Introduction to computational quantum...

Introduction to computational quantum mechanicsLecture 5: The time independent Schrodinger equation and

Ritz-Galerkin approach

Simen [email protected]

Centre of Mathematics for ApplicationsUniversity of Oslo

Seminar series in quantum mechanics at CMAFall 2009

Outline

Setting and formal results

Ritz-Galerkin/FCI

FCI: Ritz-Galerkin with Slater determinants

References

Hamiltonian, Hilbert-space and basisIn this lecture, we will use the following conventions:

I We consider N particles in d dimensions, i.e.:

H = Π−L2(Rd×S)⊗N

We usually ignore spin S; it does not modify results.I The Hamiltonian H is given by

H =N

∑k=1

H0,k + ∑(ij), i6=j

Vij

I The most general H0 is given by

H0 =−12

∇2 +W(x), W(x) smooth

ignoring irrelevant constants.

The time-independent Schrodinger equation

I Suppose the spectral decomposition of H is available:

H =∫

∞

−∞

E dP(E) = ∑Ek∈σd(H)

Ek |ψk〉〈ψk|+∫σc(H)

E |χ(E)〉〈χ(E)| dE

I Then, the time evolution is trivial to compute:

U(t) = exp(−itH) =∫

∞

−∞

e−itE dP(E).

I This – among other things – motivates the study of the eigenvalueproblem for H, i.e., finding σ(E), ψk and χ(E), formally:

Hψ= Eψ.

The spectrum of H: HVZ theoremWhat can be said about σ(H) a priori? The famous HVZ theorem (afterHunziker, van Winter and Zhislin who first proved it) says a lot:

TheoremSuppose Vij = V(xi− xj), as operator on L2(Rd), is such that

Vij(−∇2 +1)−1 is compact

Then the essential/continuous spectrum is

σc(H) = [Σ,∞).

Let a be a partition of {1, . . . ,N} into disjoint subsets. The number Σ is givenby:

Σ = min#a≥2

(Σ(A)

),

where Σ(a) = infσ(H(a)), with

H(a) = H− ∑(ij)6⊂a

Vij.

Illustration

Figure: Illustration of geometric idea behind HVZ theorem. As clusters of particlesmove far away, the interactions between the remaining ones vanish. σc(H) “should”contain points corresponding to clusters moving “slowly away” from the origin

The spectrum of H: discrete spectrum

What can we say about σd(H)? Several possibilities:I Infinitely many isolated eigenvalues < Σ, accumulating at Σ (e.g., if

Vij = 1/‖xi− xj‖)I Finitely many isolated eigenvalues < Σ (e.g., if Vij is smooth with

compact support)I No eigenvalues at all, e.g., if Vij ≥ 0 (repulsive interactions)I Eigenvalues embedded in σc(H); “resonances”I No eigenvalues of infinite multiplicity (in this Hamiltonian!)

The spectrum of H: purely discrete spectrum

The Hamiltonian may also have a purely discrete spectrum:

TheoremLet V : RNd→ R be smooth and unbounded as ‖x‖→ ∞. Then theHamiltonian

H =−12

∇2 +V(x)

has a purely discrete spectrum on the form:

σ(H) = σd(H) = {Ek}∞k=1, E1 ≤ E2 ≤ ·· · ↗+∞

(Note that ∑k ∇2k = ∇2.)

Outline


Ritz-Galerkin/FCI


References

Pet child has many names . . .

We will describe an approximation method for σ(H) called . . .I Ritz-Galerkin/Rayleigh-Ritz(-Galerkin); especially in finite element

(FE) community; continuum mechanics and studies of structuralvibrations

I Full configuration interaction (FCI): chemistry, nuclear physicsI Exact diagonalization: quantum dots, solid state physicsI Projection method, finite section method: Mathematical spectral

approximation theory

Variational characterization of σ(H)Ritz-Galerkin/FCI is based on the following well-known variationalcharacterization of the eigenvalues below σc(H):

TheoremSuppose H = H∗ is bounded from below.Define a sequence {(φk,µk)}, k = 1, · · · by:

µk = minψ∈Mk

〈ψ|Hψ〉=: 〈φk|Hφk〉 ,

whereMk = {ψ ∈ D(H) : ‖ψ‖= 1, 〈ψ|φj〉= 0, for j < k.}

Then for every k exactly one of the following holds:

1. There is j > k with µj > µk, and µk ∈ σd(H). Also, φk is an eigenvectorwith eigenvalue µk.

2. µj = µk for all j > k, and µk = infσc(H) (the bottom of the continuousspectrum.) There are then only finitely many eigenvalues below σc(H).

Ritz-Galerkin approachI Suppose {ej} is an orthonormal basis for H , such that ej ∈ D(H) for

all j.I Consider the space Hn = PnH , with

Pn :=n

∑j=1|ej〉〈ej| (orth. projection)

I Consider the compression of H onto Hn, i.e.,

Hn := PnH|Hn =n

∑j=1

n

∑k=1|ej〉〈ej|Hek〉︸︷︷︸

mat. elms.

〈ek|

I The spectrum σ(Hn) = σ(H) with

Hjk = 〈ej|Hek〉 ; n×n matrix.

