Hiroshi Nakatsuji

Nakatsuji


Director, Quantum Chemistry Research Institute (QCRI)
Visiting Professor, Waseda University
Professor Emeritus, Kyoto University

Email:h.nakatsuji/a/qcri.or.jp (replace /a/ with @)
WWW: http://www.qcri.or.jp/?lang=en

Tel: -81-75-634-3211 Fax: -81-75-634-3211

Address: Quantum Chemistry Research Institute
    Kyodai Katsura Venture Plaza, North building 107
    1-36 Goryo-Oohara, Nishikyo-ku, Kyoto, 615-8245, Japan


MAIN RESEARCH ACTIVITIES OF HIROSHI NAKATSUJI
October, 2008
     As a quantum scientist, a dream of Hiroshi Nakatsuji has been to construct highly predictive and widely applicable theories and concepts that are useful in the studies of chemistry and physics [1]. He has endeavored to make his works to be highly original, innovative, simple, intuitive, widely-applicable, and even beautiful. Further, he has tried to make up his theories starting from zero up to the useful methodologies in the fields. His main contributions are summarized as follows.

I. General Method of Solving the Schrödinger and Dirac-Coulomb Equations
     The Schrödinger equation and the relativistic Dirac-Coulomb equation represent the basic mathematical principles governing chemistry, biology, physics of matter, and related sciences [2]. So, a central theoretical theme of these sciences is to establish a general method of exactly solving these two basic equations. Thereby, we can do quite accurate predictions of the phenomena without doing experiments, and therefore, this has long been a dream of theoretical chemists and physicists, though it was believed impossible by many scientists including P. Dirac [2], H. Eyring [3] and others. Recently Nakatsuji has realized this dream! He found a general method of analytically solving the Schrödinger equation [1, 4-21], breaking the "dogma" that this equation can not be solved, which lived for over 80 years since the birth of quantum mechanics.
     He solved the singularity problem by introducing the scaled Schrödinger equation [10] and opened a new field realizing the calculations of accurate analytical solutions of the Schrödinger equation for general atoms and molecules. His free iterative complement interaction (ICI) method [10], compactly called free complement (FC) method, combined with the variation principle gave, for example, the helium ground state energy correct to 41 [15] and 43 [20] decimal figures. He further established a method to calculate the accurate analytical solutions of the Schrödinger equation without doing analytical integrations [16]. This is a very general methodology that is applicable to any atoms and molecules. Since the Schrödinger equation is the governing principle of chemistry, this means that chemistry will gradually change from "empirical" to "logical" science whose bases are firmly on the principles of quantum mechanics. Detailed explanations are as follows.
     Nakatsuji has studied since 1999 the mathematical structure of the exact wave function [4-8]. He showed that the iterative CI (ICI) method and the simplest extreme coupled cluster (SECC) method produce the wave functions having this structure and proved that the Schrödinger equation can be solved with only one [8] to singles-and-doubles [14] number of variables, in contrast to an astronomical number of variables in the full CI method. However, when we aim to solve the Schrödinger equation analytically, a big obstacle arose that was the singularity problem caused by the Coulomb potentials in atomic and molecular Hamiltonians: the integrals of the higher powers (than third) of the Hamiltonian diverge to infinity [10] and so one can not proceed further! Nakatsuji solved this problem by introducing the inverse Schrödinger equation and the scaled Schrödinger equation [10], which are equivalent to the original Schrödinger equation but do not cause the singularity problem. Between the two, the latter is simpler and easier to use. Further, he proposed the free ICI method (or the free complement (FC) method) based on the scaled Schrödinger equation, and with this method, he could calculate the world-best variational wave functions of helium [10, 15,17,18,20,21], hydrogen molecule [6], etc. For example, he calculated the energy of helium atom correct to 41-43 decimal figures [15,20] and the excited helium atom at non-relativistic limit (electron-nuclear states) [18].
A problem at this stage was that the analytical integration is not always possible for the complement functions of general atoms and molecules, because when the Hamiltonian of the system generates the complement functions through the ICI formalism, it does not care about the integratability of the functions it produces. But, Nakatsuji again solved this problem by formulating the local Schrödinger equation (LSE) method [16], which enabled the solution of the Schrödinger equation without doing analytical integrations over the complement functions involved. The method has been applied to a lot of atoms and molecules having 2 to 5 electrons [16], and further to 5 to 6 electrons. An efficient general algorithm applicable to larger atoms and molecules are now in progress.
     The relativistic Dirac-Coulomb equation was also solved similarly to the Schrödinger equation [13]. The variational collapse problem that often appeared in the relativistic field was circumvented by introducing the inverse Hamiltonian method [22,9]. The atomic balance and the approximate kinetic balance are automatically taken care in the ICI methodology as an ICI balance[19].
     Thus, the basic methodology of accurately calculating the solutions of the Schrödinger equation and the relativistic Dirac-Coulomb equation has been formulated. The remaining problems are to formulate efficient algorithms and computing, which enable the applications of this methodology to ordinary atoms and molecules appearing in chemistry and physics. Such formulations constitute a new wave in theoretical chemistry and physics toward "Accurately Predictive Quantum Science", which has much more accurate predictive power than the conventional quantum chemistry, which was essentially "an approximate science" or "theory for understanding", because people did not know how to solve the Schrödinger equation.
     Here, a word about the full CI and the contemporary state-of-the-art quantum chemistry theories: In modern quantum chemistry, one first prepares some basis functions by empiricism, forms therefrom a set of orthonormal orbitals (Hartree-Fock orbitals mostly), and then expands everything with this set of orbitals. The Hamiltonian is rewritten in a second-quantized form and the quantum chemical calculations are transformed into matrix and vector manipulations. Ultimately, the best theory is the full CI method. However, due to the incompleteness of the basis functions first prepared, the full CI solutions are usually far from the true analytical solutions of the Schrödinger equation. When we enlarge the basis functions to make it more complete, the full-CI dimension increases formidably and therefore, this route of accurate quantum chemistry must be reconsidered.
In the conventional quantum chemistry, we have basis-set nightmare, which is difficult to overcome, but our free complement method does not have it. The quantum Monte Carlo approach has an exact node nightmare, which is also difficult to overcome, but the present approach does not have it.
     Another word about the explicitly correlated wave function approach: Hylleraas, James and Coolidge, and Kolos et al. were great in imaging very accurate wave functions of helium and hydrogen molecules only by intuitions. However, for general atomic and molecular systems, it is impossible to estimate the functional form of the exact wave function only by intuitions. The free ICI (FC) formalism provides a general method of producing the exact wave functions of atoms and molecules in an analytical expansion form. The variables left undetermined in the FC method are calculated using the variation method or the LSE method.

