PHYSICAL REVIEW B 80, 045321 共2009兲
Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions
Simen Kvaal*
Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway
共Received 15 August 2008; revised manuscript received 15 May 2009; published 27 July 2009兲
We give a thorough analysis of the convergence properties of the configuration-interaction method as applied
to parabolic quantum dots among other systems, including a priori error estimates. The method converges
slowly in general, and in order to overcome this, we propose to use an effective two-body interaction well
known from nuclear physics. Through numerical experiments we demonstrate a significant increase in accuracy of the configuration-interaction method.
DOI: 10.1103/PhysRevB.80.045321
PACS number共s兲: 73.21.La, 71.15.⫺m, 02.30.Mv
I. INTRODUCTION
The last two decades, an ever-increasing amount of research has been dedicated to understanding the electronic
structure of so-called quantum dots:1 semiconductor structures confining from a few to several thousands electrons in
spatial regions on the nanometer scale. In such calculations,
one typically seeks a few of the lowest eigenenergies Ek of
the system Hamiltonian H and their corresponding eigenvectors ␺k, i.e.,
H ␺ k = E k␺ k,
k = 1, . . . ,kmax .
共1.1兲
One of the most popular methods is the 共full兲 configurationinteraction 共CI兲 method, where the many-body wave function is expanded in a basis of eigenfunctions of the harmonic
oscillator 共HO兲 and then necessarily truncated to give an approximation. In fact, the so-called curse of dimensionality
implies that the number of degrees of freedom available per
particle is severely limited. It is clear that an understanding
of the properties of such basis expansions is very important
as it is necessary for a priori error estimates of the calculations. Unfortunately, this is a neglected topic in the physics
literature.
In this article, we give a thorough analysis of the 共full兲 CI
method using HO expansions applied to parabolic quantum
dots and give practical convergence estimates. It generalizes
and refines the findings of a recent study of one-dimensional
systems2 and is applicable to, for example, nuclear systems3
and quantum chemical calculations4 as well. We demonstrate
the estimates with calculations in the d = 2 dimensional case
for N ⱕ 5 electrons, paralleling computations in the
literature.5–10
The main results are however somewhat discouraging.
The expansion coefficients of typical eigenfunctions are
shown to decay very slowly, limiting the accuracy of any
practical method using HO basis functions. We therefore propose to use an effective two-body interaction to overcome, at
least partially, the slow convergence rate. This is routinely
used in nuclear physics3,11 where the interparticle forces are
of a completely different, and basically unknown, nature. For
electronic systems, however, the interaction is well known
and simpler to analyze, but effective interactions of the
present kind have not been applied, at least to the author’s
knowledge. The modified method is seen to have conver1098-0121/2009/80共4兲/045321共16兲
gence rates of at least 1 order of magnitude higher than the
original CI method. An important point here is that the complexity of the CI calculations is not altered as no extra nonzero matrix elements are introduced. All one needs is a relatively simple one-time calculation to produce the effective
interaction matrix elements.
The HO eigenfunctions are popular for several reasons.
Many quantum systems, such as the quantum dot model considered here, are perturbed harmonic oscillators per se so that
the true eigenstates should be perturbations of the HO states.
Moreover, the HO has many beautiful properties, such as
complete separability of the Hamiltonian, invariance under
orthogonal coordinate changes, and thus easily computed
eigenfunctions so that computing matrix elements of relevant
operators becomes relatively simple. The HO eigenfunctions
are defined on the whole of Rd in which the particles live so
that truncation of the domain is unnecessary. Indeed, this is
one of the main problems with methods such as finite difference or finite element methods.12 On the other hand, the HO
eigenfunctions are the only basis functions with all these
properties.
The article is organized as follows. In Sec. II we discuss
the harmonic oscillator and the parabolic quantum dot
model, including exact solutions for the N = 2 case. In Sec.
III, we give results for the approximation properties of the
Hermite functions in n dimensions and thus also of manybody HO eigenfunctions. By approximation properties, we
mean estimates on the error 储␺ − P␺储, where ␺ is any wave
function and P projects onto a finite subspace of HO eigenfunctions, i.e., the model space. Here, P␺ is in fact the best
approximation in the norm. The estimates will depend on
analytic properties of ␺, i.e., whether it is differentiable, and
whether it falls of sufficiently fast at infinity. To our knowledge, these results are not previously published.
In Sec. IV, we discuss the full configuration-interaction
method, using the results obtained in Sec. III to obtain convergence estimates of the method as function of the model
space size. We also briefly discuss the effective interaction
utilized in the numerical calculations, which are presented in
Sec. V. We conclude with a discussion of the results, its
consequences, and an outlook on further directions of research in Sec. VI. We have also included an appendix with
proofs of the formal propositions in Sec. III.
045321-1
©2009 The American Physical Society
PHYSICAL REVIEW B 80, 045321 共2009兲
SIMEN KVAAL
The eigenvalue of ␾n共x兲 is n + 1 / 2 so that the eigenvalue
of ⌽␣共rជ兲 is
II. HARMONIC OSCILLATOR AND PARABOLIC
QUANTUM DOTS
A. Harmonic Oscillator
A spinless particle of mass m in an isotropic harmonic
potential has Hamiltonian
ប2 2 1
HHO = −
ⵜ + m␻2储rជ储2 ,
2
2m
共2.1兲
where rជ 苸 Rd is the particle’s coordinates. By choosing
proper energy and length units, i.e., ប␻ and 冑ប / m␻, respectively, the Hamiltonian becomes
1
1
HHO = − ⵜ2 + 储rជ储2 .
2
2
共2.2兲
HHO can be written as a sum over d one-dimensional harmonic oscillators, viz.,
d
冉
HHO = 兺 −
k=1
冊
1 ⳵2 1 2
+ r ,
2 ⳵ r2k 2 k
共2.3兲
so that a complete specification of the HO eigenfunctions is
given by
⌽␣1,␣2,. . .,␣d共rជ兲 = ␾␣1共r1兲␾␣2共r2兲 ¯ ␾␣d共rd兲,
共2.4兲
where ␾␣i共x兲, ␣i = 0 , 1 , . . . are one-dimensional HO eigenfunctions, also called Hermite functions. These are defined
by
2
␾n共x兲 = 共2nn ! ␲1/2兲−1/2Hn共x兲e−x /2,
n = 0,1, . . . ,
共2.5兲
where the Hermite polynomials Hn共x兲 are given by
Hn共x兲 = 共− 1兲nex
2
⳵n −x2
e .
⳵ xn
共2.6兲
⑀␣ =
共2.7兲
with H0共x兲 = 1 and H1共x兲 = 2x. The Hermite polynomial Hn共x兲
has n zeroes, and the Gaussian factor in ␾n共x兲 will eventually
subvert the polynomial for large 兩x兩. Thus, qualitatively, the
Hermite functions can be described as localized oscillations
with n nodes and a Gaussian “tail” as x approaches ⫾⬁. One
can easily compute the quantum-mechanical variance
共⌬x兲2 ª
冕
⬁
1
x2␾n共x兲2dx = n + ,
2
−⬁
共2.8兲
showing that, loosely speaking, the width of the oscillatory
region increases as 共n + 1 / 2兲1/2.
The functions ⌽␣1,. . .,␣d defined in Eq. 共2.4兲 are called
d-dimensional Hermite functions. In the sequel, we will define ␣ = 共␣1 , . . . , ␣d兲 苸 Id for a tuple of non-negative integers,
also called a multi-index; see the Appendix. Using multiindices, we may write
2
⌽␣共rជ兲 = 共2兩␣兩␣ ! ␲d/2兲−1/2H␣1共r1兲 ¯ H␣d共rd兲e−储rជ储 /2 .
共2.9兲
共2.10兲
i.e., a zero-point energy d / 2 plus a non-negative integer. We
denote by 兩␣兩 the shell number of ⌽␣ and the eigenspace
Sr共Rd兲 corresponding to the eigenvalue d / 2 + r a shell. We
define the shell-truncated Hilbert space PR共Rd兲 傺 L2共Rd兲 as
R
PR共Rd兲 ª span兵⌽␣共rជ兲兩兩␣兩 ⱕ R其 = 丣 Sr共Rd兲,
r=0
共2.11兲
i.e., the subspace spanned by all Hermite functions with shell
number less than or equal to R, or, equivalently, the direct
sum of the shells up to and including R. The N-body generalization of this space, to be discussed in Sec. III B, is a very
common model space used in CI calculations.
Since the Hermite functions constitute an orthonormal basis for L2共Rd兲, PR共Rd兲 → L2共Rd兲, in the sense that for every
␺ 苸 L2共Rd兲, limR→⬁储␺ − P␺储 = 0, where P is the orthogonal
projector on PR共Rd兲. Strictly speaking, we should use a symbol such as PR or even PR共Rd兲 for the projector. However, R
and d will always be clear from the context, so we are deliberately sloppy to obtain a concise formulation. For the
same reason, we will sometimes simply write P or PR for the
space PR共Rd兲.
An important fact is that since HHO is invariant under
orthogonal spatial transformations 共i.e., such transformation
conserve energy兲 so is each individual shell space. Hence,
each shell Sr共Rd兲, and also PR共Rd兲, is independent of the
spatial coordinates chosen.
For the case d = 1 each shell r is spanned by a single
eigenfunction, namely, ␾r共x兲. For d = 2, each shell r has degeneracy r + 1, with eigenfunctions
⌽共s,r−s兲共rជ兲 = ␾s共r1兲␾r−s共r2兲,
The Hermite polynomials also obey the recurrence formula
Hn+1共x兲 = 2xHn共x兲 − 2nHn−1共x兲,
d
+ 兩␣兩,
2
0 ⱕ s ⱕ r.
共2.12兲
The usual HO eigenfunctions used to construct manybody wave functions are not the Hermite functions ⌽␣1,¯,␣d,
however, but rather those obtained by utilizing the spherical
symmetry of the HO. This gives a many-body basis diagonal
in angular momentum. For d = 2 we obtain the so-called
Fock-Darwin orbitals given by
FD
共r, ␪兲 =
⌽n,m
冋
2n!
共n + 兩m兩兲!
册
1/2 im␪
e
兩m兩
冑2␲ Ln
2
共r2兲e−r /2 . 共2.13兲
Here, n ⱖ 0 is the nodal quantum number, counting the nodes
of the radial part, and m is the azimuthal quantum number.
