Richard Luzi Lombardini- Rovibrational Spectroscopy Calculations Using a Weyl-Heisenberg Wavelet...

ROVIBRATIONAL SPECTROSCOPY CALCULATIONS USING

A WEYL-HEISENBERG WAVELET BASIS AND

CLASSICAL PHASE SPACE TRUNCATION

by

RICHARD LUZI LOMBARDINI, B.S., M.S.

A DISSERTATION

IN

PHYSICS

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Bill Poirier Chairperson of the Committee

Greg Gellene

Wallace Glab

Thomas Gibson

Accepted

John Borrelli Dean of the Graduate School

August, 2006

ACKNOWLEDGMENTS

This was my first attempt exploring the unknowns of theoretical physics and

chemistry. Any successes that I achieved were only possible through the patience and

guidance of my advisor Bill Poirier. Encouragement was given to me unconditionally

by my parents, Barry and Leila Lombardini, that was vital to the completion of this

degree. Peers and colleagues in the chemistry department, Jason McAfee, Justin

Rajesh Rajian, and Buddhadev Maiti, especially those who were my cellmates in the

Poirier lab, Sean Xiao, Junkai Xie, Wenwu Chen, Jason Montgomery, Akbar Salam,

and Corey Trahan, made the experience enjoyable and less frustrating since we were

“all in the same boat”. Some of these calculations (Chapter IV) were done on “Jazz,”

a 350-node computing cluster operated by the Mathematics and Computer Science

Division at Argonne National Laboratory, and I am grateful for the staff at Argonne

for their technical help. Some of the staff (Srirangam Addepalli and David Chaffin)

at the Texas Tech High Performance Computing Center were very helpful with issues

involving parallel programming. Last of all, I would like to recognize and thank

my committee members (the three G’s), Greg Gellene, Wallace Glab, and Thomas

Gibson, for trying to make sense of it all.

ii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

I. MOTIVATION AND BACKGROUND . . . . . . . . . . . . . . . . . . . . 1

1.1 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Organization of Dissertation . . . . . . . . . . . . . . . . . . . . . . . 11

II. ROVIBRATIONAL SPECTROSCOPY CALCULATIONS OF Ne2 . . . . 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2.1 Phase Space Truncation . . . . . . . . . . . . . . . . . . . . . 17

2.2.2 Weylet Basis Set . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Matrix Representations and Numerical Implementation . . . . . . . . 20

2.3.1 Momentum-Symmetrized Gaussian Expansion . . . . . . . . . 20

2.3.2 Matrix Representation: 1 DOF Case (Radial Ne2) . . . . . . . 21

2.3.3 Numerical Implementation: 1 DOF Case (Radial Ne2) . . . . . 24

2.3.4 Matrix Representation: 3 DOF Case (Cartesian Ne2) . . . . . 28

2.3.5 Numerical Implementation: 3 DOF Case (Cartesian Ne2) . . . 29

2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Results for Radial Ne2 (1 DOF Case) . . . . . . . . . . . . . . 31

2.4.2 Results for Cartesian Ne2 (3 DOF Case) . . . . . . . . . . . . 33

2.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

III.CUSTOMIZED PHASE SPACE REGION OPERATORS APPLIED TO

BASIS SETS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


3.2.1 Weylets and Momentum-Symmetrized Gaussians (SG’s) . . . 50

iii

3.2.2 Phase Space Region Operator (PSRO) . . . . . . . . . . . . . 51

3.2.3 PSRO for the Harmonic Oscillator (HO) . . . . . . . . . . . . 53

3.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3.1 Morse Oscillator (1 DOF) . . . . . . . . . . . . . . . . . . . . 55

3.3.2 Morse/Harmonic Oscillator (2 DOF) . . . . . . . . . . . . . . 56

3.3.3 Harmonic Oscillator (HO) . . . . . . . . . . . . . . . . . . . . 58


3.4.1 Results for Morse Oscillator (1 DOF) . . . . . . . . . . . . . . 59

3.4.2 Results for Morse/Harmonic Oscillator (2 DOF) . . . . . . . . 61

3.4.3 Results for Harmonic Oscillator (HO) . . . . . . . . . . . . . . 62

3.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

IV.PARALLEL PREPROCESSED SUBSPACE ITERATION METHOD . . . 78

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78


4.3 Parallel and Numerical Implementation . . . . . . . . . . . . . . . . . 85


REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

APPENDICES

A. JUSTIFICATION OF SPHERICAL TRUNCATION CONDITION . . . . 107

B. EIGENFUNCTIONS OF HARMONIC OSCILLATOR PSRO . . . . . . . 109

C. EIGENVALUES OF HARMONIC OSCILLATOR PSRO . . . . . . . . . . 115

D. JUSTIFICATION OF PREPROCESSING AND SUBSPACE ITERATION

METHOD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

iv

ABSTRACT

New basis set methods are examined regarding quantum mechanical calculations

of energy levels and wave functions of bound systems. The first method (I) involves

compact orthogonal wavelets as the basis set which is subsequently truncated using

the guidance of a classical phase space picture of the system. In this dissertation, the

first application of this technique to a real molecular system (neon dimer) is presented,

and many of the technical details are developed for its use on any arbitrary system.

Although in many respects, neon dimer represents a “worst-case scenario” for the

method, it is still competitive with another state-of-the-art scheme applied to the same

system. The second method (II) greatly improves the computed accuracies of the first

through the introduction of phase space region operators, which increase the efficiency

K/N of the basis set, where N is the number of basis functions needed to calculate K

energy eigenvalues to a given level of accuracy. For one model system, the absolute

error of the computed energy levels is reduced by nearly 4 orders of magnitude, as

compared to method I. Finally, a new parallel algorithm for matrix diagonalization

(method III) is introduced, which uses a modified subspace iteration method. The

new method exhibits great parallel scalability, making it possible to determine many

thousands of accurate eigenvalues for sparse matrices of order N ≈ 106 or larger.

v

LIST OF TABLES

2.1 Gaussian Cap Parameters . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Ground and First Excited Vibrational (J = 0) Level Energies for 1

DOF radial Ne2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.3 Ground and First Excited Vibrational (J = 0) Level Energies for 3

DOF Cartesian Ne2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.4 Rovibrational Bound States of Ne2 . . . . . . . . . . . . . . . . . . . 39

3.1 Vibrational Bound Energy Levels of 1 DOF Morse Oscillator . . . . . 65

3.2 2 DOF Morse/Harmonic Oscillator . . . . . . . . . . . . . . . . . . . 66

3.3 1 DOF Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . 67




3.7 2 DOF Harmonic Oscillator (Multiple Application of PSRO) . . . . . 71

4.1 Parallel Algorithm Versus Nonparallel Direct Diagonalization . . . . . 95

vi

LIST OF FIGURES

2.1 Various Weylet Basis Sets in 1 DOF . . . . . . . . . . . . . . . . . . . 40

2.2 Plots of Symmetrized 1 DOF Weylets in Configuration Space . . . . . 41

2.3 Lennard-Jones Potential for Ne2 System . . . . . . . . . . . . . . . . 42

2.4 Modified Lennard-Jones Potentials (Gaussian Cap) . . . . . . . . . . 43

2.5 Phase Space Truncations for 1 DOF Radial Ne2 System (J = 0) . . . 44

2.6 Phase Space Truncations for 3 DOF Ne2 System . . . . . . . . . . . . 45

3.1 Classically Allowed Region of Weylets and 1 DOF Harmonic Oscillator 72

3.2 Wigner-Weyl Representations of Weylets and 1 DOF Harmonic Oscillator 73

3.3 Classically Allowed Regions of Weylets and 1 DOF Morse Oscillator . 74

3.4 Classically Allowed Region of Weylets and 2 DOF Morse/Harmonic

Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.5 Efficiency Versus N . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.6 Efficiency Versus DOF’s . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1 Node Communication Setup . . . . . . . . . . . . . . . . . . . . . . . 96

4.2 Node Communication Setup for the Last Stage . . . . . . . . . . . . . 97

4.3 Number of Eigenvalues at a Relative Accuracy Versus N for the 3 DOF

Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 98


Anisotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . 99


Isotropic Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . 100

vii

CHAPTER I

MOTIVATION AND BACKGROUND

By themselves, weakly bound or “floppy” molecular systems are very interesting in

that they defy notions of “typical” molecular behavior. However, the particular study

of clusters—i.e., 2 − 1000 monomers (either atoms or molecules) bound together by

van der Waals forces or hydrogen bonding—opens new doors of exploration of physical

chemistry at the fundamental level, with respect to the ability to control and vary the

cluster size. With the advancement of experimental techniques involving the synthesis

of clusters at variable sizes,1 in conjunction with high-resolution spectroscopy at

frequencies ranging from the microwave to the ultraviolet realm, experimentalists

have been able to work closely with theorists whose struggles are, obviously not size

control per se, but instead, severe size limitations for accurate calculations due to

the inherent anharmonicities coming from the weak bonds (to be addressed in more

detail later).

Within the realm of cluster research, one field of study involves vibrationally

induced dynamical phenomena (in hydrogen-bonded molecular complexes) which is

important for gaining a better understanding of energy transfer from reactants to

products in chemical reactions. Such investigations allow one to gain insight on the

fine line that separates time-independent spectroscopy and time-dependent chemical

dynamics. The hydrogen fluoride dimer (HF)2 is a good example; here, the main

focus lies on vibrational predissociation, more specifically, the transfer of energy from

the excited HF monomer vibration to the weak intermolecular bond, which eventually

breaks. To gain the full picture of the process, both theoretical2 and experimental3

methods traditionally used for bound states are needed in conjunction with scattering

calculation techniques4 and experimental setups5,6 conducive to measuring dissocia-

tion probabilities.

Another area of interest, for which clusters are ideally suited, is the study of

how microscopic properties evolve into macroscopic behavior, since clusters serve as a

1

type of transitional form of matter, intermediate between independent units and bulk

matter.7,8 One example is the study of first order phase transitions. At the cluster

level, systems go through phase “changes” rather than phase transitions. One key

difference is that melting and freezing points are not identical.9 For the range of tem-

peratures in between, the system can hop back and forth between phases, similar to

the dynamics of coexisting isomers. For rare gas clusters, especially those comprised

of lighter atoms (such as Ne), quantum effects can have a great influence on these

unusual “phases,” sparking the development of efficient theoretical quantum statisti-

cal techniques for many-body systems.10 In principle, as one increases the size of the

cluster, the phase changes should approach the familiar first order phase transitions

of statistical mechanics—although this bridge has never been theoretically or exper-

imentally traveled upon since typical bulk sizes (or sizes at which these macroscopic

physical properties start to appear) are so much larger than clusters which have sizes

that one can directly manipulate. The same can be said for ion solvation, which

is examined at the molecular level with cluster size solvent shells in the gas phase

(very different than in solution). Stace,11 however, argues that one can extract use-

ful thermodynamical information for the bulk counterpart from studying individual

ion-solvent intermolecular interactions.12

Strictly on the theoretical front, the anharmonic behavior of the large amplitude

motion of cluster systems, due to multiple shallow minima separated by low isomeriza-

tion and dissociation barriers on the potential surface, renders accurate spectroscopic

calculations of energy levels extremely challenging. Despite the simplification offered

by the Born-Oppenheimer approximation, in which the electronic degrees of free-

dom (DOF’s) can be effectively removed, the numerical solution of the nuclear time-

independent Schrodinger equation becomes computationally expensive very quickly

with respect to increasing cluster size, since, for the most part, the nuclear DOF’s

must be treated as being coupled. The traditional picture of treating the system

approximately as decoupled single-DOF vibrational oscillators, and subsequent nor-

2

mal mode analysis, breaks down, due to the large anharmonicity which cannot be

regarded as a small perturbation.

Using basis set methods, which is the method of choice throughout this disser-

tation, the coupling of all DOF’s translates to the need for a rather large matrix

representation (or equivalently, basis set of large size N) of the nuclear Hamilto-

nian operator, H, in order to numerically calculate accurate energy values and wave

functions of H. Diagonalization of the corresponding N × N matrix representation

H—ordinarily, the computational bottleneck—directly provides the linear expansion

coefficients of the eigenfunctions of H, in terms of the N basis functions. With suffi-

ciently large N , these numerical solutions approximate the true wave functions closely,

at least towards the bottom of the spectrum; the corresponding eigenvalues of H, in

the same sense, should be close to, yet always larger than,13 the actual H. If there is

any separability in the Hamiltonian or decoupling of DOF’s, the Hamiltonian can be

broken up into smaller parts, each of which can be represented by smaller matrices

(since there are less DOF’s) and solved separately.

The worst, but most typical, case of no decoupling may require one to be very

innovative when choosing an appropriate coordinate system and basis set, with the

purpose of making the computations tractable. For optimal coordinate systems, Bacic

and Light14 suggest four crucial criteria to consider: the coordinates need (1) bound-

ary conditions that cover all of the configuration space that the system can possibly

span, (2) to exploit the highest symmetry of the system, (3) to be orthogonal such

that the kinetic energy operator has the least possible number of cross terms, (4)

and finally, if possible, to be carefully chosen such that there is the least amount of

coupling between vibrational modes (or DOF’s). Throughout this dissertation, Carte-

sian coordinates are used (there is a slight exception in the 1 DOF case of chapter

II) which fully satisfy conditions (1) and (3); the requirements of (2) and (4) are not

dealt with, although these are worth exploring in future works. Although curvilinear

coordinates could successfully address (4), they are, in general, more difficult to use

and lack the universality of Cartesian coordinates. As discussed in Ref. [15], one does

3

not have to deal with variable coordinate limits and boundary conditions, cross terms

in the kinetic energy operator, and other system and/or coordinate specific adjust-

ments when using Cartesian coordinates. Despite the benefits, the foremost reason

these are used here is that the Weyl-Heisenberg wavelets or “weylets”, which are the

basis functions of choice throughout this dissertation, are at present only defined in

Cartesian coordinates.

This dissertation mainly focuses on basis sets and the computational techniques

that accompany them, with the expectation that the combined effort can lead to

successful calculations of weakly bound cluster systems that would otherwise not be

possible, using conventional methodologies. A key metric in any discussion of basis

set methods is the basis set efficiency K/N , where N is the number of basis functions

needed to compute K eigenvalues to a desired level of accuracy. The next section

provides a discussion of the basis set efficiency for the present basis set ideas, leading

up to a justification for the use of phase space truncated weylets.

1.1 Basis Sets

Despite their nonorthogonality, one popular choice of basis is the real Gaussian

functions in configuration (position) space. Gaussians of 1 DOF are simple in that

each function depends upon only two parameters, width and center. Also, contribut-

ing to their popularity is the convenience they provide in calculating Hamiltonian

matrix elements: the kinetic and overlap matrix elements have analytical representa-

tions, and the potential matrix elements may always be obtained using Gauss-Hermite

quadrature16—typically requiring very few quadrature points, due to the good local-

ization and lack of oscillations of the Gaussian function. The localization property

allows for two further benefits. First, in 1 DOF at least, one can effectively target

eigenstates of an arbitrary Hamiltonian below a chosen energy value. Second, the lo-

calization results in effectively sparse (most matrix elements nearly zero) Hamiltonian

and overlap matrices, allowing for the application of fast computational techniques.

4

In addition to the above inherent characteristics, the literature discusses improve-

ments upon the efficiency of the Gaussian basis set, i.e., basis optimization schemes

for selectively choosing the center and width of each Gaussian function, sometimes

guided by physical arguments. Earlier works17,18 developed accurate algorithms for

choosing parameters of a single DOF Gaussian basis, in accordance with a balance

of semiclassical (SC) arguments and basis overlap criteria. The ultimate goal was

to achieve optimal efficiency while avoiding linear dependency issues. The SC meth-

ods produce a basis of narrow Gaussian functions densely distributed in regions of

high momentum (low potential) and wide functions sparsely distributed in regions

of low momentum (high potential) in configuration space. Unfortunately, the strict

SC-Gaussian methods do not work well in more than 1 DOF, due to a lack of either

straightforward applicability17 or optimal convergence.18 Refs. [17] and [19]-[22] ad-

dress this issue and propose new approximate SC methods with19,20,22 or without17,21

additional non-SC techniques. Other sources23,24 report alternatives to SC thought

in Gaussian basis development.

The latter techniques are critical for achieving respectable efficiencies on systems

involving high energy, heavy particle dynamics. Other methods have been developed

using different basis functions such as nonlocal plane waves,25,26 configuration space

grid functions,27–29 or wave functions of solvable Hamiltonian systems.30 The cor-

responding methodologies involve either point transformations of the coordinates of

the basis functions,27,28,30 the variational principle,29 or a combination of both.25,26

Although impressive efficiencies have been achieved at low DOF’s, all of these ap-

proaches have compromising issues at high DOF’s. The methods of Refs. [25]-[29],

for instance, exhibit exponential decay of the efficiency as the number of DOF’s in-

crease, which is an inherent problem of direct product basis sets (DPB’s).14,24,29,31

Cargo and Littlejohn show that their proposed canonical transformation does not

produce efficiencies at higher DOF’s tantamount to the results obtained in their 1

DOF Morse oscillator example.30

5

In this dissertation, a different basis methodology is used—the only one currently

known that formally defeats exponential scaling. The method hinges upon a phase

space picture that is applied to both the representational basis set, and the desired

eigenstates of the target application system. It exploits the fact that the simple

phase space picture used becomes exact in the Wigner-Weyl (WW) sense,32,33 in the

large basis limit. More specifically, in order to decide upon an optimal basis set, one

must have a clear picture of how the wave functions, |θi〉, of H are represented on

phase space. The set of K orthonormal |θi〉’s that lie below some energy Emax span

a subspace of the Hilbert space that can be represented by the projection operator

ρK =K∑

i=1

|θi〉〈θi| . (1.1)

The WW phase space representation of this operator (labeled “ρK”) is a probability

distribution function that is well-contained within the classically allowed region, R,

i.e. the region enclosed by the energy surface H = Emax, where H is the classical phase

space Hamiltonian function corresponding to H, under the WW mapping. The phase

space picture presumes that Emax is quantized such that the possible volumes of Rare K(2πh)f , where f is the number of DOF’s. In other words, each |θi〉 corresponds

to a non-overlapping region of volume equal to that of a Planck cell. This picture

becomes more accurate as K increases, in which the limit of ρK approaches a uniform

value of one within R, and zero outside.34

The phase space picture, as discussed above, can be applied to any type of repre-

sentational basis set, provided the basis is orthogonal. It can also be generalized for

nonorthogonal basis sets, although in this case, the picture must be modified in subtle

ways. For instance, complex-valued or phase space Gaussians (PSG’s), at first glance,

seem to be very good candidates for basis functions, especially under the guidance of

the phase space picture. First, their average momentum and position values, which

are also the parameters that distinguish one PSG from another, are simply the cen-

ters of their real-valued WW Gaussian representations. Each PSG can be mapped to

another by Weyl-Heisenberg phase space translation operators, hence they are also

6

referred to as Weyl-Heisenberg coherent states. Second, the WW representation of

each function is well-localized within a Planck cell region around the center. How-

ever, this property does not necessarily extend to a finite collection of PSG’s on phase

space. This can be best explained by introducing the projection operator ρN which

represents the finite set of PSG’s, |φi〉, i.e.,

ρN =N∑

i,j=1

[S−1]ij|φi〉〈φj| (1.2)

where each element of the overlap matrix, S, is given by Sij = 〈φi|φj〉. Unlike

Eq. (1.1), Eq. (1.2) has cross terms due to the nonorthogonality of the PSG’s. The

magnitudes of these terms are directly related to the degree of nonorthogonality of

the PSG’s and also signify the degree of “collective” delocalization. Consequently,

ρN can have significant probability far from the centers of all N individual PSG’s,

depending upon how closely these are bunched together.

Thus, although individual PSG’s are well-localized, collectively they need not be.

This has important ramifications for the phase space truncation scheme,35–37 i.e. the

method used to restrict the representational basis in order to achieve the highest

possible efficiency. This method is extremely simple, i.e. retain only those basis

functions whose centers lie within or near ρK . In order to be effective however, this

requires that ρN be well-localized about the basis phase space centers, which in turn

requires that the basis |φi〉 be orthogonal. The ideal basis should therefore be both

localized and orthogonal—precisely the two defining properties of weylets,35–37 as will

be discussed later.

First though, we find it useful to continue our discussion of PSG’s. The set of all

PSG’s, a.k.a. “coherent states,” comprises an infinitely overcomplete family of vectors

in the Hilbert space, that nevertheless satisfies a certain resolution of the identity.38

The completeness aspect reinforces the idea of using PSG’s for the representation of

arbitrary quantum states; however, overcompleteness leads to a new drawback, i.e.

linear dependence, quite distinct from the issue of nonorthogonality and collective

nonlocality. To remedy this, one work39 suggests the use of subsets of PSG’s (in the

7

1 DOF case) with centers lying on a line or circle in phase space. The goal is to

limit the basis expansion choosing from a pool of functions grouped together by some

simple criterion. Other ideas, within the confines of single dimensional manifolds, have

ranged from the random placement of PSG centers forming an irregular polygon,40

to their selective placement along the classical energy surface H = E of the system in

question, in order to get efficient representations of wave functions at energies close

to E.41

On the other hand, a discrete rectilinear lattice of PSG’s41 would be easier to

handle than the constructs mentioned above, particularly for multi-DOF applica-

tions. The most popular lattice arrangement is the von Neumann lattice42 where

there is one PSG per rectangular (or “hypercubical”) Planck cell. This density is

“critical” with respect to providing completeness without linear dependence, and is

denoted “d=1”. Use of the von Neumann lattice has spread to the fields of condensed

matter,43,44 quantum optics,45–47 and molecular physics41 regarding the representa-

tion of arbitrary quantum states. They have also gained notice outside of quantum

mechanics, in the realm of communication theory involving signal decomposition and

transmission.48 The completeness of these functions has been well-established,49–51

and minimal expansions using truncated sets of lattice functions have been shown to

be robust45 and extremely efficient46 in representing harmonic oscillator and squeezed

states.

These findings support the use of von Neumann lattice functions as a basis for

the calculation of energy eigenvalues and wave functions. Unfortunately, Davis and

Heller41 found that the convergence of the eigenvalues was slow with respect to the

number of basis functions for arbitrary systems not resembling that of the above-

mentioned harmonic oscillator or squeezed state (which are special cases), in effect,

because of the collective nonlocality problem discussed earlier. They showed that

improved performance could be achieved by increasing the lattice density, i.e., d > 1,

but this introduces near linear dependencies into the basis, and further delocalizes ρN .

8

Most importantly, however, d > 1 implies exponential scaling of basis set efficiency

with respect to the number of DOF’s.

Poirier35,36 applied the Lowdin canonical orthogonalization procedure52 to the

set of single DOF (d = 1) von Neumann lattice PSG’s in the hopes of improving

efficiency, by introducing orthogonality into the basis. Unfortunately, the resultant

orthgonalized basis functions are no longer individually well-localized, even when

the localization is optimized. In fact, Balian53 and Low54 proved that all critically

dense lattices of states supported by the Weyl-Heisenberg group cannot satisfy the

properties of orthogonality and good phase space localization, simultaneously.