1) Deepening and Extending the Quantum Principles in Chemistry, H. Nakatsuji, Bull. Chem. Soc. Jap. 78, 1705 (2005).
2) P. A. M. Dirac, Proc. Roy. Soc. (London), A123, 714 (1929).
3) H. Eyring, J. Walter, G. E. Kimball, "Quantum Chemistry", John Wiley & Sons, Inc., New York (1944).
4) Structure of the Exact Wave Function, H. Nakatsuji, J. Chem. Phys., 113, 2949-2956 (2000).
5) Structure of the Exact Wave Function. H. Nakatsuji and E. R. Davidson, II. Iterative Configuration Interaction Method, J. Chem. Phys. 115, 2000-2006 (2001).
6) Structure of the Exact Wave Function. III. Exponential Ansatz, H. Nakatsuji, J. Chem. Phys., 115, 2465-2475 (2001).
7) Structure of the Exact Wave Function. IV. Excited Sates from Exponential Ansatz and Comparative Calculations by the Iterative Configuration Interaction and Extended Coupled Cluster Theories, H. Nakatsuji, J. Chem. Phys., 116, 1811-1824 (2002)
8) Structure of the Exact Wave Function. V. Iterative Configuration Interaction Method for Molecular Systems within Finite Basis, H. Nakatsuji and M. Ehara, J. Chem. Phys., 117, 9-12 (2002).
9) Inverse Schrödinger Equation and the Exact Wave Function, H. Nakatsuji, Phys. Rev. A 65, 052122 (2002).
10) Scaled Schrödinger Equation and the Exact Wave Function, H. Nakatsuji, Phys. Rev. Lett. 93, 030403 (2004).
11) General Method of Solving the Schrödinger Equation of Atoms and Molecules, H. Nakatsuji, Phys. Rev. A, 72, 062110 (2005).
12) Free ICI (Iterative Complements Interaction) Calculations of Hydrogen Molecule, Y. Kurokawa, H. Nakashima, and H. Nakatsuji, Phys. Rev. A, 72, 062502 (2005).
13) Analytically Solving the Dirac-Coulomb Equation for Atoms and Molecules, H. Nakatsuji and H. Nakashima, Phys. Rev. Lett., 95, 050407 (2005).
14) Iterative CI General Singles and Doubles (ICIGSD) Method for Calculating the Exact Wave Functions of the Ground and Excited States of Molecules, H. Nakatsuji, M. Ehara, J. Chem. Phys. 112, 194108 (2005).
15) Solving the Schrödinger Equation for Helium Atom and Its Isoelectronic Ions with the Free Iterative Complement Interaction (ICI) Method, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 127, 224104 (2007).
16) Solving the Schrödinger Equation of Atoms and Molecules without Analytical Integration Based on the Free Iterative-Complement-Interaction Wave Function" H. Nakatsuji, H. Nakashima, Y. Kurokawa, A. Ishikawa, Phys. Rev. Lett, 99, 240402 (2007).
17) Solving the Electron-Nuclear Schrödinger Equation of Helium Atom and Its Isoelectronic Ions with the Free Iterative-Complement-Interaction Method, H. Nakashima and H. Nakatsuji, J. Chem. Phys. 128, 154108 (2008).
18) Solving the Electron and Electron-Nuclear Schrödinger Equations for the Excited States of Helium Atom with the Free Iterative-Complement-Interaction Method, H. Nakashima, Y. Hijikata and H. Nakatsuji, J. Chem. Phys. 128, 154108 (2008).
19) Solving the Schrödinger and Dirac Equations of Hydrogen Molecular Ion Accurately by the Free Iterative Complement Interaction Method, A. Ishikawa, H. Nakashima, and H. Nakatsuji, J. Chem. Phys. 128, 124103-1-12 (2008).
20) Solving the Schrödinger Equation of Helium and Its Isoelectronic Ions with the Exponential Integral (Ei) Function in the Free Iterative Complement Interaction Method, Y. I. Kurokawa, H. Nakashima, and H. Nakatsuji, Phys. Chem. Chem. Phys. 10, 4486 (2008).
21) How accurately does the free complement wave function of a helium atom satisfy the Schrödinger equation? H. Nakashima and H. Nakatsuji, Phys. Rev. Lett. 101, 240406-1-4 (2008).
22) R. N. Hill and C. Krauthauser, Phys. Rev. Lett. 58, 83 (1987).

