JOURNAL OF CHEMICAL PHYSICS
VOLUME 113, NUMBER 22
8 DECEMBER 2000
An improved ab initio relativistic zeroth-order regular approximation
correct to order 1Õc 2
Wim Klopper,a) Joop H. van Lenthe, and Alf C. Hennumb)
Theoretical Chemistry Group, Debye Institute, Utrecht University, P. O. Box 80052, NL-3508 TB Utrecht,
The Netherlands
共Received 3 August 2000; accepted 15 September 2000兲
The equations of the original ab initio scalar-relativistic zeroth-order regular approximation
共ZORA兲 and the infinite-order regular approximation 共IORA兲 are expanded in orders of 1/c 2 . It is
shown that previous ZORA/IORA implementations in ab initio quantum chemistry programs were
not correct to order 1/c 2 , but contained imperfections leading to fictitious self-interactions. These
errors can be avoided by adding exchange-type terms 共coupling the large and small components兲 to
the relativistic ZORA correction to the Hamiltonian, yielding improved ab initio relativistic zerothand infinite-order regular approximations that are correct to order 1/c 2 . The new methods have been
tested numerically by computing the total energies, orbital energies, and static electric dipole
polarizabilities of the rare gas atoms He through Xe. © 2000 American Institute of Physics.
关S0021-9606共00兲31046-7兴
I. INTRODUCTION
where V includes the nuclear attraction (V nuc), the Coulomb
potential (J), and the exchange–correlation potential (V XC).
For a one-electron system, the KS equations reduce to the
correct one-particle CPD equations, since then—at least for
the exact KS potential—the J and V XC potentials mutually
cancel.
Whereas the introduction of the ZORA relativistic correction into KS theory seems natural and straightforward, the
situation is different for computational methods of ab
initio—that is, wave function based—quantum chemistry.
The implementation of the ZORA approach into ab initio
methods is nontrivial, especially at the correlated level.
One could, of course, try to consider only the nuclear
attraction potential—that is, insert V⫽V nuc into 共2兲. However, this leads to unphysical forces in molecules, since the
ZORA approach is not gauge invariant.2,9 The nuclei of a
neutral molecule A, at large distance from another neutral
molecule B, will induce a constant shift of the potential in
the region of B, and vice versa. This constant shift causes no
problems in a gauge-invariant computational method, but is
very problematic for the ZORA approach. Thus, one should
include at least the electrostatic Coulomb potential (J) in V
to cancel the long-range tails of the nuclear attraction.
This strategy has been followed by Faas et al.10 when
implementing the ZORA approach in the ab initio framework. Their computational scheme, however, leads to fictitious self-interactions. For example, the Coulomb potential J
does not vanish for a one-electron system, and unphysical
terms enter the ZORA operator. For a two-electron system
such as the He atom, the contribution from J is twice as large
as the contribution that is needed to reproduce the corresponding four-component 共i.e., Dirac–Fock兲 results.
Alternatively, one could employ model potentials such
as the ones that have been used successfully in KS
calculations.9,11–13 One could, for example, add van
Wüllen’s relativistic correction—which is a one-electron op-
The zeroth-order and infinite-order regular approximations 共ZORA and IORA兲1–6 have been introduced into numerical quantum chemistry to provide variationally stable
two-component computational methods for atomic and molecular electronic-structure calculations that take into account relativistic effects.
The ZORA 共or CPD兲 Hamiltonian7,8 is obtained by adding to the nonrelativistic one-particle Hamiltonian H⫽T
⫹V the relativistic correction ⌬T ZORA ,
H ZORA⫽H⫹⌬T ZORA ,
where
冉
共1兲
冊
1
V
␴•p.
⌬T ZORA⫽ ␴•p
2
2c 2 ⫺V
共2兲
Here, ␴⫽( ␴ x , ␴ y , ␴ z ) is the vector of the Pauli spin matrices, p⫽⫺i“ the momentum operator, and c the velocity of
light (c⫽137.035 9895 a 0 E h /ប). Equivalently, we can define a ZORA kinetic energy operator as
T ZORA⫽T⫹⌬T ZORA⫽ ␴•p
冉
冊
c2
␴•p.
2c 2 ⫺V
共3兲
The ZORA approach has been applied almost exclusively
within the framework of Kohn–Sham 共KS兲 density functional theory 共DFT兲. In this framework, the one-particle operator ⌬T ZORA is added to the one-particle equations
共 T ZORA⫹V 兲 ␺ i ⫽␧ i ␺ i ,
共4兲
V⫽V nuc⫹J⫹V XC ,
共5兲
a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected]
b兲
Present address: Department of Chemistry, University of Oslo, P. O. Box
1033 Blindern, N-0315 Oslo, Norway.
0021-9606/2000/113(22)/9957/9/$17.00
9957
© 2000 American Institute of Physics
9958
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
Klopper, van Lenthe, and Hennum
erator of the form 共2兲 obtained from a model potential V
⫽V model—to the one-electron Hamiltonian in Hartree–Fock
or post-Hartree–Fock ab initio methods. Also analytical energy gradients could easily be evaluated once the derivatives
of the corresponding one-electron integrals were available.
Very likely, the ab initio method would work well in practice, but one would have to resort to a DFT computer program to generate the required one-electron integrals and derivatives thereof 共to be able to compute the exchange–
correlation potential from the model density and the oneelectron integrals related to that potential兲.
The use of a DFT-type model potential and corresponding relativistic correction in ab initio methods appears inconsistent to us, however. Therefore, we wish to start from the
ab initio four-component Dirac–Fock theory for the derivation of the ZORA approximation in a rigorous ab initio
framework. The regular approximation is then derived by
eliminating the small component from the Dirac spinors.
We emphasize that the present paper is concerned only
with the Dirac–Coloumb Hamiltonian, which combines the
one-particle Dirac operator with the 共nonrelativistic兲 twoparticle interelectronic Coulomb repulsion. Two-particle
Breit or Gaunt operators are not considered and neither are
current densities in the KS framework 共a nonrelativistic approximation is used for the exchange–correlation potential兲.
For brevity, we use the short notations ‘‘Dirac–Fock’’ and
‘‘Dirac–Kohn–Sham’’ to designate the Dirac–Hartree–
Fock–Coloumb and Dirac–Kohn–Sham–Coulomb theories,
respectively.
