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JOURNAL OF CHEMICAL PHYSICS
VOLUME 112, NUMBER 22
8 JUNE 2000
Mo” ller–Plesset convergence issues in computational quantum chemistry
Frank H. Stillinger
Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974
共Received 27 December 1999; accepted 15 March 2000兲
The Hartree–Fock self-consistent-field approximation has provided an invaluable conceptual
framework and a standard computational procedure for atomic and molecular quantum theory. Its
shortcomings are significant however, and require remediation. Mo” ller–Plesset perturbation theory
offers a popular correction strategy: it formally expands eigenfunctions and eigenvalues as power
series in a coupling parameter ␭ that switches the Hamiltonian continuously between the Hartree–
Fock form (␭⫽0) and the electron-correlating ‘‘physical’’ Hamiltonian (␭⫽1). Recent high-order
Mo” ller–Plesset numerical expansions indicate that the series can either converge or diverge at ␭
⫽1 depending on the chemical system under study. The present paper suggests at least for atoms
that series convergence is controlled by the position of a singularity on the negative real ␭ axis that
arises from a collective all-electron dissociation phenomenon. Nonlinear variational calculations for
the two-electron-atom ground state illustrate this proposition, and show that series convergence
depends strongly on oxidation state 共least favorable for anions, better for neutrals, better yet for
cations兲. © 2000 American Institute of Physics. 关S0021-9606共00兲30222-7兴
interest lies at ␭⫽1, these cases 共a兲 and 共b兲, respectively,
correspond to ␭ series with radii of convergence greater than,
and less than, unity. Specifically, pattern 共b兲 is characteristic
of an energy singularity closest to the origin that lies on the
negative real ␭ axis at less than unit distance from the origin.
The purpose of the present study is to argue that this singularity stems from a multielectron autoionization phenomenon.
Beside presenting the general formalism required for this
analysis, Sec. II also offers an intuitive description of how
the eigenvalue problem is expected to evolve along the real ␭
axis. Section III bolsters this view with results from a set of
nonlinear variational calculations for 1 S ground states of
two-electron atoms; in particular these results show a twoelectron autoionization whose position on the negative ␭ axis
depends strongly on nuclear charge Z. The final Sec. IV discusses the potentially most useful next-stage calculations designed to elucidate Mo” ller–Plesset convergence issues, and
how they might be used to increase the productivity of computational quantum chemistry.
I. BACKGROUND
On account of its conceptual simplicity, computational
convenience, and adequate accuracy, the Hartree–Fock selfconsistent-field approximation has exerted a dominating influence on the evolution of atomic and molecular electronic
structure studies. Indeed its single-particle terminology 共e.g.,
orbitals, bands, sigma and pi electrons, Fermi surfaces, holes,
etc.兲 has permanently entered the general scientific vocabulary. Not surprisingly, the Hartree–Fock approximation often
serves as the starting point for more ambitious computational
procedures that attempt to come to grips with electron correlation phenomena by appending systematic corrections to
that starting point. One of these procedures was initiated by
Mo” ller and Plesset in 1934,1 and forms the subject of this
paper.
As explained in the following Sec. II, the Mo” ller–Plesset
formalism casts the electronic structure problem into the format of Rayleigh–Schrödinger perturbation theory.2 The selfconsistent-field Hartree–Fock Hamiltonian serves as the unperturbed problem, and the electronic wave function and
energy are then developed in power series in a perturbation
parameter ␭ whose increase from 0 to 1 continuously transforms and connects the independent-particle description to
the fully correlated electronic structure problem.