Wilson55 developed a simple trick for eluding the Balian-Low “no go” theorem. By

using bimodal basis functions (in the single DOF case), consisting of both positive

and negative momentum components in a symmetric fashion, and simultaneously

working on a doubly dense lattice, d = 2, he was able to construct a complete,

orthonormal lattice representation for which all basis functions decay exponentially

in phase space. Daubechies et. al.56 applied Wilson’s idea to a set of tight frame

functions, each composed of an expansion of doubly dense PSG’s. In this dissertation,

we label these types of functions as Weyl-Heisenberg wavelets, or weylets, “Weyl-

Heisenberg” because they are transformed into each other via the operators of the

Weyl-Heisenberg group, and “wavelet” because, in the 1 DOF case, there are two

parameters or quantum numbers (signifying the center of the weylet on 2-dimensional

phase space) needed to label each function in the set.

Using the ideas of Wilson and Daubechies, Poirier35–37 then derived the optimally

localized weylet basis in 1 DOF, as well as, an efficient numerical scheme for their

construction, rendering them computationally practical for multi-DOF bound state

calculations. The Poirier weylets are extremely well-suited to the phase space trunca-

tion scheme35–37 mentioned previously. The weylets are easily extended to f DOF’s,

where approximately, one can think of the weylet ρN as a group of 2f−dimensional

“blocks” (or Planck cells) that are not overlapping and are concentric with the N

individual weylets. This uniform region, R′ [with a volume of N(2πh)f ], becomes

9

more of an accurate picture of ρN as N increases, following the same principle as

that described in Ref. [34]. In practice, the truncation scheme involves keeping those

N blocks whose centers on phase space have classical energies less than some chosen

Ecut parameter which is usually chosen to be slightly larger than, if not the same as,

the energy Emax, the maximum cutoff of the K wave functions of interest. Wasted

space/inefficiency (N > K) is only manifested by those lattice blocks on the periphery

which only partially overlap R. Overall, the efficiency does not exponentially decay

as DOF’s increase, unlike DPB methods, and approaches perfection (K/N −→ 1) in

the large K and N limit since R and R′ begin to resemble each other on phase space.

Phase space truncation and maximal phase space localization are what distin-

guish weylets from the very popular DPB’s, which warrants further discussion. Like

DPB’s, the multi-DOF weylets are products of single DOF functions in each of the

coordinates, i.e.,

ϕi(q) =f∏

j=1

ϕij(qj) (1.3)

where i = (i1, i2, . . . , if ) and q = (q1, q2, . . . , qf ) (although in general, DBP’s do not

necessarily use identical functions for each DOF). For the DPB case, because basis

truncation is applied to each DOF independently, the corresponding R′ (approximate

WW region on phase space representative of the DPB set) adopts a “cylindrical”

shape, i.e.,

R′ = R′(1) ⊗R′(2) ⊗ . . .⊗R′(f) (1.4)

where eachR′(j) corresponds to 2-dimensional phase space regions representing sets of

Nj single-DOF basis functions, i.e., N =∏f

j=1 Nj. If one were to attempt to mold R′

to resemble the ρK region R which is not cylindrical, then there would be significant

wasted space in the “corners,” corresponding to extra basis functions in ρN . The

result is exponential scaling of K/N with f ,24,29 even if one optimally determines

the individual R′(j) to produce a product region R′ that most efficiently covers R.29

On the other hand, the phase space truncation scheme35–37 precisely removes those

problematic regions.

10

Despite the poor efficiency at many DOF’s, DPB’s are still very popular because

they are convenient, and in some cases necessary, e.g., for the discrete variable rep-

resentation method (DVR).57–62 The DVR method uses a configuration space grid

representation which has the tremendous advantage that potential matrix elements

can be computed without the need for costly integrations. For DPB’s that give sparse

matrices H, iterative eigensolvers can be used such as the Lanczos method—although,

as pointed out by Dawes and Carrington,63 nobody has been able to advance beyond

four-atom molecules using DPB iterative methods, which has been an ongoing area

of study for over ten years.64

The “build and prune” approach,63 an idea that has been around for over 20

years,65–67 represents a substantial improvement to DPB’s, vis-a-vis advancing theo-

retical spectroscopic analysis beyond DPB limits. Functions within the DPB set that

negligibly contribute to the target states are pruned away, resulting in a correlated

truncation of DPB functions (i.e. truncations for individual DOF’s are no longer

independent of each other). The present weylet approach with phase space trunca-

tion is thus another build and prune method. However, unlike all other strategies,

the weylet version is the only one currently known that defeats exponential scaling.

Proving their worth, the weylet calculations have already been applied successfully to

model systems up to 15 DOF’s and beyond,37 which is a record using direct matrix

diagonalization techniques.

1.2 Organization of Dissertation

There are three parts to the body of this dissertation that are each complete in

that they contain all necessary explanations and background sufficient for them to

be independent works. The first part, i.e. chapter II, describes the first application

of the phase space truncated weylets to a real molecular system, the weakly bound

neon dimer in its ground electronic state. The majority of the chapter addresses

the technical details needed to efficiently calculate the matrix representation of an

arbitrary potential up to 3 DOF’s in the weylet basis; the development of the kinetic

11

matrix is comparably insignificant because the matrix is sparse and the elements can

be found via analytical expressions.

The subsequent part (Chapter III) uses projection operators that are reflective of

the system to customize the individual weylet functions themselves (as opposed to just

their truncation), resulting in a new nonorthogonal basis representation with large

improvements in accuracy and efficiency. The mathematical ideas for the implemen-

tation of the operators are inspired by works of Bracken, Doebner, and Wood.68,69 As

a preliminary work, the new method is only applied to model systems up to 4 DOF’s

in the hopes that, with future developmental efforts, it could be used for real systems.

Chapter IV, the last part of this dissertation, presents a new iterative method for

diagonalizing large sparse symmetric matrices. We test the approach by applying it

to harmonic oscillator systems up to 6 DOF’s represented in the truncated weylet

basis. The method uses a subspace iteration method that is very suitable for paral-

lelization (more so than Lanczos) if a simple preprocessing procedure is performed

on the matrix beforehand. The ideas for the preprocessing come from single-particle

density matrix purification schemes used in ground-state electronic-structure calcu-

lations.70,71 In addition to addressing the computational performance and scalability

of the method, this chapter also reports the tremendous efficiency of the truncated

weylet basis at very large N .

12

CHAPTER II

ROVIBRATIONAL SPECTROSCOPY CALCULATIONS OF Ne2∗

2.1 Introduction

Weakly bound molecular systems exhibiting large amplitude motion or “floppy”

behavior have been of longstanding amongst theorists and experimentalists, despite

technical challenges caused by low dissociation energies, large bond lengths, and sub-

stantial anharmonicities. For theoreticians interested in computing exact rovibra-

tional spectra for such systems, these challenges manifest as extremely costly “direct”

matrix diagonalizations.16 Two basic strategies have been developed to deal with this

situation: (1) optimize the representational basis set for the particular system, in

effect directly reducing the matrix size, N ; (2) apply iterative methods which, due

to sparsity or other reasons, need not store the complete matrix. Both of these ap-

proaches are directly amenable to the most convenient and commonly used choice of

basis representation, i.e., direct product basis sets (DPB’s) where the basis functions

are separable products in the coordinates.14,31 In particular, the discrete variable

representation method (DVR)57–62 is a popular configuration space grid represen-

tation based on DPB’s. The potential-optimized DVR methods,72,73 including the

maximally-efficient variety, the phase space optimized DVR,29,74 are, as implied by

the name, examples of (1) above, whereas the sparsity of the multidimensional DVR

matrix representation of the Hamiltonian implies that these are also ideally suited to

(2) above.64,75,76

Despite such advances, all DPB and associated DVR methods are still charac-

terized by exponential scaling with respect to the number of degrees of freedom

∗Reproduced with permission of the American Institute of Physics from ”Rovibrational spec-

troscopy calculations of neon dimer using a phase space truncated Weyl-Heisenberg wavelet basis”

by R. Lombardini and B. Poirier. Journal of Chemical Physics, Vol. 124, pp. 144107 (with minor

alterations and additions). Copyright 2006 by the American Institute of Physics.

13

(DOF’s).24,29 In other words,N

K∝ ecf (2.1)

where K is the number of eigenvalues computed to a given accuracy level, and f is the

number of DOF’s. The positive exponent c can be minimized via DPB optimization,

but never reduced to 0.

Eliminating exponential scaling requires a non-DPB method. In a recent series

of papers,35–37 one of the authors has introduced a promising new non-DPB method

based on symmetrized orthogonal Weyl-Heisenberg wavelets, or “weylets.” Though

offering many of the advantages of a DPB, when combined with a phase space trun-

cation scheme (Sec. 2.2.1), the weylet representation can be shown not only to defeat

exponential scaling, but also to approach perfect efficiency (K/N → 1) in the large N

limit, regardless of dimensionality. It has already been used to extend direct matrix

eigenvalue calculations for model systems to 15 DOF’s and beyond.37 Borrowing from

the basis truncation scheme used here, another exact quantum method has recently

been developed and applied to similar model systems;63 however it does not satisfy

perfect asymptotic efficiency.

Neither of the two methods described above has been previously applied to real

molecular applications, due primarily to difficulties in representing an arbitrary po-

tential energy operator in the truncated basis representation. In this chapter, we

present an efficient numerical method for achieving this important goal in the case of

a weylet representation. Although formally the resultant weylet potential matrix is

in general dense, in practice, the optimized phase space locality of the weylet basis

ensures that many matrix elements are essentially zero. Since the multidimensional

weylet kinetic energy matrix is also formally sparse, this enables the use of sparse

iterative matrix techniques,77,78 thus greatly increasing the matrix sizes N that may

be considered. This is especially important for higher dimensional calculations, for

which it has been observed that K increases much greater than linearly with N .37 For

simplicity, however, only direct matrix diagonalization methods are employed here.

14

In this chapter, we apply the weylet method for the first time to a real molecular

application, using the new potential matrix element evaluation technique (discussed

in more detail below). In particular, the bound rovibrational energy levels of Ne2 in

its ground electonic state ( 1Σ+g )—the simplest neon cluster system—are computed

in the full 3 DOF Cartesian coordinate representation. The dimensionality is not

especially large; however, the goal is to demonstrate feasibility of the weylet approach,

in anticipation of future, more challenging applications. The calculation is, at any

rate, of non-trivial difficulty, owing to the weakly-bound and long-range character of

the interaction.

Van der Waals complex systems such as rare gas clusters have gained much atten-

tion in recent years. Statistically, clusters (even with as few as seven atoms)79 exhibit

a coexistence of phases over a range of system temperatures and energies, serving as

a prototype for bona fide bulk matter phase transitions, and also solvation. Dynam-

ically, the long-range but small-magnitude van der Waals and dispersion forces in-

volved result in very interesting behavior totally unlike traditional covalently-bonded

molecules—although they are now known to play an important ancillary role even for

covalent systems. In particular, serving as the ultimate floppy/anharmonic molecules,

clusters are not well described by the conventional equilibrium geometry/normal mode

analysis, and thus require exact quantum treatment for their elucidation. This has

motivated the development of accurate ab initio80,81 and semi-empirical82–87 poten-

tials, and a number of experimental studies.82,88,89 In particular, the Aziz potentials

(semi-empirical) have had much success in reproducing macroscopic and microscopic

properties close to experiment for dilute neon gas,85,86 as well as highly compressed

neon solid.87

From a dimensional scaling standpoint, neon clusters are also useful as a purely

computational benchmark. In particular, the near-pair-wise nature of the interaction

renders it quite convenient to expand the dimensionality simply by adding more neon

atoms without the need to develop a full-blown potential energy surface (PES) beyond

Ne2, or perhaps Ne3. However, as the main focus of the present work is to establish

15

feasibility of the weylet method for real molecular systems, only the dimer will be

considered here, using a simple Lennard-Jones (LJ) model. The resultant computed

rovibrational energy levels may be directly compared with those of a previous Carte-

sian calculation.15 Though not quite as accurate as the Aziz potential, the simple LJ

model does provide semi-quantitative accuracy for rare gas cluster systems, as amply

demonstrated by previous theoretical investigations.90–93

The Cartesian coordinate aspect of the present study bears discussion. Despite

the reduction in dimensionality obtained by separating vibrational and rotational mo-

tions, the Cartesian rovibrational approach offers many computational advantages,15

particularly with respect to dimensional scaling investigations relevant to clusters.

However, the primary motivation for the present weylet study is simply that weylet

basis sets have not yet been defined for non-Cartesian coordinates35–37 (although pre-

vious work by Johnson and coworkers94 strongly suggests that such a generalization

can be achieved). Note that the weylet approach is not limited to cluster systems;

indeed, in many respects, such systems constitute a “worst-case scenario” for a weylet

representation, owing to concave phase space regions that favor a more traditional

affine wavelet approach,94–96 and to shallow potential wells that present relatively

small regions of available phase space.

The new potential matrix element evaluation method is essentially a Gauss-

Hermite quadrature scheme,16 exploiting the fact that each weylet basis function

can be explicitly decomposed as a sum of Gaussians. Since the product of two Gaus-

sians is also a Gaussian, one can use standard Gauss-Hermite quadrature techniques

to evaluate the requisite potential energy integrations. Since a potentially large num-

ber of operations is involved, especially for the multidimensional case, a number of

“tricks” are introduced to reduce to a bare minimum the computational (CPU) ef-

fort required to set up the matrices. For simplicity, and because the dimensionality

considered here is only three, one trick that is not implemented is the sequential

summation and truncation idea described in Ref. [37].

16

The remainder of this chapter is organized as follows. Section 2.2 will briefly sum-

marize the development of the phase space truncated weylet approach, as documented

more thoroughly in Refs. [36] and [37]. Section 2.3 discusses in detail the application

of the method to both a 1 DOF radial implementation, and a full 3 DOF Carte-

sian version for dimer systems. A majority of the description involves the explicit

recipe used to generate Hamiltonian matrices in the weylet representation. Section

2.4 presents results, followed by discussion.

2.2 Theoretical Background

2.2.1 Phase Space Truncation

The starting point of the phase space truncation scheme is the uniformly mixed

ensemble—the projection operator ρK spanned by the K lowest energy eigenstates,

|φi〉, of the system Hamiltonian, i.e.,

ρK =K∑

i=1

|φi〉〈φi| . (2.2)

It can be shown29,34 that the Wigner-Weyl phase space representation32,33,97,98 of ρK

is approximately given by

ρK(q1, ..., qf ; p1, ..., pf ) ≈ Θ [Emax −H(q1, ..., qf ; p1, ..., pf )] (2.3)

(where H is the classical Hamiltonian), with the accuracy increasing in the large K

limit.34 The parameter Emax is chosen such that the associated phase space region,

R (defined by ρK 6= 0), has volume K(2π)f in h = 1 units (as will be presumed

throughout this chapter). For the present purpose, the |φi〉 are taken to be the set

of all rovibrational bound states of Ne2—i.e., Emax = 0 is the dissociation threshold

[Fig. 2.5].

The basis set chosen to represent the system Hamiltonian for a given calculation

also corresponds to its own projection operator and associated phase space region, R′,

with volume N(2π)f . The challenge is to modify the representational basis such that

the regionR′ is as small as possible, but still completely encloses the desired regionR.

17

One way to achieve this is to discard individual basis functions whose contributions

to R′ do not overlap R. This is the essence of the phase space truncation scheme.

In this capacity, the Weyl-Heisenberg wavelets of Wilson and Daubechies,55,56 as

modified by one of the authors,35–37 constitute an ideal choice of representational

basis. In essence, each wavelet basis function corresponds to a single 2f -dimensional

phase space cube of volume (2π)f , which collectively comprise a rectilinear lattice.

The underlying weylet basis set is the same for all applications; however, the resultant

phase space truncation is system-dependent, as it is determined by R. In effect, one

attempts to sculpt the region R out of the cubical “blocks” with which it intersects

[Figs. 2.1 and 2.5]. Only those blocks on the periphery, which overlap only partially

with R, lead to wasted space/inefficiency, i.e. N > K.

In practice, the above picture is complicated somewhat by technical details. For

instance, the parameter Ecut > Emax is used to effect the truncation, rather than

Emax itself, to incorporate tunneling. Moreover, the region overlap condition might

be difficult to apply in practice, and is therefore replaced with the simpler truncation

criterion Hmid < Ecut, where Hmid is the value of the classical Hamiltonian at the

center of the weylet cube.

2.2.2 Weylet Basis Set

The weylet basis derives from Weyl-Heisenberg coherent states (or phase space

Gaussians), which in 1 DOF are given by

gqp(x) = e−iqp/2eipxg(x− q). (2.4)

The values (q, p) denote the phase space center of the coherent state gqp, with the

origin-centered g00 = g representing the single fiducial or “mother” state, from which

all others are generated via phase space translation. Of particular interest is the

subset of coherent states that form a lattice on phase space,

g(d)mn(x) = e−iqmpn/2eipnxg00(x− qm), (2.5)

18

where qm = (m/a)√

2π/d and pn = na√

2π/d comprise the lattice sites, and the

indices m and n are for the moment taken to be integers. The unit-dependent quantity

a is related to the “aspect ratio” of the lattice, and d is the density of phase space

lattice sites measured in units of (2π)−1. The value d = 1 denotes “critical density,”

as in Sec. 2.2.1.

At critical density, one can construct Eq. (2.5) lattices that comprise complete

orthonormal basis sets.41,42,55,56 From among these, the optimally phase space local-

ized basis, denoted f (1)mn(x), has been identified.36 Good phase space localization of

the weylet basis is essential for efficient truncation. Unfortunately, however, even

f (1)mn(x) is insufficiently localized, due to the Balian-Low “no go” theorem.53,54 How-

ever, a solution to this dilemma has also been discovered by Wilson55,56 involving the

doubly-dense tight frame lattices, whose optimally phase space localized representa-

tive f (2)mn has also been identified,36 and found to decay exponentially quickly outside of

the corresponding square phase space region. The f (2)mn functions themselves are over-

complete, but a particular momentum-symmetrized linear combination of the f(2)m(±n)

pair yields a complete orthonormal basis with exponential decay.

The Wilson construction requires integer values of n, and an awkward special

procedure for the n = 0 case [Fig. 2.1(b)]. In addition, the construction of a suit-

able doubly-dense tight frame lattice is numerically unstable, despite improvements

by Daubechies.56 Recent developments by one of the authors35,36 alleviates both of

these major difficulties. In particular, the n = 0 problem is addressed by shifting the

lattice by one-half unit in the momentum direction, so that n is now restricted to

half-integer values. A Wilson-type construction is then applied, resulting in a com-

plete orthonormal basis for which all basis functions are treated on an equal footing

[Fig. 2.1(c)]. The second problem has been resolved via a new algorithm36 that has

been used to obtain an extremely accurate Gaussian expansion of f(2)00 , i.e.,

f(2)00 (x) =

mmax∑

m,n=−mmax

cmng(2)mn(x), (2.6)

19

with

g(2)00 (x) = g(x) = (a2/π)1/4e−(a2x2)/2. (2.7)

Several comments on Eq. (2.6) are in order. First, the meaning of the indices (m, n)

have now changed; from here on out these are regarded as summation indices. Second,

the (m,n) values are integers, and in fact, both must be even integers (the odd-valued

coefficients all vanish36). Third, although the summation is in principle infinite over

all even m and n, in practice the expansion coefficients cmn (presented in Refs. [35]

and [36]) decay exponentially in |m|+|n|, due to exponential phase space localization.

Consequently, in practice, only the truncated summation given in Eq. (2.6) need be

considered. Finally, it should be noted that the cmn, and f(2)00 (x) itself, are real-valued

and symmetric, even for finite expansions.35–37

2.3 Matrix Representations and Numerical Implementation

2.3.1 Momentum-Symmetrized Gaussian Expansion

Note that the unsymmetrized weylet fiducial state, f(x) = f(2)00 (x), as determined

from Eq. (2.6) above, does not itself belong to the phase space lattice, because the

present scheme requires a half-integer-valued momentum index. However, we can

shift both the fiducial weylet itself, and the underlying lattice of expansion Gaussians,

using Eqs. (2.4) and (2.5), and the analogs for f . This gives rise to the doubly-dense

unsymmetrized weylet lattice,

f(2)st (x) = e−istπ/2eita

√πxf(x− s

√π/a), (2.8)

for which individual lattice functions are now labeled with new indices, (s, t). Clearly,

t must be a half-integer. In contrast, s ∈ {..., ε − 1, ε, ε + 1, ...}, where ε can be any

real value. Note that the underlying lattice of expansion Gaussians is coincident with

the weylet lattice—i.e., the Gaussian centers are located at the same locations in

phase space as the weylet centers. Although the allowed index values are thus the

same for the weylet and Gaussian lattice functions, it is convenient to use a different

20

set of indices, (u, v), to label the latter. This is particularly useful for expansion

summations, in which context u = s + m and v = t + n.

In practice, the symmetrized weylet basis is most relevant, given explicitly as37

ϕst(x) =√

2 sin[ta√

πx− πt(s + 1/2)]f(x− s

√π/a). (2.9)

Note from Eq. (2.9) that ϕst(x) is manifestly real-valued, although—unlike f(2)st (x)—

asymmetric about x = (m/a)√

π. Plots of several ϕst(x)’s are presented in Fig. 2.2.

Although in principle, one can obtain ϕst(x) by first summing Gaussians [to obtain

f(2)st (x)] and then symmetrizing, it is in practice more convenient to perform these

operations in the opposite order, as the intermediate functions (and matrix elements)

are then real-valued, and there are far fewer of them (two times fewer in 1 DOF). We

thus introduce the momentum-symmetrized Gaussian (SG),

ψuv(x) =√

2 sin[va√

πx− πv(u + 1/2)]g(x− u

√π/a) (2.10)

(also referred to as a “modulated Gaussian”), in terms of which the symmetrized

weylets are obtained directly from the following summation:

ϕst(x) =mmax∑

m,n=−mmax

(−1)(n2+mt)cmnψuv(x). (2.11)

Note that Eq. (2.11) (and all subsequent equations) assume that m and n are even

integers, and t is a half-integer (always positive for momentum-symmetrized func-

tions).

2.3.2 Matrix Representation: 1 DOF Case (Radial Ne2)

We first consider a simple 1 DOF calculation—i.e., the radial Ne2 problem—in

anticipation of the extension to the higher dimensional case. The van der Waals

interaction of Ne2 is modeled by the LJ potential,

V (r) = 4ε

[(σ

r

)12

−(

σ

r

)6], (2.12)

21

where ε = 24.743267 cm−1 is the well depth, r is the separation between the two

atoms, and σ = 5.1950000 a.u. yields an equilibrium separation of re = 21/6σ =

5.8311903 a.u. (Fig. 2.3).

The only coordinate, r > 0, is radial rather than Cartesian. Although the present

weylet formulation applies only to Cartesian systems, we convert the radial Ne2 prob-

lem into a Cartesian one by extending r to −∞, making V (r) a symmetric function,

and modifying it in the vicinity of r = 0 to obviate the singularity. Physically, the

negative values of r can be thought of as the two neon atoms in opposite places as

compared to the situation with positive r. Obviously, the physics is identical for r on

either side of 0, and one can think of the extended V (r) as a symmetric double-well

potential.