The eigenvalues are
⑀n,m = 2n + 兩m兩 + 1.
共2.14兲
Thus, R = 2n + 兩m兩 is the shell number. By construction, the
Fock-Darwin orbitals are eigenfunctions of the angularmomentum operator Lz = −i ⳵ / ⳵␪ with eigenvalue m. Of
FD
as a linear combination of the
course, we may write ⌽n,m
Hermite functions ⌽共s,R−s兲, where 0 ⱕ s ⱕ R = 2n + 兩m兩, and
vice versa. The actual choice of form of eigenfunctions is
immaterial as long as we may identify those belonging to a
045321-2
PHYSICAL REVIEW B 80, 045321 共2009兲
HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
n=0
n=1
n=2
n=1
n=0
m = −4
m = −2
m=0
m=2
m=4
n=0
n=1
n=1
n=0
m = −3
m = −1
m=1
m=3
n=0
n=1
n=0
m = −2
m=0
m=2
n=0
n=0
m = −1
m=1
α = (0, 4)
α = (1, 3)
α = (2, 2)
α = (3, 1)
α = (4, 0)
α = (0, 3)
α = (1, 2)
α = (2, 1)
α = (3, 0)
shell S3
Unitarily equivalent
α = (0, 2)
α = (1, 1)
α = (0, 1)
α = (2, 0)
α = (1, 0)
n=0
m=0
α = (0, 0)
FIG. 1. Illustration of PR=4共R2兲: 共left兲 Fock-Darwin orbitals. 共right兲 Hermite functions. Basis functions with equal HO energy are shown
at same line.
given shell. The space PR=4共R2兲 is illustrated in Fig. 1 using
both Hermite functions and Fock-Darwin orbitals.
B. Parabolic quantum dots
We consider N electrons confined in a harmonic oscillator
in d dimensions. This is a very common model for a quantum dot. We comment that modeling the quantum dot geometry by a perturbed harmonic oscillator is justified by selfconsistent calculations13–15 and is a widely adopted
assumption.5,8,9,16–18
The Hamiltonian of the quantum dot is given by
共2.15兲
H ª T + U,
well understood.19–21 Here, we consider d = 2 dimensions
only, but the d = 3 case is similar. We note that for N = 2 it is
enough to study the spatial wave function since it must be
either symmetric 共for the singlet S = 0 spin state兲 or antisymmetric 共for the triplet S = 1 spin states兲. Hamiltonian 共2.15兲
becomes
1
1
␭
,
H = − 共ⵜ21 + ⵜ22兲 + 共r21 + r22兲 +
2
2
r12
where r12 = 储rជ1 − rជ2储 and r j = 储rជ j储. Introduce a set of scaled
ជ = 共rជ1 + rជ2兲 / 冑2 and rជ
center-of-mass coordinates given by R
= 共rជ1 − rជ2兲 / 冑2. This coordinate change is orthogonal and symmetric in R4. This leads to the separable Hamiltonian
where T is the many-body HO Hamiltonian, given by
1
1
␭
ជ 储 2兲 +
H = − 共ⵜr2 + ⵜR2 兲 + 共储rជ储2 + 储R
冑2储rជ储
2
2
N
T = 兺 HHO共rជk兲
共2.16兲
ជ 兲 + Hrel共rជ兲.
= HHO共R
k=1
and U is the interelectron Coulomb interactions. In dimensionless units the interaction is given by,
N
N
U ª 兺 C共i, j兲 = 兺
i⬍j
i⬍j
␭
.
储rជi − rជ j储
共2.17兲
The N electrons have coordinates rជk, and the parameter ␭
measures the strength of the interaction over the confinement
of the HO, viz.,
␭ª
冉 冊
1
e2
,
ប ␻ 4 ␲ ⑀ 0⑀
共2.18兲
where we recall that 冑ប / m␻ is the length unit. Typical values
for GaAs semiconductors are close to ␭ = 2; see, for example,
Ref. 18. Increasing the trap size leads to a larger ␭, and the
quantum dot then approaches the classical regime.1
A complete set of eigenfunctions of H can now be written on
product form, viz.,
FD
ជ 兲␺共rជ兲.
⌿共Rជ ,rជ兲 = ⌽n⬘,m⬘共R
共2.20兲
The relative coordinate wave function ␺共rជ兲 is an eigenfunction of the relative coordinate Hamiltonian given by
1
1
␭
Hrel = − ⵜr2 + r2 +
冑2r ,
2
2
共2.21兲
where r = 储rជ储. This Hamiltonian can be further separated using polar coordinates, yielding eigenfunctions on the form
␺m,n共r, ␪兲 =
eim␪
冑2␲ un,m共r兲,
共2.22兲
where 兩m兩 ⱖ 0 is an integer and un,m is an eigenfunction of the
radial Hamiltonian given by
C. Exact solution for two electrons
Before we discuss the approximation properties of the
Hermite functions, it is instructive to consider the very simplest example of a two-electron parabolic quantum dot and
the properties of the eigenfunctions since this case admits
analytical solutions for special values of ␭ and is otherwise
共2.19兲
Hr = −
1 ⳵ ⳵ 兩m兩2 1 2
␭
r + 2 + r +
冑2r .
2
2r ⳵ r ⳵ r 2r
共2.23兲
By convention, n counts the nodes away from r = 0 of un,m共r兲.
Moreover, odd 共even兲 m gives antisymmetric 共symmetric兲
wave functions ⌿共rជ1 , rជ2兲. For any given 兩m兩, it is quite easy
045321-3
SIMEN KVAAL
PHYSICAL REVIEW B 80, 045321 共2009兲
to deduce that the special value ␭ = 冑2兩m兩 + 1 yields the eigenfunction
兩cn兩 = o共n−共k+1+⑀兲/2兲.
2
u0,m = Dr兩m兩共a + r兲e−r /2 ,
共2.24兲
where D and a are constants. The corresponding eigenvalue
of Hr is Er = 兩m兩 + 2, and E = 2n⬘ + 兩m⬘兩 + 1 + Er. Thus, the
ground state 共having m = m⬘ = 0 and n = n⬘ = 0兲 for ␭ = 1 is
given by
ជ ,rជ兲 = D共r + a兲e−共r2+R2兲/2 =
⌿0共R
D
冑2 共r12 +
冑2a兲e−共r21+r22兲/2 ,
with D being a 共new兲 normalization constant.
Observe that this function has a cusp at r = 0, i.e., at the
origin x = y = 0 关where we have introduced Cartesian coordinates rជ = 共x , y兲 for the relative coordinate兴. Indeed, the partial
derivatives ⳵x␺0,0 and ⳵y␺0,0 are not continuous there, and ⌿0
has no partial derivatives 共in the distributional sense, see the
Appendix兲 of second order. The cusp stems from the famous
“cusp condition,” which in simple terms states that, for a
nonvanishing wave function at r12 = 0, the Coulomb divergence must be compensated by a similar divergence in the
Laplacian.22,23 This is only possible if the wave function has
a cusp.
ជ , rជ兲 is to
On the other hand, the nonsmooth function ⌿0共R
be expanded in the HO eigenfunctions, e.g., Fock-Darwin
orbitals. 共Recall that the particular representation for the HO
eigenfunctions is immaterial and also whether we use laboជ and rជ
ratory coordinates rជ1,2 or center-of-mass coordinates R
since the coordinate change is orthogonal.兲 For m = 0, we
have
FD
共r兲 =
⌽n,0
冑
2
2
Ln共r2兲e−r /2 ,
␲
⬁
⌿0共rជ兲 =
FD
⌽0,0
共R兲u0,0共r兲
=
FD
⌽0,0
共R兲
FD
cn⌽n,0
共r兲,
兺
n=0
共2.26兲
FD
The functions ⌽n,0
共r兲 are very smooth, as is seen by noting
2
that Ln共r 兲 = Ln共x2 + y 2兲 is a polynomial in x and y, while
u0,0共r兲 = u0,0共冑x2 + y 2兲, so Eq. 共2.26兲 is basically approximating a square root with a polynomial.
Consider then a truncated expansion ⌿0,R 苸 PR共R2兲, such
as the one obtained with the CI or coupled cluster method.24
In general, this is different from PR⌿0, which is the best
approximation of the wave function in PR共R2兲. In any case,
this expansion, consisting of the R + 1 terms such as those of
Eq. 共2.26兲 is a very smooth function. Therefore, the cusp at
r = 0 cannot be well approximated.
In Sec. III C, we will show that the smoothness properties
of the wave function ⌿ is equivalent to a certain decay rate
of the coefficients cn in Eq. 共2.26兲 as n → ⬁. In this case, we
will show that
⬁
兺 nk兩cn兩2 ⬍ + ⬁
n=0
so that
Here, k is the number of times ⌿ may be differentiated
weakly, i.e., ⌿ 苸 Hk共R2兲, and ⑀ 苸 关0 , 1兲 is a constant. For the
function ⌿0 we have k = 1. This kind of estimate directly tells
us that an approximation using only a few HO eigenfunctions necessarily will give an error depending directly on the
smoothness k.
We comment that for higher 兩m兩 the eigenstates will still
have cusps, albeit in the higher derivatives.22 Indeed, we
have weak derivatives of order 兩m兩 + 1, as can easily be deduced by operating on ␺0,m with ⳵x and ⳵y. Moreover, recall
that 兩m兩 = 1 is the S = 1 ground state, which then will have
coefficients decaying faster than the S = 0 ground state. Moreover, there will be excited states, i.e., states with 兩m兩 ⬎ 1, that
also have more quickly decaying coefficients 兩cn兩. This will
be demonstrated numerically in Sec. V.
In fact, Hoffmann-Ostenhof et al.22 showed that near r12
= 0, for arbitrary ␭ any local solution ⌿ of 共H − E兲⌿ = 0 has
the form
⌿共␰兲 = 储␰储m P
共2.27兲
冉 冊
␰
共1 + a储␰储兲 + O共储␰储m+1兲,
储␰储
共2.29兲
where ␰ = 共rជ1 , r2兲 苸 R4, and where P, deg共P兲 = m, is a hyperspherical harmonic 共on S3兲, and where a is a constant. This
also generalizes to arbitrary N, cf. Sec. III D. From this representation, it is manifest, that ⌿ 苸 Hm+1共R4兲, i.e., ⌿ has
weak derivatives of order m + 1. We discuss these results further in Sec. III D.