The elimination of the small component is achieved by
performing a Foldy–Wouthuysen14 transformation to the
Dirac–Fock operator
F Dirac–Fock⫽
冉
c ␴•p⫺K 12
V nuc⫹J⫺K 11
c ␴•p⫺K 21
⫺2c ⫹V nuc⫹J⫺K 22
2
冊
,
⫽
冉
c ␴•p
V nuc⫹J⫹V XC
c ␴•p
⫺2c ⫹V nuc⫹J⫹V XC
2
冊
In this paper we propose a new and improved ab initio
ZORA method. In the first subsection on the theory we apply
the Foldy–Wouthuysen transformation to the Dirac–Fock
operator 共6兲 and derive the ZORA Fock operator. In the second subsection we discuss our new method and point out the
differences to the former ab initio implementations. Then,
we analyze the resulting equations to leading order in 1/c 2 ,
demonstrating that the new equations are correct to order
1/c 2 while the former equations were not. Finally, in the last
subsection we discuss the gauge-dependence problem of the
共scaled兲 ZORA/IORA schemes.
A. Foldy–Wouthuysen transformation
Following the approach of Kutzelnigg,15 we use a unitary transformation U i given by
U i⫽
冉
1
冑
1
冑1⫹X †i X i
1⫹X †i X i
⫺
1
X
† i
冑1⫹X i X i
1
冑1⫹X i X †i
X †i
冊
.
共8兲
The two-component Foldy–Wouthuysen 共FW兲 orbital is then
obtained as
冉 冊 冉冊
␺ FW
i
0
␸i
⫽U i ␹ ,
i
共9兲
where ␸ i and ␹ i are the large and small components of the
four-component Dirac spinor, respectively. Inverting 共9兲
gives
共6兲
with J⫽J 11⫹J 22 . The various Coulomb and exchange operators will be explained in the next section. Here, it is sufficient to notice that the ab initio Dirac–Fock operator not
only differs from the one-particle Dirac and KS operators by
the presence of the diagonal exchange operators K 11 and
K 22 , but also by the presence of the off-diagonal exchange
operators K 12 and K 21 . Compare, for example, with
F Dirac–Kohn–Sham
II. THEORY
␸ i⫽
1
冑1⫹X †i X i
␺ FW
i
共10兲
and
␹ i ⫽X i ␸ i .
共11兲
It thus follows that
FW
␺ FW
i * ␺ i ⫽ ␸ i* ␸ i ⫹ ␹ i* ␹ i .
共12兲
⫺1
F FW
i ⫽U i F Dirac–FockU i ,
For the transformed Fock matrix,
to be block diagonal, the operator X i has to satisfy
.
共7兲
One can rationalize the ab initio ZORA method of Faas
et al.10 by omitting the operators K 22 , K 12 , and K 21 in 共6兲.
Then, a Fock operator similar to 共7兲 results, and one is automatically led to taking the potential V⫽V nuc⫹J in 共2兲.
It is the purpose of the present paper to propose and
implement an ab initio ZORA scheme that neglects K 22 , but
takes into account the off-diagonal exchange operators K 12
and K 21 that couple the large and small components of the
Dirac spinors. The explicit inclusion of these operators
avoids the aforementioned unphysical self-interaction and
enables the derivation of a ZORA approach that is equivalent
to Dirac–Fock theory to order 1/c 2 .
X i⫽
1
␧ i ⫹2c ⫺V nuc⫺J⫹K 22
2
共 c ␴•p⫺K 21兲 .
共13兲
The matrix elements of the exchange and Coulomb operators
are defined as
具 ␰ t 兩 J 兩 ␰ u 典 ⫽ 具 ␰ t 兩 J 11兩 ␰ u 典 ⫹ 具 ␰ t 兩 J 22兩 ␰ u 典
⫽
FW
兺i 共 ␰ t ␰ u兩 ␺ FW
i ␺i 兲,
FW
具 ␰ t 兩 K 兩 ␰ u 典 ⫽ 兺 共 ␰ t ␺ FW
i 兩␺i ␰u兲,
i
with
共14兲
共15兲
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
Relativistic approximation
具 ␰ t 兩 J 11兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 ␸ i ␸ i 兲 ,
F0
i
具 ␰ t 兩 J 22兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 ␹ i ␹ i 兲 ,
共16兲
i
i
具 ␰ t 兩 K 22兩 ␰ u 典 ⫽ 兺 共 ␰ t ␹ i 兩 ␹ i ␰ u 兲 ,
共17兲
i
具 ␰ t 兩 K 12兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ i 兩 ␹ i ␰ u 兲 ,
i
具 ␰ t 兩 K 21兩 ␰ u 典 ⫽ 兺 共 ␰ t ␹ i 兩 ␸ i ␰ u 兲 .
共18兲
i
Here, 兵 ␸ i 其 is the set of occupied large component functions,
兵 ␹ i 其 the set of occupied small components, and 兵 ␰ p 其 an
atomic basis function.
As in Refs. 1, 2, and 4, we apply the 共energyindependent兲 regular approximation
1
2c ⫺V nuc⫺J⫹K 22
2
共 c ␴•p⫺K 21兲 ,
共19兲
which amounts to neglecting ␧ i in 共13兲. The Fock operator
for the positive energy states resulting from the transformation within the regular approximation is
F FW⫽
1
冑1⫹X † X
⫻
冋
V nuc⫹J⫺K 11⫹ 共 c ␴•p⫺K 12兲
1
共 c ␴•p⫺K 21兲
2c 2 ⫺V nuc⫺J⫹K 22
册冑
1
1⫹X † X
.
共20兲
The term in square brackets is the ab initio ZORA operator
共cf. Refs. 10, 16–18兲
F 0 ⫽ 共 c ␴•p⫺K 12兲
1
共 c ␴•p⫺K 21兲
2c ⫺V nuc⫺J⫹K 22
2
⫹V nuc⫹J⫺K 11 .
共21兲
Expanding the square root in 共20兲 in a power series in X † X
yields
1
F FW⫽F 0 ⫺ 兵 X † X,F 0 其 ⫹ ⫹••• ,
2
共22兲
where 兵 其 ⫹ denotes the anticommutator 兵 A,B 其 ⫹ ⫽AB⫹BA.
The zeroth-order term in the expansion 共22兲 defines the
ZORA—zeroth-order regular approximation—operator. The
corresponding ZORA equation is
FW
F 0 ␺ FW
i ⫽␧ i ␺ i .