The singularity is replaced with a symmetric Gaussian “cap” in the region |r| <

rcut. When the centrifugal contribution is also included, the resultant effective poten-

tial becomes

Veff(r) =

4ε[(

σr

)12 −(

σr

)6]

+ J(J+1)2µr2 |r| ≥ rcut

αJe−βJr2+ γJ |r| < rcut

(2.13)

where J is the rotational quantum number, and µ = mNe/2 is the reduced mass

(mNe = 20.180000 a.u. is taken as the atomic mass of Ne). The parameters αJ ,

βJ , and γJ are chosen so that Veff(r) and its first two derivatives are continuous at

r = ±rcut [Fig. 2.4].

The only adjustable parameter is therefore rcut which directly affects the height

of the cap’s peak (the smaller rcut, the higher the peak). By increasing the peak’s

height, the eigenvalues converge to the correct energy values rendering the artificial

addition to be insignificant to the potential interaction of the system. The parameter

rcut needs to be close enough to 0 such that there is enough of the repulsion region

of the LJ potential to considerably damp the amplitudes of the bound states’ wave

functions leaving their transmission into the Gaussian portion to be negligible. If it is

22

too close, then the potential matrix elements become infinite. Fortunately, a balance

can be achieved which is demonstrated in Sec. 2.4. The cap parameters are listed in

Table 2.1 for the resultant rcut = 4.6 a.u. value.

The 1 DOF Hamiltonian, H, with the Cartesian modification described above,

takes on the usual kinetic-plus-potential form, T + Veff . In the symmetrized weylet

representation, therefore, the matrix elements become

〈ϕst|H|ϕs′t′〉 = 〈ϕst|T |ϕs′t′〉+ 〈ϕst|Veff |ϕs′t′〉or

[Hϕ

]s,t,s′,t′

=[Tϕ

]s,t,s′,t′

+[V ϕ

eff

]s,t,s′,t′

. (2.14)

Using Eq. (2.11), the weylet matrix elements for any observable A can be obtained

from the corresponding SG matrix elements as follows:

[Aϕ

]s,t,s′,t′

=mmax∑

m,n=−mmax

mmax∑

m′,n′=−mmax

(−1)

(n2+mt+n′

2+m′t′

)cmncm′n′

[Aψ

]u,v,u′,v′

(2.15)

For the kinetic energy contribution, the SG matrix elements are analytical:

[Tψ

]u,v,u′,v′

=πa2

4µ[h(u, v, u′, v′)− h(u, v, u′,−v′)] , (2.16)

with

h(u, v, u′, v′) = e−π4 (v2

∆+u2∆)

[1

2

(v2

+ − u2∆ +

2

π

)cos ζ − u∆v+ sin ζ

]. (2.17)

The “∆” subscript indicates the difference between the bra and ket indices, e.g.

u∆ = u − u′. The “+” subscript denotes the addition of indices, v+ = v + v′. Note

that under a change of sign of v′ [i.e., for the last term of Eq. (2.16)], v∆ becomes v+,

and vice-versa. The phase quantity ζ is given by

ζ(u, v, u′, v′) =π

2(u∆v+ + v∆). (2.18)

Note that since u∆, v∆, and v+ are always integer-valued, the trigonometric quantities

in Eq. (2.17) are always ±1 or 0.

The SG matrix elements for the potential energy can be written as

[V ψ

eff

]u,v,u′,v′

=1√π

e−π4(u∆)2 [b(u, v, u′, v′)− b(u, v, u′,−v′)] , (2.19)

23

where

b(u, v, u′, v′) =∫ ∞

−∞e−ρ2

cos[v∆

√πρ− ζ

]Veff

ρ +

√π

2u+

a

dρ, (2.20)

and ρ = ar − (√

π/2)u+.

2.3.3 Numerical Implementation: 1 DOF Case (Radial Ne2)

In this section, we present a “numerical recipe” that can be followed to imple-

ment the weylet scheme in 1 DOF. Various convergence parameters, related to basis

set expansions and quadrature integrations, are also introduced and explained. The

numerical implementation is divided into four main steps, each of which will be ad-

dressed in turn:

1. Truncate representational ϕst basis, using phase space approximation or related

means.

2. Compute all necessary Eq. (2.20) potential energy integrations.

3. Construct Hϕ, the Hamiltonian matrix in the symmetrized weylet representa-

tion.

4. Diagonalize Hϕ to compute eigenstates.

Step 1 involves the determination of which ϕst basis functions will actually be used

to represent H. For most applications, the simple phase space truncation criterion

described in Sec. 2.2.1 would be utilized (i.e., ϕst is discarded if H(qs, pt) > Ecut),

although a slightly different scheme is used in Sec. 2.4.1. To increase the basis size N

and improve upon the eigenvalue accuracy, one can simply increase Ecut. Although,

for the Ne2 system, two problems come about from this scheme. First, the surface,

H = Emax, has a tail extending out to infinity in the position direction [Fig. 2.5].

If one raises the energy parameter past a certain value, i.e., set H = Ecut where

Ecut > (a2π)/(8µ), the new surface’s tail converges right above the centers of the first

row of weylet blocks along the positive position axis. Thus, the Hmid < Ecut criteria

24

includes an infinite number of blocks. Secondly, as Ecut increases, the signature hump,

coming from the well of Veff(r), flattens along the new surface, thus not retaining the

shape of the original H = Emax surface.

In addition to the truncation convergence parameter Ecut, the determination of

the weylet basis functions themselves also presents two adjustable parameters, albeit

of lesser importance. These are the position offset ε, and the aspect ratio parameter,

a. Again, the phase space picture may be employed to select nearly optimal, or at

least suitable, values.

In Step 2, tables are constructed of all necessary integrations of the Eq. (2.20)

form. Note that apart from the phase shift quantity ζ, the integrals depend only on

the index combinations v∆ and u+. Since ζ is a multiple of π/2, it serves only to

determine the sign of the Eq. (2.20) integral, and whether a sine or cosine function

is used. Only two integration tables are therefore needed, Bs for the sine case, and

Bc for the cosine case, both of which are two-dimensional arrays. The corresponding

table elements are as follows:

[Bc]u+,v∆=

∫ ∞

−∞e−ρ2

cos(v∆

√πρ)Veff

ρ +

√π

2u+

a

dρ (2.21)

[Bs]u+,v∆=

∫ ∞

−∞e−ρ2

sin(v∆

√πρ)Veff

ρ +

√π

2u+

a

dρ. (2.22)

The new variable ρ greatly facilitates numerical evaluation of the above integrals, for

which it is natural to employ Gauss-Hermite quadrature.16 This introduces a new

convergence parameter, Q, representing the number of quadrature points used to

evaluate each integration.

The basis set truncation of Step 1, together with the expansion truncation coef-

ficient mmax, impose limits on the ranges of the u+ and v∆ values needed for the Bc

and Bs tables. Further limits may also be imposed, however, due to the fact that

many of the integrations are nearly zero, or redundant, and may thus be disregarded.

In particular, one need only consider

0 ≤ u+ ≤ u+max ; 0 ≤ v∆ ≤ v∆

max. (2.23)

25

Although in general u+ and v∆ may be positive, negative, or zero, the even/oddness

of the integrands in Eqs. (2.21) and (2.22) imply that the negative case is identical to

a corresponding positive integral (apart from a possible sign change). For Bs, the zero

case may also be ignored (either u+ = 0, v∆ = 0, or both), as Eq. (2.22) clearly results

in a vanishing integral. It can be shown that the integration absolute values decrease

with increasing v∆ and u+, thus justifying the table truncation parameters v∆max and

u+max, quite independently of basis set truncation issues. In particular, increasing v∆

leads to a more oscillatory integrand in Eqs. (2.21) and (2.22), whereas increasing u+

places the Gaussian center further towards the asymptotic region where Veff → 0.

Step 3 involves the construction of Tϕ and V ϕeff . First, however, a determination

must be made of the set of ψuv SG functions collectively required to expand the trun-

cated ϕst basis functions from Step 1. The required (u, v) values will include all of the

truncated (s, t) values, plus all those within a band of width mmax around the original

phase space region. For all calculations performed here, mmax ≤ 6 is used, which for

the equality contributes a relative error on the order of 10−6. In practice, the expo-

nential decay of cmn with respect to |m|+ |n| can be fully exploited by replacing the

“square” summation∑mmax

m,n=−mmaxin Eq. (2.11) and subsequent expressions, with the

correlated “diamond” summation,∑|m|+|n|≤mmax

.35,36 This reduces the computational

effort by a factor of two for each such summation encountered, and also reduces the

set of required (u, v) values.

Once the appropriate set of expansion SG’s, ψuv, has been established, construc-

tion of Tϕ is straightforward—i.e., one merely applies Eqs. (2.16) and (2.17), and

the Eq. (2.15) summation (with diamond truncation applied to both primed and

unprimed indices). The Veff matrix construction proceeds similarly, except that

Eqs. (2.19) and (2.20) are used, and the b(u, v, u′, v′) values of the latter are ob-

tained using tables Bc and Bs, with appropriate sign factors. Note that[V ψ

eff

]u,v,u′,v′

depends on u∆, in addition to u+ and v∆. Moreover the u∆ dependence drops off as

26

a Gaussian, meaning that[V ψ

eff

]u,v,u′,v′

can be simply set to zero except when

|u∆| ≤ u∆max, (2.24)

where u∆max is a new convergence parameter. This results in reduced computation,

and increased sparsity for the final matrix Hϕ, obtained by adding together Tϕ and

V ϕ.

The final Step 4 is the most straightforward, and for large systems, also the most

computationally expensive: the real symmetric Hamiltonian matrix Hϕ is diagonal-

ized numerically, to compute energy eigenvalues, and possibly wave functions. For

purposes of this chapter, only direct diagonalization routines are considered; how-

ever, in principle, sparse eigensolvers may also be used, as will be explored in future

publications.

For the Ne2 dimer application, collectively there are nine parameters involved:

rcut, a, ε, Ecut, Q, mmax, u+max, v∆

max, and u∆max. Convergence with respect to most of

these is fairly decoupled, although there are certain correlated combinations such as

(a,Ecut) and (Q, rcut, a), with any of the last three of the nine listed above. These

correlations become increasingly important in higher dimensions.

For example, a < 1 (depending on the units chosen) corresponds to flat, wide

weylet blocks. A larger number of these would be required to span the requisite

region of momentum space, than for narrower weylet blocks. This requires the use

of SG’s that have high indices v or v′ which reflect high frequencies of oscillation

in position space. Ultimately, this produces the need for more quadrature points Q

in the integration. In the contrapositive situation, if Q is chosen to be low, then

one needs a > 1 or tall blocks in order to avoid the need for SG’s of high frequency.

There is also a mutual dependency between Q and rcut. The choice of Q can affect the

positioning of the quadrature points along the position axis. At high Q, there can be

too many quadrature points bunched inside the region of the artificial potential cap

negatively affecting the overall accuracy of the eigenvalues. To counter this, choosing

a small rcut would limit the extent of the artificial cap along the position axis. On

27

the other side, a large rcut would require Q to be reduced. Thus, a balance between

the two parameters needs to be achieved for convergence of eigenvalues.

2.3.4 Matrix Representation: 3 DOF Case (Cartesian Ne2)

For 3 DOF Cartesian calculations of the neon dimer, the LJ potential is the same

as Eq. (2.12), except that r is taken to be r =√

x2 + y2 + z2, with (x, y, z) the three

Cartesian coordinates describing nuclear separation. As in the 1 DOF case however,

an artificial Gaussian cap is introduced in the singular region, so that the resultant

V (x, y, z) corresponds to Eq. (2.13) with J = 0. The 3 DOF Cartesian Hamiltonian

operator is therefore

H = − 1

2µ

(∂2

∂x2+

∂2

∂y2+

∂2

∂z2

)+ V (x, y, z). (2.25)

The 3 DOF orthonormal symmetrized weylet basis functions are simply products

of the corresponding 1 DOF weylets, i.e.,

ϕs1t1s2t2s3t3(x1, x2, x3) = ϕs,t(x) =3∏

j=1

ϕsjtj(xj), (2.26)

where x = (x, y, z) = (x1, x2, x3), etc. As a result, obtaining multidimensional ma-

trix elements is in principle very straightforward. In particular, given the separable

form, T =∑3

j=1 Tj, of the 3 DOF Cartesian kinetic energy, the corresponding matrix

elements are given as

[Tϕ

]s,t,s′,t′

=∑

(i,j,k)=(1,2,3)

[Tϕ

i

]si,ti,s′i,t

′i

δsjs′jδtjt′jδsks′kδtkt′k . (2.27)

where the summation is over all cyclic permutations of (1, 2, 3), δ is the Kronecker

delta function, and[Tϕ

i

]si,ti,s′i,t

′i

are 1 DOF matrix elements from Sec. 2.3.2. Note that

the sparsity of Tϕ is comparable to that of a discrete variable representation.57–62

As for the potential energy matrix V ϕ, the 3 DOF version of Eq. (2.15) applied

to V is

[V ϕ

]s,t,s′,t′

=∑

||(m,n,m′,n′)||1≤mmax

3∏

j=1

(−1)

(nj2

+mjtj+n′

j2

+m′jt′j

)

cmjnjcm′

jn′j

[V ψ

]u,v,u′,v′

,

(2.28)

28

where ||(m,n,m′,n′)||1 =∑3

j=1 |mj|+|nj|+|m′j|+|n′j| is the composite vector 1-norm.

The form of the single, correlated summation bears comment. A literal application

of Eq. (2.15) would result in six “square” summations of two indices each—i.e., a

“hypercubical” summation in 4f = 12 dimensions, involving (mmax + 1)12 summand

terms. The multidimensional generalization of diamond truncation would reduce this

to two correlated summations,37 one each for the primed and unprimed indices. How-

ever, the exponential decay of the cmn indices, together with the product form of the

summand in Eq. (2.28), imply that a further correlation across primed and unprimed

indices may also be applied. This results in the above “simplicial” summation scheme,

for which the number of summand terms is reduced by something like 12! [or (4f)!

in general], thus avoiding exponential scaling.

Equation (2.19) generalizes to a sum of eight terms,

[V ψ

]u,v,u′,v′

=(

1√π

)3

e−π4u∆·u∆

1∑

k1,k2,k3=0

(−1)k1+k2+k3 b(u,v,u′, (−1)kv′), (2.29)

where (−1)kv′ = ((−1)k1v′1, (−1)k2v′2, (−1)k3v′3), and

b(u,v,u′,v′) =∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞e−ρ·ρ

3∏

j=1

cos[v∆

j

√πρj − ζj

] V

ρ +

√π

2u+

a

dρ1dρ2dρ3.

(2.30)

The other quantities above are defined as follows: u∆ = (u − u′); v∆ = (v − v′);

u+ = (u + u′); ρ = (ax−√

π2

u+); ζj = (π/2)(u∆j v+

j + v∆j ).

2.3.5 Numerical Implementation: 3 DOF Case (Cartesian Ne2)

For the most part, the 1 DOF numerical recipe provided in Sec. 2.3.3 generalizes

in straightforward fashion to the 3 DOF case. However, insofar as the there are

substantial differences, these will be addressed here.

Regarding Step 1, there are in principle not one, but three different aspect ratio

parameters, ax, ay, and az. Given the spherical symmetry of the rovibrational Ne2

system however, it is clear all three should have the same value in this case. Similar

comments apply to the position offset parameter, ε. Regarding weylet basis trunca-

tion, the simple Hmid < Ecut would be suitable for most applications, but does not

29

work especially well in the present case, for reasons described in Sec. 2.4.2, where an

alternate prescription is described.

For Step 2, all of the time-saving tricks from Sec. 2.3.3 may be applied, as well

as additional symmetry considerations. For instance, the Eq. (2.30) integration is

invariant with respect to permutations of the three components, (x, y, z), which can

be exploited to reduce CPU time and storage. This is particularly important, given

that the 3 DOF integrations now involve Q3, rather than Q, quadrature points. Note

that each of the ζj quantities in Eq. (2.30) independently determines whether its

corresponding sinusoidal factor is a sine or a cosine. Consequently, there are in

principle not two, but 2f = 8 different integration tables analogous to Eqs. (2.21) and

(2.22). However, permutation symmetry enables us to reduce these to just (f +1) = 4

tables, Bc3 , Bc2s, Bcs2 , and Bs3 , with obvious notation, e.g.

[Bc2s]u+,v∆=

∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞e−ρ·ρ cos

(v∆

1

√πρ1

)cos

(v∆

2

√πρ2

)sin

(v∆

3

√πρ3

)

×V

ρ +

√π

2u+

a

dρ1dρ2dρ3. (2.31)

Permutation symmetry can also be applied to the individual tables themselves, re-

sulting in a f ! = 6-fold reduction in the number of Bc3 and Bs3 table elements that

must be computed, and a (f − 1)! 1! = 2-fold reduction for the Bc2s and Bcs2 tables.

Symmetry can also be used to restrict the range of relevant table values beyond

Eq. (2.23), although in this context, it is rotational rather than permutation symmetry

that is responsible. Thus, in addition to

0 ≤ u+j ≤ u+

max ; 0 ≤ v∆j ≤ v∆

max for all j, (2.32)

one may also apply “spherical” truncation,

|u+| ≤ u+max ; |v∆| ≤ v∆

max, (2.33)

where the vector lengths are now computed in the usual 2-norm sense. The validity

of Eq. (2.33) is established in Appendix A. Note that as in the 1 DOF case, the

30

u+j = 0 and/or v∆

j = 0 table elements are identically zero when j corresponds to a

sine function in the Eq. (2.30) integrand.

Finally, we comment that as in the 1 DOF case, the Eq. (2.29) integral decreases

extremely rapidly with u∆ · u∆ = |u∆|2. As discussed in Appendix A, the result is

that[V ψ

]u,v,u′,v′

may be taken to be zero except when

|u∆| ≤ u∆max. (2.34)

As in the 1 DOF case, this leads to reduced computation and increased sparsity,

although the effect is much more pronounced for higher dimensionalities.

2.4 Results and Discussion

2.4.1 Results for Radial Ne2 (1 DOF Case)

As discussed previously, the simple H(qs, pt) ≤ Ecut basis truncation criterion

works extremely well for most molecular systems,35–37 but is expected to be less ef-

ficient for Ne2. In other respects as well, weakly-bound systems with long-range

interactions present a “worst-case scenario” for the present weylet approach—thus

providing another solid (if slightly perverse) motivation for the present study. The

reasons for this are two-fold. First, the weakly-bound aspect implies that K is small

even up to the dissociation threshold, thus placing us far from the large K limit where

K/N → 1. Indeed, Ne2 has only two vibrational levels. Second, the long-range inter-

action implies concave, rather than convex, phase space regions for sufficiently high

Emax—thus favoring conventional affine wavelets over weylets.35–37,94 As mentioned

before, with Emax at the dissociation threshold as is the case here, any Ecut > Emax

results in a phase space region in the continuum, with infinite extent (although in

practice, this need not cause a difficulty.)

To ameliorate the above situation somewhat, and because the 1 DOF calculations

are so inexpensive, we have opted for a more labor-intensive, but rigorously optimal

basis truncation scheme. Specifically, a weylet is discarded if the resultant computed

vibrational eigenvalues for J = 0 agree with those of a more accurate reference cal-

31

culation15 to within some desired accuracy. Weylet block pairs are thus “whittled

away” from an initial large rectangular lattice, starting from the top and bottom

rows and working inwards towards the q axis. For the particular aspect ratio param-

eter value a = 1.6 a.u. (choice explained below), this procedure was applied to Ne2

at an accuracy level equal to 2% of the well depth, ε (≈ 0.4949 cm−1), and again for

.2% and .02% of ε. The resultant truncated basis sets are indicated schematically in

Fig. 2.5. Note that even for the present worst-case application, the resultant pattern

of blocks conforms roughly to the H(q, p) = 0 region, thus validating the phase space

truncation idea. The resultant computed eigenvalues are presented in Table 2.2.

To optimize with respect to the other two weylet basis parameters, a and ε, one

can repeat the above procedure for many different values, and determine which yields

the smallest basis size N . We have performed such an optimization for a, but have

simply taken ε = 1/2 throughout (resulting in half-integer values for both s and t

indices). For the 2% accuracy calculation, this led to the optimal choice a = 1.6 a.u.,

resulting in a basis size N = 10.

The above studies were performed primarily to assess basis efficiencies associated

with a given level of accuracy (the three benchmark values chosen above correspond

to those used in previous calculations35–37). However, we have also performed a much

more accurate reference calculation, using N = 133 weylets with regions lying within

the phase space rectangle 0 ≤ x ≤ 21 a.u., −20 ≤ p ≤ 20 a.u. This basis was used

for all J values that support bound levels (J ≤ 9), resulting in a determination of

all rovibrational bound states to an accuracy of 10−4 cm−1 or better. The results are

presented in Table 2.4, Column 3.

The remaining convergence parameters were chosen with a view towards ensuring

that these do not contribute appreciably to numerical error, even for the highest

accuracy reference calculation. In particular, the parameter values u+max = 30, v∆

max =

8, and u∆max = 5 were found to be converged to 10−7 cm−1 or better for both computed

eigenvalues. Similarly, Q = 100 led to 10−6 cm−1 convergence. The dependence on

rcut was also found to be very insensitive, with both eigenvalues changing by only

32

2× 10−4 cm−1 over the range 3.8 ≤ rcut ≤ 5.0. The value rcut = 4.6 a.u. was used for

all subsequent calculations.

2.4.2 Results for Cartesian Ne2 (3 DOF Case)

The 3 DOF Cartesian calculations of the Ne2 rovibrational states make no attempt

to exploit rotational symmetry or degeneracy—unlike a previous calculation.15 As a

consequence, the required basis size N is much too large for the optimal quantum

truncation scheme to be applied. Nevertheless, we can still exploit the results of

Sec. 2.4.1 for the 3 DOF calculation, as described below.

The first step is to define a trial phase space region in 1 DOF, |p| ≤ pmax(r), which

encloses the centers of only the non-discarded weylet pair blocks corresponding to the

optimal, 2% accurate, N = 10 basis from Fig. 2.5. The functional form

pmax(r) = χ

√1− (r − η)2

ω2e−τ |r−ξ|, (2.35)

where the five parameters allow for flexibility of the shape: χ provides the vertical

height, η the location of the ellipse center along the position axis, ω the horizontal

axis length, τ the degree of exponential decay, ξ the position of the start of the

exponential decay. The numerical assignments χ = 6.2, η = 8.6, ω2 = 24.6, τ =

0.186, and ξ = 5.4 (all in a.u.), is found to fill the bill, i.e., the above parameter

choice has been somewhat optimized to minimize the volume of the resultant phase

space region, which is presented in Fig. 2.6.

Once a suitable pmax(r) is constructed as above, radial symmetry is used to extrude

this region into the full six-dimensional phase space of the Cartesian system, by

replacing p → |p|, and r → |x|. Only those multidimensional weylets whose centers

lie within the new region are retained for the 3 DOF calculation—i.e., those that

satisfy |pt| ≤ pmax(|rs|). Using the same a and ε values as in the 1 DOF case, this

results in a 3 DOF basis of N = 3480 weylet functions.

