Hiroshi Nakatsuji
Director, Quantum Chemistry Research Institute (QCRI)
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
Kyoto Technoscience Center 16, 14 Yoshida-Kawara-Machi, Sakyou-Ku, Kyoto 606-8305, Japan
Hiroshi Nakatsuji started quantum-chemical studies in 1966 at Prof. Yonezawa’s laboratory at Kyoto university. After getting PhD, he started his own research. He tried to be as closest as possible to quantum principles. 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 [A3]. 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 below in a reverse chronological order.
I. General Method of Solving the Fundamental Equations of Quantum Mechanics - Developing Exact Quantum-Chemistry Theory
The Schrödinger equation and the relativistic Dirac-Coulomb equation represent the basic mathematical principles governing chemistry, biology, physics of matter, and related sciences. So, a central theoretical theme of these sciences is to establish a general method of exactly solving these basic equations. Thereby, we can do quite accurate predictions of the phenomena even without doing experiments. However, Paul Dirac wrote a famous paper [A4] in 1929: he stated as "The general theory of quantum mechanics is now almost complete, ….. The underlying physical lows necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. " Even so, many scientists tried to solve the Schrödinger equation, but they failed. Then, solving the Schrödinger equation was gradually believed impossible by many scientists. It continued for long. Actually, this author also read many articles expressing similar opinions.
However, this author got an inspiration in 1999 that the Schrödinger equation may be solvable along the line of his studies then. He started to study the structure of the exact wave function [1-6]. In 2003, he found a key equation to solve the Schrödinger equation, that is the scaled Schrödinger equation [7]. Based on this equation, this author could formulate the theory to solve the Schrödinger equation or the scaled Schrödinger. This theory was called free iterative configuration interaction (free ICI) theory at the beginning [7], but later renamed as free complement theory, because it is not an iterative theory. Since then, his research on solving the Schrödinger equation is developing. Some explanations are as follows.
In 1999, he initiated the study on the mathematical structure of the exact wave function [1-6]. The first subject was on the second-quantized Hamiltonian [1] and then moved to the real analytical Hamiltonian. In the first paper [1], he could exactly solve the Schrödinger equation in the second quantized form. His method was much more efficient than solving the full-CI equation. However, when he aimed to solve the Schrödinger equation analytically, a big obstacle arose. That was the divergence difficulty caused by Coulombic potentials in atomic and molecular Hamiltonians. He solved this problem by introducing the scaled Schrödinger equation [7], which has the scaling function g(r) attached to the left side of the original Schrödinger equation. It is a positive function of the electron-nucleus riA and electron-electron rij distances and therefore, all the solutions of the scaled Schrödinger equation are equal to those of the Schrödinger equation. However, with the scaled Schrödinger equation, its variational equation does not diverge in contrast to that of the original Schrödinger equation. Thus, the scaled Schrödinger equation is superior to the original Schrödinger equation: to solve the Schrödinger equation, one should start from the scaled Schrödinger equation. Actually, in 2004 [7], he could formulate the theory that leads to the exact solution of the (scaled) Schrödinger equation, called free ICI theory or free complement theory. This theory is very robust, tough theory. With this theory, he and his coworkers could calculate the world-best variational energy of the helium atom correct to 41-43 decimal figures [12,14,15,17,18,20] in comparison to 35 decimal figures of Schwartz [A6]. A word is necessary here. We admire Dr. Schwartz’s study: his inspiration on the exact wave function of the helium atom is just great, because it is very difficult to express the mathematical structure of this atom to such extent of accuracy, even though this is the second simplest atom next to the hydrogen atm.
A problem of the exact FC theory is that the variational integral method is not always possible for general atoms and molecules, for the existence of electron-electron interaction parts gij of the scaling function, because of the integration difficulty. We must apply the FC theory to all of the general atoms and molecules appearing in chemistry. Here, we know that both of the Schrödinger equation and the scaled Schrödinger equation are local equations that do not include any integration in them. So, we introduced the local Schrödinger equation (LSE) method from 2008 [13,34,48], which is the local sampling method that utilizes the fulfillment of the scaled Schrödinger equation at all the sampling points distributed all around the atoms or molecules. Since then, this method has been applied to many atoms and molecules studied in this laboratory, except for the very simple atomic systems for which integrals are available even for the functions including rij. In the LSE method, the way of the distribution of sampling points is important. This distribution is expected to be proportional to the electron density or the density matrices of the system under consideration. For this, the studies presented in Section III of this article would be useful. In some actual calculations like those published in 2018 [38], we used the Metropolis sampling method [A7]. Then, we noticed that in this method, the resultant distribution depends much on the probability factors that are rather difficult to reproduce. However, in science, the reproducibility is very important. So, we designed more rational method in which the sampling point distribution is produced directly according to the electron density distribution (or the wave function itself) of the system under consideration [48]. With this direct local sampling method, the probability dependence was much reduced, and the reproducibility of the calculations was much recovered [47,51].
