Chemical Physics Letters 408 (2005) 150–156
www.elsevier.com/locate/cplett

Relativistic conﬁguration interaction and coupled cluster
methods using four-component spinors: Magnetic shielding
constants of HX and CH3X (X = F, Cl, Br, I)
Mitsunori Kato a, Masahiko Hada a,1, Ryoichi Fukuda a, Hiroshi Nakatsuji a,b,*a

Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering, Kyoto University, Kyoto-Daigaku-Katsura,
Nishikyou-ku, Kyoto 615-8510, Japanb
Fukui Institute for fundamental Chemistry, Kyoto University, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606-8103, Japan
Received 7 January 2005; in ﬁnal form 3 March 2005
Available online 28 April 2005

Abstract
A correlation method in the relativistic theory using four-component spinors is proposed for the calculations of magnetic shielding
constants. The relativistic eﬀects are included by solving Dirac–Fock equation, and the electron-correlation eﬀects are included by the
SDCI method and the CCSD method. Some improvement on the integral transformation algorithm and the use of the direct CI method
were essential for a performance of this method. It is applied to the calculations of the magnetic shielding constants of hydrogen halides
and methyl halides. For hydrogen halides the calculated values excellently reproduce the experimental values to within 0.8 ppm.
 2005 Elsevier B.V. All rights reserved.

1. Introduction
Nuclear magnetic resonance (NMR) is one of themost popular spectroscopic procedures in chemistry
and numerous data have been accumulated. It is very
sensitive to a change of the chemical environment
around the resonant nucleus and reﬂects an interesting
electronic mechanisms of the valence electrons of molecules [1,2]. Since the magnetic shielding constantstrongly reﬂects the valence electron state in the region
close to the nucleus, where the electron has high velocity, the relativistic eﬀect is sometimes very importantand even dominant in some cases [3]. Therefore, the relativistic theory is essential for the quantitative calculation of the magnetic shielding constant.
*Corresponding author. Fax: +81 75 753 5910/383 2741.
E-mail address: hiroshi@sbchem.kyoto-u.ac.jp (H. Nakatsuji).
1
Present address: Department of Chemistry, Tokyo Metropolitan
University, Minami-osawa 1-1, Hachioji, Tokyo 192-0397, Japan.
0009-2614/$ - see front matter  2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.cplett.2005.03.147

Nakatsuji et al. [4] reported in 1995 the ﬁrst ab initio relativistic calculations of magnetic shielding constants. They adopted the ﬁnite-perturbation UHF
method and demonstrated that the spin–orbit (SO)interaction, originating from the relativistic theory, isan essential origin of the chemical shifts in the HX
and CH3X (X = F, Cl, Br, I) series. They further carried out a series of calculations for many diﬀerent molecules and conﬁrmed that the SO interaction is an
essential origin of the chemical shifts in many seriesof molecules [5–7]. They then proposed a method that
could take into account the SO term and the spin-freerelativistic (SFR) term simultaneously [8] and showedthat both terms were important and strongly coupledin the heavy nuclear systems like mercury compounds[9] and tungsten compounds [10].
Recently, the method has been improved along theline of the Douglas–Kroll–Hess (DKH) theory [11–14]
by including the relativistic magnetic interaction operator incorporated into the generalized UHF scheme