II. SAC/SAC-CI Method for Studying Chemistries of Excited and Ionized States [*]
1. Single-reference theory
     Nakatsuji and Hirao proposed the SAC (symmetry adapted cluster) method for the ground states of closed and open-shell electronic structures [1-3]. The SAC method is a kind of coupled cluster method that takes all the excitation operators to be symmetry adapted, so that no spin-contamination problem arises and it leads to a stable convergence to the spin-eigen function. Then, Nakatsuji proposed in 1978 the SAC-CI (configuration interaction) method [4-6] to describe accurately and efficiently the electronic structures of the excited, ionized and electron-attached states of molecules (ground and excited states of singlet to triplet spin multiplicities). The coding of the SAC/SAC-CI method was completed in singles and doubles approximation for all of these states in the same year and applied to water et cetera to examine the accuracy of the method. In those days, no full-CI calculations existed and only SDTQ CI results were available for water with very limited basis set, but the SAC/SAC-CI results were satisfactorily quite accurate, reproducing these SDTQ-CI results for both ground and excited states. Outside Japan, a CCD coding was reported by John Pople around this year [7], but the SAC/SAC-CI theory and coding [1-6] correspond to the CCSD[8]/CCLRT[9] or EOM-CC[10] theory. But, such codes were developed much later in the West World.
     The SAC/SAC-CI method was applied to the valence and Rydberg singlet and triplet excitations and the ionization of various molecules from small to relatively large molecules like porphyrins and gave very accurate descriptions of these various electronic states [11-13]. The methods were also used to study the inner-valence ionization spectra and their satellites [13], the excitation and ionization spectra and the hyper-fine splitting constants of radicals [14]. These studies opened a reliable ab initio methodology for studying excited, ionized, and open-shell molecules with the SAC-CI calculations.
     The original SAC-CI code was for ordinary single-electron excited or ionized states (SAC-CI SD-R), but it was extended later to multi-electron excitation and ionization (shake-up) states (SAC-CI general-R) [15] and further to include high-spin electronic states of quartet-to-septet spin multiplicities [16]. The accuracy of the SAC/SAC-CI method was confirmed by comparing the results with the experimental results and also with the full-CI results when they were available [17,18]. The geometries of the excited states and the courses of the photochemical reactions are studied efficiently when the forces acting on the constituent nuclei of the systems are available. For this purpose, the energy gradient method was implemented in the SAC/SAC-CI code [19,20], so that we can calculate the forces acting on the constituent nuclei for every ground and excited state of singlet to septet spin-multiplicities of both single and multi-electron excitation natures. This enables us to calculate the geometries of molecules in excited and ionized states and to study the dynamics of molecular systems in their ground, excited and ionized states, which are difficult to study by experimental methods alone.
     The SAC/SAC-CI method was implemented in Gaussian 03 in the spring of 2003 and is now widely used worldwide not only in universities and institutes, but also in industries. The improvements and the extensions of the SAC/SAC-CI code on Gaussian are consistently done in our laboratory to explore the SAC-CI world in chemistry. The method has been applied in our laboratory to various chemistries involving more than 170 molecules in the ground and excited states of organic, inorganic, and surface molecular systems. The SAC-CI methodology related with Gaussian is summarized in the WEB [21].
     The SAC/SAC-CI method is applicable to the ground, excited, ionized, and electron-attached states of valence, Rydberg, inner-valence, and inner-core energy regions in a same good accuracy and therefore has opened a new field called theoretical fine spectroscopy, which in conjunction with the experimental fine spectroscopy, opens a new dimension of spectroscopy and dynamics in chemistry [22-24]. By calculating the potential energy surfaces of the ground, excited and ionized states, fine vibrational structures of the spectra are also studied by the SAC-CI method. This is true not only for the main excitation and ionization peaks, but also for many satellite peaks accompanying to the main peaks, which are due to multi-electron processes. The open-reference (OR-)SAC/SAC-CI method has recently been developed and used to study efficiently the inner-core excitations and their satellites [25].
     The SAC-CI methodology has made it also applicable to relatively large molecules like porphyrins and biologically important molecular systems. So, this method is very useful for the study of photo-biology. For example, the spectra and the electron transfer pathways of the photosynthetic reaction centers of Rhodopseudomonas Viridis [26-28] and Rhodobactor Sphaeroides [29], photosynthetic bacteria, were clarified with this method. In combination with the QM/MM method where QM is SAC-CI, this method has been very powerful for studying the color tuning mechanism in retinal proteins [30]. Similar approach is also possible by using the SAC-CI/ONIOM method recently incorporated in Gaussian [31].
     Surface photochemistry is also an interesting field to which the SAC-CI method has been applied. With the help of the Dipped Adcluster Model (DAM) explained below, we can describe the electronic structures of molecules adsorbed on a metal surface [32,33]. Large low-field shifts of the spectra of the adsorbates in comparison with the gas-phase spectra are well described by a combination of the DAM and the SAC-CI method. By combining the experimental and SAC-CI theoretical surface spectroscopies, we can not only identify the adsorbate species, but also clarify the electronic structures of the adsorbates, for which experiments alone are very difficult in reality. More details will be explained below together with the DAM.
     The direct algorithm was introduced recently to accelerate the calculations and to increase the accuracy of the SAC and SAC-CI program [34]. So far, the direct algorithm was introduced only to the singles and doubles part, and the introduction to the general R part is in progress.
     Extensions of the SAC/SAC-CI methodology to truly giant molecular systems such as molecular crystals, polymers, and biological systems are important for investigating photo-electronic processes in giant molecular systems. Giant SAC/SAC-CI theory and its code have been completed recently [35], realizing the study of giant molecular systems without loss of accuracy. In such a giant systems, exact satisfaction of the size extensivity and the size intensivity are important, because if not satisfied, the error soon becomes incredible. Interesting applications are now in progress.