The scaling function g(r) of the FC theory is of special importance in our theory of solving the Schrödinger equation. It was introduced to prevent the divergence difficulty of the variational equation due to the collisions of the charged particles. For very small special systems, they are almost automatically introduced from the coordinate system or intuitions, but for general cases, its mathematical forms must be studied from physical considerations [46,52]. The “correct” behavior of the g function should be as follows: it must be close to r near the origin (r=0), but as r increases, the g function should become unity, because there the interparticle collision does never occur. Such class of g functions was called correct g functions and several functional examples were presented and their roles and behaviors were studied [46,52]. When some g functions behave like r near the origin, but does not become unity at infinity, such functions were called “reasonable”. The “approximate” functions like for the use for the Gaussian bases, the recommended forms were also investigated [46]. These classifications are important to design the roles of the g functions themselves.
The FC theory has been applied to many systems since 2004. Though they are small so far, (the largest was formaldehyde, H2CO molecule [38]), we are developing the method applicable to larger molecules of variety. Because the FC theory is exact, it must produce the exact wave function for any system, but for the present computer systems, we have to design efficient formalism of the theory considering the most important factors. The exactness and the efficiency of our theory has already been proven by applying it to real systems as shown from the many references given below. Two prominent such proofs may be picked up here. One is the application to the simplest helium (He) atom: we have published the most accurate upper and lower bound energies as explained above in 2007 and 2008 [12-18]. The other is the FC theory calculations of the potential “energy curves of the lower nine valence states of the Li2 molecule [47]. For this molecule, the experimental potential energy curves were reported by many authors for the seven valence states from 1986 to 2009 [A8]. When we compare the potential energy curves of nine valence states obtained by the experiments and by our FC theory, they just completely overlapped to each other both on the absolute energy diagrams [47]. The differences were small and within the experimental accuracies. This was again a surprising result that shows the exactness of the FC theory, or more precisely, the exactness of the Schrödinger or the scaled Schrödinger equation. A note here is that the potential energy curves of the nine valence states of the Li2 molecule were published by ourselves using the SAC/SAC-CI theory in 1985 in the Canadian Journal of Chemistry to the festschrift volume of the 65th birthday of Late Prof. Camille Sandorfy [A9]. Many experimental papers were published from the next year, 1986, but the reference to our pioneering theoretical result was very few.
When the FC is going to be applied to very large molecular systems, the size dependence of the theory becomes important. Particularly, in quantum chemistry, exchange interactions are time-consuming. We have studied this problem and found that the exchange interactions in large molecular systems decay quickly and therefore, we can relax the antisymmetry rule in large systems [35], which may be a big benefit since the fulfilment of the antisymmetry rule of electrons is sometimes very time-consuming. This concept was called inter-exchange theory. Combined with this theory, the exact FC theory was applied to “helium fullerene” He60, an imaginary system. We could have shown the stability of this system and gave the geometry of this fullerene, which is the largest system so far the FC theory had applied to, but probably an imaginary system. However, interelectron exchange interactions are very important and essential in chemistry, like for formations of chemical bonds, etc, it would be important to study its decaying behaviors and take this into account when we study big systems. As the exchange interactions decay rather quickly against the distance, and therefore, at large separations, the interactions are exchangeless and so rather classical. With this simplification, the computational cost would be saved much for very large systems.
Thus, some important basic methodology of accurately calculating the solutions of the Schrödinger equation has been established, by introducing the scaled Schrödinger equation. The remaining problem is to formulate more efficient method than the present one. In the conventional quantum chemistry, we have the basis-set nightmare, which is difficult to overcome, but our free complement method does not have it. It gives the solution of the (scaled) Schrödinger equation from any initial function, and then we are currently extending the applicable fields by extending the initial functions of this theory. The free complement formalism provides a general method of producing the exact wave functions of atoms and molecules in an analytical expansion form. The parameters left undetermined in the FC method are calculated using the variational method or the LSE method.
