Schrödinger equation (SE) is the most fundamental equation in chemistry. If its exact solution can be obtained, we can accurately predict all chemical phenomena. Therefore, solving the SE has invaluable significance in both pure and applied sciences. Recently, Nakatsuji has realized this dream: a general method for solving the analytical solution of the SE has been developed. In addition, Nakatsuji proposed Scaled Schrödinger equation which is equivalent to the SE. With these breakthroughs, we succeeded first in the world to solve analytical solution of the SE for many-electron systems, which was the long-cherished dream of theoretical chemist. Since our method is based on general theory, it is applicable to large systems. Our recent calculations established new world records regarding the precision of the solutions in many systems so far applied.

In our recent study, we extended our method to calculate heavy elements in which the relativistic effect become important. Using the Dirac-Coulomb equation (DCE) as the fundamental equation, we re-formulated our method and succeeded to obtain the analytical solution of DCE first in the world. In addition, since our method is applicable to the time-dependent SE, we can also study electron dynamics with very high accuracy.

This method has a great possibility for bringing a paradigm shift in quantum chemistry, and its future development is highly appreciated.

The SAC/SAC-CI method is an accurate electronic-structure theory and applicable to study wide-range of molecular sciences. The SAC/SAC-CI method was proposed by Nakatsuji in 1978. Since then, the method has been recognized as the most advanced method.

The function of the SAC-CI program published in Gaussian03 package is summarized in the figure. The SAC-CI program is applicable to wide-range of chemistry. We are continuously developing new functions and algorithms, which will also be released in the forthcoming version of Gaussian. Our method would manifest more and more important contributions in the field of the excited-states chemical reaction and fine theoretical spectroscopy.

As shown in the figures, our recent study has realized significantly-fine theoretical spectroscopy which highly resemble to the experimental one. In addition, our method is able to predict the structure of molecules in their excited state. The scope of the application has been broadened to core-electron processes and metal surfaces.

Out target is now expanding to the photochemistry of the photo-functional molecules. As an example, the figure compares the experimental spectrum with the SAC-CI theoretical one including the thermal effect of the internal bond rotation.

Photosynthesis, vision, and bioluminescence are one of the most intriguing research subjects, for they are life phenomenons which explicitly use the light, excited states. Photosynthesis of the green plants is very important reaction producing the energy on the globe. We have solved some enigmatic questions regarding to the photo-induced electron-transfer in the reaction center. For vision, we are studying the color-tuning mechanism of the retinal proteins. We are also studying the molecular mechanism of the bioluminescence of the fireflies and jerryfishes.

The oxygen-fixation process is essential for the living systems. We are studying the mechanism of the oxygen-molecule binding by through that observed in oxyheme and oxyhemocyanin.

We are developing SAC/SAC-CI method applicable to giant systems like molecular crystals and biological systems. Based on the view point that the entire system is the aggregate of local systems, truly large systems become our computational target.

As applications of this method, we want to clarify the excited state of the molecular crystal, the role of protein in biological reactions, and the solvation effect in chemical reactions which sometime becomes essentially important.

Solid catalysts have long been called as "Magic stone". Even though they are practically very important, their essential part of the catalytic mechanism has not still unveiled in many reactions. We have revealed the catalytic mechanism of various metal surfaces with our theoretical method.

Figure shows concept of the Dipped-Adcluster Model (DAM) for studying the catalytic metal surfaces. With this model, we are studying the mechanism of the catalytic metal-surfaces and photocatalytic reactions.

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 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.

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.

The relativistic effects, spin-orbit effect in particular, are very important for the chemical shifts of molecules including heavy elements. We have developed a method to calculate these relativistic effects. We showed for the first time that the spin-orbit effect is the dominant origin of the proton and ¹³C chemical shifts in the HX and CH₃X (X=F, Cl, Br, I) series of molecules, respectively. We performed later the Dirac-Fock four-spinor calculations of various molecules in the magnetic field. The relativistic effects and the electron correlation effects couple strongly, so that they must be calculated at the same time.

We have long been advocating to the world about the importance of the relativistic effect in chemical phenomenon involving the heavy elements. Relativity is indispensable to modern quantum chemistry. Including both the electron-correlation effect and relativistic effect by using the Dirac equation as the fundamental equation, we are developing relativistic quantum chemistry as an ultimate theory.

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. The second-order density matrix depends only on 4 electron coordinates at maximum. 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).

Nakatsuji presented in 1976 a basic equation called density equation (or later called contracted Schrödinger equation, again in the West World) which is equivalent to the Schrödinger equation in the space of the density-matrix. In 1996, by improving Valdemoro's theory, Nakatsuji and Yasuda firstly solved the density equation for real molecules. In 2001, Nakatsuji and Nakata further invented a variational method for directly solving the second-order density matrices of molecules using positive semi-definite programming (SDP) algorithm. This method was further shown to be stable under the multi-reference and strong-correlation situations.

Nakatsuji proposed a conceptual force model, called ESF (electrostatic force) model for molecular geometries and chemical reactions. 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.

The multiple bond between metal and carbon is of great interest as a typical bonding mode in organometallic chemistry. The formation/breaking of these bonds is a key step in many organometallic reactions. There are two types of metal carbene complexes: Fisher- and Schrock-types. The reactivity of these complexes are different.

In early 1980's, Nakatsuji and co-worker performed SCF-MO calculations. The reactivity of these metal-carbon multiple bonds are understood in a unified form on the basis of the frontier orbital theory. They would never be explained by the charge-controlled mechanism.