Modern advances in computing power have enabled numerical studies to carry out Mo” ller–Plesset expansions to
high order, at least for some atomic and molecular systems
of modest size. Although basis set adequacy always remains
a significant concern, one can safely assume that behavior
patterns exhibited by published Mo” ller–Plesset series for energy eigenvalues are at least qualitatively correct. These patterns appear to fall into two categories: 共a兲 convergent 共examples are BH, CH2 兲 and 共b兲 divergent with even–odd sign
alternation in the high-order series coefficients 共observed for
Ne, HF, H2O兲.3 In view of the fact that the physical state of
0021-9606/2000/112(22)/9711/5/$17.00
II. FORMALISM
In the interest of maximum clarity, the following will
concentrate on the case of 2n electrons with equal numbers
of up and down spins, i.e., a spin singlet state. The corresponding self-consistent-field Hamiltonian, to be denoted by
H(0), consists strictly of a sum of 2n identical operators for
the 2n electrons:
2n
H共 0 兲 ⫽
兺
j⫽1
冋
N
⫺ 共 1/2兲 “ 2j ⫺
兺
k⫽1
册
Z k / 兩 r j ⫺Rk 兩 ⫹V共scf兲
.
j
共2.1兲
Here k indexes the N nuclei, with charges Z k and positions
Rk . Vscf
j is the self-consistent-field operator for electron j,
including both Coulomb and exchange portions. The ‘‘physi9711
© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 112, No. 22, 8 June 2000
Frank H. Stillinger
cal’’ Hamiltonian, H(1), replaces the sum of self-consistentfield operators with the pairwise sum of all 2n 2 ⫺n electron–
electron Coulomb repulsions. More generally, let H(␭) be
the Hermitian operator that linearly interpolates between
these two cases:
2n
H共 ␭ 兲 ⫽
兺
j⫽1
冋
N
⫺ 共 1/2兲 “ 2j ⫺
兺
k⫽1
Z k / 兩 r j ⫺Rk 兩
册
2n
⫹ 共 1⫺␭ 兲 V共scf兲
j ⫹␭
兺
j⬍1
1/兩 r j ⫺rl 兩 .
共2.2兲
For each electronic state of interest, two ␭-dependent
energy functions need to be distinguished. The first is the
corresponding eigenvalue of H(␭), to be denoted by E(␭),
associated with normalized wave function ␺ (1,...,2n,␭):
H共 ␭ 兲 ␺ 共 1,...,2n,␭ 兲 ⫽E 共 ␭ 兲 ␺ 共 1,..,2n,␭ 兲 .
共2.3兲
The second, W(␭), is the expectation value of the physical
Hamiltonian H(1) in the state described by ␺ (1,...,2n,␭):
W 共 ␭ 兲 ⫽ 具 ␺ 共 1,...,2n,␭ 兲 兩 H共 1 兲 兩 ␺ 共 1,...,2n,␭ 兲 典 .
共2.4兲
It is this latter quantity whose power series represents the
Mo” ller–Plesset expansion:
W 共 ␭ 兲 ⫽W 0 ⫹W 1 ␭⫹W 2 ␭ 2 ⫹•••• .
共2.5兲
In general E(␭) and W(␭) are not equal. The obvious
exception occurs at the physical value of the coupling constant,
E 共 1 兲 ⫽W 共 1 兲 .
共2.6兲
4
The Rayleigh–Ritz variational principle requires for the
ground electronic state 共or indeed for the lowest-energy state
of any given symmetry兲 that W(␭) must pass through its
absolute minimum at ␭⫽1, an attribute not shared by E(␭).
It is traditional to express the wave function at ␭⫽0 as a
2n⫻2n Slater determinant whose elements are space–spin
orbitals. However, this is not necessary in view of the fact
that Hamiltonian H(␭) is spin independent for all ␭. Instead,
we can confine attention to any one spin–space component,
say that for electrons 1,...,n with spins down and n
⫹1,...,2n with spins up, and consider just the position-space
dependence of ␺ (␭⫽0). Let ␸ 1 (r)¯ ␸ n (r) be an appropriate orthonormal set of position-space orbitals. Then we can
set
␺ 共 r1 ¯r2n ,␭⫽0 兲 ⫽D 共 r1 ¯rn 兲 D 共 rn⫹1 ¯r2n 兲 ,
共2.7兲
where D is an n⫻n determinant
D 共 r1 ¯rn 兲 ⫽ 共 n! 兲
⫺1/2
det关 ␸ i 共 r j 兲兴 .
共2.8兲
共c兲
共e兲
V共scf兲
j ⫽V j ⫺V f .