The purpose of this research is to develop “Exact Quantum Chemistry Theory” that makes exact predictions of chemical phenomena possible only from the theoretical side, like the exact solutions of the Newton’s low makes space travelling possible.
A1) Deepening and Extending the Quantum Principles in Chemistry, H. Nakatsuji, Bull. Chem. Soc. Jap. 78, 1705 (2005).
A2) M. Kaku, The God Equation, The Quest for a Theory of Everything, Stuart Krichevsky Literary Agency, Inc., New York (2021).
A3) S. Weinberg, Dreams of a Final Theory, Pantheon, New York, 1992.
A4) P. A. M. Dirac, Proc. Roy. Soc. (London), A123, 714 (1929).
A5) H. Eyring, J. Walter, G. E. Kimball, "Quantum Chemistry", John Wiley & Sons, Inc., New York (1944).
A6) C. Schwartz, Int. J. Mod. Phys. E 15, 877 (2006).
A7) N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087.
A8) The potential energy curve of the ground the X 1Σg+ ground state was observed by the laser-induced fluorescence Fourier-transform spectrometry: B. Barakat, R. Bacis, F. Carrot, S. Churassy, P. Crozet, F. Martin, and J. Verges, Chem. Phys. 102, 215 (1986). Then, the potential curves of many valence-excited states were reported. For more details, refer to Reference 47 below.
A9) Cluster Expansion of the Wavefunction. Potential Energy Curves of the Ground, Excited, and Ionized States of Li2, H. Nakatsuji, J. Ushio, T. Yonezawa, Can. J. Chem., 63, 1857 (1985) (the festschrift volume of the 65th birthday of Late Prof. Camille Sandorfy).
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1) Structure of the Exact Wave Function, H. Nakatsuji, J. Chem. Phys., 113, 2949 (2000).
2) Structure of the Exact Wave Function. H. Nakatsuji, E. R. Davidson, II. Iterative Configuration Interaction Method, J. Chem. Phys. 115, 2000 (2001).
3) Structure of the Exact Wave Function. III. Exponential Ansatz, H. Nakatsuji, J. Chem. Phys., 115, 2465 (2001).
4) 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 (2002)
5) Structure of the Exact Wave Function. V. Iterative Configuration Interaction Method for Molecular Systems within Finite Basis, H. Nakatsuji, M. Ehara, J. Chem. Phys., 117, 9 (2002).
6) Inverse Schrödinger Equation and the Exact Wave Function, H. Nakatsuji, Phys. Rev. A 65, 052122 (2002).
7) Scaled Schrödinger Equation and the Exact Wave Function, H. Nakatsuji, Phys. Rev. Lett. 93, 030403 (2004).
8) General Method of Solving the Schrödinger Equation of Atoms and Molecules, H. Nakatsuji, Phys. Rev. A, 72, 062110 (2005).
9) Free ICI (Iterative Complements Interaction) Calculations of Hydrogen Molecule, Y. Kurokawa, H. Nakashima, H. Nakatsuji, Phys. Rev. A, 72, 062502 (2005).
10) Analytically Solving the Dirac-Coulomb Equation for Atoms and Molecules, H. Nakatsuji, H. Nakashima, Phys. Rev. Lett., 95, 050407 (2005).
11) 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).
12) 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).
13) 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).
14) Solving the Electron-Nuclear Schrödinger Equation of Helium Atom and Its Isoelectronic Ions with the Free Iterative-Complement-Interaction Method, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 128, 154108 (2008).
15) 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, H. Nakatsuji, J. Chem. Phys. 128, 154108 (2008).
16) Solving the Schrödinger and Dirac Equations of Hydrogen Molecular Ion Accurately by the Free Iterative Complement Interaction Method, A. Ishikawa, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 128, 124103 (2008).
17) 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, H. Nakatsuji, Phys. Chem. Chem. Phys. 10, 4486 (2008).
18) How accurately does the free complement wave function of a helium atom satisfy the Schrödinger equation? H. Nakashima, H. Nakatsuji, Phys. Rev. Lett. 101, 240406 (2008).