III. APPROXIMATION PROPERTIES OF HERMITE
SERIES
共2.25兲
using the fact that these are independent of ␪. Thus,
共2.28兲
A. Hermite functions in one dimension
In this section, we consider some formal mathematical
propositions whose proofs are given in the Appendix, and
discuss their importance for expansions in HO basis functions.
The first proposition considers the one-dimensional case,
and the second considers general multidimensional expansions. The treatment for one-dimensional Hermite functions
is similar, but not equivalent to, that given by Boyd25 and
Hille.26
We stress that the results are valid for any given wave
function—not only eigenfunctions of quantum dot
Hamiltonians—assuming only that the wave function decays
exponentially as 兩x兩 → ⬁. In the Appendix, more general conditions are also considered.
The results are stated in terms of weak differentiability of
the wave function, which is a generalization of the classical
notion of a derivative. The space Hk共R兲 傺 L2共R兲 is roughly
defined as the 共square-integrable兲 functions ␺共x兲 having k
共square-integrable兲 derivatives ⳵m
x ␺共x兲 and 0 ⱕ m ⱕ k. Correspondingly, the space Hk共Rn兲 傺 L2共Rn兲 consists of the functions whose partial derivatives of total order ⱕk are square
integrable. For wave functions of electronic systems, it turns
out that k times continuous differentiability implies k + 1
times weak differentiability.22 The order k of differentiability
is not always known, but an upper or lower bound can often
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PHYSICAL REVIEW B 80, 045321 共2009兲
HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
be found through analysis. It is however important, that the
Coulomb singularity implies that k is finite.
For the one-dimensional case, we have the following
proposition:
Proposition 1 共Approximation in one dimension兲.
Let k ⱖ 0 be a given integer. Let ␺ 苸 L2共R兲 be exponentially decaying as 兩x兩 → ⬁ and given by
⬁
␺共x兲 = 兺 cn␾n共x兲,
⬁
rk p共r兲⬍ + ⬁.
兺␣ 兩␣兩k兩c␣兩2 = 兺
r=0
Again, we notice that the latter implies that
p共r兲 = o共r−共k+1兲兲.
共3.1兲
⬁
nk兩cn兩2 ⬍ ⬁.
兺
n=0
兩cn兩 = o共n−共k+1兲/2兲,
共3.3兲
which shows that the more ␺共x兲 can be differentiated, the
faster the coefficients will fall off as n → ⬁. Moreover, let
R
␺R = PR␺ = 兺n=0
cn␾n. Then
冉兺 冊
⬁
储 ␺ − ␺ R储 =
兩cn兩
2
1/2
,
共3.4兲
n=R+1
which gives an estimate of how well a finite basis of Hermite
functions will approximate ␺共x兲 in the norm. We already
notice that for low k = 2, which is typical, the coefficients fall
off as o共n−3/2兲, which is rather slowly.
In the general n-dimensional case, the wave function ␺
苸 L2共Rn兲 has an expansion in the n-dimensional Hermite
functions ⌽␣共x兲 and ␣ 苸 In given by
␺共x兲 = 兺 c␣⌽␣共x兲 =
␣
兺
␣1¯␣n
c␣1¯␣n␾␣1共x1兲 ¯ ␾␣n共xn兲.
In order to obtain useful estimates on the error, we need to
define the shell weight p共R兲 by the overlap of ␺共x兲 with the
single shell SR, i.e.,
p共R兲 = 储P共SR兲␺储2 =
兺
␣,兩␣兩=R
兩c␣兩2 ,
共3.5兲
where P共SR兲 is the projection onto the shell. Thus,
⬁
储␺储2 =
兺 p共R兲.
共3.6兲
冉兺 冊
⬁
储共1 − P兲␺储 =
p共r兲
1/2
.
共3.10兲
r=R+1
In applications, we often observe a decay of nonintegral
order; i.e., there exists an ⑀ 苸 关0 , 1兲 such that we observe
共3.2兲
We notice that the latter implies that
共3.9兲
Moreover, for the shell-truncated Hilbert space PR, the approximation error is given by
n=0
where ␾n共x兲 is given by Eq. 共2.5兲. Then ␺ 苸 Hk共R兲 if and
only if
共3.8兲
p共r兲 = o共r−共k+1+⑀兲兲.
共3.11兲
This does not, of course, contradict the results. To see this,
we observe that if ␺ 苸 Hk共Rn兲 but ␺ 苸 Hk+1共Rn兲, then p共r兲
must decay at least as fast as o共r−共k+1兲兲 but not as fast as
o共r−k+2兲. Thus, the actual decay exponent can be anything
inside the interval 关k + 1 , k + 2兲.
Consider also the case where ␺ 苸 Hk共Rn兲 for every k, i.e.,
we can differentiate it 共weakly兲 as many times we like. Then
p共r兲 decays faster than r−共k+1兲, for any k ⱖ 0, giving so-called
exponential convergence of the Hermite series. Hence, functions that are best approximated by Hermite series are rapidly decaying and very smooth functions ␺. This would be
the case for the quantum dot eigenfunctions if the interparticle interactions were nonsingular.
B. Many-body wave functions
We now discuss N-body eigenfunctions of the HO in d
dimensions, including spin, showing that we may identify
the expansion of such with 2N expansions in Hermite functions in n = Nd dimensions, i.e., 2N expansions in HO eigenfunctions of imagined spinless particles in n = Nd dimensions. Each expansion corresponds to a different spin
configuration.
Each particle k = 1 , . . . , N has both spatial degrees of freedom rជk 苸 Rd and a spin coordinate ␶k 苸 兵⫾1其, corresponding
to the z-projection Sz = ⫾ ប2 of the electron spin. The configuration space can thus be taken as two copies X of Rd; one for
each spin value, i.e., X = Rd ⫻ 兵⫾1其 and xk = 共rជk , ␶k兲 苸 X are
the coordinates of particle k.
For a single particle with spin, the Hilbert space is now
L2共X兲, with basis functions given by
R=0
⌽̂i共x兲 = ⌽␣共rជ兲␹␴共␶兲,
For the one-dimensional case, we of course have p共R兲
= 兩cR兩2.
Proposition 2 共Approximation in n dimensions兲.
Let ␺ 苸 L2共Rn兲 be exponentially decaying as 储x储 → ⬁ and
given by
where i = i共␣ , ␴兲 is a new generic index and where ␹␴ is a
basis function for the spinor space C2.
Ignoring the Pauli principle for the moment, the N-body
Hilbert space is now given by
␺共x兲 = 兺 c␣⌽␣共x兲.
␣
Then ␺ 苸 Hk共Rn兲 if and only if
共3.7兲
H共N兲 = L2共X兲N ⬅ L2共RNd兲 丢 共C2兲N ,
共3.12兲
共3.13兲
i.e., each wave function ␺ 苸 H共N兲 is equivalent to 2N spincomponent functions ␺共␴兲 苸 L2共RNd兲 and ␴ = 共␴1 , . . . , ␴N兲
苸 兵⫾1其N. We have
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SIMEN KVAAL
␺共x1, . . . ,xN兲 = 兺 ␺共␴兲共␰兲␹␴共␶兲,
␴
R
␰ ⬅ 共rជ1, . . . ,rជN兲, 共3.14兲
储PR␺储2 = 兺 p共r兲.
共3.21兲
r=0
where ␶ = 共␶1 , . . . , ␶N兲, and where ␹␴共␶兲 = ␦␴,␶ are basis funcN
tions for the N-spinor space C2 , being eigenfunctions for Sz,
i.e., corresponding to a given configuration of the N spins.
The ␴th component function ␺共␴兲共␰兲 苸 L2共RNd兲 is a function of Nd variables, and by considering the Nd-dimensional
HO as the sum of N HOs in d dimensions, it is easy to see
that a basis for the L2共RNd兲 is given by the functions
As should be clear now, studying approximation of Hermite
functions in arbitrary dimensions automatically gives the
corresponding many-body HO approximation properties
since the many-body eigenfunctions can be seen as 2N component functions and since the shell-truncated Hilbert space
transfers to a many-body setting in a natural way.
⌽␤共␰兲 ⬅ ⌽␣1共rជ1兲 ¯ ⌽␣N共rជN兲
C. Two electrons revisited
共3.15兲
where ␰ = 共rជ1 , . . . , rជN兲, and where ␤ = 共␣1 , . . . , ␣N兲 is an
Nd-component multi-index. Correspondingly, a basis for the
complete space H共N兲 = L2共X兲N is given by the functions
⌽̂i1¯iN共␰, ␶兲 ⬅ ⌽␣1共rជ1兲 ¯ ⌽␣N共rជN兲␹␴共␶兲,
␺ 共␴兲共 ␰ 兲 = 兺
␤
=
兺
␣1¯␣N
共␴兲
c␣1¯␣N⌽␣1共rជ1兲
¯ ⌽␣N共rជN兲,
and we define the ␴th shell weight p共␴兲共r兲 as before, i.e.,
p共␴兲共r兲 ⬅
兺 兩c␤共␴兲兩2 .
兺
i1¯iN
具⌽̂i1¯iN, ␺典␦兩␤兩,r = 兺 p共␴兲共r兲
␴
再
PAS,R = span ⌿i1,. . .,iN:i1 = i ⬍ ¯ ⬍ iN,
兺k 兩␣k兩 ⱕ R
冎
,
共3.19兲
which is precisely the computational basis used in many CI
calculations. 共See however also the discussion in Sec. IV.兲
We stress that PAS,R is independent of the actual one-body
HO eigenfunctions used. The shell weight of ␺ 苸 HAS共X兲 is
now given by
p共r兲 =
兺
i1¯iN
and
2
2
⳵xj⳵k−j
y ␺ 苸 L 共R 兲,
具⌿i1¯iN, ␺典␦兩␤共i1¯iN兲兩,r ,
共3.20兲
0 ⱕ j ⱕ k,
共3.23兲
where x = r cos共␪兲 and y = r sin共␪兲. Then, by Lemma 4 in the
Appendix, 共a†x 兲 j共a†y 兲k−j␺ 苸 L2共R2兲 for 0 ⱕ j ⱕ k as well.
The function ␺共r , ␪兲 was expanded in Fock-Darwin orbitals, viz.,
⬁
FD
␺共r, ␪兲 = 兺 cn⌽n,m
共r, ␪兲.