The above basis was anticipated to yield rovibrational energy levels within the

desired 2% accuracy level, but in fact fell somewhat short of this goal. To improve

33

accuracy, the phase space region was enlarged via simultaneous variation of χ, ω, and

τ . Also, in practice it was found that including the interior weylet functions (with

centers in the potential cap region) substantially improves accuracy, at the cost of

adding only around 1200–2500 additional basis functions. Table 2.3 indicates the

resulting convergence of the two pure vibrational level energies. The largest of these

calculations (N = 24 392) computed all 125 rovibrational states to within the desired

2% tolerance (using the fully converged 1 DOF calculations of Sec. 2.4.1 and Ref.

[15] as reference). The computed energies are presented in Table 2.4. For the above

calculations, all of the remaining parameters were converged to within 10−2 cm−1—

i.e., still substantially smaller than the basis truncation error. The values u∆max = 4,

v∆max = 6, Q = 17, and rcut = 4.9 a.u. were used for all calculations (except for the

last row where u∆max = 5 and Q = 18 were increased by one for better convergence),

whereas u+max was varied as per Table 2.3, Column 5.

Of the four computational steps described in Sec. 2.3.3, Step 3 was generally found

to be the bottleneck, due to the Eq. (2.28) summation. The choice mmax = 6 results

in 2625 summand terms per potential matrix element. In comparison, mmax = 4

and mmax = 2 require 313 and 25 summand terms, respectively. As the computed

eigenvalues were found to differ only by around 0.02cm−1 between mmax = 6 and

mmax = 4, the latter was used for the results presented in Tables 2.3 and 2.4. To

improve numerical performance, some effort was made towards finding the “leanest”

(i.e., least time consuming) calculation that computes all vibrational levels to within

the 2% tolerance. Using region parameters χ = 7.5, η = 8.6, ω2 = 35.0, τ =

0.160, and ξ = 5.4, and shaving off some of the high-momentum weylets in the

potential cap area, a suitable N = 10 896 basis was obtained. With the additional

parameter choices u+max = 19, v∆

max = 4, u∆max = 3, Q = 10, and mmax = 2, the total

time (for all steps) required on a Compaq Alpha 1200 MHz CPU was found to be 1.2

hours, with 99/125 rovibrational levels computed to within the strict 2% tolerance,

and the remaining 26 to a comparable level of accuracy.

34

2.4.3 Discussion

For reasons discussed in Secs. 2.1 and 2.4.1, the present Ne2 application presents

a “worst-case scenario” (apart from the low dimensionality) for the weylet basis ap-

proach. By any measure, even using more established optimized methodologies,15 the

3 DOF Cartesian Ne2 system presents a very challenging numerical calculation. It is

therefore reassuring to discover that the weylet approach is nevertheless competitive.

In comparing with Ref. [15] for instance, one finds that the basis sizes and especially

CPU times required to compute most of the 125 rovibrational states are greatly re-

duced; however, the computed eigenvalue errors are also substantially larger, to the

extent that two-to-three digits of accuracy are lost.

For the lean 3 DOF calculation, the resultant basis efficiency K/N = 99/10 896 ≈0.01 is not especially large. The corresponding efficiency value for the 3 DOF isotropic

harmonic oscillator system, for instance (Table IV. of Ref. [37]), is around 0.25—

although the comparison is somewhat biased against the Ne2 case because of the

way that accurate eigenvalues are counted. In any event, there are two important

causes that underlie this “efficiency gap”: (1) small and concave-shaped phase space

region; (2) large singularity “hole” in the potential cap region, due to use of Carte-

sian coordinates, and to large Ne2 equilibrium separation. Note that quadrature error

can definitely be ruled out as a major cause (Sec. 2.4). In practice, very few molec-

ular applications should exhibit such low weylet efficiencies in 3 DOF’s, since any

modification to the above (i.e. deeper well-depth, shorter-range interaction, smaller

equilibrium separation, or use of non-Cartesian coordinates) would greatly improve

performance. Efficiencies for typical systems—especially deeply-bound systems at

energies substantially below dissociation—are likely to be much closer to harmonic

oscillator values.37

Future efforts will investigate other molecular systems more amenable to the

weylet approach, including those at higher dimensionalities. In this regard, even

larger neon clusters such as Ne3 are much more favorable than Ne2, apart from the

increased dimensionality. The reason is that for purposes of studying solvation, or

35

the liquid-solid phase change, the energy range of interest extends only up to the first

isomerization threshold—i.e., the energy of just one bond—which for NeN>2, is far

below the dissociation threshold.

Note that the simple Gaussian quadrature scheme as employed here—though

shown to be remarkably effective for the three-dimensional application considered—

nevertheless reintroduces exponential scaling, in that the total number of quadrature

points grows as Qf . The present procedure, however, is unnecessarily wasteful, in that

no attempt is made to remove the “corner” region quadrature points. This could eas-

ily be achieved via spherical truncation, as can be justified using an argument similar

to that of Appendix A. We did not bother to do so here because the integral evalua-

tions comprised only a small fraction of the total CPU time. At high dimensionalities,

quadrature integral evaluation could in principle become the computational bottle-

neck, although the spherical truncation remedy described above—which incidentally,

is applicable even for non-spherically-symmetric potentials—reduces CPU effort ex-

ponentially with increasing f . Alternatively, if the desired accuracy level is not too

high, Monte Carlo integration techniques could be used—applied in phase space so

as to replace oscillatory “modulated” Gaussians with ordinary Gaussians. Thus, the

CPU cost associated with matrix initialization need not become the bottleneck at

large dimensionalities.

Further improvements and modifications to the weylet method will also be ex-

plored, e.g. non-Cartesian coordinates, sparse matrix methods, and those discussed

in Sec. IV C of Ref. [37]. Clearly, however, the most important area for improvement

is increasing the level of accuracy that can be obtained with the weylet approach.

Using projection operator methods (to customize weylet basis functions for given ap-

plications), we have recently taken large strides in this area, as will be reported in

future publications.

36

Table 2.1. Parameter values for the Gaussian cap of Eq. (2.13), with rcut = 4.6 a.u.

J αJ (a.u.) βJ (a.u.) γJ (10−4 a.u.)

0 3.1600 0.3758 -1.0607

1 3.1562 0.3757 -1.0406

2 3.1485 0.3756 -1.0005

3 3.1370 0.3753 -0.9404

4 3.1218 0.3750 -0.8603

5 3.1029 0.3746 -0.7601

6 3.0805 0.3741 -0.6400

7 3.0547 0.3736 -0.4998

8 3.0257 0.3729 -0.3397

9 2.9935 0.3722 -0.1597

Table 2.2. Ground and first excited vibrational (J = 0) level energies for 1 DOFradial Ne2, computed from quantum truncated weylet basis (Fig. 2.5)for three different target accuracy levels, measured in units of the welldepth. The literature energy values (Ref. [15]) are −14.0245 cm−1 and−2.6834 cm−1, respectively.

% well depth basis size N ground (cm−1) excited (cm−1)

2% 10 -13.8710 -2.3519

0.2% 26 -13.9789 -2.6346

0.02% 36 -14.0200 -2.6786

37

Table 2.3. Convergence of computed ground and first excited vibrational (J = 0)level energies for 3 DOF Cartesian Ne2, with respect to increasing phasespace region volume. Columns 1–3: parameter values used to specifyphase space region in Eq. (2.35) (the values η = 8.6 and ξ = 5.4 are heldconstant). Column 4: resultant 3 DOF phase-space-truncated basis size,N (including weylets in potential cap region). Column 5: u+

max value re-quired for convergence to within 10−2 cm−1. Columns 6 and 7: computedlevel energies; ‘*’ indicates those lying within desired 2% error tolerance.

χ ω2 τ N u+max ground (cm−1) excited (cm−1)

6.2 24.6 0.186 5200 18 −13.0457 −0.1094

6.8 29.4 0.174 7464 19 −13.6977∗ −1.4897

7.4 34.2 0.162 10 992 20 −13.8204∗ −1.8764

7.7 36.6 0.156 12 768 20 −13.8581∗ −2.0180

8.0 39.0 0.150 15 120 20 −13.8718∗ −2.1568

8.9 46.2 0.132 24 392 22 −14.0288∗ −2.3599∗

38

Table 2.4. All 125 rovibrational bound states of Ne2, as computed using the 3 DOFweylet basis of Table 2.3, Row 6. Parentheses indicate numerical degen-eracies. Last two columns: corresponding reference 1 DOF results fromSec. 2.4.1 (Column 3) and Ref. [15] (Column 4).

J E (cm−1) E (Sec. 2.4.1) E (Ref. [15])

0 -14.029(1) -14.0245 -14.0245

-2.360(1) -2.6834 -2.6834

1 -13.695(2), -13.674(1) -13.7213 -13.7214

-2.152(2), -2.145(1) -2.4922 -2.4922

2 -13.042(2), -13.035(2), -13.026(1) -13.1165 -13.1165

-1.746(2), -1.740(2), -1.736(1) -2.1143 -2.1143

3 -12.091(1), -12.089(1), -12.086(2), -12.080(3) -12.2129 -12.2129

-1.149(1), -1.147(1), -1.146(2), -1.142(2), -1.5595 -1.5595

-1.137(1)

4 -10.884(1), -10.854(2), -10.853(1), -10.849(3), -11.0148 -11.0148

-10.846(2)

-0.378(1), -0.377(2), -0.370(1), -0.367(2), -0.8455 -0.8452

-0.362(2), -0.359(1)

5 -9.400(2), -9.398(1), -9.348(2), -9.347(1), -9.5286 -9.5286

-9.346(2), -9.335(3)

6 -7.664(2), -7.628(2), -7.626(1), -7.584(1), -7.7628 -7.7628

-7.579(1), -7.574(2), -7.561(1), -7.559(2),

-7.556(1)

7 -5.616(2), -5.611(1), -5.601(1), -5.595(2), -5.7290 -5.7290

-5.566(1), -5.544(2), -5.540(1), -5.530(2),

-5.528(1), -5.519(2)

8 -3.302(1), -3.300(2), -3.294(1), -3.270(2), -3.4427 -3.4427

-3.267(2), -3.264(1), -3.255(2), -3.242(1),

-3.239(3), -3.237(2)

9 -0.739(2), -0.736(1), -0.733(1), -0.726(2), -0.9264 -0.9264

-0.725(2), -0.720(1), -0.719(1), -0.718(2),

-0.713(2), -0.709(1), -0.707(1), -0.698(2),

-0.697(1)

39

6? -� rrrrr rrrrr rrrrr rrrrr rrrr qp

(a)?

6? -� r r r r rr r r r r r r r rr r r r r r r rr r r r r r r r rr r r r r r r r rr r r r r r r r rr r r r r r r rr r r r r r r r rr r r r r r r r r qp

(b)??

6? -� rrrrrr

rrrrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrrrr

rrrrrr

rrrrrrrr

rrrrrrrr q

p(c)

??Figure 2.1. Schematic indicating phase space partitionings associated with various

weylet basis sets in 1 DOF. Dots represent phase space centers (q, p) forunsymmetrized [i.e., f(x)-type] weylet functions. A single symmetrized[ϕ(x)-type] weylet is indicated by the ‘?’: (a) critically dense weylets,

? = f(1)22 (x); (b) doubly-dense Wilson-Daubechies weylets, ? = ϕ22(x);

(c) doubly-dense weylets of present work, ? = ϕ 32

32(x).

40

- 10 - 5 5 10 15 20 25

- 1.5

- 1.0

- 0.5

0.5

1.0

1.5

PSfrag replacements�

x

- 10 - 5 5 10 15 20 25

- 1.5

- 1.0

- 0.5

0.5

1.0

1.5


x

- 10 - 5 5 10 15 20 25

- 1.5

- 1.0

- 0.5

0.5

1.0

1.5


x

t = 1=2

t = 3=2

t = 5=2

Figure 2.2. Plots of six different symmetrized 1 DOF weylets, ϕst(x) vs. x, fora = 0.5 a.u. Each plot corresponds to a different t value as indicated,with both s = 1/2 (left) and s = 11/2 (right) weylets represented. Largert values are associated with increasingly oscillatory behavior.

41

6 7 8 9 10 11

10

30

- 20

- 30

- 10

20PSfrag replacementsV

r(cm�1)

(a.u.)

Figure 2.3. The Lennard-Jones potential used for the neon dimer.

42

- 10 - 5 5 10

PSfrag replacementsV

r

(cm�1)

(a.u.)6� 1054� 1052� 105

Figure 2.4. Lennard-Jones potential (solid) and Eq. (2.13) modified potentials forrcut = ±4.9 a.u. (dashed) and rcut = ±4.7 a.u. (dotted), emphasizingthe singular region.

43

2.5 5 7.5 10 12.5 15 17.5- 2.5

- 5

- 7.5

- 10

2.5

5

7.5

10

PSfrag replacements (a.u.)(a.u.)

qp

Figure 2.5. Schematic indicating optimal quantum truncation of a = 1.6 a.u. weyletbasis functions used to compute bound vibrational (J = 0) level energiesfor 1 DOF radial Ne2 system. Thick solid/dashed/dotted lines enclosebasis used to achieve 2%/0.2%/0.02% well-depth error tolerance. Thethin solid line encloses the phase space region corresponding to all vibra-tional states up to the dissociation threshold, i.e. H(q, p) ≤ Emax = 0.

44

2.5 5 7.5 10 12.5 15 17.5- 2.5

- 5

- 7.5

- 10

2.5

5

7.5

10

PSfrag replacements (a.u.)(a.u.)

rp

Figure 2.6. Schematic indicating phase space regions used to truncate 3 DOF a =1.6 a.u. weylet basis functions to compute rovibrational bound states ofNe2. Innermost solid line encloses the phase space region correspond-ing to the dissociation threshold (Fig. 2.5). Next concentric solid linecorresponds to Eq. (2.35) with parameters from Table 2.3, Row 1, andoutermost concentric solid line to Table 2.3, Row 6. Inclusion of the capregion weylets corresponds to the dashed line extensions.

45

CHAPTER III

CUSTOMIZED PHASE SPACE REGION OPERATORS APPLIED TO BASIS

SETS

3.1 Introduction

For molecular bound state calculations, the choice of basis directly determines

the computational effort in solving the quantum Hamiltonian. More specifically,

the efficiency, K/N , where N represents the number of basis functions needed to

calculate K eigenvalues at a desired accuracy, has a strong dependence on the degree

of correlation between the basis functions and the target system. This relationship

has prompted research in basis optimization which focuses on the maximization of

the efficiency and thus, ultimately, the reduction of CPU effort and memory usage.

Symmetrized Weyl-Heisenberg wavelets, or “weylets,” comprise a type of univer-

sal orthonormal basis that can be effectively used to represent any bound molecular

system with an efficiency approaching perfection (K/N −→ 1) in the large N and

K limit, regardless of system dimensionality. Although the weylet basis functions

themselves are universal, the truncation of the basis set, achieved using a phase

space truncation scheme,35–37 is what tailors the method to individual systems. In

contrast, all direct product basis sets (DPB’s),14,31 even those for which individual

basis functions are optimized via self-consistent field or other techniques,29,99–101 ex-

hibit exponential reduction in efficiency with f , the number of degrees of freedom

(DOF’s).24,29 Consequently, the weylet approach has been applied to direct matrix

eigenvalue calculations for model systems of 15 DOF’s and beyond,37 far beyond what

would be feasible using a DPB. Recently, the method was successfully applied to a

real molecular system, Ne2 (in Cartesian coordinates).102 Although Ne2 presents a

“worst-case scenario” for the weylet method, in that f and K are small, and includes

states near the dissociation threshold, it is still competitive with other state-of-the-art

exact quantum dynamics methods.15

46

Phase space truncation of the weylet basis is effective because individual weylet

functions have good phase space localization, and are orthogonal. Achieving both

properties together is nontrivial,53,54 but can be achieved using a momentum-symmetrization

modification first introduced by Wilson55 and Daubechies et al.56 In 1 DOF, one starts

with a 2x overcomplete set of coherent states (CS’s) which are derived from phase

space Gaussians and arranged on a doubly-dense lattice (i.e., two CS’s per Planck

cell) on phase space. Provided the lattice of CS’s constitutes a tight frame, a partic-

ular linear combination of positive and negative momentum CS pairs then yields a

complete 1 DOF orthonormal weylet basis, |ϕi〉 (the general f DOF case is addressed

in Sec. 3.2.1). Poirier35–37 later refined the approach by constructing maximally phase

space localized weylets, in a computationally tractable manner for quantum dynamic

calculations.

To understand phase space truncation, we must first introduce two projection

operators:

ρN =N∑

i=1

|ϕi〉〈ϕi| (3.1)

and

ρK = Θ[Emax − H(q, p)] . (3.2)

The Wigner-Weyl (WW) phase space representation32,33,97,98 of ρK (simply labeled

as “ρK”) [see Fig. 3.2(a)] is a function that oscillates about unity in the classically

allowed region (QC) of phase space where H ≤ Emax, and is damped exponentially

outside this region. It can be shown that

ρK ≈ ρQCK = Θ[Emax −H(q, p)] (3.3)

in the large K limit, i.e., ρK can be associated with the classically allowed region of

phase space, R.34

Similarly, each single DOF weylet |ϕi〉 can be associated with a (momentum-

symmetrized) pair of blocks (corresponding to the CS pairs mentioned ealier), cen-

tered on the lattice sites, which on truncation, comprise a region R′ [see Fig. 3.1(a)].

47

If the block size is small (i.e., N and K large), then R′ closely resembles R and

N ≈ K, as the region volumes are proportional to basis size. In any event, it is the

blocks in the boundary of R′ that are the leading cause of inefficiency, since they

overlap R only partially. This effect is more pronounced at larger dimensionalities

for a given K value, so that in practice, the limiting difficulty of the weylet method

is the level of computed accuracy that can be achieved, rather than dimensionality

per se.

The main purpose of this chapter is to address this limitation by customizing the

individual weylet functions (i.e., not just their truncation) for particular applications.

Consider that the basis ρK |ϕi〉 = |ϕ′i〉, rather than |ϕi〉 itself, can in principle result

in an exact calculation of the lowest K eigenvalues. In the phase space picture,

this projection effectively transforms Fig. 3.1(a) to Fig. 3.1(b) which is seen to yield

R = R′ even when N and K are not large. Note that the peripheral basis functions

are most affected by the projection transformation.

In practice, the above picture is complicated by additional concerns. First, the

|ϕ′i〉 are not orthogonal—though this is easily remedied via orthogonalization52 or

direct solution of the generalized eigenvalue problem

H~v = ES~v , (3.4)

where H and S are the Hamiltonian and overlap matrices, respectively, in the nonorthog-

onal basis representation, and (E,~v) is the (eigenvalue,eigenvector) pair. More im-

portantly, |ϕ′i〉 might in principle be linearly dependent or nearly so, and in fact this

is almost certain to cause numerical instabilities if |ϕi〉 is a “random” basis or even

DPB, at sufficiently large f or N . Use of phase space truncated weylets as the start-

ing basis, |ϕi〉, almost completely alleviates this difficulty, however. Finally, ρK is not

known a priori—though ρQCK is known, and is likely to constitute a worthy substitute.

Fortunately, the mathematical development of ρQCK , or what we call the “phase space

region operator” (PSRO), and its action on arbitrary functions, has been extensively

studied by Bracken, Doebner, and Wood.68,69 With their insights, we have developed

48

very efficient projected basis sets, ρQCK |ϕi〉 = |ϕ(1)

i 〉 (the “(1)” superscript will be ex-

plained later), which are shown to greatly increase the accuracy levels that can be

obtained in weylet calculations.

A second idea explained in this chapter involves the use of momentum-symmetrized

Gaussians (SG’s), |ψi〉, rather than weylets, |ϕi〉, as the initial basis. SG’s centered on

the same lattice sites as their weylet counterparts span nearly the same subspace, and

individual SG’s are nearly orthogonal due to the momentum symmetrization.35–37,41

Moreover, the PSRO projected subspace of the SG’s and weylets are much closer still,

as compared to the unprojected case. Although |ϕ(1)i 〉 is still anticipated to be more

efficient than |ψ(1)i 〉, the latter are far more convenient to work with numerically.

Our results indicate impressive improvements in efficiency for PSRO-modified

weylets, |ϕ(1)i 〉, and SG’s, |ψ(1)

i 〉, on a wide range of model systems and dimension-

alities. The most noticeable improvements are in cases where N and K are small,

as expected. Moreover, the efficiencies of the two projected basis sets are nearly the

same, with SG’s actually more efficient than weylets in some cases, making them a

competitive basis if one can develop inexpensive techniques for the PSRO modifica-

tion. We have also found that multiple applications of the PSRO results in further

increases in efficiency for both basis sets, up to a point. The rationale here is that

higher powers of ρQCK are more nearly idempotent, and therefore presumably closer to

the exact projection operator ρK .

The remainder of this chapter is organized as follows. The next section presents a

brief description of the weylets and SG’s in the single and general f DOF cases (3.2.1).

Also, the theoretical development of the PSRO is discussed (3.2.2) and applied to the

special case of the harmonic oscillator (3.2.3). Section 3.3 provides the details of

the numerical application of the PSRO to both basis sets. Section 3.4 presents the

results, followed by a discussion (3.4.4) of all the data presented, and possible future

developments.

49


3.2.1 Weylets and Momentum-Symmetrized Gaussians (SG’s)

A complete analysis of the construction of the weylets and SG’s are documented

in Refs. [35]-[37] and Ref. [102], respectively; thus, only the mathematical form of the

basis functions will be presented in this chapter, along with a brief description. First,

the SG’s for the single DOF case in h = 1 units (as will be presumed throughout this

chapter) have the form:

ψst(q) =

(4a2

π

)1/4

cos[ta√

π(q − (s + 1/2)

√π/a

)]e−a2(q−s

√π/a)2/2 . (3.5)

The previous index i is replaced with the two indices, s and t, signifying lattice sites

(qs,±pt) where the unsymmetrized pair of phase space Gaussians are centered. The

lattice sites are specified by qs = (s/a)√

π and pt = ta√

π, with the parameter a

related to the “aspect ratio” of the lattice. Momentum symmetrization requires t to

be restricted to positive half integers, i.e., t = {0.5, 1.5, 2.5, . . .}, but s = {. . . , ε −1, ε, ε + 1, . . .} for any real ε.

The 1 DOF weylet functions can themselves be expanded into SG’s,

ϕst(q) =mmax∑

m,n=−mmax

(−1)(n2+mt)cmnψs+m,t+n(q) (3.6)

where m and n are even integers and cmn are coefficients listed in Refs. [35] and [36].

The cmn decay exponentially with respect to |m|+ |n|; thus, in practice, one can apply

a correlated “diamond” summation,∑|m|+|n|≤mmax

.35,36 In this chapter, the bound on

the summation is chosen to be mmax ≤ 6 outside of which all cmn have magnitudes

less than 10−6.