*http://qcri.or.jp/sacci/
1) Cluster Expansion of the Wavefunction. Pseudo-Orbital Theory Applied to Spin Correlation, H. Nakatsuji and K. Hirao, Chem. Phys. Lett., 47(3), 569-571 (1977).
2) Cluster Expansion of the Wavefunction. Symmetry-Adapted-Cluster (SAC) Expansion, Its Variational Determination, and Extension of Open-Shell Orbital Theory, H. Nakatsuji and K. Hirao, J. Chem. Phys., 68(5), 2053-2065 (1978).
3) Cluster Expansion of the Wavefunction. Pseudo-Orbital Theory Based on the SAC Expansion and Its Application to the Spin Density of Open-Shell Systems, H. Nakatsuji and K. Hirao, J. Chem. Phys., 68(9), 4279-4291 (1978).
4) Cluster Expansion of the Wavefunction. Excited States, H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978).
5) Cluster Expansion of the Wavefunction. Electron Correlations in Ground and Excited States by SAC (Symmetry-Adapted-Cluster) and SAC-CI Theories, H. Nakatsuji, Chem. Phys. Lett., 67(2,3), 329-333 (1979).
6) Cluster Expansion of the Wavefunction. Calculation of Electron Correlations in Ground and Excited States by SAC and SAC-CI Theories, H. Nakatsuji, Chem. Phys. Lett., 67(2,3), 334-342 (1979).
7) J. A. Pople, R. Krishnan, H. B. Schlegel and J. S. Binkley, Int. J. Quantum Chem. 14, 545 (1978).
8) G. D. Purvis and R. J. Bartlett, J. Chem. Phys. 76, 1910 (1982).
9) H. Koch and P. Jørgensen, J. Chem. Phys., 93, 3333 (1990).
10) J. F. Stanton and R. J. Bartlett, J. Chem. Phys., 98, 7029. (1993)
11) Electronic Structures of Ground, Excited, Ionized, and Anion States Studied by the SAC/SAC-CI Theory, H. Nakatsuji, Acta Chimica Hungarica, Models in Chemistry, 129(5), pp.719-776 (1992).
12) SAC-CI Method: Theoretical Aspects and Some Recent Topics, H. Nakatsuji, in Computational Chemistry - Reviews of Current Trends, Vol. 2, p. 62-124 (1997).
13) Cluster Expansion of the Wavefunction. Valence and Rydberg Excitations, Ionizations, and Inner-Valence Ionizations of CO2 and N2O Studied by the SAC and SAC-CI Theories, H. Nakatsuji, Chem. Phys., 75, 425 (1983)
14) Cluster Expansion of the Wavefunction. Spin and Electron Correlations in Doublet Radicals Studied by the SAC and SAC-CI Theories, H. Nakatsuji, K. Ohta, and T. Yonezawa, J. Phys. Chem., 87, 3068 (1983).
15) Description of Two- and Many-Electron Processes by the SAC-CI Method, H. Nakatsuji, Chem. Phys. Lett., 177(3), 331-337 (1991).
16) SAC-CI Method Applied to High-Spin Multiplicity, H. Nakatsuji and M. Ehara, J. Chem. Phys., 98(9), 7179-7184 (1993).
17) SAC-CI and Full CI Calculations for the Singlet and Triplet Excited States of H2O, H. Nakatsuji, K. Hirao, and Y. Mizukami, Chem. Phys. Lett., 179(5,6), 555-558 (1991).
18) Outer- and Inner-Valence Ionization Spectra of N2 and CO: SAC-CI (general-R) Spectra Compared with the Full-CI One, M. Ehara and H. Nakatsuji, Chem. Phys. Lett., 282(5,6) 347-354 (1998).
19) Analytical Energy Gradient of the Ground, Excited, Ionized and Electron-Attached States Calculated by the SAC/SAC-CI Method, T. Nakajima and H. Nakatsuji, Chem. Phys. Lett., 280 (1,2) 79-84 (1997).
20) Analytical Energy Gradients of the Excited, Ionized and Electron-Attached States Calculated by the SAC-CI General-R Method, M. Ishida, K. Toyoda, M. Ehara and H. Nakatsuji, Chem. Phys. Lett., 347, 493-498 (2001).
21) http://qcri.or.jp/sacci/
22) SAC-CI General-R Study of the Ionization Spectrum of HCl, M. Ehara, P. Tomasello, J. Hasegawa, and H. Nakatsuji, Theor. Chem. Acc., 102 (1-6), 161-164 (1999).
23) Electronic Excitation Spectra of Furan and Pyrolle: Revisit by the SAC-CI Method, J. Wan, J. Meller, M. Hada, M. Ehara, and H. Nakatsuji, J. Chem. Phys., 113(18), 7853-7866 (2000).
24) Fine Theoretical Spectroscopy Using SAC-CI General-R Method: Outer- and Inner-Valance Ionization Spectra of CS2 and OCS, M. Ehara, M. Ishida, and H. Nakatsuji, J. Chem. Phys., 117, 3248-3255 (2002).
25) Inner-shell ionizations and satellites studied by the OR-SAC/SAC-CI method Y. Ohtsuka and H. Nakatsuji, J. Chem. Phys. in press.
26) Excited States and Electron Transfer Mechanism in the Photosynthetic Reaction Center of Rhodopseudomonas Viridis: SAC-CI Study, H. Nakatsuji, J. Hasegawa, and K. Ohkawa, Chem. Phys. Lett., 296 (5,6), 499-504 (1998).
27) Excited States of the Photosynthetic Reaction Center of Rhodopseudomonas Viridis: SAC-CI Study, J. Hasegawa, K. Ohkawa, H. Nakatsuji, J. Phys. Chem. B, 102 (50), 10410-10419 (1998).
28) Mechanism and Unidirectionality of the Electron Transfer in the Photosynthetic Reaction Center of Rhodopseudomonas Viridis: SAC-CI Theoretical Study, J. Hasegawa and H. Nakatsuji, J. Phys. Chem. B, 102 (50), 10420-10430 (1998).
29) Mechanism and Excited States and Electron Transfer Mechanism in the Photosynthetic Reaction Center of Rhodobactor Sphaeroides: SAC-CI Theoretical Study, J. Hasegawa and H. Nakatsuji, Chemistry Letters. 34, 1242-1243 (2005).
30) Mechanism of color-tuning in retinal proteins: SAC-CI and QM/MM study, K. Fujimoto, J. Hasegawa, S. Hayashi, S. Kato, H. Nakatsuji, Chem. Phys. Lett., 414, 239-242 (2005).
31) Y. Ohtuka, H. Nakatsuji, and K. Morokuma, Symposium on Molecular Science, 2P098, Tokyo, Sept. 27-30, 2005.
32) Dipped Adcluster Model for Chemisorptions and Catalytic Reactions on a Metal Surface, H. Nakatsuji, J. Chem. Phys., 87(8), 4995-5001 (1987).
33) Dipped Adcluster Model for Chemisorption and Catalytic Reactions, H. Nakatsuji, Progress in Surface Science, Vol. 54, p. 1-68 (1997).
34) Formulation and Implementation of the Direct Algorithm for the SAC/SAC|CI Method, J. Chem. Phys. 128, 094105-1-14 (2008).
35) SAC/SAC-CI Methodology extended to Giant Molecular Systems: Ring Molecular Crystals, H. Nakatsuji, T. Miyahara and R. Fukuda, J. Chem. Phys. 126, 084105-1-18 (2007).

