i.e., a device in which a number of interactingelectrons are confined in a quasi-zero dimensional area usingsemi-conductor techniques, and are interesting due to their

possible use as components of future quantum computingtechnologies.3,17 We comment, that modelling the quantumdot geometry by a perturbed harmonic oscillator is justifiedby self-consistent calculations,18–20 and is a widely adoptedassumption.8,12,13,21–23

The Hamiltonian of the quantum dot is given by

H := T +U, (6)

whereT is the many-body HO Hamiltonian, andU is theinter-electron Coulomb interactions. Thus, in dimensionlessunits,

U :=N

∑i< j

C(i, j) =N

∑i< j

λ

‖~r i −~r j‖.

The parameterλ measures the strength of the interaction overthe confinement of the HO, viz,

λ :=1

hω

(e2

4πε0ε

),

where we recall that√

h/mω is the length unit. Typical valuesfor GaAs semiconductors are close toλ = 2, see for exampleRef. 23. Increasing the trap size leads to a largerλ, and thequantum dot then approaches the classical regime.3

D. Exact solution for two electrons

We are shortly prepared for the analysis of the approxi-mation properties of the HO eigenfunctions. However, it isinstructive to consider the very simplest example of a two-electron parabolic quantum dot, since this case admits analyt-ical solutions for special values ofλ.24,25 Here, we considerd = 2 dimensions only, but thed = 3 case is similar. We note,that forN = 2 it is enough to study the spatial wave function,since it must be either symmetric (for the singletS = 0 spinstate) or anti-symmetric (for the tripletS = 1 spin states). TheHamiltonian (6) becomes

H =−12(∇2

1 +∇22)+

12(r2

1 + r22)+

λ

r12, (7)

5

where r12 = ‖~r1 − ~r2‖ and r j = ‖~r j‖. Introduce a set ofscaled centre of mass coordinates given by~R= (~r1 +~r2)/

√2

and~r = (~r1−~r2)/√

2. This coordinate change is orthogonaland symmetric inR4. This leads to the manifestly separableHamiltonian

H = −12(∇2

r +∇2R)+

12(‖~r‖2 +‖~R‖2)+

λ√2‖~r‖

= HHO(~R)+ Hrel(~r).

A complete set of eigenfunctions ofH can now be written onproduct form, viz,

Ψ(~R,~r) = ΦFDn′,m′(~R)ψ(~r).

The relative coordinate wave functionψ(~r) is an eigenfunctionof the relative coordinate Hamiltonian given by

Hrel =−12

∇2r +

12

r2 +λ√2r, (8)

where r = ‖~r‖. This Hamiltonian can be further separatedusing polar coordinates, yielding eigenfunctions on the form

ψm,n(r, θ) =eimθ√

2πun,m(r),

where|m| ≥ 0 is an integer andun,m is an eigenfunction of theradial Hamiltonian given by

Hr =− 12r

∂

∂rr∂

∂r+|m|2

2r2 +12

r2 +λ√2r.

By convention,n counts the nodes away fromr = 0 of un,m(r).Moreover, odd (even)m gives anti-symmetric (symmetric)wave functionΨ(~r1,~r2). For any given|m|, it is quite easy todeduce that the special valueλ=

√2|m|+1 yields the eigen-

function

u0,m = Dr |m|(a+ r)e−r2/2,

whereD anda are constants. The corresponding eigenvalueof Hr is Er = |m|+2, andE = 2n′+ |m′|+1+ Er . Thus, theground state (havingm= m′ = 0,n= n′ = 0) forλ= 1 is givenby

Ψ0(~R,~r) = D(r +a)e−(r2+R2)/2

=D√2(r12+

√2a)e−(r2

1+r22)/2,

with D being a (new) normalization constant.Observe that this function has a cusp atr = 0, i.e., at the

origin x = y= 0 (where we have introduced Cartesian coordi-nates~r = (x,y) for the relative coordinate). Indeed, the partialderivatives∂xψ0 and∂yψ0 are not continuous there, andΨ0has no partial derivatives (in the distributional sense, see Ap-pendix B) of second order. The cusp stems from the famous“cusp condition” which simply states that, for a non-vanishingwave function atr12 = 0, the Coulomb divergence must be

compensated by a similar divergence in the Laplacian.26,27

This is only possible if the wave function has a cusp.On the other hand, the non-smooth functionΨ0,0(~R,~r) is to

be expanded in the HO eigenfunctions, e.g., Fock-Darwin or-bitals. (Recall, that the particular representation for the HOeigenfunctions are immaterial – also whether we use lab co-ordinates~r1,2 or centre-of-mass coordinates~Rand~r, since thecoordinate change is orthogonal.) Form= 0, we have

ΦFDn,0(r) =

√2π

Ln(r2)e−r2/2,

using the fact that these are independent ofθ. Thus,

Ψ0(~r) = ΦFD0,0(R)u0,0(r) = ΦFD

0,0(R)∞

∑n=0

cnΦFDn,0(r), (9)

The functionsΦFDn,0(r) are very smooth, as is seen by noting

that Ln(r2) = Ln(x2 + y2) is a polynomial inx andy, whileu0,0(r) = u0,0(

√x2 +y2), so Eqn. (9) is basically approximat-

ing a square root with a polynomial.Consider then a truncated expansionΨ0,R∈PR(R2), such

as the one obtained with the CI or coupled cluster method.7 Ingeneral, this is different fromPRΨ0, which is the best approx-imation of the wave function inPR(R2). In any case, thisexpansion, consisting of theR+1 terms like those of Eqn. (9)is a very smooth function. Therefore, the cusp atr = 0 cannotbe well approximated.

In Section III C, we will show that the smoothness proper-ties of the wave functionΨ is equivalent to a certain decayrate of the coefficientscn in Eqn. (9) asn→ ∞. In this case,we will show that

∞

∑n=0

nk|cn|2 <+∞,

so that

|cn|= o(n−(k+1+ε)/2). (10)

Here,k is the number of timesΨ may be differentiated weakly,i.e.,Ψ ∈ Hk(R2), andε ∈ [0,1) is a constant. For the functionΨ0 we havek = 1. This kind of estimate directly tells us thatan approximation using only a few HO eigenfunctions neces-sarily will give an error depending directly on the smoothnessk.

We comment, that for higher|m| the eigenstates will stillhave cusps, albeit in the higher derivatives.27 Indeed, we haveweak derivatives of order|m|+1, as can easily be deduced byoperating onψ0,m with ∂x and∂y. Moreover, recall that|m|= 1is theS = 1 ground state, which then will have coefficients de-caying faster than theS = 0 ground state. Moreover, there willbe excited states, i.e., states with|m| > 1, that also have morequickly decaying coefficients|cn|. This will be demonstratednumerically in Sec. V.

In fact, Hoffmann-Ostenhofet al.27 have shown that nearr12 = 0, for arbitraryλ any local solutionΨ of (H−E)Ψ = 0has the form

Ψ(ξ) = ‖ξ‖mP

(ξ

‖ξ‖

)(1+a‖ξ‖)+O(‖ξ‖m+1),

6

whereξ= (~r1, ~r2) ∈R4, and whereP, deg(P) = m, is a hyper-spherical harmonic (onS3), and wherea is a constant. Thisalso generalizes to arbitraryN, cf. Sec. III D. From this repre-sentation, it is manifest, thatΨ ∈ Hm+1(R4), i.e.,Ψ has weakderivatives of orderm+1. We discuss these results further inSec. III D.

III. APPROXIMATION PROPERTIES OF HERMITESERIES

A. Hermite functions in one dimension

We will now derive approximation properties for the Her-mite functions, cf. Eqns. (1) and (4). We first consider theone-dimensional case, and then turn to the general case inSec. III B. The multidimensional approach has, to the author’sknowledge, not been published previously. The treatment forone-dimensional Hermite functions is similar, but not equiv-alent to, that given by Boyd28 and Hille.29 See Appendix Bfor definitions ofL2-spaces, weak derivatives and the SobolevspacesHk, used extensively throughout this section.

Recall that the Hermite functionsφn∈ L2(R), wheren∈N0is a non-negative integer, are defined by

φn(x) = (2nn!√π)−1/2Hn(x)e−x2/2, (11)

whereHn(x) is the usual Hermite polynomial.A well-known method for finding the eigenfunctions of

HHO in one dimension involves writing

HHO = a†a+12,

where the ladder operatora is given by

a :=1√2(x+∂x),

with Hermitian adjointa† given by

a† :=1√2(x−∂x).

The name “ladder operator” comes from the important formu-las

aφn(x) =√

nφn−1(x) (12)

a†φn(x) =√

n+1φn+1(x), (13)

valid for all n. This can easily be proved by using the re-currence relation (3). By repeatedly acting onφ0 with a† wegenerate every Hermite function, viz,

φn(x) = n!−1/2(a†)nφ0(x). (14)

As Hermite functions constitute a complete, orthonormalsequence inL2(R), anyψ ∈ L2(R) can be written as a seriesin Hermite functions, viz,

ψ(x) =∞

∑n=0

cnφn(x),

where the coefficientscn are uniquely determined bycn =〈φn,ψ〉.

An interesting fact is that the Hermite functions are alsoeigenfunctions of the Fourier transform with eigenvalues(−i)n, as can easily be proved by induction by observingfirstly that the Fourier transform ofφ0(x) is φ0(k) itself, andsecondly that the Fourier transform ofa† is −ia† (acting onthe variabley). It follows from completeness of the Hermitefunctions, that the Fourier transform defines a unitary operatoron L2(R).

We now make a simple observation, namely that

(a†)kφn(x) = Pk(n)1/2φn+k(x), (15)

wherePk(n) = (n+k)!/n! > 0 is a polynomial of degreek forn≥ 0. MoreoverPk(n+1) > Pk(n) andPk+1(n) > Pk(n).

We now prove the following proposition:

Proposition 1 (Hermite series in one dimension)Let ψ ∈ L2(R). Then

1. a†ψ ∈ L2(R) if and only if aψ ∈ L2(R) if and only if∑∞

n=0n|cn|2 <+∞, where cn = 〈φn,ψ〉

2. a†ψ ∈ L2(R) if and only if xψ,∂xψ ∈ L2(R)

3. (a†)k+1ψ ∈ L2(R) implies (a†)kψ ∈ L2(R)

4. (a†)kψ ∈ L2(R) if and only if

∞

∑n=0

nk|cn|2 <+∞. (16)

5. (a†)kψ ∈ L2(R) if and only if x j∂k− jx ψ ∈ L2(R) for 0≤

j ≤ k

Proof: We have

‖a†ψ‖2 =∞

∑n=0

(n+1)|cn|2 = ‖ψ2‖+‖aψ‖2,

from which statement 1 follows. Statement 2 follows fromthe definition ofa†, and thata†ψ ∈ L2 impliesaψ ∈ L2 (since‖aψ‖ ≤ ‖a†ψ‖), which again impliesxψ,∂xψ ∈ L2. Statement3 follows from the monotone behaviour ofPk(n) as functionof k. Statement 4 then follows. By iterating statement 2 andusing[∂x, x] = 1 and statement 3, statement 5 follows.♦

The significance of the conditiona†ψ ∈ L2(R) is that thecoefficientscn of ψ must decay faster than for a completelyarbitrary wave function inL2(R). Moreover,a†ψ ∈ L2(R) isthe same as requiring∂xψ ∈ L2(R), andxψ ∈ L2(R). Propo-sition 1 also generalizes this fact for(a†)kψ ∈ L2(R), givingsuccessfully quicker decay of the coefficients. In all cases,the decay is expressed in an average sense, through a growingweight function in a sum, as in Eqn. (16). Since the terms inthe sum must converge to zero, this implies a pointwise fasterdecay, as stated in Eqn. (17) below.

We comment here, that the partial derivatives must be un-derstood in the weak, or distributional, sense: Even thoughψmay not be everywhere differentiable in the ordinary sense,

7

it may have a weak derivative. For example, if the classicalderivative exists everywhere except for a countable set (andif it is in L2), this is the weak derivative. Moreover, if thisderivative has a jump discontinuity, there are no higher orderweak derivatives. See Appendix B for a further discussion.

Loosely speaking, sincexψ(x) ∈ L2 ⇔ ∂yψ(y) ∈ L2, whereψ(y) is the Fourier transform, point 2 of Prop. 1 is a combinedsmoothness condition onψ(x) andψ(y). Point 5 is a general-ization to higher derivatives, but is difficult to check in generalfor an arbitraryψ. On the other hand, it is well-known thatthe eigenfunctions of many Hamiltonians of interest, such asthe quantum dot Hamiltonian (6), decay exponentially fast as‖~r‖→∞. For such exponentially decaying functions overR1,xkψ∈ L2(R) for all k≥ 0, i.e.,ψ(y) is infinitely differentiable.We then have the following proposition:

Proposition 2 (Exponential decay in 1D)Assume that xkψ ∈ L2(R) for all k≥ 0. Then a sufficient cri-terion for (a†)mψ ∈ L2(R) is ∂m

xψ ∈ L2(R), i.e., ψ ∈ Hm(R).In fact, xk∂m′

x ψ ∈ L2 for all m′ <m and all k≥ 0.

Proof: We prove the Proposition inductively. We note that,since∂m

xψ ∈ L2 implies∂m−1x ψ ∈ L2, the proposition holds for

m−1 if it holds for a givenm. Moreover, it holds trivially form= 1.

Assume then, that it holds for a givenm, i.e., thatψ ∈ Hm

implies xk∂m− jx ψ ∈ L2 for 1≤ j ≤ m and for allk (so that, in

particular,(a†)mψ ∈ L2). It remains to prove, thatψ ∈ Hm+1

implies xk∂mxψ ∈ L2, since then(a†)m+1ψ ∈ L2 by statement

5 of Proposition 1. We compute the norm and use integrationby parts, viz,

‖xk∂mxψ‖2 =

∫R

x2k∂kxψ

∗(x)∂kxψ(x)

= −2k〈∂mxψ, x

2k−1∂m−1x ψ〉

−〈∂m+1x ψ, x2k∂m−1

x ψ〉 <+∞.

The boundary terms vanish. Therefore,xk∂mxψ ∈ L2 for all k,

and the proof is complete. ♦The proposition states that for the subset ofL2(R) consist-

ing of exponentially decaying functions, the approximationproperties of the Hermite functions willonly depend on thesmoothness properties ofψ. Moreover, the derivatives up tothe penultimate order decay exponentially as well. (The high-est order derivative may decay much slower.)

From Prop. 1 and Prop. 2 we extract the following impor-tant characterization of the approximating properties of Her-mite functions ind = 1 dimensions:

Proposition 3 (Approximation in one dimension)Let k≥ 0 be a given integer. Let ψ ∈ L2(R) be given by

ψ(x) =∞

∑n=0

cnφn(x).

Then ψ ∈ Hk(R) if and only if

∞

∑n=0

nk|cn|2 < ∞.

The latter implies that

|cn|= o(n−(k+1)/2). (17)

Let ψR = PRψ= ∑Rn=0cnφn. Then

‖ψ−ψR‖=

(∞

∑n=R+1

|cn|2)1/2

.

This is the central result for Hermite series approximationin L2(R1). Observe that Eqn. (17) implies that the error‖ψ−ψR‖ can easily be estimated. See also Prop. 6 and commentsthereafter.

Now a word on pointwise convergence of the Hermite se-ries. As the Hermite functions are uniformly bounded,28 viz,

|φn(x)| ≤ 0.816 ∀ x∈ R,

the pointwise error inψR is bounded by

|ψ(x)−ψR(x)| ≤ 0.816∞

∑n=R+1

|cn|.

Hence, if the sum on the right hand side is finite, the conver-gence is uniform. If the coefficientscn decay rapidly enough,both errors can be estimated by the dominating neglected co-efficients.

B. Analysis of multidimensional expansions

We now consider expansions of functions inL2(Rn). Tostress thatRn may be other than the configuration space ofa single particle, we use the notationx = (x1, · · · , xn) ∈ Rn

instead of the previous~r ∈ Rd. For N electrons ind spatialdimensions,n = Nd.

Recall that the Hermite functions overRn are indexed bymulti-indicesα= (α1, · · · ,αn) ∈In, and that

Φα(x) := φα1(x1) · · ·φαn(xn).

Now, to each spatial coordinatexk define the ladder operatorsak := (xk+∂k)/

√2. These obey[a j ,ak] = 0 and[a j ,a

†k] = δ j,k,

as can easily be verified. Leta be a formal vector of the ladderoperators, viz,

a := (a1,a2, . . . ,an).

For the first Hermite function, we have

Φ0(x) := π−n/4e−‖x‖2/2.

By using Eqn. (14) and Eqn. (A4), we may generate all otherHermite functions, viz,

Φα(x) := α!−1/2(a†)αΦ0(x). (18)

Given two multi-indicesα andβ, we define the polynomialPα(β) by

Pβ(α) :=(α+β)!α!

=D

∏j=1

Pβ j (α j),

8

whereP is defined for integers as before.Since the Hermite functionsΦα constitute a basis, anyψ ∈

L2(Rn) can be expanded as

ψ(x) = ∑α

cαΦα(x), ‖ψ‖2 = ∑α

|cα|2, (19)

where the sum is to be taken over all multi-indicesα ∈ In.Now, letβ ∈In be arbitrary. By using Eqn. (13) in each spa-tial direction we compute the action of(a†)β onψ:

(a†)βψ = (a†1)β1 · · ·(a†

n)βn ∑

α

cαΦα

= ∑α

cαn

∏k=1

(αk +βk)!1/2

αk!1/2 Φα+β

= ∑α

cαPβ(α)1/2Φα+β.

Similarly, by using Eqn. (12) we obtain

aβψ = aβ11 · · ·aβn

n ∑α

cαΦα

= ∑α

cα+β

n

∏k=1

(αk +βk)!1/2

αk!1/2 Φα

= ∑α

cα+βPβ(α)1/2Φα,

Computing the square norm gives

‖(a†)βψ‖2 = ∑α

Pβ(α)|cα|2. (20)

and

‖aβψ‖2 = ∑α

Pβ(α)|cα+β|2. (21)

The polynomialPβ(α) > 0 for all β,α ∈ In, and Pβ(α+α′) > Pβ(α) for all non-zero multi-indicesα′ 6= 0. There-fore, if (a†)βψ ∈ L2(Rn) then aβψ ∈ L2(Rn). However, theconverse is not true forn > 1 dimensions, as the norm inEqn. (21) is independent of infinitely many coefficientscα,while Eqn. (20) is not. (This should be contrasted with theone-dimensional case, whereaψ ∈ L2(R) wasequivalenttoa†ψ ∈ L2(R).) On the other hand, as in then = 1 case,the conditiona†

kψ ∈ L2(Rn) is equivalent to the conditionsxkψ ∈ L2(Rn) and∂kψ ∈ L2(Rn).

We are in position to formulate a straightforward general-ization of Prop. 1. The proof is easy, so we omit it.