共3.24兲
n=0
FD
Recall, that the shell number N for ⌽n,m
was given by
N = 2n + 兩m兩. Thus, the shell weight p共N兲 is in this case simply
共3.18兲
since the shell number 兩␤兩 = 兺兩␣k兩 does not depend on the
spin configuration of the basis function.
Including the Pauli principle to accommodate proper
wave-function symmetry does not change these considerations. The basis functions ⌽̂i1¯iN are antisymmetrized to
become Slater determinants ⌿i1¯iN 共see, for example, Ref.
27 for details兲, which is equivalent to consider the projection
HAS共N兲 = PASH共N兲 of the unsymmetrized space onto the antisymmetric subspace. Moreover, the projections PR and PAS
commute so that the shell-truncated space is given by
共3.22兲
where f共r兲 decayed exponentially fast as r → ⬁. Assume now
that ␺ 苸 Hk共R2兲, i.e., that all partial derivatives of ␺ of order
k exists in the weak sense, viz.,
共3.17兲
兩␤兩=r
We may then apply the analysis from Sec. III A to each of
the spin-component functions and note that the total shell
weight is
p共r兲 ⬅
␺共r, ␪兲 = eim␪ f共r兲,
共3.16兲
where ik = i共␣k , ␴k兲. Notice, that the HO energy and hence the
shell number 兩␤兩 only depends on ␤ = 共␣1 , . . . , ␣N兲.
The functions ␺共␴兲 may be expanded in the functions ⌽␤,
i.e.,
c␤共␴兲⌽␤共␰兲
We return to the exact solutions of the two-electron quantum dot considered in Sec. II C. Recall, that the wave functions were on the form
p共N兲 = 兩c共N−兩m兩兲/2兩2,
N ⱖ 兩m兩,
共3.25兲
and p共N兲 = 0 otherwise. From Proposition 2, we have
⬁
兺
Nk p共N兲 ⬍ + ⬁,
共3.26兲
N=兩m兩
which yields
兩cn兩 = o共n−共k+1+⑀兲/2兲,
0 ⱕ ⑀ ⬍ 1,
共3.27兲
as claimed in Sec. II C.
D. Smoothness properties of many-electron wave functions
Let us mention some results, mainly due to HoffmannOstenhof et al.,22,28 concerning smoothness of many-electron
wave functions. Strictly speaking, their results are valid only
in d = 3 spatial dimensions since the Coulomb interaction in
d = 2 dimensions fails to be a Kato potential, the definition of
which is quite subtle and out of the scope for this article.28
On the other hand, it is reasonable to assume that the results
will still hold true since the analytical results of the N = 2
case is very similar in the d = 2 and d = 3 cases: the eigenfunctions decay exponentially with the same cusp singularities at the origin.19,20
Consider the Schrödinger equation 共H − E兲␺共␰兲 = 0, where
␰ = 共␰1 , ¯ , ␰Nd兲 = 共rជ1 , ¯ , rជN兲 苸 RNd and where ␺共␰兲 is only
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HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
assumed to be a solution locally. 共A proper solution is of
course also a local solution.兲 Recall that ␺ has 2N spin components ␺共␴兲. Define a coalesce point ␰CP as a point where at
least two particles coincide, i.e., rជ j = rជᐉ and j ⫽ ᐉ. Away from
the set of such points, ␺共␴兲共␰兲 is real analytic since the interaction is real analytic there. Near a ␰CP, the wave function
has the form
␺共␴兲共␰ + ␰CP兲 = rk P共␰/r兲共1 + ar兲 + O共rk+1兲,
共3.28兲
where r = 储␰储, P is a hyperspherical harmonic 共on the sphere
SNd−1兲 of degree k = k共␰CP兲, and where a is a constant. It is
immediately clear that ␺共␴兲共␰兲 is k + 1 times weakly differentiable in a neighborhood of ␰CP. However, at K-electron coalesce points, i.e., at points ␰CP where K different electrons
coincide, the integer k may differ. Using exponential decay
of a proper eigenfunction, we have ␺共␴兲 苸 Hmin共k兲+1共RNd兲.
Hoffmann-Ostenhof et al.22,28 also showed that symmetry restrictions on the spin components due to the Pauli principle
induces an increasing degree k of the hyperspherical harmonic P, generating even higher order of smoothness. A general feature is that the smoothness increases with the number
of particles.
However, their results in this direction are not general
enough to ascertain the minimum of the values for k for a
given wave function, although we feel rather sure that such
an analysis is possible. Suffice it to say that the results are
clearly visible in the numerical calculations in Sec. V.
Another interesting direction of research has been undertaken by Yserentant,29 who showed that there are some very
high order mixed partial derivatives at coalesce points. It
seems unclear, though, if this can be exploited to improve the
CI calculations further.
instead of the global shell number 共or energy兲 is also
common.6,8 For obvious reasons, PR共N兲 is often referred to
as an “energy cut space,” while MR共N兲 is referred to as a
“direct product space.”
As in Sec. III A, PR is the orthogonal projector onto the
model space PR共N兲. We also define QR = 1 − PR as the projector onto the excluded space PR共N兲⬜. The discrete eigenvalue
problem is then
共PRHPR兲␺h,k = Eh,k␺h,k,
The CI method becomes, in principle, exact as R → ⬁.
Indeed, a widely used name for the CI method is “exact
diagonalization,” being somewhat a misnomer as only a very
limited number of degrees of freedom per particle are
achievable.
It is clear that
PR共N兲 傺 MR共N兲 傺 PNR共N兲
Eh − E ⱕ 关1 + ␯共R兲兴共1 + K␭兲具␺,QRT␺典,
where K is a constant, and where ␯共R兲 → 0 as R → ⬁. Using
T⌽␤ = 共Nd / 2 + 兩␤兩兲⌽␤ and Eq. 共3.8兲, we obtain
⬁
The basic problem is to determine a few eigenvalues and
eigenfunctions of the Hamiltonian H in Eq. 共2.15兲, i.e.,
The CI method consists of approximating eigenvalues of H
with those obtained by projecting the problem onto a finitedimensional subspace Hh 傺 H共N兲. As such, it is an example
of the Ritz-Galerkin variational method.30,31 We comment
that the convergence of the Ritz-Galerkin method is not simply a consequence of the completeness of the basis
functions.30 We will analyze the CI method when the model
space is given by
再
冎
Hh = PRHAS共N兲 = PR共N兲 = span ⌿i1,. . .,iN: 兺 兩␣k兩 ⱕ R ,
k
k
冉 冊
Nd
+ r p共r兲.
2
共4.6兲
rk p共r兲 ⬍ + ⬁
兺
r=0
共4.7兲
implying that rp共r兲 = o共r−k兲. We then obtain, for k ⬎ 1,
具␺,共1 − PR兲T␺典 = o共R−共k−1兲兲 + o共R−k兲.
共4.8兲
For k = 1 共which is the worst case兲, we merely obtain convergence, 具␺ , 共1 − PR兲T␺典 → 0 as R → ⬁. We assume, that R is
sufficiently large, so that the o共R−k兲 term can be neglected.
Again, we may observe a slight deviation from the decay,
and we expect to observe eigenvalue errors on the form
Eh − E ⬃ 共1 + K␭兲R−共k−1+⑀兲 ,
used in Refs. 10 and 32, for example, although other spaces
also are common. 共We drop the subscript “AS” from now
on.兲 The space
MR共N兲 ª span兵⌿i1,. . .,iN:max兩␣k兩 ⱕ R其,
兺
r=R+1
⬁
共4.1兲
k = 1, . . . ,kmax .
共4.5兲
Assume now, that ␺共␴兲 苸 Hk共RNd兲 for all ␴ so that according
to proposition 2, we will have
A. Convergence analysis using HO eigenfunction basis
H ␺ k = E k␺ k,
共4.4兲
so that studying the convergence in terms of PR共N兲 is sufficient. In our numerical experiments we therefore focus on
the energy cut model space. A comparison between the convergence of the two spaces is, on the other hand, an interesting topic for future research.
Using the results in Refs. 30 and 33 for nondegenerate
eigenvalues for simplicity, we obtain an estimate for the error
in the numerical eigenvalue Eh as
具␺,QRT␺典 =
IV. CONFIGURATION-INTERACTION METHOD
共4.3兲
k = 1, . . . ,kmax .
where 0 ⱕ ⑀ ⬍ 1.
As for the eigenvector error 储␺h − ␺储 共recall that ␺h
⫽ PR␺兲, we mention that
储␺h − ␺储 ⱕ 关1 + ␩共R兲兴关共1 + K␭兲具␺,共1 − PR兲T␺典兴1/2 ,
共4.2兲
i.e., a cut in the single-particle shell numbers 共or energy兲
共4.9兲
共4.10兲
where ␩共R兲 → 0 as R → ⬁.
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PHYSICAL REVIEW B 80, 045321 共2009兲
SIMEN KVAAL
The effective two-body interaction Ceff共i , j兲 is now given
B. Effective interaction scheme
Effective interactions have a long tradition in nuclear
physics, where the bare nuclear interaction is basically unknown and highly singular and where it must be renormalized and fitted to experimental data.3 In quantum chemistry
and atomic physics, the Coulomb interaction is of course
well known so there is no intrinsic need to formulate an
effective interaction. However, in lieu of the in general low
order of convergence implied by Eq. 共4.9兲, we believe that
HO-based calculations such as the CI method in general may
benefit from the use of effective interactions.
A complete account of the effective interaction scheme
outlined here is out of scope for the present article, but we
refer to Refs. 2, 11, and 34–36 for details as well as numerical algorithms.
Recall, that the interaction is given by
N
N
i⬍j
i⬍j
U = 兺 C共i, j兲 = 兺
␭
,
储rជi − rជ j储
共4.11兲
a sum of fundamental two-body interactions. For the N = 2
problem we have in principle the exact solution since Hamiltonian 共2.19兲 can be reduced to a one-dimensional radial
equation, e.g., the eigenproblem of Hr defined in Eq. 共2.21兲.
This equation may be solved to arbitrarily high precision
using various methods, for example using a basis expansion
in generalized half-range Hermite functions.37 In nuclear
physics, a common approach is to take the best two-body CI
calculations available, where R = O共103兲, as “exact” for this
purpose.
