Ó 2005 The Chemical Society of Japan

Bull. Chem. Soc. Jpn., 78, 1705–1724 (2005)

1705

Award Accounts
The Chemical Society of Japan Award for 2003

Deepening and Extending the Quantum Principles in Chemistry
Hiroshi Nakatsuji
Department of Synthetic Chemistry and Biological Chemistry, Graduate School of Engineering,
Kyoto University, Katsura, Nishikyo-ku, Kyoto 615-8510
Fukui Institute for Fundamental Chemistry, Kyoto University,
34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606-8103
Received January 12, 2005; E-mail: hiroshi@sbchem.kyoto-u.ac.jp

A brief summary of the author’s research projects for deepening and extending the quantum principles in chemistry
is given. First, the structure of the exact wave function that is the solution of the Schrödinger equation was clariﬁed and amethod of calculating it starting from an approximate wave function is given. The singularity problem intrinsic to atoms
and molecules was overcome, and a general method of solving the Schrödinger equation in an analytical form has been
established. Quantum chemistry of excited states and ionized states is essential since it is diﬃcult with experiments alone
to explore this ﬁeld due to the short lives of these species. The SAC-CI theory developed in the author’s groups oﬀers apowerful method for investigating the chemistry of excited and ionized states. The energy gradient method incorporated
into this program is particularly useful for studying geometries and reactions involving these states. The catalysis on ametal surface is a magic process that must be clariﬁed with the help of quantum theory. The author studied it using his
dipped adcluster model (DAM) that includes the eﬀects of the free electrons of the bulk metal. The epoxidation reactionsof ethylene and propylene were studied with this method. The NMR chemical shifts include much information about thevalence electrons of molecular systems. The author clariﬁed that the major electronic mechanisms of the chemical shifts
could be attributed to the intrinsic atomic properties that are related to the position of the element in the periodic table.
For the chemical shifts induced by heavy ligands, the relativistic eﬀect is sometimes dominant. The spin-orbit eﬀect ismost important for light resonant nuclei. For heavier resonant nuclei, other spin-free relativistic eﬀects and the electron
correlation eﬀects are also important and they strongly couple with each other, implying an existence of something‘‘unexpected’’ in the chemistry of heavy elements. Finally, the strategy of the author’s study in quantum chemistry is
shortly described.

Chemistry is considered to be the results of the realizationof several principles, among which the quantum principles
are of particular importance since they describe the motionsof electrons that govern the structures, reactions, and properties of atoms, molecules, molecular systems, and biologicalsystems.1 Therefore, one of the most fruitful methodologies
would be to deeply understand and cultivate these quantum
principles and to realize them in chemistry and biology.
Putting this belief into his basic philosophy, the author has
pursued his researches in chemistry. Here, on the occasion of
the Chemical Society of Japan Award of 2003 being presented
to him, some of his achievements are summarized as references for the successors. The topics are mainly collected fromthe author’s studies after 1990, since he was also presented the
Physical Chemistry Award of the Chemical Society of Japanfor 1990.
The basic trends in theoretical chemistry are threefold:more-and-more-accurate predictions, variety-of-states applicability, and larger-and-larger-system applicability. Since the
Schrödinger equation and its relativistic counterpart, which

may be called the Dirac equation for brevity, are the basic
principles governing the whole of chemistry,1 the moststraightforward way of achieving the ﬁrst purpose of theoretical chemistry is to construct a general method of solving theseequations as accurately as possible. This has long been a dreamof theoretical chemists and physicists, and likewise, that of theauthor’s research life. Recently, this dream has been realized
essentially after several years of theoretical considerations on
the structure of the exact wave function that started from1999.2–8 By the exact wave function, I mean the solution of
the Schrödinger equation. The ﬁrst topic of this account is amethod of solving the Schrödinger equation in an analytical
form8 based on the studies reported in this series of articles.2–8
Two typical examples of applications to helium atom and hydrogen molecule are given from among several applications to
atoms and molecules. The results are compared with the best
quality wave functions reported in the literature.
The second topic of this account is related to the second andthird trends of theoretical chemistry. We have developed atheory called SAC (symmetry adapted cluster)9,10/SAC-CI

Published on the web October 4, 2005; DOI 10.1246/bcsj.78.1705

1706 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTS(conﬁguration interaction)11–14 theory in 1977–78 as an accurate electronic-structure theory for ground, excited, ionized,
and electron-attached states of molecules. The calculations
based on the ﬁrst version of the program applied to the groundand excited states of Be and H2 O were presented in 1978 at
Hiroshima in the Symposium on Molecular Science.15 Since
then many developments were done in both theories and programs, mainly in the Nakatsuji laboratory of Kyoto University.
The SAC/SAC-CI theory9–14 and their extensions16–21 havebeen applied to the molecular spectroscopy and photochemistry of a variety of molecules of various sizes and have been
established as a reliable and useful theory.22–27 In 1985, the
SAC-CI program was released as SAC85 program system23,28
and in 1991, the SAC-CI general R method was introduced,
enabling the calculations of many electron excitation processes.13 The energy gradient method developed later29–35 has
made it possible to calculate the forces acting on the constituent nuclei of molecules in any electronic states dealt with by
the SAC-CI method and has made it possible to study equilibrium geometries and dynamics of molecules in various electronic states. More recently, we have collaborated with Dr. M.
Frisch of Gaussian Inc.34,35 and incorporated our SAC-CI code
into the Gaussian03 suit of programs and published it worldwide.36 It was nice that a method born in our laboratory has
grown up to become a tool that is useful to solve chemical
problems not only in universities and institutes, but also inchemical and physical industries worldwide.
The SAC-CI method has been applied to various ﬁelds ofphotochemistry. The applications in the initial stage were to
examine the accuracy of the new theory37–40 and to explore
some new ﬁelds.41–46 The theory has been shown to be particularly useful to molecular spectroscopies of electronic excitations and ionizations of organic and inorganic molecules andtransition metal complexes.37–39,41,43,46–66 It was also shownto give reliable values of the hyperﬁne splitting constants ofdoublet radicals.44,67,68 It has been applied to the collision induced absorption spectra69 and the reaction dynamics of excited states.70,71 A reﬁnement of the program based on the SACCI general-R method13 enabled us to calculate some multielectron processes, like the shake-up states involved in ionization spectra, as quantitatively as can be done for the ordinary
single-electron processes. This has opened up a new ﬁeld,
called theoretical ﬁne spectroscopy: the SAC-CI theoretical
spectra have become almost as quantitative as the experimental spectra,72–81 enabling very ﬁne spectroscopic researches incollaboration with the experimentalists.
The perturbation selections adopted in the SAC-CI code43
have made the calculations of larger systems possible. We
have calculated the excitation spectra of many diﬀerent porphyrin compounds,82–92 and even performed the geometry
optimizations of the ground and excited states of free baseporphin.93 Based on these experiences, we have applied the
SAC-CI method to the assignment of the electronic spectraof the whole of the photosynthetic reaction centers of Rhodopseudomonus viridis, a bacteria.94–96 We have further utilized
the SAC-CI theory to clarify the origin of the unidirectionality
and the eﬀectiveness of the electron transfer in this photosynthetic reaction center.97–100 The SAC-CI theory has further
been applied to surface catalytic reactions of Pd and Pt involv-

ing H2 and acetylene.101–103 It was then combined with the dipped adcluster model (DAM) for chemisorptions and catalytic
reactions on a metal surface104–108 and applied to the electrontransfer and surface photo-luminescent reactions of halogen
molecules on alkali metal surfaces.108,109 It was also applied
to the studies of O2 chemisorptions on an Ag surface110–112
and of the surface spectroscopy of NO adsorbed on a Pt
surface.113
The coupled cluster linear response theory (CC-LRT) andthe equation of motion coupled cluster (EOM-CC) method
are very similar to the SAC-CI theory. In the calculations of
the ground-state properties with the coupled-cluster method,
Monkhorst114 described a linear response formulation of theexcitation energy from a consideration of the poles of the dynamic polarizability, but this is not a theory for the excitedstate, because the wave functions of the excited states were
not formulated explicitly. Mukherjee115 gave a more extensive
formulation of the coupled cluster linear response theory (CCLRT). Hirao derived in 1983 the CC-LRT equation startingfrom the SAC-CI theory.116 The applications of the CC-LRT
were not done before Koch, Jørgensen, and others reformulated it and applied to some molecular systems.117,118 The equation of motion coupled cluster (EOM-CC) method published in
1989119,120 was essentially equivalent to the SAC-CI method:
the basic theories were identical and the diﬀerences were onlyin the ways of approximations in actual calculations. When thesame approximations are done, these three theories give identical results,26 which constitutes a numerical proof, if necessary, of the equivalence between SAC-CI, CC-LRT, and EOMCC theories.
The third topic of this account is also related to the second
trend of theoretical chemistry. It is related to the chemistry ona metal surface. A metal surface is an inﬁnite system andshows interesting chemistry, like catalysis, when it adsorbs
atoms and molecules. Since this is a really huge system, somemodeling is necessary to study it theoretically. This is certainly
an attractive ﬁeld to which theory must make some contributions. In 1987, the author proposed the dipped adcluster model(DAM) for catalysis and surface reactions on a metal surface.104–108 This model has been applied to various chemistries
on metal surfaces.104–112,121–131 As stated above, a combined
use of the DAM with the SAC-CI method was very eﬀective
to explore the photochemistry on the metal surface.107–112
We explain here only one example of applications to catalytic
reactions, epoxidation reaction of oleﬁns on a silver surface.121–131
The last topic of this account is the electronic theory of
NMR chemical shifts132–154 and the importance of the relativistic eﬀect in chemistry.143–154 The NMR chemical shifts areprobably one of the most frequently used properties of molecules in analytical chemistry. We want to note that they involve valuable information about the valence electrons ofmolecules and molecular systems.132,133,138 The major electronic mechanisms of the chemical shifts can be attributed tothe intrinsic atomic electronic structures that are characterized
by the positions of the resonant atoms in the periodic table.138
Further, when the resonant atom is bonded with heavy elements, the relativistic eﬀects are important and sometimes become the dominant origin of the chemical shift.143–154 The

Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji

heavy element NMR is quite interesting since the spin-dependent and spin-free relativistic eﬀects are both important andthey couple strongly with the correlation eﬀects. Though therelativistic eﬀects are important in various aspects of chemistry, we limit ourselves here only to that eﬀect in the NMR
chemical shifts.
A brief summary and an overview will then follow and I
would like to close this article with acknowledgements tomy teachers, colleagues, and students whose supports were
essential for realizing these researches.
Method of Solving the Schrödinger Equation
Theory. The Schrödinger equation (SE) has truly quantitative predictive power in chemistry.1 Therefore, the theory
that enables us to solve the SE has huge scientiﬁc and practical
merits. For atoms and molecules, the SE is written in timeindependent form as
H

¼E ;

þ

X

A

i

i> j

1=rAB ¼ ke þ Vne þ Vee þ Vnn ;

ð2Þ

A>B

where ke , Vne , and Vee are electron kinetic operator, nuclearelectron attraction operator, and electron–electron repulsion
operator, respectively. Vnn is the nuclear–nuclear repulsion
term which is essentially a constant under the Born–
Oppenheimer approximation and so will not be consideredhereafter. Up to quite recently, the only practical method ofsolving the SE was the full-CI method.155 However, the number of variables involved is astronomical and so its practical
merits are quite limited. Furthermore, even if we could get
the full-CI solution, it is actually far from the true solutionof the SE, because the basis sets used are far from complete.
The basis set expansion method most common in modern
computational chemistry must be reconsidered.
As one can clearly see from Eq. 1, the exact wave function
is an analytical entity in a mathematical sense. However,there is no general method of solving the SE in an analyticalmust be a functional of H.
form. Eq. 1 also implies that
In a series of papers starting from 2000,2–8 the author has investigated the structure of the exact wave function. He considered two questions. What mathematical structures does the exact wave function have? Is a general theory possible that constructs the exact wave function systematically? As a result, theICI (iterative conﬁguration or composite interaction) and
SECC (simplest extreme coupled cluster) methods have been
proposed as general methods of calculating the exact wave
function. The details have already been published in the literature and interested readers are recommended to read the original papers.2–8 The basic ideas may brieﬂy be summarized asfollows.
The SE deﬁnes the exact wave function and is valid only forthe exact wave function. We have some equations that areequivalent to the SE. The variational principle

ð3Þ

in the domain of theis used to calculate the best possible
and when this domain is general enough it gives
function
the exact wave function. On the other hand, the H-square equation,h jðH  EÞ2 j i ¼ 0;

ð4Þis valid only when is exact and is strictly identical to the SE.
Comparing Eqs. 3 and 4, we notice that when a including
only one variable C satisﬁes@¼ ðH  EÞ ;
ð5Þ@Cthen we can say that this has a structure of the exact wave
function, because the variational best of this satisﬁes the Hsquare equation. There are two types of ’s that satisfy Eq. 5.
One is the simplest ICI (SICI) given by the recursion formula:ð1Þ

is an exact wave function, E is its energy, and H is
where
the Hamiltonian given byXXXX
i =2 ZA =rAi þ
1=ri jH¼
ih jH  Ej i ¼ 0;

1707

nþ1 ¼ ½1 þ Cn ðH  En Þ

n;

ð6Þ

with n being an iteration number. It satisﬁes Eq. 5 at convergence. The other is the simplest ECC (SECC) given by¼ exp½CðH  EÞ

0:

ð7Þ

A diﬀerent argument shows that C of Eq. 7 may be minus
inﬁnity for the exact wave function.156
The SICI and SECC can be generalized as follows. We diND
P
vide the Hamiltonian H into ND pieces as H ¼
HI . Then,
I¼1

we can generalize the ICI method by introducing the ND variables {CI;n }, ICIND:!
ND
X
CI;n HIð8Þ
nþ1 ¼ 1 þn;I¼1
3
which is also guaranteed to be exact at convergence.
P r þWhen
we use the second quantized Hamiltonian, H ¼  p ar a p þprP rs þ þ
w pq ar as aq a p , ND may be taken to be the sum of generalpqrs

singles and doubles (GSD), and so the following ICI-GSD:


X
X
r þ
rs þ þ
¼
1
þ
c
a
a
þ
c
a
a
a
a
p
q
p
nþ1
n ; ð9Þp r
pq r sprpqrs

is also guaranteed to be exact at convergence. Similarly, we
can deﬁne ECC and ECC-GSD (or CCGSD), but they are
not guaranteed to be exact, though some contradictory arguments exist in the literature.157–160
Figure 1 shows the convergence process of the minimal basis SICI calculations of some small molecules. Each iteration
step of ICI is variational and so the energy monotonically converges from above to the full-CI results. For C2 H4 , for example, the full CI result obtained using 107 952 variables is reproduced by the SICI using only one variable, though the convergence speed is rather slow. Figure 2 shows the ICI-GSD calculations.161 Here, the convergence is very fast. Actually, ICIGSD is a special case where the singularity problem described
below can be circumvented,161 and therefore is worth studied
for practical utility. However, all these results are anyway of
the full-CI accuracy, which is usually quite inaccurate in com-

1708 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTS

diﬃculties.
The ICI and SECC theories are among the theories that include such higher powers of the Hamiltonian. This is easily
seen from the secular equation of the SICI method.3 In general,
as stated in the Introduction, the exact wave function must be afunctional of the Hamiltonian applied to some function 0 , say¼ f ðHÞ 0 and the function f ðHÞ includes, in general, higher powers of Hamiltonian in the expression of the Maclaurin
expansion. So, this is an essential diﬃculty that must be encountered in any theory describing the true analytical solutionof the SE. This diﬃculty must be solved by going back to the
SE itself.
The author has proposed two diﬀerent ways of solution.6,8First, he introduced the inverse Schrödinger equation (ISE),665

70

Ecorr (%)

75

80

85

90
CO2

95

100

O3
C2H 4

HCN
CH3F

0

HCHO

5

10

15

20

25

30

Iteration

Fig. 1. Convergence of the SICI calculations with the minimal STO-6G basis sets.
99.0

H 1

¼ E1 ;

which is equivalent to the original SE. At the singular points,
H 1 is zero and does not have singularity. Some nice featuresof the theory based on the ISE were demonstrated, but one
problem was how to write H 1 in a closed form. More recently, he proposed the scaled Schrödinger equation (SSE)8gðH  EÞ99.2

Ecorr (%)

99.6

O3

HCN

CH3F

0

1

ð12Þ

r!r0

HCHO
100

ð11Þ

lim gV 6¼ 0;

C2H4

99.8

¼ 0;

to solve the singularity problem. The scaling function g is afunction of electron coordinates and does not commute with
the Hamiltonian. It is everywhere positive and non-zero,
except at the singular points, and even at the singular points
r0 , it must satisfy

CO2

99.4

ð10Þ

2

3

4

5

6

7

Iteration

Fig. 2. Convergence of the ICI-GSD calculations with theminimal STO-6G basis sets.

parison with the true solution of the SE.
Now, let us consider a method of solving the SE in an analytical form for atoms and molecules. Then, we immediately
encounter a diﬃculty called the singularity problem. The
Hamiltonians for atoms and molecules given by Eq. 2 havesingularities in the nuclear attraction and electron–electron repulsion terms. In the SE given by Eq. 1, the rhs, E , has no
singularity: E is a constant and is a continuous, single-valued, and integrable ﬁnite function (this condition is called Qcondition). So, the singularity caused by the potentials in the
Hamiltonian must be canceled out within the lhs of the SE(for example, in hydrogen atom, the attractive force must be
canceled out, for stability, by the centrifugal force inducedby the electronic motion). The cusp condition ﬁrst derived
by Kato162 is easily derived by this cancellation idea. However, this complete cancellation occurs only for the exact wave
function: for approximate wave functions, some singularities
remain in H and cause a problem. For example, the integrals
h jH n j i do not exist for n  3, if is approximate.8 Then,the theory including higher powers of the Hamiltonian has

so as not to eliminate the information of the Hamiltonian itself
at the singular points: the singularity also contains important
information about the system. In Eq. 12, V is the Coulombic
potential of Eq. 2. One choice of g would be g ¼ 1=ðVne þ
Vee Þ. A diﬀerent choice would be g ¼ 1=ðVne Vee Þ. Anotherchoice, g ¼ r=ðr þ aÞ for the potential 1=r is also a preferable
choice where a is a constant.
The SSE has the variational principle and the H-square theorem as the original SE, and therefore we can formulate the
ICI and ECC methods based on the SSE; the simplest ICI of
the SSE is written, for example, as,nþ1 ¼ ½1 þ Cn gðH  En Þ

n:

ð13Þ

Because of the existence of the scaling function, the integrals
involved in this ICI do not diverge.
An analytical nature of the ICI calculation is explained asfollows. We start with some guess function 0 , which may
be determinants including Slater, Gaussian, or mixed functions, but anyway it is an analytical function. The Hamiltonianis just a sum of the diﬀerentiation operator (kinetic operator)
and the multiplicative operators (potential operator). g is amultiplicative function. Then, nþ1 is an analytical function
with unknown coeﬃcients {Cn }. We note that the diﬀerentiation is always possible and easy.
The role of the ICI method is twofold: one role is to iteratively construct the exact wave function starting from 0
and the other role is to keep the number of the variables small
and ﬁxed. A useful extension of the ICI method is obtained byrelaxing the second property. Namely, we take all of the M

Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji

independent functions fi gðnÞ included in the ICI wave function given by Eq. 13 and we write the next step as
nþ1 ¼

M
X

ci i ;

ð14Þi

by giving an independent coeﬃcient ci to every i . We callthis relaxed method the free ICI method. In principle, the free
ICI method converges faster than the ordinary ICI method. The
functions fi gðnÞ may be called complement functions because
they are the components of the complete functions that are
guaranteed to express the exact wave function at convergence(complete ICI). Thus, ICI here is an abbreviation of independent complements interaction.
The free ICI method thus gives an analytical function having the structure of the exact wave function with unknown coeﬃcients [Eq. 14]. It has some merits over the conventional
ICI [e.g. Eq. 13]. One is the faster convergence. Second, each
iteration is independent, i.e., does not depend on the results ofthe former iterations. This means no accumulation of errors
and further means that you can start from your desired iteration
step of a desired accuracy. Third, in the free ICI method, the gfunction can be chosen more freely. Even if some divergent
functions are generated due to some limited choice of g, youtojust throw out such divergent functions and keep yoursatisfy the Q-condition.
Thus, we have established a general method of constructing
an analytical function that has the structure of the exact wave
function with linear unknown coeﬃcients. When these unknown coeﬃcients are determined by some appropriate method, the free ICI function is guaranteed to become the solutionof the SE. You can make it as accurate as you desire by increasing the iteration number.
Now, how do we calculate these unknown coeﬃcients? One
is to use the variational principle by calculating the necessary
integrals. In the present formalism based on the SSE, the singularity problem does not occur. Though diﬀerentiations arealways possible, integrations are not always possible. For thisreason, the choice of the g-function is not completely free, if
one wants to do analytical integration. When analytical integration is diﬃcult, one may use the Monte Carlo integration
method. However, it is generally diﬃcult to get highly accurate integrals through the Monte Carlo integration method.
Nevertheless, for most problems, chemical accuracy is the accuracy to be required, which is not diﬃcult even with theMonte Carlo integration method.
In the ﬁeld of quantum Monte Carlo, various methods havealready been developed. In particular, in the variational Monte
Carlo, the largest problem was the lack of a theory that provides an accurate trial function that is guaranteed to have thestructure of the exact wave function.163 Now, the free ICI
method provides one to an arbitrary accuracy: you can jump
into large n in the free ICI to get trial functions of highaccuracy. For general applications of the present theory tomolecules without symmetry, a Monte Carlo type method
would be a good choice and such methodologies are underdevelopment in our laboratory.
Now a few words about the excited states. As easily understood, the ICI process generates the basis functions not only
for the ground state but also for the excited states of the same

1709

symmetry. Since we usually start from 0 that approximate
the ground state, the ICI process simulates ﬁrst the groundstate, but at the same time, it generates the basis functions appropriate for the excited states. Thus, the second higher-energy
solution is an approximation to the ﬁrst excited state. For getting fast convergence, you may include in 0 the functions
that mimic the excited states as well as the ground state.
Calculation of the Analytical Solutions of the SE. The
ICI method has been applied to calculate both full CI solutions7,161 and analytical solutions8,164–168 of the SE. The former
examples were given above in Figs. 1 and 2. Since the analytical solutions are far more accurate than the full CI solutions, our major concern here is how eﬃciently our ICI method
gives the analytical solution of the SE. We have already performed the applications quite successfully to hydrogen atom,8
helium atom,8,164 hydrogen molecule,8,165 Hooke’s two-particle atoms,166 lithium, beryllium, and boron atoms,167 andHeHþ ,168 and others. The results are superior or comparable
to the already existing best analytical solutions for these systems, showing the accuracy and the usefulness of the ICI method for the analytical solutions of the SE. Among these, we give
here a brief account of the results obtained for the helium atomand the hydrogen molecule.
Table 1 shows the free ICI result for the helium atom. The g
factor is g ¼ 1=ne ee and 0 ¼ expðsÞ with s ¼ r1 þ r2 .
The exponent  was optimized at each n. As n increases, theenergy decreases. Since ICI is variational at each n, the energy
decreases monotonically and approaches the exact value169
from above. The chemical accuracy, milli-hartree accuracy,
is obtained at the second iteration with 26 independent functions. The energy becomes exact up to 9 decimal ﬁgures at
n ¼ 7. The wave function generated automatically by the ICI
method is diﬀerent from the existing one. It is similar to theHylleraas functions170 but includes both positive and negative
powers of s (in Hylleraas functions, only positive powers of s
were included). It is in some sense similar to the Kinoshita
function,171 but diﬀerent in details. This negative power terms
are important for a rapid convergence.
We have examined diﬀerent choices of the g factor and 0(the result of Table 1 is due to the simplest choice). Table 2
shows the result of the free ICI calculations with g ¼
1  1=ne þ 1=ee and 0 ¼ ð1 þ s1=2 þ u1=2 Þ expðsÞ. With
this reﬁned choice, the convergence rate is faster and at the 8th
iteration the energy agrees to the best Schwartz result169 up to
Table 1. Free ICI Energy of Helium AtomaÞ
Iteration, n
0
1
2
3
4
5
6
7
Best valuebÞ

Mn b Þ
1
6
26
74
159
291
481
738

Ritz energy
2.847 656 250
2.901 577 012
2.903 708 675
2.903 723 901
2.903 724 347
2.903 724 373
2.903 724 376
2.903 724 377
2.903 724 377

Optimal 
1.68751.67281.88032.03302.19982.33072.4862(2.68)a) g ¼ V1N V1e and 0 ¼ expðsÞ. b) Number of independentfunctions. c) Ref. 169.

1710 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTS

was chosen as g ¼ 1  1=V1  1=V2 þ 1=V12 , where Vi is
the nuclear electron attraction potential for electron i and V12
is the electron repulsion potential. We see from Table 3 thateven from diﬀerent 0 ’s we will get similar converged energy
values by the ICI method. The chemical accuracy is obtained
after two or three iterations. The energy shown at the bottomof Table 3 is the best value, in the variational sense, so far
obtained in the literature.172
Table 4 shows the history of the hydrogen molecule wave
function. The present best result shown in Table 3 is given
at the bottom of the table. It is comparable with and probably
lower than the result obtained by Cencek and Rychlewski using Gaussian functions.172 Table 4 also shows the historical result of James and Coolidge,173 the full-CI result with extended
basis,174 and the landmark result175 of Kolos and Wolniewicz
studies.176 The Hartree–Fock result is also cited for reference.177 We note again that the wave functions generated by
the ICI method were diﬀerent from those existing in the literature, like those of James and Coolidge173 and Kolos and
Wolniewicz.175,176 This is an origin of the eﬃciency of thepresent free ICI method. More details will be described
elsewhere in the literature.165
Based on the above results for helium and hydrogen molecule and some results not given here, we can conclude thatthe method of solving the SE in an analytical form is now

18 decimal ﬁgures. You can get as accurate a result as you desire by increasing n. The convergence rate is dependent on the
choice of g function and 0 , and more detailed examinations
will be given elsewhere in the literature.164
Table 3 is a summary of the free ICI result for the hydrogenmolecule.165 The 0 is chosen to be 0 ¼ exp½ð 1 þ
max
Pi 2 Þð i Þ, where  i ¼ ðriA þ riB Þ=R and  ¼ r12 . The interi¼0

nuclear distance R is 1.4011 au and  ¼ 1:1. The g function
Table 2. Free ICI Energy of Helium AtomaÞ
Iteration
0
1
2
3
4
5
6
7
8

Dimension
3
18
74
186
377
668
1080
1634
2351

EnergybÞ /au
2.890 468 719 627
2.903 711 336 015
2.903 724 340 354
2.903 724 376 978
2.903 724 377 033 224
2.903 724 377 034 108
2.903 724 377 034 119 316
2.903 724 377 034 119 587 358
2.903 724 377 034 119 597 725

1=2a) g ¼ 1  V1N þ V1e ,þ u1=2 Þ expðsÞ. b)
0 ¼ ð1 þ s
Bestvalue by Schwartz169 (dimension,
10257):
2.903 724 377 034 119 598 311.

