JOURNAL OF CHEMICAL PHYSICS
VOLUME 109, NUMBER 19
15 NOVEMBER 1998
Electron intracule densities and Coulomb holes from energy-derivative
two-electron reduced density matrices
Jerzy Cioslowskia) and Guanghua Liu
Department of Chemistry and Supercomputer Computations Research Institute, Florida State University,
Tallahassee, Florida 32306-3006
~Received 30 March 1998; accepted 14 August 1998!
Application of the energy-derivative formalism to two-electron reduced density matrices produces
a robust approach to the approximate evaluation of electron intracule densities I(R) and Coulomb
holes in atoms and molecules. The versatility of this approach, which makes routine calculations of
correlated I(R) feasible at any level of electronic structure theory, is demonstrated by results of
selected MP2 calculations. The MP2/(20s10p10d) values of I(0) are within 10% of their ‘‘exact’’
counterparts in systems such as H2 , He, Li1 , Be21 , Li, and Be. Quantitative reproduction of the
exact I(R) is found to be contingent upon the inclusion of Gaussian primitives with high angular
momenta in the basis sets. © 1998 American Institute of Physics. @S0021-9606~98!31443-9#
pansions such as CISD.11 However, these densities are not
size-consistent and are thus grossly inaccurate for larger systems.
Being an expectation value of a two-electron operator,
I(R) is a first-order, two-electron response property amenable to the energy-derivative formalism that has been employed with great success in calculations of energy
gradients12 and, more recently, in conjunction with the extended Koopmans’ theorem.13 In this paper, a combination of
this formalism with the recently introduced approach to the
evaluation of I(R) and its derivatives that minimizes the
computational effort per grid point14 is described. The versatility of this new method, which makes routine calculations
of correlated electron intracule densities feasible at any level
of electronic structure theory, is demonstrated by results of
selected MP2 calculations. The basis set dependence of the
computed I(R) is also discussed.
I. INTRODUCTION
Analysis of electron–electron interactions in Coulombic
systems is greatly facilitated by the introduction of quantities
that, while retaining the relevant characteristics of the electronic wave function C, depend on a single threedimensional vector R. Among those quantities, the electron
intracule density I(R), defined as1,2
I ~ R! 5 ^ C u
d @~ r j 2ri ! 2R# u C & ,
(
i. j
~1!
is particularly useful. The difference
DI ~ R! [I ~ R! 2I HF~ R!
~2!
between I(R) and its uncorrelated counterpart I HF(R) derived from the corresponding Hartree–Fock wave function,
which constitutes a generalization of the Coulomb hole,2,3 is
also of much interest. Since DI(R) differs appreciably from
zero only at small interelectron distances R[ u Ru , topological analysis of I HF(R) ~and of the respective extracule density! provides valuable insights into two-electron aspects of
bonding in molecules.4,5 On the other hand, understanding of
electron correlation effects calls for reliable estimates of
DI(R).
Until very recently, the lack of efficient algorithms for
the computation of I(R) has hampered its widespread use in
quantum chemistry. In particular, only very few calculations
of highly accurate electron intracule densities from explicitly
correlated wave functions have been reported so far.3,6–8 Although capable of producing I(R) with proper electron–
electron coalescence cusps,9,10 such calculations are too expensive to be carried out routinely. In principle, approximate
electron intracule densities can also be obtained from wave
functions produced by truncated configuration interaction ex-
II. THEORY
In electronic structure theories based upon a singledeterminantal reference state, the total energy is given by15
E5
1E c @ $ e p % , $ ^ pq uu rs & % # ,
~3!
where $ h pq % are matrix elements of the core Hamiltonian and
$ ^ pq uu rs & % are the antisymmetrized two-electron repulsion
integrals, both computed in the basis set of canonical
Hartree–Fock spinorbitals $ f p % with the occupancies $ n p %
~equal 0 or 1!. The orbital energies $ e p % are the diagonal
elements of the Fock matrix
F pq 5h pq 1 ~ 1/2!