Proposition 4 (General Hermite expansions)Let ψ ∈ L2(Rn), with coefficients cα as in Eqn. (19), andlet β ∈ In be arbitrary. Assume (a†)βψ ∈ L2(Rn). Then(a†)β

′ψ ∈ L2(Rn) and aβ

′ψ ∈ L2(Rn) for all β′ ≤ β. Moreover,

the following points are equivalent:

1. (a†)βψ ∈ L2(Rn)

2. For all multi-indices γ ≤ β, xγ∂β−γψ ∈ L2(Rn).

3. ∑ααβ|cα|2 <+∞

We observe, that as we obtained forn = 1, condition 2 is acombined decay and smoothness condition onψ, and that thiscan be expressed as a decay-condition on the coefficients ofψin the Hermite basis by 3.

Exponential decay ofψ ∈ L2(Rn) implies that thatxγψ ∈L2(Rn) for all γ ∈ In. We now generalize Proposition 2 tothen-dimensional case.

Proposition 5 (Exponentially decaying functions)Assume that ψ ∈ L2(Rn) is such that for all γ ∈ In, xγψ ∈L2(Rn). Then, a sufficient criterion for (a†)βψ ∈ L2(Rn) is∂βψ ∈ L2(Rn). Moreover, for all µ ≤ β, we have xγ∂β−µψ ∈L2(Rn) for all γ ∈In such that γk = 0 whenever µk = 0, i.e.,the partial derivatives of lower order than β decay exponen-tially in the directions where the differentiation order is lower.

Proof: The proof is a straightforward application of then = 1 case in an inductive proof, together with the followingelementary fact concerning weak derivatives: If 1≤ j < k≤ n,and if x jψ(x) and∂kψ(x) are inL2(Rn), then, by the productrule,∂k(x jψ(x)) = x j(∂kψ(x)) ∈ L2(Rn). Notice, that Prop. 2trivially generalizes to a single index inn dimensions, i.e., toβ = βkek, since the integration by parts formula used is validin Rn as well. Similarly, the present Proposition is valid inn−1 dimensions, it is valid forβ= (0,β2, · · · ,βn).

Assume that our statement holds forn−1 dimensions. Wemust prove that it then holds inn dimensions. Assume then,that∂βψ ∈ L2(Rn). Let φ = ∂βψ ∈ L2. Moreover,∂β1

1 φ ∈ L2.Sinceψ is exponentially decaying, and by the product rule,xγ1

1 φ ∈ L2 for all γ1 ≥ 0. By Prop. 2,xγ11 ∂

β1−µ11 φ ∈ L2 for all

γ1 and 0< µ1 ≤ β1. Thus,xγ11 ∂

β−e1µ1ψ ∈ L2. Thus, the resultholds as long asµ = e1µ1; or equivalentlyµ = ekµk for anyk. To apply induction, letχ = xγ1

1 ∂β1−µ11 ψ ∈ L2. Note that

∂βχ ∈ L2 andxγχ ∈ L2 for all γ = (0,γ2, · · · ,γn). But by theinduction hypothesis,xγ∂β−µχ∈ L2 for all µ≤ β and allγ suchthat γk = 0 if µk = 0. This yields, using the product rule, thatxγ∂β−µψ∈ L2 for all µ≤ β and allγ such thatγk = 0 if µk = 0,which was the hypothesis forn dimensions, and the proof iscomplete. Notice, that we have proved that(a†)βψ ∈ L2 as aby-product. ♦

In order to generate a simple and useful result for approxi-mation inn dimensions, we consider the case whereψ decaysexponentially, andψ ∈ Hk(Rn), i.e.,∂βψ(x) ∈ L2(Rn) for allβ ∈ In with |β| = k. In this case, we may also generalizeEqn. (17). For this, we consider the shell-weightp(r) definedby

p(r) := ∑α∈In, |α|=r

|cα|2, (22)

wherecα = 〈Φα,ψ〉. Then,‖ψ‖2 = ∑∞r=0 p(r). Moreover, ifP

projects onto the shell-truncated Hilbert spacePR(Rn), then

‖Pψ‖2 =R

∑r=0

p(r).

9

Proposition 6 (Approximation in n dimensions)Let ψ ∈ L2(Rn) be exponentially decaying and given by

ψ(x) = ∑α

cαΦα(x).

Then ψ ∈ Hk(Rn), k≥ 0, if and only if

∑α

|α|k|cα|2 =∞

∑r=0

rkp(r) <+∞. (23)

The latter implies that

p(r) = o(r−(k+1)). (24)

Moreover, for the shell-truncated Hilbert space PR, the ap-proximation error is given by

‖(1−P)ψ‖=

(∞

∑r=R+1

p(r)

)1/2

. (25)

Proof: The only non-trivial part of the proof concernsEqn. (23). Sinceψ is exponentially decaying and sinceψ∈Hk

if and only if ∂βψ ∈ L2 for all β, |β| ≤ k, we know that∑αα

β|cα|2 < +∞ for all β, |β| ≤ k. Since|α|k is a polyno-mial of orderk with terms of typeaβαβ, aβ ≥ 0 and|β| = k,we have

∑α

|α|k|cα|2 = ∑β, |β|=k

aβ∑α

αβ|cα|2 <+∞.

On the other hand, sinceaβ ≥ 0 and the sum overβ has finitelymany terms,∑α |α|k|cα|<+∞ implies∑αα

β|cα|2 <+∞ for allβ, |β| = k, and thusψ ∈ Hk sinceψ was exponentially decay-ing. ♦

In applications, we often observe a decay of non-integralorder, i.e., there exists anε ∈ [0,1) such that we observe

p(r) = o(r−(k+1+ε)). (26)

This does not, of course, contradict our analysis. To see this,we observe that ifψ ∈ Hk(Rn) but ψ /∈ Hk+1(Rn), thenp(r)must decay at least as fast aso(r−(k+1)) but as fast aso(r−k+2).Thus, the decay exponent can be anything inside the interval[k+1,k+2).

Consider also the case whereψ ∈ Hk(Rn) for everyk, i.e.,we can differentiate it (weakly) as many times we like. Thenp(r) decays faster thanr−(k+1), for anyk≥ 0, giving so-calledexponential convergence of the Hermite series. Hence, func-tions that are best approximated by Hermite series are rapidlydecayingandvery smooth functionsψ.

C. Two electrons revisited

To clarify how the preceding discussion can be applied, wereturn to the exact solutions of the two-electron quantum dotconsidered in Sec. II D. Recall, that the wave functions wereon the form

ψ(r, θ) = eimθ f (r),

where f (r) decayed exponentially fast asr → ∞. Assumenow, thatψ ∈ Hk(R2), i.e., that all partial derivatives ofψ oforderk exists in the weak sense, viz,

∂ jx∂

k− jy ψ ∈ L2(R2), 0≤ j ≤ k,

wherex = r cos(θ) andy = r sin(θ). Then, by Proposition 5,(a†

x)j(a†

y)k− jψ ∈ L2(R2) for 0≤ j ≤ k as well.

The functionψ(r, θ) was expanded in Fock-Darwin orbitals,viz,

ψ(r, θ) =∞

∑n=0

cnΦFDn,m(r, θ).

Recall, that the shell numberN for ΦFDn,m was given byN =

2n+ |m|. Thus, the shell-weightp(N) is in this case simply

p(N) = |c(N−|m|)/2|2, N≥ |m|,

andp(N) = 0 otherwise. From Prop. 6, we have

∞

∑N=|m|

Nkp(N) <+∞,

which yields

|cn|= o(n−(k+1+ε)/2), 0≤ ε < 1,

as claimed in Sec. II D.

D. The analytic properties of many-electron wave functions

As we now have established the impact of weak regularityof the eigenfunctions on the approximation properties of theHO eigenfunctions for arbitrary number of particlesN andspatial dimensionsd, we mention some results, mainly dueto Hoffmann-Ostenhofet al.,27,30 concerning smoothness ofmany-electron wave functions. Strictly speaking, their resultsare valid only ind = 3 spatial dimensions, since the Coulombinteraction ind = 2 dimensions fails to be a Kato potential,the definition of which is quite subtle and out of the scope forthis article. See, however, Ref. 30. On the other hand, it is rea-sonable to assume that the results will still hold true, since theanalytical results of theN = 2 case is very similar in thed = 2andd = 3 cases: The eigenfunctions decay exponentially withthe same cusp singularities at the origin.24,25

Consider the Schrödinger equation(H−E)ψ(ξ) = 0, whereξ = (ξ1, · · · , ξNd) = (~r1, · · · ,~rN) ∈ RNd, and whereψ(ξ) is as-sumed only to be a solution locally, i.e., it may fail to satisfythe boundary conditions. (A proper solution is of course alsoa local solution.) Recall, thatψ has 2N spin-components. De-fine a coalesce pointξCP as a point where at least two particlescoincide, i.e.,~r j =~r`, j 6= `. Away from the set of such points,ψ(ξ) is real analytic, since the interaction is real analytic there.Near aξCP, the wave function has the form

ψ(ξ+ ξCP) = rkP(ξ/r)(1+ar)+O(rk+1),

10

wherer = ‖ξ‖, P is a hyper-spherical harmonic (on the sphereSNd−1) of degreek = k(ξCP), and wherea is a constant. Itis immediately clear, thatψ(ξ) is k+ 1 times weakly differ-entiable in a neighborhood ofξCP. However, atK-electroncoalesce points, i.e., at pointsξCP where K different elec-trons coincide, the integerk may differ. Using exponentialdecay of a proper eigenfunction, we haveψ∈Hmin(k)+1(RNd).Hoffmann-Ostenhofet al. also showed, that symmetry restric-tions on the spin-components due to the Pauli principle in-duces an increasing degreek of the hyper-spherical harmonicP, generating even higher order of smoothness. A generalfeature, is that the smoothness increases with the number ofparticles.

However, their results in this direction are not generalenough to ascertain theminimumof the values fork for a givenwave function, although we feel rather sure that such an anal-ysis is possible. This is, however, left for future work. Sufficeit to say, that the results are clearly visible in the numericalcalculations in Sec. V.

Another interesting direction of research has been under-taken by Yserentant,31 who showed that there are some veryhigh order mixed partial derivatives at coalesce points. Itseems unclear, though, if this can be exploited to improve theCI calculations further.

IV. THE CONFIGURATION INTERACTION METHOD

A. Convergence analysis using HO eigenfunction basis

The basic problem is to determine a few eigenvalues andeigenfunctions of the HamiltonianH in Eqn. (6), i.e.,

Hψk = Ekψk, k = 1, · · · ,kmax.

The CI method consists of approximating eigenvalues ofHwith those obtained by projecting the problem onto a finite-dimensional subspaceHh⊂H (X,N), whereX = Rd×±1as in Sec. II B. Thus, it is identical to the Ritz-Galerkin vari-ational method.15,32,33We comment, that the convergence ofthe Ritz-Galerkin method isnot simply a consequence of thecompleteness of the basis functions.15 We will analyze the CImethod when the model space is given by

Hh = PRH (X,N) = PR(N)

= span

Ψi1,··· ,iN : ∑

k

|αk| ≤ R

,

used in Refs. 11,34, for example, although other spaces alsoare common. The spacePR(N) given by

MR(N) := span

Ψi1,··· ,iN : ∀k, |αk| ≤ R,

i.e., a cut in the single-particle shell numbers instead of theglobal shell number is also common.9,12 For obvious reasons,PR(N) is often referred to as an “energy cut space”, whileMR(N) is referred to as a “direct product space”. See alsoRef. 4 for a further discussion.

As in Sec. III B, PR is the orthogonal projector onto themodel spacePR(N). We also defineQR = 1−PR as the pro-jector onto the excluded spaceH ⊥

h . The discrete eigenvalueproblem is then

(PRHPR)ψh,k = Eh,kψh,k, k = 1, · · · ,kmax.

Recall from Sec. II B, that eachψ ∈H (N) can be viewedas 2N component wave functionsψ(σ), eachσ correspondingto a different configuration of theN spin projections. In termsof Hermite functions inn = Nd dimensions, we write

ψ(σ)(~r1, · · · ,~rN) := ∑α

c(σ)α Φα(~r1, · · · ,~rN)

with corresponding shell-weightsp(r)(σ), viz,

p(r)(σ) := ∑α; |α|=r

|c(σ)α |.

The CI method becomes, in principle, exact asR→ ∞. In-deed, a widely-used name for the CI method is “exact diag-onalization.” This is somewhat a misnomer in light of theanalysis presented here and the fact that only a very limitednumber of degrees of freedom per particle is achievable.

It is clear that

PR(N)⊂MR(N)⊂PNR(N), (27)

so that studying the convergence in terms ofPR(N) is suffi-cient. In our numerical experiments we focus on the energycut model space. A comparison between the convergence ofthe two spaces is, on the other hand, an interesting topic forfuture research.

Using the results in Refs. 15,35 for non-degenerate eigen-values, we obtain an estimate for the error in the numericaleigenvalueEh as

Eh−E≤ [1+ν(R)](1+ Kλ)〈ψ,QRTψ〉, (28)

whereK is a constant, and whereν(R)→ 0 asR→ ∞. UsingTΦα = (Nd/2+ |α|)Φα and Eqn. (23), we obtain

〈ψ,QRTψ〉=∞

∑r=R+1

(Nd2

+ r

)∑σ

p(σ)(r).

Assume now, thatψ(σ) ∈ Hk(RNd) for all σ, so that accord-ing to Prop. 6, we will have

∞

∑r=0

rkp(σ)(r) <+∞

for all 2N valuesσ, implying thatrp(σ)(r) = o(r−k). We thenobtain, fork> 1,

〈ψ,(1−PR)Tψ〉= o(R−(k−1))+o(R−k).

For k = 1 (which is the worst case), we merely obtain〈ψ,(1−PR)Tψ〉 → 0 asR→ ∞. We assume, thatR is suf-ficiently large, so that theo(R−k) term can be neglected.

11

Again, we may observe a slight deviation from the decay,confer discussion after Prop. 6. Thus, we expect to observeeigenvalue errors on the form

Eh−E∼ (1+ Kλ)R−(k−1+ε), (29)

where 0≤ ε < 1.As for the eigenvector error‖ψh − ψ‖ (recall thatψh 6=

PRψ), we mention that

‖ψh−ψ‖ ≤ [1+η(R)] [(1+ Kλ)〈ψ,(1−PR)Tψ〉]1/2 ,

whereη(R)→ 0 asR→ ∞.

B. Effective interaction scheme

Effective interactions have a long tradition in nuclearphysics, where the bare nuclear interaction is basically un-known and highly singular, and where it must be renormal-ized and fitted to experimental data.1 In quantum chemistryand atomic physics, the Coulomb interaction is well-known sothere is no intrinsic need to formulate an effective interaction.However, in lieu of the in general low order of convergenceimplied by Eqn. (29), we believe that HO-based calculationsof the CI and coupled cluster type in general may benefit fromthe use of effective interactions.

A complete account of the effective interaction scheme out-lined here is out of scope for the present article, but we refer toRefs. 4,14,36,37 for details as well as numerical algorithms.

Recall, that the interaction is given by

U =N

∑i< j

C(i, j) =N

∑i< j

λ

‖~r i −~r j‖,

a sum of fundamental two-body interactions. For theN = 2problem we have, in principle, the exact solution, since theHamiltonian (7) can be reduced to a one-dimensional radialequation, e.g., the eigenproblem ofHr defined in Eqn. (8).This equation may be solved to arbitrarily high precision usingvarious methods, for example using a basis expansion in gen-eralized half-range Hermite functions.38 In nuclear physics, acommon approach is to take the best two-body CI calculationsavailable, whereR= O(103), as “exact” for this purpose.

We now define the effective Hamiltonian forN = 2 as aHermitian operatorHeff defined only withinPR(N = 2) thatgivesK = dim[PR(N = 2)] exacteigenvaluesEk of H, andK approximate eigenvectorsψeff,k. Of course, there are in-finitely many choices for theK eigenpairs, but by treatingU = λ/r12 as a perturbation, and “following” the unperturbedHO eigenpairs (λ = 0) through increasing values ofλ, onemakes the eigenvalues unique.4,39 The approximate eigenvec-tors ψeff,k ∈ PR(N = 2) are chosen by minimizing the dis-tance to the exact eigenvectorsψk ∈H (N = 2) while retain-ing orthonormality.36 This uniquely definesHeff for the two-body system. In terms of matrices, we have

Heff = U diag(E1, · · · ,EK)U†, (30)

whereX andY are unitary matrices defined as follows. LetU be theK×K matrix whosek’th column is the coefficientsof PRψk. Then the singular value decomposition ofU can bewritten

U = XΣY†,

whereΣ is diagonal. Then,

U := XY†.

The columns ofU are the projectionsPRψk “straightened out”to an orthonormal set. Eqn. (30) is simply the spectral decom-position ofHeff. Although different in form than most imple-mentations in the literature (e.g., Ref. 14), it is equivalent.

The effective two-body interactionCeff(i, j) is now givenby

Ceff(1,2) := Heff−PRT PR,

which is defined only withinPR(N = 2).TheN-body effective Hamiltonian is defined by

Heff := PRT PR+N

∑i< j

Ceff(i, j), (31)

wherePR projects ontoPR(N), and thusHeff is definedonlywithin PR(N). The diagonalization ofHeff(N) is equivalentto a perturbation technique where a certain class of diagramsis summed to infinite order in the full problem.37

Rigorous mathematical treatment of the convergence prop-erties of the effective interaction is, to the author’s knowl-edge, not available. Effective interactions have, however, en-joyed great success in the nuclear physics community, and westrongly believe that we soon will see sufficient proof of theimproved accuracy with this method. Indeed, in Sec. V we seeclear evidence of the accuracy boost when using an effectiveinteraction.

V. NUMERICAL RESULTS

A. Code description

We now present numerical results using the fullconfiguration-interaction method forN = 2–5 electronsin d = 2 dimensions. We will use both the “bare” Hamilto-nianH = T +U and the effective Hamiltonian (31).

Since the Hamiltonian commutes with angular momentumLz, the latter taking on eigenvaluesM ∈ Z, the Hamiltonianmatrix is block diagonal. (Recall, that the Fock-Darwin or-bitals ΦFD

n,m are eigenstates ofLz with eigenvaluem, so eachSlater determinant has eigenvalueM = ∑N

k=1mk.) Moreover,the calculations are done in a basis of joint eigenfunctions fortotal electron spinS2 and its projectionSz, as opposed to theSlater determinant basis used for convergence analysis. Suchbasis functions are simply linear combinations of Slater deter-minants within the same shell, and further reduce the dimen-sionality of the Hamiltonian matrix.12 The eigenfunctions of

12

H are thus labeled with the total spinS = 0,1, · · · , N2 for even

N andS = 12 ,

32 , · · · ,

N2 for oddN, as well as the total angular

momentumM = 0,1, · · · . (−M produce the same eigenvaluesasM, by symmetry.) We thus splitPR(N) (or MR(N)) intoinvariant subspacesPR(N,M,S) (MR(N,M,S)) and performcomputations solely within these, viz,

PR(N) =⊕M,S

PR(N,M,S).

