Introduction to computational quantum mechanics
Lecture 5: The time independent Schrödinger equation and
Ritz-Galerkin approach
Simen Kvaal
simen.kvaal@cma.uio.no
Centre of Mathematics for Applications
University of Oslo
Seminar series in quantum mechanics at CMA
Fall 2009
Outline
Setting and formal results
Ritz-Galerkin/FCI
FCI: Ritz-Galerkin with Slater determinants
References
Outline
Setting and formal results
Ritz-Galerkin/FCI
FCI: Ritz-Galerkin with Slater determinants
References
Hamiltonian, Hilbert-space and basis
In 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
N
H=
∑ H0,k + ∑
k=1
I
Vij
(ij), i6=j
The most general H0 is given by
1
H0 = − ∇2 + W(x),
2
ignoring irrelevant constants.
W(x) smooth
The time-independent Schrödinger equation
I
Suppose the spectral decomposition of H is available:
Z ∞
E dP(E) =
H=
−∞
I
∑
Ek |ψk i hψk | +
Z
Ek ∈σd (H)
σc (H)
E |χ(E)i hχ(E)| dE
Then, the time evolution is trivial to compute:
Z ∞
U(t) = exp(−itH) =
e−itE dP(E).
−∞
I
This – among other things – motivates the study of the eigenvalue
problem for H, i.e., finding σ(E), ψk and χ(E), formally:
Hψ = Eψ.
The spectrum of H: HVZ theorem
What can be said about σ(H) a priori? The famous HVZ theorem (after
Hunziker, van Winter and Zhislin who first proved it) says a lot:
Theorem
Suppose 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 given
by:
Σ = min Σ(A) ,
#a≥2
where Σ(a) = inf σ(H(a)), with
H(a) = H −
∑
(ij)6⊂a
Vij .
Illustration
Figure: Illustration of geometric idea behind HVZ theorem. As clusters of particles
move 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/kxi − xj k)
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:
Theorem
Let V : RNd → R be smooth and unbounded as kxk → ∞. Then the
Hamiltonian
1
H = − ∇2 + V(x)
2
has a purely discrete spectrum on the form:
σ(H) = σd (H) = {Ek }∞
k=1 ,
(Note that ∑k ∇2k = ∇2 .)
E1 ≤ E2 ≤ · · · % +∞
Outline
Setting and formal results
Ritz-Galerkin/FCI
FCI: Ritz-Galerkin with Slater determinants
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 structural
vibrations
I
Full configuration interaction (FCI): chemistry, nuclear physics
I
Exact diagonalization: quantum dots, solid state physics
I
Projection method, finite section method: Mathematical spectral
approximation theory
Variational characterization of σ(H)
Ritz-Galerkin/FCI is based on the following well-known variational
characterization of the eigenvalues below σc (H):
Theorem
Suppose H = H ∗ is bounded from below.
Define a sequence {(φk , µk )}, k = 1, · · · by:
µk = min hψ|Hψi =: hφk |Hφk i ,
ψ∈Mk
where
Mk = {ψ ∈ D(H) : kψk = 1, hψ|φj i = 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 eigenvector
with eigenvalue µk .
2. µj = µk for all j > k, and µk = inf σc (H) (the bottom of the continuous
spectrum.) There are then only finitely many eigenvalues below σc (H).
Ritz-Galerkin approach
I
I
Suppose {ej } is an orthonormal basis for H , such that ej ∈ D(H) for
all j.
Consider the space Hn = Pn H , with
n
Pn := ∑ |ej i hej |
(orth. projection)
j=1
I
Consider the compression of H onto Hn , i.e.,
n
Hn := Pn H|Hn = ∑
n
j |Hek i hek |
∑ |ej i he
| {z }
j=1 k=1
mat. elms.
I
The spectrum σ(Hn ) = σ(H) with
Hjk = hej |Hek i ;
I
n × n matrix.
One hopes that, in some sense,
σ(Hn ) −→ σ(H),
as n → ∞
Convergence of Ritz-Galerkin
Recall the variational characterization:
µk = min hψ|Hψi =: hφk |Hφk i ,
ψ∈Mk
where
Mk = {ψ ∈ D(H) : kψk = 1, hψ|φj i = 0, for j < k.}
(n)
I
The eigenvalues µk of H are obtained by replacing D(H) with H\ .
I
Thus, we immediately obtain:
(n)
µk ≤ µk ,
I
for all k ≤ n and n.
Conclusion: The eigenvalues are approximated from above. If
(n)
#σd (H) = m < ∞ (at least), the n − m remaining µk converges to
Σ = inf σc (H).