19) Solving non-Born-Oppenheimer Schrödinger equation for hydrogen molecular ion and its isotopomers using the free complement method, Y. Hijikata, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 130, 024102 (2009)
20) How Does the Free Complement Wave Function Become Accurate and Exact Finally for the Hydrogen Atom Starting from the Slater and Gaussian Initial Functions and for the Helium Atom on the Cusp Conditions? H. Nakatsuji, H. Nakashima, Intern. J. Quantum Chem. 109, 2248 (2009)
21) Free Complement Method for Solving the Schrödinger Equation: How Accurately Can We Solve the Schrödinger Equation, H. Nakatsuji, H. Nakashima, Advances in the Theory of Atomic and Molecular Systems, 19, 47 (2009)
22) LiH potential energy curves for ground and excited states with the free complement local Schrödinger equation method, A. Bande, H. Nakashima, H. Nakatsuji, Chem. Phys. Lett. 496, 347-350 (2010)
23) Solving the Schrödinger and Dirac equations for hydrogen atom in the universe’s strongest magnetic fields with the free complement method, H. Nakashima, H. Nakatsuji, Astrophys. J. 725, 528 (2010)
24) Relativistic free complement method for correctly solving the Dirac equation with the applications to hydrogen isoelectronic atoms, H. Nakashima, H. Nakatsuji, Theoret. Chem. Acc. (Pekka Pyykko issue), 129, 567 (2011).
25) Full configuration-interaction calculations with the simplest iterative complement method: Merit of the inverse Hamiltonian, H. Nakatsuji, Phys. Rev. A. 84, 062507 (2011).
26) Accurate solutions of the Schrödinger and Dirac equations of H2+, HD+, and HT+:With and without Born-Oppenheimer approximation and under magnetic field, A. Ishikawa, H. Nakashima, H. Nakatsuji, Chem. Phys. 401, 62 (2012).
27) Discovery of a General Method of Solving the Schrödinger and Dirac Equations That Opens a Way to Accurately Predictive Quantum Chemistry, H. Nakatsuji, Acc. Chem. Res., 45, 1480-1490 (2012). This article was ranked 4 in Most Read Articles in July 2012.
28) Solving the non-Born-Oppenheimer Schrödinger equation for hydrogen molecular ion with the free complement method II: Highly-accurate electronic, vibrational, and rotational excited states, H. Nakashima, Y. Hijikata, H. Nakatsuji, Astrophys. J. 770, 144 (2013).
29) Efficient antisymmetrization algorithm for the partially correlated wave functions in the free complement - local Schrödinger equation method, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 139, 044112 (2013).
30) General coalescence conditions for the exact wave functions: Higher-order relations for two-particle systems, Yusaku I. Kurokawa, Hiroyuki Nakashima, Hiroshi Nakatsuji, J. Chem. Phys. 139, 044114 (2013).
31) Non-Born-Oppenheimer potential energy curve: Hydrogen molecular ion with highly accurate free complement method, H. Nakashima, H. Nakatsuji, J. Chem. Phys. 139, 074105 (2013).
32) General coalescence conditions for the exact wave functions. II. Higher-order relations for many-particle systems, Yusaku I. Kurokawa, Hiroyuki Nakashima, Hiroshi Nakatsuji, J. Chem. Phys. 140, 214103 (2014).
33) Solving the Schrödinger Equations of Some Organic Molecules with Superparallel Computer TSUBAME, Hiroshi Nakatsuji, Hiroyuki Nakashima, TSUBAME e-Science J., 11, 8-12, 24-29 (2014).
34) Free-complement local-Schrödinger-equation method for solving the Schrödinger equation of atoms and molecules: Basic theories and features, Hiroshi Nakatsuji, Hiroyuki Nakashima, J. Chem. Phys., 142, 084117 (2015)
35) Solving the Schrödinger equation of molecules by relaxing the antisymmetry rule: Inter-exchange theory, Hiroshi Nakatsuji, Hiroyuki Nakashima, J. Chem. Phys., 142, 194101 (2015)
36) General coalescence conditions for the exact wave functions: Higher-order relations for Coulombic and non-Coulombic systems, Yusaku I. Kurokawa, Hiroyuki Nakashima and Hiroshi Nakatsuji, Advances in Quantum Chemistry, 73, 59-79 (2016)
37) Solving the Schrödinger equation of atoms and molecules: Chemical-formula theory, free-complement chemical-formula theory, and intermediate variational theory, Hiroshi Nakatsuji, Hiroyuki Nakashima, and Yusaku I. Kurokawa, J. Chem. Phys., 149, 114105 (2018).