2. Multi-reference theory
     The SAC theory is a single reference theory and so breaks down when quasi-degenerate situations take place, like in the course of bond-breaking of a homo-polar bond. Nakatsuji presented in 1985 the MR(multi-reference)-SAC theory [1] and the exponential generation (EG) idea of the wave functions [2] for describing both dynamic and non-dynamic correlations. The latter idea was generalized in 1991 to the EGCI theory [3] and the MEG-WF (mixed exponentially generated wave function) theory [4], both for ground, excited, ionized and electron attached states, and later extended to high-spin multiplicities [5]. These theories are the generalizations of the SAC/SAC-CI theory to the quasi-degenerate situations and has successfully been applied to the ground and excited states of quasi-degenerate cases [6].

1) Multireference Cluster Expansion Theory: MR-SAC Theory, H. Nakatsuji, J. Chem. Phys., 83, 713-722 (1985).
2) Exponentially Generated Wave Functions, H. Nakatsuji, J. Chem. Phys., 83(11), 5743-5748 (1985).
3) Exponentially Generated Configuration Interaction Theory. Descriptions of Excited, Ionized, and Electron Attached States, H. Nakatsuji, J. Chem. Phys., 94(10), 6716-6727 (1991).
4) Mixed-Exponentially Generated Wave Function Method for Ground, Excited, Ionized, and Electron Attached States of a Molecule, H. Nakatsuji, J. Chem. Phys., 95(6), 4296-4305 (1991).
5) EGCI Method Applied to High-Spin Multiplicity, H. Nakatsuji and M. Ehara, J. Chem. Phys., 99(3), 1952-1961 (1993).
6) Exponentially Generated Wave Functions and Excited States of Benzene, H. Nakatsuji, Theoret. Chim. Acta., 71(2,3), 201-229 (1987).