Convergenve in Hausdorff metric
But we can say more:
Theorem
If H = H ∗ is bounded from below and we have have
σc (H) = [Σ, ∞) or empty ,
then
σ(H) → σ(H) in Hausdorff metric.
Convergence in Hausdorff metric: illustration
Figure: Convergence of discrete spectrum in Hausdorff metric, meaning that the
convergence is ordinary for the discrete points in σ(H), while the other discrete
eigenvalues fill up the continuous spectrum, with smaller and smaller distance among
them.
Simple example: one-particle quantum dot
I
We consider a simple d = 1 problem, N = 1 particle.
I
Hamiltonian:
H=−
I
1 ∂2
+ W(x);
2 ∂x2
2 /2σ2
Basis functions:
ej (x) = Nn Hn (x)e−x
I
W(x) = −V0 e−x
2 /2
(Hermite functions)
Matrix elements evaluated using Gaussian quadrature, matrix
diagonalized in Matlab for an increasing number of basis functions.
Simple example: plot of eigenvalues
Convergence history of the approximate eigenvalues
10
8
6
µ(n)
k
4
2
0
−2
−4
−6
−8
0
50
100
150
n
200
250
(n)
Figure: Convergence with increasing n of eigenvalues µk towards µk . Seemingly,
σc (H) = [0, ∞), which in fact can be proven. Note extremely rapid convergence in
this case for the bound states, i.e., for σd (H).
Outline
Setting and formal results
Ritz-Galerkin/FCI
FCI: Ritz-Galerkin with Slater determinants
References
Brief outline
1. Given an N-body Hamiltonian
H/Hilbert space HN
2. Use subset of Slater-determinant
basis BN
3. Construct matrix
4. Diagonalize to obtain approximation
to σd (H).
Figure: Typical structure of an
N = 3 matrix
A little repetition does little harm
I
N-particle Hilbert space:
HN = Π− H1⊗N = H1 ∧ · · · ∧ H1
{z
}
|
N factors
I
Single-particle orthonormal “orbitals”:
H1 = Span{φj : j = 1, 2, · · · }
I
Slater determinants constitute a basis for HN :
ΦSD
j1 ,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 = ΦSD
j1 ,··· ,jN : j1 < j2 < · · · < jN
I
Introduce a cut parameter L and consider:
BNL := ΦSD
j1 ,··· ,jN : ji ≤ L ⊂ BN
I
There are of course other ways to truncate the basis, but this is the most
common.
I
Note:
#BNL
L
=
N
⇒ ”Curse of dimensionality”
More repetition
I
The Hamiltonian can be written:
H = ∑ hα,β c†α cβ +
αβ
I
1
∑ vαβγδ c†α c†β cδ cγ
2 αβγδ
Here, hαβ is a “one particle matrix element”:
hαβ = hφα , H0 φβ i ,
(inner product in one-particle space)
and vαβγδ is a “two-particle matrix element”. (See earlier lecture for
more)
Building matrix elements
Recall two basic facts:
Theorem
Even though N-body Hilbert space consists of Nd-dimensional functions, the
matrix elements can be computed solely with low-dimensional integrals.
Theorem
The matrix obtains a sparsity structure due to the Slater–Condon rules: The
Hamiltonian only “changes two single-particle functions” when acting on
ΦSD . Therefore, matrix element between Slater determinants are zero
whenever 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. What
about the speed of convergence?
I Depends on the analytic properties of the eigenfunction ψk . Typical
question: “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 functions
I
For electronic problems, Vij ∼ 1/kxi − xj k, and typically
EkL − Ek ≤ C1 L−ν1 , withν1 > 0.
I
In nuclear physics, Vij is, in fact, smooth, and we obtain
EkL − Ek ≤ C2 e−ν2 L , withν2 > 0.
Outline
Setting and formal results
Ritz-Galerkin/FCI
FCI: Ritz-Galerkin with Slater determinants
References
References
Simon, B.
Schrödinger operators in the 20th century
J. Math. Phys 41, p. 3523
2000
Hansen, A.C.
On the approximation of spectra of linear operators on Hilbert spaces
J. Func. Anal. 254, pp. 2092–2126
2008
Babuska, I. and Osborn, J.E.
Finite Element-Galerkin Approximation of the Eigenvalues and
Eigenvectors of Selfadjoint Problems
Math. Comp. 52, pp. 275–297
1989
S.K.
Analysis of many-body methods for quantum dots
PhD thesis
2009