Dipole and quadrupole polarizabilities and shielding factors of beryllium
from exponentially correlated Gaussian functions
Jacek Komasa
*
Quantum Chemistry Group, Department of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60-780 Poznan
´
, Poland
共Received 25 July 2001; published 14 December 2001兲
Dynamic dipole and quadrupole polarizabilities as well as shielding factors of the beryllium atom in the
ground state were computed at real frequencies by using the variation-perturbation method. The zeroth- and the
first-order wave functions were expanded in many-electron basis of exponentially correlated Gaussian func-
tions. The 1600-term expansion of the unperturbed wave function yielded the ground-state energy accurate to
1cm
⫺ 1
. The first-order wave functions were expanded in very large bases 共4800 and 4400 terms兲. The
nonlinear parameters of the first-order correction functions were optimized with respect to both the static and
dynamic polarizabilities, and with respect to the excited-state energies. The procedure employed ensures a high
accuracy of determination of dynamic properties in a wide range of frequencies and correct positions of the
transition poles. Test calculations, performed on He and Li, confirmed the ability of this method to obtain the
atomic properties with very high accuracy. The final values of the static properties of Be were 37.755e
2
a
0
2
E
H
⫺ 1
and 300.96e
2
a
0
4
E
H
⫺ 1
for the dipole and quadrupole polarizabilities, respectively, and 1.4769 for the quadrupole
shielding factor. The convergence of the atomic properties with the size of the expansion of both the zeroth-
and first-order functions was checked. Thanks to very high accuracy of the unperturbed wave function and the
efficient method of construction of the first-order wave functions, the dynamic polarizability results presented
in this work are of benchmark quality. As a by-product of this project, a set of the most accurate upper bounds
to the energies of
1
P and
1
D states of Be was obtained.
DOI: 10.1103/PhysRevA.65.012506 PACS number共s兲: 32.10.Dk, 31.25.⫺v
I. INTRODUCTION
One of the most important aspects of our knowledge of
many-electron systems is the ability to predict their behavior
in external fields. Particular attention of theoreticians is
drawn to the polarizabilities, which can be relatively easily
modeled mathematically and are involved in a variety of
physical phenomena. These response properties are often
linked to the optical properties of matter, scattering processes
or interatomic interactions. In particular, the frequency-
dependent polarizabilities enter the formulas defining second
refractive virial coefficient, Verdet constant, van der Waals
coefficients, refractive index, etc. A growing interest in the
accurate knowledge of polarizability of atomic gases is ob-
served. Such theoretically predicted polarizability, if suffi-
ciently accurate, might serve to calibrate measuring appara-
tus for various experiments 关1,2兴 and to independently
estimate fundamental constants of physics and chemistry
关3–6兴.
The dipole polarizability of an atom (
␣
1
) corresponds to a
dipole moment induced in the atom interacting with an ex-
ternal electric field (F
1
). Similarly, quadrupole polarizability
(
␣
2
) is related to a quadrupole moment induced by an exter-
nal electric field gradient (F
2
). There are two other quanti-
ties closely related to the polarizabilities, namely, the dipole
(
␥
1
) and quadrupole (
␥
2
) shielding factors. They give a pic-
ture of dipole and quadrupole moments induced in the elec-
tron charge distribution by pertinent nuclear moments. Alter-
natively,
␥
1
and
␥
2
can be treated as parameters describing
the change in the field and field gradient, respectively, expe-
rienced by the nucleus, resulting from the electron cloud
shielding (
␥
⬎ 0) or antishielding (
␥
⬍ 0) 关7兴. Although the
physical nature of the shielding factors slightly differs from
that of polarizabilities, they are mathematically closely
coupled and in this work they are studied together.
For the beryllium atom the values of these properties have
not been experimentally determined yet and we have to rely
on the theoretical predictions. In cases like this, it is crucial
to have an access to reliable reference values. The aim of this
study was to supply such benchmark values of the dynamic
dipole and quadrupole polarizabilities and shielding factors.
For an accurate description of polarizability, the electron cor-
relation has to be taken into consideration at a very high
level. Additionally, good description of the outer, energeti-
cally less important, region of the electron density distribu-
tion is indispensable. These requirements are met by very
flexible, explicitly correlated wave functions employed in
this paper.