Table 3. Free ICI Energy of Hydrogen Molecule with Diﬀerent Initial Functions(R ¼ 1:4011 au)
Initial function
max i0

Number of
iteration (n)
1
2
3
4
5
6

4

1
2
3
4

5

1
2
3
4

Number offunctions
5
30
114
343
832
1788
Best value
33
260
951
2441
Best value
41
346
1276
3246
Best value

Energy/au
1.164 409 776 802 47
1.172 712 604 472 60
1.174 434 056 534 60
1.174 475 391 331 89
1.174 475 917 716 33
1.174 475 930 732 94
1.174 475 931 397 74
1.173 489 787 488 80
1.174 475 797 933 38
1.174 475 931 085 68
1.174 475 931 391 16
1.174 475 931 397 74
1.173 497 602 544 99
1.174 475 872 960 00
1.174 475 931 318 44
1.174 475 931 397 74
1.174 475 931 397 74

Table 4. History of Hydrogen Molecule Wave Function
Type of wave function
Hartree–FockFull CI (30s29p12d9f)
Hylleraas type
Hylleraas type
Gaussian functions
Free ICI(extended Hylleraas type)

Reference
SundholmaÞ
Röhse et al.bÞ
James–CoolidgecÞ
WolniewiczdÞ
Cencek–RychlewskieÞ

H–H distance/au
1.4
1.4
1.4
1.40111.4011

Total energy/au
1.133 629 573
1.174 285
1.173 539f Þ
1.174 475 930 742
1.174 475 931 39

Present

1.4011

1.174 475 931 397 74a) Ref. 177. b) Ref. 174. c) Ref. 173. d) Ref. 175. e) Ref. 172. f) Recalculated by the present author.

Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji

established. The free ICI method seems to be remarkably
eﬃcient. We have many prospects for works to be done in
the future and we are essentially very optimistic about them.
SAC-CI Method Applied to Photochemistry
and Photobiology
The SAC/SAC-CI theory itself is exact and yet very simple,
so that it compactly describes the physics of the electronic
structure of the ground and excited states of molecules. It
has the following merits.
1. The ground and excited states calculated by the SAC/
SAC-CI theory satisfy the orthogonality and the Hamiltonianorthogonality to each other, which are the important necessary
requirements for any theory dealing with diﬀerent electronic
states at the same time.
2. The theory satisﬁes size-extensivity and size-intensivity
requirements that are important when theories are applied tolarge molecular systems.
3. The theory satisﬁes the self-consistency requirement thatis important to avoid depending on the choice of the referencefunctions, which are usually Hartree–Fock.
4. It describes not only one-electron excited states from theground state, but also multi-electron excited states like those
appearing as satellite states in ionization spectra.13
5. This theory covers a very wide range of diﬀerent electronic states, from singlet to septet spin multiplicities, withthe same good accuracy. Conceptually, the theory utilizes anapproximate transferability of electron correlations between
ground and excited states. Figure 3 shows the states that canbe studied by the SAC-CI code in Gaussian03 program.36 It
covers ground, excited, ionized, and electron-attached statesof valence, Rydberg, inner-valence, and inner-core states.
6. We can directly compare the energies and the propertiesof diﬀerent electronic states. This is a very important merit of
the SAC/SAC-CI method and is very useful in actual chemical
studies.
7. The energy-gradients, which are the forces acting on theconstituent nuclei, can be calculated for any states shown in
Fig. 3 for both one- and multi-electron excited states.29–35
Using these quantities, we can also do geometry optimizations
for any of these states.

SAC/SAC-CI program
Accurate correlation theory for ground and exited states

SAC

singlet closed-shell (ground state)

SAC-CI
singlet excited states
triplet ground & excited states
ionized states (doublet ground & excited states)
electron-attached states(doublet ground & excited states)
quartet to septet ground & excited states

Energy Gradient (Force acting on nuclei)
Dynamics involving ground and excited states

Subject: Chemistry and Physics involving these states(www.sbchem.kyoto-u.ac.jp/nakatsuji-lab)

Fig. 3. Chemistry and physics covered by the SAC-CI
program on Gaussian03.

1711

8. Chemistry and physics are investigated with the same
methodology from ﬁne accurate calculations of small molecules to less accurate but reliable calculations of large molecules and molecular systems. The SAC-CI method has been
useful for studying ﬁne spectroscopy of molecules, for studying many diﬀerent electronic states of diﬀerent natures andqualities at the same time, like those appearing in molecules
including transition metals and in the spectra of wide energy
range. It has also been useful for studying relatively large
molecules like those appearing in the photo-electronic processes in biological systems.
9. In combination with the dipped adcluster model, which isa model for the catalysis and the adsorbate on a metal surface,
we can study catalytic reactions of the adsorbates on a metal
surface and the surface photochemistry.
Theories of the SAC and SAC-CI methods were explainedin some detail in the original papers9–14 and the earlier review
papers.22–27 Here, I will focus only on some recent applications
to the ﬁne spectroscopy and the electronic processes in thephotosynthetic reaction center of Rhodopseudomonus viridis.
When one studies electronic states in a wide energy range,
one ﬁnds that they involve not only one-electron excited states
but also multi-electron processes like shake-up states in ionization spectroscopy, which are generated by simultaneous ionization–excitation processes. In 1991, the author showed that
the SAC-CI theory is able to describe such excitation and ionization processes very accurately when not only singles anddoubles (SD) but also higher excitation operators are includedin the linked operators of the SAC-CI method.13 This method
was called SAC-CI general-R method in comparison with the
SAC-CI SD-R method. This generalization was done actually
as a by-product in the generalizations of the SAC-CI method toquasi-degenerate and multi-reference electronic states.16–21
The SAC-CI general-R method enables us to calculate excitation and ionization spectra over a wide energy region with highaccuracy; the method opened a new ﬁeld called theoretical ﬁne
spectroscopy.
Figure 4 shows the photoelectron spectrum of HCl in valence and inner-valence energy region (10–35 eV).73 The
peaks in the outer-valence region are essentially due to the
Koopmans one-electron process, but in the inner-valence region the peaks are due to the shake-up processes, i.e., the simultaneous ionization–excitation processes. So, the numberof the peaks is larger than the number of the molecular orbitals. In order to describe these photo-electron processes, thetheory must be able to describe both one-electron and multielectron processes with the same good accuracy. The SACCI general-R method can do this and gives a theoretical spectrum which is almost indistinguishable from the experimental
spectrum178 as shown in Fig. 4.
Figure 5 shows the inner-valence ionization spectrum of
H2 S. The experimental spectrum is due to Brion et al.179 andthe theoretical one is due to Ehara et al.75 The experimentalists
attributed seven peaks in this energy region, while the theoretists obtained much more peaks by the SAC-CI method. Actually, due to the SAC-CI general-R calculations, there existed
a much larger number of states in the inner-valence energy region than those shown in the spectrum but they did not haveintensities that were high enough to be written down. Howev-

1712 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

2π-1

AWARD ACCOUNTS(a) X-ray excited
photoelectron

2

HCl

5σ-1

1

6
3

10

15

20

25

5

30

7

35

R

40

2 (b) SAC-CI general-R

0.6

INTENSITY

4

V

X 1/5

V

V

R

R
R

0.4

X 1/5
6

1
0.2

3 4

5

7

0.0
10

15

20

25

30

35

40

BINDING ENERGY (eV)Fig. 4. HCl photoelectron spectra (experimental spectrum
from Ref. 178 and theoretical SAC-CI spectrum from
Ref. 73).

er, these states are expected to play important roles in the energy-relaxation processes and other dynamic processes in thisenergy region. Note that the experimental and theoretical spectra are again quite similar showing the reliability and usefulness of the SAC-CI method. If necessary, you can calculate,for example, the equilibrium geometries of these two-electronstates by the SAC-CI program.
The accuracy and the reliability of the SAC-CI method illustrated above are examples from theoretical ﬁne spectroscopy. This ﬁeld is now being developed intensively in our laboratory mainly by Prof. M. Ehara. In collaborations with experimentalists, entirely new dimensions of spectroscopy are being
developed that were impossible in the past. Entirely new ﬁeldsof spectroscopy are being opened with the aids of the very
advanced spectroscopic tools and of the simultaneous use of
the SAC-CI method for the analyses and understanding of
the natures of the spectra.
The energy-gradient method is a useful tool for studying
molecular geometries and chemical reactions. The energy gradients are the forces acting on the constituent nuclei of the system. The nuclei of the system move toward these forces andreach the equilibrium geometry or the product of the chemical
reaction. This author presented a simple, intuitive force concept for molecular geometries and reactions, called electrostatic (Hellmann–Feynman) force theory (ESF theory).180–183 This
theory is very predictive and works better than the Walsh model184 and the VSEPR theory.185 Based on this concept, we understand that the electron-cloud ﬂow is the cause of the chemical reaction, a systematic movement of the nuclear coordinates, which the author called electron-cloud preceding.182,183
When we can calculate these forces for every state of theground, excited, ionized, and electron-attached states of singlet
to septet spin-multiplicities of one- and many-electron proc-