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156[15,16]. We have ﬁgured out that the eﬀect of the socalled Ôpicture changeÕ is so important for the Hg chemical shifts in mercury dihalides that it is not able toreproduce the experimental results without this correction [16]. In the framework of the density functional theory, Wolﬀ et al. [17] made use of the zeroth-orderregular approximation to incorporate the relativistic effects into the calculations of the NMR shielding of Hg,
W, and Pb.
The theory using four-component spinors is necessary to consider the exact relativistic eﬀects. Nakatsuji
and coworkers [18,19] calculated the NMR chemical
shifts using the Dirac–Fock (DF) theory. The DF calculations showed that the second-order DKH theory
slightly breaks down even in moderately heavy elements, such as Te [19]. Quiney et al. [20] used thesum-over-state method in the four-component formalism to calculate the magnetic shielding constants inwater. Visscher et al. [21] studied magnetic shielding
and spin–spin coupling constants of HX (X = F, Cl,
Br, and I) using the relativistic random phaseapproximation.
In addition to the relativistic eﬀect, the correlation effect is important for precise calculations of magnetic
shielding constants. Nakatsuji and coworkers [22] suggested the relativistic coupled cluster (CCSD) theory in
the DKH level, and applied to the proton and carbon
shielding constants of HX (X = F, Cl, Br, and I), H2X(X = O, S, Se, and Te), and CH3X (X = F, Cl, Br, and
I). They concluded that the coupling of the relativistic
eﬀect and the electron correlation is signiﬁcant for thecompounds including heavy elements. Visscher et al.
[23] applied the relativistic CCSD and CCSD(T) theories
to some properties of hydrogen halides and examined
the relativistic eﬀect by the four-component correlation
methods [24].
In this Letter, we describe the relativistic singles anddoubles conﬁguration interaction (SDCI) method andthe relativistic CCSD method using four-component
spinors. The reference function is written with the single determinant that is constructed from the spinors
belonging to the so-called Ôelectron stateÕ, because ourinterest is focused only on electron behaviors. This theory is applied to the calculations of the magnetic
shielding constants of hydrogen halides and methyl
halides.

2. Theory
Our starting point is the no-pair Dirac–Coulomb–
Breit (DCB) Hamiltonian described by Sucher [12] andMittleman [25]. In the presence of the magnetic vector
potential, the eﬀective many-body Hamiltonian is given
by

H DCB
ðB 0 Þ ¼
þ

X

151

"
0 2

ca  ðPi þ Ai Þ þ b c þ

i

þ Lþ

!
1X 1
Lþ ;
2 i6¼j rij

X#
V n ðiÞ

n

ð1Þ

where c is the speed of light and a and b 0 are the Diracmatrices. Vn is the nuclear attraction term due to the nucleus n. L+ is the product multiplied n times by K+(i),i = 1, . . ., n, which is the projection operator onto thespace spanned by the positive-energy eigenfunctions of
the matrix DF equation. Throughout this study, atomic
units are used. The Breit interaction was not consideredhere.
The vector potential Ai arising from a uniform external ﬁeld B0 and the nuclear magnetic moment of the nth
nucleus ln is given by1
1 X
Ai ¼ B0  ðri  dÞ  2
l  rGn ;
ð2Þ
2
c n n
where
Z
wn ðRn ÞdR.
Gn ¼
jr  Rj

ð3Þ

Here, the position of the nth nucleus is N and
Rn = |R  N|. The weight function wn(Rn) of the Gaussian nucleus model is given byg 3=2ð4Þ
wn ðRn Þ ¼ n
exp gn R2n .
p
The nuclear exponent gn was taken from [26].
Molecular DF spinors are obtained from the Dirac–
Fock equation"X1
ca  P þ b0 c2 þ
V n þ cB0t fðr  dÞ  agt
2
n
#
1X

l frGn  agt þ J  K /DF¼ ei /DFð5Þii ;c n ntwhere J and K are Coulomb and exchange operators,
respectively. In Eq. (5), the nuclear magnetic moment
term is included explicitly. In the SCF level of calculations, the Hellmann–Feynman theorem is satisﬁed andutilized, but in this study the calculations are done in
the post-SCF levels of SDCI and CCSD, where the Hellmann–Feynman theorem is not satisﬁed. This is why the
perturbation of the nuclear magnetic moment must be
considered.
We are interested only in the positive energy solutionsof Eq. (5), which is the so-called Ôelectronic stateÕ. The
zero-order wave function in the electron-correlation calculations is approximated by a single Slater determinantof the molecular DF spinors belonging to the Ôelectronic
stateÕ.
For the electron-correlation calculations of relativistic treatment, we used SDCI method and CCSD