38) Solving the Schrödinger equation of atoms and molecules with the free-complement chemical-formula theory: First-row atoms and small molecules, Hiroshi Nakatsuji, Hiroyuki Nakashima, Yusaku I. Kurokawa, J. Chem. Phys., 149, 114106 (2018)
39) Solving the Schrödinger equation of hydrogen molecule with the free complement-local Schrödinger equation method: Potential energy curves of the ground and singly excited singlet and triplet states, Σ, Π, Δ, and Φ, H. Nakashima, and H. Nakatsuji, J. Chem. Phys., 149, 244116 (2018).
40) Solving the Schrödinger equation with the free-complement chemical-formula theory. Variational study of the ground and excited states of Be and Li atoms, H. Nakatsuji, H. Nakashima, J. Chem. Phys., 150, 044105 (2019)
41) Solving the Schrödinger Equation of Hydrogen Molecule with the Free-Complement Variational Theory: Essentially Exact Potential Curves and Vibrational Levels of the Ground and Excited States of Σ symmetry, Y. I. Kurokawa, H. Nakashima, H. Nakatsuji, Phys. Chem. Chem. Phys., 21, 6327 (2019)
42) Inverse Hamiltonian method assisted by the complex scaling technique for solving the Dirac-Coulomb equation: Helium isoelectronic atoms, Hiroyuki Nakashima, Hiroshi Nakatsuji, Chem. Phys. Lett. 749, 137447 (2020)
43) Solving the Schrödinger equation of atoms and molecules using one- and two-electron integrals only, Hiroshi Nakatsuji, Hiroyuki Nakashima, Yusaku I. Kurokawa, Phys. Rev. A. 101, 062508 (2020)
44) Solving the Schrödinger Equation of Hydrogen Molecule with the Free-Complement Variational Theory: Essentially Exact Potential Curves and Vibrational Levels of the Ground and Excited States of Π symmetry, Y. I. Kurokawa, H. Nakashima, H. Nakatsuji, Phys. Chem. Chem. Phys. 22, 13489 (2020)
45) Free complement sij-assisted rij theory: Variational calculation of the quintet state of a carbon atom, H. Nakashima, H. Nakatsuji, Phys. Rev. A., 102, 052835 (2020).
46) Accurate scaling functions of the scaled Schrödinger equation, Hiroshi Nakatsuji, Hiroyuki Nakashima, and Yusaku I. Kurokawa, J. Chem. Phys. 156, 014113 (2022).
47) Potential curves of the lower nine states of Li2 molecule: Accurate calculations with the free complement theory and the comparisons with the SAC/SAC-CI results, Hiroshi Nakatsuji, Hiroyuki Nakashima, J. Chem. Phys. 157, 094109 (2022).
48) Direct local sampling method for solving the Schrödinger equation with the free complement- local Schrödinger equation theory, H. Nakatsuji, H. Nakashima, Chem. Phys. Lett. 806, 140002 (2022).
49) Solving the Schrödinger equation of a planar model H4 molecule, H. Nakashima, H. Nakatsuji, Chem. Phys. Lett., 815, 140359 (2023).
50) Gaussian functions with odd power of r produced by the free complement theory, Y. I. Kurokawa, H. Nakatsuji, J. Chem. Phys., 159, 024103 (2023).
51) Potential Energy Curves of the Low-Lying Five 1Σ+ and 1Π States of a CH+ Molecule Based on the Free Complement - Local Schrödinger Equation Theory and the Chemical Formula Theory, Hiroyuki Nakashima, Hiroshi Nakatsuji, J. Chem. Theory Comput., 19, 6733 (2023).
52) Accurate Scaling Functions of the Scaled Schrödinger Equation. II. Variational Examination of the Correct Scaling Functions with the Free Complement Theory Applied to the Helium Atom, Hiroshi Nakatsuji, Hiroyuki Nakashima, J. Chem. Theory Comput., 20, 3749 (2024).
53) Exact Theory Applied to the Lithium Atom, Hiroshi Nakatsuji, Hiroyuki Nakashima, J. Chem. Theory Comput., ***, *** (2024).
II. SAC/SAC-CI Method for Studying Chemistries of Excited, Ionized, and Electron Attached 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. Jgensen, 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).
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).
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)
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).