冑1⫹X † X
FW
†
␺ FW
i ⫽␧ i 冑1⫹X X ␺ i ,
共23兲
As an alternative to taking into account only the zeroth-order
FW
to infinite order by
term, we can solve F FW␺ FW
i ⫽␧ i ␺ i
writing the eigenvalue equation as
共24兲
or, by virtue of 共10兲,
F 0 ␸ i ⫽␧ i 共 1⫹X † X 兲 ␸ i .
具 ␰ t 兩 K 11兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ i 兩 ␸ i ␰ u 兲 ,
X i ⬇X⫽
1
9959
共25兲
This equation is referred to as IORA—infinite-order regular
approximation—equation.4
It is important to notice that the two-component orbitals
that solve the ZORA equation 共23兲 are Foldy–Wouthuysen
functions 共Schrödinger or ZORA picture兲, whereas the twocomponent orbitals that solve the IORA equation 共25兲 correspond to the large components of the four-component Dirac
spinors 共Dirac or IORA picture兲. In the IORA case, we must
explicitly construct the small components ␹ i ⫽X ␸ i 共or,
equivalently, the Foldy–Wouthuysen functions ␺ FW
i
⫽ 冑1⫹X † X ␸ i ) in order to compute the one-electron density
and the Coulomb potential J 共cf. Ref. 4兲.
B. Improved ab initio ZORA
By omitting all spin–orbit terms in 共21兲, we obtain the
scalar-relativistic, one-component version of ZORA/IORA,
F 0 共 scalar兲 ⫽ 共 cp⫺K12兲 •
1
共 cp⫺K21兲
2c ⫺V nuc⫺J⫹K 22
2
⫹V nuc⫹J⫺K 11 .
共26兲
In the course of the present work, we have implemented a
scalar-relativistic, one-component ZORA/IORA approach
into the SORE program19 to test our new method numerically.
Therefore, we focus on the scalar-relativistic variant in the
following.
In the IORA picture, we make the approximation
F IORA⬇ 共 cp⫺K12兲 •
1
共 cp⫺K21兲
2c ⫺V nuc⫺J 11
2
⫹V nuc⫹J⫺K 11 ,
共27兲
that is, we neglect the operators K 22 and J 22 in the denominator. The computation of the corresponding terms would
involve two-electron integrals of the type 具 p␰ p 兩 J 22
⫺K 22兩 p␰ q 典 —that is, over four small components from a
computational viewpoint—while their contributions are negligible for our purposes.
The ZORA approach requires a slightly different Fock
operator. In the sense of a neglect of picture change, the
of the ZORA Fock
Foldy–Wouthuysen eigenfunctions ␺ FW
i
operator 共23兲—or 共26兲—take the place of the large components in IORA. We again neglect K 22 and write
F ZORA⬇ 共 cp⫺K12兲
1
共 cp⫺K21兲
2c ⫺V nuc⫺J 11
2
⫹V nuc⫹J 11⫺K 11 .
共28兲
It is not an approximation to write J as J 11 , because the
‘‘large component’’ ␸ i is in fact a Foldy–Wouthuysen function. Accordingly, J 11 yields the full Coulomb potential by
9960
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
Klopper, van Lenthe, and Hennum
virtue of 共14兲. The neglect of picture change occurs in the
operators K 11 , K21 , and K12—the exchange operator K 22 is
removed from the denominator, but not J 22 .
Furthermore, we approximate X as
X⫽
c
2c ⫺V nuc⫺J 11
2
p.
共29兲
Within the scalar-relativistic ZORA/IORA framework,
the exchange and Coulomb matrix elements take the form
具 ␰ t 兩 J 22兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 X• ␸ i X␸ i 兲 ,
共30兲
具 ␰ t 兩 K 22兩 ␰ u 典 ⫽ 兺 共 ␰ t X• ␸ i 兩 X␸ i ␰ u 兲 ,
共31兲
具 ␰ t 兩 K12兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ i 兩 X␸ i ␰ u 兲 ,
共32兲
具 ␰ t 兩 K21兩 ␰ u 典 ⫽ 兺 共 ␰ t X␸ i 兩 ␸ i ␰ u 兲 .
共33兲
i
i
i
i
The scalar-relativistic ZORA/IORA equations represent effective equations for the large component orbitals 兵 ␸ i 其 while
the corresponding small components are given by 兵 X␸ i 其 .
The total IORA energy is obtained by inserting the
IORA large and small components into the expression for the
Dirac-Fock expectation value,
E Dirac-Fock⫽
1
兺i ␧ i ⫺ 2 兺i 具 ␸ i兩 J 11⫺K 11兩 ␸ i 典
⫺
⫺
兺i 具 ␸ i兩 J 22兩 ␸ i 典 ⫹ 兺i 具 ␸ i兩 K 12兩 ␹ i 典
1
2
兺i 具 ␹ i兩 J 22⫺K 22兩 ␹ i 典 .
共34兲
兺i 具 ␸ i兩 K12•X兩 ␸ i 典 ⫺ 兺i 具 ␸ i兩 J 22兩 ␸ i 典 .
E ZORA⫽
兺i
⫹
共35兲
and
respectively, with
⫽
␧ scaled
i
␧i
1⫹ 具 ␸ i 兩 X† •X兩 ␸ i 典
共38兲
.
Notice that the ZORA and scaled ZORA total energies do
not contain terms with J 22 , as J 11 already represents the full
Coulomb potential in these cases.
Matrix elements that depend on X␸ i —that is, on the
small component—are evaluated by expanding X␸ i in the
basis 兵 p␸ p 其 ,
X␸ i ⬇
兺p x pi p␸ p
共39兲
with
x pi ⫽
兺q S ⫺1
pq 具 p• ␸ q 兩 X␸ i 典
⫽c
1
兺q S ⫺1
pq 具 p• ␸ q 兩
2c 2 ⫺V
兩 p␸ i 典 .