III. Wave Mechanics without Wave - Directly Solving the Second-Order Density Matrix without Using the Wave Function
     When the wave function of a molecule is given, we can calculate the properties of the molecule by applying the operators on the wave function. Energy and other electronic properties are such properties. Since any basic operators of properties include only one- and two-electron operators, all the properties of molecules can be calculated if the exact second-order density matrices are given. Therefore, we may construct quantum mechanics using the second-order density matrix as a basic variable instead of the wave function (Wave Mechanics without Wave). The second-order density matrix depends only on 4 electron coordinates at maximum, but the wave function depends on the coordinates of all the electrons involved. However, there was no deterministic equation of the density matrix, like the Schrödinger equation for the wave function.
     Nakatsuji presented in 1976 [1,2] a basic equation called density equation (or later (around 1985) called "contracted" Schrödinger equation, again in the West World) which is equivalent to the Schrödinger equation in the space of the density-matrix. However, it took 20 years before this density equation was solved for real molecules. In 1992, Valdemoro presented a clever method of approximating higher-order density matrices in terms of the lower-order ones [3]. By improving her theory with the use of the Green function method, Nakatsuji and Yasuda firstly solved in 1996 [4] the density equation for real molecules, i.e., the electronic structures of several molecules were calculated directly from the second-order density matrices without using the wave functions [5]. This method was further used to calculate the potential energy curves and vibrational frequencies of molecules [6].
     Nakatsuji further invented with Nakata in 2001 [7] a variational method for directly solving the second-order density matrices of molecules using positive semi-definite programming (SDP) algorithm. It was shown that the so-called P-, Q-, and G-conditions of the density matrices, together with some other trivial properties, constitute a good approximation of the N-representability condition. The variational energies constrained with these necessary conditions overshot the true full-CI energies by only a few percentages. This method was further shown to be stable under the multi-reference and strong-correlation situations [8]. However, this SDP algorithm is very time-consuming and further even a few percent of the correlation energy may be very important for chemical accuracy.

1) Equation for the Direct Determination of the Density Matrix, H. Nakatsuji, Phys. Rev., A14, 41 (1976).
2) Equation for the Direct Determination of the Density Matrix: Time- Dependent Density Equation and Perturbation Theory, H. Nakatsuji, Theor. Chem. Acc. 102, 97-104 (1999).
3) C. Valdemoro, Phys. Rev. A 45, 4462 (1992).
4) Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation, H. Nakatsuji and K. Yasuda, Phys. Rev. Lett., 76, 1039-1042 (1996).
5) Direct Determination of the Quantum-Mechanical Density Matrix Using the Density Equation. II., K. Yasuda and H. Nakatsuji, Phys. Rev. A 56, 2648-2657 (1997).
6) Density Equation Theory in Chemical Physics, H. Nakatsuji, in Many-electron Densities and Reduced Density Matrices, edited by J. Cioslowski, Kluwer Academic, New York 2000, pp85-116.
7) Variational Calculations of Fermion Second-Order Reduced Density Matrices by Semi- definite Programming Algorithm, M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, J. Chem. Phys., 114, 8282-8292 (2001).
8) Density Matrix Varitional Theory: Application to the Potential Energy Surfaces and Strongly Correlated Systems, M. Nakata, M. Ehara, and H. Nakatsuji, J. Chem. Phys., 116, 5432-5439 (2002).

IV. Dipped Adcluster Model (DAM) for Surface-Molecule Interactions and Reactions
     Metal surfaces show various important chemical properties, among others, the catalyses of many important reactions. When you describe surface-molecule interactions, some modeling is necessary because a surface is an infinite system. However, the cluster model that is very often utilized as a model of a surface is very crude for a metal surface, because it can not describe the effects of the free band electrons that are characteristic of a metal surface.
     The dipped adcluster model (DAM) proposed by Nakatsuji in 1987 [1-4] is a theoretical model of the adsorbate on a metal surface and includes the effects of transfer of bulk-metal electrons and of the image force of a metal surface. The adcluster, which is a combined system of the admolecule plus cluster, borrows easily some electrons from the bulk metal and the equilibrium is described using the chemical potential of the metal surface. The DAM has been of crucial importance for clarifying the mechanisms of various catalytic reactions such as O2 chemisorption on an Ag surface [5-6], the epoxidation reactions of olefins on a silver surface [7-11], and methanol synthesis on a Cu/Zn surface [12-14]. It also describes the surface photochemistry of the adsorbates in combination with the SAC-CI method. For example, the harpooning, chemiluminescence, and electron emission in the course of halogen chemisorption on alkali metal surfaces [15] and the surface spectroscopy of NO on Pt(111) surface [16] were described successfully by combining the DAM with the SAC-CI method.