共2.9兲
The first of these is just an r-space-function multiplier,
n
兺
k⫽1
冕
ds ␸ 2k 共 s兲 / 兩 r j ⫺s兩 ,
n
V共j e 兲 • f 共 r j 兲 ⫽
兺
k⫽1
␸ k共 rj 兲
冕
ds ␸ k 共 s兲 f 共 s兲 / 兩 r j ⫺s兩 .
共2.11兲
The effect of the self-consistent-field essentially is to
provide a static negative charge cloud that is spatially distributed according to the extension of the orbitals comprised
in ␺ (␭⫽0). The exchange operators reduce the magnitude
of the corresponding repulsion somewhat, but only to a partial extent. As ␭ increases from 0 to 1 in H(␭), Eq. 共2.2兲, this
diffuse repulsion continuously switches off while being replaced by explicit electron-pair repulsions. Formally extending ␭ to even larger positive values greater than 1 causes the
self-consistent field to convert to a diffuse attraction surrounding the nuclei, while electron pairs become even more
repulsive. For very large positive ␭ we can expect the 2n
electrons to concentrate around configurations that represent
a compromise between these competing strong attractive and
repulsive interactions.
As ␭ moves from the origin along the negative real axis,
interaction roles are reversed in comparison with the ␭⬎1
regime. The self-consistent field becomes ever more repulsive, overcoming the direct nuclear attractions, thereby diminishing the capacity of the electrons to remain bound in
the neighborhood of the nuclei. But now the explicit
electron–electron pair interactions have become attractive.
As a result the 2n electrons in isolation from nuclei have the
capacity to form their own bound state whose energy would
have the form (␭⬍0)
⫺A 共 n,n 兲 ␭ 2 ,
共2.12兲
where A is a suitable positive constant. This last expression
共2.12兲 locates an autoionization threshold at which the 2n
electrons spontaneously leave the neighborhood of the nuclei, together as a bound composite particle, tunneling
through a repulsive barrier due to the magnified selfconsistent field. The negative ␭ value at which this occurs
can be identified by equating eigenvalue E(␭) to quadratic
expression 共2.12兲. Just as an analogous autoionization threshold for the Z ⫺1 expansion of the two-electron-atom ground
state creates a wave function and energy singularity in that
context,5,6 so too can we expect the same for ␺共␭兲, E(␭), and
W(␭). Hence we propose that this collective autoionization
phenomenon determines the Mo” ller–Plesset convergence radius.
III. SIMPLE ILLUSTRATIVE EXAMPLE
Self-consistency requires that the orbitals ␸ i which compose eigenfunction ␺ (␭⫽0) both determine the operators
and also minimize W(0). Each of the V共scf兲
resolves
V共scf兲
j
j
into Coulomb 共c兲 and exchange 共e兲 portions,
V共j c 兲 ⫽2
while the second is an integral operator with the property
共2.10兲
In order to provide support for the qualitative ideas expressed at the end of Sec. II, we now set up and carry out a
simple nonlinear variational calculation. While it is desirable
eventually to use a more sophisticated and precise calculation, the following example will suffice for present purposes.
Specifically we consider the general two-electron atom
共nuclear charge Z兲 in its singlet ground state, for which the
spatial wave function is symmetric under electron interchange. Our task is to estimate E(␭) and W(␭), and for this
purpose we introduce the following two-parameter correlated
variational wave function 共the nucleus is at the origin兲:
J. Chem. Phys., Vol. 112, No. 22, 8 June 2000
␺ ␯ 共 r1 ,r2 ,␭ 兲 ⫽C 共 ␣ , ␤ 兲 exp关 ⫺ ␣ 共 r 1 ⫹r 2 兲 ⫺ ␤ r 12兴 .
Perturbation convergence
9713
共3.1兲
The normalizing constant has the value
C共 ␣,␤ 兲⫽
冋
8 ␣ 3共 ␣ ⫹ ␤ 兲 5
␲ 共 8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2 兲
2
册
1/2
.