Fig. 5. Inner-valence ionization spectra of H2 S (experimental spectrum from Ref. 179 and SAC-CI theoretical
spectrum from Ref. 75).

esses, we can certainly do a lot of chemistry. The Gaussian03
SAC-CI program can do this.36
Acetylene is a simple molecule whose geometries are observed not only for the one-electron excited state (A state)
but also for the two-electron excited state (C state).186 We haveapplied the SAC-CI geometry-optimization method to thismolecule and obtained the result shown in Table 5.35 The geometries of the excited states are trans, in contrast to being linear in the ground state. We have applied both SD-R and general-R SAC-CI methods to the excited states. We see from
Table 5 that the geometries of all the ground and excited states
are rather well reproduced by both SD-R and general-R methods, but the adiabatic excitation energy of the two-electron excited state (C state) is calculated correctly only by the generalR method: the SD-R result diﬀers by as much as 2.3 eV fromthe experimental value. For the one-electron excited state (Astate), both SD-R and general-R methods give reasonableagreement with the experiment.
Free base porphin (FBP) is a key compound in biologicalchemistry and its excited states appear in many photo-biological processes, like photosynthesis. The SAC-CI method has
successfully been applied to the spectroscopy of many porphyrins, hemes, and related compounds, showing its applicability
to biologically oriented photochemistry.82–92 Recently, we
have performed the geometry optimization of FBP in its
ground and excited states using the SAC-CI method93 andfound that the changes in the geometrical parameters underthe excitations are very small. The structural relaxation energies in the excited states were also very small. These propertiesof porphyrins are merits when the molecules are used as photofunctional compounds in biological systems.93
Photo-electronic processes in biology are very attractive
ﬁeld that may be elucidated by the SAC-CI method. Photosynthesis is one of the most attractive phenomena since it constitutes the basis of the existence of life on this globe. After conﬁrming that the SAC-CI theory was able to describe the photo-Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji

1713

Table 5. Spectroscopic Constants of the Ground and Excited States of Acetylene35

CCH

Excitation
level
—

RCH

/A
1.068
1.063

RCC

/A
1.219
1.203

/degree
180
180

Te
/eV
—
—

SD-R
General-R
Exptl.aÞ

1
1

1.098
1.097
1.097

1.377
1.385
1.375

122.6
121.7
122.5

5.485
5.329
5.232

SD-R
General-R
Exptl.aÞ

2
2

1.105
1.111
1.14

1.634
1.643
1.65

103.8
103.0
103

10.098
7.844
7.723

State

MethodX1 Ag(linear)

SAC
Exptl.aÞ

A1 Au(trans)C01 Ag(trans)a) Ref. 184.

chemistry of porphyrins,82–93 we studied the photo-electron
processes involved in the photosynthesis in a photosynthetic
bacteria, Rhodopseudomonus viridis.94–100 The crystalline
structure of the reaction center of this bacteria was alreadyknown.187 The absorption spectra of the reaction center as awhole were observed by Breton188 but were not assigned byany reliable theory. So, we applied our SAC-CI theory. The
photon-energy-assisted long-distance electron transfer couples
with the proton transfer in the reaction center and thus the photon energy accumulates as a concentration diﬀerence of protons on the two sides of the membrane. The electron transfer
pathway in the reaction center was already known to be very
eﬃcient and unidirectional, but the reasons were not yet established. We wanted to clarify the reasons of the unidirectionality and eﬀectiveness and the roles of proteins in this photoelectron process.
In the reaction center of Rhodopseudomonus viridis, elevenchromophores are involved and the proteins are essentiallytransparent in the energy region we are interested in. We calculated the absorption spectra of all the chromophores, placing
them in the ﬁeld of proteins and the other chromophores,
which were approximated by the electrostatic model putting
a point-charge on each atom of protein or chromophore.
Figure 6 shows the absorption spectrum and the linear dichroism spectrum of Rhodopseudomonus viridis observed experimentally and calculated by the SAC-CI theory. The assignments of the observed peaks were done using both theoreticaland experimental information available on that date. This was
the ﬁrst systematic assignment of the spectra with a reliable abinitio theory including electron correlations. The average discrepancy between theory and experiment was 0.13 eV (about
3 kcal/mol). It would constitute a basis for future photochemical studies of the reaction center. For more details, we refer tothe original papers and accounts.94–96
The rate of electron transfer (ET) is proportional to thesquare of the transfer integral Hif between the initial and theﬁnal states of the ET.189 This factor is called the electronic factor. The nuclear factor involves thermodynamic factors189 andis therefore more complex to calculate with ab initio theories.
We studied the nature of this photon-assisted ET in thereaction center by calculating the electronic factor. Jortner
et al.190 studied the ET using the semi-empirical theory by assuming that the transfer integral was proportional to the overlap between the LU (lowest unoccupied) MO’s of the two

Fig. 6. Absorption and linear dichroism spectra of thephotosynthetic reaction center of Rhodopseudomonus
viridis; (a) SAC-CI theoretical excitation spectrum,95 (b)experimental absorption and linear dichroism spectra,188
and (c) SAC-CI theoretical linear dichroism spectrum.95

1714 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

kcal/mol

P+BH-

30

P+(12A)H-(12A)

P+B-H

P*BH
1.7
P*(22A)
P+(12A)B-(12A) 0.7

15.9
10.4

20

AWARD ACCOUNTS

2.5
12.8

hν

M-branch

PBH

P +BH-

P+(12A)B-(12A)

25.2(21)
24.4

10
0

P+B-H

P+(12A)H-(12A)

104.4(72)
123.3

1.0
1.9

in protein (exptl.)
in gas phase

L-branch

Fig. 7. Electronic factors for the electron transfers from the special pair (P) to bacteriochlorophyll (B) and to bacteriopheophytin(H) of the photosynthetic reaction center of Rhodopseudomonus viridis.

^^

chromophores of the initial and ﬁnal states. When we calculated the overlap integrals between the LUMO’s of the special
pair (P) and bacteriochlorophyll (B) with the ab initio methods,
the integral was larger for the M-region than for the L-region,in contrast to the actual ET in the L-region. This implies that
the ET is not an overlap property: it is a more local property as
shown below. We then calculated the transfer integrals of theET pathways (1) and (2),ðAÞP BH    ð1Þ --- Pþ B H    ð2Þ --- Pþ BH ;using the SAC-CI wave functions of the excited and ionizedstates of P, the neutral and anion wave functions of B andthe neutral and anion wave functions of bacteriopheophytin(H).94,97 The results are summarized in Fig. 7.
The rhs of Fig. 7 represents the L-branch and the lhs theM-branch. The ﬁrst event is the excitation of P giving P .
From P , the transfer integral to the L-branch B is 25.2 au
while to the M-branch B the transfer integral is 1.7 au: theL-branch selectivity is well reproduced. The value in parentheses is the experimentally estimated value191 and so we see thatthe theoretical SAC-CI value is in good agreement. The value
in the denominator is the SAC-CI value without the protein environment; we can see that the eﬀects of proteins represented
by the point-charge model are small. The next step from B to
H is again preferable in the L-branch than in the M-branch: nonarrow path exists in the next step of the L-branch. Competitive to this path, another pathway is from B to P (from B to P
is up-hill), which is called a charge recombination process. In
comparison with the electronic factor from B to H, 104.4 au,
the charge recombination process is only 1.0 au, and so is almost forbidden. This is in good agreement with the experimentally known facts. Thus, the SAC-CI method nicely reproduced
the experimentally known unidirectionality and the eﬀectiveness of the ET in the reaction center.
Now that the theory has reproduced the experimentally
known facts, we can analyze and understand the origin of
the phenomena by examining the transfer integrals involved.94,97 The origin of the L-branch selectivity is attributed
to the biological structure of the reaction center. Figure 8
shows the analysis of the transfer integral for the Pþ B H !
Pþ BH process. The center-to-center distance between B
and H is almost the same in both the L- and the M-branches.
However, the Mulliken bond-order type analysis of the transfer
integral showed that the transfer integral between B and H is

P+B-H P+BHL-Branch

BM
I

BL
10.7Å

I

Center to center

10.6Å

Center to center

5.0Å

5.5Å

I

1-C to 1-C

1-C to 1-C

I

M-BranchHM

HL

LUMOs of BM-HM and BL-HL
Fig. 8. LUMO’s and the geometries of the BM –HM and
BL –HL that explain the origin of the diﬀerence in the electronic factors of the transfer integrals between M and L
branches.