共40兲
c
⫺1
Here, V c ⫽V nuc⫹J 11 and S ⫺1
pq is a matrix element of S ,
which is the inverse of the metric S pq ⫽ 具 p• ␸ p 兩 p␸ q 典 . We
represent the potential V c in this basis and obtain the matrix
representation of 1/(2c 2 ⫺V c ) from
1
2c 2 ⫺V c
兩 p␸ u 典 ⬇ 共 S1/2V⫺1 S1/2兲 tu ,
共41兲
where V⫺1 is the inverse of the matrix
V pq ⫽
⫺1/2
2
S ⫺1/2
兺
pr 具 p• ␸ r 兩 2c ⫺V c 兩 p␸ s 典 S sq .
r,s
共42兲
Accordingly,
Matrix elements involving K12 and K21 are computed as
具 ␸ t 兩 K12•
⬇
兺i 具 ␸ i兩 J 11⫺K 11兩 ␸ i 典
兺i 具 ␸ i兩 K12•X兩 ␸ i 典
共37兲
r,s
Similarly, the total energies of the ZORA and scaled ZORA
approaches are obtained from the expressions
1
␧ i⫺
2
兺i 具 ␸ i兩 K12•X兩 ␸ i 典 ,
具 ␸ t 兩 X† •X兩 ␸ u 典 ⬇ 兺 x rt* S rs x su ⫽c 2 共 S1/2V⫺2 S1/2兲 tu . 共43兲
1
兺i ␧ i ⫺ 2 兺i 具 ␸ i兩 J 11⫺K 11兩 ␸ i 典
⫹
1
⫺ 兺 具 ␸ i 兩 J 11⫺K 11兩 ␸ i 典
兺i ␧ scaled
i
2 i
⫹
具 p• ␸ t 兩
The last term in 共34兲 involves two-electron integrals of the
type (SS 兩 SS) over small components only and these integrals are neglected within the ZORA/IORA framework.
Hence, we compute the IORA total energy as
E IORA⫽
E scaled–ZORA⫽
共36兲
具 ␸ t 兩 p•
⬇
1
p兩 ␸ u 典
2c 2 ⫺V c
1
兩 p␸ u 典 ,
具 ␸ t 兩 K12• 兩 p␸ p 典 S ⫺1
兺
pq 具 p• ␸ q 兩
2c 2 ⫺V c
p,q
共44兲
1
K 兩␸ 典
2c 2 ⫺V c 21 u
1
兩 p␸ p 典 S ⫺1
具 p• ␸ t 兩 2
兺
pq 具 p• ␸ q 兩 K21兩 ␸ u 典 ,
2c ⫺V c
p,q
共45兲
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
具 ␸ t 兩 K12•
⬇
Relativistic approximation
(0)
(0)
(0)
(1)
(0)
(0)
␧ (1)
i ⫽ 具 ␸ i 兩 X • 共 V nuc⫹J 11 ⫺␧ i 兲 X ⫹J 11
1
K 兩␸ 典
2c ⫺V c 21 u
2
兺
(1)
(0)
(0)
⫺K 11
⫹J 22
⫺2K12
•X(0) 兩 ␸ (0)
i 典.
具 ␸ t 兩 K12• 兩 p␸ p 典 S ⫺1
pq
p,q,r,s
1
⫺1
⫻ 具 p• ␸ q 兩 2
兩 p␸ r 典 S rs
具 p• ␸ s 兩 K21兩 ␸ u 典 .
2c ⫺V c
i
共53兲
The operators are expanded as
共46兲
All operators that occur in our new approach can
be constructed from two-electron integrals of the type
( ␰ a ␰ b 兩 p• ␰ c p␰ d ), where 兵 ␰ a 其 is the basis set of atomic orbitals. These integrals are needed anyway to build the matrix
representation of J 11 in the 兵 p␸ p 其 basis,
具 p• ␸ t 兩 J 11兩 p␸ u 典 ⫽ 兺 共 p• ␸ t p␸ u 兩 ␸ i ␸ i 兲 ,
9961
共47兲
(0)
(0) (0)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ (0)
具 ␰ t 兩 K12
i 兩X ␸i ␰u兲,
共54兲
(0)
(0)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 ␸ (0)
具 ␰ t 兩 J 11
i ␸i 兲,
共55兲
i
i
(1)
(0)
(0) (1)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 ␸ (1)
具 ␰ t 兩 J 11
i ␸ i 兲 ⫹ 兺 共 ␰ t␰ u兩 ␸ i ␸ i 兲 ,
i
i
共56兲
(0)
(0) (0)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␰ u 兩 X(0) • ␸ (0)
具 ␰ t 兩 J 22
i X ␸i 兲,
i
具 p• ␸ t 兩 K21兩 ␸ u 典 ⫽ 兺 共 p• ␸ t X␸ i 兩 ␸ i ␸ u 兲
共57兲
(0)
(0)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ (0)
具 ␰ t 兩 K 11
i 兩␸i ␰u兲,
i
i
⬇
兺i 兺s x si共 p• ␸ t p␸ s兩 ␸ i ␸ u 兲 ,
共48兲
i
具 ␸ t 兩 K12• 兩 p␸ u 典 ⫽ 兺 共 ␸ t ␸ i 兩 X• ␸ i p␸ u 兲
兺i 兺s x si*共 ␸ t ␸ i兩 p• ␸ s p␸ u 兲 ,
and so on. This yields
共49兲
(1)
⫽
E IORA
具 ␸ t 兩 J 22兩 ␸ u 典 ⫽ 兺 共 ␸ t ␸ u 兩 X• ␸ i X␸ i 兲
* x si 共 ␸ t ␸ u 兩 p• ␸ r p␸ s 兲 .
x ri
兺i 兺
r,s
共50兲
The extra computational cost needed to build four operators
instead of one is insignificant.
C. Perturbation expansion in 1Õ c2
(1)
E DPT
⫽
(0)
(0)
(0)
⫽E scaled–ZORA
⫽E IORA
⫽E Hartree–Fock
E ZORA
1
共51兲
(0) (0)
(0)
(0) (0)
兺i 兵 ␧ (1)
i ⫺ 具 ␸ i 兩 J 22 ⫺K12 •X 兩 ␸ i 典 其
⫺2 Re
(0) (0) (0)
兵 共 ␸ (1)
兺
i ␸i 兩␸ j ␸ j 兲
i, j
(0) (0) (0)
⫺ 共 ␸ (1)
i ␸ j 兩␸ j ␸ i 兲其,
with X ⫽ p and
(0)
1
2
(0)
(0) (0)
兺i 具 ␹ (0)
i 兩 V nuc⫹J 11 ⫺␧ i 兩 ␹ i 典
⫺
(0) (0)
兺i 具 ␸ (0)
i 兩 K 12 兩 ␹ i 典 ,
共60兲
1
(0)
␹ (0)
i ⫽ ␴•p␸ i ,
2
For the first-order energy 共i.e., of order 1/c 2 ) we obtain the
expression
(1)
⫽
E IORA
共59兲
with
(0) (0) (0)
兵 共 ␸ (0)
兺i ␧ (0)
i ⫺
i ␸i 兩␸ j ␸ j 兲
2 兺
i, j
(0) (0) (0)
⫺ 共 ␸ (0)
i ␸ j 兩␸ j ␸ i 兲其.