We now define the effective Hamiltonian for N = 2 as a
Hermitian operator Heff defined only within PR共N = 2兲 that
gives K = dim关PR共N = 2兲兴 exact eigenvalues Ek of H, and K
approximate eigenvectors ␺eff,k. Of course, there are infinitely many choices for the K eigenpairs, but by treating U
= ␭ / r12 as a perturbation, and “following” the unperturbed
HO eigenpairs 共␭ = 0兲 through increasing values of ␭, one
makes the eigenvalues unique.2,38 The approximate eigenvectors ␺eff,k 苸 PR共N = 2兲 are chosen by minimizing the distance to the exact eigenvectors ␺k 苸 H共N = 2兲 while retaining
orthonormality.35 This uniquely defines Heff for the two-body
system. In terms of matrices, we have
Heff = Ũdiag共E1, . . . ,EK兲Ũ† ,
共4.12兲
where X and Y are unitary matrices defined as follows. Let U
be the K ⫻ K matrix whose kth column is the coefficients of
PR␺k. Then the singular value decomposition of U can be
written as
U = X ⌺ Y†,
共4.13兲
where 兺 is diagonal. Then,
Ũ ª XY † .
共4.14兲
The columns of Ũ are the projections PR␺k “straightened
out” to an orthonormal set. Equation 共4.12兲 is simply the
spectral decomposition of Heff. Although different in form
than most implementations in the literature 共e.g., Ref. 11兲, it
is equivalent.
by
Ceff共1,2兲 ª Heff − PRTPR ,
共4.15兲
which is defined only within PR共N = 2兲.
The N-body effective Hamiltonian is defined by
N
Heff ª PRTPR + 兺 Ceff共i, j兲,
共4.16兲
i⬍j
where PR projects onto PR共N兲, and thus Heff is defined only
within PR共N兲. The diagonalization of Heff共N兲 is equivalent to
a perturbation technique where a certain class of diagrams is
summed to infinite order in the full problem.34 In implementations, Eqs. 共4.12兲 and 共4.16兲 are treated in COM coordinates, utilizing block diagonality of both H and Heff; see Ref.
36 for details.
We comment that unlike the bare Coulomb interaction,
the effective two-body interaction Ceff corresponds to a nonlocal potential due to the “straightening out” of truncated
eigenvectors. Rigorous mathematical treatment of the convergence properties of the effective interaction is, to the author’s knowledge, not available. Effective interactions have,
however, enjoyed great success in the nuclear physics community, and we strongly believe that we soon will see sufficient proof of the improved accuracy with this method. Indeed, in Sec. V we see clear evidence of the accuracy boost
when using an effective interaction.
V. NUMERICAL RESULTS
A. Code description
We now present numerical results using the full
configuration-interaction method for N = 2 – 5 electrons in d
= 2 dimensions. We will use both the “bare” Hamiltonian
H = T + U and effective Hamiltonian 共4.16兲.
Since the Hamiltonian commutes with angular momentum
Lz, the latter taking on eigenvalues M 苸 Z, the Hamiltonian
matrix is block diagonal. 共Recall that the Fock-Darwin orbitFD
are eigenstates of Lz with eigenvalue m, so each
als ⌽n,m
N
mk.兲 Moreover,
Slater determinant has eigenvalue M = 兺k=1
the calculations are done in a basis of joint eigenfunctions
for total electron spin S2 and its projection Sz as opposed to
the Slater determinant basis used for convergence analysis.
Such basis functions are simply linear combinations of Slater
determinants within the same shell and further reduce the
dimensionality of the Hamiltonian matrix.8 The eigenfunctions of H are thus labeled with the total spin S = 0 , 1 , . . . , N2
for even N and S = 21 , 23 , . . . , N2 for odd N, as well as the total
angular momentum M = 0 , 1 , . . .. 共−M produce the same eigenvalues as M, by symmetry.兲 We thus split PR共N兲 共or
MR共N兲兲 into invariant subspaces PR共N , M , S兲共MR共N , M , S兲兲
and perform computations solely within these.
The calculations were carried out with a code similar to
that described by Rontani et al. in Ref. 8. Table I shows
comparisons of the present code with that of Table IV of Ref.
8 for various parameters using the model space
MR共N , M , S兲. Table I also shows the case ␭ = 1, N = 2, M
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HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
TABLE I. Comparison of current code and Ref. 8. Figures from the latter have varying number of
significant digits. We include more digits from our own computation for reference.
R=5
R=6
N
␭
M
2S
Current
Ref. 8
Current
2
1
2
0
0
1
1
1
0
0
2
0
0
0
0
2
1
1
3
0
4
5
5
3.013626
3.733598
4.143592
8.175035
11.04480
11.05428
23.68944
23.86769
21.15093
29.43528
3.7338
4.1437
8.1755
11.046
11.055
23.691
23.870
21.15
29.44
3.011020
3.731057
4.142946
8.169913
11.04338
11.05325
23.65559
23.80796
21.13414
29.30898
2
4
4
6
5
2
4
= 0, and S = 0, whose exact lowest eigenvalue is E0 = 3; cf.
Sec. II C. We note that there are some discrepancies between
the results in the last digits of the results of Ref. 8. The
spaces MR共N兲 were identical in the two approaches; i.e., the
number of basis functions and the number of nonzero matrix
elements produced are crosschecked and identical.
We have checked that the code also reproduces the results
of Refs. 9, 10, and 39, using the PR共N , M , S兲 spaces. Our
code is described in detail elsewhere36 where it is also demonstrated that it reproduces the eigenvalues of an analytically
solvable N-particle system40 to machine precision.
B. Experiments
For the remainder, we only use the energy cut spaces
PR共N , M , S兲. Figure 2 shows the development of the lowest
eigenvalue E0 = E0共N , M , S兲 for N = 4, M = 0 , 1 , 2 and S = 0 as
function of the shell truncation parameter R, using both
Hamiltonians H and Heff. Apparently, the effective interaction eigenvalues provide estimates for the ground-state eigenvalues that are better than the bare interaction eigenvalues. This effect is attenuated with higher N, due to the fact
that the two-electron effective Coulomb interaction does not
take into account three- and many-body effects which become substantial for higher N.
We take the Heff eigenvalues as “exact” and graph the
relative error in E0共N , M , S兲 as function of R on a logarithmic scale in Fig. 3, in anticipation of the relation
ln共Eh − E兲 ⬇ C + ␣ ln R,
␣ = − 共k − 1 + ⑀兲.
3.7312
4.1431
21.13
29.31
Current
Ref. 8
3.009236
3.729324
4.142581
8.166708
11.04254
11.05262
23.64832
23.80373
21.12992
29.30251
3.7295
4.1427
8.1671
11.043
11.053
23.650
23.805
21.13
29.30
14.6
14.5
14.4
14.3
14.2
共5.1兲
The graphs show straight lines for large R, while for small R
there is a transient region of nonstraight lines. For N = 5,
however, ␭ = 2 is too large a value to reach the linear regime
for the range of R available, so in this case we chose to plot
the corresponding error for the very small value ␭ = 0.2,
showing clear straight lines in the error. The slopes are more
or less independent of ␭, as observed in different calculations.
In Fig. 4 we show the corresponding graphs when using
the effective Hamiltonian Heff. We estimate the relative error
Ref. 8
as before, leading to artifacts for the largest values of R due
to the fact that there is a finite error in the best estimates for
the eigenvalues. However, in all cases there are clear, linear
regions, in which we estimate the slope ␣. In all cases, the
slope can be seen to decrease by at least ⌬␣ ⬇ −1 compared
to Fig. 3, indicating that the effective interaction indeed accelerates the CI convergence by at least an order of magnitude. We also observe, that the relative errors are improved
by an order of magnitude or more for the lowest values of R
shown, indicating the gain in accuracy when using small
model spaces with the effective interaction.
Notice, that for symmetry reasons only even 共odd兲 R for
even 共odd兲 M yields increases in basis size
dim关PR共N , M , S兲兴, so only these values are included in the
plots. To overcome the limitations of the two-body effective
interaction for higher N, an effective three-body interaction
could be considered, and is hotly debated in the nuclear
physics community. 共In nuclear physics, there are also more
complicated three-body effective forces that need to be
included.41兲 However, this will lead to a huge increase in
E0
3
R=7
14.1
14
13.9
13.8
13.7
13.6
6
8
10
12
14
R
16
18
20
22
FIG. 2. Eigenvalues for N = 4, S = 0, and ␭ = 2 as function of R
for H 共solid兲 and Heff 共dashed兲. M = 0,1, and 2 are represented by
squares, circles, and stars, respectively.
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SIMEN KVAAL
Relative error for N=2, λ = 2
1
10
M=0, S=1, α = −1.2772
M=0, S=3, α = −2.1716
M=2, S=1, α = −1.3093
M=2, S=3, α = −2.5417
M=0, S=0, α = −1.0477
0
10
M=0, S=2, α = −2.1244
M=3, S=0, α = −3.1749
−1
10
−2
10
M=3, S=2, α = −4.0188
Relative error
Relative error
Relative error for N=3, λ=2
−1
10
−2
10
−3
10
−3
10
−4
10
−4
10
−5
10
−6
10
−5
6
(a)
8
10
12
14
16
18
20
R
10
Relative error for N=4, λ = 2
−1
6
(b)
8
12
14
16
18
20
R
Relative error for N=5, λ = 0.2
−2
10
10
10
M=0, S=0, α = −1.4233
M=0, S=4, α = −2.8023
M=3, S=2, α = −1.5327
M=3, S=4, α = −3.2109
−2
10
M=0, S=1, α = −1.5150
M=0, S=5, α = −3.6563
M=3, S=1, α = −1.8159
M=3, S=5, α = −4.2117
−3
Relative error
Relative error
10
−3
10
−4
10
−4
10
(c)
6
8
10
12
14
16
18
20
R
(d)
6
8
10
12
14
16
18
20
R
FIG. 3. Plot of relative error using the bare interaction for various N, M, and S. Clear o共R␣兲 dependence in all cases.
memory consumption due to extra nonzero matrix elements.
At the moment, there are no methods available that can generate the exact three-body effective interaction with sufficient precision.
We stress that the relative error decreases very slowly in
general. It is a common misconception, that if a number of
digits of E0共N , M , S兲 is unchanged between R and R + 2, then
these digits have converged. This is not the case, as is easily
seen from Fig. 3. Take for instance N = 4, M = 0 and S = 0, and
␭ = 2. For R = 14 and R = 16 we have E0 = 13.844 91 and E0
= 13.841 53, respectively, which would give a relative error
estimate of 2.4⫻ 10−4, while the correct relative error is
1.3⫻ 10−3.