II. METHOD
The dipole and quadrupole polarizabilities appear as ex-
pansion coefficients in the expression for the energy change
caused by an electric field and electric field gradient 关8兴.
Formally, the polarizability is defined as a second derivative
of the perturbation-dependent energy 关9兴
␣
⫽⫺
冉
2
E
共
F
兲
F
2
冊
F
→0
共2.1兲
and can be related to the second-order perturbation energy by
*
Electronic address: komasa@man.poznan.pl
PHYSICAL REVIEW A, VOLUME 65, 012506
1050-2947/2001/65共1兲/012506共11兲/$20.00 ©2001 The American Physical Society65 012506-1
␣
⫽⫺2E
(2)
. 共2.2兲
In the notation incorporated hereinafter, ⫽ 1 for dipole and
⫽ 2 for quadrupole properties.
We shall work in the nonrelativistic infinite nuclear mass
framework. The mass polarization and the relativistic correc-
tions 关5,6,10兴 to the dipole polarizability are the subject of
our current study and will be presented separately. The
atomic units are employed throughout this paper. In particu-
lar, ប⫽ 1 and the electron mass m⫽ 1 are assumed,
␣
1
and
␣
2
are expressed in units of e
2
a
0
2
E
H
⫺ 1
and e
2
a
0
4
E
H
⫺ 1
, respec-
tively, and the energy is expressed in the Hartree energy
(E
H
). Both
␥
1
and
␥
2
are dimensionless.
If the total wave function ⌿
⫽ ⌿
(0)
⫹ F
⌿
(1)
⫹ ••• 共as-
sumed real and normalized兲 satisfies the Hellmann-Feynman
theorem 关11–13兴, then
␣
and
␥
can be expressed conve-
niently as single integrals 关14兴:
␣
1
⫽⫺2
冕
⌿
1
(1)
冉
兺
i⫽ 1
n
y
i
冊
⌿
(0)
d
, 共2.3兲
␣
2
⫽⫺4
冕
⌿
2
(1)
冉
兺
i⫽ 1
n
y
i
z
i
冊
⌿
(0)
d
, 共2.4兲
␥
1
⫽⫺2
冕
⌿
1
(1)
冉
兺
i⫽ 1
n
y
i
r
i
3
冊
⌿
(0)
d
, 共2.5兲
␥
2
⫽⫺4
冕
⌿
2
(1)
冉
兺
i⫽ 1
n
y
i
z
i
r
i
5
冊
⌿
(0)
d
. 共2.6兲
The Hellmann-Feynman theorem allows
␥
1
to be determined
a priori. For an n-electron atom with a nucleus of charge Z
关14,15兴,
␥
1
⫽
n
Z
. 共2.7兲
For this fact,
␥
1
was recommended as a useful tool for as-
sessment of the quality of approximated wave functions in-
volved in Eqs. 共2.3兲 and 共2.5兲.
A. The ansatz
There are three different functions involved in the formu-
las 共2.3兲–共2.6兲. ⌿
(0)
is the unperturbed or zeroth-order wave
function of the atom; ⌿
1
(1)
and ⌿
2
(1)
are the first-order cor-
rection functions resulting from the dipole and quadrupole
perturbation, respectively. In this paper, all three functions
are expressed in the form of antisymmetrized linear combi-
nations of n-electron basis functions,
k
(
)
(
⫽ 0or1兲,
⌿
˜
(
)
共
r,
兲
⫽ A
ˆ
冉
⌶
n,S,M
S
共
兲
兺
k⫽ 1
K
(
)
c
k
(
)
k
(
)
共
r
兲
冊
, 共2.8兲
where ⌶
n,S,M
S
(
)isann-electron spin function 共e.g.,
⌶
4,0,0
(
)⫽
␣␣
⫺
␣␣
⫺
␣␣
⫹
␣␣
for the four-
electron singlet state兲, and where, in general, the linear coef-
ficients c
k
(1)
of the expansion of the first-order function de-
pend on the light frequency; r isa3n-element vector of
electron position coordinates and
represents n spin vari-
ables. The tilde over ⌿ is used to distinguish between the
exact wave function and its approximation.