For the general f DOF case, the SG’s and weylets are products of the 1 DOF

functions:

ψs,t(q) =f∏

j=1

ψsjtj(qj) (3.7)

50

ϕs,t(q) =f∏

j=1

ϕsjtj(qj) (3.8)

where q = (q1, q2, . . . , qf ), s = (s1, s2, . . . , sf ), and t = (t1, t2, . . . , tf ). Each weylet

of Eq. (3.8) is approximately represented by groups of 2f blocks, with centers at

(qs1 , p±t1 , qs2 , p±t2 , . . . , qsf, p±tf ), each of volume (π)f [see Fig. 3.1(a)]. Thus, the set

of 2fN blocks, R′, has a total volume of N(2π)f , and similarly, the set of K target

eigenstates, R, has a total volume of K(2π)f . The SG’s of Eq. (3.7) follow the same

design, except that individual functions correspond to phase space “spheres”, rather

than blocks. The spheres overlap slightly, reflecting nonorthogonality of the SG basis,

and also leading to a somewhat lower efficiency of the SG’s compared to the weylets.

3.2.2 Phase Space Region Operator (PSRO)

In the f DOF case, the action of an arbitrary smooth operator A on the basis

function |ϑs,t〉 is

(Aϑs,t)(q) =∫〈q|A|q′〉〈q′|ϑs,t〉dfq′ (3.9)

where ϑs,t represents either the weylets, ϕs,t, or SG’s, ψs,t. The term 〈q|A|q′〉 in

Eq. (3.9) is known as the “configuration kernel” of A, and can be represented by the

expression:

〈q|A|q′〉 =1

(2π)f

∫A [(q + q′)/2,p] eip·(q−q′)dfp (3.10)

where A is the result of the WW mapping of A.32,33,97,98

The observable of interest is the PSRO, i.e.,

A(q,p) = ρQCK (q,p) = Θ [Emax −H(q,p)]

= Θ

Emax −

f∑

j=1

p2j/(2mj)− V (q)

(3.11)

for a system with a quantum Hamiltonian in the kinetic-plus-potential form (H =

T + V ) in Cartesian coordinates. Using Eqs. (3.9) and (3.10), the PSRO-modified

51

basis functions become

(ρQCK ϑs,t)(q) =

1

(2π)f

∫ρQC

K [(q + q′)/2,p] eip·(q−q′)〈q′|ϑs,t〉dfp dfq′ . (3.12)

Further simplification is possible for f = 168,69 as shown below. First, Eq. (3.11)

can be rewritten as

ρQCK (q, p) = Θ

[Emax − p2/(2m)− V (q)

]

=

1 xmin ≤ q ≤ xmax and − pmax(q) ≤ p ≤ pmax(q)

0 otherwise(3.13)

where pmax(q) =√

2m [Emax − V (q)]. The parameters xmin and xmax define the bound-

aries where pmax(q) is real and pmax(xmin) = pmax(xmax) = 0. At Emax equal to the

dissociation of the bound system, one or both of the parameters extend to infinity,

and, in practice, finite bounds need to be chosen when using ρQCK in computations.

Plugging Eq. (3.13) into (3.10), the 1 DOF configuration kernel of the PSRO can be

reduced to:

〈q|ρQCK |q′〉 =

1(2π)

∫ pmax[(q+q′)/2]−pmax[(q+q′)/2] e

ip(q−q′)dp xmin ≤ q+q′2≤ xmax

0 otherwise

=

sin

[(q−q′) pmax( q+q′

2)

]

π(q−q′) 2xmin ≤ (q + q′) ≤ 2xmax

0 otherwise. (3.14)

Finally, placing Eq. (3.14) into Eq. (3.9), the 1 DOF version of Eq. (3.12) simplifies

to

(ρQCK ϑst)(q) =

∫ 2xmax−q

2xmin−q

sin[(q − q′) pmax(

q+q′2

)]

π(q − q′)ϑst(q

′)dq′. (3.15)

52

3.2.3 PSRO for the Harmonic Oscillator (HO)

Consider the multidimensional isotropic harmonic oscillator (HO), where the masses

and frequencies are all equal to unity in atomic units, i.e., mj = ωj = 1 a.u. for

j = 1, . . . , f so that H(q, p) = (1/2)∑f

j=1(p2j + q2

j ). The spherical symmetry of this

system renders an exact analytical solution of Eq. (3.12) possible. The QC phase

space region R = {(q,p) | 0 ≤ ∑fj=1

[p2

j + q2j

]≤ 2Emax} [where ρQC

K (q, p) = 1] is a

2f−dimensional hypersphere centered at the phase space origin.

The operator ρQCK can always be written in the form

ρQCK =

∑

i

wi|Φi〉〈Φi| (3.16)

where wi are the eigenvalues of ρQCK and |Φi〉 the corresponding eigenfunctions. We

then have

(ρQCK ϑs,t)(q) =

∑

i

wi〈Φi|ϑs,t〉Φi(q) . (3.17)

For the isotropic HO system, the wi’s and corresponding |Φi〉’s are known analytically,

as are the overlaps 〈Φi|ϑs,t〉.As shown in Refs. [68] and [69], the eigenfunctions of ρQC

K are also those of ρK ,

i.e., the HO eigenstates (see Appendix B),

|Φi〉 = |n〉 = |n1〉 ⊗ |n2〉 ⊗ . . .⊗ |nf〉 (3.18)

where nj is a nonnegative integer representing the quantum excitation of the jth DOF

of the HO. The eigenvalues can be determined by

wn(Emax) = 〈n|ρQCK |n〉 . (3.19)

Using the WW formalism, Eq. (3.19) becomes

wn(Emax) =∫

ρQCK (q,p) Wn(q,p)dfqdfp

=∫

RWn(q,p)dfqdfp (3.20)

53

where Wn(q,p) represents the WW phase space representation32,33,97,98 of the pure

state density operator of each HO eigenfunction, i.e., |n〉〈n|. The analytical expression

is

Wn(q,p) =f∏

j=1

Wnj(qj, pj) (3.21)

where Wnj(qj, pj) =

(−1)nj

πLnj

[2(q2j + p2

j)]e−(q2

j +p2j ) (3.22)

and Lnjis a Laguerre polynomial of degree nj.

The above equations imply that the eigenvalues wn(Emax) depend only upon nS =∑f

j=1 nj instead of n itself, i.e., wn(Emax) = wnS(Emax) which is proven in Refs. [68]

and [69] (see Appendix C). Upon integration of Eq. (3.20), a recurrence relationship

can be derived:69

wnS+1(Emax)− wnS(Emax) =

(−1)nS+1 (nS)!

2f−1(nS + f)!e−2Emax(4Emax)

fLfnS

(4Emax) (3.23)

and w0(Emax) =1

(f − 1)!

∫ 2Emax

0tf−1e−tdt

= P (f, 2Emax) (3.24)

where LfnS

is the associated Laguerre polynomial, and P (f, 2Emax) is the incomplete

gamma function. The closed form of Eq. (3.23) is

wnS(Emax) = P (f, 2Emax) +

nS−1∑

k=0

(−1)k+1 k!

2f−1(k + f)!e−2Emax(4Emax)

fLfk(4Emax) (3.25)

for nS > 0. As shown previously, the radius of the hyperspherical regionR is√

2Emax.

Given that R has both a volume V2f = K(2π)f and

V2f =(2πEmax)

f

f !, (3.26)

54

one can derive a useful direct relation between Emax and K:

Emax = (Kf !)1/f (3.27)

which in practice is more useful than working with Emax.

Finally, the PSRO used in the calculation of the K HO eigenvalues and eigenfunc-

tions in the decomposed form of Eq. (3.16) is

ρQCK =

∑

nS ≤ nmax

wnS(Emax)|n〉〈n| (3.28)

where the sum above includes all states |n〉 that have nS ≤ nmax. The nonnegative

integer parameter nmax is chosen such that a desired level of convergence is achieved

in the final calculation.

3.3 Numerical Implementation

3.3.1 Morse Oscillator (1 DOF)

For 1 DOF realistic potentials, we resort to explicit numerical integration of

Eq. (3.15). Although the application to large K and N is computationally expensive,

the small problem sizes of this study are appropriate as the focus is to determine

whether this method can achieve significant increases in efficiency. Future projects

may involve the development of new time and memory saving techniques or approxi-

mations (similar to Sec. 3.3.2) to enhance application of this PSRO method to larger

problems.

We choose to examine the Morse oscillator, for which

H(q, p) =p2

2+ D(e−2κq − 2e−κq) . (3.29)

The parameters, D = 12.000 and κ = 0.2041241 are chosen so that R has a volume

of 48π a.u. at dissociation energy Emax = 0, thus signifying that there are 24 bound

states (K = 24). For comparison, the eigenvalues wi or energy values of the bound

states can be analytically determined:103

wi = −D + κ√

2D(i− 1

2

)− κ2

2

(i− 1

2

)2

(3.30)

55

where i = 1, . . . , 24.

The PSRO-modified weylets and SG’s are numerically computed using Mathemat-

ica. A set of points for q spaced at 0.02 a.u. increments between boundaries X(1)min

and X(1)max are used to define each of the N modified basis functions, ϑ

(1)st (q), using

Eq. (3.15). The boundaries, X(1)min and X(1)

max span slightly beyond the PSRO region be-

cause each projected function, ϑ(1)st (q), extends outside the [xmin,xmax] range, although

the extension does decay rapidly. In practice, xmin−X(1)min and X(1)

max−xmax are chosen

to be approximately between 1 and 2 a.u. which allows sufficient convergence of the

elements of H and S.

For functions resulting from the application of the PSRO operator p > 1 times,

ϑ(p)st (q) = [(ρQC

K )p ϑst](q), one needs to determine beforehand all appropriate bound-

aries for all modified basis functions from ϑ(1)st to ϑ

(p)st . This is done in a reverse fashion

by first choosing sufficient boundaries, X(p)min and X(p)

max, for ϑ(p)st (q), i.e., xmin − X

(p)min

and X(p)max−xmax are between 1 and 2 a.u., and then using the following equations for

the 1 ≤ b < p limits:

X(b)min =

X(p)min + (p− b)(xmin − xmax) (p− b) even

−X(p)max + (p− b)(xmin − xmax) + xmax + xmin (p− b) odd

(3.31)

and

X(b)max =

X(p)max + (p− b)(xmax − xmin) (p− b) even

−X(p)min + (p− b)(xmax − xmin) + xmax + xmin (p− b) odd

.(3.32)

One finds that the functions of lower modification need larger boundaries, i.e., X(b)min <

X(b′)min and X(b)

max > X(b′)max for b < b′.

3.3.2 Morse/Harmonic Oscillator (2 DOF)

For systems where f > 1, the Eq. (3.12) integrations are rather costly, even for

f = 2, for which four-dimensional integrals are needed. In such cases however, one

56

may apply a separable PSRO modification to greatly reduce the computational cost.

In this section, we apply this idea to the 2 DOF Morse/HO system,

H(x′, y′, px, py) =1

2(p2

x + p2y) +

(x′)2

2+ D(e−2κy′ − 2e−κy′ + 1) , (3.33)

which becomes coupled via rotation of the coordinates:

x = x′ cos 10o + y′ sin 10o (3.34)

and

y = −x′ sin 10o + y′ cos 10o . (3.35)

Instead of using the ρQCK (x, y, px, py) PSRO which is coupled, one applies the sep-

arable approximation ρQCKx

(x, px)ρQCKy

(y, py) where KxKy ≥ K. Since the ϑs,t basis is

also separable, one obtains

(ρQCK ϑs,t)(x, y) =

∫ 2xmax−x

2xmin−x

sin[(x− x′) pxmax(

x+x′2

)]

π(x− x′)ϑsxtx(x

′)dx′ ×

∫ 2ymax−y

2ymin−y

sin[(y − y′) pymax(

y+y′2

)]

π(y − y′)ϑsyty(y

′)dy′

= ϑ(1)sxtx(x)ϑ

(1)syty(y) (3.36)

where pxmax(x) =√

2[Emax − Vx(x)] and pymax(y) =√

2[Emax − Vy(y)]. Any of a

number of techniques may be used to obtain suitable 1 DOF marginal potentials (Vx

and Vy), with the primary criterion being that ρQCKx

ρQCKy

resemble ρQCK as closely as

possible. For this chapter, we use the method of Ref. [74].

The marginal potentials, Vx and Vy, resulting from the optimization74 can be found

by simply minimizing the original potential with respect to y and x, respectively,19,104

i.e.,

Vx(x) = min[V (x, y)]y (3.37)

57

and

Vy(y) = min[V (x, y)]x . (3.38)

No adjustments need to be made for the above equations, since min[V (x, y)] = 0.

The classically allowed region corresponding to the separable PSRO has a “cylin-

drical” shape composed of the product of 2 two-dimensional phase space regions,

Rx × Ry. This region contains “corners” not present in the nonzero region of ρQCK .

Thus, the separable PSRO is different from, and less effective than ρQCK , in the sense

that it fails to smooth out all of the peripheral lattice states due to the wasted space

of the corners.

3.3.3 Harmonic Oscillator (HO)

For the HO system, one does not need to bother with numerical integrations

needed for the realistic cases presented in Secs. 3.3.1 and 3.3.2. Instead, an analytical

representation of Eq. (3.12), as presented in Sec. 3.2.3, can be used; thus, one can

avoid the numerical errors coming from the computationally expensive integrations.

Ultimately, we want to represent the HO Hamiltonian in the PSRO-modified SG

|ψ(1)s,t 〉 or weylet |ϕ(1)

s,t〉 basis. Using Eq. (3.28), the orthonormality property of the

HO eigenfunctions, i.e., 〈n|n′〉 = δn1n′1δn2n′2 . . . δnf n′f, and the eigenvalue relationship

H|n〉 = (nS +f/2)|n〉, the Hamiltonian matrix elements in the modified SG basis are

given by

[Hψ(1)

]s,t,s′,t′

= 〈ψ(1)s,t | H |ψ(1)

s′,t′〉

=∑nS≤nmax

[wnS

(Emax)]2

(nS +

f

2

) f∏

j=1

(〈ψsjtj |nj〉〈nj|ψs′jt′j〉

).(3.39)

To solve Eq. (4.2), we also need the overlap matrix

[Sψ(1)

]s,t,s′,t′

= 〈ψ(1)s,t |ψ(1)

s′,t′〉

58

=∑

nS ≤ nmax

[wnS

(Emax)]2 f∏

j=1

(〈ψsjtj |nj〉〈nj|ψs′jt′j〉

). (3.40)

Similar equations apply for the weylet basis. The overlaps are given explicitly as

follows:

〈ψst|n〉 =πn/2

√2n−1n!

e−π4(s2+t2) Re

[(s + it)ne−i(π/2)t(s+1)

](3.41)

and

〈ϕst|n〉 =mmax∑

m,n=−mmax

(−1)(n2+mt)cmn〈ψst|n〉 (3.42)

In the above expressions, we have chosen the parameters a = 1 a.u., and ε = 1/2.

The generalization for |ψ(p)s,t 〉 is obtained by replacing [wnS

(Emax)]2 with [wnS

(Emax)]2p

in Eqs. (3.39) and (3.40).


3.4.1 Results for Morse Oscillator (1 DOF)

For the 1 DOF Morse oscillator system, we optimized our calculations for the

lowest K = 6 eigenstates. These are sufficiently far below dissociation that the QC

eigenstate region R (Emax = −6.7500 a.u.) is convex, suitable for the SG’s and

weylets, yet the last few eigenstates are high enough to clearly exhibit anharmonic

behavior (note the shape of R in Fig. 3.3). The N = 9 basis functions are chosen to

sufficiently cover R and are each modified by the PSRO ρQC6 . In practice, a suitable

basis size N depends strongly on K. If N is insufficiently larger than K, then the

PSRO is ineffective at computing all K desired states to sufficient accuracy. On the

other hand, if N is too large, then the overlap matrix S is ill-conditioned (eigenvalues

of S are too small) preventing generalized eigenvalue routines from solving Eq. (4.2).

59

The weylet basis functions were chosen as in Fig. 3.3. The selection is very similar

to what would be obtained via the phase space truncation criterion used in Secs. 3.4.3

and 3.4.2. We chose a = 1.8282 a.u. such that the heights of the lattice cells equal the

maximum extent of R in the positive and negative momentum direction. Also, the

position (horizontal) shift of the rectangles are adjusted (ε = −0.0273 a.u.) so that

the right edge of the block furthest along the position axis (in the positive direction)

corresponds to the right boundary xmax = 5.3058 a.u. of R. The left boundary

xmin = −2.4871 a.u. of R extends slightly further than one basis function; thus, we

added a ninth basis function on the negative side for sufficient coverage.

We considered both the SG’s and weylets as basis sets, as well as, their PSRO-

modified versions, up to p = 3. The triple modified functions, ϑ(3)st (q), lie between

chosen boundaries of X(3)min = −4 and X(3)

max = 7 a.u., sufficiently outside of the PSRO

bounds, xmin and xmax. By Eqs. (3.31) and (3.32), the bounds of the b < p functions

are X(2)min = −11.9743, X

(1)min = −19.5859, X(2)

max = 14.6116, and X(1)max = 22.5859 a.u.

We report in Table 3.1 the absolute errors between the calculated and analytical values

for all basis set types except for the p = 3 case, since these show no improvement

over the p = 2 case.

In both the weylet and SG case, the greatest reduction in the absolute error from

the unmodified to the modified case occurs in the lowest eigenstates. For example,

a near-four-order-of-magnitude improvement in accuracy is shown for the lowest two

eigenvalues of the p = 2 weylets, |ϕ(2)st 〉, and SG’s, |ψ(2)

st 〉, relative to the correspond-

ing unmodified basis sets, |ϕst〉 and |ψst〉. The |ϕ(1)st 〉 and |ψ(1)

st 〉 sets show accuracy

improvements ranging from 1-3 orders of magnitude for all 6 targeted eigenvalues.

In general, both of the SG’s and weylets follow the same pattern vis-a-vis the PSRO

modification, and exhibit comparable accuracies for the same p value, even in the

unmodified case, p = 0. This advocates strongly in favor of using SG basis functions

for practical applications. The main conclusion, though, is that PSRO modification is

extremely effective for either basis set, with the largest relative error for the targeted

K = 6 energies being only around 3× 10−5 for p = 2. Table 3.1 also indicates errors

60

for the remaining N −K = 3 eigenvalues, which show only small improvements and

sometimes loss of accuracy by the PSRO modification, thus demonstrating the ability

of the method to single out just the desired K eigenvalues from the others.

3.4.2 Results for Morse/Harmonic Oscillator (2 DOF)

We developed a separable PSRO that corresponds to a product region, Rx ×Ry,

where both components are projected regions onto the (x, px) and (y, py) axes,74

respectively, of R, the classically allowed region of the lowest K = 14 eigenstates

(Emax = 5.0700 a.u.) of the 2 DOF Morse/HO system. We applied the PSRO to

each of the SG basis functions (N = 49) selected by the basis truncation criterion

H(qs, pt) ≤ Ecut = 9.0000 a.u.35–37 In parts (a) and (b) of Fig. 3.4, the inner curves

(solid line) denote Rx and Ry, respectively, with boundaries of xmin = −3.1911 and

xmax = 3.1797 a.u. for the former and ymin = −2.4942 and ymax = 5.0820 a.u. for

the latter. The projections of R′, the classically allowed region of the SG basis set

(actually, more reflective of the weylets), are highlighted by the dotted lines and, in

a fashion similar to PSRO regions, are labeled as R′x and R′

y. The parameters of the

SG basis set were chosen to produce optimal computed eigenvalues for the unmodified

basis set and are listed as follows: ax = ay = 0.7979, εx = 0.4064, and εy = 0.0926

a.u.

The analytical eigenvalues of the Morse/harmonic oscillator (column 2 of Ta-

ble 3.2) are equal to the sum of the Morse [Eq. (3.30)] and 1 DOF harmonic eigenval-

ues, exactly like the uncoupled case [Eq. (3.33)], since the coupling is only due to a

rotation on the (x, y) plane. Improvements up to two orders of magnitude are shown

for the modified versus unmodified case. Note that there is not a clear divorcing of

the first 14 and the remaining 6 reported in Table 3.2, as there is in Table 3.1; al-

though, the accuracy improvements diminish rapidly beyond the K = 14 cutoff. This

behavior is due to the artificial separability used in the 2 DOF PSRO. The method

is nevertheless quite effective at improving the accuracy of the desired eigenvalues.

61

3.4.3 Results for Harmonic Oscillator (HO)

The isotropic HO up to 4 DOF’s was solved using 6 different basis sets: |ϕs,t〉,|ϕ(1)

s,t〉, |ϕ(2)s,t〉, |ψs,t〉, |ψ(1)

s,t 〉, and |ψ(2)s,t 〉 (Tables 3.3-3.6). In all cases, the basis truncation

criterion H(qs, pt) ≤ Ecut35–37 was used to obtain a representational basis of size N .

We also chose K = N , i.e., we set the volume of the PSRO region R to equal that of

the basis region R′ [Fig. 3.1(a)].

In Fig. 3.2, we can see, pictorially, how the PSRO affects the weylets in the 1

DOF case. Part (a) shows the WW phase space representation of the projection

operator containing the lowest 6 HO eigenstates. Part (b) gives the same setup for

the 6 weylets chosen as a basis to solve the HO Hamiltonian [the QC region of (b)

is the collective region of blocks in Fig. 3.1(a)], and part (c) represents the same

weylet set under a single PSRO modification. The modified weylets show concentric

ring-like features much like the HO eigenstates resulting in a considerable amount

of improvement in efficiency over the unmodified set (Table 3.3)—although, one can

still observe residual rectilinear weylet features in the slight box-like cornering of the

rings.

The analytical eigenvalues of the isotropic HO are (nS + f/2), with degeneracy

deg(nS, f) =

(nS + f − 1

f − 1

)(3.43)

for f > 1 (the eigenvalues are of course nondegenerate for f = 1). They are compared

to the actual calculated eigenvalues using the different basis sets, and Tables 3.3-3.7

report how many of the computed values fall within relative accuracies of 2 × 10−2,

2×10−3, 2×10−4, and so on (which correspond to error tolerances of (0.2)f , (0.02)f ,

(0.002)f , . . . , respectively.37) Table 3.7 stands apart from the other HO tables in

that it reports the effects of basis set efficiencies as one increases p, the power of the

PSRO operator.

62

For this analysis, we define the efficiency to be L/N , where L is the number

of eigenvalues actually computed to a certain relative accuracy as opposed to the

number of targeted eigenstates, K, used to defined the PSRO (K = N for the HO

case). This efficiency for f = 2 at 2× 10−4 relative accuracy is plotted against N in

Fig. 3.5 which demonstrates similar efficiency curves (apart from a shift) for all six

basis types. Although |ϕ(p)s,t〉 is now substantially more efficient than |ψ(p)

s,t 〉, the effect

is still no greater than that of the PSRO modification itself, in that |ϕs,t〉 and |ψ(1)s,t 〉

are comparable. Note the large increase in efficiency from N = 176 to 2 340; this

“efficiency cliff”37 shifts to the right with increasing dimensionality, and is the leading

cause of accuracy restrictions on weylet calculations at higher dimensionalities. The

fact that the PSRO improvements are comparable on and off the cliff is thus highly

encouraging.

Fig. 3.6 clearly shows that the efficiencies (2 × 10−2 accuracy) of all basis types

do not decay exponentially as the DOF’s increase (N ≈ 10 000 is held constant).