(0)
(0) (0)
兺i 具 ␸ (0)
i 兩 K12 • 兩 X ␸ i 典 ,
(1)
(1)
and K 11
cancel with the
as the terms with the operators J 11
(0)
. This firsttwo-electron integrals in 共52兲, and so does J 22
order energy is equal to the first-order energy of relativistic
direct perturbation theory 共DPT兲, which is given by 共see, for
example, Refs. 20, 21兲
It is obvious that the zeroth-order total ZORA/IORA energies are equal to the total nonrelativistic Hartree–Fock energy
⫽
(0)
(0)
(0) (0)
兺i 具 X(0) • ␸ (0)
i 兩 V nuc⫹J 11 ⫺␧ i 兩 X ␸ i 典
⫺
i
⬇
i
共58兲
i
⬇
(1)
(0)
(0) (1)
兩 ␰ u 典 ⫽ 兺 共 ␰ t ␸ (1)
具 ␰ t 兩 K 11
i 兩␸ i ␰ u 兲⫹ 兺 共 ␰ t␸ i 兩␸ i ␰ u 兲,
共52兲
(0) (0)
(0) (0) (0) (0)
具 ␸ (0)
i 兩 K 12 兩 ␹ i 典 ⫽ 兺 共 ␸ i ␸ j 兩 ␹ j ␹ i 兲 .
j
共61兲
The first-order DPT energy is correct to order 1/c 2 , and so is
the IORA approach. This would not be true however, if the
K12 and K21 operators had been neglected in 共27兲.
A former point of concern was the fact that the term
(0)
contributed to the first-order energy of a onecontaining J 11
electron system. When describing a one-electron system
within the DPT or our new ZORA/IORA framework, the
(0) (0)
兩 ␹ 典 cancels exactly the Coulomb
extra term ⫺ 具 ␸ (0) 兩 K 12
(0) (0) (0)
term 具 ␹ 兩 J 11 兩 ␹ 典 .
9962
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
Klopper, van Lenthe, and Hennum
TABLE I. Zeroth- and first-order contributions 共in E h ) to the DPT, ZORA, scaled ZORA, and IORA energies
of rare gas atoms, computed with the Ahlrichs TZV basis set 共Ref. 23兲.
Atom
E (0)
(0)
具 ␹ (0)
i 兩 V nuc兩 ␹ i 典
(0) (0)
⫺␧ (0)
i 具␹i 兩␹i 典
(0) (0)
具 ␹ (0)
i 兩 J 11 兩 ␹ i 典
(0) (0)
⫺ 具 ␸ (0)
i 兩 K 12 兩 ␹ i 典
He
Ne
Ar
Kr
Xe
⫺2.859 90
⫺128.541
⫺526.803
⫺2752.00
⫺7232.04
⫺0.000 28
⫺0.282
⫺3.445
⫺63.28
⫺344.91
0.000 07
0.082
1.027
19.25
106.17
0.000 16
0.073
0.686
9.54
43.83
⫺0.000 08
⫺0.017
⫺0.115
⫺1.06
⫺3.87
We note in passing that it is obvious that the DPT and
IORA first-order energies both are gauge invariant as they
contain the difference V nuc⫺␧ (0)
i . This is consistent with the
observation that the gauge dependence of the IORA approach is O(c ⫺4 ).4
The zeroth-order ZORA and scaled ZORA energies are
equal to the Hartree–Fock energies, but we find that the firstorder ZORA energy is different from its IORA counterpart.
For ZORA we obtain
(1)
⫽
E ZORA
(0)
(0) (0)
兺i 具 X(0) • ␸ (0)
i 兩 V nuc⫹J 11 兩 X ␸ i 典
⫺
兺i
(0)
(0) (0)
具 ␸ (0)
i 兩 K12 • 兩 X ␸ i 典 .
共62兲
共63兲
This follows directly from the expansion of the scaled orbital
energies:
1 (0)
(0) 2 (0)
兵 ␧ scaled
其 (1) ⫽␧ (1)
i
i ⫺ 4 ␧i 具␸i 兩p 兩␸i 典,
共64兲
where we have used that
1
1⫹ 具 ␸ i 兩 X† •X兩 ␸ i 典
⫽1⫺ 具 ␸ i 兩 X† •X兩 ␸ i 典
⫹ 具 ␸ i 兩 X† •X兩 ␸ i 典 2 ⫺••• .
In the preceding section, we have found that the ZORA
approach suffers from a gauge dependence of order 1/c 2 ,
which can be estimated by the expression (⌬/2c 2 ) 具 T 典 . The
IORA equation is not gauge invariant either and it appears
that the corresponding error is of order 1/c 4 .4 When we add
a constant ⌬ to both the potential V nuc and the orbital energy
␧ i , the terms of order 1/c 2 on both sides of 共25兲 cancel. To
order 1/c 4 , however, the difference between the left- and
right-hand sides amounts to
⫺p•
Thus, in the ZORA approach, the terms containing the
zeroth-order orbital energies are missing and already the
first-order ZORA energy is gauge dependent. When we add a
constant ⌬ to the potential V nuc , the first-order ZORA energy is increased incorrectly by (⌬/2c 2 ) 具 T 典 .
In contrast, the first-order scaled ZORA energy is indeed
gauge invariant, simply because the first-order scaled ZORA
energy is equivalent to the first-order IORA energy,
(1)
(1)
E scaled–ZORA
⫽E IORA
.
D. Gauge dependence
共65兲
TABLE II. First-order DPT and scalar-relativistic 共scaled兲 ZORA/IORA
total energies 共in E h ) of rare gas atoms, computed with the Ahlrichs TZV
basis set 共Ref. 23兲.
Atom
DPT
IORA
Scaled ZORA
ZORA
He
Ne
Ar
Kr
Xe
⫺2.860 03
⫺128.686
⫺528.650
⫺2787.56
⫺7430.82
⫺2.860 03
⫺128.686
⫺528.634
⫺2786.62
⫺7420.83
⫺2.860 03
⫺128.686
⫺528.633
⫺2786.53
⫺7419.61
⫺2.860 10
⫺128.768
⫺529.653
⫺2805.34
⫺7523.02
⌬ 共 ⌬⫹2␧ i 兲
p.