As the many-electron basis functions
k
, the exponen-
tially correlated Gaussian 共ECG兲 functions of Singer 关16兴 are
employed:
k
共
r
兲
⫽ ⌳
m
k
exp
关
⫺
共
r⫺ s
k
兲
A
k
共
r⫺ s
k
兲
T
兴
, 共2.9兲
with s
k
restricted to zero—the natural choice for the position
of the nucleus. The remaining nonlinear parameters are or-
ganized in the form of positive definite symmetric n⫻ n ma-
trices A
k
. T superscript means a vector transposition. The
preexponential factor ⌳
m
k
⫽ 1 for the unperturbed wave
function, ⌳
m
k
⫽ y
m
k
for ⌿
˜
1
(1)
, and ⌳
m
k
⫽ y
1
z
m
k
for ⌿
˜
2
(1)
,y
i
,
and z
i
are the Cartesian components of the ith electron po-
sition vector. Such a choice of the basis functions ensures
respectively S,P, and D symmetry of the atomic wave func-
tions. The m
k
subscript labels the electrons. An experience
has shown that the restriction of m
k
to a single electron leads
to erroneous convergence. In this work, all possible values
1⭐m
k
⭐n were used and spread out uniformly over all basis
functions. Their presence makes possible using only a single
spin function without loss of completeness.
The ECG wave functions have been proved to work very
well for few-electron systems yielding in many cases the best
variational energies available in the literature: H
2
关17–19兴,
HeH
⫹
,H
3
⫹
关18兴,H
3
关20,21兴,He
2
⫹
, LiH 关22兴,He
2
关23,24兴,
HeHHe
⫹
关25兴,Be关26,27兴, e
⫹
LiH 关28兴. Also many accurate
expectation values of beryllium atom in position and mo-
mentum space come from the ECG calculations 关24,29,30兴.
In the present paper, we extend this list by energies of a few
lowest excited states and the second-order properties of Be.
B. Variation-perturbation method
For the harmonic, monochromatic perturbation of an an-
gular frequency
, the stationary-state first-order functions
are represented by 关31兴
⌿
(1)
共
r,t
兲
⫽ ⌿
⫹
(1)
共
r
兲
exp
关
⫺ i
共
E
(0)
⫺
兲
t
兴
⫹ ⌿
⫺
(1)
共
r
兲
exp
关
⫺ i
共
E
(0)
⫹
兲
t
兴
. 共2.10兲
The plus and minus components, ⌿
⫾
(1)
, can be obtained from
the solution of the first-order perturbation equations 共assum-
ing ⌿
(0)
is known兲
共
H
(0)
⫺ E
(0)
⫾
兲
⌿
⫾
(1)
⫽⫺O
ˆ
⌿
(0)
, 共2.11兲
where H
(0)
and E
(0)
are the unperturbed Hamiltonian and
energy, respectively, and O
ˆ
1
⫽ 兺
i⫽ 1
n
y
i
for the dipole and O
ˆ
2
⫽ 兺
i⫽ 1
n
y
i
z
i
for the quadrupole polarizability. Equations
共2.11兲 can be solved variationally, i.e., by minimization of
the Hylleraas functional 关32,33兴,
JACEK KOMASA PHYSICAL REVIEW A 65 012506
012506-2
J
关
⌿
⫾
(1)
兴
⫽
冕
⌿
⫾
(1)
共
H
(0)
⫺ E
(0)
⫾
兲
⌿
⫾
(1)
d
⫹ 2
冕
⌿
⫾
(1)
O
ˆ
⌿
(0)
d
, 共2.12兲
with respect to the parameters of the first-order function.
⌿
⫾
(1)
determined in this way enter the pertinent expressions
for
␣
⫾
and
␥
⫾
, Eqs. 共2.3兲–共2.6兲. Finally, the frequency-
dependent polarizabilities and shielding factors are obtained
from
␣
(
)⫽
␣
⫹
(
)⫹
␣
⫺
(
) and a similar equation for
␥
(
).
As shown by, e.g., Kolker and Michels 关34兴, when ⌿
(1)
is
expanded in the complete set of the unperturbed Hamilto-
nian’s eigenfunctions, one arrives at the spectral representa-
tion of
␣
(
). From this point of view, the first-order wave
function involves an infinite number of excited states, includ-
ing the continuum, so that it is not a trivial task to generate
such a wave function with an accuracy comparable to those
attainable for the unperturbed systems even if the wave func-
tion includes explicitly the electron correlation factor.