(r n r~ ^ pr uu qr & 1 ^ rp uu rq & ! 5 d pq e p .
~4!
The correlation energy E c is a function of $ e p % and
a!
Author to whom correspondence should be addressed. Electronic mail:
[email protected], home page: http://www.scri.fsu.edu/;jerzy
0021-9606/98/109(19)/8225/7/$15.00
n p n q ^ pq uu pq &
(p n p h pp 1 ~ 1/2! (
pq
$ ^ pq uu rs & % .
8225
© 1998 American Institute of Physics
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8226
J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
TABLE I. Parameters of the even-tempered basis sets employed in the present calculations.a
Gaussian primitives
Element
H
He
Li
Be
F
Ne
Na
s
p
d
0.008 376 78/2.030 106 87
0.098 902 88/2.097 546 84
0.020 830 85/2.191 634 37
0.035 350 97/2.201 404 30
0.190 539 83/2.196 941 85
0.235 645 57/2.195 391 45
0.021 136 68/2.367 709 06
0.012 283 75/2.587 416 99
0.156 619 80/2.756 483 31
0.034 362 77/3.114 908 86
0.059 988 15/3.107 700 28
0.139 235 56/2.476 781 62
0.168 643 05/2.486 480 63
0.255 518 55/2.420 313 63
0.012 283 75/2.587 416 99
0.156 619 80/2.756 483 31
0.034 362 77/3.114 908 86
0.059 988 15/3.107 700 28
0.139 235 56/2.476 781 62
0.168 643 05/2.486 480 63
0.255 518 55/2.420 313 63
Values of a/values of b; see Eq. ~29!.
a
G pq 5n p d pq 1
Consider a one-electron perturbation described by a Hermitian operator V̂ 1 with matrix elements $ ^ f p u V̂ 1 u f q & % .
Since the Hartree–Fock wave function is variational, the
first-order derivative of E with respect to the perturbation
strength x equals
x
E [ ~ ] E/ ]x ! u x 50 5
(p
n p h xpp 1E xc @ $ e p % , $ ^ pq uu rs & % # .
~5!
The expression for E cx @ $ e p % , $ ^ pq uu rs & % # reads12,13
3
5
(p
(
pqrs
x
E x 5 ~ 1/2!
B pqrs ~ ^ p q uu rs & 1 ^ pq uu rs &
~6!
F xpq 5 ~ 1/2!
where both $ A p % and $ B pqrs % ,
A p[ ] E c / ] e p
~7!
E xc 5
~8!
The derivatives $ e xp and $ f xp % are readily available from
the coupled perturbed Hartree–Fock ~CPHF! equations,16
which relate them to $ F xpq % through perturbation-independent
tensors $ C pqr % and $ D pqrs % ,
qr
Since the matrices
and f xp 5
$ F xpq %
and
x
D pqrs F qr
fs .
(
qrs
$ h xpq %
(
pq
B pqrs ~ ^ f p f q u V̂ 12u f r f s &
~14!
Consequently @compare Eqs. ~9!–~11!#,
E x 5 ~ 1/2!
(
pqrs
~ G pr n q d qs 1n p d pr G qs 2n p d pr n q d qs 12B pqrs !
3 ~ ^ f p f q u V̂ 12u f r f s & 2 ^ f p f q u V̂ 12u f s f r & !
~15!
or
are the same as
Eqs. ~5!, ~8!, and ~9! yields
G pq ^ f p u V̂ 1 u f q & .
(
pqrs
2 ^ f p f q u V̂ 12u f s f r & ! .
~9!
$ ^ f p u V̂ 1 u f q & % for one-electron perturbations, combining
E x5
~13!
(p A p e xp 12 pqrs
( B pqrs~ ^ p x q uu rs & 1 ^ pq uu r x s & !