mostly due to the single contribution from 1-C of B to 1-C
of H and that of the L-branch is much larger than that of the
M-branch. The reason is the distance between these atoms: that
 shorter than that of the M-branch.
in the L-branch is 0.5 A
Since the transfer integral depends exponentially on the dis diﬀerence causes an order of magnitude diftance, this 0.5 A
ference, as shown in Fig. 8. The ET is a very local phenomenon: it occurs from 1-C of B to 1-C of H and not by face-toface overlap between the two LUMO’s. The same is true for
the ET from P BH to Pþ B H. Thus, the origin of the L-branch
selectivity is attributed to the structural factor.
The origin of the forbidden charge recombination process is
attributed to something like the orbital symmetry control.94,97
Figure 9 explains the origin. The ET from P to B is essentiallyfrom the LUMO of P to the LUMO of B, while the back-transfer is from the LUMO of B to the HOMO of P, since the pathfrom the LUMO of B to the LUMO of P is up-hill. As seenfrom Fig. 9, the LUMO of PM is localized in the downward
ring in the vicinity of BL , so that the transfer integral between
the edge carbon of this ring of PM and the edge carbon of theneighboring ring of BL becomes large (this value is much larg-Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji(a) LUMO(P) and LUMO (B)PMPL(b) HOMO(P) and LUMO (B)PM

PLTransferBL

is still expanding vividly in our laboratory. For getting higher
eﬃciency, the direct algorithm currently being developed in
our laboratory by Dr. Fukuda is very promising. The SACCI applications to truly gigantic systems would soon become
possible based on the new framework of the theory now inprogress in our laboratory.

Recombination

BL

Fig. 9. The origin of the forbidden charge recombination
process.

er than the corresponding value of the PL –BM pair due to the which is the origin of the L-branch selecshortness by 0.5 A,
94,97). On the other hand, the HOMO of PM is localized intivitythe region far away from BL , so that the back-ET from BL to
PM is diﬃcult. Thus, the charge recombination from B to P
is ruled out, due to the diﬀerent localization natures of the
HOMO and LUMO of the special pair.
In the ET from P to B to H, the protein eﬀect was small,but in the next step of the ETs from H to menaquinone(MQ) and from MQ to ubiquinone (UQ), the protein residues
play important roles.98–100 We sketch here only the importantresults; for the details you refer to the original papers.98–100
For the ET from HL to MQ, the one-bridge mechanism throughTrp M250 is important. For the ET from MQ to UQ, the twobridge mechanism through His M217 and His L190 is by far
more important than the other possible pathways.98
In Rhodopseudomonus viridis, the cytochrome subunit is
attached to the reaction center like a cap; an electron is supplied by cytochrome c2 to this unit and transferred through it
to Pþ to neutralize it. The docking site of cytochrome c2 to thissubunit is believed to be close to the top heme c554 and the ET
occurs from c554 to Pþ . We studied the ET pathway fromc554 to Pþ .99 In the cytochrome c unit, four hemes, c554,
c556, c552, and c559 are arranged toward P. We calculated
the transfer integrals along this pathway, examining the possible roles of the protein residues near the pathway. For the ET
from c554 to c559, the ET was shown to be direct and no protein residues were found to play a bridge role. However, for
the ET from c559 to Pþ , two protein residues were found important. The one-bridge pathway c559–Tyr162–Pþ is most important, but the two-bridge pathway c559–Phe253–Tyr162–Pþis also important, reducing the transfer integral due to theabove one-bridge pathway. The biological implication ofsuch reducing role of the two-bridge pathway c559–Phe253–
Tyr162–Pþ is worth studying and the mutation experiment
would be interesting.
Figure 10 shows a summary of the ET pathways studied in
our laboratory. This study showed a high potentiality of the
SAC-CI method in biological photochemistry. There are many
important photo-electron processes in biology. Visions due torhodopsin is another example.192 This ﬁeld is now being actively developed in our laboratory, mainly by Dr. Hasegawa.
Before closing this chapter, we note that the SAC-CI world

1715

Catalytic Reactions on a Metal Surface
Catalysis is one of the most attractive phenomena in chemistry. The reactions that never occur in other conditions
smoothly proceed, for example, on metal surfaces. To clarify
the origin of this magic is a challenging subject for theoreticalchemists. These reactions occur at the interface between ﬁnite(adsorbates) and inﬁnite (surface) systems; for the latter, somemodeling is necessary for theoretical studies. The model that ismost frequently used in theoretical calculations is the clustermodel (CM), but this model is not adequate for a metal surface
because it does not include the eﬀects of free electrons of thebulk metal that are characteristic to a metal surface. In 1987,
the author presented the dipped adcluster model (DAM) for
chemisorptions and catalytic reactions on a metal surface.104
This model is able to include the eﬀects of the free bulk electrons characterizing the metal surface. The admolecule, which
is a combined system of the admolecule plus cluster, can easily
borrow some electrons from a metal surface and further, theadmolecule can be stabilized by the image force exerted fromthe surface.
Figure 11 explains the concept of the DAM. The adcluster, acombined system of the admolecule and a metal cluster, is dipped onto the electron bath of the free electrons of the metal
surface and electron exchange is permitted between the adcluster and the surface. The equilibrium is established when thechemical potential of the admolecule becomes equal to thechemical potential of the metal surface, .
dEðnÞ
ð15Þ
¼ ;
dn
where E is the energy of the adcluster and n is the number ofelectrons and spins transferred into the adcluster. Since theadcluster is an imaginary partial system, the number of theelectrons, n need not to be an integer and so we have a quantum chemistry with a non-integer number of electrons. Someinteresting physics exists behind the n dependence of the energy E and in the realization of the equilibrium condition. For
more details of the DAM, we refer to the original paper andthe review article.104–109
The DAM has been applied to study O2 chemisorptions on
palladium and silver surfaces,104,110,111 and to clarify the ‘‘harpooning’’ electron transfer from halogen molecule to alkalimetal surface and the succeeding electron transfer and surface
luminescence phenomena.108,109 It has also been applied to theepoxidation reactions and oxidation reactions of oleﬁns on asilver surface,121–127 the OH chemisorptions and reactions on
an Ag surface,128 and the hydrogenation reactions of carbon
dioxide giving methanol on a Cu/Zn surface,129–131 where
the promoting eﬀect of Zn was studied in some detail. The
DAM has also been used, in combination with the SAC-CI
method, for studying the spectroscopy of the molecules adsorbed on a metal surface.107–113 Without the DAM, the large1716 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTSElectron transfer route in the reaction center
Special pair to pheophytin: SAC-CI study
P∗BH
P+ BH−
P+B−H
P+B− Hkcal/mol30

P+(12A)H−(12A)

15.9

20

P∗(22A)

P+(12A)B−(12A)

P+(12A)B−(12A)
P+(12A)H−(12A)

25.2

1.7

P+ BH −

hν

104.4

1.0

10
0

M–branch

L–branch
PBH

L-side unidirectionality: Biostructural origin
cytochrome C2
cytochrome c2

c554

from cytochrome C2
to special pair

c556
c552

c–559

PHE230c559

PHE253
TYR162

special pairthrough TYR162

special pairdirect transferfrom pheophytin tomenaquinone
L−Bphefrom menaquinone to ubiquinone
His L190

Ala M216

His M264

4.78
3.64

His M217

UQ

MQ

Glu M232

3.86

MQ
3.30

Trp M250

His L230

through His M217 and His L190

through Trp M250

Fig. 10. Summary of the electron-transfer pathway in the photosynthetic reaction center of Rhodopseudomonus viridis.

Dipped Adcluster Model (DAM)

Fig. 11. Concept of the dipped adcluster model (DAM).

lower-energy shifts of the peaks in the UV and photoelectron
spectra of the adsorbed molecules from those of the gas-phasewere not explained,113 to be able to explain such shifts may beconsidered as an experimental justiﬁcation of the DAM. An
approach that combines the experimental surface spectroscopic
measurements and the theoretical DAM/SAC-CI method
would become a useful methodology for identifying the species adsorbed on a metal surface: the experimental results
alone are insuﬃcient for doing this identiﬁcation.
Here, we only brieﬂy review the DAM studies of the partial
oxidations of oleﬁns.121–127 Ethylene is epoxidized on a silver
surface to give ethylene oxide with a selectivity of about
85%.193 We studied two reaction modes, one being an attackof ethylene to molecular superoxide O2  on a silver surface
and the other being an attack to atomic oxygen O on a silver
surface. Since the former reaction seemed to be the main reaction course, we explain only this course here. Figure 12 showsthe eﬀect of the silver surface, i.e., the energy diagram for thereactions with and without the silver surface. Figure 13 showsBull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji

80

Without Ag surface

1.43

HH
1.31

Energy (kcal/mol)

60

O

H
H
C

C

H
H
O

O 1.82

C

1.49

C

O

1.49

HH

1.49

C H
H

O

O

C H
H
1.47

1.48

EO
+

1.37

O

40
20

n=0

52.9

43.8

39.7

TS forapproachIntermediate

C2H4

1717

38.0

+
O2

0
-20

Reactant

TS fordissociation

Product

Energy (kcal/mol)

With Ag surface

0
26.9

-20

11.3
0.1

-40

Adsorbed
oxygenReactant

TS forIntermediateapproach

10.2

1.7

TS forProductdissociation

Fig. 12. Energy diagram showing the eﬀect of the silver surface for the epoxidation of ethylene.