152

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156

method. Since the relativistic eﬀects are already taken
into account at the SCF level, the post-SCF method
can be similar to the non-relativistic one. But in thecase of the relativistic calculation, especially with
four-component spinors, the computational cost is so
big that its application is limited to a small system.
Therefore, we implement direct SDCI algorithm [27]and an eﬃcient integral transformation algorithm,
which are written in the following section in somedetail.
The magnetic shielding constant rn,tu (t, u = x, y, z) is
given by Ramsey [28] aso2 E 
rn;tu ¼;
ð6ÞoB0t oln;u B ¼l ¼00tn;uwhere E is the total energy. In this Letter, we calculate
the isotropic term given by
rn;xx þ rn;yy þ rn;zz
.
ð7Þ
rn ¼
3

3. Computational details
The step to transform one-electron and two-electron
atomic integrals to molecular ones is costly even in the
non-relativistic theory. In the relativistic theory, fourcomponent spinors are used and the cost is bigger thanthat in the non-relativistic theory. We propose a new
eﬃcient scheme of the integral transformation. The
number of one-electron integrals is small and its treatment is easy. Therefore, we concentrate on the two-electron integrals. The molecular spinors are designed forthe symmetry adapted KramersÕ restricted calculation.
However, we have to employ the unrestricted DF calculation because magnetic ﬁeld breaks the KramersÕ
degeneracy. Thus, we design an eﬃcient scheme of integral transformation for unrestricted four-component
spinors.
A scalar basis set consists of linear combinations ofprimitive cartesian Gaussian basis functions, gLu ; gSv ,where the superscripts ÔLÕ and ÔSÕ represent large andsmall components, respectively,
X
X
cLup gLu ; gSq ¼
cSvq gSv .
ð8Þ
gLp ¼
u

v

Sa
The transformation coeﬃcients, d La
pl and d ql , is deﬁned as complex numbers. One-electron and two-electron integrals are calculated and stored using the basisof the form in Eq. (8)Z
 X X Y YYYr s kt u ¼ gXr ð1ÞgXs ð1Þr1
12 gt ð2Þgu ð2Þ ds
Z
YY gXr ð1ÞgXu ð1Þr1
12 gt ð2Þgs ð2Þ ds;

X ; Y ¼ L or S.

ð10Þ

But the molecular spinors are expanded with the basisof the form in Eq. (9)010 La 1vl
0
B vLb C X B 0 C
X
B
C
B
C
jii ¼
lC Lli B l C þ
C Ssi B Sa C
@
@ 0 A
vs A
s
vSb
0
s
1
0 P La L 1
00
d pl gp
C
C
B p
B
0
C
P Lb L C
X B
X B
P
C
C
B
B
d
g
Sa
SC
LB
SB
pl
p
C
d
g
¼
C li B p
C si Bql q C.
Cþ
C
C
B
B q
s
l
A
@
@ P Sb S A
0
d ql gq0qð11Þ

Therefore, a transformation from the form in Eq. (8)to the form in Eq. (9) is needed. The four-component
spinors in Eq. (9) are adapted for the point group symmetry. There is no restriction for the coeﬃcients, C in
the unrestricted DF calculation. Therefore, the computation scheme is rather straightforward. A simple idea
is that four atomic bases are transformed into four spinors at once
 L L L Lr s kt u ! RL S L kT L U L
20 La 1 0 La 10 La 1 0 La 13
  vt
vu
vs
vr

7
 C
6B vLb C B vLb CB
B
Lb
Lb C
vt C
6B r C B s CB
B vu C 7
B
C
¼ 6B
CB
C 
C7 ; . . . ;CB
4@ 0 A @ 0 AB
@ 0 A7
0
@
A
5



0
0
0
0
ð12Þ
Generally, the integral transformation can be divided
into four stages
X½iSkTU  ¼
C ri ½RSkTU ;r

The four-component spinor is constructed with thisscalar basis set in the following way:

½ijkTU  ¼

0 P La L 1
d pl gp
0 La 1vl
B p
C
P Lb L C
B vLb C BCdg
B l C BLplpC;vl ¼ B
C¼B
p
B
C@ 0 A B
C
@
A
0
0
0

½ijkkU  ¼

0

1

0
1
0
B
C
0
C
B 0 C BP Sa S C
B
C B
S
C.
d
g
vs ¼ B Sa C ¼ B
qlqC@ vs A B
B q
C@ P Sb S A
Savsd ql gq
0

q

ð9Þ

X

C sj ½iSkTU ;sXC tk ½ijkTU ;

ð13Þ

t

½ijkkl ¼

X

C ul ½ijkkU .

uwhere i, j, k, l are molecular four-component spinors expressed by Eq. (9). The ﬁrst step is most costly because
the number of [RSiTU] is the biggest of other integrals

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156

and in addition, four-component spinors are used. Here,
we note that the two latter spinors, t and u, are not changed during the ﬁrst two steps and we implemented sothat these spinors are not transformed during the ﬁrst
two steps
 L L L Lr s kt u ! RL S L ktL uL ; . . . ;XC ri ½RSktu;½iSktu ¼
ð14Þ
r
X
C sj ½iSktu;½ijktu ¼

153

Table 1
Molecular geometries used in the calculationsa
X

F
Cl
Br
I
a

HX

CH3X

rHX

rCX

rCH

hHCX

0.91751.27471.41411.60461.39071.78591.94342.1417

1.09791.0944
1.09301.0939108.81108.43107.69107.66

Bond lengths in Å and angles in degrees.

s






ijktL uL ! ijkT L U L ; . . . ;XC tk ½ijkTU ;½ijkkU  ¼
t½ijkkl ¼

X

ð15Þ

C ul ½ijkkU .

u

The number of operations in the ﬁrst-half transformation is reduced to 1/4 with using Eq. (14) because |TU]
includes four elements. The core memory required forthe step is also reduced to 1/4. The eﬃcient use of memory further accelerated computations. The CPU time forthe integral transformation step was shown in Table 4.
The computations with Eq. (14) were approximately
six times faster than those with Eq. (13). In the fourcomponent method, the ﬁrst-half transformation ismuch more costly than the second-half transformation;
therefore, the present scheme has worked very
eﬃciently.
The direct CI method for the relativistic theory was
implemented. The direct method diminished the disk
I/O of matrix elements which was a bottleneck of CI
calculations.

tions were included. For Br and I, DyallÕs [32] sets wereused. The molecular geometries used are collected inTable 1, which are taken from [33]. The gauge center
was located on the halogen atom.
In the calculations of 1H magnetic shielding constants
eight occupied electrons were correlated. In the calculations of CH3F and CH3Cl, all 18 and 26 electrons,
respectively, and in the calculations of CH3Br andCH3I, 18 and 24 electrons, respectively, were correlated.
The virtual orbitals with energies below 2 hartree were
included, except for CH3F and CH3Cl for which all virtual orbitals were correlated. The number of active orbitals were then 32 for HF, 32 for HCl, 42 for HBr, 48 for
HI, 76 for CH3F, 76 for CH3Cl, 44 for CH3Br, and 44
for CH3I.
We chose the ﬁeld strength of ln so that the perturbation energies due to the applied B0 and ln come to thesame magnitude: we used the ﬁeld strengths that change
the energy around 106 hartree. The accuracy of thenumerical diﬀerentiations was conﬁrmed with these ﬁeld
strengths. The ﬁeld strengths used in this study werethen B0 = 0.005, ln = 50.0 for HF and HCl, B0 =0.001, ln = 10.0 for HBr, and B0 = 0.0005, ln = 10.0
for HI, all in a.u.