The slopes in Fig. 3 vary greatly, showing that the eigenfunctions indeed have varying global smoothness, as predicted in Sec. III D. For 共N , M , S兲 = 共5 , 3 , 5 / 2兲, for example,
␣ ⬇ −4.2, indicating that ␺ 苸 H5共R10兲. It seems, that higher S
gives higher k, as a rule of thumb. Intuitively, this is because
the Pauli principle forces the wave function to be zero at
coalesce points, thereby generating smoothness.
VI. DISCUSSION AND CONCLUSION
We have studied approximation properties of Hermite
functions and harmonic oscillator eigenfunctions. This in
turn allowed for a detailed convergence analysis of numerical methods such as the CI method for the parabolic quantum
dot. Our main conclusion is, that for wave functions ␺
苸 Hk共Rn兲 falling off exponentially as 储x储 → ⬁, the shellweight function p共r兲 decays as p共r兲 = o共r−k−1兲. Applying this
to the convergence theory of the Ritz-Galerkin method, we
obtained estimate 共4.5兲 for the error in the eigenvalues. A
complete characterization of the upper bound on the differentiability k, i.e., in ␺共s兲 苸 Hk, as well as a study of the constant K in Eq. 共4.9兲, would complete our knowledge of the
convergence of the CI calculations.
We also demonstrated numerically that the use of a twobody effective interaction accelerates the convergence by at
least an order of magnitude, which shows that such a method
should be used whenever possible. On the other hand, a rigorous mathematical study of the method is yet to come.
Moreover, we have not investigated to what extent the in-
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HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
Relative error for N=2, λ = 2
Relative error for N=3, λ = 2
1
10
M=0, S=1, U , α = −4.1735
M=0, S=0, Ueff
0
10
eff
M=0, S=3, U , α = −4.7914
0
M=0, S=2, U
10
eff
eff
M=3, S=1, U , α = −4.4535
M=3, S=0, U
eff
eff
10
−5
10
Relative error
Relative error
M=3, S=3, U , α = −4.7521
−1
M=3, S=2, Ueff
eff
−2
10
−3
10
−4
10
−10
10
−5
10
−6
10
6
(a)
8
10
12
14
16
18
20
R
Relative error for N=4, λ = 2
1
10
6
(b)
8
10
10
18
eff
M=0, S=5, Ueff, α = −4.4925
M=3, S=0, U , α = −4.7816
M=3, S=1, Ueff, α = −3.6192
eff
eff
−1
20
eff
M=0, S=4, U , α = −5.3978
M=3, S=4, U , α = −6.2366
M=3, S=5, U , α = −5.0544
eff
−3
10
Relative error
10
Relative error
16
M=0, S=1, U , α = −3.8123
eff
0
14
Relative error for N=5, λ = 0.2
−2
10
M=0, S=0, U , α = −4.6093
12
R
−2
10
−3
10
−4
10
−4
10
−5
10
−5
10
(c)
6
8
10
12
14
16
18
20
R
(d)
7
9
11
R
13
15
17 18
FIG. 4. Plot of relative error using an effective interaction for various N, M, and S. Clear o共R␣兲 dependence in all cases, but notice
artifacts when R is large, due to errors in most correct eigenvalues. The R = 5 case does not contain enough data to compute the slopes with
sufficient accuracy.
crease in convergence is independent of the interaction
strength ␭. This, together with a study of the accuracy of the
eigenvectors, is an obvious candidate for further investigation.
The theory and ideas presented in this article should in
principle be universally applicable. In fact, Figs. 1–3 of Ref.
11 clearly indicate this, where the eigenvalues of 3He as
function of model space size are graphed both for bare and
effective interactions, showing some of the features we have
discussed.
Other interesting future studies would be a direct comparison of the direct product model space MR共N兲 and our
energy cut model space PR共N兲. Both techniques are common, but may have different numerical characteristics. Indeed, dim关MR共N兲兴 grows much quicker than dim关PR共N兲兴,
while we are uncertain of whether the increased basis size
yields a corresponding increased accuracy.
We have focused on the parabolic quantum dot first because it requires relatively small matrices to be stored, due to
conservation of angular momentum, but also because it is a
widely studied model. Our analysis is, however, general, and
applicable to other systems as well, e.g., quantum dots
trapped in double wells, finite wells, and so on. Indeed, by
adding a one-body potential V to the Hamiltonian H = T + U
we may model other geometries, as well as adding external
fields.42–44
ACKNOWLEDGMENTS
The author wishes to thank M. Hjorth-Jensen 共CMA兲 for
useful discussions, suggestions, and feedback and also G. M.
Coclite 共University of Bari, Italy兲 for mathematical discussions. This work was financed by CMA through the Norwegian Research Council.
APPENDIX: MATHEMATICAL DETAILS
1. Multi-indices
A very handy tool for compact and unified notation, when
the dimension n of the underlying measure space Rn is a
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PHYSICAL REVIEW B 80, 045321 共2009兲
SIMEN KVAAL
parameter, are multi-indices. The set In of multi-indices are
defined as n tuples of non-negative indices, viz, ␣
= 共␣1 , . . . , ␣n兲, where ␣k ⱖ 0.
We define several useful operations on multi-indices as
follows. Let u be a formal vector of n symbols. Moreover, let
␾共␰兲 = ␾共␰1 , ␰2 , . . . , ␰n兲 be a function. Then, define
兩␣兩 ⬅ ␣1 + ␣2 + ¯ + ␣n ,
共A1兲
␣ ! ⬅ ␣1 ! ␣2 ! ¯ ␣n ! ,
共A2兲
␣ ⫾ ␤ ⬅ 共␣1 ⫾ ␤1, . . . , ␣n ⫾ ␤n兲,
共A3兲
u␣ ⬅ u1␣1u2␣2, . . . ,un␣n ,
共A4兲
⳵ ␣␾ 共 ␰ 兲 ⬅
⳵ ␣1 ⳵ ␣1
⳵␰1␣1 ⳵␰2␣2
¯
⳵ ␣d
⳵␰d␣1
␾共␰兲.
共A5兲
In Eq. 共A3兲, the result may not be a multi-index when we
subtract two indices, but this will not be an issue for us.
Notice, that Eq. 共A5兲 is a mixed partial derivative of order
兩␣兩. Moreover, we say that ␣ ⬍ ␤ if and only if ␣ j ⬍ ␤ j for all
j. We define ␣ = ␤ similarly. We also define “basis indices” e j
by 共e j兲 j⬘ = ␦ j,j⬘. We comment, that we will often use the notation ⳵x ⬅ ⳵⳵x and ⳵k ⬅ ⳵␰⳵ k to simplify notation. Thus,
⳵␣ = 共⳵1␣1, . . . , ⳵n␣n兲,
共A6兲
consistent with Eq. 共A4兲.
2. Weak derivatives and Sobolev spaces
We present a quick summary of weak derivatives and related concepts needed. The material is elementary and superficial but probably unfamiliar to many readers, so we include
it here. Many terms will be left undefined; if needed, the
reader may consult standard texts, e.g., Ref. 45.
The space L2共Rn兲 is defined as
再
L2共Rn兲 ⬅ ␺:Rn → C:
冕
Rn
冎
兩␺共␰兲兩2dn␰ ⬍ + ⬁ ,
冕
Rn
␺1共␰兲ⴱ␺2共␰兲dn␰ ,
冕
Rn
Hk共Rn兲 ⬅ 兵␺ 苸 L2:⳵␣␺ 苸 L2,
冕
Rn
␾共␰兲v共␰兲dn␰ , 共A9兲
∀ ␣ 苸 I n,
兩␣兩 ⱕ k其.
共A10兲
The Sobolev space is also a Hilbert space with the inner
product
共 ␺ 1, ␺ 2兲 ⬅
兺
␣,兩␣兩ⱕk
具⳵␣␺1, ⳵␣␺2典,
共A11兲
and this is the main reason why one obtains a unified theory
of partial differential equations using such spaces.
The space Hk共Rn兲 for n ⬎ 1 is a big space. There are some
exceptionally ill-behaved functions there; for example, there
are functions in Hk that are unbounded on arbitrary small
regions but still differentiable. 共Hermite series for such functions would still converge faster than, e.g., for a function
with a jump discontinuity.兲 For our purposes, it is enough to
realize that the Sobolev spaces offer exactly the notion of
derivative we need in our analysis of the Hermite function
expansions.
3. Proofs of propositions
We will now prove the propositions given in Sec. III and
also discuss these results on mathematical terms. Recall that
the Hermite functions ␾n 苸 L2共R兲, where n 苸 N0 is a nonnegative integer, are defined by
␾n共x兲 = 共2nn ! 冑␲兲−1/2Hn共x兲e−x /2 ,
2
共A12兲
where Hn共x兲 is the usual Hermite polynomial.
A well-known method for finding the eigenfunctions of
HHO in one dimension involves writing
共A8兲
where the asterisk denotes complex conjugation.
The classical derivative is too limited a concept for the
abstract theory of partial differential equations, including the
Schrödinger equation. Let C⬁0 be the set of infinitely differentiable functions, which are nonzero only in a ball of finite
radius. Of course, C⬁0 傺 L2共Rn兲. Let ␺ 苸 L2共Rn兲, and let ␣
共⳵␣␾共␰兲兲␺共␰兲dn␰ = 共− 1兲兩␣兩
then ⳵␣␺ ⬅ v 苸 L2 is said to be a weak derivative, or distributional derivative, of ␺. In this way, the weak derivative is
defined in an average sense, using integration by parts.
The weak derivative is unique 共up to redefinition on a set
of measure zero兲, obeys the product rule, chain rule, etc.
It is easily seen that if ␺ has a classical derivative v
苸 L2共Rn兲 it coincides with the weak derivative. Moreover, if
the classical derivative is defined almost everywhere 共i.e.,
everywhere except for a set of measure zero兲, then ␺ has a
weak derivative.