The calculations were carried out with a code similar to thatdescribed by Rontaniet al. in Ref. 12. Table I shows compar-isons of the present code with that of Table IV of Ref. 12 forvarious parameters using the model spaceMR(N,M,S). Ta-ble I also shows the caseλ = 1, N = 2, M = 0, andS = 0,whose exact lowest eigenvalue isE0 = 3, cf. Sec. II D. Wenote that there are some discrepancies between the results inthe last digits of the results of Ref. 12. The spacesMR(N)were identical in the two approaches, i.e., the number of basisfunctions and the number of non-zero matrix elements pro-duced are cross-checked and identical.

We have checked that the code also reproduces the resultsof Refs. 11,13,40, using thePR(N,M,S) spaces. Our codewill be published and described in detail elsewhere,41 whereit will also be demonstrated that it reproduces the eigenval-ues of an analytically solvableN-particle system42 to machineprecision.

B. Experiments

For the remainder, we only use the energy cut spacesPR(N,M,S). Figure 2 shows the development of the low-est eigenvalueE0 = E0(N,M,S) for N = 4, M = 0,1,2 andS = 0 as function of the shell truncation parameterR, usingboth HamiltoniansH andHeff. Apparently, the effective in-teraction eigenvalues provide estimates for the ground stateeigenvalues that are better than the bare interaction eigenval-ues. This effect is attenuated with higherN, due to the factthat the two-electron effective Coulomb interaction does nottake into account three- and many-body effects which becomesubstantial for higherN.

We take theHeff-eigenvalues as “exact” and graph the rel-ative error inE0(N,M,S) as function ofR on a logarithmicscale in Fig. 3, in anticipation of the relation

ln(Eh−E)≈C+α lnR, α=−(k−1+ ε). (32)

The graphs show straight lines for largeR, while for smallRthere is a transient region of non-straight lines. ForN = 5,however,λ = 2 is too large a value to reach the linear regimefor the range ofRavailable, so in this case we chose to plot thecorresponding error for the very small valueλ= 0.2, showingclear straight lines in the error. The slopes are more or lessindependent ofλ, as observed in different calculations.

In Fig. 4 we show the corresponding graphs when usingthe effective HamiltonianHeff. We estimate the relative erroras before, leading to artifacts for the largest values ofR dueto the fact that there is a finite error in the best estimates for

6 8 10 12 14 16 18 20 2213.6

13.7

13.8

13.9

14

14.1

14.2

14.3

14.4

14.5

14.6

R

E0

FIG. 2: Eigenvalues forN = 4, S = 0, λ = 2 as function ofR for H(solid) andHeff (dashed).M = 0,1 and 2 are represented by squares,circles and stars, resp.

the eigenvalues. However, in all cases there are clear, linearregions, in which we estimate the slopeα. In all cases, theslope can be seen to decrease by at least∆α ≈ −1 comparedto Fig. 3, indicating that the effective interaction indeed accel-erates the CI convergence by at least an order of magnitude.We also observe, that the relative errors are improved by anorder of magnitude or more for the lowest values ofR shown,indicating the gain in accuracy when using small model spaceswith the effective interaction.

Notice, that for symmetry reasons only even (odd)R foreven (odd)M yields increases in basis size dim[PR(N,M,S)],so only these values are included in the plots.

To overcome the limitations of the two-body effective inter-action for higherN, an effective three-body interaction couldbe considered, and is hotly debated in the nuclear physicscommunity. (In nuclear physics, there are also more compli-cated three-body effective forces that need to be included.43)However, this will lead to a huge increase in memory con-sumption due to extra nonzero matrix elements. At the mo-ment, there are no methods available that can generate the ex-act three-body effective interaction with sufficient precision.

We stress, that the relative error decreases very slowly ingeneral. It is a common misconception, that if a number ofdigits of E0(N,M,S) is unchanged betweenRandR+2, thenthese digits have converged. This is not the case, as is easilyseen from Fig. 3. Take for instanceN = 4, M = 0 andS = 0,andλ = 2. ForR= 14 andR= 16 we haveE0 = 13.84491andE0 = 13.84153, respectively, which would give a relativeerror estimate of 2.4×10−4, while the correct relative error is1.3×10−3.

The slopes in Fig. 3 vary greatly, showing that the eigen-functions indeed have varying global smoothness, as pre-dicted in Sec. III D. For(N,M,S) = (5,3,5/2), for example,α ≈ −4.2, indicating thatψ ∈ H5(R10). It seems, that higher

13

TABLE I: Comparison of current code and Ref. 12

R= 5 R= 6 R= 7N λ M 2S Current Ref. 12 Current Ref. 12 Current Ref. 122 1 0 0 3.013626 3.011020 3.009236

2 0 0 3.733598 3.7338 3.731057 3.7312 3.729324 3.72951 2 4.143592 4.1437 4.142946 4.1431 4.142581 4.1427

3 2 1 1 8.175035 8.1755 8.169913 8.166708 8.16714 1 1 11.04480 11.046 11.04338 11.04254 11.043

0 3 11.05428 11.055 11.05325 11.05262 11.0534 6 0 0 23.68944 23.691 23.65559 23.64832 23.650

2 4 23.86769 23.870 23.80796 23.80373 23.8055 2 0 5 21.15093 21.15 21.13414 21.13 21.12992 21.13

4 0 5 29.43528 29.44 29.30898 29.31 29.30251 29.30

6 8 10 12 14 16 18 2010

−6

10−5

10−4

10−3

10−2

10−1

100

101

R

Rel

ativ

e er

ror

Relative error for N=2, λ = 2

M=0, S=0, α = −1.0477

M=0, S=2, α = −2.1244

M=3, S=0, α = −3.1749

M=3, S=2, α = −4.0188

6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

10−1

R

Rel

ativ

e er

ror

Relative error for N=3, λ=2

M=0, S=1, α = −1.2772M=0, S=3, α = −2.1716M=2, S=1, α = −1.3093M=2, S=3, α = −2.5417

6 8 10 12 14 16 18 20

10−4

10−3

10−2

10−1

R

Rel

ativ

e er

ror


M=0, S=0, α = −1.4233M=0, S=4, α = −2.8023M=3, S=2, α = −1.5327M=3, S=4, α = −3.2109

6 8 10 12 14 16 18 20

10−4

10−3

10−2

R

Rel

ativ

e er

ror

Relative error for N=5, λ = 0.2

M=0, S=1, α = −1.5150M=0, S=5, α = −3.6563M=3, S=1, α = −1.8159M=3, S=5, α = −4.2117

FIG. 3: Plot of relative error using the bare interaction for variousN, M andS. Clearo(Rα) dependence in all cases.

14

6 8 10 12 14 16 18 20

10−10

10−5

100

R

Rel

ativ

e er

ror


M=0, S=0, U

eff

M=0, S=2, Ueff

M=3, S=0, Ueff

M=3, S=2, Ueff

6 8 10 12 14 16 18 20

10−6

10−5

10−4

10−3

10−2

10−1

100

101

R

Rel

ativ

e er

ror


M=0, S=1, U

eff, α = −4.1735

M=0, S=3, Ueff

, α = −4.7914

M=3, S=1, Ueff

, α = −4.4535

M=3, S=3, Ueff

, α = −4.7521

6 8 10 12 14 16 18 20

10−5

10−4

10−3

10−2

10−1

100

101

R

Rel

ativ

e er

ror


M=0, S=0, U

eff, α = −4.6093

M=0, S=4, Ueff

, α = −5.3978

M=3, S=0, Ueff

, α = −4.7816

M=3, S=4, Ueff

, α = −6.2366

7 9 11 13 15 17 18

10−5

10−4

10−3

10−2

R

Rel

ativ

e er

ror

Relative error for N=5, λ = 0.2

M=0, S=1, U

eff, α = −3.8123

M=0, S=5, Ueff

, α = −4.4925

M=3, S=1, Ueff

, α = −3.6192

M=3, S=5, Ueff

, α = −5.0544

FIG. 4: Plot of relative error using an effective interaction for variousN, M andS. Clearo(Rα) dependence in all cases, but notice artifactswhenR is large, due to errors in most correct eigenvalues. TheR= 5 case does not contain enough data to compute the slopes with sufficientaccuracy.

S gives higherk, as a rule of thumb. Intuitively, this is be-cause the Pauli principle forces the wave function to be zeroat coalesce points, thereby generating smoothness.

VI. DISCUSSION AND CONCLUSION

We have studied approximation properties of Hermite func-tions and harmonic oscillator eigenfunctions. This in turn al-lowed for a detailed convergence analysis of numerical meth-ods such as the CI method for the parabolic quantum dot.Our main conclusion, is, that for wave functionsψ ∈ Hk(Rn)falling off exponentially as‖x‖ → ∞, the shell-weight func-

tion p(r) decays asp(r) = o(r−k−1). Applying this to theconvergence theory of the Ritz-Galerkin method, we obtainedthe estimate (28) for the error in the eigenvalues. A completecharacterization of the upper bound on the differentiabilityk, i.e., inψ(s) ∈ Hk, as well as a study of the constantK inEqn. (29), would complete our knowledge of the convergenceof the CI calculations.

We also demonstrated numerically, that the use of a two-body effective interaction accelerates the convergence by atleast an order of magnitude, which shows that such a methodshouldbe used whenever possible. On the other hand, a rig-orous mathematical study of the method is yet to come.

The theory and ideas presented in this article should in prin-

15

ciple be universally applicable. In fact, Figs. 1–3 of Ref. 14clearly indicates this, where the eigenvalues of3He as functionof model space size are graphed both for bare and effective in-teractions, showing some of the features we have discussed.

We would like to point out, that coupled-cluster methodshave eigenvalue errors bounded by Eqn. (29) as well.2 A fur-ther study of coupled-cluster calculations using effective inter-actions for parabolic quantum dots could be a very interestingventure, both with respect to the study of the coupled clustermethods, but also for the effective interactions.

Other interesting future studies would be a direct compar-ison of the direct product model spaceMR(N) and our en-ergy cut model spacePR(N). Both techniques are com-mon, but may have different numerical characteristics. In-deed, dim[MR(N)] grows much quicker than dim[PR(N)],while we are uncertain of whether the increased basis sizeyields a corresponding increased accuracy.

We have focused on the parabolic quantum dot, firstly be-cause it requires relatively small matrices to be stored, due toconservation of angular momentum, but also because it is awidely studied model. Our analysis is, however, general, andapplicable to other systems as well, e.g., quantum dots trappedin double-wells, finite wells, and so on. Indeed, by adding aone-body potentialV to the HamiltonianH = T +U we maymodel other geometries, as well as adding external fields.44–46

As long as the potential is sufficiently weak, the differentia-bility of the wave function should not be affected.

APPENDIX A: MULTI-INDICES

A very handy tool for compact and unified notation whenthe dimensionn of the underlying measure spaceRn is aparameter, are multi-indices. The setIn of multi-indicesare defined asn-tuples of non-negative indices, viz,α =(α1, · · · ,αn).

We define several useful operations on multi-indices as fol-lows. Let u be a formal vecor ofn symbols. Moreover, letφ(ξ) = φ(ξ1, ξ2, · · · , ξn) be a function. Then, define

|α| := α1 +α2 + · · ·+αn (A1)

α! := α1!α2! · · ·αn! (A2)

α±β := (α1±β1, · · · ,αn±βn) (A3)

uα := uα11 uα2

2 · · ·uαnn , (A4)

∂αφ(ξ) :=∂α1

∂ξα11

∂α1

∂ξα22· · · ∂

αd

∂ξα1d

φ(ξ) (A5)

In Eqn. (A3), the result may not be a multi-index when wesubtract two indices, but this will not be an issue for us. No-tice, that Eqn. (A5) is a mixed partial derivative of order|α|.Moreover, we say thatα < β if and only if α j < β j for all j.We defineα = β similarly. We also define “basis indices”ejby (ej) j′ = δ j, j′ . These relations will be used extensively inSec. III B. We comment, that we will often use the notation∂x := ∂

∂x and∂k := ∂∂ξk

to simplify notation. Thus,

∂α = (∂α11 , · · · ,∂αn

n ),

consistent with Eqn. (A4).

APPENDIX B: WEAK DERIVATIVES AND SOBOLEVSPACES

We present a quick summary of weak derivatives and re-lated concepts needed. The material is elementary and superfi-cial, but probably unfamiliar to many readers, so we include ithere. Many terms will be left undefined; if needed, the readermay consult standard texts, e.g., Refs. 6,47.

The spaceL2(Rn) is defined as

L2(Rn) :=ψ : Rn → C :

∫Rn|ψ(ξ)|2dnξ <+∞

, (B1)

where the Lebesgue integral is more general than the Riemann“limit-of-small-boxes” integral. It is important that we iden-tify two functionsψ andψ1 differing only at a setZ ∈ Rn

of measure zero. Examples of such sets are points ifn≥ 1,curves ifn≥ 2, and so on, and countable unions of such. Forexample, the rationals constitute a set of measure zero inR.Under this assumption,L2(Rn) becomes a Hilbert space withthe inner product

〈ψ1,ψ2〉 :=∫

Rnψ1(ξ)∗ψ2(ξ)dnξ,

where the asterisk denotes complex conjugation.The classical derivative is too limited a concept for the ab-

stract theory of partial differential equations, including theSchrödinger equation. LetC ∞

0 be the set of infinitely differ-entiable functions which are non-zero only in a ball of finiteradius. Of course,C ∞

0 ⊂ L2(Rn). Let ψ ∈ L2(Rn), and letα ∈ In be a multi-index. If there exists av ∈ L2(Rn) suchthat, for allφ ∈ C ∞

0 ,∫Rn

(∂αφ(ξ))ψ(ξ)dnξ = (−1)|α|∫

Rnφ(ξ)v(ξ)dnξ,

then∂αψ := v∈ L2 is said to be aweak derivative, or distri-butional derivative, ofψ. In this way, the weak derivative isdefined in an average sense, using integration by parts.

The weak derivative is unique (up to redefinition on a set ofmeasure zero), obeys the product rule, chain rule, etc.

It is easily seen, that ifψ has a classical derivativev ∈L2(Rn) it coincides with the weak derivative. Moreover, ifthe classical derivative is defined almost everywhere (i.e., ev-erywhere except for a set of measure zero), thenψ has a weakderivative.

The Sobolev spaceHk(Rn) is defined as the subset ofL2(Rn) given by

Hk(Rn) :=ψ ∈ L2 : ∂αψ ∈ L2, ∀α ∈In, |α| ≤ k

.

(B2)The Sobolev space is also a Hilbert space with the inner prod-uct

(ψ1,ψ2) := ∑α, |α|≤k

〈∂αψ1,∂αψ2〉,

and this is the main reason why one obtains a unified theoryof PDE using such spaces.

16

The spaceHk(Rn) for n> 1 is a big space – there are someexceptionally ill-behaved functions there, for example thereare functions inHk that are unbounded on arbitrary small re-gions but still differentiable. (Hermite series for such func-tions would still converge faster than, e.g., for a function witha jump discontinuity!) For our purposes, it is enough to realizethat the Sobolev spaces offer exactly the notion of derivativewe need in our analysis of the Hermite function expansions.

ACKNOWLEDGMENTS

The author wishes to thank Prof. M. Hjorth-Jensen (CMA)for useful discussions, suggestions and feedback, and alsoDr. G. M. Coclite (University of Bari, Italy) for mathemati-cal discussions. This work was financed by CMA through theNorwegian Research Council.

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York, USA, 1995).

3 Geometry of effectiveHamiltonians

This paper appeared in volume 78 of Physical Review C, 2008.

PHYSICAL REVIEW C 78, 044330 (2008)

Geometry of effective Hamiltonians

Simen Kvaal*

Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway(Received 13 August 2008; published 31 October 2008)

We give a complete geometrical description of the effective Hamiltonians common in nuclear shell-modelcalculations. By recasting the theory in a manifestly geometric form, we reinterpret and clarify several points.Some of these results are hitherto unknown or unpublished. In particular, commuting observables and symmetriesare discussed in detail. Simple and explicit proofs are given, and numerical algorithms are proposed that improveand stabilize methods commonly used today.

DOI: 10.1103/PhysRevC.78.044330 PACS number(s): 21.30.Fe, 21.60.−n

I. INTRODUCTION

Effective Hamiltonians and interactions are routinely usedin shell-model calculations of nuclear spectra [1–3]. Thepublished mathematical theory of the effective Hamiltonian iscomplicated and usually focuses on perturbation theoreticalaspects, diagram expansions, etc. [4–12]. In this article,we recast and reinterpret the basic elements of the theorygeometrically. We focus on the geometric relationship betweenthe exact eigenvectors |ψk〉 and the effective eigenvectors|ψeff

k 〉, both for the usual non-Hermitian Bloch-Brandoweffective Hamiltonian [1,4,5,9] and for the Hermitian effectiveHamiltonian [6,11,13,14], which we dub the canonical effec-tive Hamiltonian because of its geometric significance. Thisresults in a clear geometric understanding of the decouplingoperator ω (defined in Sec. III C), and a simple proof and char-acterization of the Hermitian effective Hamiltonian in termsof subspace rotations, in the same way as the non-HermitianHamiltonian is characterized by subspace projections.

The goal of effective interaction theory is to devise aHamiltonian Heff in a model space P of (much) smallerdimension m than the dimension n of Hilbert space H,with m exact eigenvalues of the original Hamiltonian H =H0 + H1, where H1 is usually considered as a perturbation.The model space P is usually taken as the span of a feweigenvectors |ek〉mk=1 of H0, i.e., the unperturbed Hamiltonianin a perturbational view.

Effective Hamiltonians in A-body systems must invariablybe approximated (otherwise there would be no need forHeff), usually by many-body perturbation theory, but sincesuch approaches suffer from bad convergence properties ingeneral [15], a subcluster approximation approach has beenproposed and is in wide use today [3,6]. In this case, theexact a-body canonical effective Hamiltonian is computed,where a < A. From this, one extracts an effective a-bodyinteraction and applies it to the A-body system. We present anew algorithm for computing the exact effective Hamiltonian,which is useful in the subcluster approach to Heff . It isconceptually and computationally simpler than the usualone, which relies on both matrix inversion and square root[3,11], because the only nontrivial matrix operation is the

*[email protected]

singular value decomposition (SVD). This algorithm is asimple by-product of the geometric approach to analyzingthe effective Hamiltonians. We comment that in the subclusterapproach, it is invariably assumed that the a-body Hamiltonianis exactly diagonizable, so its exact eigenpairs are available.Thus, algorithms for the subcluster approach to Heff are notperturbative in nature.