The results of the weylets and the SG’s are very similar in these plots: the weylets

are slightly better than the SG’s in the unmodified case, and for both the single and

double PSRO-modified sets, the results are essentially identical (modified weylets are

not shown). These similarities are only true at the 2× 10−2 accuracy, whereas higher

accuracies display significant improvement of the weylets over SG’s, as shown in the

previous analysis. The most important result is that the efficiency of the modified

functions decreases significantly slower than the unmodified with increasing f ; thus,

the benefits of PSRO modifications are expected to be substantially greater at larger

dimensionality.

In general, the PSRO projection operator significantly improves the efficiency of

both the SG’s and weylets as demonstrated by all tables and plots. Also, the modifi-

cations produce bases that allow for the calculation of a large number of eigenvalues

at extremely high accuracies, otherwise not attainable. For example, in the 2 DOF

case (Table 3.4), there are L = 757 (N = 20 864) eigenvalues that fall within the

relative accuracy of 2×10−12 using the |ϕ(2)s,t〉 basis. Even the less efficient basis |ψ(2)

s,t 〉

63

is impressive at L = 305 at similar conditions. A particularly surprising result is to

be found in Table 3.7, for which over 1 000 eigenvalues are computed to 2 × 10−7

accuracy, using the |ϕ(11)s,t 〉 basis of only N = 2 340 functions. Thus, the PSRO to the

11 power is very close to the actual projection operator ρK .

3.4.4 Discussion

In conclusion, the PSRO modification of the weylets and SG’s produce new

nonorthogonal bases with efficiencies far better than the unmodified sets—especially

useful in cases where the latter are limited in accuracy, or when K and N are small.

For realistic 1 DOF systems, one can successfully apply the PSRO to either the weylet

or SG basis via numerical integration to gain large improvements in accuracy, which

for the most part, is not computationally expensive. However, the same computations

do become a serious problem at larger dimensionalities. The favorable symmetry of

the model HO system allowed the analyses of higher DOF calculations, which would

otherwise not be possible, showing that efficiency and accuracy improvements get

even better as one increases f . This promising result provides motivation to explore

algorithms that would reduce computations involving real systems of multiple DOF’s,

for example, the separable PSRO of Sec. 3.3.2. For large basis sizes N and number

of PSRO modifications p, the computational effort becomes expensive, but is clearly

worth exploring further. One idea for possible future works involves the application

of Monte Carlo integration methods to the phase space integration in Eq. (3.12).

64

Table 3.1. The absolute differences between computed and analytical [Eq. (3.30)]values (in a.u.) for the lowest 9 energy levels of the 1 DOF Morse potential,using various basis sets of N = 9 functions in each case.

State index |ϕst〉 |ϕ(1)st 〉 |ϕ(2)

st 〉 |ψst〉 |ψ(1)st 〉 |ψ(2)

st 〉1 0.087 78 0.000 26 0.000 02 0.104 36 0.000 32 0.000 02

2 0.218 67 0.004 05 0.000 04 0.230 72 0.004 04 0.000 03

3 0.341 36 0.003 13 0.000 19 0.350 87 0.003 69 0.000 18

4 0.462 69 0.006 34 0.000 39 0.468 68 0.006 09 0.000 40

5 0.605 69 0.020 51 0.000 28 0.620 50 0.020 14 0.000 29

6 0.381 79 0.013 20 0.000 08 0.397 10 0.014 95 0.000 11

7 0.587 65 0.143 43 0.083 23 0.573 73 0.144 73 0.083 07

8 1.808 86 1.645 99 1.130 88 1.745 77 1.669 11 1.126 17

9 2.624 91 3.237 55 3.074 34 2.668 52 3.128 43 3.064 19

65

Table 3.2. Lowest 20 eigenvalues of the 2 DOF Morse/HO system (in a.u.). Analyti-cal eigenvalues are in column 2, whereas columns 3 and 4 present absolutedifferences between computed (with indicated basis of N = 49 functions)and analytical values.

State index exact |ψs,t〉 |ψ(1)s,t 〉

1 0.994 79 0.110 71 0.007 23

2 1.953 12 0.160 49 0.007 92

3 1.994 79 0.137 23 0.019 68

4 2.869 79 0.182 10 0.011 07

5 2.953 12 0.152 55 0.013 31

6 2.994 79 0.287 10 0.047 11

7 3.744 79 0.109 67 0.003 07

8 3.869 79 0.264 59 0.025 60

9 3.953 12 0.228 14 0.007 93

10 3.994 79 0.512 73 0.130 02

11 4.578 12 0.276 80 0.022 38

12 4.744 79 0.233 47 0.016 40

13 4.869 79 0.498 17 0.048 03

14 4.953 12 0.468 67 0.063 08

15 4.994 79 0.786 11 0.368 03

16 5.369 79 0.612 01 0.049 81

17 5.578 12 0.495 62 0.042 01

18 5.744 79 0.461 22 0.042 82

19 5.869 79 0.608 98 0.093 21

20 5.953 12 0.786 84 0.336 79

66

Table 3.3. Results for the 1 DOF isotropic HO system computed using six differentbasis sets. The values under the different basis columns indicate thenumber of eigenvalues computed to relative accuracy indicated in column3.

Ecut (a.u.) N accuracy |ϕst〉 |ϕ(1)st 〉 |ϕ(2)

st 〉 |ψst〉 |ψ(1)st 〉 |ψ(2)

st 〉4.0 6 2× 10−2 2 5 6 2 6 6

2× 10−3 0 3 5 0 3 5

2× 10−4 0 1 2 0 0 3

2× 10−5 0 0 1 0 0 1

110.0 108 2× 10−2 84 97 104 81 97 104

2× 10−4 53 75 88 48 54 83

2× 10−6 31 47 50 22 37 47

2× 10−8 17 20 29 7 18 20

2× 10−10 4 6 13 0 3 5

2× 10−12 0 1 2 0 0 0

177.0 180 2× 10−2 144 157 172 139 156 172

2× 10−4 105 118 131 97 108 121

2× 10−6 74 93 98 61 83 92

2× 10−8 52 60 75 35 54 58

2× 10−10 28 35 53 17 28 36

2× 10−12 6 24 27 7 13 21

2× 10−14 0 7 7 1 5 5

67

Table 3.4. Number of accurately computed eigenvalues for the 2 DOF isotropic HOsystem using six different basis sets (consult Table 3.3 for further details).

Ecut (a.u.) N accuracy |ϕs,t〉 |ϕ(1)s,t〉 |ϕ(2)

s,t〉 |ψs,t〉 |ψ(1)s,t 〉 |ψ(2)

s,t 〉20.0 176 2× 10−2 73 139 154 71 140 154

2× 10−3 31 69 113 16 67 111

2× 10−4 3 29 49 0 8 42

2× 10−5 0 0 8 0 0 5

70.0 2 340 2× 10−2 1 375 1 900 2 097 1 341 1 899 2 099

2× 10−4 510 882 1 133 363 575 919

2× 10−6 32 279 355 37 146 223

2× 10−8 1 23 45 1 6 11

2× 10−10 0 0 1 0 0 0

140.0 9 956 2× 10−2 6 730 8 167 8 826 6 557 8 153 8 818

2× 10−4 3 597 4 968 5 740 2 966 3 748 4 788

2× 10−6 1 906 2 563 2 874 1 097 1 806 2 279

2× 10−8 712 913 1 218 280 672 773

2× 10−10 120 235 416 46 94 134

2× 10−12 4 29 56 0 0 1

2× 10−14 0 0 1 0 0 0

205.0 20 864 2× 10−2 14 917 17 406 18 661 14 569 17 356 18 647

2× 10−4 9 037 11 440 13 053 7 808 9 221 11 103

2× 10−6 5 416 7 043 7 710 3 679 5 260 6 170

2× 10−8 2 738 3 320 3 959 1 500 2 618 2 990

2× 10−10 998 1 424 1 844 620 881 1 129

2× 10−12 47 637 757 81 142 305

2× 10−14 0 15 24 0 3 5

2× 10−16 0 2 2 0 0 168



s,t〉 |ψs,t〉 |ψ(1)s,t 〉 |ψ(2)

s,t 〉9.0 176 2× 10−2 50 137 141 50 140 141

2× 10−3 1 22 49 0 25 52

2× 10−4 0 3 19 0 3 19

21.5 1 928 2× 10−2 661 1 443 1 595 622 1 445 1 594

2× 10−3 185 482 851 89 452 851

2× 10−4 18 118 256 4 35 171

2× 10−5 1 1 20 0 0 3

40.0 9 552 2× 10−2 4 067 7 188 7 985 3 884 7 195 7 994

2× 10−3 1 728 3 095 4 446 1 122 2 934 4 355

2× 10−4 517 1 295 1 905 211 568 1 317

2× 10−5 119 267 502 28 103 196

2× 10−6 18 66 105 0 12 29

2× 10−7 0 9 17 0 0 1

2× 10−8 0 0 1 0 0 0

69



s,t〉 |ψs,t〉 |ψ(1)s,t 〉 |ψ(2)

s,t 〉6.5 144 2× 10−2 15 90 90 15 90 90

2× 10−3 0 5 14 0 5 20

2× 10−4 0 0 4 0 0 4

13.0 1 616 2× 10−2 372 1 174 1 212 355 1 168 1 212

2× 10−3 25 118 388 5 99 411

2× 10−4 0 1 16 0 0 11

22.0 12 720 2× 10−2 3 679 9 283 10 098 3 473 9 300 10 090

2× 10−3 632 1 938 4 098 282 1 827 4 045

2× 10−4 45 350 637 1 46 311

2× 10−5 0 1 1 0 0 0

70

Table 3.7. Number of accurately computed eigenvalues for the 2 DOF isotropic HOsystem at Ecut = 70.0 a.u. (N = 2 340) using modified weylet basiswith the PSRO applied from 3 to 11 times. Beyond p = 11, numericaldifficulties arise in the diagonalization of the Hamiltonian matrix.

accuracy 3 5 7 9 11

2× 10−2 2 161 2 204 2 218 2 226 2 242

2× 10−3 1 772 1 933 2 039 2 088 2 121

2× 10−4 1 343 1 633 1 778 1 870 1 956

2× 10−5 792 1 142 1 439 1 610 1 719

2× 10−6 441 690 986 1 218 1 385

2× 10−7 246 392 627 835 1 056

2× 10−8 71 142 302 507 689

2× 10−9 17 28 106 256 417

2× 10−10 1 4 24 86 223

2× 10−11 0 0 1 28 92

71

2 4- 2- 4q ( a.u.)

- 4

- 2

2

4

p ( a.u.)

(a)

2 4- 2- 4q ( a.u.)

- 4

- 2

2

4

p ( a.u.)

(b)Figure 3.1. Classically allowed (QC) region for the weylets and HO PSRO in the

1 DOF case. The basis truncation criterion H(qs, pt) ≤ Ecut = 4.0 a.u.gives the weylet basis of N = 6 functions (Table 3.3). In part (a), theQC region of each weylet is represented by two squares each with volumeπ a.u. symmetrically placed about the q axis. The two squares with smallunfilled circles represent the weylet |ϕ1/2,3/2〉. The larger unfilled circle

is the QC region R corresponding to the PSRO ρQCK (K = N = 6). The

same weylets modified by the exact projection ρK are approximatelygiven in part (b).

72

02

4

- 2- 4

024

- 2- 4

0.5

02

4

- 2- 4q (a.u.) p (a.u.)

�6(q;p) (a.u.)(a)

02

4

- 2- 4

024

- 2- 4

0.5

02

4

- 2- 4q (a.u.) p (a.u.)

�6(q;p) (a.u.)(b)

02

4

- 2- 4

024

- 2- 4

0.5

02

4

- 2- 4q (a.u.) p (a.u.)

�6(q;p) (a.u.)(c)

Figure 3.2. Wigner-Weyl representation (1 DOF case) of projection operators con-sisting of three different sets of N = 6 (a) HO eigenstates; (b) weylets;(c) PSRO-modified weylets. Different than what is stated in Sec. 3.1, allplots oscillate about (1/2π), instead of unity, due to a discrepant normal-ization. In part (b), there are quantum interference fringes that emergefrom the momentum symmetrization. The modified weylets in part (c)do not have these fringes and adopt the ring-like pattern of the true HOeigenstates, rendering them a more efficient basis for representing theHO system (see Table 3.3).

73

2 4- 2- 4 6q ( a.u.)

2

4

- 2

- 4

p ( a.u.)

Figure 3.3. Classically allowed region for the 1 DOF weylets and Morse PSRO. Theegg-shaped region R represents the 1 DOF Morse system at Emax =−6.7500 a.u. There are K = 6 eigenstates at that energy (area inside is12π a.u.). Filled circles indicate the centers of the N = 9 basis functionsused for the calculation of the 6 corresponding eigenvalues (Table 3.1).

74

2 4- 2- 4x ( a.u.)

2

4

- 2

- 4

px ( a.u.)

(a)

2 4 6 8 10- 2- 4y ( a.u.)

2

4

- 2

- 4

py ( a.u.)

(b)Figure 3.4. Classically allowed region of the separable PSRO for the 2 DOF

Morse/HO system. Rx and Ry are shown in parts (a) and (b), respec-tively, as the inner “solid” curve. R′

x and R′y, reflective of the basis set

(N = 49) of SG’s, are also shown in parts (a) and (b) outlined by the“dotted” lines.

75

0 10000 20000N

0

0.2

0.4

0.6

Effi

cien

cy L

/N

Figure 3.5. Efficiency versus N for the 2 DOF HO at 2 × 10−4 relative accuracy(Table 3.4). The “solid” lines correspond to the weylets either unmodi-fied (circles), single PSRO-modified (squares), or double PSRO-modified(triangles). The “dashed” lines represent the SG’s.

76

2 3 4DOF

0.2

0.4

0.6

0.8

1

Effi

cien

cy L

/N

Figure 3.6. Efficiency versus DOF’s at N ≈ 10 000 held constant for the HO system.The three data points for each line, starting from the left 2 DOF point,represent basis sizes of N = 9 956, 9 552, and 12 720, with L at relativeaccuracy of 2 × 10−2. The labels for the plots are the same as that inFig. 3.5. Only the SG’s are plotted for the modified cases since they arevery similar to the weylets.

77

CHAPTER IV

PARALLEL PREPROCESSED SUBSPACE ITERATION METHOD

4.1 Introduction

For iterative eigenvalue solvers, a common goal is to create a subspace invariant

under the N × N matrix A in question (assumed to be real, symmetric, and sparse

throughout this chapter). Although this subspace is spanned by a basis of select

eigenvectors of A, one need not have any prior knowledge of the eigenvectors or the

corresponding eigenvalues in order to iteratively converge towards the target invariant

subspace (ISUB). In the physics/chemistry community, the ISUB is often represented

as a density or projection matrix ρ, a uniformly mixed ensemble of the appropriate

eigenvectors.

In this paper, an N × d rectangular matrix containing column vectors (not nec-

essarily eigenvectors) spanning the ISUB of dimension d, will be referred to as S. If

the column vectors are orthonormal, then one can easily project out a smaller d× d

real and symmetric matrix C (known as a Rayleigh matrix), i.e.,

STAS = C (4.1)

where T designates the transpose. The advantage gained is that the much smaller C

matrix (d ¿ N) can then be numerically diagonalized using direct standard eigen-

value techniques to obtain the eigenvalues of A that correspond to the eigenvectors of

A contained within the ISUB. Alternatively, one might choose not to orthogonalize

successive column vectors of S. In this case, one must solve the generalized eigenvalue

problem

C~x = λM~x , (4.2)

where C comes from Eq. (4.1) (no longer referred to as a Rayleigh matrix), M = ST S

is the overlap matrix, and (λ, ~x) is the (eigenvalue,eigenvector) pair.

In practice, one is often unable to obtain an exact ISUB, but rather an approximate

ISUB through numerical means. For example, a common technique is the Lanczos

78

method where the approximate ISUB (or Krylov subspace) is

Kw = span(~b, A~b, A2~b, . . . , Aw−1~b) , (4.3)

where ~b is an initial column vector (usually random) and w is the dimension of the

subspace. If one orthonormalizes the vectors spanning Kw in between successive

matrix-vector products, and then combines the vectors after all w − 1 iterations

to make a new N × w matrix, Kw−1 (subscript denotes the number of iterations),

then the Rayleigh matrix, C, is simply found via Eq. (4.1) with Kw−1 replacing S.

Particular to the Lanczos method, C has the favorable property of being tridiagonal.

Not only is the Rayleigh matrix easier to diagonalize as a result, but a clever algorithm

can be implemented requiring minimal storage of only four vectors throughout the

iterations: the growing main and adjacent diagonal of C and two adjacent column

vectors of Kw−1.78

Mathematically, the Lanczos algorithm above produces orthogonal vectors span-

ning the Krylov subspace; due to finite numerical precision however, in practice,

orthogonality is compromised after successive iterations, leading to computed eigen-

values with extra multiplicities, known as “spurious” eigenvalues.105 An occasional

re-orthogonalization of all vectors in the Krylov subspace can remedy this, and there

are methods such as selective106 and partial re-orthogonalization107,108 that do this

efficiently. Unfortunately, all of the column vectors of Kw−1 need to be stored, instead

of just the four mentioned above, which may thus become a major issue.

Wu and Simon have addressed the storage problem using two separate strategies.

First, they have developed a thick-restart Lanczos method109 where after the memory

is completely filled by the growing Krylov subspace, the Ritz vectors (or eigenfunctions

of the Rayleigh matrix at that point) are calculated, and all are used to develop a

starting point for a new and more accurate Krylov subspace, which replaces the old.

Second, they have developed a parallel algorithm, PLANSO,110 where the Lanczos

vectors are uniformly and conformally mapped among compute nodes, and all of

the linear algebra operations needed in the Lanczos algorithm are parallelized to

79

accommodate the distribution. Popular sparse and parallel packages can be used to

interface with PLANSO in order to handle sparse operations such as matrix-vector

multiplication.

In principle, there are numerous strategies one might develop for parallelizing the

Lanczos method. For the most part, however, these only address the parallelization

of the linear operations required of individual Lanczos iterations, i.e., multiple vec-

tors are non-parallelizable, owing to the sequential nature of the Krylov subspace

methods. Block Lanczos methods111 [as opposed to “vector” Lanczos methods of

Eq. (4.3)], where each iteration involves a group of orthonormal vectors instead of

a single vector, does offer a design conducive to parallelization at the vector level,

although the effectiveness of the parallelization is limited by the size of the blocks,

which in practice is relatively small. In order to fully take advantage of the bene-

fits offered by parallelization at the vector level, one must completely eliminate the

sequential aspect of the iterations. Thus, we chose a different iterative method that

offers this, the subspace iteration (SI) method, where the approximate ISUB is

Zd = span(Ar~b1, Ar~b2, . . . , A

r~bd) . (4.4)

The set of column vectors (~b1,~b2, . . . ,~bd) must, at least, be linearly independent or else

all of the vectors spanning Zd (which comprise the N × d matrix Zr) might converge

to the same eigenvector of A as r increases.

In practice, the total number of matrix-vector products of the SI method, r × d,

exceeds that of the vector Lanczos, w−1, when achieving similar eigenvalue accuracies

of A, but parallelizing Zr such that each column vector is calculated on separate nodes,

effectively reduces the number of matrix-vector products to r, which is considerably

smaller than w− 1. Parallelization of the block Lanczos methods can achieve similar

savings as the SI method only if the block size of the former is the same as d, although

this introduces problematic memory issues for the latter method since its approximate

ISUB has a large w × d dimension.

80

The SI method also offers other key advantages over both vector and block Lanczos

methods. For example, one has better control over the dimension of the approximate

ISUB. In other words, one directly chooses d in the SI case, and independently ad-

justs the accuracy via the iteration number r, whereas in the vector and block Lanczos

cases, the dimensionality and number of iterations are both dependent upon the same

parameter w. In practice, good iteration number and dimension parameter values re-

quire a careful balance of factors. Another advantage of the SI method over Lanczos,

in particular the vector Lanczos, is that the former does not become inefficient in

cases where the eigenvalue spectrum is degenerate or nearly-degenerate. This situa-

tion is common in spectroscopic rovibrational calculations, particular when there is

symmetry.15,112,113 The block Lanczos methods do correctly account for degeneracy

as long as the individual block size dimensions are greater than or equal to the extent

of the degeneracy. Last, in the SI method, the approximate ISUB approaches that of

the exact ISUB when not considering numerical error issues, i.e.,

Zr −→ S as r −→∞ , (4.5)

which is not the case for both the vector and block Lanczos methods.

Although the SI method is clearly a natural candidate for parallelization, the in-

corporation of vector orthogonalization (either after every matrix-vector product, or

just occasionally) does require substantial internode communication. In this chapter,

we present a way to preprocess A such that a satisfactory ISUB can be iteratively

computed without the need for costly orthogonalizations. Communication is only re-

quired after the final iteration, in order to calculate the elements of C using Eq. (4.1)

(Zr replaces S), as well as the elements of M (column vectors of Zr are nonorthogo-

nal) and to initialize both C and M on a single node for direct diagonalization using

Eq. (4.2). These preprocessing ideas are inspired by single-particle density matrix pu-

rification (DMP) schemes used in ground-state electronic-structure calculations.70,71

In the DMP context, the matrices are often sufficiently small as to allow direct matrix-

matrix multiplications to be performed, e.g., PRISM (Parallel Research on Invariant

81

Subspace Methods).114,115 For many SI applications, however, e.g. quantum dynam-

ics, the matrices involved are large and sparse, and therefore not amenable to direct

matrix-matrix products. Instead, our approach uses matrix-vector products, which

are far less costly, and also enable efficient sparse matrix techniques to be employed.

In this chapter, we apply the new preprocessed SI method to several model sys-

tems: isotropic (3 and 6 degrees of freedom or DOF’s) and anisotropic (3 DOF)

uncoupled harmonic oscillators (HO’s). We represent the Hamiltonian operator H

of these systems using a type of Weyl-Heisenberg wavelet (or “weylet”) basis chosen

under the guidance of a phase space truncation scheme35–37 which gives N×N sparse

matrix representations, H, of the system. In all cases, the eigenvalues of H are cal-

culated from an approximate ISUB of dimension d = 6 000, and a determination is

made of how many of these fall within a certain relative accuracy level. Although the

6 000 × 6 000 matrix C is diagonalized directly, the expectation is that N À 6 000,

so that the overall calculation is still extremely efficient.

In practice, we find that towards the lower end of the spectrum of the 6 000,

the eigenvalues match those of H with increasing accuracy. The greatest errors are

towards the high end of the spectrum, as is typical for subspace methods. However,

the SI method enables one to improve the accuracy of all d = 6 000 eigenvalues,

in principle to arbitrary precision, simply by increasing r, because of Eq. (4.5). In

practice, we find limitations on the maximum value of r that can be applied. In such

cases, we can still improve the accuracy of the desired eigenvalues arbitrarily, simply

by increasing d (at the expense of using more memory on the nodes), as per other

subspace methods.

The new SI method is also found to be extremely scalable. With respect to

storage requirements, the iteration vectors are found to require most of the memory,

but these can be evenly distributed over all available nodes. With respect to CPU

operations, the bottleneck is the iterative generation of the ISUB vectors, which

exhibits near perfect parallel speedup, as it requires no internode communication.

The subsequent C matrix creation and initialization steps also parallelize efficiently,

82

although communication is involved. Having applied the new method in this paper

to matrices as large as N ≈ 106, we find it to be very effective even in its present

incarnation, although there is still ample room for future improvement and fine tuning.