共3.2兲
Parameters ␣共␭兲 and ␤共␭兲 are to be determined by minimizing
具 ␺ ␯ 兩 H共 ␭ 兲 兩 ␺ ␯ 典 ⭓E 共 ␭ 兲
共3.3兲
at each ␭.
Negative real ␭ permits formation of a bound ‘‘dielectron.’’ In its free state this composite particle has wave function (␭⬍0)
共 兩 ␭ 兩 3 /8␲ 兲 1/2 exp共 ⫺ 兩 ␭ 兩 r 12/2兲
共3.4兲
and binding energy
FIG. 1. Coupling constant 共␭兲 dependence of variational parameters ␣ and ␤
in trial wave function ␺ ␯ , Eq. 共3.1兲, for the helium atom ground state (Z
⫽2).
具 ␺ ␯ 兩 H共 1 兲 兩 ␺ ␯ 典 ⫽ 关共 ␣ ⫹ ␤ 兲 / 共 8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2 兲兴
⫺ 兩 ␭ 兩 2 /4.
共3.5兲
Accurate numerical solutions are available for the
Hartree–Fock approximation to the two-electron ground
state.7 In principle they could be used to construct the singleparticle operators V(c)
and V(e)
j
j . However, that would be
‘‘numerical overkill’’ given the modest objective of our elementary variational strategy. Instead we exploit two simplifying approximations. First we use the best effective-charge,
single-exponential 1s orbital for each nuclear charge Z:8
␸ 0 共 r 兲 ⫽ 共 ␣ 30 / ␲ 兲 1/2 exp共 ⫺ ␣ 0 r 兲 ,
共3.6兲
␣ 0 ⫽Z⫺5/16.
In this approximation V(c)
is a simple closed-form r-space
j
potential
⫻ 关 8 ␣ 3 ⫹7 ␣ 2 ␤ ⫹4 ␣␤ 2 ⫹ ␤ 3 ⫹5 ␣ 2
⫹4 ␣␤ ⫹ ␤ 2 ⫺4Z ␣ 共 4 ␣ ⫹ ␤ 兲兴 .
共3.10兲
By substituting the ␣共␭兲 and ␤共␭兲 variational results into this
expression one obtains the corresponding W(␭).
IV. NUMERICAL RESULTS
Figure 1 shows ␣共␭兲 and ␤共␭兲 computed for the neutral
helium atom, Z⫽2. The corresponding energy functions
E(␭) and W(␭) appear in Fig. 2, which also contains the
free dielectron binding energy curve, Eq. 共3.5兲. Numerically
it appears possible to locate normalizable wave functions in
this approximation for
V共j c 兲 ⫽ 共 2/r j 兲关 1⫺exp共 ⫺2 ␣ 0 r j 兲兴 ⫺2 ␣ 0 exp共 ⫺2 ␣ 0 r j 兲 .
共3.7兲
The second simplifying assumption involves the following
modification of the exchange operator:
冕
冕
V共j e 兲 • f 共 r j 兲 ⫽ ␸ 0 共 r j 兲
→ f 共 rj 兲
ds ␸ 共 s 兲 f 共 s兲 / 兩 r j ⫺s兩
ds ␸ 0 共 s 兲 ␸ 0 共 s 兲 / 兩 r j ⫺s兩
⬅ 共 1/2兲 V共j c 兲 • f 共 r j 兲 .
共3.8兲
Note that this is exact when f ⫽ ␸ 0 . Thus we assume
V共j c 兲 ⫺V共j e 兲 ⬵ 共 1/2兲 V共j c 兲
共3.9兲
for illustrative purposes; this is equivalent to the restricted
Hartree approximation.9
Subject to these simplifications, the Appendix contains
the matrix elements needed for the E(␭) variational minimization, each of which has a rational algebraic form. These
have been used to obtain ␣共␭兲 and ␤共␭兲 numerically along
the real ␭ axis. Note that the H(1) matrix element has a
relatively simple form:10
FIG. 2. Variationally determined energy curves E(␭) and W(␭) for the
helium atom ground state. The free dielectron binding energy ⫺␭ 2 /4 for
␭⬍0 has been included to locate the dielectron ionization singularity, its
intersection with E(␭).