In the past, many accurate results were obtained within
the variation-perturbation approach in connection with the
explicitly correlated wave functions. For example, Glover
and Weinhold 关35兴 employed the Hylleraas-type wave func-
tions in their work devoted to rigorous lower and upper
bounds to the dynamic polarizability of two-electron atoms.
Sims and Rumble 关36兴 used this type of wave function in the
variation-perturbation calculations of static polarizability of
four-electron atoms. The Kołos-Wolniewicz wave function
was applied to both static 关37,38兴 and dynamic 关39–44兴 di-
pole polarizability of H
2
in the ground and excited states. In
principle, ⌿
(1)
should be optimized for each frequency sepa-
rately. So far, however, only the wave functions with opti-
mized linear parameters have been reported in literature. Ex-
ceptionally, simple adjustments of the nonlinear parameters
with respect to the static polarizability have been performed
关35兴. Only very recently, Cencek et al. 关6兴 fully optimized
first- and second-order ECG wave functions of He but also
with respect to the static properties. In the present paper,
much more flexibility was added to both the method and the
wave functions, as the nonlinear parameters of ⌿
˜
(1)
were
optimized with respect to the static and dynamic polarizabil-
ities and also with respect to the lowest excitation energies.
The optimization algorithm was similar to that applied to the
unperturbed wave function 关26兴, but the goal function was
either
␣
(
) or the excited-state energy. The nonlinear opti-
mization, although time consuming, was crucial for obtain-
ing accurate results. More about our optimization scheme
can be found in Refs. 关20,24,26,45兴.
C. Construction of the first-order wave function
At the absorption frequency the dynamic polarizabilities
exhibit discontinuities or poles. In practical calculations the
poles appear at the frequencies
l
⫽ E
l
⫺ E
0
(l⫽ 1,2,...),
where E
0
is the unperturbed state energy obtained from ⌿
˜
(0)
and is assumed to be known with very high accuracy. E
l
are
consecutive eigenvalues obtained by diagonalization of the
Hamiltonian H
(0)
in the basis of the perturbation correction
wave function ⌿
˜
(1)
of the appropriate symmetry. As men-
tioned above, it is a common practice to optimize this basis
set with respect to the static polarizability, not the energies
E
l
. As a consequence, the
␣
(0) values are recovered with
reasonable accuracy but when the frequency departs from
zero the accuracy of the
␣
(
) curve drops rapidly and,
additionally, the positions of the poles appear too high on the
frequency scale. Examples of such a tendency can be found
even in the most accurate calculations employing explicitly
correlated wave functions 关39–44兴.
The procedure described below, based on the variational-
ity of both the energy and the polarizability, allows this de-
ficiency to be eliminated. It relies on the observation that if
we merge a basis set of the length K
(1)
, whose nonlinear
parameters were optimized with respect to the static polariz-
ability, with a basis set of the length K
l
, optimized with
respect to the energy of the lth eigenvalue of an appropriate
symmetry, then the resulting basis set of the size K
F
⫽ K
(1)
⫹ K
l
gives
␣
(K
F
)⭓
␣
(K
(1)
) and simultaneously E
l
(K
F
)
⭐E
l
(K
l
), i.e., the combined basis set deteriorates neither the
polarizability nor the excited state energy obtained from the
separate basis sets. In this way we can generate a basis set
that combines the advantages of its components and yields
both accurate static polarizability and the position of the
pole.
In general, the first-order correction wave function can be
constructed by merging many basis sets, each optimized with
different goal functions. In the present paper, the nonlinear
parameters of the final expansion of ⌿
˜
(1)
were generated in
several separate optimization steps and the length K
F
of the
final expansion was a sum of the sizes of the basis sets em-
ployed in these steps: K
F
⫽ 兺
j
K
(1)
(
j
)⫹ 兺
l
K
l
. In the first
group of steps, the nonlinear parameters were optimized
variationally with respect to the polarizability by using the
Hylleraas functional, Eq. 共2.12兲, at selected frequencies
j
(j⫽ 0,1,...).Thesize of the basis sets optimized in these
steps was labeled K
(1)
(
j
) with explicit dependence on the
frequency in order to emphasize that the optimization was
performed not only for the static polarizability but also at
some frequencies from the range 0⬍
j
⬍ E
1
⫺ E
0
共in this
range the Hylleraas variational principle is valid 关31兴兲. The
second group of the steps generates the K
l
-term expansions
with nonlinear variational parameters optimum with respect
to the energy of the lth root of the Hamiltonian diagonalized
with the function of appropriate symmetry: ⫽ 1 for P states
or ⫽ 2 for states of D symmetry. The final basis set of the
size K
F
obtained in the above procedure was not optimized
any further. Optimization of this basis would improve the
selected goal quantity but deteriorate the rest of the features
of the
␣
(
) function.