The SI method is considered to be a variation of the “power method,” for which,

instead of focusing on the attainment of one eigenvector, one desires to find a group

of eigenvectors corresponding to some region of the eigenvalue spectrum. For the

SI case presented in the previous section [Eq. (4.4)], the fully converged Zr (r −→∞) would be the space spanned by the d eigenvectors of A that have the largest

eigenvalue moduli. Traditional SI methods orthogonalize the ISUB vectors after each

matrix-vector product via a QR decomposition or a modified Gram-Schmidt. This

prevents Zd from losing dimensionality, i.e., the new Zr with orthogonal column

vectors maintains full rank d. In addition, the projected matrix C found by Eq. (4.1)

is a Rayleigh matrix, and the eigenvalues are found via direct standard eigenvalue

techniques instead of using Eq. (4.2).

This approach is preferable when r and d are small, otherwise it is too compu-

tationally expensive. Our strategy, on the other hand, is to preprocess A such that

the orthogonalizations may be avoided altogether, thus making it possible to easily

parallelize the complete algorithm. We are not currently able to eliminate the rank

problems completely; however, using the present preprocessing scheme, we can ac-

curately retrieve a significant portion of the eigenvalue spectrum of A. Also, since

our chief interest lies in calculating rovibrational energies of small molecules (A now

becomes the Hamiltonian matrix H), our preprocessing scheme focuses on the lower,

most accurate portion of the eigenvalue spectrum.

There are two steps in the preprocessing. First, the rows and columns of H are

reordered so as to place diagonal elements of H in ascending order, moving from

the top-left corner towards the bottom-right. The second step, commonly known as

83

“scaling”, is the following simple adjustment of H:

H ′ = γ(µI − H) + I (4.6)

where γ = min[

1

εmax − µ,

1

µ− εmin

].

The N × N matrix I is the identity, and µ is known as the “chemical potential”

in DMP papers71 (the significance will be addressed later). The parameters εmax

and εmin are the approximate largest and smallest eigenvalues, respectively, found via

Gershgorin’s formulas,116

εmax = max[Hii +

∑

j 6=i

|Hij|]

i(4.7)

and

εmin = min[Hii −

∑

j 6=i

|Hij|]

i. (4.8)

No eigenvalues of H stray outside of the above boundaries.

The eigenvalues of H ′ range between 0 and 2, with those between 1 and 2 cor-

responding to the desired eigenvalues of H below µ. The µ parameter should be

chosen so that d (dimension of ISUB) eigenvalues of H ′ lie within the latter range.

In practice, one first decides on d, and then determines a suitable µ value after a few

trial runs starting from a reasonable initial guess. The matrix H ′ = A is then used

to obtain the approximate ISUB via Eq. (4.4) with the initial vectors ~bi chosen to

be the orthonormal unit vectors ~zi (with the ith component equal to 1, and all other

components 0). The approximate ISUB can therefore be written (H ′)rZ0 = Zr where

r is the number of iterations, and Z0 = (~z1, ~z2, . . . , ~zd). Mathematically, as r −→ ∞,

Zr −→ S, as mentioned before. Only matrix-vector products are performed so that

one has only to deal with the d column vectors of matrix Zr and the sparse matrix

H ′. In contrast, matrix-matrix products of H ′ would lead to a dense N ×N matrix

that is too large to store on each node. The rationale here is that the Zr contribution

from the subspace of eigenvectors corresponding to the eigenvalues of H ′ (the original

matrix H has the same eigenvectors) between 0 and 1 will dissipate exponentially

84

with respect to r. By the same token, the [1, 2] contribution becomes increasingly

prominent with increasing r (see Appendix D).

Since the column vectors of Zr are not orthogonal, then the eigenvalues are ob-

tained by Eqs. (4.1) and (4.2). This requires that the overlap matrix M be positive

definite, which is true formally if Zr is of full rank, but can cause numerical instabili-

ties if the smallest M eigenvalues are too close to zero. The reordering of H discussed

earlier, together with the choice ~bi = ~zi, is designed to ameliorate this difficulty. It

is certainly found to be very successful in this regard; however, one is still limited in

the number of iterations, r, that can be used without numerical instabilities arising

in the generalized eigenvalue solution routines. This limitation, in turn, restricts the

accuracy that can be achieved for a given value of d.

4.3 Parallel and Numerical Implementation

In this section, we describe the numerical algorithm in detail, including paral-

lelization. The implementation can be broken down into 7 sequential steps as follows:

1. Construct H using the truncated weylet basis.

2. Preprocess H to get the new matrix H ′.

3. Distribute d vectors ~zi of Z0 across nodes; duplicate H ′ across nodes.

4. Perform r iterations in parallel, in order to compute Zr = (H ′)rZ0.

5. Calculate the elements of C and M while keeping the d Zr vectors distributed

among the nodes.

6. Send all of the C and M matrix elements to a single node.

7. Solve the generalized eigenvalue problem of Eq. (4.2) on a single node.

In this study, we only look at separable systems, e.g., isotropic and anisotropic

uncoupled HO where the Hamiltonian operator is

H = (1/2)f∑

j=1

(p2j/mj + mjω

2j q

2j ) (4.9)

85

(all of the masses are set to unity, i.e., mj = 1 for j = 1, . . . , f). The 1 DOF weylet

lattice basis (h = 1 assumed throughout chapter) used to represent H is

ϕst(q) =∑

|m|+|n|≤ 6

(−1)(n2+mt)cmnψuv(q) (4.10)

where ψuv(q) = (4a2/π)(1/4) cos[va√

π(q−(u+1/2)

√π/a

)]e−a2(q−u

√π/a)2/2 , (4.11)

u = s + m, v = t + n, m and n are even integers, cmn are coefficients with values

reported in Ref. [36], a is related to the “aspect ratio,” s (half-integer) is the position

index of the weylet block on phase space, and t (positive half-integer) is the momen-

tum parameter. For f DOF’s, the basis consists of products of the 1 DOF functions,

i.e.,

ϕs,t(q) =f∏

j=1

ϕsjtj(qj) (4.12)

where q = (q1, q2, . . . , qf ), s = (s1, s2, . . . , sf ), and t = (t1, t2, . . . , tf ).

Step 1 , the creation of the Hamiltonian matrix H, is very quick since the matrix

elements are analytical. The 1 DOF kinetic energy matrix has elements derived from

〈ϕst|p2|ϕs′t′〉 =∑

|m|+|n|≤ 6

∑

|m′|+|n′|≤ 6

(−1)(n2+mt+n′

2+m′t′)cmncm′n′〈ψuv|p2|ψu′v′〉 (4.13)

with

〈ψuv|p2|ψu′v′〉 =πa2

2[h(u, v, u′, v′)− h(u, v, u′,−v′)] , (4.14)

and

h(u, v, u′, v′) = e−(π/4)(u2∆+v2

∆)

[1

2

(v2

+ − u2∆ +

2

π

)

× cos ζ − u∆v+ sin ζ

]. (4.15)

The “∆” and “+” subscripts indicate the difference and addition, respectively, of the

bra and ket indices, e.g., u∆ = u− u′ and v+ = v + v′. Note that under a change of

sign of v′ [i.e., for the last term of Eq. (4.14)], v∆ becomes v+ and vice-versa. The

phase quantity ζ is given by

ζ(u, v, u′, v′) =π

2(u∆v+ + v∆) , (4.16)

86

and since u∆, v∆, and v+ are always integers, the trigonometric quantities in Eq. (4.15)

are always ±1 or 0. The 1 DOF potential energy matrix elements are obtained from

〈ψuv|q2|ψu′v′〉 =π

2a2[τ(u, v, u′, v′)− τ(u, v, u′,−v′)] , (4.17)

and

τ(u, v, u′, v′) = e−(π/4)(u2∆+v2

∆)

[1

2

(u2

+ − v2∆ +

2

π

)

× cos ζ + v∆u+ sin ζ

]. (4.18)

With regard to the full f DOF matrix representation, the separability of H results

in a sparse matrix H, especially for large f . The full f DOF kinetic energy matrix is

given by

(1/2)〈ϕs,t|p21 + p2

2 + . . . + p2f |ϕs′,t′〉 = (1/2)

∑

(i,j,...,k)=(1,2,...,f)

(〈ϕsiti|p2

i |ϕs′it′i〉

×δsjsj′δtjtj ′ . . . δsksk

′δtktk′

)(4.19)

where the summation is over all cyclic permutations of (1, 2, . . . , f), δ is the Kronecker

delta function, and 〈ϕsiti|p2i |ϕs′it

′i〉 is defined in Eqs. (4.13)-(4.15). The potential

energy matrix elements adhere to a similar sparse form as Eq. (4.19) except that q’s

are put in the place of p’s.

To maximize storage efficiency and to speed up linear algebra operations needed

later, the row-indexed sparse storage mode16 is used for H. The nonzero elements

of H are stored directly in a 1-dimensional array (SA) accompanied by another 1-

dimensional array (IJA) of integer parameters containing the matrix positions of the

nonzero elements. Both of the arrays contain G + 1 elements, where G is the number

of nonzero elements (the extra storage slot is needed for the storage mode).

The good phase space localization of the weylets, along with the orthonormality,

allows for a phase space truncation scheme which defeats exponential scaling. In

87

other words, the efficiency K/N , where N represents the number of basis functions

(order of H) needed to calculate K eigenvalues at a desired accuracy, does not decay

exponentially with respect to f , the number of DOF’s of the system in question.35–37

More specifically, the truncation involves selecting only those weylets, whose center

coordinates on phase space when plugged into the classical Hamiltonian expression,

produce values Hmid below some energy cutoff Ecut. The justification of the scheme is

based upon the quasiclassical approximation which treats the set of weylets as a lattice

of 2f -dimensional blocks partitioning phase space, and the target K eigenstates of H

are represented as a uniform region. This approximation improves as both K and N

increase.34

Due to the sparsity of H, the preprocessing in step 2 is very quick, as well, and

the sparsity is preserved exactly in the new matrix H ′ (which fully replaces H in the

1-dimensional arrays). The reordering of H and the calculation in Eq. (4.6) using

the Gershgorin’s formulas in Eqs. (4.7) and (4.8) take full advantage of the sparsity

and the row-indexed sparse storage mode. This allows the step to be an insignificant

contribution to the total CPU time required.

In step 3, the 1-dimensional array SA(1 : G + 1) containing the elements of H ′,

along with the integer array IJA(1 : G + 1) are copied onto each of the g compute

nodes. The strategy is that the iterations, i.e., H ′Zr−1 = Zr, needed to create the

ISUB will be done in parallel by distributing the column vectors of Z0 as equally as

possible over the g nodes. Simultaneously, each node performs matrix-vector products

with H ′ and its assigned vectors. In practice, instead of apportioning Z0 at the start

of the process, one can simply divvy up the first d columns of H ′ which is equivalent

to Z1.

For clarity, we assign a number y for each node ranging from 0 to g − 1. For

communication purposes, as shown in Fig. 4.1, the nodes are arranged in a loop, with

the column vectors stored sequentially around the loop. If d > g, then the distribution

will continue looping around in the same manner. With this arrangement, node y

will have either αy = [[d/g]] + 1 vectors ([[ ]] denotes “greatest integer smaller than,”

88

also known as the “floor” function) if y + 1 ≤ mod(d, g) = d − g[[d/g]], otherwise

αy = [[d/g]] vectors.

Step 4 involves the r matrix-vector products between H ′ and the αy vectors on each

node y done in parallel. Multiple trial runs are needed to find the “largest” number

of iterations, r, the point before the numerical generalized eigenvalue solver fails (or

eigenvalues of M are too close to zero) due to loss of rank. One can incorporate a

singular value decomposition method if one exceeds r; although, one finds that no

accuracy is gained beyond the largest iteration point.

After the r iterations are completed, step 5, the construction of matrices C and

M , is effected. The simplest method would be to transfer all of the dense vectors of Zr

to a single node; although, it is more than likely that each node does not have enough

memory to handle Zr. Instead, an efficient communication scheme is employed with

[[g/2]] stages.

Before any communication in step 5, two important steps are implemented. First,

H ′ is “unpreprocessed” back to H which is used for the calculation of C. Second,

all matrix elements of C and M corresponding to the local set of αy vectors are

calculated. For example, in Fig. 4.1, node y = 0 contains column vectors ~z(r)

1 , ~z(r)

6 ,

and ~z(r)

11 of Zr. Matrix elements (1, 1), (6, 6), (11, 11), (6, 1), (11, 1), and (11, 6),

where the first and second numbers in each set (i, j) is the row and column of C and

M , are obtained by the matrix products (~z(r)

i )TH~z(r)

j and (~z(r)

i )T~z(r)

j , respectively.

Note that since C and M are symmetric, we only need to calculate those elements in

the lower (or upper) triangular portion of the matrices.

In the first stage of the communication, node y sends αy vectors, one at a time, to

node mod(y + 1, g), and node y receives αmod(y−1,g) vectors from node mod(y − 1, g).

Immediately, after each vector is received, all of the possible matrix elements between

the transferred vector and the local vectors belonging to the receiving node are calcu-

lated and stored. For example, in Fig. 4.1, node 1 will receive ~z(r)

1 from node 0 and

will calculate elements (2,1), (7,1), and (12,1), followed by the immediate deletion of

89

~z(r)

1 in order to save memory. Vectors ~z(r)

6 and ~z(r)

11 from node 0 undergo the same

process.

The second stage involves node y sending αy vectors to mod(y+2, g) and receiving

αmod(y−2,g) vectors from node mod(y − 2, g). In general, one can conclude for stage v

that y sends αy vectors to mod(y + v, g) and receives αmod(y−v,g) vectors from node

mod(y − v, g). When dealing with an odd number of nodes, this generalization is

true for all [[g/2]] stages. On the other hand, for even g, the generalization is true

except for the last stage g/2. As shown in Fig. 4.2, the nodes in each of the g/2 pairs

take turns in sending a single vector to the other. In the first “substage”, one vector

from one of the nodes in the pair will be sent to the other and will be combined

with all vectors present on the receiving end for the calculation of all possible matrix

elements. Next, the sender and receiver switch roles, and one vector is sent in the

other direction, to be combined with vectors that have not been sent in previous

substages. Fig. 4.2 explicitly shows the pattern for the d = 13 and g = 6 case.

It is important that we tally all of the arrays needed for each node in this algorithm

such that, based on the known limitations of memory on each node, we can determine

the minimum bound of g—any number of nodes larger than or equal to the minimum

of g can be used for the successful handling of the target problem. Each node will

contain SA(1 : G + 1) and IJA(1 : G + 1) which should not be a significant memory

consumer due to the sparsity of H; although, the size of G (number of nonzero

elements), in our chosen model problems, does grow at a slightly faster than linear

rate with N and could overstep the memory limitations for very large N . In the

future, we may investigate ways to distribute these arrays among the nodes. At the

present, however, the largest memory consumer is the set of vectors that make up Zr

which we will denote as the 2-dimensional array vec(1 : N, 1 : αy), specific for the

node y. Obviously, as one increases g, then αy is reduced; thus, we have direct control

of the size of vec by varying g. Each node also needs an additional column vector

recvec(1 : N) which is the vector that each node receives during the aforementioned

communication steps for the purpose of calculating matrix elements of C and M .

90

Last, elements from the two matrices need to be stored on each node. Fortunately,

these are also distributed almost evenly among the nodes, though, in step 6, all such

elements will be transferred to one node.

For step 5, a safe shift algorithm117 is used on each node to implement the commu-

nication in a timely fashion. There are two key elements in the method that add flow

control to the message passing, which is important, especially when communicating

large messages. First, a nonblocking receive command in the code is posted before

the send command. This is known as “preposting” the message, i.e., the nodes are

ready to receive any message before anything is sent. Second, a small message is sent

in the reverse direction which is known as a “permission to send” (PTS) message.

The receiver sends a small PTS message to the sender after the nonblocking receive

command (or the preposting) which opens up the pathway for the communication of

the actual data.

Finally, step 7 is performed, i.e., the generalized eigenvalue problem [Eq. (4.2)].

This is done on one node using the LAPACK subroutine DSYGV. In the future, if

we want to recover a large number of eigenvalues, i.e., d is large, then we will need

to incorporate a parallel dense linear algebra solver, such as specified in the PRISM

project.114,115


First, we considered the f = 3 DOF isotropic case, where ω1 = ω2 = ω3 = 1

[Eq. (4.9)]. For the Hamiltonian matrix H of order N = 36 083, we chose d =

6 000. For comparison, the exact eigenvalues of the HO Hamiltonian operator, H,

are[( ∑f

j=1 nj

)+f/2

]where nj is the nonnegative integer signifying the energy level

of the jth DOF of the HO. The degeneracy for each level is

deg(nS, f) =

nS + f − 1

f − 1

(4.20)

for f > 1 where nS =∑f

j=1 nj; the eigenvalues are nondegenerate for f = 1.

91

Table 4.1 reports the number of eigenvalues, K, out of the lowest d = 6 000,

computed to various relative accuracies (2×10−2, 2×10−3, 2×10−4, etc., correspond to

error tolerances of (0.2)f , (0.02)f , (0.002)f ,. . ., in a.u. as per Ref. [37]). Eigenvalues

for both Eq. (4.2) (computed using the proposed parallel algorithm of Sec. 4.3) and

for the 36 083× 36 083 matrix H (the LAPACK DSYEV subroutine) are considered,

with errors taken relative to the exact analytical eigenvalues of H (described above).

At relative accuracies 2 × 10−5 or better, there is a perfect match between the two

calculations, indicating that the proposed method introduces no substantial error

beyond those of H itself. For relative accuracies of 2 × 10−4 and above, the small

discrepancies indicate that the r value chosen (r = 37) is insufficiently large to achieve

exact agreement for the highest eigenvalues. The value r = 37 is the largest that can

be used without encountering numerical instabilities in the eigensolver routines due to

the loss of full rank. Though quite small, this value nevertheless sufficiently converged,

with respect to achieving nearly one full accuracy of H itself throughout the spectrum.

Lanczos, for instance, would not achieve anywhere near the performance of Table 4.1.

The 3 DOF system is studied further as shown in Fig. 4.3. The numbers of

eigenvalues, K, that have relative accuracies of 2 × 10−2, 2 × 10−4, 2 × 10−6, and

2×10−8 with regard to the eigenvalues of H, are plotted against N for fixed d = 6 000.

As N increases, the largest iteration number r increases, as well, e.g., at N = 21 976,

36 083, 49 840, 104 912, 207 320, and 416 840, the values of r are 33, 37, 43, 59,

75, and 95, respectively. The 2 × 10−2 relative accuracy curve is basically flat at

all values of N , and nearly equal to the full d = 6 000. Clearly, a large basis size

N is not required to compute the desired eigenvalues to this low level of accuracy.

However, the method becomes increasingly useful for higher accuracy calculations,

for which much larger N values are required, but CPU effort (since d is still 6 000)

increases only modestly. In general, K increases with N as expected, but only up

to a point, beyond which K is essentially flat, due to the limitations on r. For the

higher accuracy curves, the point at which the curve flattens is at a larger N . For

92

example, at 2 × 10−6 and 2 × 10−8 the flattening does not occur until N = 104 912

and 207 320, respectively.

We also looked at the 3 DOF anisotropic case where ω1 =√

2, ω2 =√

3, and

ω3 =√

5 a.u. [Eq. (4.9)]. For the isotropic case, the aspect ratios for each DOF

(a1, a2, and a3) are all unity, but in the anisotropic case, for optimal efficiency of the

weylet basis, a1 =√

ω1, a2 =√

ω2, and a3 =√

ω3. The eigenvalues of the Hamiltonian

operator, H, are non-degenerate and equal to[ ∑f

j=1 ωj(nj + 1/2)].

Comparing between Figs. 4.3 and 4.4, we note that, although the isotropic and

anisotropic cases demonstrate similar patterns with respect to the different relative

accuracies, the isotropic calculation is slightly more accurate. For example, at 2×10−4

and 2×10−6 the curves in the isotropic case reach a maximum number of eigenvalues

at around K = 4 000 and 2 700, respectively; whereas, the same curves in the

anisotropic case flatten out at K = 3 500 and 2 400. However, we have found that

incorporating coupling into the problem does not affect the efficiency of the weylet

basis. More specifically, we did a small study (data not reported) on a coupled

anisotropic system [obtained by adding −ε(q1q2 + q1q3 + q2q3) to Eq. (4.9) with f = 3

and the coupling parameter ε = 0.1] and found scarcely any difference in the number

of accurately computed eigenvalues. The coupling does reduce sparsity somewhat,

however, resulting in increased computation time.

We have also applied the parallel algorithm to the 6 DOF isotropic uncoupled HO

system at fixed d = 6 000. Fig. 4.5 indicates that two of the curves (2 × 10−3 and

2 × 10−4) have monotonic behavior over the N range considered, i.e., it would be

fruitful to consider basis sets even larger than N = 106 here, which is not surprising,

though we have not done so. With a C of order d = 6 000 representing H at

N = 977 789, we were able to extract an impressive 4 419 eigenvalues at 2 × 10−3

relative accuracy and 1 184 at 2× 10−4.

Another strategy for increasing accuracy that we have not considered explicitly

is to increase d. With larger d, more nodes will be needed in order to accommodate

increased memory requirements; although, it also turns out that the largest number

93

of iterations, r, is smaller. While this reduces CPU effort somewhat, it also nullifies

to some extent the gain in accuracy. The loss of rank problem might also be remedied

by the addition of some useful steps in the algorithm. We feel that incorporating an

orthogonalization scheme at strategic moments during the iteration step 4 could help

Zr to achieve full rank. This would lead to C more accurately reflecting H. This

scheme would require some costly communication similar to that of step 5, although

this would not be required after every iteration but possibly after every 10 or 20

iterations or so. For Hamiltonian matrices of size N larger than what was considered

in this chapter, we believe that this modification of the proposed method would be

worth investigating.

94

Table 4.1. Number of accurately computed eigenvalues, K, out of the lowest d =6 000 that fall within some relative accuracy (column 1) for the 3 DOFisotropic HO (N = 36 083) using either Eq. (4.2) at r = 37 (column 2)or the direct diagonalization of H (column 3). By comparing columns2 and 3, one can assess the additional error introduced by the parallelalgorithm.

accuracy parallel direct

2× 10−2 5 654 6 000

2× 10−3 4 790 6 000

2× 10−4 3 577 4 008

2× 10−5 1 653 1 653

2× 10−6 585 585

2× 10−7 59 59

2× 10−8 6 6

95

�� @@I

@@R ��

-

0

1

2 3

4

1,6,11

2,7,12

3,8,13 4,9

5,10

0

1

2 3

4

1,6,11

2,7,12

3,8,13 4,9

5,10

��

��

��

�� AA

AA

AA

AAK

HHHHHHHHj��*�

�� @@I

@@R ��

-

�

0 5

1

2 3

4

1,7,13 6,12

2,8

3,9 4,10

5,11

0 5

1

2 3

4

1,7,13 6,12

2,8

3,9 4,10

5,11

?