4. Calculation
5. Results and discussion
We calculated the magnetic shielding constants of 1H
in HX (X = F, Cl, Br, I) and 13C in CH3X (X = F, Cl,
Br, I). In the calculation of hydrogen halides, uncontracted Csizmadia and coworkers [29] (6s3p), (9s6p),(14s11p5d), and (15s11p6d) sets were used for thelarge-components of F, Cl, Br, and I, respectively. For
the large-component of hydrogen, the uncontracted
Huzinaga–DunningÕs (4s) plus their ﬁrst-order basis
function (FOBFs) [30] (4p) was used: improving basis
functions by adding the FOBFs is a systematic method
for reducing the gauge origin dependence. The smallcomponent basis sets were the ﬁrst derivatives of thelarge-component basis functions so as to satisfy the condition of Ôkinetic balanceÕ. In the calculation of methyl
halides, we used the same basis functions for hydrogen.
For the large-component of C, F, and Cl, DunningÕs ccpVDZ [31] sets were used, and for the small-component
the ﬁrst derivatives of the large-component basis func-

5.1. 1H Magnetic shielding constants of HX (X = F, Cl,
Br, I)
The calculations of 1H magnetic shielding constants
were carried out for HX (X = F, Cl, Br, I) using SDCI
and CCSD methods. Table 2 shows the calculated 1H
magnetic shielding constants. The calculated values in
the quasi-relativistic theory and non-relativistic theory
were taken from [22], and the experimental values weretaken from [33]. All experimental values are the resultsof gas-phase NMR measurements with respect to CH4.
The absolute value of CH4 was estimated by a measurement of the rotational constant of CH4, and then the
NMR chemical shifts of CH3X were converted into
the absolute values with this value.
Fig. 1 shows the relativistic eﬀects on the 1H magnetic
shielding constants: the vertical axis is the diﬀerence

154

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156

Table 2
1
H magnetic shielding constants of HX (in ppm)
Non-relativistica

HF
HCl
HBr
HI

Quasi-relativistica

Experimentalb

Relativistic

SCF

MP2

SCF

CCSD

SCF

SDCI

CCSD

28.83
30.76
31.09
31.77

29.39
30.96
31.20
31.81

29.06
31.88
36.87
48.01

29.87
32.90
36.39
44.29

27.17
30.07
34.09
45.10

28.47
31.12
34.19
44.03

28.53
31.14
34.20
43.92

28.50 ± 0.2c
31.06
34.96
43.86

a

Ref. [22].
Ref. [33].
c
The rovibrational contribution at T = 300 K was estimated at 0.52 ppm.

b

between the experimental and calculated values, and d,n, and m represent the results of the non-relativistic,
quasi-relativistic, and relativistic methods, respectively.
The results of the relativistic method agree somewhat
badly with the experimental values in HF and HCl,but agree fairly well in HBr and HI. This behavior is
opposite to the behaviors of the non-relativistic andthe quasi-relativistic method, which agree better withthe experimental values in HF and HCl than in HBr
and HI. It is expected that as the atom becomes heavier,the relativistic eﬀect becomes larger, so the result of therelativistic method is most reasonable.
Fig. 2 shows the electron-correlation eﬀects on the 1H
magnetic shielding constants: the vertical axis is the difference between the experimental and calculated values,
and m, ¤, d, and n represent the result of the quasi-relativistic SCF, the quasi-relativistic CCSD, the relativistic SCF, and the relativistic CCSD method,respectively. The relativistic CCSD values obtained inthis study reproduce well the experimental values, andthe biggest error is only 0.76 ppm in HBr. It is seen from
Table 2 that the SDCI method already includes the electron-correlation eﬀects appropriately. The results of therelativistic SCF theory for HF and HCl, which agree
somewhat badly, are corrected excellently at the CCSD
level in close agreement with the experimental values.
This behavior is opposite to the behavior of the quasirelativistic CCSD method, which disagrees for HF and
HCl. It is expected that the electron-correlation eﬀects
would also increase as the atom becomes heavier, andso the present study expresses appropriately the coupling between the electron-correlation eﬀects and the relativistic eﬀects.
The relativistic SCF result of the same series usinglarger basis sets was reported by Visscher et al. [21].
Their results of HBr and HI are around 2 ppm larger
than ours. The basis set convergence is not good in the
single gauge-origin calculation and the basis set should
be systematically improved. We use the FOBF
method for improving basis set. The SCF calculation