The article is organized as follows. In Sec. II we introducesome notation and define the singular value decomposition oflinear operators and the principal angles and vectors betweentwo linear spaces. In Sec. III we define and analyze theBloch-Brandow and canonical effective Hamiltonians. Themain part consists of a geometric analysis of the exacteigenvectors and forms the basis for the analysis of the effectiveHamiltonians. We also discuss the impact of symmetries ofthe Hamiltonian, i.e., conservation laws. In Sec. IV we giveconcrete matrix expressions and algorithms for computing theeffective Hamiltonians; and in the canonical case, it is, to thebest of our knowledge, previously unknown. In Sec. V we sumup and briefly discuss the results and possible future projects.

II. TOOLS AND NOTATION

A. Linear spaces and operators

We shall use the Dirac notation for vectors, inner products,and operators, in order to make a clear, basis-independentformulation. By F ,G, etc., we denote (finite dimensional)Hilbert spaces, and vectors are denoted by kets, e.g., |ψ〉, asusual. Our underlying Hilbert space is denoted byH, with n =dim(H). In general, n is infinite. We shall, however, assume itto be finite. Our results are still valid in the infinite dimensionalcase if H is assumed to have a discrete spectrum and at leastm linearly independent eigenvectors.

We are also given a Hamiltonian H , a linear, Hermitianoperator (i.e., H = H †) on H. Its spectral decomposition isdefined to be

H =n∑

k=1

Ek|ψk〉〈ψk|.

Thus, Ek and |ψk〉 are the (real) eigenvalues and (orthonormal)eigenvectors, respectively.

0556-2813/2008/78(4)/044330(9) 044330-1 ©2008 The American Physical Society

SIMEN KVAAL PHYSICAL REVIEW C 78, 044330 (2008)

We are also given a subspace P ⊂ H, called the modelspace, which in principle is arbitrary. Let |ek〉mk=1 be anorthonormal basis, for definiteness, viz.,

P := span|ek〉 : k = 1, . . . , m.Let P be its orthogonal projector, i.e.,

P : H → P, P =m∑

j=1

|ej 〉〈ej |, m = dim(P) n.

The basis |ej 〉mj=1 is commonly taken to be eigenvectors forH0.

The orthogonal complement of the model space, Q = P⊥,has the orthogonal projector Q = 1 − P , and is called theexcluded space.

This division of H into P and Q transfers to operators inH. These are in a natural way split into four parts, viz., for anarbitrary operator A,

A= (P + Q)A(P + Q) =PAP + PAQ + QAP + QAQ,

(1)

where PAP maps the model space into itself, QAP maps Pinto Q, and so forth. It is convenient to picture this in a blockform of A, viz.,

A =[

PAP PAQ

QAP QAQ.

]

B. Singular value decomposition

A recurrent tool in this work is the singular value decompo-sition (SVD) of an operator A : X → Y . Here, p = dim(X )and q = dim(Y) are arbitrary. Then there exists orthonormalbases |xk〉pk=1 and |yk〉qk=1 of X and Y , respectively, andr = min(p, q) non-negative real numbers σk with σk σk+1

for all k, such that

A =r∑

k=1

σk|yk〉〈xk|.

This is the SVD of A, and it always exists. It may happen thatsome of the basis vectors do not participate in the sum, eitherif p = q or if σk = 0 for some k.

The vectors |xk〉 are called right singular vectors, while|yk〉 are called left singular vectors. The values σk arecalled singular values, and A is one-to-one and onto (i.e.,nonsingular) if and only if σk > 0 for all k, and p = q. Theinverse is then

A−1 =r∑

k=1

1

σk

|xk〉〈yk|,

as easily verified.A recursive variational characterization of the singular

values and vectors is the following [16]:

σk = max|u〉∈X , 〈u|u〉=1〈u|uj 〉=0, j<k

max|v〉∈Y, 〈v|v〉=1〈v|vj 〉=0, j<k

Re〈v|A|u〉

=:〈vk|A|uk〉. (2)

The latter equality implicitly states that the maximum isactually real. The SVD is very powerful, as it gives aninterpretation and representation of any linear operator A as asimple scaling with respect to one orthonormal basis, and thentransformation to another. The singular vectors are not unique,but the singular values are.

C. Principal angles and vectors

Important tools for comparing linear subspaces F and Gof H are the principal angles and principal vectors [17,18].The principal angles generalize the notion of angles betweenvectors to subspaces in a natural way. They are also calledcanonical angles. Assume that

p = dim(F) q = dim(G) 1.

(If p < q, we simply exchange F and G.) Then, q principalangles θk ∈ [0, π/2], with θk θk+1 for all k, and the leftand right principal vectors |ξk〉 ∈ F and |ηk〉 ∈ G are definedrecursively through

cos θk = max|ξ 〉∈F , 〈ξ |ξ 〉=1〈ξ |ξj 〉=0, j<k

max|η〉∈G, 〈η|η〉=1〈η|ηj 〉=0, j<k

Re〈ξ |η〉

=:〈ξk|ηk〉. (3)

Again, the last equality implicitly states that the maximumactually is real. One sees that θk is the angle between |ξk〉 ∈ Fand |ηk〉 ∈ G.

It is evident from Eqs. (2) and (3) that the principal anglesand vectors are closely related to the SVD. Indeed, if weconsider the product of the orthogonal projectors PF and PGand compute the SVD, we obtain

PFPG =p∑

k=1

|ξk〉〈ξk|q∑

j=1

|ηk〉〈ηk| =q∑

k=1

cos θk|ξk〉〈ηk|,

where we extended the orthonormal vectors |ξk〉qk=1 withp − q vectors into a basis for F , which is always possible.This equation in particular implies the additional orthogonalityrelation 〈ξj |ηk〉 = δj,k cos θk on the principal vectors.

The principal vectors constitute orthonormal bases thatshould be rotated into each other if the spaces were to bealigned. Moreover, the rotations are by the smallest anglespossible.

III. EFFECTIVE HAMILTONIANS

A. Similarity transforms

The goal of the effective Hamiltonian is to reproduceexactly m of the eigenvalues, and (necessarily) approximatelym of the eigenvectors. We shall assume that the first m

eigenpairs (Ek, |ψk〉), k = 1, . . . , m, defines these. We definethe space E as

E := span|ψk〉 : k = 1, . . . , m.The orthogonal projector P ′ onto E is

P ′ =m∑

k=1

|ψk〉〈ψk|. (4)

044330-2

GEOMETRY OF EFFECTIVE HAMILTONIANS PHYSICAL REVIEW C 78, 044330 (2008)

We denote by (Ek, |ψeffk 〉), k = 1, . . . , m, the effective

Hamiltonian eigenvalues and eigenvectors. Of course, the|ψeff

k 〉 ∈ P must constitute a basis for P , but not necessarilyan orthonormal basis. Geometrically, we want |ψeff

k 〉 to be asclose as possible to |ψk〉, i.e., we want E to be as close to P aspossible.

Let |ψeffk 〉 be the bi-orthogonal basis, i.e., 〈ψeff

j |ψeffk 〉 = δj,k ,

so that

P =m∑

k=1

|ek〉〈ek| =m∑

k=1

∣∣ψeffk

⟩⟨ψeff

k

∣∣.The spectral decomposition of Heff becomes

Heff =m∑

k=1

Ek

∣∣ψeffk

⟩⟨ψeff

k

∣∣.Since Heff is to have eigenvalues identical to m of those of H ,and since Heff operates only in P , we may relate Heff to H

through a similarity transform, viz.,

Heff = PHP = P (e−SHeS)P, (5)

where exp(S)exp(−S) = I . Any invertible operator has alogarithm, so Eq. (5) is completely general.

Now, Heff = PHP is an effective Hamiltonian only if theBloch equation

QHP = Qe−SHeSP = 0 (6)

is satisfied [8], since P is then invariant under the action of H .The eigenvectors of Heff are now given by∣∣ψeff

k

⟩ = e−S |ψk〉 ∈ P, k = 1, . . . , m. (7)

Thus, an effective Hamiltonian can now be defined for everyS such that Eq. (6) holds. It is readily seen that H = H † ifand only if S is skew-Hermitian, i.e., that S† = −S. There isstill much freedom in the choice of exponent S. Indeed, givenany invertible operator A in P, A−1HeffA is a new effectiveHamiltonian with the same effective eigenvalues as Heff , and|ψeff〉 = A−1exp(−S)|ψ〉.

B. Geometry of the model space

We will benefit from a detailed discussion of the spaces Eand P before we discuss the Bloch-Brandow and canonicaleffective Hamiltonians in detail.

Since dim(P) = dim(E) = m, the closeness of the effectiveand exact eigenvectors can be characterized and measured bythe orientation of E relative to P in H, using m canonicalangles θk and principal vectors |ηk〉 ∈ E and |ξk〉 ∈ P . Recallthat cos θk = 〈ξk|ηk〉 and that the angles θk ∈ [0, π/2] werethe smallest possible such that the principal vectors are theorthonormal bases of P and E that are closest to each other.

We now define the unitary operator Z = exp(G) that rotatesP into E according to this description, i.e., we should haveZ|ξk〉 = |ηk〉. In Fig. 1 the plane spanned by |ηk〉 and |ξk〉 ifθk > 0 is depicted. Recall that 〈ξj |ηk〉 = cos θj δj,k . Note that|ξk〉 = |ηk〉 if and only if θk = 0, and the plane degenerates intoa line. If θk > 0, the vector |χk〉 is defined so that it together

FIG. 1. Plane spanned by |ξk〉 and |ηk〉, and action of projectorsP and Q on |ηk〉.with |ξk〉 is an orthonormal basis for the plane, viz.,

|ηk〉 = P |ηk〉 + Q|ηk〉 = cos(θk)|ξk〉 + sin(θk)|χk〉, (8)

where

|χk〉 = Q|ηk〉〈ηk|Q|ηk〉1/2 . (9)

Thus, |χk〉 ∪ |ξk〉 is an orthonormal basis forP ⊕ E , whosedimension is 2m − nz, where nz is the number of θk = 0. Theset |χk〉 : θk > 0, k = 1, . . . , m is an orthonormal basis forQE which contains Q|ψk〉 for all k = 1, . . . , m.

The operator Z is now defined as a rotation in P ⊕ E , i.e.,by elementary trigonometry,

Z|ξk〉:= |ηk〉,(10)

Z|ηk〉:= 2 cos θk|ηk〉 − |ξk〉.In terms of the orthonormal basis, we obtain a manifest planarrotation for each k, i.e.,

Z|χk〉 = cos θk|χk〉 − sin θk|ξk〉,(11)

Z|ξk〉 = sin θk|χk〉 + cos θk|ξk〉.On the rest of the Hilbert space,H (P ⊕ E), Z is the identity.The operator Z implements the so-called direct rotation [19]of P into E . From Eq. (11) we obtain

Z:= I +m∑

k=1

[cos(θk) − 1](|χk〉〈χk| + |ξk〉〈ξk|)(12)

+m∑

k=1

sin(θk)(|χk〉〈ξk| − |ξk〉〈χk|).

It is instructive to exhibit the Lie algebra element G ∈ su(n)such that Z = exp(G) ∈ SU(n). Since we have Eq. (11), it iseasy to do this. Indeed, taking the exponential of

G =m∑

k=1

θk(|χk〉〈ξk| − |ξk〉〈χk|), (13)

by summing the series for sin(θ ) and cos(θ ), we readily obtainZ = exp(G), the desired result. Moreover, observe that thekth term in Eq. (13) commutes with the j th term, so, exp(G)is exhibited as a sequence of commuting rotations using thecanonical angles θk .

044330-3


C. Bloch-Brandow effective Hamiltonian anddecoupling operator

For reference, we review some properties of the Bloch-Brandow effective Hamiltonian, which we denote by H BB

eff[1,4,5,9]. The effective eigenvectors |ψeff

k 〉 are defined by∣∣ψeffk

⟩:= P |ψk〉 := |Pψk〉. (14)

Since |Pψk〉 are the orthogonal projections of |ψk〉 ontoP , wededuce that the Bloch-Brandow effective eigenvectors are theclosest possible to the exact model space eigenvectors. In thissense, the Bloch-Brandow effective Hamiltonian is the optimalchoice.

It is obvious that H BBeff is non-Hermitian, as rejecting the

excluded space eigenvector components renders the effectiveeigenvectors non-orthonormal, i.e.,⟨

ψeffj |ψeff

k

⟩ = δj,k − 〈ψj |Q|ψk〉 = δj,k.

In terms of similarity transforms, we obtain H BBeff by setting

S = ω, the so-called decoupling operator or correlation oper-ator [1,11]. It is defined by ω = QωP and the equation

ωP |ψk〉 := Q|ψk〉. (15)

Again, for this to be a meaningful definition, |Pψk〉mk=1 mustbe a basis for P .

Since ω2 = 0, exp(±ω) = 1 ± ω, and Eq. (7) becomes

e−ω|ψk〉 = (1 − ω)|ψk〉 = (1 − Q)|ψk〉 = |Pψk〉.For H BB

eff we thus obtain

H BBeff = Pe−ωHeωP = PH (P + ω). (16)

After this initial review, we now relate ω to the geometry ofE andP . The SVD of ω is readily obtainable by expanding theprincipal vectors |ηk〉mj=1 in the m eigenvectors |ψk〉mk=1,sets, which both constitute a basis for E , and inserting intoEq. (15). We have

Q|ηk〉 =m∑

j=1

Q|ψj 〉〈ψj |ηk〉

=m∑

j=1

ωP |ψj 〉〈ψj |ηk〉 = ωP |ηk〉,

that is,

ω (cos θk|ξk〉) = sin θk|χj 〉.The result is

ω =m∑

k=1

tan θk|χk〉〈ξk|, (17)

which is the SVD of ω. The operator ω is thus exhibited as anoperator intimately related to the principal angles and vectorsof P and E : it transforms the principal vectors of P into anorthonormal basis for QE , with coefficients determined by thecanonical angles θk . Using Eq. (8) we obtain an alternativeexpression, viz.,

ω + P =m∑

k=1

1

cos θk

|ηk〉〈ξk|. (18)

D. Canonical effective Hamiltonian

Hermitian effective Hamiltonians have independentlybeen introduced by various authors since 1929, when VanVleck [13,14,20] introduced a unitary transformation H =exp(−S)H exp(S) to decouple the model space to secondorder in the interaction. In 1963, Primas [21] considered anorder by order expansion of this H using the Baker-Campbell-Hausdorff formula and commutator functions to determineS, a technique also used in many other settings in whicha transformation is in a Lie group, see, e.g., Ref. [22] andreferences therein. This approach was elaborated by Shavittand Redmon [7], who were the first to mathematically connectthis Hermitian effective Hamiltonian to H BB

eff , as in Eq. (27)below. In the nuclear physics community, Suzuki [23] hasbeen a strong advocate of Hermitian effective interactions andthe a-body subcluster approximation to the A-body effectiveinteraction [3,11,23]. Hermiticity in this case is essential.

Even though a Hermitian effective Hamiltonian is notunique because of the non-uniqueness of S = −S†, the variousHermitian effective Hamiltonians put forward in the literatureall turn out to be equivalent [6]. In the spirit of Klein and Shavitt[6,7] we employ the term “canonical effective Hamiltonian”since this emphasizes the “natural” and geometric nature of theHermitian effective Hamiltonian, which we denote by H c

eff .Recall the spectral decomposition

H ceff =

m∑k=1

Ek

∣∣ψeffk

⟩⟨ψeff

k

∣∣,where the (orthonormal) effective eigenvectors are now de-fined by the following optimization property: The effectiveeigenvectors |ψeff

k 〉 are the closest possible to the exacteigenvectors |ψk〉 while still being orthonormal. Thus, wherethe Bloch-Brandow approach globally minimizes the distancebetween the eigenvectors, at the cost of non-orthonormality,the canonical approach has the unitarity constraint on thesimilarity transformation, rendering H c

eff Hermitian.Given a collection = |φ1〉, . . . , |φm〉 ⊂ P of m vectors,

which are candidates for effective eigenvectors, we define thefunctional S[] by

S[]:=m∑

k=1

‖|φk〉 − |ψk〉‖2

=m∑

k=1

‖Q|ψk〉‖2 +m∑

k=1

‖P |ψk〉 − |φk〉‖2

= m +m∑

k=1

‖|φk〉‖2 − 2 Rem∑

k=1

〈ψk|P |φk〉. (19)

In the last equality, we have used ‖|ψk〉‖ = 1. The effectiveeigenvectors are now minimizers of S[].

The global minimum, when ⊂ P is allowed to varyfreely, is seen to be attained for |φk〉 = |Pψk〉, the Bloch-Brandow effective eigenvectors. However, the canonical ef-fective eigenvectors are determined by minimizing S[] overall orthonormal sets , which then becomes equivalent tomaximizing the last term in Eq. (19), i.e., the overlaps∑

k Re〈ψk|P |φk〉 under the orthonormality constraint.

044330-4


We will now prove the striking fact that the solution is givenby ∣∣ψeff

k

⟩ = |φk〉 = e−G|ψk〉, (20)

where the unitary operator Z := exp(G) ∈ SU(n) is the ro-tation (12). Equation (20) should be compared with Eq. (7).Thus, the exact eigenvectors are simply the direct rotations ofthe effective eigenvectors from the model space into E .

Let us expand |ψk〉 ∈ E and |φk〉 ∈ P in the principal vectorbases, viz.,

|ψk〉 =m∑

j=1

|ηj 〉〈ηj |ψk〉,

|φk〉 =m∑

j=1

|ξj 〉〈ξj |φk〉.

Using 〈ηj |ξk〉 = δj,k cos θj , we compute the sum A :=∑mk=1 〈ψk|P |φk〉 as

A =m∑

k,j,=1

〈ψk|ηj 〉〈ηj |ξ〉〈ξ|φk〉

=m∑

j,k=1

cos θj 〈ξj |φk〉〈ψk|ηj 〉

Now,

A =m∑

j=1

cos θjuj,j ,

where uj,k is a unitary matrix, which implies |uj,j | 1.Moreover, uj,j = 1 for all j if and only if uj,k = δj,k , whichthen maximizes A, and also ReA. Thus,

m∑k=1

〈ξj |φk〉〈ψk|η〉 = δj,,

i.e.,

〈ξj |φk〉 = 〈ηj |ψk〉 = 〈ξj |Z†|ψk〉,from which Eq. (20) follows, since |ξk〉mk=1 is a basis for P ,and the proof is complete.

The similarity transform in Eq. (5) is thus manifest, withS = G, viz.,

H ceff = PZ†HZP = Pe−GHeGP. (21)

Moreover, QHP = PHQ = 0, verifying that the direct rota-tion in fact block-diagonalizes H .

E. Computing |ψ effk 〉

Assume that |Pψk〉 := P |ψk〉, k = 1, . . . , m are available,as would be the case in the subcluster approach to the effectiveHamiltonian. The effective eigenvectors |ψeff

k 〉 are then givenby a basis change F , i.e., the operator F : P → P defined by

F |Pψk〉 := ∣∣ψeffk

⟩.