8c 4
共66兲
Thus, the error ␦ in the energy shift of the scalar-relativistic
IORA total energy can be estimated as
␦ ⬇⫺
⌬
4c 4
兺i 共 ⌬⫹2␧ i 兲 具 ␸ i兩 T 兩 ␸ i 典 .
共67兲
In this equation, ␧ i is the orbital energy associated with
⌬⫽0. A gauge dependence of order 1/c 4 might seem negligible, but we note that the noninvariance error increases with
the nuclear charge as Z 4 . Thus, the relevant gauge dependence is proportional to (Z/c) 4 and can easily become problematic for calculations on molecules containing heavy atoms (Z⬎50).
III. NUMERICAL RESULTS
A. Computational details
The one-component calculations have been performed
with a modified version of the SORE program19 on the SGI
Origin 2000 of the Chemistry Department of Utrecht University. We have compared our results with full four-component
Dirac–Fock calculations, which were carried out with the
22
DIRAC program on the SGI Origin 2000 of the University
of Bergen, Norway.
TABLE III. Hartree–Fock repulsion and MP2 correlation contributions 共in
kelvin units兲 to the interaction energy of the Ne dimer, as computed with the
C1D2 basis set at the fixed internuclear separation of R Ne–Ne⫽6.1a 0 . Results
obtained by Faas et al. 共Ref. 17兲 are given in parentheses.
Nonrelativistic
ZORA
Scaled ZORA
First-order DPT
Hartree–Fock
⌬MP2共FULL兲
15.66 共15.7兲
15.80 共15.8兲
15.65 共15.7兲
15.65
⫺38.76 (⫺38.7)
⫺38.79 (⫺38.8)
⫺39.04 (⫺39.0)
⫺38.79
⌬MP2共FC兲
⫺38.68
⫺38.70
⫺38.95
⫺38.71
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
Relativistic approximation
9963
TABLE IV. Static electric dipole polarizability 关in (ea 0 ) 2 /E h ] of rare gas atoms, computed with the uncontracted and augmented Ahlrichs TZV basis set 共Ref. 23兲. The basis set has been augmented with diffuse
functions and a set of polarization functions 共see the text兲.
Atom
Hartree–Fock
Dirac–Fock
DPT
IORA
Scaled ZORA
ZORA
He
Ne
Ar
Kr
Xe
1.316 75
2.3594
10.704
15.64
26.6
1.316 55
2.3625
10.720
15.64
26.5
1.316 55
2.3624
10.718
15.61
26.3
1.316 55
2.3624
10.719
15.62
26.3
1.316 55
2.3624
10.718
15.61
26.3
1.316 56
2.3710
10.840
16.32
29.4
The static electric dipole polarizability was computed by
applying finite perturbation theory. Electric field strengths of
0.0, ⫾0.001, ⫾0.002, ⫾0.003, and ⫾0.004 E h /(ea 0 ) were
applied and the computed energies were fitted to fourth-order
polynomials in the field strength. Note that a calculation with
a finite electric field along the z axis, for example, involves
one-electron integrals of the type 具 p• ␰ t 兩 z 兩 p␰ u 典 .
We have used the Ahlrichs TZV basis sets,23 except for
the computation of the electric dipole polarizabilities and the
interaction energy of the Ne dimer. Only the spherical–
harmonic components of the basis sets were used.
The Ahlrichs TZV basis sets for He, Ne, Ar, Kr, and Xe
are contracted sets of the form 3s, 5s3p, 5s4p, 6s5 p2d,
and 8s7 p5d, respectively. For the computation of the
electric dipole polarizability, we have augmented the corresponding primitive sets with both diffuse functions and
sets of polarization functions. Two diffuse sets were
added to every shell of the TZV basis by geometrical
extrapolation, and l -type polarization functions were
added with exponents obtained by applying the recipe
␨ l ⫽ ␨ l ⫺1 (2 l ⫹3)/(2 l ⫹1) to the exponents ␨ l ⫺1 of
the most diffuse functions of the ( l ⫺1) shell of the
primitive, doubly augmented TZV basis 共cf. Ref. 24兲.
The uncontracted and augmented TZV sets used for
the computation of the electric dipole polarizability
read 7s4p, 13s8p4d, 16s11p5d, 19s15p8d3 f , and 21s
17p11d6 f , respectively, for He through Xe.
B. Discussion
Various contributions to the first-order relativistic DPT
energies of the atoms He through Xe are displayed in Table
I. DPT, IORA, and scaled ZORA are equivalent at the
TABLE V. Gauge dependence of the 共scaled兲 ZORA/IORA methods. The
positive constant ⌬ that was added to the one-electron potential V nuc was
chosen such that the total energy was supposed to shift by exactly 100 E h .
The deviation 共in E h 兲 from this exact shift is displayed in the table, as
obtained from calculations with the Ahlrichs TZV basis set 共Ref. 23兲.
ZORA
Atom Predicted a Observed
He
Ne
Ar
Kr
Xe
a
0.0038
0.034
0.078
0.20
0.36
0.0038
0.034
0.077
0.20
0.34
IORA
Predicted b
Observed
⫺0.000 0049 ⫺0.000 0052 ⫺0.000 0005
0.000 034
0.000 021 ⫺0.000 016
0.000 29
0.000 24
0.000 18
0.0028
0.0024
0.0018
0.010
0.009
0.006
Predicted from the expression (⌬/2c 2 ) 具 T 典 .
Estimated from Eq. 共67兲.
b
Observed
Scaled ZORA
O(c ⫺2 ) level, and the entries of all columns of Table I contribute to the DPT, IORA, and scaled ZORA relativistic en(0)
ergies. The ⫺␧ i 具 ␹ (0)
i 兩 ␹ i 典 terms, however, do not contribute
to the parent, unscaled ZORA approach. These terms enter
the ZORA approach only through the scaling 共scaled ZORA兲
or through the renormalization procedure 共IORA兲. Accordingly, we find that the unscaled ZORA total energies are
lower than the DPT, IORA, and scaled ZORA total energies
(0)
by about the ⫺␧ i 具 ␹ (0)
i 兩 ␹ i 典 terms 共Table II兲. For He and Ne,
which are well described by first-order perturbation theory,
the unscaled ZORA energies are 0.000 07 and 0.082 E h ,
respectively, lower than the DPT, IORA, and scaled ZORA
total energies. The latter are virtually identical. The O(c ⫺2 )
error of unscaled ZORA appears to be equally important for
all atoms He through Xe. For these atoms, the magnitude of
(0)
the missing ⫺␧ i 具 ␹ (0)
i 兩 ␹ i 典 contribution amounts to about
(0)
(0)
30% of the 具 ␹ i 兩 V nuc兩 ␹ i 典 term.