Energy (kcal/ mol)

60

Ag

40

20

C 3H 6
+
O2
+
Ag 2

H

1.43
C3 H6
+
1.38
O
O
2.34
Ag

H C
H

C

1.48

O

Ag

C

H C

O 1.41

1.48
O

1.78
O

1.52H
H
C H
1.41

H

H

H
C H
H
1.51

H
1.48
Ag

Ag

69.4

Ag

Ag

26.9

O 1.37

3.8
−40

1.51
H
H C
1.48

O

O

Ag

Reactant

Adsorbed
oxygenTS forapproach

Ag

PRO
+
Ag2O−

28.3

28.8

13.7

C

O

1.55
H H H
H
C H

C

Ag

0

−20

1.52

1.51
H H
C HH
1.43

Intermediate

C

H
H C

1.61
Ag

39.5
H

1.51

O

1.44

H
H
C H
1.51

65.3

O

Ag

CH3CH 2 CHO+
Ag2 O −

Ag

TS for dissociationor H–migration

Intermediate

Product

Fig. 13. Energy diagram, the reaction intermediates, the transition states, and the products along the course of the ethylene epoxidation reaction on a silver surface.

the more detailed energy diagram, the reaction intermediates,
the transition states, and the products along the course of theoxidation reaction on a silver surface.
From Fig. 12, you immediately see how large the eﬀect of
the silver surface is. Without silver, the barrier of the reaction
is too high. Figure 13 shows two reaction courses, one leading
to epoxide and the other leading to acetaldehyde, which leads
further to a complete oxidization. Two pathways branch fromthe intermediate adduct. Only the pathway leading to the epoxide is energetically feasible. The barrier to acetaldehyde intermediate is high (71.5 kcal/mol), and therefore this course isforbidden. Thus, the DAM explains the catalytic activity ofthe silver surface and the selectivity towards the epoxide.

The atomic oxygen on a silver surface was shown to be lessselective. The molecular superoxide O2  is an active speciesof epoxidation on a silver surface.
When we replace ethylene with propylene, the selectivity ofthe epoxidation reaction is much reduced: from 85% to only
2.5%.194 Most propylene is converted up to complete oxidation. So, the catalysis design leading to propylene oxide is ofmuch practical merit. Figure 14 shows the reaction energy diagram for propylene calculated in a similar way to Fig. 13 forethylene.121 You see in Fig. 14 that there are certainly two similar routes leading to propylene oxide and aldehyde, the latterroute being energetically forbidden due to its high energy barrier. However, for propylene, there is another route that starts

1718 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

Energy (kcal/mol)

60

Ag

40

20

AWARD ACCOUNTS

C3H6
+
O2
+
Ag 2

H C
H

C

1.48

O

Ag

2.32

1.51

H
C H
H

1.48
2.31

Ag

10.8

13.7

C

H C
H

1.78

Ag

H

1.51

1.51

O

2.34
Ag

H

1.43

C3H6
+
1.38
O
O

H

1.48

69.4

Ag

Ag

C

H C

1.43

O

O

H
C H
H

O

1.52 H
H
H
C H
1.41

H
H C

O 1.41

H
C

H
H
H

O
+
Ag2O−

Ag

Ag

C

0

−20

28.8

26.9

H
H

C1

C2

1.38

1.47
Oa
2.33
Ag1

Ob

H

1.40

Ha

1.17

Ob 1.50
Oa
2.27
Ag1

CH3CH2CHO + Ag 2O −

18.1

Ag2

Ha
1.66

0.953
Oa

o

Ag 2

Ha

Ha

0.95
Ob

Oa

95.1o

2.24
Ag 1

42.6

4.22

Ob
84.9

2.32

65.3

H
H

87.5

1.532
Oa

1.41

H
C 2 C3H C1

0.954

102.5o

Ag2

39.5

2.8

H
C3 H

Ha

28.3

3.8

H 1.38

1.46

−40

29.7

Ag 2

Ob

Ag1

1.47
H
H
H
C C3 HH C1 2
1.36

4.09

2.36

2.10

2.19
Ag1

0.96

Ag2

Ag1

Ag 2

Fig. 14. Energy diagram, the reaction intermediates, the transition states, and the products along the course of the propylene
epoxidation reaction on a silver surface.

from an attack of superoxide to a proton of methyl group thatis absent in ethylene. You see that this additional route is also
very smooth and lies just below the epoxidation route; in thelast stage, this additional route is much more favorable thanthe other routes and it ﬁnally leads to complete oxidation of
propylene. Therefore, without any particular artiﬁce, the epoxidation of propylene is diﬃcult with the silver catalysis. However, since we know from Fig. 14 that there is certainly thepath on the silver surface leading to propylene oxide, it would
be nice if we could invent an artiﬁce to block the route involving the attack of superoxide to the hydrogen of methyl group.
Actually, if such hydrogens do not exist, as is the case instyrene, the epoxidation reaction occurs smoothly, like it doesfor ethylene.
The electronic structures and the reactions of the surface adsorbates are quite interesting subjects that are of importancefrom both scientiﬁc and industrial points of view. However,studies with only experimental means are rather diﬃcult in thisﬁeld and so the collaborations between theory and experiment
would be very desirable for getting deeper understanding of
the surface phenomena.
Relativistic Quantum Chemistry and the Electronic
Theory of NMR Chemical Shifts
Relativity becomes more and more important as the system
involves heavier elements. The eﬀects are of particular importance for the molecules in an external magnetic ﬁeld and so are
reﬂected sensitively in the NMR chemical shifts of molecules
when they includes heavy elements in the neighborhood of theresonant nuclei. When molecules do not include heavy elements, the relativistic eﬀects are small and the chemical shifts
are expressed by a sum of the diamagnetic term and paramagnetic term. The former is dependent basically only on the molecular geometry and the nuclear species involved. The variety
of NMR chemical shifts is mainly caused by the paramagnetic

term.
Since 1984, we have studied the electronic mechanisms of
the NMR chemical shifts for various resonant nuclei on the
periodic table.132–154 Since chemical shift is a local propertyof valence electrons near the resonant nuclei,132 the primarymechanism of the chemical shift is an atomic property thatis dependent solely on the position of the element in the periodic table.138 A detailed account of the electronic mechanismof the chemical shift was given in Refs. 132 and 138. It may be
brieﬂy summarized as follows. Since the NMR chemical shift
is a measure of the angular momentum of electrons induced
around the resonant nuclei by the external magnetic ﬁeld,
the p- and d-electronic structure of the resonant nuclei in themolecule is the central issue. (An s-electron does not have angular momentum.) To rotate, a p-electron has to move fromsay px to py (rotation within xy plane) and this rotation is expressed in the paramagnetic term by the excitation of an electron from the px -dominant occupied MO i (whose coeﬃcient is
Cxi ) to the py -dominant unoccupied MO a (whose coeﬃcient is
Cya  ), so that the strength of this rotation is proportional to thedensity of this transition, say Cxi Cya  and the facility of thisrotation is inversely dependent on the excitation energy of thistransition Ei!a .
The chemical shifts of the dn metal complexes, Ti, Nb, Mo,
and Mn complexes, are due to the d–d mechanism. Both thehigher-occupied and the lower-unoccupied MO’s of thesecomplexes are characterized by the metal d orbitals and the rotation induced by the external magnetic ﬁeld is expressed by atransition from the occupied d orbital to the unoccupied d orbital. The facility of the rotation (or transition) would be inversely proportional to the excitation energy of the magneticallyallowed transitions. We have actually calculated the magnetically allowed d–d transition energies of several Mo complexes by the SAC-CI method and found a beautiful linearity
between the observed chemical shifts and the inverses of theBull. Chem. Soc. Jpn., 78, No. 10 (2005)