5.0
Difference between theory and experiment (ppm)

Difference between theory and experiment (ppm)

5.0

0.0

-5.0

-10.0

non-relativistic
quasi-relativistic
relativistic

SCF(quasi-relativistic)
CCSD(quasi-relativistic)
SCF(relativistic)
CCSD(relativistic)

4.0

3.0

2.0

1.0

0.0

-1.0

-2.0

-15.0
HF

HCl

HBr

HI

Fig. 1. The diﬀerence between theory and experiment for the 1H
magnetic shielding constants of HX at the SCF level.

HF

HCl

HBr

HI

Fig. 2. The diﬀerences between theory and experiment for the 1H
magnetic shielding constants of HX at the CCSD level.

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156

155

overestimates the relativistic eﬀect. The coupling between the electron-correlation eﬀect and relativistic effect, which reduces the 1H shielding constants in heavy
halides, supports our results and does not contradict
with the ﬁndings of Visscher et al. [21].

severe to include these basis functions up to the SDCI
level because of the heavy demands on computers.

5.2. 13C Magnetic shielding constants of CH3X (X = F,
Cl, Br, I)

We have performed the relativistic electron-correlation calculations of the magnetic shielding constants
and chemical shifts. The relativistic eﬀect is consideredat the Dirac–Fock level, and the electron correlation effect is considered at the SDCI and the CCSD levels. An
improvement in the integral transformation step and the
direct CI algorithm were necessary to perform this calculation. We have applied the method to the calculations of the 1H magnetic shielding constants of HX(X = F, Cl, Br, I) and 13C chemical shifts of CH3X
(X = F, Cl, Br, I). For the HX series, the calculated values reproduced excellently the experimental values: thediﬀerences were within 0.8 ppm. For the CH3X series,
the results were not so satisfactory mainly because we
could not use large enough basis set for severe demandof the relativistic correlation method on computational
facilities.

The calculations of 13C NMR chemical shifts werecarried out for CH3X (X = F, Cl, Br, I) using the SDCI
method and the results are summarized in Table 3 together with the experimental value. The reference molecule is CH3F. All the experimental values except forCH3F are due to the NMR measurements in neat liquid
and that of CH3F is due to the gas-phase NMR
measurement.
The SCF results reproduce the experimental values
qualitatively, but the agreement is not satisfactory.
Unfortunately, the electron-correlation eﬀect did not
improve these results, in particular for CH3I. We thinkthat the reason of the deviation at the SCF level is thelack of the basis functions, especially tight s-type functions, because in the quasi-relativistic study of the
NMR chemical shifts, the Fermi-Contact term was themost important origin [4–10] and the tight s-type functions are necessary to describe the Fermi-Contact term
appropriately. The deviation at the SDCI level would
also be due to the lack of the basis functions, especiallytight p-type functions, because these are necessary to describe the reorganization of the orbitals in the correlation of core electrons. For the CH3X series, it was

6. Conclusion

Acknowledgment
This work has been supported by the Grant for
Creative Scientiﬁc Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

References

Table 313
C NMR chemical shifts of CH3X (in ppm)
Experimentala

Relativistic

CH3F
CH3Cl
CH3Br
CH3I
a

SCF

SDCI

0
39.34
51.48
77.680
43.25
45.41
82.620
51.3066.8098.50Ref. [33].

Table 4Computational time for the integral transformation step

CH3F
CH3Cl
CH3Br
a
b

BasisaActive space

336
364
488

18 · 76
26 · 76
18 · 44

CPU timeb (h)Eq. (13)

Eq. (14)

26
67
200

4
13
32

The number of spinor basis.
COMPAQ XP1000 workstation (2 GB memory).