The Sobolev space Hk共Rn兲 is defined as the subset of
2
L 共Rn兲 given by
共A7兲
where the Lebesgue integral is more general than the Riemann “limit-of-small-boxes” integral. It is important that we
identify two functions ␺ and ␺1 differing only at a set Z
苸 Rn of measure zero. Examples of such sets are points if
n ⱖ 1, curves if n ⱖ 2, and so on, and countable unions of
such. For example, the rationals constitute a set of measure
zero in R. Under this assumption, L2共Rn兲 becomes a Hilbert
space with the inner product
具 ␺ 1, ␺ 2典 ⬅
苸 In be a multi-index. If there exists a v 苸 L2共Rn兲 such that,
for all ␾ 苸 C⬁0 ,
1
HHO = a†a + ,
2
共A13兲
where the ladder operator a is given by
a⬅
1
冑2 共x + ⳵x兲,
with Hermitian adjoint a† given by
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HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
a† ⬅
1
冑2
共x − ⳵x兲.
共A15兲
The name “ladder operator” comes from the important formulas
a␾n共x兲 = 冑n␾n−1共x兲
共A16兲
a†␾n共x兲 = 冑n + 1␾n+1共x兲,
共A17兲
valid for all n. This can easily be proven by using recurrence
relation 共2.7兲. By repeatedly acting on ␾0 with a† we generate every Hermite function, viz.,
␾n共x兲 = n!−1/2共a†兲n␾0共x兲.
共A18兲
As Hermite functions constitute a complete orthonormal
sequence in L2共R兲, any ␺ 苸 L2共R兲 can be written as a series
in Hermite functions, viz.,
⬁
␺共x兲 = 兺 cn␾n共x兲,
共A19兲
n=0
where the coefficients cn are uniquely determined by
cn = 具␾n , ␺典.
An interesting fact is that the Hermite functions are also
eigenfunctions of the Fourier transform with eigenvalues
共−i兲n, as can easily be proven by induction by observing first
that the Fourier transform of ␾0共x兲 is ␾0共k兲 itself, and second
that the Fourier transform of a† is −ia† 共acting on variable k兲.
It follows from completeness of the Hermite functions that
the Fourier transform defines a unitary operator on L2共R兲.
We now make a simple observation, namely, that
共a†兲k␾n共x兲 = Pk共n兲1/2␾n+k共x兲,
共A20兲
where Pk共n兲 = 共n + k兲 ! / n ! ⬎ 0 is a polynomial of degree k for
n ⱖ 0. Moreover Pk共n + 1兲 ⬎ Pk共n兲 and Pk+1共n兲 ⬎ Pk共n兲.
We now prove the following lemma:
Lemma 1 共Hermite series in one dimension兲.
Let ␺ 苸 L2共R兲. Then 共1兲 a†␺ 苸 L2共R兲 if and only if a␺
⬁
苸 L2共R兲 if and only if 兺n=0
n兩cn兩2 ⬍ +⬁, where cn = 具␾n , ␺典; 共2兲
†
2
a ␺ 苸 L 共R兲 if and only if x␺ , ⳵x␺ 苸 L2共R兲; 共3兲 共a†兲k+1␺
苸 L2共R兲 implies 共a†兲k␺ 苸 L2共R兲; 共4兲 共a†兲k␺ 苸 L2共R兲 if and
only if
⬁
兺 nk兩cn兩2 ⬍ + ⬁,
The significance of the condition a†␺ 苸 L2共R兲 is that the
coefficients cn of ␺ must decay faster than for a completely
arbitrary wave function in L2共R兲. Moreover, a†␺ 苸 L2共R兲 is
the same as requiring ⳵x␺ 苸 L2共R兲, and x␺ 苸 L2共R兲. Lemma 1
also generalizes this fact for 共a†兲k␺ 苸 L2共R兲, giving successfully quicker decay of the coefficients. In all cases, the decay
is expressed in an average sense, through a growing weight
function in a sum, as in Eq. 共A21兲. Since the terms in the
sum must converge to zero, this implies a pointwise faster
decay, as stated in Eq. 共A25兲 below.
We comment here that the partial derivatives must be understood in the weak, or distributional, sense: even though ␺
may not be everywhere differentiable in the ordinary sense, it
may have a weak derivative. For example, if the classical
derivative exists everywhere except for a countable set 共and
if it is in L2兲, this is the weak derivative. Moreover, if this
derivative has a jump discontinuity, there are no higher order
weak derivatives.
Loosely speaking, since x␺共x兲 苸 L2 ⇔ ⳵y␺ˆ 共y兲 苸 L2, where
ˆ␺共y兲 is the Fourier transform, point 1 of Lemma 1 is a combined smoothness condition on ␺共x兲 and ␺ˆ 共y兲. Point 共A21兲 is
a generalization to higher derivatives but is difficult to check
in general for an arbitrary ␺. On the other hand, it is well
known that the eigenfunctions of many Hamiltonians of interest, such as quantum dot Hamiltonian 共215兲, decay exponentially fast as 兩x兩 → ⬁. For such exponentially decaying
functions over R1, xk␺ 苸 L2共R兲 for all k ⱖ 0; i.e., ␺ˆ 共y兲 is infinitely differentiable. We then have the following lemma:
Lemma 2 共Exponential decay in 1D兲.
Assume that xk␺ 苸 L2共R兲 for all k ⱖ 0. Then a sufficient
2
m
criterion for 共a†兲m␺ 苸 L2共R兲 is ⳵m
x ␺ 苸 L 共R兲, i.e., ␺ 苸 H 共R兲.
m
⬘
In fact, xk⳵x ␺ 苸 L2 for all m⬘ ⬍ m and all k ⱖ 0.
Proof: we prove the proposition inductively. We note that,
m−1
2
2
since ⳵m
x ␺ 苸 L implies ⳵x ␺ 苸 L , the proposition holds for
m − 1 if it holds for a given m. Moreover, it holds trivially for
m = 1.
Assume then that it holds for a given m, i.e., that ␺
2
苸 Hm implies xk⳵m−j
x ␺ 苸 L for 1 ⱕ j ⱕ m and for all k 关so that,
† m
2
in particular, 共a 兲 ␺ 苸 L 兴. It remains to be proven that ␺
2
† m+1
苸 Hm+1 implies xk⳵m
␺ 苸 L2 by statex ␺ 苸 L since then 共a 兲
ment 5 of Lemma 1. We compute the norm and use integration by parts, viz.,
共A21兲
n=0
2
and 共5兲 共a†兲k␺ 苸 L2共R兲 if and only if x j⳵k−j
x ␺ 苸 L 共R兲 for
0 ⱕ j ⱕ k.
Proof: we have
2
储xk⳵m
x ␺储 =
冕
x2k⳵kx␺ⴱ共x兲⳵kx␺共x兲
R
2k−1 m−1
2k m−1
= − 2k具⳵m
⳵x ␺典 − 具⳵m+1
x ␺,x
x ␺,x ⳵x ␺典 ⬍ + ⬁.
⬁
储a†␺储2 = 兺 共n + 1兲兩cn兩2 = 储␺2储 + 储a␺储2 ,
共A22兲
n=0
from which statement 1 follows. Statement 2 follows from
the definition of a† and that a†␺ 苸 L2 implies a␺ 苸 L2 共since
储a␺储 ⱕ 储a†␺储兲, which again implies x␺ , ⳵x␺ 苸 L2. Statement 3
follows from the monotone behavior of Pk共n兲 as function of
k. Statement 4 then follows. By iterating statement 2 and
䊏
using 关⳵x , x兴 = 1 and statement 3, statement 5 follows.
2
The boundary terms vanish. Therefore, xk⳵m
x ␺ 苸 L for all k,
and the proof is complete.
䊏
The proposition states that for the subset of L2共R兲 consisting of exponentially decaying functions, the approximation
properties of the Hermite functions will only depend on the
smoothness properties of ␺. Moreover, the derivatives up to
the penultimate order decay exponentially as well. 共The
highest order derivative may decay much slower.兲
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From Lemmas 1 and 2 we extract the following important
characterization of the approximating properties of Hermite
functions in d = 1 dimensions:
Proposition 1 共Approximation in one dimension兲.
Let k ⱖ 0 be a given integer. Let ␺ 苸 L2共R兲 be given by
⬁
␺共x兲 = 兺 cn␾n共x兲.
共A23兲
n=0
2
⌽0共x兲 ⬅ ␲−n/4e−储x储 /2 .
By using Eqs. 共A4兲 and 共A18兲, we may generate all other
Hermite functions, viz.,
⌽␣共x兲 ⬅ ␣!−1/2共a†兲␣⌽0共x兲.
共A32兲
Given two multi-indices ␣ and ␤, we define the polynomial P␣共␤兲 by
D
共␣ + ␤兲!
= 兿 P␤ j共␣ j兲,
P ␤共 ␣ 兲 ⬅
␣!
j=1
Then ␺ 苸 H 共R兲 if and only if
k
⬁
兺 nk兩cn兩2 ⬍ ⬁.
共A24兲
兩cn兩 = o共n−共k+1兲/2兲.
共A25兲
n=0
The latter implies that
R
cn␾n. Then
Let ␺R = PR␺ = 兺n=0
冉兺 冊
⬁
储 ␺ − ␺ R储 =
共A31兲
兩cn兩2
1/2
.
共A26兲
n=R+1
This is the central result for Hermite series approximation in
L2共R1兲. Observe that Eq. 共A25兲 implies that the error
储␺ − ␺R储 can easily be estimated. See also proposition 2 and
comments thereafter.
Now a word on pointwise convergence of the Hermite
series. As the Hermite functions are uniformly bounded,25
viz.,
兩␾n共x兲兩 ⱕ 0.816
∀ x 苸 R,
where P is defined for integers as before.
Since the Hermite functions ⌽␣ constitute a basis, any
␺ 苸 L2共Rn兲 can be expanded as
␺共x兲 = 兺 c␣⌽␣共x兲,
⌽␣共x兲 ⬅ ␾␣1共x1兲 ¯ ␾␣n共xn兲.
␣
n
= 兺 c␣ 兿
a
共A35兲
Similarly, by using Eq. 共A16兲 we obtain
a␤␺ = a1␤1 ¯ an␤n 兺 c␣⌽␣
␣
= 兺 c ␣+␤ 兿
For the first Hermite function, we have
␣
共A28兲
Now, to each spatial coordinate xk define the ladder operators
ak ⬅ 共xk + ⳵k兲 / 冑2. These obey 关a j , ak兴 = 0 and 关a j , a†k 兴 = ␦ j,k, as
can easily be verified. Let a be a formal vector of the ladder
operators, viz.,
a ⬅ 共a1,a2, . . . ,an兲.
k=1
共␣k + ␤k兲!1/2
⌽␣+␤ = 兺 c␣ P␤共␣兲1/2⌽␣+␤ .