1
(p A p e xp 12 pqrs
( B pqrs~ ^ p x q uu rs & 1 ^ pq uu r x s & ! .
x
e xp 5 ( C pqr F qr
(r n r~ ^ f p f ru V̂ 12u f q f r & 2 ^ f p f ru V̂ 12u f r f q &
and
are perturbation independent. Thanks to the antisymmetry of
the tensor $ B pqrs % and the double-bar integrals, Eq. ~6! can
be simplified to
E xc 5
n p n q ~ ^ f p f q u V̂ 12u f p f q &
(
pq
1 ^ f r f p u V̂ 12u f r f q & 2 ^ f r f p u V̂ 12u f q f r & ! ,
and B pqrs [ ] E c / ] ^ pq uu rs & ,
~11!
2 ^ f p f q u V̂ 12u f q f p & ! 1E xc @ $ e p % , $ ^ pq uu rs & % # , ~12!
x
1 ^ pq uu r x s & 1 ^ pq uu rs x & ! ,
(w ~ D rq* pw ^ ws uu tu & 1D t pqw ^ rs uu wu & ! .
Now consider a two-electron perturbation described by a
with
matrix
elements
Hermitian
operator
V̂ 12
$ ^ f p f q u V̂ 12u f r f s & % [ $ ^ f p (1) f q (2) u V̂ 12u f r (1) f s (2) & % . In
this case @compare Eqs. ~3!–~5! and ~8!#
E xc @ $ e p % , $ ^ pq uu rs & % #
A p e xp 1
(r A r C rpq 12 rstu
( B rstu
~10!
where the energy-derivative one-electron reduced density
matrix $ G pq % has the elements12,13
E x5
(
pqrs
g pqrs ^ f p f q u V̂ 12u f r f s & ,
~16!
where the energy-derivative two-electron reduced density
matrix $ g pqrs % has the elements12,13
g pqrs 52B pqrs 1 ~ 1/2!~ G pr n q d qs 1n p d pr G qs 2n p d pr n q d qs !
2 ~ 1/2!~ G ps n q d qr 1n p d ps G qr 2n p d ps n q d qr ! .
~17!
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J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
8227
FIG. 1. Isotropic Coulomb holes in ~a! H2 , ~b! He, ~c! Li1 , and ~d! Be21 .
The exact reduced density matrices are known to satisfy
three important sum rules, namely:15,17
(p G pp 5N,
~18!
g pqpq 5 ~ 1/2! N ~ N21 ! 5M ,
(
pq
~19!
and
g pq pq 5 ~ 1/2! N ~ N21 ! 1 ( ~ N2n p ! A p 12 ( B pqpq ,
(
pq
p
pq
~23!
and
G pq h pq 1 ( g pqrs ^ pq u rs &
(
pq
pqrs
5E2E c @ $ e p % , $ ^ pq uu rs & % #
G pq h pq 1 ( g pqrs ^ pq u rs & 5E,
(
pq
pqrs
~20!
where N and M are the numbers of electrons and electron
pairs, respectively. It is straightforward to demonstrate that
for the energy-derivative matrices $ G pq % and $ g pqrs % the sum
rules ~18! and ~20! follow from the transformation properties
of the function E c @ $ e p % , $ ^ pq uu rs & % # . Since12,13
G pp 5n p 1A p ,
~21!
Eqs. ~18!–~20! can be rewritten as
(p G pp 5N1 (p A p ,
~22!
1
(p A p e p 1 pqrs
( B pqrs ^ pq uu rs & .
~24!
Because E xc @ $ e p % , $ ^ pq uu rs & % # must be invariant with respect
to a uniform shift of orbital energies, i.e., it must conform to
the condition
(p ] E c / ] e p 5 (p A p 50,
~25!
the sum rule ~18! is satisfied by virtue of Eq. ~22!. Moreover,
since E c @ $ e p % , $ ^ pq uu rs & % # must scale linearly upon simulta-
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8228
J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
FIG. 2. Isotropic Coulomb holes in ~a! Li and ~b! Be.