The final basis set constructed in this way has the follow-
ing advantages over the basis generated in a single step. 共i兲 It
yields improved polarizabilities and excited-state energies
without the time-consuming optimization of large basis sets.
共ii兲 As the final basis set contains the basis functions of sev-
eral excited states, it ensures that the subsequent poles of the
DIPOLE AND QUADRUPOLE POLARIZABILITIES AND . . . PHYSICAL REVIEW A 65 012506
012506-3
␣
(
) curves are extremely accurate—their positions corre-
spond to the excitation energies of the states obtained from
the K
F
-term expansions, i.e., are only a fraction of milliHar-
tree in error. 共iii兲 Forcing the correct position of the poles
and optimization at
⬎ 0 ensures that the high accuracy of
␣
is preserved in a wide range of frequencies.
III. RESULTS AND DISCUSSION
A. Test calculations
The approach described in Sec. II C was tested on helium
and lithium atoms, for which exact values of energies and
polarizabilities are available from the literature. In Table I,
appropriate variational energies and static polarizabilities ob-
tained using the variation-perturbation method from the ECG
wave functions are confronted with the other most accurate
energies available in the literature and with the polarizability
values computed with the practically exact Hylleraas wave
functions by using the sum over state procedure 关5,46–51兴.
For the present calculations, the unperturbed ground-state
wave function of He was taken from the work of Cencek and
Kutzelnigg 关17兴. The 600-term ECG expansion gives the
ground-state energy with 12⫻ 10
⫺ 12
E
H
of error.
The dipole polarizability first-order expansion was as-
sembled from 660-term
␣
1
(0)-optimized ECG wave func-
tion and 610-term ECG basis optimized with respect to 2
1
P
state energy. The size of the final basis set was K
1
F
⫽ 1270 and
␣
1
(0)⫽ 1.383 192154 obtained in this procedure differs
from the exact value 关5,6兴 in ninth significant figure. The
dipole shielding factor differs from unity by less than 10
⫺ 7
.
The 2
1
P state energy computed in the final basis set is only
1 nanoHartree in error. The first pole of the dynamic polar-
izability curve is located 共with the same error as the energy兲
at
2
1
P
⫽ 0.779 881291.
The dynamic dipole polarizability of He was confronted
with the rigorous upper and lower bounds given by Glover
and Weinhold 关35兴 for frequencies up to the second reso-
nance. Though none of the components of ⌿
˜
1
(1)
was opti-
mized at
⬎ 0, the
␣
1
(
) curve fits perfectly those bounds.
Figures 1 and 2 show two curves constructed from Glover
and Weinhold’s data:
␣
1
ub
(
)⫺
␣
1
av
(
), and
␣
1
lb
(
)
⫺
␣
1
av
(
), compared with the
␣
1
(
)⫺
␣
1
av
(
), curve ob-
tained in this work. The ‘‘ub’’ and ‘‘lb’’ superscripts mean
the rigorous upper and, respectively, lower bound curve, and
‘‘av’’ is an arithmetic average of them.
For the quadrupole polarizability, ⌿
˜
2
(1)
was built of three
600-term ECG basis sets: one set optimized with respect to
␣
2
(0) and two sets with respect to 3
1
D and 4
1
D state en-
ergies. The final 1800-term expansion recovered
␣
2
(0)
⫽ 2.445 083016 with a relative error of 3⫻ 10
⫺ 8
. The poles
of the
␣
2
(
) function are located at
3
1
D
⫽ 0.848 103644
and
4
1
D
⫽ 0.872 4445 with all quoted figures being exact.
TABLE I. Results of the test calculations.