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J. Chem. Phys., Vol. 112, No. 22, 8 June 2000
Frank H. Stillinger
V. DISCUSSION
FIG. 3. Variationally determined energy curves E(␭) and W(␭) for the
hydride anion ground state (Z⫽1), along with the free dielectron binding
energy ⫺␭ 2 /4 for ␭⬍0.
⫺2.10⭐␭
共 Z⫽2 兲 .
共4.1兲
However, notice in Fig. 2 that E(␭) intersects the curve for
the dielectron binding energy at the ionization threshold
critical value
␭ c 共 Z⫽2 兲 ⬵⫺1.33;
共4.2兲
if the variational wave function had been sufficiently flexible
to describe the dielectron ionization explicitly and realistically, no normalizable solution would have been found for
␭⬍␭ c . In view of the fact that the predicted magnitude
兩 ␭ c (Z⫽2) 兩 exceeds unity, the Mo” ller–Plesset series for the
helium atom ground state is predicted to be absolutely convergent.
Analogous calculations have been performed for Z⫽3
(Li⫹) and Z⫽4 (Be⫹⫹). The critical thresholds for these
cases are the following:
␭ c 共 Z⫽3 兲 ⬵⫺2.52,
␭ c 共 Z⫽4 兲 ⬵⫺3.70.
共4.3兲
Evidently the ionization singularity moves farther from the
origin as Z increases, which would be reflected as a more
rapid convergence of the Mo” ller–Plesset series.
The situation is drastically different for Z⫽1 (H ⫺ ). The
corresponding energy curves appear in Fig. 3. The E(␭) and
dielectron binding energy functions intersect very close to
the origin:
␭ c 共 Z⫽1 兲 ⬵⫺0.08,
共4.4兲
indicative of a strongly divergent Mo” ller–Plesset series.
Although the variational calculations presented above
are admittedly crude and are restricted to two-electron
atomic ground states, it is reasonable to suppose that they
present qualitatively correct patterns. In particular they lead
to the proposition that the Mo” ller–Plesset series for W(␭),
Eq. 共2.4兲, has a radius of convergence determined by the
presence of a singularity on the negative real ␭ axis. Furthermore this singularity arises from a wave function–distorting
phenomenon whereby the electrons are expelled from the
region of the nucleus as a free dielectron complex. The variational calculations also indicate that for a fixed number of
electrons the singularity position depends strongly on nuclear
charge Z: larger Z moves the singularity farther from the
origin of the complex ␭ plane and improves the convergence
rate of the Mo” ller–Plesset energy power series.
Obviously it is desirable to strengthen the case by repeating the two-electron ground-state calculations in a more
precise manner. One area for improvement involves the ␭
⫽0 Hartree–Fock orbitals, approximated crudely in the
present study by a single exponential with effective nuclear
charge. A result of this approximation is that the variational
calculations at ␭⫽0 do not quite replicate a product of
simple exponentials; instead ␤共0兲 has a small negative value,
and ␣共0兲 slightly exceeds effective charge Z⫺5/16, for all Z
values investigated. These minor discrepancies would be
eliminated upon insertion of a correct Hartree–Fock solution
for H(0).
At the same time it is also desirable to employ a more
flexible, and thus potentially more accurate, variational wave
function. For the two-electron ground state considered above
it would be advantageous to work with linear combinations
of Gaussian functions that are appropriately symmetrized:
2
␺ ␯ 共 r1 ,r2 兲 ⫽ 兺 A i exp共 ⫺a i r 21 ⫺b i r 22 ⫺c i r 12
兲.
i
共5.1兲
The full set of parameters 兵 A i ,a i ,b i ,c i 其 could in principle
be treated as independent variables 共subject to ␺ ␯ normalization兲, but for practical reasons might be linked into contracted subsets. In any case sufficient flexibility should remain to describe the formation and ionization of the
dielectron complex at negative ␭.
Using a Gaussian basis would remove another source of
imprecision in the present calculations, the replacement 共3.8兲
of exchange operators with their Coulomb operator analogs.