Using the principal vector basis, we obtain

F |Pψk〉 = FP

m∑j=1

|ηj 〉〈ηj |ψk〉

= F

m∑j=1

cos θj |ξj 〉〈ηj |ψk〉

:=m∑

j=1

|ξj 〉〈ξj |ψeffk 〉

=m∑

j=1

|ξj 〉〈ηj |ψk〉,

from which we get the SVD

F :=m∑

k=1

1

cos θk

|ξk〉〈ξk|

= (ω†ω + P )1/2, (22)

where we have used Eq. (18). From Eq. (22) we see that F

is symmetric and positive definite. Moreover, smaller anglesθk means F is closer to the identity, consistent with E beingcloser to P .

Let |Pψk〉 now be given in the orthonormal “zero order”basis |ek〉mk=1 for P , i.e., we have the basis change operatorU given by

U :=m∑

k=1

|Pψk〉〈ek|, (23)

which transforms from the given basis to the Bloch-Brandoweffective eigenvectors. In terms of the principal vector basis,

U =m∑

j=1

cos θj |ξj 〉〈ηj |∑

k

|ψk〉〈ek|

=:m∑

j=1

cos θj |ξj 〉〈yj |, (24)

which is, in fact, the SVD, since the last sum over k is a unitarymap from P to E . In the operator U U † this basis-dependentfactor cancels, viz.,

U U † =m∑

k=1

|Pψk〉〈Pψk|

=m∑

k=1

cos2 θk|ξk〉〈ξk|,

that is,

F = (U U †)−1/2.

If we seek |ψeffk 〉 in the basis |ek〉mk=1 as well, we let V be the

corresponding basis change operator, i.e.,

V := FU = (U U †)−1/2U . (25)

Equation (25) shows that |ψeffk 〉 is obtained by “straightening

out” |Pψk〉, and that this depends only on the latter vectors.This is, in fact, an alternative to the common Gram-Schmidt

044330-5


orthogonalization used in mathematical constructions andproofs. It was first introduced by Lowdin [24] under the name“symmetric orthogonalization,” and the so-called Lowdinbases are widely used in quantum chemistry, where non-orthogonal basis functions are orthogonalized according toEq. (25). It seemingly requires both inversion and matrixsquare root but is easily computed using the SVD. CombiningEqs. (22) and (24) gives

V =m∑

k=1

|ξk〉〈yk|, (26)

so that if the SVD (24) is available, V is readily computed.Equation (26) is easily expressed in terms of matrices, but wedefer the discussion to Sec. IV.

F. Shavitt’s expression for exp(G)

Shavitt and Redmon [7] proved that

G = tanh−1(ω − ω†) (27)

gives the Lie algebra element for the unitary operatorZ = exp(G). The quite complicated proof was done usingan expansion of the similarity transform using the Baker-Campbell-Hausdorff formula.

It may be clear now that in the present context we obtainthe result simply as a by-product of the treatment in Sec. III Band the SVD (17) of ω, given in terms of the principal vectorsand angles. We prove this here.

The function tanh−1(z) is defined by its (complex) Taylorexpansion about the origin, i.e.,

tanh−1(z) =∞∑

n=0

z2n+1

2n + 1. (28)

The series converges for |z| < 1. Moreover,

tanh−1(z) = 1

2ln

(1 + z

1 − z

), (29)

also valid for |z| < 1. For z := ω − ω† we compute

z =m∑

k=1

µk(|χk〉〈ξk| − |ξk〉〈χk|), µk := tan(θk).

Using orthogonality relations between |ξk〉 and |χk〉, we obtain

z2n+1 = (−1)nm∑

k=1

µ2n+1k (|χk〉〈ξk| − |ξk〉〈χk|)

= i

m∑k=1

(−iµk)2n+1(|χk〉〈ξk| − |ξk〉〈χk|).

Using i tanh−1(−iz) = tan−1(z), we sum the series (28) to

G = tanh−1(z) =m∑

k=1

θk(|χk〉〈ξk| − |ξk〉〈χk|),

which is identical to Eq. (13). The series does not convergefor θk π/4, but the result is trivially analytically continuedto arbitrary 0 θk π/2.

We now turn to the effective Hamiltonian. It is common[2,3,11] to compute H c

eff in terms of ω directly, using thedefinition (29) of tanh−1(z), which implies

e± tanh−1(z) =√

1 ± z

1 ∓ z= 1 ± z√

1 − z2

Upon insertion into H = exp(−G)Hexp(G), we obtain

H = 1 − ω + ω†√

1 + ω†ω + ωω†H

1 + ω − ω†√

1 + ω†ω + ωω†.

Projecting onto P , the effective Hamiltonian becomes

Heff = (P + ω†ω)−1/2(P + ω†)H (P + ω)(P + ω†ω)−1/2.

(30)

By using the Bloch equation (6) for the Bloch-Brandoweffective Hamiltonian, we may eliminate QHQ from theabove expression for Heff , yielding

H ceff = (P + ω†ω)1/2H (P + ω)(P + ω†ω)−1/2. (31)

This expression is commonly implemented in numericalapplications [3,25]. By comparing with Eqs. (22) and (16),we immediately see that

H ceff = FH BB

eff F−1, (32)

which gives H ceff as a similarity transform of H BB

eff . Inthemselves, Eqs. (31) and (32) are not manifestly Hermitian,stemming from the elimination of QHQ. An implementationwould require complicated matrix manipulations, including amatrix square root. It is therefore better to compute H c

eff using

H ceff =

∑k

Ek

∣∣ψeffk

⟩⟨ψeff

k

∣∣,together with Eq. (26), where the most complicated operationis the SVD of the operator U given by Eq. (23). In Sec. IV wegive a concrete matrix expression for H c

eff .

G. Commuting observables

Great simplifications arise in the general quantum problemif continuous symmetries of the Hamiltonian can be identified,i.e., if one can find one or more observables S such that[H, S] = 0. Here, we discuss the impact of such symmetriesof H on the effective Hamiltonian Heff ; both in the Bloch-Brandow and the canonical case. We point out the importanceof choosing a model space that is an invariant of S as well, i.e.,[S, P ] = 0. In fact, we prove that this is the case if and only if[Heff, S] = 0, i.e., Heff has the same continuous symmetry.

Let S = S† be an observable such that [H, S] = 0, i.e., H

and S have a common basis of eigenvectors. We shall assumethat |ψk〉nk=1 is such a basis, viz.,

H |ψk〉 = Ek|ψk〉,(33)

S|ψk〉 = sk|ψk〉.In general, there will be degeneracies in both Ek and sk .

We now make the important assumption that

[S, P ] = 0, (34)

044330-6


which is equivalent to

S = PSP + QSQ.

Under the assumption (34), we have

S|Pψk〉 = PS|ψk〉 = sk|Pψk〉,so that the Bloch-Brandow effective eigenvectors are stilleigenvectors of S with the same eigenvalue sk . Moreover, aswe assume that |Pψk〉mk=1 is a (non-orthonormal) basis forP , this not possible if [S, P ] = 0. Thus, [H BB

eff , S] = 0 if andonly if [S, P ] = 0 [in addition to the assumption (33).]

The assumption (34) also implies that [S, ω] = 0, whereω = QωP is the decoupling operator. We prove this bychecking that it holds for all |ψk〉. For k m,

ωS|ψk〉 = skQ|ψk〉 = SQ|ψk〉 = Sω|ψk〉, (35)

while for k > m we need to expand P |ψk〉 in |Pψj 〉, j m,viz.,

P |ψk〉 =m∑

j=1

|Pψj 〉〈Pψj |P |ψk〉, k > m,

and use Eq. (35). Furthermore, [S, ω†]† = [ω†, S] = 0. Itfollows that

[S, (ω − ω†)n] = 0, n = 0, 1, . . . ,

and, by Eq. (28), that

[S, eG] = [S, e−G] = 0.

This gives

S∣∣ψeff

k

⟩ = Se−G|ψk〉 = sk

∣∣ψeffk

⟩.

Again, since |ψeffk 〉mk=1 is a basis for P , this holds if and

only if [S, P ] = 0. Accordingly, [H ceff, S] = 0 if and only if

[S, P ] = 0 [and the assumption (33).]The importance of this fact is obvious. If one starts with

a Hamiltonian that conserves, say, angular momentum, andcomputes the effective interaction using a model space thatis not an invariant for the angular momentum operator, i.e.,not rotationally symmetric, then the final Hamiltonian will nothave angular momentum as a good quantum number.

One possible remedy if [P, S] = 0 is to define the effectiveobservable Seff := P exp(−G)Sexp(G)P (which in the com-muting case is equal to PSP ) which obviously commutes withHeff and satisfies

Seff

∣∣ψeffk

⟩ = sk

∣∣ψeffk

⟩.

This amounts to modifying the concept of rotational symmetryin the above example.

The assumptions (33) and (34) have consequences also forthe structure of the principal vectors |ξk〉 ∈ P and |ηk〉 ∈ E .Indeed, we write

E =⊕

s

Es ,

P =⊕

s

Ps ,

where the sum runs over all distinct eigenvalues sk, k =1, . . . , m of S, and where Es (Ps) is the corresponding

eigenspace, i.e.,

Es := span|ψk〉 : S|ψk〉 = s|ψk〉,Ps := span

∣∣ψeffk

⟩: S

∣∣ψeffk

⟩ = s∣∣ψeff

k

⟩.

The eigenspaces are all mutually orthogonal, viz.,Es ⊥ Es ′ ,Ps ⊥Ps ′ , and Es ⊥Ps ′ , for s = s ′. The definition (3)of the principal vectors and angles can then be written

cos(θk) = maxs

max|ξ 〉∈Ps , 〈ξ |ξ〉=1〈ξj |ξ 〉=0, j<k

max|η〉∈Es , 〈η|η〉=1〈ηj |η〉=0, j<k

Re〈ξ |η〉

=:〈ξk|ηk〉.Thus, for each k, there is an eigenvalue s of S such that

S|ξk〉 = s|ξk〉,S|ηk〉 = s|ηk〉,

showing that the principal vectors are eigenvectors of S if andonly if [S, P ] = 0, [S,H ] = 0, and the assumption (33).

The present symmetry considerations imply that modelspaces obeying as many symmetries as possible should befavored over less symmetric model spaces, since these othermodel spaces become less “natural” or “less effective” inthe sense that their geometry is less similar to the originalHilbert space. This is most easily seen from the fact thatprincipal vectors are eigenvectors for the conserved observableS. This may well have great consequences for the widely usedsubcluster approximation to the effective Hamiltonian in no-core shell model calculations [3,6,26], where one constructsthe effective Hamiltonian for a system of a particles inorder to obtain an approximation to the A > a-body effectiveHamiltonian. The model space in this case is constructed indifferent ways in different implementations. Some of thesemodel spaces may therefore be better than others due todifferent symmetry properties.

IV. MATRIX FORMULATIONS

A. Preliminaries

Since computer calculations are invariably done usingmatrices for operators, we here present matrix expressionsfor H c

eff and compare them to those usually programed in theliterature, as well as expressions for H BB

eff and ω.Recall the standard basis |ek〉nk=1 ofH, where the |ek〉mk=1

constitute a basis for P . These are usually eigenvectors of theunperturbed “zero order” Hamiltonian H0, but we will notuse this assumption. As previously, we also assume withoutloss that the eigenpairs we wish to approximate in Heff are(Ek, |ψk〉)mk=1.

An operator A : H → H has a matrix A ∈ Cn×n associated

with it. The matrix elements are given by Ajk = 〈ej |A|ek〉 suchthat

A =n∑

j,k=1

|ej 〉〈ejA|ek〉〈ek|| =n∑

j,k=1

|ej 〉Ajk〈ek|.

Similarly, any vector |φ〉 ∈ H has a column vector φ ∈ Cn

associated with it, with φj = 〈ej |φ〉. We will also view dualvectors, e.g., 〈ψ |, as row vectors.

044330-7


The model space P and the excluded space Q are conve-niently identified with C

m and Cn−m, respectively. Also note

that PAP,PAQ, etc., are identified with the upper left m × m,upper right m × (n − m), etc., blocks of A as in Eq. (1). Weuse a notation inspired by FORTRAN and MATLAB and write

PAP = A(1 : m, 1 : m), PAQ = A(1 : m,m + 1 : n),

and so forth.We introduce the unitary operator U as

U =n∑

k=1

|ψk〉〈ek|,

i.e., a basis change from the chosen standard basis to theeigenvector basis. The columns of U are the eigenvectorcomponents in the standard basis, i.e.,

Ujk = ψk,j = 〈ej |ψk〉,and are typically the eigenvectors returned from a computerimplementation of the spectral decomposition, viz.,

H = UEU†, E = diag(E1, . . . , En). (36)

The SVD is similarly transformed to matrix form. The SVDdefined in Sec. II B is then formulated as follows: for any ma-trix A ∈ C

q×r there exist matrices X ∈ Cq×p[p = min(q, r)]

and Y ∈ Cr×p, such that X†X = Y†Y = Ip (the identity matrix

Cp×p), and a non-negative diagonal matrix ∈ R

p×p suchthat

A = X Y†.

Here, = diag(σ1, . . . , σp), σk being the singular values.The columns of X are the left singular vectors’ components,

i.e., Xj,k = 〈ej |xk〉, and similarly for Y and the right singularvectors. The difference between the two SVD formulations isthen purely geometric, as the matrix formulation favors thestandard bases in X and Y .

The present version of the matrix SVD is often referred toas the “economic” SVD, since the matrices X and Y may beextended to unitary matrices over C

q and Cr , respectively, by

adding singular values σk = 0, k > m. The matrix is then aq × r matrix with “diagonal” given by σk . This is the “full”SVD, equivalent to our basis-free definition.

B. Algorithms

Let the m eigenvectors |ψk〉 be calculated and arranged ina matrix U, i.e., ψk = U(1 : n, k) (where the subscript doesnot pick a single component). Consider the operator U definedin Eq. (23), whose matrix columns are the Bloch-Brandoweffective eigenvectors |Pψk〉 in the standard basis, viz.,

U = U(1 : m, 1 : m).

The columns of the matrix of V = (U U †)−1/2U are thecanonical effective eigenvectors ψeff

k . The SVD (24) can bewritten as

U = X Y†,

which gives

UU† = X 2X†.

Since kk = cos θk > 0, we obtain

(UU†)−1/2 = X −1X†,

which gives, when applied to U,

V = (UU†)−1/2U = XY†.

Thus, we obtain the canonical effective eigenvectors bytaking the matrix SVD of U = U(1 : m, 1 : m) and multiplyingtogether the matrices of singular vectors. As efficient androbust SVD implementations are almost universally available,e.g., in the LAPACK library, this makes the canonical effectiveinteraction much easier to compute than that in Eq. (31), viz.,

Hceff = VE(1 : m, 1 : m)V†.

This version requires one SVD computation and three matrixmultiplications, all with m × m matrices, one of which isdiagonal. Equation (31) requires, on the other hand, severalmore matrix multiplications, inversions, and the square rootcomputation. The Bloch-Brandow effective Hamiltonian issimply calculated by

HBBeff = UE(1 : m, 1 : m)U−1.

For the record, the matrix of ω is given by

ω = U(m + 1 : n, 1 : m)U−1,

although we have no use for it when using the SVD-basedalgorithm. It may be useful, though, to be able to computethe principal vectors for P and E . For this, one may computethe SVD of ω or of PP′ = UU(1 : n, 1 : m)†; the latter givescos θk, |ξk〉, and |ηk〉 directly in the standard basis as singularvalues and vectors, respectively.

V. DISCUSSION AND OUTLOOK

We have characterized the effective Hamiltonians com-monly used in nuclear shell-model calculations in termsof geometric properties of the spaces P and E . The SVDand the principal angles and vectors were central in theinvestigation. While the Bloch-Brandow effective Hamiltonianis obtained by orthogonally projecting E onto P , therebyglobally minimizing the norm-error of the effective eigen-vectors, the canonical effective Hamiltonian is obtained byrotating E into P using exp(−G), which minimizes thenorm-error while retaining orthonormality of the effectiveeigenvectors. Moreover, we obtained a complete descriptionof the decoupling operator ω in terms of the principal anglesand vectors defining exp(G).

An important question is whether the present treatmentgeneralizes to infinite dimensional Hilbert spaces. Our analysisfits into the general assumptions in the literature, being thatn = dim(H) is large but finite, or at least that the spectrumof H is purely discrete. A minimal requirement is that H hasm eigenvalues, so that E can be constructed. In particular,the SVD generalizes to finite rank operators in the infinitedimensional case and is thus valid for all the operatorsconsidered here even when n = ∞.

Unfortunately, H has almost never a purely discretespectrum. It is well known that the spectrum in general

044330-8


has continuous parts and resonances embedded in these,and a proper theory should treat these cases as well asthe discrete part. In fact, the treatments of Heff in theliterature invariably glosses over this. It is an interestingfuture project to develop a geometric theory for the effectiveHamiltonians that incorporates resonances and continuousspectra.

The geometrical view simplified and unified the availabletreatments in the literature somewhat and offered further in-sights into the effective Hamiltonians. Moreover, the symmetryconsiderations in Sec. III G may have significant bearing onthe analysis of perturbation expansions and the properties ofsubcluster approximations to H c

eff .Indeed, it is easy to see that if we have a complete set of

commuting observables (CSCO) [27] for H0, and the sameset of observables form a CSCO for H1, all eigenvalues andeigenfunctions of H (z) = H0 + zH1 are analytic in z ∈ C,implying that the Rayleigh-Schrodinger perturbation seriesfor H = H0 + H1 converges (i.e., at z = 1) [15]. Intuitively,the fewer commuting observables we are able to identify, themore likely it is that there are singularities in Heff(z), theso-called intruder states. The Rayleigh-Schrodinger seriesdiverges outside the singularity closest to z = 0 [15]; and innuclear systems, this singularity is indeed likely to be close toz = 0. On the other hand, resummation of the series can beconvergent and yield an analytic continuation of Heff outsidethe region of convergence [28]. To the best of our knowledge,there is no systematic treatment of this phenomenon in theliterature, except for some iterative procedures [29,30] andattempts at using Pade approximants [31]. On the contrary, to

be able to do such a resummation consistently is sort of the“holy grail” of many-body perturbation theory. A geometricstudy of the present kind to many-body perturbation theory anddiagram expansions may yield a step closer to this goal, as wehave clearly identified the impact of commuting observableson the principal vectors of E and P .

We have also discussed a compact algorithm in terms ofmatrices to compute H c

eff , relying on the SVD. This algorithm,useful in the subcluster approach to the effective Hamiltonian,is, to the best of our knowledge, previously unpublished. Sincerobust and fast SVD implementations are readily available,e.g., in the LAPACK library, and since few other matrixmanipulations are needed, it should be preferred in computerimplementations.

As stressed in the Introduction, the algorithms presentedare really only useful if we compute the exact effectiveHamiltonian, as opposed to a many-body perturbation the-oretical calculation, and if we know what exact eigenpairsto use, such as in a subcluster approximation. In this case,one should analyze the error in the approximation, i.e., theerror in neglecting the many-body correlations in H c

eff . Inthe perturbative regime, some results exist [6]. We believethat the geometric description may facilitate a deeper analysis,and this is an interesting idea for future work.