Property Reference Basis size Value
He
E(1
1
S) Exact 关46兴 ⫺ 2.903 724 377 034 119 598 3
ECG 600 ⫺ 2.903 724 377 022
␣
1
(0) Exact 关5兴 1.383 192 174 455共1兲
ECG 1270 1.383 192 154
␥
1
(0) Exact 1.0000000
ECG 1270 0.9999999
E(2
1
P) Exact 关47兴 ⫺ 2.123 843 086 498 094(5)
ECG 1270 ⫺ 2.123 843 085 6
␣
2
(0) Exact 关48兴 2.445 083 101共2兲
ECG 1800 2.445 083 016
␥
2
(0) Exact N/A
ECG 1800 0.407 681 0
E(3
1
D) Exact 关47兴 ⫺ 2.055 620 732 852 246(6)
ECG 1800 ⫺ 2.055 620 732 38
E(4
1
D) Exact 关47兴 ⫺ 2.031 279 846 178 687(7)
ECG 1800 ⫺ 2.031 279 817
Li
E(2
2
S) Exact 关49兴 ⫺ 7.478 060 323 650 3(71)
ECG 1536 ⫺ 7.478 060 314 3
␣
1
(0) Exact 关48兴 164.111共2兲
ECG 3700 164.11171
␥
1
(0) Exact 1.00000
ECG 3700 0.99973
E(2
2
P) Exact 关49兴 ⫺ 7.410 156 531 763(42)
ECG 3700 ⫺ 7.410 156 22
E(3
2
P) 关51兴 ⫺ 7.337 149 02
ECG 3700 ⫺ 7.337 149 032 2
E(4
2
P) 关51兴 ⫺ 7.311 883 30
ECG 3700 ⫺ 7.311 864
␣
2
(0) Exact 关48兴 1423.266共5兲
ECG 2800 1423.282
␥
2
(0) Exact N/A
ECG 2800 0.7385
E(3
2
D) Exact 关50兴 ⫺ 7.335 523 541 10(43)
ECG 2800 ⫺ 7.335 519
E(4
2
D) 关51兴 ⫺ 7.311 184 77
ECG 2800 ⫺ 7.310 40
FIG. 1. Projection of the ECG dynamic dipole polarizability of
He (⫹ ) on the area allowed by the Glover-Weinhold rigorous
bounds 共solid lines兲关35兴 at frequencies up to the first excitation
2
1
P
⫽ 0.779 881 291.
JACEK KOMASA PHYSICAL REVIEW A 65 012506
012506-4
The quadrupole shielding factor
␥
2
(0)⫽ 0.407 681 0 is the
most accurate estimation of this quantity in literature. Previ-
ous estimations of
␥
2
(0) come from late fifties: 0.424 关7兴,
0.416 关52兴, and 0.413 关53兴.
In the case of lithium atom, ⌿
˜
(0)
was chosen as 1536-
term ECG expansion of Cencek 关54兴, which yields the
ground-state energy with an error of 9.3⫻ 10
⫺ 9
E
H
.
Four basis sets of P symmetry were combined to get the
final 3700-term ECG expansion of ⌿
˜
1
(1)
: 1200-term basis
optimized with respect to
␣
1
(0), one 1300-term
2
2
P-optimized basis set, and two 600-term basis sets opti-
mized with respect to the energy of the 3
2
P and 4
2
P states.
The final dipole polarizability
␣
1
(0)⫽ 164.111 71 agrees
perfectly with Yan et al. result 164.111(2) 关48兴共see Table I兲.
The dipole shielding factor is equal to 0.999 73. The excita-
tion energy to 2
2
P state is 0.06 7904 1E
H
with 3
⫻ 10
⫺ 7
E
H
of error. For the next two excited
2
P states no
exact calculations are available in literature. The most accu-
rate to date are those by Pestka and Woz
´
nicki 关51兴. For 3
2
P
state the present ECG calculations give the variational upper
bound to the energy that is 0.01
E
H
lower than the energy
cited in Ref. 关51兴. For the 4
2
P state their energy is by
20
E
H
lower that the ECG energy. The appropriate positions
of the dipole polarizability poles are
3
2
P
⫽ 0.140 911 3 and
4
2
P
⫽ 0.166 20.