H. Nakatsuji−20

Relativistic effect

HI

−10

Exptl (ppm)transition energies.134,135
The chemical shifts of the Cu, Ag, Zn, and Cd complexes
are due to the d-hole and p-electron mechanisms.58 These elements have d10 s1{2 p0 conﬁgurations. The chemical bonds between metal and ligands are mainly due to the s-electrons.
However, since an s electron has no angular momentum, theinduced angular momentum must come from d- and/or p-electrons. For the Cu complexes, the d–s level spacing is smaller
than the s–p one, so that the d-orbital can more easily participate in the chemical bond than the p-orbital. Therefore, as theligand is electron-withdrawing, the chemical shift increases,
because the hole produced in the d-shell gives rise to the angular momentum. We call this d-hole mechanism. For the Cd andZn complexes, the s–p level spacing is smaller than the d–sone, so that the p-orbital can more easily participate in thebond than the d-orbital. Therefore, the chemical shift increases
as the ligand is more electron-donating, since the number of pelectrons near the metal increases. We call this mechanism pelectron mechanism. Note that between the d-hole and p-electron mechanisms, the ligand-electronegativity dependence ofthe chemical shift is reversed. For Ag complexes, the s–p
and d–s spacings are close, so the p-electron and d-hole mechanisms are competitive: the behaviors of the Ag chemical
shifts are more complex than those in the other cases.
The Ga and In complexes are interesting examples where
the chemical shifts are mainly governed by the diamagnetic
term: the chemical shifts are dependent solely on the molecular
structure of the complexes, like the metal–ligand distances andligand nuclear charges.139,145 These elements have s2 p1 conﬁguration. The metal p-electron is usually withdrawn almost
completely to the ligand sites, because the ligands are usuallymore electronegative than the Ga and In metals. Therefore, only the s-electrons are left near the metal atoms but the s-electrons have no angular momentum and therefore the paramagnetic term is small. So, for the Ga and In complexes, the chemical shifts are due to the dia-mechanism.
We have investigated the mechanism of the chemical shifts
for a variety of metal complexes and found that the primarymechanism of the chemical shift is an intrinsic property of
the central resonant metal atom itself and therefore can beunderstood from the position of the element on the periodic
table.132–142 However, this is not the case when we examine
the chemical shift induced by the halogen ligands. We have
two diﬀerent dependences: normal halogen dependence andinverse halogen dependence. The above (non-relativistic)
study could explain the inverse halogen dependence, but not
the normal one. There were many examples of the normal
dependence, but the origin was not clear until we showed it
to be due to the relativistic eﬀect.143
Figure 15 shows the proton NMR chemical shifts of thehydrogen halides. It shows a typical normal halogen dependence.143 When we calculate it by non-relativistic theory, we
get a dot result: we get essentially no chemical shift by changing the halogen from a light one to a heavy one. However,when we calculate it with the relativistic method, which we invented by modifying the UHF method,143 we could very nicely
reproduce the observed trends—normal halogen dependence.
Thus, the proton chemical shifts in HX series are entirely
due to the relativistic eﬀect. This proton NMR experiment

1719

HBr
HCl

0

Non-rela
Relativistic

HF
10
10

−10

0

−20

Theory (ppm)Fig. 15. Importance of the relativistic eﬀect on the proton
NMR chemical shifts of the hydrogen halides.

actually observes a relativistic eﬀect! Among the relativistic
eﬀects, the spin-orbit eﬀect was most important.
A similar importance of a relativistic eﬀect, spin-orbit eﬀect
among others, was observed in many ab-initio theoretical
calculations published from our laboratory,143–154 and now it
is generally accepted that the normal halogen dependence isdue to the relativistic eﬀect, though the inverse halogen dependence is understood mainly from the electronegativity of theligand.
Among the relativistic eﬀects, the spin-orbit eﬀect of thedirectly bonded heavy elements was most important for thechemical shifts of the relatively light nuclei. However, forthe chemical shift of heavy resonant nuclei, like mercury
and lead, other spin-free relativistic eﬀects were also very important, the correlation eﬀects were also shown to be very important and furthermore, the couplings between these eﬀects
were also very important.148,149,151–154 Therefore, when we calculate the chemical shifts of heavy resonant nuclei, we have toinclude all the relativistic eﬀects and the correlation eﬀects atthe same time. For this purpose, we have developed a straightforward method: ﬁrst we solve the four-component Dirac–
Fock equation and then solve the coupled-cluster equation
based on the Dirac–Fock solution.154 However, at this moment, the computer time and the required memory are toolarge to perform calculations for heavier resonant nuclei withthe use of satisfactorily large basis sets. We then have also developed two-component Douglass–Kroll–Hess transformation
for magnetic shielding calculations.152,153 Then, we may use
less computationally demanding electron-correlation method
like Meller–Plesset perturbation method. Such a combinationis certainly practical at this moment, though the more straightforward method described above will certainly become orthodox in the near future. It is important to open a new ﬁeld in
relativistic quantum chemistry.
Summary
Quantum principles in chemistry give us a basis of imagination for understanding this wonderful world. The author has
tried to describe some branches of chemistry in terms of theconcepts of the quantum mechanics. When he started quantum
chemistry, the excited states were something very diﬃcult to
calculate by the quantum chemical method. The SAC-CI theo-1720 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTS

ry he introduced certainly has made it much easier. It has realized the quantum chemistry involving a variety of electronic
states, singlet to septet ground and excited states, ionized statesand electron-attached states. The energy-gradient technique
has made this method even more useful. It has opened new
ﬁelds like theoretical ﬁne spectroscopy and biological photochemistry. Through the use of the SAC-CI code in the Gaussian suit of programs, this theory would provide a useful method to solve chemical and physical problems in science andindustry worldwide.
The magic of catalysis on a metal surface has stimulated theauthor’s imagination and brought about a model, called dipped
adcluster model (DAM) of the adsorbate on a metal surface,
including the roles of the free electrons characteristic to themetals. The DAM has been useful to describe electronic structures of the adsorbates on a metal surface, so that it was usefulfor studying catalytic reactions, electron transfers, and photoelectronic processes on a metal surface. It will provide a usefultheoretical spectroscopic tool for identifying surface adsorbates on metal surfaces by combining it with the SAC-CI
method.
The NMR chemical shifts are probably the most frequently
observed properties of molecules in analytical chemistry. The
author showed that these quantities include rich chemistry reﬂecting ﬁne information about the valence electrons of molecules. He clariﬁed the origins of the trends observed in the
NMR chemical shifts of various resonant nuclei and showedthat the principal mechanism of the chemical shifts is an intrinsic electronic property of the resonant atom itself and therefore
can be understood from the position of the element on the
periodic table. Further, when the element is connected withheavy elements, the NMR chemical shift is shown to be inﬂuenced very sensitively by their relativistic eﬀects. For example, the proton chemical shift of the hydrogen halide series
was shown to be due entirely to the relativistic eﬀect, in particular the spin-orbit eﬀect, of the heavier halogens. The normal halogen dependence was attributed to the relativistic effect. For heavy element NMR, both spin-dependent and spinfree relativistic eﬀects are important, the electron correlationeﬀects are also important, and further, they strongly couple
with each other, making its chemistry rich, though the calculations are rather complex and time consuming. This suggests
the existence of something very interesting in the chemistryof heavy elements.
Doing theoretical chemistry, the author has felt more andmore strongly the importance of developing a general theoryof solving the Schrödinger equation without approximation.
Quantum chemistry must be truly predictive not only qualitatively but also quantitatively. Since 1999, he could have a
stream of ideas that allowed him to realize this dream. He ﬁrst
investigated the structure of the exact wave function and developed the method of constructing the exact wave function
starting from an approximate one. An obstacle on the way tothe exact analytical solutions for atoms and molecules was
the singularity problem intrinsic to the Coulombic nuclear
and electron potentials, but this was solved by introducing
the scaled Schrödinger equation. His method has immediately
been applied to several atoms and molecules with success: thefree ICI method based on the scaled Schrödinger equation is

now safely said to be an established method of solving the
Schrödinger equation of atoms and molecules in an analyticalform for both ground and excited states. An extension to therelativistic case has been ﬁnished195 and an extension to thetime-dependent case is now in progress. We expect many ﬂowers and fruits along this orthodox route of quantum chemistry.
The author would like to express his thanks to his parents
and his wife for their support of his risky way of life. This riskhas been diminished by the guidances of his teachers: Professors Junji Furukawa, Takayuki Fueno, Teijiro Yonezawa,
Hiroshi Kato, and Ken-ichi Fukui. Professors Jeremy Israel
Musher and Robert G. Parr have encouraged him very muchwith their scientiﬁc inspirations and optimism. The author
would also like to deeply acknowledge his colleagues, Drs.
Kimihiko Hirao, Masahiko Hada, Hiromi Nakai, Masahiro
Ehara, Zhen-ming Hu, Junya Hasegawa, and Ryoichi Fukuda,
for their active collaborations with him. This author ﬁnally
thanks his students for introducing a lot of interesting problems.
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H. Nakatsuji
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1724 Bull. Chem. Soc. Jpn., 78, No. 10 (2005)

AWARD ACCOUNTS

Hiroshi Nakatsuji was born in 1943 in Suita, Osaka, Japan, and received his Ph.D. in 1971 from
Kyoto University, where he became an instructor soon afterwards. He did postdoctoral studies
with Prof. J. I. Musher (Yeshiva University) and Prof. R. G. Parr (Univ. North Carolina). He became a full professor at the Faculty of Engineering, Kyoto University in 1990. He was a Visiting
Professor of Tokyo Institute of Technology, 1989–1990, a Concurrent Professor of the Universityof Tokyo, 1996–1998, Trustee, Institute for Fundamental Chemistry, 1997–2000, Director, Fukui
Institute for Fundamental Chemistry, Kyoto University, since 2004. Since 1993, he is a memberof the International Academy of Quantum Molecular Science (France) and since 1994, he is a
Board of Director, International Society for Theoretical Chemical Physics. Since 2000, he is
an Editor, Journal of Computational Chemistry. He was awarded the Physical Chemistry Awardof the Chemical Society of Japan for 1990 and Chemical Society of Japan Award for 2003.