ACKNOWLEDGMENTS

The author wishes to thank Prof. M. Hjorth-Jensen, CMA,for helpful discussions. This work was funded by CMAthrough the Norwegian Research Council.

[1] D. Dean, T. Engeland, M. Hjorth-Jensen, M. Kartamyshev, andE. Osnes, Prog. Part. Nucl. Phys. 53, 419 (2004).

[2] E. Caurier, G. Martinez-Pinedo, F. Nowacki, A. Poves, andA. P. Zuker, Rev. Mod. Phys. 77, 427 (2005).

[3] P. Navratil, J. P. Vary, and B. R. Barrett, Phys. Rev. C 62, 054311(2000).

[4] C. Bloch, Nucl. Phys. 6, 329 (1958).[5] B. Brandow, Rev. Mod. Phys. 39, 771 (1967).[6] D. Klein, J. Chem. Phys. 61, 786 (1974).[7] I. Shavitt and L. T. Redmon, J. Chem. Phys. 73, 5711 (1980).[8] I. Lindgren, J. Phys. B 7, 2441 (1974).[9] P. Ellis and E. Osnes, Rev. Mod. Phys. 49, 777 (1977).

[10] I. Lindgren and J. Morrison, Atomic Many-Body Theory(Springer, New York, 1982).

[11] K. Suzuki, Prog. Theor. Phys. 68, 246 (1982).[12] T. Kuo, Springer Lect. Notes Phys. 144, 248 (1981).[13] J. Van Vleck, Phys. Rev. 33, 467 (1929).[14] Kemble, The Fundamental Principles of Quantum Mechanics

with Elementary Applications (McGraw-Hill, New York, 1937).[15] T. Schucan and H. Weidenmuller, Ann. Phys. (NY) 76, 483

(1973).

[16] A. Bjorck and G. Golub, Math. Comput. 27, 579 (1973).[17] A. Knyazev and M. Argentati, SIAM J. Sci. Comput. 23, 2008

(2002).[18] G. Golub and C. van Loan, Matrix Computations (Johns

Hopkins, Baltimore, MD, 1989).[19] C. Davis and W. M. Kahan, SIAM J. Numer. Anal. 7, 1 (1970).[20] O. Jordahl, Phys. Rev. 45, 87 (1934).[21] H. Primas, Rev. Mod. Phys. 35, 710 (1963).[22] S. Blanes, F. Casas, J. Oteo, and J. Ros, J. Phys. A: Math. Theor.

31, 259 (1998).[23] K. Suzuki, Prog. Theor. Phys. 68, 1627 (1982).[24] P.-O. Lowdin, J. Chem. Phys. 18, 365 (1950).[25] G. Hagen, M. Hjorth-Jensen, and N. Michel, Phys. Rev. C 73,

064307 (2006).[26] J. Da Providencia and C. Shakin, Ann. Phys. (NY) 30, 95

(1964).[27] A. Messiah, Quantum Mechanics (Dover, New York, 1999).[28] P. Schaefer, Ann. Phys. (NY) 87, 375 (1974).[29] F. Andreozzi, Phys. Rev. C 54, 684 (1996).[30] K. Suzuki and S. Lee, Prog. Theor. Phys. 64, 2091 (1980).[31] H. M. Hofmann, Ann. Phys. (NY) 113, 400 (1978).

044330-9

4 Open source FCI code forquantum dots and effectiveinteractions

This paper is submitted to Physical Review E.

Open source FCI code for quantum dots and effective interactions

Simen Kvaal1,∗

1Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway(Dated: October 15. 2008)

We describe OPENFCI, an open source implementation of the full configuration-interaction method (FCI) fortwo-dimensional quantum dots with optional use of effective renormalized interactions. The code is written inC++ and is available under the Gnu General Public License. The code and core libraries are well documentedand structured in a way such that customizations and generalizations to other systems and numerical methodsare easy tasks. As examples we provide a matrix element tabulation program and an implementation of a simplemodel from nuclear physics, in addition to the quantum dot application itself.

PACS numbers: 02.60.-x, 95.75.Pq, 73.21.La, 24.10.CnKeywords: full configuration interaction; open source; C++; quantum dots; effective interactions

I. INTRODUCTION

Quantum dots, nanometre-scale semiconductor devicesconfining a varying number of electrons, have been studiedintensely in the last two decades. Quantum dots are fabricatedusing essentially macroscopic tools, for example etching tech-niques, but the resulting confinement allows for quantum me-chanical behaviour of the electrons. Many of the parametersare directly controllable, thereby justifying the term “artificialatoms” or “designer atoms”. These considerations explain theimmense research activity on these systems. For a general in-troduction, see Ref. [1] and references therein.

A very common model is that of a parabolic quantum dot, inwhichN electrons are confined in an isotropic harmonic oscil-lator potential ind spatial dimensions, whered is determinedby the semi-conductor environment. Electronic structure cal-culations on the parabolic dot and similar systems are of-ten carried out using the full configuration-interaction method(FCI), also called exact diagonalization [1]. The Hamiltonianis then projected onto a finite-dimensional subspace of theN-electron Hilbert space and diagonalized. Care is taken in or-der to exploit dynamical and discrete symmetries of the exactproblem, such as conservation of angular momentum and totalelectron spin, in order to block-diagonalize the Hamiltonianmatrix and reduce the computational complexity.

In this article, we describe OPENFCI, a recently devel-oped open source C++ code implementing the FCI methodfor quantum dots [2]. The code has a generic framework in theshape of library functions, thereby allowing easy customiza-tion and extension to other systems and methods, e.g., three-dimensional quantum dots or the nuclear no-core shell model.

OPENFCI implements a renormalization of the two-bodyinteractions, a technique widely used in nuclear no-core shellmodel calculations. This allows for accelerated convergencewith respect to Slater determinant basis size [3, 4]. To theauthor’s knowledge, no other available code provides such ef-fective interactions for quantum dot systems. The code canbe easily modified to create effective interactions for almost

∗Electronic address:[email protected]

many-body problem using a harmonic oscillator basis.The code is developed in a Linux environment using Gnu

C++, and is readily portable to other environments and com-pilers. The Fortran 77 libraries LAPACK and ARPACK are re-quired, but as these are available on a wide range of platforms,portability should not be affected. OPENFCI is released underthe Gnu General Public License (Gnu GPL) [5] and is docu-mented using Doxygen [6]. As an open source project, thecode can freely be used and modified.

The article is organized as follows: In Section II, the FCImethod is introduced in the context of the parabolic quantumdot, where we also discuss the reduction of the Hamiltonianmatrix by means of commuting operators and configurationalstate functions. In Section III we discuss the effective two-body interaction. As the technique is likely to be unfamiliarto most readers outside the nuclear physics community, this isdone in some detail. In Section IV we discuss the organizationand use of OPENFCI. We also give some results from exam-ple runs, and in particular an analytically solvable non-trivialmodel due to Johnson and Payne is considered [7], where theonly modification of the parabolic quantum dot is the interac-tion. Finally, we conclude our article in Section V.

Two appendices have been provided, Appendix A detailingthe heavily-used centre-of-mass transformation and AppendixB discussing the exact numerical solution of the two-electronquantum dot needed for the effective interaction scheme.

II. FCI METHOD

A. Hamiltonian in occupation number formalism

We considerN electrons trapped in an isotropic harmonicoscillator potential ind spatial dimensions. The electronsinteract via the Coulomb potential given byU(r i j ) = λ/r i j ,wherer i j = ‖~r i −~r j‖ is the inter-particle distance andλ is aconstant. The quantum dot Hamiltonian then reads

H :=N

∑i=1

H0(i)+N

∑i< j

U(r i j ), (1)

2

where the second sum runs over all pairs 1≤ i < j ≤ N, andwhereH0(i) is the one-body Hamiltonian defined by

H0(i) :=−12

∇2i +

12‖~r i‖2.

The interaction strengthλ is given by

λ=√

m∗

ωh3

1ε

e2

4πε0, (2)

whereε is the dielectric constant of the semiconductor bulk,e2/4πε0 ≈ 1.440 eV·nm, andω = h/m∗a2, a being the trapsize and length unit, andm∗ being the effective electron mass.Typical values for GaAs quantum dots areε = 12.3, m∗ =0.067 electron masses, anda = 20 nm, yieldingλ = 2.059.The energy unit ishω, in this casehω= 2.84 meV.

Choosing a complete setφα(x)α∈A of single-particle or-bitals (wherex = (~r , s) denotes both spatial and spin de-grees of freedom, andα = (a,σ) denotes both generic spa-tial quantum numbersa and spin projection quantum numbersσ=±1), H can be written in occupation number form as

H = ∑a,b

∑σ

haba†

a,σab,σ+12 ∑

abcd∑στ

uabcda†

a,σa†b,τad,τac,σ, (3)

wherea†α (aα) creates (destroys) a particle in the orbitalφα(x).

These operators obey the usual anti-commutation relations

aα,a†β= δα,β, aα,aβ= 0. (4)

For a review of second quantization and occupation numberformalism, see for example Ref. [8]. The single-particle or-bitals are chosen on the form

φ(a,σ)(x) := ϕa(~r)χσ(s),

where ϕa(~r) are spinless orbitals andχσ(s) = δσ,s arespinor basis functions corresponding to the eigenstates of thespin-projection operatorSz with eigenvaluesσ/2.

It is important, that since the single-particle orbitalsϕa(~r)a∈A are denumerable, we may choose an ordering onthe setA, such thatA can in fact be identified with a rangeof integers,A u 0,1,2, · · · ,L/2. In mostab initio systemsL is infinite, since the Hilbert space is infinite-dimensional.Similarly, α = (a,+1) is identified with even integers, andα = (a,−1) with odd integers, creating an ordering of thesingle-particle orbitalsφα(x), andα is identified with an in-teger 0≤ I(α)≤ L.

The single-particle matrix elementshab and the two-particle

elementsuabcd are defined by

hab := 〈ϕa|H0|ϕb〉=

∫ϕa(~r)H0ϕb(~r) ddr,

and

uabcd := 〈ϕaϕb|U(~r12)|ϕcϕd〉

= λ

∫ϕa(~r1)ϕb(~r2)

1r12

ϕc(~r1)ϕd(~r2) ddr1ddr2, (5)

respectively.The spatial orbitalsϕa(~r) are usually chosen as eigenfunc-

tions ofH0, so thathab = δa,bεa.

The basis functions forN-particle Hilbert space are Slaterdeterminants|Φα1,α2,··· ,αN〉 defined by

|Φα1,···αN〉 := a†α1

a†α2· · ·a†

αN|−〉,

where |−〉 is the zero-particle vacuum. In terms of single-particle orbitals, the spatial representation is

Φα1,··· ,αN(x1, · · · , xN) =1√N!

∑p∈SN

(−)|p|N

∏i=1

φαp(i)(xi),

whereSN is the group of permutations ofN symbols. TheSlater determinants are anti-symmetric with respect to per-mutations of bothxi andαi , so that the orbital numbersαimust all be distinct to give a nonzero function. Each or-bital is then occupied by at most one particle. Moreover,for a given setαiN

i=1 of orbitals, one can createN! distinctSlater determinants that are linearly dependent. In order toremove this ambiguity, we choose only orbital numbers suchthat I(αi) < I(α j) wheneveri < j.

It follows, that there is a natural one-to-one correspondencebetween Slater determinants withN particles and integersbwhose binary representations haveN bits set. (If|A|= L < ∞,the integers are limited to 0≤ b < 2L.) Each bit positionkcorresponds to an orbitalφα(x) throughk = I(α), and the bitis set if the orbital is occupied. Creating and destroying par-ticles in |Φα1,··· ,αN〉 simply amounts to setting or clearing bits(possibly obtaining the zero-vector if a particle is destroyed orcreated twice in the same orbital), keeping track of the possi-ble sign change arising from bringing the setαi on orderedform using Eqn. (4). Note that the vacuum|−〉 corresponds tob = 0, which isnot the zero vector, but the single state withzero particles.

B. Model spaces

The FCI calculations are done in a finite-dimensional sub-spaceP of the N-particle Hilbert space, called the modelspace. The model space has a basisB of Slater determinants,andP has the orthogonal projectorP given by

P := ∑|Φb〉∈B

|Φb〉〈Φb|. (6)

The configuration-interaction method in general now amountsto diagonalizing (in the sense of finding a few of the lowesteigenvalues of) the, in general, large and sparse matrixPHP.The only approximation we have made is the truncation of theN-particle Hilbert space.

The model spaceP is seen to be a function of the singleparticle orbitalsϕa(~r), whom we choose to be the eigenfunc-tions of H0, i.e., harmonic oscillator eigenfunctions. Thesemay be given on several equivalent forms, but it is convenientto utilize rotational symmetry ofH0 to create eigenfunctions

3

−1 1 2 3 4−2−3−4

0

1

2

3

4

R

m

0

n = 0

n = 0 n = 0

n = 0

n = 0 n = 0

n = 0

n = 0n = 1

n = 1 n = 1

n = 1n = 1 n = 2

.

.

.

n = 0

FIG. 1: Structure of single-particle orbitals of the two-dimensionalharmonic oscillator. Angular momentum and shell number/energyon axes, and nodal quantum numbern at each orbital. Orbitaln = 0,m=−3 in shellR= 3 is occupied by two electrons for illustration.

of the projection of the angular momentumLz. In d = 2 di-mensions we obtain the Fock-Darwin orbitals defined in polarcoordinates by

ϕn,m(r, θ) =1√π

eimθr |m|L|m|n (r2)e−r2/2. (7)

HereLkn(x) = (−1)n[n!/(n+ |m|)!]1/2Lk

n(x) is the normalizedgeneralized Laguerre polynomial. The factor(−1)n is forconvenience, see Appendix A 1. The harmonic oscillator en-ergy is 2n+ |m|+ 1 and the eigenvalue ofLz = −i∂/∂θ is m.All eigenfunctions with the same energy 2n+ |m|+1=: R+1span a single-particleshell. The single-particle orbitals areillustrated in Fig. 1.

For a Slater determinant|Φα1,··· ,αN〉, we have

N

∑i=1

H0(i)|Φα1,··· ,αN〉= E0α1,··· ,αN

|Φα1,··· ,αN〉

with

E0α1,··· ,αN

:=N

∑i=1

(Ri +1),

whereRi = 2ni + |mi |, and

N

∑i=1

Lz(i)|Φα1,··· ,αN〉= M|Φα1,··· ,αN〉,

whereM = ∑Ni=1mi .

To complete our definition ofP, we let

B = BR =

|Φα1,··· ,αN〉 :

N

∑i=1

Ri ≤ R

, (8)

whereR is called the energy cut, for obvious reasons. AsR→∞, the whole Hilbert space is spanned, and the eigenpairs ofPHPconverge to those ofH.

C. Configurational state functions and block diagonality

In order to reduce the complexity of the computations, weneed to exploit symmetries ofH. First of all, [H,Lz] = 0, andit is obvious that also[H,Sz] = 0, where the spin projectionoperatorSz is given by

Sz :=12 ∑

a,σσa†

a,σaa,σ.

The Slater determinants are eigenvectors of bothLz and Szwith eigenvaluesM andsz = ∑N

i=1σi/2, respectively. We ob-tain a natural splitting of the model spaceP into subspaceswith constant angular momentumM and spin projectionsz,viz,

P =⊕M,sz

PM,sz, P = ∑M

∑sz

PM,sz.

The diagonalization ofH can thus be done within eachspacePM,sz separately, amounting to diagonalizing individ-ual blocksPM,szHPM,sz.

The Hamiltonian (3) also commutes with total electron spinS2, [S2,Sz] = 0, given by

S2 := S2z +

12(S+S−+S−S+),

with

S± := ∑a

a†a±aa∓ ,

so that a common basis forSz and S2 would lead to evensmaller matrix blocks.

The eigenvalues ofS2 are on the forms(s+ 1), where0≤ 2s≤ N is an odd (even) integer for odd (even)N. For ajoint eigenfunction ofSz andS2, called a configurational statefunction (CSF),|sz| ≤ s. The Slater determinants are, how-ever, not eigenfunctions ofS2, but such can be constructedby taking linear combinations of a small number Slater deter-minants. For details on this algorithm, see Ref. [9]. Sufficeit to say here, thatS2 only couples Slater determinants withidentical sets of doubly occupied orbitals (meaning thatφ(a,+)andφ(a,−) are both occupied, as in Figure 1) and singly occu-pied orbitals (meaning that only one ofφ(a,+) andφ(a,−) areoccupied). It is easy to see thatS2 does not couple Slater de-terminants inPM,sz to anotherPM′,s′z. Thus, we obtain thesplitting

PM,sz =⊕

s

PM,sz,s.

We stress that all the mentioned operators commute with eachother, viz,

[H,Ωi ] = [Ωi ,Ω j ] = 0,

with Ωi ∈ Lz,Sz,S2. If a modified problem breaks, say,rotational symmetry, such that[H,Lz] 6= 0, we may still splitthe model space into to the eigenspaces ofSz andS2.

4

D. Matrix elements of Coulomb interaction

The remaining ingredient in the FCI method is the Coulombmatrix elementsuab

cd defined in Eqn. (5). These can be calcu-lated by first expandingLk

n(x) in powers ofx using

Lkn(x)≡

n

∑m=0

(−1)m (n+k)!(n−m)!(k+m)!m!

xm,

and evaluating the resulting integral term-by-term by analyt-ical methods [10]. The resulting expression is a seven-foldnested sum, which can be quite time-consuming, especiallyif a large number of Fock-Darwin orbitals occurs in the basisB. Moreover, the terms are fractions of factorials with alter-nating signs, which is a potential source of loss of numericalprecision.

We therefore opt for a more indirect approach, giving a pro-cedure applicable to a wide range of potentialsU(r12) in ad-dition to the Coulomb potential. Moreover, it can be general-ized to arbitrary spatial dimensionsd. The approach is basedon directly transforming the product functionsϕa(~r1)ϕb(~r2) tothe centre-of-mass system, where the interactionU(r12) onlyacts on the relative coordinate, and then transforming back tothe lab system. This reduces the computational cost to a dou-bly nested sum, as well as the pre-computation of the centre-of-mass transformation and the relative coordinate interactionmatrix. Both can be done exactly using Gaussian quadrature.The transformations to and from the centre of mass frame areunitary transformations, which are stable and will not magnifyround-off errors.

In Appendix A we provide the details of the centre-of-mass transformation. One then obtains the following prescrip-tion for the interaction matrix elementsuab

cd: Let a = (µ1, ν1),b= (µ2, ν2), c= (µ3, ν3), andd= (µ4, ν4) be the circular quan-tum number equivalents of the usual polar coordinate quan-tum numbersni andmi . Due to conservation of angular mo-mentum, we assumem1 +m2 = m3 +m4; otherwise, the ma-trix elementuab

cd = 0. Define M = µ1 + µ2, M′ = µ3 + µ4,N = ν1 + ν2, andN′ = ν3 + ν4. Sinceuab

cd is linear inλ, wesetλ= 1 without loss of generality. Now,

uabcd =

M

∑p=p0

T(M)p,µ2T(M′)

p′,µ4

N

∑q=q0

T(N)q,ν2T(N′)

q′,ν4C|p−q|

n,n+s, (9)

wheren = min(p,q), s= M′−M, p′ = p+ M′−M, q′ = q+N′−N. Moreover,p0 = max(M′−M,0) andq0 = max(N′−N,0).