For the calculation of the quadrupole properties a 1000-
term
␣
2
(0)-optimized basis was combined with 1200-term
3
2
D-optimized and 600-term 4
2
D-optimized basis sets. The
final 2800-term ⌿
˜
2
(1)
gave
␣
2
(0)⫽ 1423.282 compared to
1423.266(5)e
2
a
0
4
E
H
⫺ 1
obtained by Yan et al. 关48兴. For un-
known reasons these two results differ by more or less three
times their estimated error bar. The value of
␥
2
(0)⫽ 0.7385,
which can be compared with 0.7156 estimated by Mahapatra
and Rao 关55兴.
Encouragingly the high accuracy of the test results sup-
ports the assertion that the above-described method of con-
struction of ⌿
˜
(1)
from the ECG functions has a potential
capability of yielding accurate results also for larger systems
including beryllium atom.
B. Convergence of the static properties of beryllium
In principle, the Hylleraas functional yields polarizabil-
ities that are lower bounds to the exact values. However, the
computed polarizabilities would represent rigorous lower
bounds only if an exact ⌿
(0)
was used in solving the
variation-perturbation equations. It is known that the Hyller-
aas functional 共2.12兲 is very sensitive to the quality of the
unperturbed wave function. The leading errors in
␣
are of
the second order in the error of ⌿
˜
⫾
(1)
, but only of the first
order in the error of ⌿
˜
(0)
关35,56兴. Therefore, particular effort
was put on the construction of the wave function describing
the unperturbed atom.
The unperturbed beryllium atom wave functions were
generated variationally for many expansion lengths, K
(0)
⫽ 50,...,1600. The wave functions with K
(0)
⭐1200 were
exactly those of Ref. 关26兴. The only new wave function with
K
(0)
⫽ 1600 yields the nonrelativistic energy of
⫺ 14.667 355 536 E
H
, which is the lowest variational en-
ergy of the ground-state beryllium to date. The estimated
error of the energy is less than 1 cm
⫺ 1
. Weinhold presented
the formula 关57兴 for the rigorous lower bound to polarizabil-
ity even when both wave functions are only approximate.
This formula becomes equivalent to the Hylleraas result in
the limit S→1, where S⫽
具
⌿
˜
(0)
兩
⌿
(0)
典
. The value of S,a
measure of quality of ⌿
˜
(0)
, can be estimated using the Eck-
art’s 关58兴 or the Weinberger’s 关59兴 inequality. For the 1600-
term ECG wave function, the first one yields S
⭓0.999 991 0, the second—stronger criterion— S
⭓0.999 992 0. Even the Weinberger’s bound is known to
give too weak an estimation 关60兴, and the true overlap is still
closer to unity. Very high accuracy of the ⌿
˜
(0)
applied in the
final calculations allows the error originating from the unper-
turbed wave function to be minimized and in practice the
variationality of the functional 共2.12兲 is preserved with good
precision.
Table II illustrates the influence of the choice of ⌿
˜
(0)
on
the static properties evaluated with well-optimized 1200-
term first-order wave function. For the smallest expansions
the dipole polarizability decreases with the growing basis
size. Only beginning with K
(0)
⫽ 100 it converges monotoni-
cally to the final value yielding five stable digits. It is seen
that
␣
1
(0) obtained even with the smallest K
(0)
differs from
that obtained with K
(0)
⫽ 1600 by less than 0.25%. The quad-
rupole polarizability behaves more regularly and grows
monotonically in the whole range of K
(0)
displayed in Table
II, yielding four converged figures. Although, for
␣
2
(0) the
convergence is slightly slower than in the dipole case, al-
ready K
(0)
⫽ 150 yields
␣
2
(0) within 1% of that obtained
with K
(0)
⫽ 1600.
The value of
␥
1
(0), which for the neutral Be atom is
known a priori to be equal 1, was evaluated using exactly
the same zeroth- and first-order wave function as
␣
1
. Devia-
tions of the computed
␥
1
(0) from unity can be seen as a
rough measure of quality of the pair of wave functions in-
volved in the computations. From Table II, we see that be-
ginning with K
(0)
⫽ 300,
␥
1
(0) grows monotonically to-
wards 1. Less regular is the behavior of
␥
2
(0) which
FIG. 2. As in Fig. 1 but at frequencies between the first and
second excitation
3
1
P
⫽ 0.848 596 1.
DIPOLE AND QUADRUPOLE POLARIZABILITIES AND . . . PHYSICAL REVIEW A 65 012506
012506-5