In the former ab initio 共scaled兲 ZORA implementation
(0) (0)
by Faas et al.,10 the ⫺ 具 ␸ (0)
i 兩 K 12 兩 ␹ i 典 terms were neglected
共last column of Table I兲. This introduced a large error for
light elements such as He and Ne. For He, the former scaled
ZORA relativistic correction amounted to ⫺0.000 05 E h instead of the correct value of ⫺0.000 13 E h . Thus, the relative error was ca. 60%. For Ne, the error of the former implementation was already much smaller, but still significant 共ca.
10%兲. Fortunately, because the former method was applied
mainly to heavy systems (Z⬎50)—the relative error decreased rapidly with increasing nuclear charge. For Ar, Kr,
and Xe, the relative errors were only about 6%, 3%, and 2%,
respectively.
Errors of about 10% in relativistic corrections to absolute energies of light systems (Z⬇10) might have serious
TABLE VI. Scalar-relativistic scaled ZORA orbital energies 共in E h ) of rare
gas atoms, computed with the Ahlrichs TZV basis set 共Ref. 23兲.
1s
2s
2p
3s
3p
3d
4s
4p
4d
5s
5p
He
Ne
Ar
Kr
Xe
⫺0.9169
⫺32.82
⫺1.935
⫺0.8485
⫺119.3
⫺12.43
⫺9.595
⫺1.285
⫺0.5895
⫺530.6
⫺72.14
⫺63.85
⫺11.20
⫺8.433
⫺3.760
⫺1.184
⫺0.5233
⫺1274.0
⫺201.7
⫺182.6
⫺42.75
⫺36.14
⫺25.88
⫺8.355
⫺6.133
⫺2.682
⫺0.9979
⫺0.4556
9964
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
consequences for the calculation of energy differences or
molecular properties. To test this, we have reinvestigated the
Ne dimer by performing calculations with the original C1D2
basis of van Mourik25 at the fixed internuclear separation of
6.1 a 0 共Table III兲. For this test case, Faas et al. report a
relativistic correction to the interaction enery of ⫺0.3 K at
the scaled ZORA–MP2共FULL兲/C1D2 level.17
We note that the first-order DPT value is as much as one
order of magnitude smaller (⫺0.04 K兲 than the ZORA result
previously reported by Faas et al.17 Our DPT results show
that the Hartree–Fock repulsion is decreased by 0.01 K and
that the magnitude of the MP2 correlation contribution is
increased by 0.03 K. However, it appears that the error in
Faas’ interaction energy is not related to the missing
(0) (0)
⫺ 具 ␸ (0)
i 兩 K 12 兩 ␹ i 典 terms, since our new scaled ZORA–
MP2共FULL兲/C1D2 results coincide with the results obtained
by Faas et al.17
Following Faas et al., we performed our unscaled
ZORA–MP2 calculations by simply using the unscaled
ZORA orbital coefficients and energies in a standard MP2
calculation. In the scaled ZORA case, we used the scaled
orbital energies in conjunction with the unscaled ZORA orbitals 共the ZORA orbitals are not changed by the scaling
procedure兲.
We find that the scaled ZORA result is in full agreement
with the first-order DPT result at the Hartree–Fock level. It
disagrees, however, with the first-order DPT result at the
MP2 level, where the correlation contribution is overestimated by the scaled ZORA approach. This contribution is,
on the other hand, indeed correctly reproduced by the unscaled ZORA approach. The gauge dependence spoils the
unscaled ZORA results at the Hartree–Fock level, but nevertheless, the unscaled ZORA approach yields appropriate
orbitals and orbital energies for post-Hartree–Fock calculations. We expect that a strictly atomic 共and thus gauge independent兲 unscaled ZORA method would fully reproduce the
DPT–MP2 results.26
The scaled ZORA approach appears to overestimate the
MP2 correlation contribution significantly. The scaling procedure appears to yield too small orbital energy gaps. As the
scaling factor is largest for the 1s core orbitals, the bulk of
the overestimation could be due to the inclusion of these
orbitals into the correlation treatment, but the frozen-core
共FC兲 results show that this is not the case.
Since the scaled ZORA–MP2 correlation contributions
are too large compared to their Dirac–Fock–MP2 reference
values,17 our first-order DPT value seems more reliable than
the scaled ZORA result obtained by Faas et al.,17 and we
conclude, contrary to earlier presumptions,27 that relativistic
effects do not contribute as much as some tenth of a kelvin to
the well depth of the Ne dimer.
Table IV displays the static electric dipole polarizability
of the atoms He through Xe as computed in the uncontracted
and augmented TZV basis. The computation of the polarizability provides insight into the gauge dependence of the
共scaled兲 ZORA/IORA approaches by investigating how the
energies of these methods respond to a finite external electric
field in comparison with the full four-component Dirac–
Fock treatment. The unscaled ZORA results differ most from
Klopper, van Lenthe, and Hennum
TABLE VII. Scalar-relativistic IORA orbital energies 共in E h ) of rare gas
atoms, computed with the Ahlrichs TZV basis set 共Ref. 23兲.
1s
2s
2p
3s
3p
3d
4s
4p
4d
5s
5p
He
Ne
Ar
Kr
Xe
⫺0.9169
⫺32.82
⫺1.934
⫺0.8484
⫺119.2
⫺12.42
⫺9.593
⫺1.285
⫺0.5894
⫺530.2
⫺72.09
⫺63.81
⫺11.19
⫺8.429
⫺3.759
⫺1.184
⫺0.5232
⫺1273.0
⫺201.5
⫺182.4
⫺42.73
⫺36.12
⫺25.88
⫺8.352
⫺6.131
⫺2.682
⫺0.9977
⫺0.4556
the Dirac–Fock values as the unscaled ZORA variant suffers
most from the gauge dependence. It is found that the unphysical gauge dependence of this variant leads to a significant overestimation of the polarizability. The IORA and
scaled ZORA approaches appear to perform much better and
to coincide with the first-order DPT results for the light atoms He, Ne, and Ar.
The gauge dependence has also been investigated by
adding a constant ⌬ to the potential V nuc . The results of such
calculations are displayed in Table V and compared to the
estimates discussed in Sec. II D. We find that the theoretical
predictions provide a useful estimate of the gauge dependence. Furthermore, we observe that the gauge dependence
of the scaled ZORA approach is roughly 30% smaller than
the dependence of the IORA approach.