1) Dipped Adcluster Model for Chemisorptions and Catalytic Reactions on a Metal Surface, H. Nakatsuji, J. Chem. Phys., 87(8), 4995-5001 (1987).
2) Dipped Adcluster Model for Chemisorptions and Catalytic Reactions on a Metal Surface: Image Force Correction and Applications to Pd-O2 Adclusters, H. Nakatsuji, H. Nakai, and Y. Fukunishi, J. Chem. Phys., 95(1), 640-647 (1991).
3) Theoretical Model Studies for Surface-Molecule Interacting Systems, H. Nakatsuji, Intern. J. Quantum Chem., Symp.26, 725-736 (1992).
4) Dipped Adcluster Model for Chemisorption and Catalytic Reactions, H. Nakatsuji, Progress in Surface Science, Vol. 54, p. 1-68 (1997).
5) Theoretical Study on Molecular and Dissociative Chemisorptions of an O2 Molecule on an Ag Surface:Dipped Adcluster Model Combined with SAC-CI Method, H. Nakatsuji and H. Nakai, Chem. Phys. Lett., 174(3,4), 283 (1990).
6) Dipped Adcluster Model Study for Molecular and Dissociative Chemisorption of O2 on an Ag Surface, H. Nakatsuji and H. Nakai, J. Chem. Phys. 98(3), 2423-2436 (1993).
7) Mechanism of the Partial Oxidation of Ethylene on an Ag Surface: Dipped Adcluster Model Study, H. Nakatsuji, K. Ikeda, Y. Yamamoto, and H. Nakai, Surf. Sci., 384, 315-333 (1997).
8) Theoretical Studies on the Catalytic Activity of Ag Surface for the Oxidation of Olefins, H. Nakatsuji, Z. M. Hu, and H. Nakai, Intern. J. Quantum. Chem., 65, 839-855 (1997).
9) Electron Transfer and Back-Transfer in the Partial Oxidation of Ethylene on an Ag Surface: Dipped Adcluster Model Study, H. Nakatsuji, K. Takahashi, and Z.M Hu, Chem. Phys. Lett., 277(5,6), 551-557 (1997).
10) Activation of O2 on Cu, Ag, and Au Surfaces for the Epoxidation of Ethylene: Dipped Adcluster Model Study, H. Nakatsuji, Z. M. Hu, H. Nakai and K. Ikeda, Surf. Sci., 387 328-341 (1997)
11) Oxidation Mechanism of Propylene on an Ag Surface: Dipped Adcluster Model Study, Z. Hu, H. Nakai, and H. Nakatsuji, Surf. Sci., 401(3), 371-391 (1998).
12) Active Sites for Methanol Synthesis on a Zn/Cu(100) Catalyst, Z. M. Hu, and H. Nakatsuji, Chem. Phys. Lett., 313 (1,2), 14-18 (1999).
13) Mechanism of the Hydrogenation of CO2 to Methanol on a Cu(100) Surface: Dipped Adcluster Model Study, Z. M. Hu, K. Takahashi, and H. Nakatsuji, Sur. Sci., 442,(1), 90-106 (1999).
14) Mechanism of Methanol Synthesis on Cu(100) and Zn/Cu(100) Surfaces: Comparative Dipped Adcluster Model Study, H. Nakatsuji and Zhen-Ming Hu, Intern. J. Quantum Chem., 77, 341-349 (2000).
15) Dipped Adcluster Model and SAC-CI Method Applied to Harpooning, Chemiluminescence, and Electron Emission in Halogen Chemisorption on Alkali Metal Surface, H. Nakatsuji, R. Kuwano, H. Morita and H. Nakai, J. Mol. Catalysis, 82, 211-228 (1993).
16) Theoretical Surface Spectroscopy of NO on the Pt(111) Surface with the DAM (Dipped Adcluster Model) and the SAC-CI Method, H. Nakatsuji, N. Matsumune, and K. Kuramoto, J. Chem. Theo. Comp. 1, 239-247 (2005).

V. Electronic Mechanisms and the Relativistic Effects in NMR Chemical Shifts
     NMR chemical shifts are very widely used in analytical chemistry but it is not well known that they involve a lot of information about the electronic structure of molecules. A purpose of the study of Nakatsuji is to clarify the electronic mechanisms of the metal chemical shifts and to offer the means for understanding the natures of bonding in the metal complexes.
     Nakatsuji and his collaborators presented a methodology for studying the electronic mechanisms of the metal chemical shifts and studied the electronic structures of the Ag, Cd, Cu, Zn, complexes through their chemical shifts [1]. Then, they studied the electronic mechanisms of the chemical shifts of various resonant nuclei such as Mn, Sn, Ti, Mo, Nb, Ga, In, Se, Ge, As, Sb, and Xe. Since the chemical shift measures the angular momenta of electrons induced around the resonant nuclei by the applied magnetic field, the p- and/or d-orbital electronic structures of the metal complexes are reflected to their chemical shifts. Nakatsuji has shown that the primary mechanisms of the metal chemical shifts are the intrinsic properties of the resonant nuclei characterized by their positions in the periodic table [2].
     He also recognized that the relativistic effects, spin-orbit effect in particular, are very important for the chemical shifts of molecules including heavy elements, and developed a method to calculate these relativistic effects. From the ab initio calculations he showed for the first time that the spin-orbit effect is the dominant origin of the proton and C13 chemical shifts in the HX and CH3X (X=F, Cl, Br, I) series of molecules, respectively [3]. He performed later the Dirac-Fock four-spinor calculations of various molecules in the magnetic field [4]. Later, he showed that the relativistic effects and the electron correlation effects couple strongly, so that they must be calculated at the same time [5].