The result of V(e) operating on any Gaussian term can be put
into closed form, so approximation 共3.8兲 becomes unnecessary. It is also clear that more than two electrons should be
considered, and polyatomic molecules as well, to provide a
more comprehensive view of Mo” ller–Plesset convergence issues in quantum chemistry.
Two further matters deserve mention. The first is the
nature of the free ‘‘multelectron’’ in its ground and excited
states. Locating singularity-associated thresholds generally
will require determining the energy of these ‘‘selfgravitating’’ units composed of given numbers of down-spin
and up-spin electrons. The other matter is the precise mathematical characterization of the E(␭) and W(␭) singularities
J. Chem. Phys., Vol. 112, No. 22, 8 June 2000
Perturbation convergence
themselves; if the analogous 1/Z-expansion threshold singularity for two electrons is any indication, this mathematical
problem will require a deep analysis.5 Understanding these
singularities and how they dominate high-order series coefficients should suggest how best to sum partial series into a
closed form, including cases that formally diverge, to provide reliable energy estimates for systems of chemical interest.
9715
where
R⫽
␣
关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴共 ␣ 2 ⫺ ␤ 2 兲 2
⫺
⫹
␤
␣ 0 共 2 ␣ ⫹ ␣ 0 兲共 ␣ 2 ⫺ ␤ 2 兲 2
4 ␣ 2␤
4 ␣␤ 2
⫹
␣ 0 共 2 ␣ ⫹ ␣ 0 兲共 ␣ 2 ⫺ ␤ 2 兲 3 关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴共 ␣ 2 ⫺ ␤ 2 兲 3
2␣2␤
2␣␤2
⫹ 2
⫹
␣0共2␣⫹␣0兲2共␣2⫺␤2兲2 关共␣⫹␣0兲2⫺␤2兴2共␣2⫺␤2兲2
ACKNOWLEDGMENTS
The author is pleased to acknowledge helpful discussions with Dr. K. Raghavachari and Dr. C. White.
⫺
2 共 ␣ ⫹ ␣ 0 兲 ␣␤
,
2
␣ 0 共 2 ␣ ⫹ ␣ 0 兲 2 关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴 2
共A6兲
APPENDIX
Owing to the special choice 共3.1兲 of variational wave
function, to the restricted Hartree simplification 共3.8兲–共3.9兲,
and to the fact that only one nucleus is present, all of the
matrix elements needed to evaluate E(␭) and W(␭) can be
expressed as rational algebraic functions of the variational
parameters ␣ and ␤. The kinetic energy has the form
具K典⫽
⫽
冕 冕
dr1
dr2 ␺ ␯ 共 r1 ,r2 兲关 ⫺ 共 “ 21 ⫹“ 22 兲 /2兴 ␺ ␯ 共 r1 ,r2 兲
共 ␣ ⫹ ␤ 兲共 8 ␣ 3 ⫹7 ␣ 2 ␤ ⫹4 ␣␤ 2 ⫹ ␤ 3 兲
.
8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2
S⫽
⫹
冕 冕
dr1
⫽⫺
具 V e典 ⫽
冕 冕
dr1
dr2 ␺ 2␯ /r 12⫽
具 H共 1 兲 典 ⫽ 具 K 典 ⫹ 具 V N 典 ⫹ 具 V e 典 .
共A4兲
In order to carry out the variational calculation for arbitrary ␭ that leads to evaluation of ␣共␭兲, ␤共␭兲, E(␭), and
W(␭), one also requires a general expression for the Hartree
self-consistent-field matrix element. The effective-charge approximation, Eq. 共3.6兲 above, also produces an algebraic matrix element, though rather more complicated than before:
⫽
冕 冕
dr1
⫹
dr2 ␺ 2␯ 关共 V共1c 兲 ⫹V共2c 兲 兲 /2兴
4 ␣ 共 ␣ ⫹ ␤ 兲共 4 ␣ ⫹ ␤ 兲
16␣ 3 共 ␣ ⫹ ␤ 兲 5
⫺
8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2
8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2
⫻ 兵 R⫹ ␣ 0 S⫹4 ␣ 0 共 ␣ ⫹ ␣ 0 兲 ␣␤ T 其 ,
共A5兲
␤
关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴 2 共 ␣ 2 ⫺ ␤ 2 兲 3
.