neous multiplication of $ e p % and $ ^ pq uu rs & % by a real constant ~an operation that corresponds to change of the energy
unit!, the identity
(p ~ ] E c / ] e p ! e p 1 pqrs
( ~ ] E c / ] ^ pq uu rs & ! ^ pq uu rs &
5
(p A p e p 1 pqrs
( B pqrs ^ pq uu rs &
5E c @ $ e p % , $ ^ pq uu rs & % #
~26!
and thus also the sum rule ~20! holds @compare Eq. ~24!#. On
the other hand, the condition
B pqpq 52 ( B pqpq 2 ( n p A p 50
(p ~ N2n p ! A p 12 (
pq
pq
p
~27!
is rarely satisfied. Consequently, the two-electron reduced
density matrices produced by the energy-derivative formalism are usually not properly normalized.13 Moreover, since
the eigenvalues of neither $ G pq % nor $ @ N/2# 21 g pqrs % are
guaranteed to belong to the @0, 1# interval, these matrices are
FIG. 3. Isotropic Coulomb holes in ~a! F2 , ~b! Ne, and ~c! Na1 .
in general not N representable ~i.e., derivable from an Nelectron wave function!.
With both $ G pq % and the tensor $ B pqrs % being available
at several levels of theory from the contemporary quantumchemical software, approximate electron intracule densities
can be readily computed by forming $ g pqrs % on the fly and,
after transformation to the basis set of Gaussian primitives,
inputting it into the recently developed two-step integral
evaluation algorithm.14 For closed-shell systems, additional
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J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
8229
TABLE II. Intracule densities at R50, effective numbers of electron pairs, and electron–electron repulsion
energies in selected atoms and ions.a
System
M
W
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
22
1.298310
7.88331023
4.08831023
2.48631023
2.74531023
1.0000
1.0271
1.0438
1.0457
1.0000
0.3955
0.3639
0.3500
0.3469
0.3110
He
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
1.90631021
1.60631021
1.30031021
1.18431021
1.06431021
1.0000
1.0049
1.0106
1.0111
1.0000
1.0258
0.9992
0.9680
0.9624
0.9458
Li1
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
7.70231021
6.93531021
6.08931021
5.78531021
5.338 31021
1.0000
1.0021
1.0049
1.0051
1.0000
1.6517
1.6248
1.5871
1.5808
1.5677
Be21
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
1.9913100
1.8453100
1.6793100
1.6203100
1.5253100
1.0000
1.0011
1.0028
1.0030
1.0000
2.2771
2.2498
2.2087
2.2020
2.1909
Li
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
7.84231021
7.07231021
6.21031021
5.90031021
5.42631021
3.0000
3.0020
3.0054
3.0057
3.0000
2.2810
2.2540
2.2175
2.2114
2.1895
Be
HF/(20s)
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’
2.0993100
1.9523100
1.7743100
1.7113100
1.6073100
6.0000
6.0043
6.0564
6.0605
6.0000
4.4892
4.4565
4.4062
4.3956
4.3787
F
HF/(20s10p)
MP2/(20s10p)
MP2/(20s10p10d)
2.9943101
2.7963101
2.7443101
45.0000
45.0914
45.1230
44.6477
43.9100
43.7476
Ne
HF/(20s10p)
MP2/(20s10p)
MP2/(20s10p10d)
4.2543101
4.0053101
3.9353101
45.0000
45.0467
45.0695
54.0397
53.4856
53.3017
Na1
HF/(20s10p)
MP2/(20s10p)
MP2/(20s10p10d)
5.8453101
5.5383101
5.4473101
45.0000
45.0289
45.0465
63.2266
62.7646
62.5590
H
2
Level of theory
I(0)
All quantities listed in @a.u.#; ‘‘exact’’ values taken from Refs. 7 and 10.
a
savings in computational effort can be realized by converting
$ g pqrs % to the respective spinless reduced density matrix.
III. TEST CALCULATIONS
The formalism described in the preceding section of this
paper has been implemented in the GAUSSIAN 94 suite of
programs.18 In order to assess its performance, several test
calculations employing the MP2 correlation energy
function,15
E MP2
@ $ e p % , $ ^ pq uu rs & % #
c
5 ~ 1/4!