␣k!1/2
␣
n
共A29兲
共A30兲
共A34兲
共a†兲␤␺ = 共a†1兲␤1 ¯ 共a†n兲␤n 兺 c␣⌽␣
共A27兲
Hence, if the sum on the right-hand side is finite, the convergence is uniform. If the coefficients cn decay rapidly enough,
both errors can be estimated by the dominating neglected
coefficients.
We now consider expansions of functions in L2共Rn兲. To
stress that Rn may be other than the configuration space of a
single particle, we use the notation x = 共x1 , . . . , xn兲 苸 Rn instead of rជ 苸 Rd. For N electrons in d spatial dimensions, n
= Nd.
Recall that the Hermite functions over Rn are indexed by
multi-indices ␣ = 共␣1 , . . . , ␣n兲 苸 In and that
␣
where the sum is to be taken over all multi-indices ␣ 苸 In.
Now, let ␤ 苸 In be arbitrary. By using Eq. 共A17兲 in each
spatial direction we compute the action of 共a†兲␤ on ␺,
⬁
兺 兩cn兩.
n=R+1
储␺储2 = 兺 兩c␣兩2 ,
␣
the pointwise error in ␺R is bounded by
兩␺共x兲 − ␺R共x兲兩 ⱕ 0.816
共A33兲
k=1
共␣k + ␤k兲!1/2
⌽␣
␣k!1/2
= 兺 c␣+␤ P␤共␣兲1/2⌽␣ ,
␣
共A36兲
Computing the square norm gives
储共a†兲␤␺储2 = 兺 P␤共␣兲兩c␣兩2 .
共A37兲
储a␤␺储2 = 兺 P␤共␣兲兩c␣+␤兩2 .
共A38兲
␣
and
␣
The polynomial P␤共␣兲 ⬎ 0 for all ␤ , ␣ 苸 In, and P␤共␣ + ␣⬘兲
⬎ P␤共␣兲 for all nonzero multi-indices ␣⬘ ⫽ 0. Therefore, if
共a†兲␤␺ 苸 L2共Rn兲 then a␤␺ 苸 L2共Rn兲. However, the converse is
not true for n ⬎ 1 dimensions as the norm in Eq. 共A38兲 is
independent of infinitely many coefficients c␣ while Eq.
共A37兲 is not. 关This should be contrasted with the onedimensional case, where a␺ 苸 L2共R兲 was equivalent to a†␺
苸 L2共R兲.兴 On the other hand, as in the n = 1 case, the condition a†k ␺ 苸 L2共Rn兲 is equivalent to the conditions xk␺
苸 L2共Rn兲 and ⳵k␺ 苸 L2共Rn兲.
We are in position to formulate a straightforward generalization of Lemma 1. The proof is easy, so we omit it.
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HARMONIC OSCILLATOR EIGENFUNCTION EXPANSIONS,…
Lemma 3 共General Hermite expansions兲.
Let ␺ 苸 L2共Rn兲, with coefficients c␣ as in Eq. 共A34兲, and
let ␤ 苸 In be arbitrary. Assume 共a†兲␤␺ 苸 L2共Rn兲. Then
共a†兲␤⬘␺ 苸 L2共Rn兲 and a␤⬘␺ 苸 L2共Rn兲 for all ␤⬘ ⱕ ␤. Moreover,
the following points are equivalent:
共1兲 共a†兲␤␺ 苸 L2共Rn兲; 共2兲 for all multi-indices ␥ ⱕ ␤,
␥ ␤−␥
x ⳵ ␺ 苸 L2共Rn兲; and 共3兲 兺␣␣␤兩c␣兩2 ⬍ +⬁.
We observe, that as we obtained for n = 1, condition 2 is a
combined decay and smoothness condition on ␺ and that this
can be expressed as a decay condition on the coefficients of
␺ in the Hermite basis by 3.
Exponential decay of ␺ 苸 L2共Rn兲 as 储x储 → ⬁ implies that
␥
x ␺ 苸 L2共Rn兲 for all ␥ 苸 In. We now generalize Lemma 2 to
the n-dimensional case.
Lemma 4 共Exponentially decaying functions兲.
Assume that ␺ 苸 L2共Rn兲 is such that for all ␥ 苸 In, x␥␺
苸 L2共Rn兲. Then, a sufficient criterion for 共a†兲␤␺ 苸 L2共Rn兲 is
⳵␤␺ 苸 L2共Rn兲. Moreover, for all ␮ ⱕ ␤, we have x␥⳵␤−␮␺
苸 L2共Rn兲 for all ␥ 苸 In such that ␥k = 0 whenever ␮k = 0; i.e.,
the partial derivatives of lower order than ␤ decay exponentially in the directions where the differentiation order is
lower.
Proof: the proof is a straightforward application of the n
= 1 case in an inductive proof, together with the following
elementary fact concerning weak derivatives: if 1 ⱕ j ⬍ k
ⱕ n, and if x j␺共x兲 and ⳵k␺共x兲 are in L2共Rn兲, then, by the
product rule, ⳵k关x j␺共x兲兴 = x j关⳵k␺共x兲兴 苸 L2共Rn兲. Notice, that
Lemma 2 trivially generalizes to a single index in n dimensions, i.e., to ␤ = ␤kek, since the integration by parts formula
used is valid in Rn as well. Similarly, the present Lemma is
valid in n − 1 dimensions if it holds in n dimensions, as it
must be valid for ¯␤ = 共0 , ␤2 , . . . , ␤n兲.
Assume that our statement holds for n − 1 dimensions. We
must prove that it then holds in n dimensions. Assume then,
¯
that ⳵␤␺ 苸 L2共Rn兲. Let ␾ = ⳵␤␺ 苸 L2. Moreover, ⳵1␤1␾ 苸 L2.
Since ␺ is exponentially decaying, and by the product rule,
x1␥1␾ 苸 L2 for all ␥1 ⱖ 0. By Lemma 2, x1␥1⳵1␤1−␮1␾ 苸 L2 for all
␥1 and 0 ⬍ ␮1 ⱕ ␤1. Thus, x1␥1⳵␤−e1␮1␺ 苸 L2. Thus, the result
holds as long as ␮ = e1␮1, or equivalently ␮ = ek␮k for any k.
¯
To apply induction, let ␹ = x1␥1⳵1␤1−␮1␺ 苸 L2. Note that ⳵␤␹
苸 L2 and x¯␥␹ 苸 L2 for all ¯␥ = 共0 , ␥2 , . . . , ␥n兲. But by the in¯
¯ ⱕ ¯␤ and all ¯␥ such
duction hypothesis, x¯␥⳵␤−␮¯ ␹ 苸 L2 for all ␮
¯ k = 0. This yields, using the product rule, that
that ¯␥k = 0 if ␮
x␥⳵␤−␮␺ 苸 L2 for all ␮ ⱕ ␤ and all ␥ such that ␥k = 0 if ␮k
= 0, which was the hypothesis for n dimensions, and the
proof is complete. Notice, that we have proved that 共a†兲␤␺
苸 L2 as a by-product.
䊏
p共r兲 ⬅
S. M. Reimann and M. Manninen, Rev. Mod. Phys. 74, 1283
共2002兲.
2
S. Kvaal, M. Hjorth-Jensen, and H. M. Nilsen, Phys. Rev. B 76,
085421 共2007兲.
3
M. Hjorth-Jensen, T. Kuo, and E. Osnes, Phys. Rep. 261, 125
共1995兲.
兺
␣苸In, 兩␣兩=r
兩c␣兩2 ,
共A39兲
⬁
p共r兲. Moreover, if P
where c␣ = 具⌽␣ , ␺典. Then, 储␺储2 = 兺r=0
projects onto the shell-truncated Hilbert space PR共Rn兲, then
R
储P␺储 = 兺 p共r兲.
共A40兲
2
r=0
Proposition 2 共Approximation in n dimensions兲.
Let ␺ 苸 L2共Rn兲 be exponentially decaying and given by
␺共x兲 = 兺 c␣⌽␣共x兲.
共A41兲
␣
Then ␺ 苸 Hk共Rn兲, k ⱖ 0, if and only if
⬁
rk p共r兲 ⬍ + ⬁.
兺␣ 兩␣兩 兩c␣兩 = 兺
r=0
k
2
共A42兲
The latter implies that
p共r兲 = o共r−共k+1兲兲.
共A43兲
Moreover, for the shell-truncated Hilbert space PR, the approximation error is given by
冉兺 冊
⬁
储共1 − P兲␺储 =
p共r兲
1/2
.
共A44兲
r=R+1
Proof: the only nontrivial part of the proof concerns Eq.
共A42兲. Since ␺ is exponentially decaying and since ␺ 苸 Hk if
and only if ⳵␤␺ 苸 L2 for all ␤, 兩␤兩 ⱕ k, we know that
兺␣␣␤兩c␣兩2 ⬍ +⬁ for all ␤, 兩␤兩 ⱕ k. Since 兩␣兩k is a polynomial
of order k with terms of type a␤␣␤, a␤ ⱖ 0, and 兩␤兩 = k, we
have
兺␣ 兩␣兩k兩c␣兩2 = 兺
␤, 兩␤兩=k
a␤ 兺 ␣␤兩c␣兩2 ⬍ + ⬁.
␣
共A45兲
On the other hand, since a␤ ⱖ 0 and the sum over ␤ has
finitely many terms, 兺␣兩␣兩k兩c␣兩 ⬍ +⬁ implies 兺␣␣␤兩c␣兩2
⬍ +⬁ for all ␤, 兩␤兩 = k, and thus ␺ 苸 Hk since ␺ was exponentially decaying.
䊏
4 T.
*simen.kvaal@cma.uio.no
1
In order to generate a simple and useful result for approximation in n dimensions, we consider the case where ␺ decays exponentially, and ␺ 苸 Hk共Rn兲, i.e., ⳵␤␺共x兲 苸 L2共Rn兲 for
all ␤ 苸 In with 兩␤兩 = k. In this case, we may also generalize
Eq. 共A25兲. For this, we consider the shell weight p共r兲 defined
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