Orbital energies of the atoms He through Xe are displayed in Tables VI, VII, and VIII. We find little difference
between the scaled ZORA and IORA orbital energies. For
Xe, we have also listed the orbital energies as obtained from
the uncontracted, doubly augmented TZV basis 共the f-type
functions do not contribute without an external electric field兲.
With this uncontracted basis, we could perform comparative
Dirac–Fock calculations with the DIRAC program.
TABLE VIII. Orbital energies 共in E h ) of the Xe atom computed at the
scalar-relativistic 共scaled兲 ZORA/IORA levels with the uncontracted, doubly augmented TZV basis set (21s17p11d).
1s
2s
2p
3s
3p
3d
4s
4p
4d
5s
5p
ZORA a
Scaled ZORA a
IORA a
Dirac–Fockb
⫺1331.0
⫺204.4
⫺183.1
⫺43.15
⫺36.17
⫺25.80
⫺8.442
⫺6.139
⫺2.668
⫺1.011
⫺0.4567
⫺1278.0
⫺202.6
⫺181.6
⫺43.04
⫺36.08
⫺25.74
⫺8.436
⫺6.135
⫺2.667
⫺1.011
⫺0.4567
⫺1279.0
⫺202.5
⫺181.5
⫺43.02
⫺36.07
⫺25.74
⫺8.432
⫺6.135
⫺2.667
⫺1.010
⫺0.4567
⫺1277.0
⫺202.5
⫺181.7
⫺43.01
⫺36.10
⫺25.74
⫺8.430
⫺6.140
⫺2.667
⫺1.010
⫺0.4576
The scalar-relativistic total energies are: E ZORA⫽⫺7564.1 E h ,
E scaled-ZORA⫽⫺7444.3 E h , and E IORA⫽⫺7448.1 E h .
b
Spin–orbit averaged Dirac–Fock orbital energies computed with the same
basis. The total energy is E Dirac–Fock⫽⫺7446.3 E h .
a
J. Chem. Phys., Vol. 113, No. 22, 8 December 2000
IV. SUMMARY
We have implemented the ZORA approach at the onecomponent independent-particle Hartree–Fock level, accounting for off-diagonal exchange-type terms that couple
the large and small components. This is necessary to become
correct to order 1/c 2 —that is, to show the same leading term
in 1/c as the parent Dirac–Fock method. To this point, we
emphasize that these off-diagonal exchange operators do not
occur in Kohn–Sham calculations 共with nonrelativistic approximations for the exchange–correlation potential兲. Thus,
the current improvement of the ZORA implementation has
no bearing on KS implementations. Numerical tests on the
rare gas atoms He through Xe show that the new implementation yields satisfactory results for the whole range of
nuclear charges investigated (2⭐Z⭐54).
Future work will be concerned with the implementation
of the extra K 12-dependent terms into our strictly atomic
ZORA methods that include spin–orbit terms—that is, a
two-component method16—and electron correlation effects.26
ACKNOWLEDGMENTS
A grant of computing time from the Research Council of
Norway is gratefully acknowledged 共Grant No. NN1118K兲.
The research of W.K. has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.
Relativistic approximation
1
9965
E. van Lenthe, E. J. Baerends, and J. G. Snijders, J. Chem. Phys. 99, 4597
共1993兲.
2
E. van Lenthe, E. J. Baerends, and J. G. Snijders, J. Chem. Phys. 101,
9783 共1994兲.
3
E. van Lenthe, A. van der Avoird, and P. E. S. Wormer, J. Chem. Phys.
108, 4783 共1998兲.
4
K. G. Dyall and E. van Lenthe, J. Chem. Phys. 111, 1366 共1999兲.
5
E. van Lenthe, A. Ehlers, and E. J. Baerends, J. Chem. Phys. 110, 8943
共1999兲.
6
E. van Lenthe and E. J. Baerends, J. Chem. Phys. 112, 8279 共2000兲.
7
C. Chang, M. Pélissier, and P. Durand, Phys. Scr. 34, 394 共1986兲.
8
J.-L. Heully, I. Lindgren, E. Lindroth, S. Lundqvist, and A.-M.
Mårtensson-Pendrill, J. Phys. B 19, 2799 共1986兲.
9
C. van Wüllen, J. Chem. Phys. 109, 392 共1998兲.
10
S. Faas, J. G. Snijders, J. H. van Lenthe, E. van Lenthe, and E. J. Baerends, Chem. Phys. Lett. 246, 632 共1995兲.
11
C. van Wüllen, J. Comput. Chem. 20, 51 共1999兲.
12
P. H. T. Philipsen, E. van Lenthe, J. G. Snijders, and E. J. Baerends, Phys.
Rev. B 56, 13556 共1997兲.
13
F. Wang, G. Hong, and L. Li, Chem. Phys. Lett. 316, 318 共2000兲.
14
L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 共1950兲.
15
W. Kutzelnigg, Z. Phys. D: At., Mol. Clusters 15, 27 共1990兲.
16
S. Faas, J. H. van Lenthe, A. C. Hennum, and J. G. Snijders, J. Chem.
Phys. 113, 4052 共2000兲.
17
S. Faas, J. H. van Lenthe, and J. G. Snijders, Mol. Phys. 98, 1467 共2000兲.
18
S. Faas, Ph.D. thesis, Utrecht University, The Netherlands, 2000.
19
W. Klopper, SORE program 共unpublished兲.
20
W. Kutzelnigg, J. Chem. Phys. 110, 8283 共1999兲.
21
W. Klopper, J. Comput. Chem. 18, 20 共1997兲.
22
T. Saue, T. Enevoldsen, T. Helgaker, H. J. Aa. Jensen, J. Lærdahl, K.
Ruud, J. Thyssen, and L. Visscher, DIRAC, a relativistic ab initio electronic
structure program, Release 3.1, 1998.
23
R. Ahlrichs and K. May, Phys. Chem. Chem. Phys. 2, 943 共2000兲.
24
W. Klopper and W. Kutzelnigg, J. Chem. Phys. 94, 2020 共1991兲.
25
T. van Mourik, Ph.D. thesis, Utrecht University, The Netherlands, 1994.
26
J. H. van Lenthe, S. Faas, and J. G. Snijders, Chem. Phys. Lett. 328, 107
共2000兲.
27
J. van de Bovenkamp and F. B. van Duijneveldt, Chem. Phys. Lett. 309,
287 共1999兲.