[1] H. Nakatsuji, in: J.A. Tossell (Ed.), Nuclear Magnetic Shieldings
and Molecular Structure, NATO ASI Series C-386, Kluwer
Academic Publishers, Dordrecht, The Netherlands, 1993, p. 263.
[2] H. Nakatsuji, K. Kanda, K. Endo, T. Yonezawa, J. Am. Chem.
Soc. 106 (1984) 4653.
[3] P. Pyykkö, Adv. Quantum Chem. 11 (1978) 353.
[4] H. Nakatsuji, H. Takashima, M. Hada, Chem. Phys. Lett. 233(1995) 95.
[5] H. Takashima, M. Hada, H. Nakatsuji, Chem. Phys. Lett. 235(1995) 13.
[6] H. Nakatsuji, T. Nakajima, M. Hada, H. Takashima, S. Tanaka,
Chem. Phys. Lett. 247 (1995) 418.
[7] H. Nakatsuji, M. Hada, T. Tejima, T. Nakajima, M. Sugimoto,Chem. Phys. Lett. 249 (1996) 284.
[8] C.C. Ballard, M. Hada, H. Kaneko, H. Nakatsuji, Chem. Phys.
Lett. 254 (1996) 170.
[9] H. Nakatsuji, M. Hada, H. Kaneko, C.C. Ballard, Chem. Phys.
Lett. 255 (1996) 195.
[10] M. Hada, H. Kaneko, H. Nakatsuji, Chem. Phys. Lett. 261 (1996)7.
[11] M. Douglas, N.M. Kroll, Ann. Phys. (N. Y.) 82 (1974) 89.
[12] J. Sucher, Phys. Rev. A 22 (1980) 348.
[13] B.A. Hess, Phys. Rev. A 32 (1985) 756.
[14] B.A. Hess, Phys. Rev. A 33 (1986) 3742.
[15] R. Fukuda, M. Hada, H. Nakatsuji, J. Chem. Phys. 118 (2003)
1015.

156

M. Kato et al. / Chemical Physics Letters 408 (2005) 150–156[16] R. Fukuda, M. Hada, H. Nakatsuji, J. Chem. Phys. 118 (2003) 1027.
[17] S.K. Wolﬀ, T. Ziegler, E. van Lenthe, E. Baerends, J. Chem.
Phys. 110 (1999) 7698.
[18] M. Hada, Y. Ishikawa, J. Nakatani, H. Nakatsuji, Chem. Phys.
Lett. 310 (1999) 342.
[19] M. Hada, R. Fukuda, H. Nakatsuji, Chem. Phys. Lett. 321 (2000)
452.
[20] H.M. Quiney, H. Skaane, I.P. Grant, Chem. Phys. Lett. 290(1998) 473.
[21] L. Visscher, T. Enevoldsen, T. Saue, H.J.A. Jensen, J. Oddershede, J. Comput. Chem. 20 (1999) 1263.
[22] K. Ishimura, R. Fukuda, M. Hada, H. Nakatsuji (submitted).
[23] L. Visscher, K.G. Dyall, T.J. Lee, Int. J. Quantum Chem. Symp.
29 (1995) 411.

[24] L. Visscher, J. Styszynski, W.C. Nieuwpoort, J. Chem. Phys. 105(1996) 1987.
[25] M.H. Mettleman, Phys. Rev. A 24 (1981) 1167.
[26] O. Visser, P.J.C. Aerts, D. Hegarty, W.C. Nieuwpoort, Chem.
Phys. Lett. 134 (1987) 34.
[27] B. Roos, Chem. Phys. Lett. 15 (1972) 153.
[28] N.F. Ramsey, Phys. Rev. 78 (1950) 699.
[29] R. Poirier, R. Kari, I.G. Csizmadia, Handbook of Gaussian Basis
Sets, Elsevier, Amsterdam, 1985.
[30] M. Sugimoto, H. Nakatsuji, J. Chem. Phys. 102 (1995) 285.
[31] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.
[32] K.G. Dyall, Theor. Chem. Acc. 99 (1998) 366.
[33] J. Vaara, K. Ruud, O. Vahtras, H. Ågren, J. Jokisaari, J. Chem.
Phys. 109 (1998) 1212.