HHHHHHHHj��*

6

HHHHHHHHY ��

0 5

1

2 3

4

1,7,13 6,12

2,8

3,9 4,10

5,11JJJ

JJ

JJ

JJJ]

�

�

-�

Figure 4.1. Node communication setup, with the first and second columns repre-senting the g = 5 and 6 case, respectively, both with d = 13, where grepresents the number of nodes and d is the dimension of the approximateISUB, Zr. The numbers inside the boxes (representing nodes) indicatewhich column vectors of the N ×d matrix Zr are stored accordingly, andthe numbers outside are the labels y of the nodes. The arrows designatethe communication direction in each stage (first stage is in the top row).For the g = 6 case, the last stage has communication in both directionsbetween the pairs of nodes which is further explained in Fig. 4.2.

96

0 3

1 7 13 4 10m��

-

1 7 13 4 10m��

�

1 7 13 4 10m � ��

-

1 7 13 4 10� ��

� ��

�

1 4

2 8 5 11m��

-

2 8 5 11m m�

2 8 5 11m � ��

-

2 5

3 9 6 12m��

-

3 9 6 12m m�

3 9 6 12m � ��

-

Figure 4.2. Node communication setup for the last stage (g = 6) with d = 13 (fromexample in Fig. 4.1). The pairs of nodes {0, 3}, {1, 4}, and {2, 5}, gothrough a two-way communication process. The circles highlight thosespecific vectors that are sent in the direction of the arrow and involvedin the element calculations of C and M . For example, in the first step ofthe {0, 3} node pair, ~z

(r)1 is sent followed by the calculation of elements

(4, 1) and (10, 1).

97

0 1 2 3 4N x 100,000

0

2000

4000

6000

Num

ber

of E

igen

valu

es

Figure 4.3. Number of eigenvalues, K, at a relative accuracy versus N for the 3DOF isotropic HO (d = 6 000). In general, the number of accurateeigenvalues increases with the growth of N . The solid line represents thenumber of eigenvalues with a relative accuracy of 2 × 10−2, dotted line2 × 10−4, dashed line 2 × 10−6, and the long dashed line represents themost accurate eigenvalues at 2× 10−8.

98

0 1 2 3 4N x 100,000

0

2000

4000

6000

Num

ber

of E

igen

valu

es

Figure 4.4. Number of eigenvalues, K, at a relative accuracy versus N for the 3 DOFanisotropic HO (d = 6 000). The set up is the same as that reported inFig. 4.3.

99

0 2 4 6 8 10N x 100,000

0

2000

4000

6000

Num

ber

of E

igen

valu

es

Figure 4.5. Number of eigenvalues, K, at a relative accuracy versus N for the 6DOF isotropic HO (d = 6 000). The solid line represents the number ofeigenvalues at 2 × 10−2 relative accuracy. The dotted and dashed linereflect higher accuracies at 2× 10−3 and 2× 10−4, respectively.

100

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106

APPENDIX A

JUSTIFICATION OF SPHERICAL TRUNCATION CONDITION

To justify the rotational symmetry implicit in Eqs. (2.33) and (2.34), only the un-

symmetrized Gaussian representation will be addressed (for simplicity); however, from

Eq. (2.10), it is clear that the conclusions drawn here also apply to the symmetrized

case. The doubly dense unsymmetrized 3 DOF Gaussians are given by

guv(x) =

(a2

π

)3/4

e−i π2u·vei

√πav·xe

−a2

2

(x−u

√π

a

)2

, (A.1)

from which the potential matrix elements are found to be

[V g

]u,v,u′,v′

=

(a2

π

)3/2

e−i π2(u·v−u′·v′)e−

π4(u∆)2

∫V (x)e−i

√πax·v∆e

−a2

(x−

√π

2

u+a

)2

dx.

(A.2)

By taking the absolute value of the Eq. (A.2) integrand, one obtains an upper limit

on the absolute value of the integral:

∣∣∣∣[V g

]u,v,u′,v′

∣∣∣∣ ≤(

a2

π

)3/2

e−π4(u∆)2

∫V (x)e

−a2

(x−

√π

2

u+a

)2

dx. (A.3)

Since the potential V (x) = V (x · x) is rotationally symmetric, it is obvious that

Eq. (A.3) is invariant with respect to any rotation of the vectors u∆ or u+; thus, a

spherical truncation is applicable for these parameters.

For v∆, the rotational invariance can be deduced from the momentum space rep-

resentation, in which the 3 DOF Gaussians are found to be

guv(p) =1

(πa2)3/4ei π

2u·ve−i

√π

au·pe−

12a2 (p−v

√πa)

2

. (A.4)

The potential energy operator is represented as a function of the momentum Lapla-

cian, i.e. V = V (−∇2p). For simplicity, we consider only the first-order −∇2

p term,

although similar conclusions can be drawn for all other orders as well. The upper

limit of the −∇2p matrix element integral is found to be

∣∣∣〈g(2)uv |(−∇2

p)|g(2)u′v′〉

∣∣∣ ≤ 1

(πa2)3/2e−

π4(v∆)2

∫e− 1

a2

(p−

√π

2av+

)2

107

×ζ[(p− v′a

√π),u′

]dp, (A.5)

where ζ [p∆,u′] =

√(1

a4(p∆)2 − 3

a2− π

a2(u′)2

)2

+π

a6(u′ · p∆)2. (A.6)

Since ζ is invariant with respect to simultaneous rotation of all vectors, the same

must be true of the integrand in Eq. (A.5), and also the entire right hand side. Thus,

spherical truncation in v∆ may also be applied.

108

APPENDIX B

EIGENFUNCTIONS OF HARMONIC OSCILLATOR PSRO

This section proves that the PSRO ρQCK associated with 2f -dimensional hyper-

spheres centered at the origin have the same eigenfunctions as the quantum HO

Hamiltonian. The first part of the proof describes a specific symplectic matrix and

its corresponding operator formalism known as a metaplectic operator. Next, it is

shown that the PSRO and its Wigner-Weyl (WW) phase space representation ρQCK are

invariant under this metaplectic/symplectic transformation which ultimately leads to

the conclusion of the eigenfunctions of the PSRO.

Consider a subgroup of SO(2f) that consists of the set of 2f -dimensional rotations

R(Θ) where Θ = (θ1, θ2, . . . , θf ) that all have a block diagonal matrix form, i.e.,

R(Θ) = R(θ1)⊕ R(θ2)⊕ . . .⊕ R(θf )

=

R(θ1) 0

R(θ2). . .

0 R(θf )

(B.1)

where R(θi) =

cos(θi) sin(θi)

−sin(θi) cos(θi)

and θi ∈ [0, 2π).

If we use the subgroup to act on 2f -dimensional phase space, i.e.,

R(Θ)~z = ~z ′ (B.2)

109

and ~z =

q1

p1

...

qf

pf

,

then R(Θ) can be thought of as a composition of f 2-dimensional counterclockwise

rotations about the origin each rotating a pair of phase space axes (qj, pj) by θj where

j = 1, . . . , f .

One important property of these rotation matrices is that they are symplectic,

i.e., R(Θ) ∈ Sp (2f,R), since they satisfy the equation:

[R(Θ)

]Tβ R(Θ) = β (B.3)

where β =

0 1 0

−1 0. . .

0 1

0 −1 0

and[R(Θ)

]Tis the transpose of R(Θ). Symplectic matrices are known in classical

mechanics as transformations of canonical coordinates that leave the Poisson bracket

invariant. This property carries over to quantum mechanics where the commutator

of the operators corresponding to the canonical coordinates, i.e,

[(~z)i, (~z)j]Q = i[β

]ij

(B.4)

110

where ~z =

q1

p1

...

qf

pf

,

are invariant under these transformations. Thus, one can write

[(~z)′i, (~z)′j]Q = [(~z)i, (~z)j]Q (B.5)

where (~z)′i =[R(Θ)

]ij

(~z)j

and repeated indices are implied to be summed.

For every symplectic matrix there corresponds a unitary operator which in our

case will be denoted as U [R(Θ)] parameterized by R(Θ) where

U [R(Θ)] (~z)i U [R(Θ)]−1 =[R(Θ)

]ij

(~z)j . (B.6)

These unitary operators are known as metaplectic operators and are thoroughly re-

viewed in Ref. [118] . Based on the definition of R(Θ) in (B.1), the corresponding

metaplectic operator is

U [R(Θ)] = ei2Θ(q2+p2)

= ei2θ1(q2

1+p21) ⊗ e

i2θ2(q2

2+p22) ⊗ . . .⊗ e

i2θf (q2

f+p2f ) . (B.7)

The last expression can be verified by plugging it into (B.6) and using the Baker-

Hausdorff lemma. For example,

111

U [R(Θ)] qj U [R(Θ)]−1 = ei2θj(q

2j +p2

j ) qj e−i2θj(q

2j +p2

j )

= qj +(iθj/2)

1![q2

j + p2j , qj]Q

+(iθj/2)2

2![q2

j + p2j , [q

2j + p2

j , qj]Q]Q + . . .

= qj cos(θj) + pj sin(θj) (B.8)

which is in agreement with the right hand side of (B.6). One can also substitute qj

with pj and get

U [R(Θ)] pj U [R(Θ)]−1 = −qj sin(θj) + pj cos(θj) (B.9)

also satisfying the symplectic/metaplectic relationship.

Let us consider the case where f = 1 and the rotation parameter for the single

DOF is very small, i.e., |θ| ¿ 1. The rotation matrix can be written approximately

as

R(θ) =

cos(θ) sin(θ)

−sin(θ) cos(θ)

= eiθJ

≈ 1 + iθJ (B.10)

where J =

0 −i

i 0

is the generator of the SO(2) group. The corresponding infinitesimal metaplectic

operator is

U [R(θ)] = ei2θ(q2+p2)

112

≈ 1 +i

2θ(q2 + p2) (B.11)

which approximately transforms a 1 DOF PSRO by the equation

U [R(θ)] ρQCK (q, p) U [R(θ)]−1 ≈ ρQC

K (q, p) +i

2θ[q2 + p2, ρQC

K (q, p)]Q . (B.12)

Our goal is to show how the WW representation of the corresponding PSRO,

ρQCK (q, p), transforms via (B.12). Let’s first look at the WW transform in two forms

relating the PSRO and its WW representation:

ρQCK (q, p) =

1

2π

∫〈q − 1

2q′| ρQC

K (q, p) |q +1

2q′〉 eiq′pdq′ (B.13)

ρQCK (q, p) =

1

2π

∫〈p− 1

2p′| ρQC

K (q, p) |p +1

2p′〉 e−ip′qdp′ (B.14)

where the matrix elements inside the integrals are configuration kernels defined in

Eq. (3.10). Using (B.13) and (B.14), one can easily find the WW representations

corresponding to [q2, ρQCK (q, p)]Q and [p2, ρQC

K (q, p)]Q, respectively:

[q2, ρQCK (q, p)]Q −→ 2iq

∂

∂pρQC

K (q, p) (B.15)

[p2, ρQCK (q, p)]Q −→ −2ip

∂

∂qρQC

K (q, p) . (B.16)

Thus, the righthand side of Eq. (B.12)

ρQCK (q, p) +

i

2θ[q2 + p2, ρQC

K (q, p)]Q −→ ρQCK (q, p) + pθ

∂

∂qρQC

K (q, p)− qθ∂

∂pρQC

K (q, p)

≈ ρQCK (q + pθ, p− qθ)

= ρQCK (

[1 + iθJ

]1j

~zj,[1 + iθJ

]2j

~zj )

≈ ρQCK (

[R(θ)

]1j

~zj,[R(θ)

]2j

~zj ) (B.17)

where ~z =

q

p

.

113

In the 1 DOF case, we find that the infinitesimal metaplectic transformation of the

PSRO is equivalent to the infinitesimal symplectic transformation of the coordinates

of the corresponding WW representation. If we now consider the rotation parameter θ

to be finite, then applying N infinitesimal rotations repeatedly each with parameter

θ/N (where N is large) is the same as applying the actual metaplectic operator

U [R(θ)] and symplectic matrix R(θ) since

U [R(θ)] = limN→∞

(1 +

i

2

θ

N(q2 + p2)

)N

= ei2θ(q2+q2) (B.18)

R(θ) = limN→∞

(1 + i

θ

NJ

)N

= eiθJ . (B.19)

Thus, our argument can be extended to transformations that are not infinitesimal.

Also, it is obvious that the relationship is valid when carried over to the f DOF case

dealing with the rotations of (B.1) and (B.7); therefore, we can conclude that

U [R(Θ)] ρQCK (q, p) U [R(Θ)]−1 −→

ρQCK

( [R(Θ)

]1j

(~z)j,[R(Θ)

]2j

(~z)j, . . . ,[R(Θ)

](2f)j

(~z)j

). (B.20)

In general, this mapping is true for any symplectic/metaplectic pair, and a formal

proof of this can be seen in Ref. [119] .

For the HO case, ρQCK (q,p) is a 2f -dimensional hypersphere centered at the origin

which is invariant under rotations along the center, i.e.,

ρQCK

( [R(Θ)

]1j

(~z)j,[R(Θ)

]2j

(~z)j, . . . ,[R(Θ)

](2f)j

(~z)j

)= ρQC

K (q,p) . (B.21)

Since the WW transform is isomorphic, one can deduce from (B.20) and (B.21) that

U [R(Θ)] ρQCK (q, p) U [R(Θ)]−1 = ρQC

K (q, p) ; (B.22)

thus, U [R(Θ)] and ρQCK (q, p) share the same eigenfunctions. From (B.7), we see

that the eigenfunctions of U [R(Θ)] are the HO eigenfunctions |n〉; therefore, the

eigenfunctions of ρQCK (q, p) are |n〉, as well.

114

APPENDIX C

EIGENVALUES OF HARMONIC OSCILLATOR PSRO

This section proves the relationship wn(Emax) = wnS(Emax) where nS =

∑fj=1 nj

and provides an analytical expression of the eigenvalue. The following derivation is

very similar to the proof presented in Ref. [69]; although, there are slight deviations.

First, we want to find an analytical expression for the single DOF eigenvalue

w(1)n (Emax) which will be useful later in the proof. We start by plugging Eqs. (3.21)

and (3.22) into (3.20) for the 1 DOF case, i.e.,

w(1)n (Emax) =

∫

RWn(q, p)dqdp

=(−1)n

π

∫

RLn[2(q2 + p2)]e−(q2+p2)dqdp (C.1)

where R is a disk of radius√

2Emax. The next logical step is to introduce polar

coordinates r =√

q2 + p2 and φ such that the region of integration or disk R =

{(r, φ) | 0 ≤ r ≤ √2Emax, 0 ≤ φ ≤ 2π}; thus,

w(1)n (Emax) =

(−1)n

π

∫ 2π

0

∫ √2Emax

0Ln(2r2)e−r2

rdrdφ. (C.2)

After integrating out the angle and substituting 2r2 with the variable t, Eq. (C.2)

simplifies to

w(1)n (Emax) =

(−1)n

2

∫ 4Emax

0Ln(t)e−t/2dt. (C.3)

From Ref. [120] , there is a useful Laguerre polynomial identity:

Ln(t) =d

dt[Ln(t)− Ln+1(t)]. (C.4)

After plugging this equation into (C.3) and integrating by parts, we arrive at the

expression

115

w(1)n (Emax) =

(−1)n

2e−2Emax [Ln(4Emax)− Ln+1(4Emax)] +

1

2

((−1)n

2

×∫ 4Emax

0Ln(t)e−t/2dt +

(−1)n+1

2

∫ 4Emax

0Ln+1(t)e

−t/2dt)

=(−1)n

2e−2Emax [Ln(4Emax)− Ln+1(4Emax)]

+1

2

(w(1)

n (Emax) + w(1)n+1(Emax)

)(C.5)

which simplifies into the recurrence relationship

w(1)n+1(Emax) = w(1)

n (Emax) + (−1)n+1e−2Emax [Ln(4Emax)− Ln+1(4Emax)]. (C.6)

Given that w(1)0 = 1− e−2Emax [look at (C.3) where L0(t) = 1], the closed form of the

last equation is

w(1)n (Emax) = 1− 2e−2Emax

[ n−1∑

k=0

(−1)kLk(4Emax) +(−1)n

2Ln(4Emax)

](C.7)

for n > 0.

Let’s now go back to the f DOF case where (C.1) is now

wn(Emax) =∫

RWn(q,p)dfqdfp

=∫

R

f∏

j=1

(−1)nj

πLnj

[2(q2j + p2

j)]e−(q2

j +p2j )dqjdpj. (C.8)

We can use polar coordinates r = (r1, . . . , rf ) and φ = (φ1, . . . , φf ) where qj = rj cosφj

and pj = rj sinφj. The region of integration or hypersphere R = {(r, φ) | 0 ≤ r1 ≤√

2Emax, 0 ≤ r2 ≤√

2Emax − r21, . . . , 0 ≤ rf ≤

√2Emax − r2

1 − r22 − · · · − r2

f−1, 0 ≤φ1 ≤ 2π, 0 ≤ φ2 ≤ 2π, . . . , 0 ≤ φf ≤ 2π}; thus, (C.8) can be written as

116

wn(Emax) = 2f (−1)n1+···+nf

∫ √2Emax

0Ln1(2r

21)e

−r21r1

∫ √2Emax−r21

0Ln2(2r

22)e

−r22r2 · · ·

×∫ √2Emax−r2

1−···−r2f−1

0Lnf

(2r2f )e

−r2f rf drf · · · dr2dr1 (C.9)

where the all of the angles have been integrated to give (2π)f .

In order to solve for an analytical representation of this integral, we will first look

at the rf−1 and rf contributions, the two innermost integrals, i.e.,

22(−1)nf−1+nf

∫ √2Emax−r21−···−r2

f−2

0Lnf−1

(2r2f−1)e

−r2f−1rf−1

×∫ √2Emax−r2

1−···−r2f−1

0Lnf

(2r2f )e

−r2f rf drfdrf−1. (C.10)

From Eq. (C.2), we notice that we have a 1 DOF eigenvalue for the rf contribution:

2(−1)nf−1

∫ √2Emax−r21−···−r2

f−2

0Lnf−1

(2r2f−1)e

−r2f−1rf−1

×w(1)nf

[Emax − (1/2)(r21 − · · · − r2

f−1)] drf−1. (C.11)

Replacing the eigenvalue with (C.7) and substituting 2r2f−1 with the variable u, the

previous expression becomes

(−1)nf−1

2

∫ 2(2Emax−r21−···−r2

f−2)

0Lnf−1

(u)e−u/2

(1− 2e−(2Emax−r2

1−···−r2f−2−u/2)

×[ nf−1∑

k=0

(−1)kLk[2(2Emax − r21 − · · · − r2

f−2)− u]

+(−1)nf

2Lnf

[2(2Emax − r21 − · · · − r2

f−2)− u]])

du. (C.12)

117

Noticing that the first term is also a 1 DOF eigenvalue and using the Laguerre iden-

tity120

∫ t

0Lm(u)Ln(t− u)du = Lm+n(t)− Lm+n+1(t), (C.13)

we get the expression

w(1)nf−1

(E ′max)− (−1)nf−1 e−2E′max

( nf−1∑

k=0

(−1)k(Lnf−1+k(4E

′max)

−Lnf−1+k+1(4E′max)

))− (−1)nf−1+nf

2e−2E′max

(Lnf−1+nf

(4E ′max)

−Lnf−1+nf+1(4E′max)

)(C.14)

where E ′max = Emax− (1/2)(r2

1−· · ·− r2f−2) . Replacing the first term with (C.7) and

then combining summations, one can recognize, using (C.7) in reverse, that (C.14)

simplifies to

1

2

(w

(1)nf−1+nf

[Emax − (1/2)(r2

1 − · · · − r2f−2)

]

+w(1)nf−1+nf+1

[Emax − (1/2)(r2

1 − · · · − r2f−2)

]). (C.15)

Finally, plugging (C.15) into our original expression (C.9), we get

wn(Emax) = 2f−2(−1)n1+···+nf−2

∫ √2Emax

0Ln1(2r

21)e

−r21r1

×∫ √2Emax−r2

1

0Ln2(2r

22)e

−r22r2 · · ·

∫ √2Emax−r21−···−r2

f−3

0Lnf−2

(2r2f−2)

×e−r2f−2rf−2

1

2

(w

(1)nf−1+nf

[Emax − (1/2)(r2

1 − · · · − r2f−2)

]

+w(1)nf−1+nf+1

[Emax − 1

2(r2

1 − · · · − r2f−2)

])drf−2 · · · dr2dr1 . (C.16)

118

Using steps (C.11)-(C.15) for each eigenvalue term, we can eliminate the innermost

integral (rf−2 contribution) to give the simplified expression

1

22

(w

(1)nf−2+nf−1+nf

(E ′′max) + 2w

(1)nf−2+nf−1+nf+1(E

′′max)

+w(1)nf−2+nf−1+nf+2(E

′′max)

)(C.17)

where E ′′max = Emax − (1/2)(r2

1 − · · · − r2f−3). If one repeats this process in order to

eliminate all of the integrals in Eq. (C.9), the final equation becomes

wn(Emax) =1

2f−1

( f−1∑

k=0

(f − 1

k

)w

(1)nS+k(Emax)

). (C.18)

Note that wn(Emax) depends on nS; thus, we can write wn(Emax) = wnS(Emax).

119

APPENDIX D

JUSTIFICATION OF PREPROCESSING AND SUBSPACE ITERATION

METHOD

The symmetric and real matrix, H ′, can be diagonalized via some orthogonal

similarity transformation, i.e.,

H ′ = F ΛFT , (D.1)

where the columns of F are the orthonormal eigenvectors of H ′, and Λ is a diagonal

matrix containing the corresponding eigenvalues. To simplify the proof, we arrange

the columns of F such that the eigenvalues in Λ are listed in descending order, starting

from the top-left corner (H ′ is similarly reordered). Thus, Λ can be rewritten as

Λ =

λu

λl

(D.2)

where λu is the d×d diagonal matrix that contains the eigenvalues ranging from 1 to

2 and λl is the N −d×N −d diagonal matrix with the remaining eigenvalues ranging

from 0 to 1.

Since FTF = I, then

(H ′)rZ0 = F ΛrFTZ0

= F

(λu)r

(λl)r

FTZ0 . (D.3)

We can rewrite Eq. (D.3) as

(H ′)rZ0 = F

V d×d

W (N−d)×d

(D.4)

120

where each row of V d×d and W (N−d)×d is a multiple of a diagonal element of (λu)r

and (λl)r, respectively. We can also write F in a split form, i.e.,

F =[

FN×du F

N×(N−d)l

](D.5)

where the column eigenvectors of FN×du and F

N×(N−d)l correspond to the eigenvalues

in λu and λl, respectively.

With Eqs. (D.4) and (D.5), a useful expression can be realized:

(H ′)rZ0 = FN×du V d×d + F

N×(N−d)l W (N−d)×d . (D.6)

Since the elements in (λl)r approach 0 as r −→∞, then

(H ′)rZ0 −→ FN×du V d×d (D.7)

as the iteration number r increases. Since V d×d has full rank, then the space spanned

by the column vectors of FN×du V d×d is the same as that spanned by FN×d

u . Thus,

after many iterations, one approaches an ISUB spanned by the eigenvectors that

correspond to eigenvalues of H ′ that range between 1 and 2. These eigenvectors are

exactly the same as those of the original matrix H with eigenvalues less than µ. This

derivation is based upon ideas obtained from Ref. [121] .

121

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