Here, T(N) are centre-of-mass transformation coefficientsdefined in Appendix A, while the relative coordinate interac-tion matrix elementsC|m|

n,n′ , n,n′ ≥ 0, are defined by

C|m|n,n′ := 〈ϕn,m(r, θ)|U(

√2r)|ϕn′,m(r, θ)〉

= 2∫ ∞

0r2|m|L|m|n (r2)L|m|n′ (r2)U(

√2r)e−r2

rdr. (10)

Depending onU(r12), the integral is best computed using gen-eralized half-range Hermite quadrature (see Appendix B and

Ref.[11]) or Gauss-Hermite quadrature. Weights and abscissafor quadratures are conveniently computed using the Golub-Welsch algorithm [12], which only depends on the ability tocompute the coefficients of the three-term recursion relationfor the polynomial class in question, as well as diagonalizinga symmetric tri-diagonal matrix.

Let p(r) be a polynomial, and letα < 2 andβ be non-negative constants. Then

U(r12) = rα12p(r12)e−βr212 (11)

admit exact evaluations using generalized half-range Gauss-Hermite quadrature. The Coulomb potential, Gaussian poten-tials, and the parabolic interaction−λr2

12/2 of the analyticallysolvable model treated in Sec. IV C belong to this class of po-tentials.

In the case ofα= 1 andp(r) = q(r2) (i.e., an even polyno-mial), the integral is more convenient to evaluate using stan-dard Gauss-Hermite quadrature. The Coulomb interactionfalls into this class.

Of course, one may letp(r) be a non-polynomial functionas well and still obtain very good results, as long asp(r) iswell approximated with a polynomial, e.g., is smooth.

III. EFFECTIVE INTERACTIONS

A. Motivation

The FCI calculations converge relatively slowly as func-tion of the model space parameterR [3], as the error∆E inthe eigenvalue behaves likeo(R−k) in general, wherek =O(1). This behaviour comes from the singular nature of theCoulomb interaction.

In Ref. [3], numerical results using an effective interac-tion were presented. This method is widely used in no-core shell model calculations in nuclear physics, where thenucleon-nucleon interaction is basically unknown but highlysingular [4]. This so-called sub-cluster effective interactionscheme replaces the Coulomb interaction (or another inter-action)U(r i j ) = λ/r i j with a renormalized interactionU(i, j)obtained by a unitary transformation of the two-body Hamil-tonian that decouples the model spaceP and its complement[13]. Therefore, the two-body problem becomesexactin a fi-nite number of harmonic oscillator shells. Loosely speaking,the effective interaction incorporates information about the in-teraction’s actionoutsidethe model space. In general,U(i, j)is non-linear inλ and not a local potential.

Using the renormalizedU(i, j), the many-body system doesnot become exact, of course, butU(i, j) will perform betterthan the bare interaction in this setting as well. To the au-thor’s knowledge, there exists no rigorous mathematical treat-ment with respect to this, but it has nevertheless enjoyed greatsuccess in the nuclear physics community [4, 14, 15], and ournumerical experiments unambiguously demonstrate that theconvergence of the FCI method is indeed improved drastically[3], especially forN ≤ 4 particles. We stress that the cost ofproducingU(i, j) is very small compared to the remaining cal-culations.

5

B. Unitary transformation of two-body Hamiltonian

We now describe the unitary transformation of the two-body Hamiltonian (i.e., Eqn. (1) or (3) withN = 2) that de-couplesP and its complement. This approach dates back asfar as 1929, when Van Vleck introduced such a generic uni-tary transformation to de-couple the model space to first orderin the interaction [16, 17].

Let P be given by Eqn. (6), and letD = dim(P). The ideais to find a unitary transformationH = Z†HZ of H such that

(1−P)H P = 0,

i.e.,H is block diagonal. This implies thatHeff defined by

Heff := PH P

has eigenvalues identical toD of those of thefull operatorH.SinceD is finite,Heff is called an effective Hamiltonian.

SelectingZ is equivalent to selecting a set of effectiveeigenpairs(Ek, |Ψeff

k 〉)Dk=1, whereEk is an eigenvalue ofH

and |Ψeffk 〉D

k=1 ⊂ P are the effective eigenvectors; an or-thonormal basis forP. It is clear thatZ is not unique, sincethere are many ways to pickD eigenvalues ofH, and for eachsuch selection any unitaryD×D matrix would yield an eigen-vector set.

However, some choices are more natural than others, sincethe eigenvectors and eigenvalues are usually continuous func-tions of λ. We then select theD eigenvaluesEk(λ) that de-velop adiabatically fromλ = 0. For the corresponding effec-tive eigenvectors|Ψeff

k (λ)〉, we choose the orthonormal set thatminimizes the distance to the exact eigenvectors|Ψk〉D

k=1,i.e.,

|Ψeffk 〉D

k=1 := argmin|Ψ′

k〉Dk=1

D

∑k=1

‖|Ψk〉− |Ψ′k〉‖2, (12)

where the minimization is taken over orthonormal sets only.The effective eigenvectors also turn out to be continuous func-tions ofλ, soHeff will also be continuous.

Let U is theD×D matrix whose columns containP|Ψk〉in the chosen basis, and letV be the corresponding matrixcontaining|Ψeff

k 〉. Clearly,V is unitary, whileU only approx-imately so. Equation (12) can then be written

V := argminU ′

trace[(U−U ′)(U−U ′)†], (13)

where the minimum is taken over all unitary matrices. IfUhas singular value decomposition given by

U = XΣY†, (14)

the solutionV is given by

V := XY†. (15)

If E = diag(E1, · · · ,ED) is the diagonal matrix whose ele-ments are the chosen eigenvalues, we have

Heff = VEV†.

See Ref. [13] for a thorough discussion of the above prescrip-tion for Heff.

Having computed the two-bodyHeff, we define the effectiveinteractionU(1,2) by

U(1,2) := Heff−P2

∑i=1

H0(i)P,

which gives meaningsolely in the model space. In secondquantization,

U(1,2) :=12 ∑

abcd∑στ

uabcda†

aσa†bτadτacσ,

and theN-bodyHeff becomes (cf. Eqn. (1))

Heff =N

∑i=1

H0(i)+N

∑i< j

U(i, j),

with occupation number formalism form (cf. Eqn. (3))

Heff = ∑a,b

∑σ

haba†

a,σab,σ+12 ∑

abcd∑στ

uabcda†

a,σa†b,τad,τac,σ. (16)

Now, Heff is well-defined in the space ofN-body Slater deter-minants where no pairs of occupied orbitals constitute a two-body state outside the two-particle model space, since thenthe matrix element ˜uab

cd would be undefined. A little thoughtshows us that ifU(1,2) was computed in a two-body energycut space with parameterR, Heff is well-defined on the many-body model space with the same cutR.

C. A comment concerning the choice of model space

The two-body problem is classically integrable, i.e., thereexists 2d−1 constants of motionΩi , such that their quantummechanical observables commute withH and each other, viz,

[H,Ωi ] = [Ωi ,Ω j ] = 0, for all i, j.

Indeed, the centre-of-mass harmonic oscillatorHC defined inEqn. (18) below and the corresponding centre-of-mass an-gular momentum provides two constants, while total angularmomentumLz provides a third.

Using the model spaceP defined by an energy cut, wehave

[P,Ωi ] = 0

as well, which is equivalent [13] to

[Heff,Ωi ] = 0, (17)

so thatHeff is integrable as well. In particular,U(1,2) isblock-diagonal with respect toΩi .

If we consider the commonly encountered model spaceP ′

defined by the Slater determinant basisB′ given by

B′ := |Φα1,··· ,αN〉 : max(Ri)≤ R

6

instead of Eqn. (8), we will have

[P′,HC] 6= 0,

as is easily verified. Indeed,P ′ is not an invariant subspaceof the centre-of-mass transformationT defined in AppendixA. Thus,[H′

eff,HC] 6= 0, so that the centre-of-mass energy nolonger is a constant of motion! The symmetry-breaking ofthe effective Hamiltonian in this case is problematic, since inthe limit λ→ 0, the exact eigenfunctions that develop adia-batically are not all either in the model space or in the com-plement. The adiabatic continuation of the eigenpairs startingout inP is thus not well-defined.

We comment, that the model spaceP ′ is often used inboth no-core shell model calculations and quantum dot cal-culations, but the effective interaction becomes, in fact, ill-behaved in this case.

D. Solution of the two-body problem

What remains for the effective interaction, is the compu-tation of the exact eigenpairs(Ek, |Ψk〉)D

k=1. We must alsosolve the problem of following eigenpairs adiabatically fromλ= 0.

For the two-body Coulomb problem, analytical solutionsare available only for very special values forλ [18]. These areuseless for our purpose, so we must use numerical methods.

A direct application of the FCI method using Fock-Darwinorbitals with a largeR′ > R will converge slowly, and there isno device in the method for following eigenvalues adiabati-cally. As the eigenvalues may cross, selecting, e.g., the lowesteigenvalues will not work in general.

For the two-body problem, the Pauli principle leads to asymmetric spatial wave function for the singlets = 0 spinstate, and an anti-symmetric wave function for the triplets= 1spin states. For the spatial part, we exploit the integrability ofthe system as follows. Define centre-of-mass coordinates by

~R :=1√2(~r1 +~r2)

and

~r :=1√2(~r1−~r2).

Using these coordinates, the two-body Hamiltonian becomes

H = H0(~R)+[H0(~r)+U(

√2r;λ)

]=: HC + Hrel (18)

wherer12 =√

2r :=√

2‖~r‖. We have introduced the param-eterλ explicitly in the potential in this equation.H is clearlyseparable, and the centre-of-mass coordinate HamiltonianHCis a trivial harmonic oscillator, while the relative coordinateHamiltonian can be written as

Hrel :=−12

∇2 +12

r2 +U(√

2r;λ),

where in polar coordinates~r = (r cosθ, r sinθ) we have

∇2 =1r∂

∂rr∂

∂r+

∂2

∂θ2 .

Applying separation of variables again, the eigenfunctions ofHrel can be written

ψn,m(~r) :=eimθ√

2πun,m(r)

wheren is the nodal quantum number.un,m(r) satisfies

K|m|un,m(r) = µn,mun,m(r) (19)

where

K|m| :=− 12r

∂

∂rr∂

∂r+

m2

2r2 +12

r2 +U(√

2r;λ). (20)

Equation (19) is an eigenvalue problem in the Hilbert spaceL2([0,∞), rdr), where the measurerdr is induced by the polarcoordinate transformation. Although it is natural to try andsolve the radial problem using Fock-Darwin orbitals, this willconverge slowly. The solution to this problem is to use a ra-dial basis of generalized half-range Hermite functions [11]. InAppendix B this is laid out in some detail.

Equation (19) is a one-dimensional equation, so there willbe no degeneracy in the eigenvaluesµm,n for fixed m. In par-ticular, the eigenvalues as function of the interaction strengthλ will not cross, and will be continuous functions ofλ. Wethus haveµm,n < µm,n+1 for all n, wheren is the nodal quan-tum number.

At λ = 0 we regain the harmonic oscillator eigenvalues2n+ |m|+ 1. Correspondingly, the eigenfunctionsψm,n(r, θ)approaches the Fock-Darwin orbitalsϕm,n(r, θ), i.e., the har-monic oscillator eigenfunctions. For the radial part,

limλ=0

um,n(r) = g|m|n (r) :=√

2r |m|L|m|n (r2)e−r2/2.

Reintroducing spin, the full eigenfunctionsΨ =Ψn1,m1,n2,m2 are on the form

Ψ(x1, x2) = ϕn1,m1(~R)eim2θ

√2π

un2,m2(r)χs,sz,

wheres= 0 for oddm2, ands= 1 for evenm2, and|sz| ≤ s isan integer.

Let Ri = 2ni + |mi | be the shell numbers for the centre-of-mass coordinate and relative coordinate, respectively. TheeigenvalueE = En1,m1,n2,m2 is

E = R1 +1+µn2,m2,

with limit

E−→λ→0

R1 +R2 +2,

which is the harmonic oscillator eigenvalue.As the centre-of-mass coordinate transformation conserves

harmonic oscillator energy, atλ = 0, the eigenfunctions that

7

are in the model space are exactly those obeyingR1 + R2 ≤R. Turning on the interaction adiabatically, the eigenpairs wemust choose for the effective Hamiltonian at a givenλ areexactly those withR1 +R2 ≤ R.

The model-space projectionPΨ needed in Eqns. (12) and(13) is now given by

PΨ(x1, x2) = ϕn1,m1(~R)eim2θ

√2π

[P|m2|

R−R1un2,m2(r)

]χs,sz,

where

P|m|R :=n

∑n=0

|g|m|n 〉〈g|m|n |, n =⌊

R−|m|2

⌋, (21)

wherebxc is the integer part ofx. This operator thus projectsonto then+1 first radial basis functions with given|m|.

Due to Eqn. (17), the unitary operatorZ can be decomposedinto its action on blocks defined by tuples ofn1,m1 andm2[13]. The minimization (12) can then be applied on block-per-block basis as well. Each sub-problem is equivalent tothe calculation of an effective HamiltonianKeff of the radialproblem for a givenm2 andn.

To this end, let ¯n andm= m2 be given. LetU be the(n+1)× (n+1) matrix whose elements are given by

Un,k = 〈g|m|n |um,n〉, 0≤ n,k≤ n,

i.e., the model space projections of the exact eigenvectors withthe lowest eigenvalues. LetU = XΣY† be the singular valuedecomposition, and letV = XY†. Then,

Keff = Vdiag(E0, · · · ,En)V†

and

Cn,|m| := Keff−diag(|m|+1,2+ |m|+1, · · · ,2n+ |m|+1)

is the(n1,m1,m2)-block of the effective interaction. If we re-turn to Eqn. (9), the effective interaction matrix elements ˜uab

cd

are now given by replacing the matrix elementsC|p−q|n,n+s by the

matrix elementsCn,|p−q|n,n+s , where

n =⌊

R−R1−|m|2

⌋, R1 = N+ M− (p+q),

whereN, M, p andq are defined immediately after Eqn. (9).

IV. CODE ORGANIZATION AND USE

A. Overview

The main program is calledQDOT, and processes a textualconfiguration file with problem parameters before proceedingwith the diagonalization of the Hamiltonian. Eventually, itwrites the resulting data to a MATLAB /GNU OCTAVE com-patible script for further processing.

As a C++ library as well as stand-alone application, OPEN-FCI is organized in several namespaces, which logically sep-arate independent units. There are three main namespaces:manybody, gauss, andquantumdot. Put simply,manybodyprovides generic tools for many-body calculations, such as oc-cupation number formalism, Slater determinants and CSFs,while gauss provides tools for orthogonal polynomials andGaussian quadrature. These namespaces are independent ofeach other, and are in no way dependent on the particularquantum dot model. On the other hand,quantumdot synthe-sizes elements from the two former into a quantum dot FCIlibrary. In QDOT, the main work is thus processing of the con-figuration file.

Two other namespaces are also defined, beingsimple_sparse and simple_dense, which are, re-spectively, simple implementations of sparse and densematrices suitable for our needs. We will not go into details inthe present article.

It should be clear that extending and customizingQDOT isa relatively easy task. The applicationQDOT is provided as atool with a minimum of functionality, and the interested willalmost certainly desire to further develop this small applica-tion.

In order to help with getting started on such tasks,some stand-alone demonstration applications are provided, allbased on the core classes and functions. These include an in-teraction matrix element tabulatorTABULATE , and a simpleprogramPAIRING for studying the well-known pairing Hamil-tonian [19], which we will not discuss further here. Finally,there is a small interactive console-based Slater determinantdemonstration programSLATER_DEMO as well. These appli-cations will also serve as indicators of the flexibility of OPEN-FCI.

OPENFCI does not yet support parallel computation onclusters of computers, using for example the Message PassingInterface [20]. Future versions will almost certainly be paral-lelized, but the present version in fact competes with parallelimplementations of the standard FCI method with respect toconvergence due to the effective interaction implemented, seeSec. IV C. The simple structure of OPENFCI also allows userswith less resources to compile and run the code.

B. Core functionality

The manybody namespace currently contains four mainclasses: Slater, CsfMachine, NChooseKBitset, andMatrixMachine. These will probably form the backbone ofany manybody computation with OPENFCI.

The classSlater provides Slater determinants, creationand annihilation operators, and so on. It is based on the stan-dard template library’s (STL)bitset class, which providesgeneric bit set manipulations. The classNChooseKBitsetprovides means for generating sets ofk objects out ofn possi-ble represented as bit-patterns, i.e., bit patterns correspondingto Slater determinants in the basisB or B′. This results in aSTL vector<Slater> object, which represent Slater deter-minant bases in OPENFCI.

8

The classCsfMachine is a tool for converting a basis ofSlater determinants into a basis of configurational state func-tions. These are represented asvector<csf_block> objects,wherecsf_block is astruct containing a few CSFs associ-ated with the same set of Slater determinants [9].

A CSF basis is again input for the classMatrixMachine,which is a template class, and generates a sparse matrixPAPof an operatorA, whereP projects onto the basis. It alsohandles bases of pure Slater determinants as they are triviallydealt with in the CSF framework. The template parameter toMatrixMachine is a class that should provide the matrix ele-mentshαβ , uαβγδ , etc, of the generic operator given by

A = ∑αβ

hαβa†αaβ+

12 ∑αβγδ

uαβγδa†αa†

βaδaγ

+13! ∑

αβ···vαβγδεζ a†

αa†βa

†γaζaεaδ.

Notice, that the indices are generic orbitals, and not assumedto be on the form(a,σ) as in Eqn. (3).

Currently, only one-, two-, and three-body operators are im-plemented. The reason is, that the matrix elements arenotcomputed by directly applying the sum of creation- and anni-hilation operators to Slater determinants, since this approach,however natural, is very inefficient. Instead, we apply Wick’stheorem directly [8] on the matrix elements known to be notidentically zero.

In the gauss namespace, several functions are de-fined which computes sequences of orthogonal polyno-mials via recurrence relations and weights and abscissafor Gaussian quadratures based on these. The lat-ter is done using the Golub-Welsh algorithm, whichonly depends on being able to compute the coeffi-cients of the recurrence relation [12]. The most impor-tant functions are perhapscomputeLaguerrePolys() andcomputeGenHalfGaussHermite(), which computes a se-quence of generalized Laguerre polynomials evaluated at agiven set of points and quadrature rules for generalized half-range Hermite functions, respectively.