共A8兲
The variational eigenvalue estimate E(␭) is obtained as follows:
共 ␣ ⫹ ␤ 兲共 5 ␣ 2 ⫹4 ␣␤ ⫹ ␤ 2 兲
.
8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2
共A3兲
Equation 共3.10兲 above results from substituting Eqs. 共A1兲,
共A2兲, and 共A3兲 into the expression
具 V H典 ⫽
␣
␤
⫺ 2
2
2 2
2 3⫹
␣0共2␣⫹␣0兲 共␣ ⫺␤ 兲 关共␣⫹␣0兲 ⫺␤2兴3共␣2⫺␤2兲2
共A2兲
Direct Coulomb repulsion between the two electrons generates the matrix element:
共A7兲
␣⫹␣0
␣
⫺ 2
2
2
2 3⫹ 3
␣0共2␣⫹␣0兲 关共␣⫹␣0兲 ⫺␤ 兴 ␣ 0 共 2 ␣ ⫹ ␣ 0 兲 3 共 ␣ 2 ⫺ ␤ 2 兲 2
共A1兲
dr2 ␺ ␯2 共 ⫺Z/r 1 ⫺Z/r 2 兲
4Z ␣ 共 ␣ ⫹ ␤ 兲共 4 ␣ ⫹ ␤ 兲
.
8 ␣ 2 ⫹5 ␣␤ ⫹ ␤ 2
␣␤
,
␣ 20 共 2 ␣ ⫹ ␣ 0 兲 2 关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴 2
␣⫹␣0
T⫽⫺ 3
␣0共2␣⫹␣0兲3关共␣⫹␣0兲2⫺␤2兴2
Nuclear attractions, proportional to Z, lead to the following:
具 V N典 ⫽
共 ␣⫹␣0兲␣
共 ␣⫹␣0兲␤
⫹
关共 ␣ ⫹ ␣ 0 兲 2 ⫺ ␤ 2 兴 2 共 ␣ 2 ⫺ ␤ 2 兲 2 ␣ 20 共 2 ␣ ⫹ ␣ 0 兲 2 共 ␣ 2 ⫺ ␤ 2 兲 2
E 共 ␭ 兲 ⫽ min 关 具 K 典 ⫹ 具 V N 典 ⫹␭ 具 V e 典 ⫹ 共 1⫺␭ 兲 具 V H 典 兴 .
共␣,␤兲
共A9兲
C. Mo” ller and M. S. Plesset, Phys. Rev. 46, 618 共1934兲.
R. Courant and D. Hilbert, Methods of Mathematical Physics 共Interscience, New York, 1953兲, Vol. 1, pp. 343–350; K. Gottfried, Quantum
Mechanics, Vol. I Fundamentals 共W. A. Benjamin, New York, 1966兲, pp.
357–365.
3
J. Olsen, O. Christiansen, H. Koch, and P. Jo” rgensen, J. Chem. Phys. 105,
5082 共1996兲.
4
See R. Courant and D. Hilbert, Ref. 2, pp. 175 and 176.
5
J. D. Baker, D. E. Freund, R. N. Hill, and J. D. Morgan, Phys. Rev. A 41,
1247 共1990兲.
6
I. A. Ivanov, Phys. Rev. A 52, 1942 共1995兲.
7
C. C. Roothaan, L. M. Sachs, and A. W. Weiss, Rev. Mod. Phys. 32, 186
共1960兲; C. A. Stuart, J. F. Toland, and P. G. Williams, Proc. R. Soc.
London, Ser. A 388, 229 共1983兲.
8
L. Pauling and E. B. Wilson, Introduction to Quantum Mechanics
共McGraw-Hill, New York, 1935兲, pp. 184–185.
9
D. R. Hartree, The Calculation of Atomic Structures 共Wiley, New York,
1957兲.
10
F. H. Stillinger, J. Chem. Phys. 45, 3623 共1966兲, Appendix.
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2