(
abi j
^ i j uu ab &^ ab uu i j & ~ e i 1 e j 2 e a 2 e b ! 21 ,
~28!
where the indices i and j ~a and b! refer to the occupied
~virtual! spinorbitals, were carried out for atoms and ions in
their ground states. The even-tempered basis sets,19 consisting of uncontracted s, p, and d Gaussian primitives with the
exponents $ z i % given by
z i 5 ab i21 ,
i51,... ,
~29!
were used ~Table I!. For He, Li, Li1 , Be, and Be21 , the
(20s), (20s10p), and (20s10p10d) basis sets with the
‘‘regular’’ exponents taken from the original paper were employed, whereas for H2 the exponents were scaled down
from those recommended for the hydrogen atom in order to
account for the diffuseness of electron density in the anion.
The (20s10p) and (20s10p10d) basis sets for F2 , Ne, and
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8230
J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
TABLE III. Radial maxima of intracule densities. Numbers of strongly correlated electron pairs, and their
repulsion energies in selected atoms and ions.a
System
Level of theory
R max
I(R max)
M corr
W corr
H2
MP2/(20s)
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’ ~Ref. 8!
‘‘exact’’ ~Ref. 10!
0.1198
0.9201
1.0803
0.8761
0.9272
7.88531023
5.43231023
5.14131023
4.05331023
4.031023
5.68531025
1.70731022
2.52931022
n/a
n/a
7.11531024
2.71931022
3.34431022
n/a
n/a
He
MP2/(20s10p)
MP2/(20s10p10d)
‘‘exact’’ ~Ref. 8!
‘‘exact’’ ~Ref. 10!
0.0366
0.1663
0.1778
0.1937
1.30031021
1.20231021
1.09431021
1.16531021
2.65931025
2.31131023
n/a
n/a
1.09131023
2.08131022
n/a
n/a
All quantities listed in @a.u.#.
a
Na1 involved the unscaled parameters a and b. In all cases,
the numbers of primitives were chosen by examining the
convergence of the computed quantities with respect to the
basis set sizes.
The isotropic Coulomb holes D(R),
are too high by between ca. 0.5% (Be21 ) and 12% (H2 ).
The MP2/(20s10p10d) estimates of I(0) and W in the
lithium and beryllium atoms are of comparable accuracy.
The effective numbers of electron pairs M,
M 54 p
D ~ R ! 54 p R DI ~ R ! ,
2
~30!
computed for the two-electron species by subtracting the
HF/(20s) intracule densities from those obtained at the
MP2/(20s), MP2/(20s10p), and MP2/(20s10p10d) levels
of theory are compared in Fig. 1 with their ‘‘exact’’ counterparts derived from 491-term Hylleraas-type wave
functions.10,20 The extent and the depth of the Coulomb hole
in the helium atom @Fig. 1~b!# are reproduced surprisingly
well with the (20s10p) and (20s10p10d) basis sets, whereas
the MP2/(20s) D(R) is far too shallow. Similar trends are
observed for the Coulomb holes in the Li1 and Be21 cations
@Figs. 1~c! and 1~d!#. As expected, the accuracy of the MP2
approximation steadily improves with increasing nuclear
charge. On the other hand, the MP2 formalism is less successful in reproducing the exact Coulomb hole in H2 @Fig.
1~a!#.
The ‘‘exact’’ Coulomb holes in the lithium and beryllium atoms ~computed with a variational Monte Carlo
method10,21 and thus less accurate for larger values of R! are
again well approximated by their MP2/(20s10p) and
MP2/(20s10p10d) counterparts ~Fig. 2!. For the 10-electron
species, highly accurate wave functions are not currently
available, making quantitative assessment of the computed
MP2/(20s10p) and MP2/(20s10p10d) D(R) ~Fig. 3! difficult. As expected, augmentation of the basis set with polarization functions deepens the Coulomb holes in F2 , Ne, and
Na1 , indicating improved description of electron correlation.