I One hopes that, in some sense,

σ(Hn)−→ σ(H), as n→ ∞

Convergence of Ritz-Galerkin

Recall the variational characterization:

µk = minψ∈Mk

〈ψ|Hψ〉=: 〈φk|Hφk〉 ,

whereMk = {ψ ∈ D(H) : ‖ψ‖= 1, 〈ψ|φj〉= 0, for j < k.}

I The eigenvalues µ(n)k of H are obtained by replacing D(H) with H\.

I Thus, we immediately obtain:

µk ≤ µ(n)k , for all k ≤ n and n.

I Conclusion: The eigenvalues are approximated from above. If#σd(H) = m < ∞ (at least), the n−m remaining µ(n)

k converges toΣ = infσc(H).

Convergenve in Hausdorff metric

But we can say more:

TheoremIf H = H∗ is bounded from below and we have have

σc(H) = [Σ,∞) or empty ,

thenσ(H)→ σ(H) in Hausdorff metric.

Convergence in Hausdorff metric: illustration

Figure: Convergence of discrete spectrum in Hausdorff metric, meaning that theconvergence is ordinary for the discrete points in σ(H), while the other discreteeigenvalues fill up the continuous spectrum, with smaller and smaller distance amongthem.

Simple example: one-particle quantum dot

I We consider a simple d = 1 problem, N = 1 particle.I Hamiltonian:

H =−12∂2

∂x2 +W(x); W(x) =−V0e−x2/2σ2

I Basis functions:

ej(x) = NnHn(x)e−x2/2 (Hermite functions)

I Matrix elements evaluated using Gaussian quadrature, matrixdiagonalized in Matlab for an increasing number of basis functions.

Simple example: plot of eigenvalues

0 50 100 150 200 250

−8

−6

−4

−2

0

2

4

6

8

10

n

µ k(n)

Convergence history of the approximate eigenvalues

Figure: Convergence with increasing n of eigenvalues µ(n)k towards µk. Seemingly,

σc(H) = [0,∞), which in fact can be proven. Note extremely rapid convergence inthis case for the bound states, i.e., for σd(H).

Outline


Ritz-Galerkin/FCI


References

Brief outline

1. Given an N-body HamiltonianH/Hilbert space HN

2. Use subset of Slater-determinantbasis BN

3. Construct matrix4. Diagonalize to obtain approximation

to σd(H).

Figure: Typical structure of anN = 3 matrix

A little repetition does little harmI N-particle Hilbert space:

HN = Π−H ⊗N1 = H1∧·· ·∧H1︸︷︷︸

N factors

I Single-particle orthonormal “orbitals”:

H1 = Span{φj : j = 1,2, · · ·}

I Slater determinants constitute a basis for HN :

ΦSDj1,j2,··· ,jN (x1,x2, · · · ,xN) = φj1 ∧φj2 ∧·· ·∧φjN

I We could also express these using creation operators:

ΦSD = c†

j1· · ·c†

jN Φ0, c†j : HN →HN+1,

where Φ0 is a trivial Slater determinant with zero particles.

FCI Slater determinant subset

I The complete basis:

BN ={

ΦSDj1,··· ,jN : j1 < j2 < · · · < jN

}I Introduce a cut parameter L and consider:

BLN :=

{Φ

SDj1,··· ,jN : ji ≤ L

}⊂BN

I There are of course other ways to truncate the basis, but this is the mostcommon.

I Note:

#BLN =

(LN

)⇒ ”Curse of dimensionality”

More repetition

I The Hamiltonian can be written:

H = ∑αβ

hα,βc†αcβ+

12 ∑αβγδ

vαβγδc†αc†

βcδcγ

I Here, hαβ is a “one particle matrix element”:

hαβ = 〈φα,H0φβ〉 , (inner product in one-particle space)

and vαβγδ is a “two-particle matrix element”. (See earlier lecture formore)

Building matrix elements

Recall two basic facts:

TheoremEven though N-body Hilbert space consists of Nd-dimensional functions, thematrix elements can be computed solely with low-dimensional integrals.

TheoremThe matrix obtains a sparsity structure due to the Slater–Condon rules: TheHamiltonian only “changes two single-particle functions” when acting onΦSD. Therefore, matrix element between Slater determinants are zerowhenever they have > 2 different indices ji

A convergence result for quantum dot systems

We have considered the fact that FCI does converge as L increases. Whatabout the speed of convergence?

I Depends on the analytic properties of the eigenfunction ψk. Typicalquestion: “How smooth is it?”

I This in turns depends on the Hamiltonian H.I The speed of convergence with respect to L depends on the

approximation properties of φj, the single-particle functionsI For electronic problems, Vij ∼ 1/‖xi− xj‖, and typically

ELk −Ek ≤ C1L−ν1 ,withν1 > 0.

I In nuclear physics, Vij is, in fact, smooth, and we obtain

ELk −Ek ≤ C2e−ν2L,withν2 > 0.

Outline


Ritz-Galerkin/FCI


References

ReferencesSimon, B.Schrodinger operators in the 20th centuryJ. Math. Phys 41, p. 35232000

Hansen, A.C.On the approximation of spectra of linear operators on Hilbert spacesJ. Func. Anal. 254, pp. 2092–21262008

Babuska, I. and Osborn, J.E.Finite Element-Galerkin Approximation of the Eigenvalues andEigenvectors of Selfadjoint ProblemsMath. Comp. 52, pp. 275–2971989

S.K.Analysis of many-body methods for quantum dotsPhD thesis2009

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