Finally, the quantumdot namespace defines classes andfunctions that combined define the quantum dot problem. TheclassRadialPotential encapsulates potentials on the form(11). It also computes effective interaction blocksCn,|m|. TheclassQdotHilbertSpace provides means for generating thebasesB andB′, utilizing conservation of angular momen-tum, using a fast, custom made algorithm independent ofNChooseKBitset. The classQdotFci sews everything to-gether and is basically a complete solver for the FCI methodwith effective interactions.

C. Sample runs

A basic configuration file forQDOT is shown in Fig. 2.Varying the parameterslambda, R, S and the number of parti-clesA [23], and runningQDOT each time, we produce a tableof ground state energies, shown in Table I. By changing the

# ** Simple configuration file for qdot **

# -- model space parameters --

A = 4 # number of particles

M = 0 # angular momentum

S = 0 # total spin * 2

R = 15 # cut of model space

# -- interaction parameters --

lambda = 1.0 # interaction strength

# pot_p = -0.5 0 # uncomment

# pot_p_iseven = yes # these lines

# pot_alpha = 0 # for Johnson &

# pot_beta = 0 # Payne model

# -- computational parameters --

use_energy_cut = yes

nev = 50 # no. eigenpairs

use_veff = no

matlab_output = results.m

FIG. 2: Simple configuration file forQDOT

TABLE I: Some ground state eigenvalues produced byQDOT for N =3,4 electrons withλ= 2. Both the bare and the effective interactionare used

N = 3, M = 0, s= 12 N = 4, M = s= 0

R E0 E0,eff E0 E0,eff6 9.02370 8.96523 13.98824 13.88832

10 8.97698 8.95555 13.86113 13.8328014 8.96800 8.95465 13.84491 13.8284818 8.96411 8.95444 13.83923 13.8276122 8.96191 8.95435 13.83626 13.8273026 8.96049 8.9543030 8.95950 8.95428

parameteruse_veff we turn on and off the effective inter-action. The corresponding effective interaction ground statesare also shown in the table. Notice, that with the effectiveinteraction we obtain the same precision as the bare interac-tion, but with much smaller model spaces. This indicates thatOPENFCI can produce results that in fact compete with par-allel implementations of the standard FCI method, even in itspresent serial form.

In Table II we compare the ground state energies reportedin Ref. [9] with the alternative model spaceP ′ to the cor-responding values produced byQDOT, also usingP ′. ThisTable also appears in Ref. [3], and serves as a check of thevalidity of the calculations.

By uncommenting the lines following the definition oflambda, we override the default Coulomb interaction, andproduce a configuration file for the analytically solvablemodel given by Johnson and Payne [7], where the Coulomb

9

TABLE II: Comparison of current code and Ref. [9], taken from Ref. [3]

R= 5 R= 6 R= 7N λ M 2s Current Ref. 9 Current Ref. 9 Current Ref. 92 1 0 0 3.013626 3.011020 3.009236

2 0 0 3.733598 3.7338 3.731057 3.7312 3.729324 3.72951 2 4.143592 4.1437 4.142946 4.1431 4.142581 4.1427

3 2 1 1 8.175035 8.1755 8.169913 8.166708 8.16714 1 1 11.04480 11.046 11.04338 11.04254 11.043

0 3 11.05428 11.055 11.05325 11.05262 11.0534 6 0 0 23.68944 23.691 23.65559 23.64832 23.650

2 4 23.86769 23.870 23.80796 23.80373 23.8055 2 0 5 21.15093 21.15 21.13414 21.13 21.12992 21.13

4 0 5 29.43528 29.44 29.30898 29.31 29.30251 29.30

interaction is replaced by the parabolic interaction

U(r12) =−12λr2

12.

If λ is sufficiently small, all the eigenvalues of this model areon the form

E j,k = 1+ j +(k+ N−1)√

1−Nλ, j,k≥ 0. (22)

Since the potential is smooth, the eigenfunctions are allsmooth, implying exponential convergence with respect toR[3]. We therefore expect very accurate eigenvalues even withmoderateR. In Table III we show the first eigenvalues alongwith the error computed forN = 4 electrons withλ= 1/8. Thecomputations are done in theM = s = sz = 0 model spacewith R = 10 andR = 15. Some duplicates exist, and theyare included for illustration purposes. It is evident, that theeigenvalues become very accurate with increasingR; a clearindication of the correctness of the implementation.

V. CONCLUSION AND OUTLOOK

We have presented OPENFCI, an open source full configu-ration interaction implementation for quantum dots and sim-ilar systems. OPENFCI also implements a renormalized ef-fective interaction widely used in nuclear no-core shell modelcalculations, and we demonstrated that such interactions areindeed useful in the quantum dot calculations as well.

OPENFCI is easy to extend and adapt. Possible ap-plications are computations on systems with more generalsymmetry-breaking geometries and ind = 3 spatial dimen-sions. Also, a generalization of the CSF part of the code tohandle isobaric spin would allow us to handle nuclear sys-tems.

There is one more symmetry of the HamiltonianH that canbe exploited, namely that of conservation of centre-of-massmotion, which would further reduce the block sizes of thematrices. We exploited this symmetry for the effective inter-action, but it is a fact that it is a symmetry for the full Hamil-tonian as well. Using the energy cut model spaceP we maytake care of this symmetry in a way similar to the CSF treat-ment [21].

TABLE III: Results from diagonalizing the Johnson and Paynemodel. Many digits are included due to comparison with exact re-sults and high precision

R= 10 R= 15E ∆E E ∆E

4.535550207816 1.63·10−5 4.535533958447 5.25·10−8

5.950417930316 6.70·10−4 5.949751427847 3.96·10−6

5.950417930316 6.70·10−4 5.949751427847 3.96·10−6

5.950417930316 6.70·10−4 5.949751427847 3.96·10−6

5.951592166603 1.84·10−3 5.949760599290 1.31·10−5

6.243059891817 4.19·10−4 6.242642740293 2.05·10−6

6.243059891817 4.19·10−4 6.242642740293 2.05·10−6

6.535776573577 2.43·10−4 6.535534873729 9.68·10−7

6.535776573577 2.43·10−4 6.535534873729 9.68·10−7

6.535776573577 2.43·10−4 6.535534873729 9.68·10−7

7.375904323762 1.19·10−2 7.364103882564 1.43·10−4

7.375904323762 1.19·10−2 7.364103882564 1.43·10−4

7.375904323762 1.19·10−2 7.364103882564 1.43·10−4

7.375904323762 1.19·10−2 7.364103882564 1.43·10−4

7.375904323762 1.19·10−2 7.364103882564 1.43·10−4

7.393706556283 2.97·10−2 7.364440927813 4.80·10−4

7.393706556283 2.97·10−2 7.364440927813 4.80·10−4

7.393706556283 2.97·10−2 7.364440927813 4.80·10−4

7.410720999386 4.68·10−2 7.364876152101 9.15·10−4

7.665921446569 9.07·10−3 7.656945606956 9.14·10−5

As mentioned, we have not parallelized the code at the timeof writing, but it is not difficult to do so. A future version willalmost certainly provide parallelized executables, for exampleusing the Message Passing Interface [20].

APPENDIX A: CENTRE OF MASS TRANSFORMATION

1. Cartesian coordinates

In this appendix, we derive the centre-of-mass (COM)transformation utilized in Eqn. (9) for the interaction matrixelementsuab

cd.The one-dimensional harmonic oscillator (HO) Hamilto-

nian(p2x+ x2)/2 is easily diagonalized to yield eigenfunctions

10

on the form

φn(x) = (2nn!√π)−1/2Hn(x)e−x2/2

= (n!)−1/2Anxφ0(x), (A1)

where Ax := (x− ipx)/√

2 is the raising operator in thex-coordinate, and whereφ0(x) = π−1/4exp(−x2/2). The eigen-values aren+1/2.

Using separation of variables, the two-dimensional HOH0in x1 and x2 is found to have eigenfunctions on the formΦn1,n2(x1, x2) := φn1(x1)φn2(x2) and eigenvaluesn1 +n2 +1.Define the raising operatorsAxi := (xi − ipxi )/

√2, so that

Φn1,n2(~x) := (n1!n2!)−1/2An1x1

An2x2

Φ0,0(~x), (A2)

where the non-degenerate ground state is given by

Φ0,0(x1, x2) =1√π

e−(x21+x2

2)/2. (A3)

Note that this can equally well describe two (distinguishableand spinless) particles in one dimension.

To this end, we introduce normalized COM frame coordi-nates by [

ξ1ξ2

]=

1√2

[1 11 −1

][x1x2

]=: F

[x1x2

](A4)

The matrixF is symmetric and orthogonal, i.e.,FT F = F2 =1, transforming a set of Cartesian coordinates into another.The operatorH0 is invariant under this transformation, so theeigenfunctions have the same form with respect to these coor-dinates, viz,

Φ′n1,n2

(ξ1, ξ2) := φn1(ξ1)φn2(ξ2)

= (n1!n2!)−1/2An1ξ1

An2ξ2

Φ′0,0, (A5)

whereAξi = (ξi − ip′i)/√

2 are the raising operators with re-spect to the COM coordinates, andp′i are the correspondingmomentum components.

Define the operatorT by

Tψ(x1, x2) := ψ(ξ1, ξ2) = ψ

(x1 + x2√

2,x1− x2√

2

), (A6)

so that

TΦn1,n2 := Φ′n1,n2

. (A7)

Since T maps eigenfunctions in the two frames onto eachother,T must be a unitary operator, and the invariance ofH0under the coordinate transformation is the same as[H0,T] = 0,i.e., that energy is conserved. This in turn means thatT isblock diagonal with respect to each shellR= n1 +n2, viz,

TΦR−n2,n2 =R

∑n=0

〈ΦR−n,n|Φ′R−n2,n2

〉ΦR−n,n

=:R

∑n=0

T(R)n2,nΦR−n,n, (A8)

where T(R) is the (R+ 1)× (R+ 1) transformation matrixwithin shell R. It is real, symmetric, and orthogonal. Nu-merically, the matrix elements are conveniently computed us-ing two-dimensional Gauss-Hermite quadrature of sufficientlyhigh order, producing exact matrix elements.

In a two-dimensional setting, the two-particle harmonic os-cillator becomes a 4-dimensional oscillator. Let~r i = (xi ,yi),i = 1,2, be the particles’ coordinates, and leta = (m1,n2) andb = (m2,n2) to compress the notation a little. An eigenfunc-tion is now on the form

Ψa,b(~r1,~r2) := Φa(~r1)Φb(~r2)= CAm1

x1An1

y1Am2

x2An2

y2Ψ0,0(~r1,~r2) (A9)

whereC = (m1!n1!m2!n2!)−1/2.The COM coordinate transformation now acts in thex andy

directions separately, viz,F acts onxi andyi to yield the COMcoordinatesξi and ηi : [ξ1, ξ2]T = F[x1, x2]T and [η1,η2]T =F[y1,y2]T . The induced operatorT again conserves energy.Let M = m1 +m2 andN = n1 +n2. It is readily verifiable thatthe COM frame transformation becomes

TΨM−m2,N−n2,m2,n2 := Ψ′M−m2,N−n2,m2,n2

=M

∑p=0

T(M)m2,p

N

∑q=0

T(N)n2,pΨM−p,N−q,p,q. (A10)

Note that the shell number isR= N+ M, which is conservedby T.

2. Centre of mass transformation for Fock-Darwin orbitals

Consider a Fock-Darwin orbitalϕn,m(~r) in shell R= 2n+|m| with energyR+1. It is straightforward but somewhat te-dious to show that these can be written in terms of co-calledcircular raising operatorsB+ andB− [22] defined by[

B+B−

]:=

1√2

[1 i1 −i

][AxAy

]. (A11)

Letting µ = n+ max(0,m) and ν = n+ max(0,−m) (whichgivesµ,ν≥ 0) one obtains

ϕn,m(~r) = (µ!ν!)−1/2Bµ+Bν−Φ0,0(~r), (A12)

which should be compared with Eqn. (A2). Moreover,R =µ+ ν and m = µ− ν, giving energy and angular momen-tum, respectively. We comment that this is the reason for thenon-standard factor(−1)n in the normalization of the Fock-Darwin orbitals in Eqn. (7).

Let a two-particle HO state be given by

Ψµ1,ν1,µ2,ν2 := ϕn1,m1(~r1)ϕn2,m2(~r2),= CBµ1

1+Bν11−Bµ2

2+Bν22−Ψ0,0(~r1,~r2), (A13)

whereµi = ni + max(0,mi) and νi = ni + max(0,−mi), andwhere C = (µ1!ν1!µ2!ν2!)−1/2. We will now prove that,in fact, when applying the centre-of-mass transformation to

11

Eqn. (A13), we obtain an expression on the same form asEqn. (A10) viz,

TΨM−µ2,N−ν2,µ2,ν2 := Ψ′M−µ2,N−ν2,µ2,ν2

=M

∑p=0

T(M)µ2,p

N

∑q=0

T(N)ν2,qΨM−p,N−q,p,q, (A14)

whereM := µ1 +µ2 andN := ν1 +ν2.To this end, we return to the raising operatorsAξi andAηi ,

and express them in terms ofAxi andAyi . By using Eqn. (A4),we obtain [

Aξ1

Aξ2

]= F

[Ax1

Ax2

], (A15)

and similarly forAηi in terms ofAyi . In terms of the raisingoperators, the COM transformation becomes

Ψ′m1,n1,m2,n2

= C(Ax1 + Ax2)m1 (Ay1 + Ay2)

n1

× (Ax1 −Ax2)m2 (Ay1 −Ay2)

n2 Ψ0,0,(A16)

where

C = (2n1+n2+m1+m2n1!n2!m1!m2!)−1/2, (A17)

and we have used Eqn. (A9), but in the analogous COM case.Expanding the powers using the binomial formula (and thefact that the raising operators commute), we obtain a linearcombination of the individual eigenfunctions, which must beidentical to Eqn. (A10).

Let the COM circular ladder operators be defined by[B′j+B′j−

]:=

1√2

[1 i1 −i

][Aξ j

Aη j

]. (A18)

Using Eqn. (A15), we obtain that the circular raising operatorstransformin the same wayas the Cartesian operators whengoing to the COM frame, i.e.,[

B′1+B′2+

]= F

[B1+B2+

], (A19)

and similarly forB′i− and terms ofBi−. Using Eqn. (A13) inthe COM case, we obtain

Ψ′µ1,ν1,µ2,ν2

= C(B1+ + B2+)µ1 (B1−+ B2−)ν1

× (B1+−B2+)µ2 (B1−−B2−)ν2 Ψ0,0,(A20)

with

C = (2µ1+ν1+µ2+ν2µ1!ν1!µ2!ν2!)−1/2. (A21)

Eqn. (A20) is on the same form as Eqn. (A16). Again, byexpanding the powers using the binomial formula (and thatthe raising operators commute), we obtain a linear combina-tion with coefficients identical to those of the expansion ofEqn. (A16). It then follows that Eqn. (A14) holds.

APPENDIX B: NUMERICAL TREATMENT OF RADIALPROBLEM

We now briefly discuss the numerical method used for solv-ing the radial problem (19), i.e., the eigenvalue problem forthe operatorK|m| defined in Eqn. (20). This is an eigenprob-lem in the Hilbert spaceL2([0,∞), rdr), where the measurerdris induced by the polar coordinate transformation. The innerproduct on this space is thus given by

〈 f |g〉=∫ ∞

0f (r)g(r)rdr. (B1)

Let the Fock-Darwin orbitals be given by

ϕn,m(r, θ) =eimθ√

2πg|m|n (r), (B2)

with radial part

g|m|n (r) :=√

2L|m|n (r2)r |m|exp(−r2/2) (B3)

Thus,

〈g|m|n |g|m|n′ 〉= δn,n′ ,

and these functions form an orthonormal sequence inL2([0,∞), rdr) for fixed |m|.

In the electronic case,U(√

2r) = λ/√

2r has a singularityat r = 0 which gives rise to a cusp in the eigenfunctionum,n(r)at r = 0, or in one of its derivatives. Away fromr = 0, theeigenfunction is smooth. These considerations are also truefor more general potentials smooth forr > 0.

Diagonalizing the matrix ofK with respect to the truncatedbasisg|m|n (r)n

n=0 will give eigenpairs converging slowly withrespect to increasing ¯n due to the non-smoothness ofun,m(r)at r = 0. This is easily seen for them = 0 case andλ = 1,which has the exact ground state

u0,0(r) =(

r +1√2

)e−r2/2,

a polynomial of odd degree multiplied by a Gaussian. Thecusp atr = 0 is evident. However,

g0n(r) =

√2Ln(r2)e−r2/2,

which are allevenpolynomials. It is clear, that ¯n must be largeto resolve the cusp ofu0,0(r).

The eigenproblem is best solved using a basis of general-ized half-range Hermite functionsf j(r) [11], which will re-solve the cusp nicely. These functions are defined by

f j(r) := P j(r)exp(−r2/2),

where P j(r) are the orthonormal polynomials defined byGram-Schmidt orthogonalization of the monomialsrk with re-spect to the weight functionr exp(−r2). Thus,

〈 f j | f j′〉=∫ ∞

0f j(r) f j′(r)rdr = δ j, j′ .

12

The fundamental difference betweenf j(r) andg|m|n (r) is thatthe latter containsonly even (odd) powers ofr for even (odd)|m|. Both sets constitute orthonormal bases, butf j(r) will ingeneral have better approximation properties.

Moreover, since deg(r |m|L|m|n (r2)) = 2n+ |m|,

g|m|n (r) =2n+|m|

∑j=0

〈 f j |g|m|n 〉 f j(r) (B4)

gives the Fock-Darwin orbitals as afinite linear combinationof the generalized half-range Hermite functions, while theconverse is not possible.

Computing the matrix ofK with respect to f j(r) jj=0 and

diagonalizing will give eigenpairs converging exponentiallyfast with respect to increasingj. The resulting eigenfunc-tions’ expansion ing|m|n are readily computed using Eqn. (B4),whose coefficients〈 f j |g|m|n 〉 can be computed numerically ex-

actly using Gaussian quadrature induced byPJ(r), for J suf-ficiently large.

The basis sizej to use in the diagonalization depends onhow many eigenfunctions ¯n we desire. We adjustj semi-empirically, noting that 2¯n+ |m| is sufficient to resolveg|m|n ,and assuming that the exact eigenfunctions are dominatedby the latter. We then add a fixed numberj0 to get j =2n+ |m|+ j0, and numerical experiments confirm that thisproduces eigenvalues that indeed have converged within de-sired precision.

ACKNOWLEDGMENTS

The author wishes to thank Prof. M. Hjorth-Jensen (CMA)for helpful discussions, suggestions and feedback. Thiswork was financed by CMA through the Norwegian ResearchCouncil.

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(2000).[5] Seehttp://www.gnu.org/.[6] Seehttp://www.doxygen.org/.[7] N. F. Johnson and M. C. Payne, Phys. Rev. Lett.67, 1157

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