Inspection
of
Table
II
reveals
that
the
MP2/(20s10p10d) level of theory yields values of I(0) that
are within 10% of their ‘‘exact’’ counterparts in two-electron
systems. The corresponding electron–electron repulsion energies W,
W54 p
E
`
0
I ~ R ! R dR,
~31!
E
`
0
I ~ R ! R 2 dR,
~32!
are consistently greater than ~1/2! N (N21), reflecting the
incorrect normalization of $ g pqrs % ~see the previous section
of this paper!. The difference between M and the actual number of the electron pairs is also positive for the other systems
under study, as expected from the inequality
g pq pq 5 ~ 1/2! N ~ N21 ! 1 ~ 1/2! ( ^ i j uu ab &^ ab uu i j &
(
pq
abi j
3~ e i 1 e j 2 e a 2 e b ! 22 . ~ 1/2! N ~ N21 ! ~33!
that follows from Eqs. ~23! and ~28!.
Although intracule densities with electron–electron coalescence cusps can be obtained only from explicitly correlated wave functions involving terms linear in the interelectron distance9 or from the Hiller–Sucher–Feinberg
identity,10 the formalism described in this paper is in principle capable of producing I(R) with minima at R50. Indeed, maxima at R max.0 are observed in some of the computed densities ~Table III!. The values of R max and I(R max),
the numbers of the strongly correlated electron pairs M corr ,
and their repulsion energies W corr ,
M corr54 p
E
W corr54 p
E
R max
0
I ~ R ! R 2 dR
and
R max
0
I ~ R ! R dR,
~34!
appear to be very sensitive to the quality of basis sets employed in their computation.
IV. CONCLUSIONS
Application of the energy-derivative formalism to twoelectron reduced density matrices produces a robust approach to the approximate evaluation of electron intracule
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J. Chem. Phys., Vol. 109, No. 19, 15 November 1998
J. Cioslowski and G. Liu
densities and Coulomb holes in atoms and molecules. Being
valid at any level of theory based upon single-determinantal
reference state, this approach makes it possible to tailor the
accuracy and computational cost of I(R) to the requirements
of a given research project and the limitations of the available computer hardware. Thanks to this versatility, routine
calculations of correlated electron intracule densities are now
feasible for a wide spectrum of chemical systems.
Test calculations of I(R) in atoms and ions reveal the
impressive accuracy of Coulomb holes computed within the
MP2 approximation. However, quantitative reproduction of
the exact I(R) and DI(R) is found to be contingent upon the
use of large basis sets including Gaussian primitives with
high angular momenta. This observation suggests that the
results of previous CISD studies employing small basis sets11
should be viewed with caution.
The approximation
g pqrs ' ~ 1/2!~ G pr G qs 2G ps G qr !
~35!
has been recently proposed in conjunction with investigations of spin contamination in the Kohn–Sham theory.22 Test
calculations invoking this approximation in place of Eq. ~17!
reveal its gross inaccuracy. For this reason, Eq. ~35! does not
constitute a viable alternative to the present formalism.
ACKNOWLEDGMENTS
This work was partially supported by the National Science Foundation under Grant No. CHE-9632706 and Gaussian Inc. under a software development grant.
A. J. Coleman, Int. J. Quantum Chem. S1, 457 ~1967!.
A. J. Thakkar, ‘‘Extracules, intracules, correlation holes, potentials, coefficients and all that,’’ in Density Matrices and Density Functionals, edited
by R. Erdahl and V. H. Smith, Jr. ~Reidel, New York, 1987!, pp. 553–581.
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C. A. Coulson and A. H. Neilson, Proc. Phys. Soc. London 78, 831
~1961!.
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J. Cioslowski and G. Liu, J. Chem. Phys. 105, 8187 ~1996!.
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X. Fradera, M. Duran, and J. Mestres, J. Chem. Phys. 107, 3576 ~1997!.
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