1) Theoretical Study of the Metal Chemical Shift in Nuclear Magnetic Resonance. Ag, Cd, Cu, and Zn Complexes, H. Nakatsuji, K. Kanda, K. Endo, and T. Yonezawa, J. Am. Chem. Soc., 106, 4653 (1984).
2) Electronic Mechanisms of Metal Chemical Shifts from Ab Initio Theory, H. Nakatsuji, in Nuclear Magnetic Shieldings and Molecular Structure, Ed. by J. A. Tossell, NATO ASI Series, C386, Reidel, Dordrecht, pp. 263-278 (1993).
3) Spin-Orbit Effect on the Magnetic Shielding Constant Using Ab Initio UHF Method, H. Nakatsuji, H. Takashima, and M. Hada, Chem. Phys. Lett., 233, 95-101 (1995).
4) Relativistic Study of Nuclear Magnetic Shielding Constants: Hydrogen Halides, C. C. Ballard, M. Hada, H. Kaneko, and H. Nakatsuji, Chem. Phys. Lett., 254, 170-178 (1996).
5) Relativistic Configuration Interaction and Coupled Cluster Methods Using Four-Component Spinors: Magnetic Shielding Constants of HX and CH3X (X = F, Cl, Br, I), M. Kato, M. Hada, R. Fukuda, H. Nakatsuji, Chem. Phys. Lett., 408, 150-156 (2005)

VI. Force Concept of Molecular Geometry Based on the Electrostatic (Hellmann- Feynman) Theorem
     Nakatsuji proposed a conceptual force model, called ESF (electrostatic force) model for molecular geometries and chemical reactions [1]. The Hellmann-Feynman theorem (electrostatic theorem) states that the force acting on the nucleus A is the vector sum of the repulsive electrostatic forces due to the other nuclei B in the molecule and the attractive forces exerted on the nucleus A from the negatively charged electron cloud of the molecule. The ESF model predicts the molecular geometry and the course of the chemical reaction by considering the vector sum of these forces on the constituent nuclei. For example, let us take ammonia NH3 as an example. It has a lone-pair density and three bond-electron densities. The former pulls the nitrogen nucleus toward the centroid of the lone-pair density and the three bond densities pull the nitrogen nucleus along the bonds: the vector sum of the bond forces is just opposite to the force due to the lone-pair density. Since the centroid of the lone-pair is closer to the nitrogen nucleus than the centroid of the bond density, the force due to the lone-pair density is larger than that due to the bond density. So, the balancing between these two kinds of forces is done by making the HNH angle smaller than the tetrahedral angle: the vector sum of the bond-forces increases and balances with the lone-pair force. As may be understood from this example, the ESF model was very intuitive and predictive for the geometries of molecules and was more useful than the VSEPR theory and the Walsh model. You can easily predict the shapes of molecules with the ESF model [2]. The model can also be applied to the geometries of molecules in their excited and ionized states.
     Then, he characterized the common behaviors of molecular electron density (like electron-cloud preceding and incomplete following) under nuclear rearrangement processes (like molecular vibrations and chemical reactions) [3]. In the course of the chemical reaction, electron-cloud preceding generally occurs: the preceded electron cloud pulls the nuclei toward the direction of the chemical reaction. The HOMO and LUMO play a central role in this electron-cloud preceding. In the molecular vibration process around the equilibrium geometry, the electron-cloud incomplete following takes place and pulls back the nuclei toward the equilibrium structure. Thus, the motion of electron cloud is less than that of the nuclei and bent bond often appears in the vibrational process.
     The Hellmann-Feynman force is only poorly calculated when LCAO-MO approximation is introduced because the centers of the AOs are fixed on the nuclear positions. Nakatsuji showed that when the first derivatives of the AOs are included in the basis set, the SCF calculations satisfy the Hellmann-Feynmann theorem [4]. He further used this method to calculate the second derivatives of the potential energy surfaces [5].

1) Electrostatic Force Theory for a Molecule and Interacting Molecules I. Concept and Illustrative Applications, H. Nakatsuji, J. Am. Chem. Soc., 95(2), 345 (1973).
2) Force Models for Molecular Geometry, H. Nakatsuji and T. Koga, in The Force Concept in Chemistry, B. M. Deb, Ed. (Van Nostrand Reinhold, New York,1981), Chap.3, pp. 137-217.
3) Common Natures of the Electron Cloud of the System Undergoing Change in Nuclear Configuration, H. Nakatsuji, J. Am. Chem. Soc., 96(1), 24 (1974).
4) Force in SCF Theories, H. Nakatsuji, K. Kanda, and T. Yonezawa, Chem. Phys. Lett., 75(2), 340 (1980).
5) Force in SCF Theories. Second Derivative of Potential Energy, H. Nakatsuji, K. Kanda, and T. Yonezawa, J. Chem. Phys., 77, 1961 (1982).