I am interested in mathematical study of molecules. Mathematically speaking, this subject corresponds to the N-body problem of the Schrödinger equation, i.e. the problem for many yet finite number of particles. Some qualitative properties of the solutions to the the Schrödinger equation can be proved without any approximation. For example we know some qualitative structures of the spectrum of the N-body Hamiltonian, exponential decay and the cusp condition of the eigenfunctions, complete classification of the long-time asymptotic behaviors of the solutions to the Schrödinger equation, and the relation between the asympotic behaviors of the solutions to the time-dependent Schrödinger equation and the generalized eigenfunctions of the Hamiltonian (if we do not regard the results concerned with asymptotic behaviors as assertions about approximation. Note that in real experiments the time and distances are always finite values and it would be hopeless to give quantitative error bounds of the differences between the asymptotic behaviors and the values for finite time and distances.).

Except for a few exactly solvable simple systems, for quantitative study we would need approximations. The most fundamental idea for such approximations in the study of molecules would be to distinguish between nuclei and electrons. The difference between nuclei and electrons in the Hamiltonian are the differences of their charges and mass. The difference of the mass gives different roles for nuclei and electrons. As the nuclei move slowly the electrons move rapidly around the nuclei and change their bound states adiabatically. The approximation in accordance with this idea is called Born-Oppenheimer approximation. Mathematically it is justified as the asymptotics of solutions as the small parameter which is the ratio of the mass of nuclei and electrons tends to 0.

According to the Born-Oppenheimer approximation the motion of nuclei is governed by the effective potential which is the sum of the nuclear repulsion potential and eigenvalues (energy levels) of the electronic Hamiltonian as functions of the nuclear positions. There exist several electronic states corresponding to different energy levels of electrons. Actually there are interactions between the different electronic states. One of my main interests is to evaluate the electronic energy levels (i.e. eigenvalues of the electronic Hamiltonian) accurately with rigorous quantitative error bounds. Upper bounds of the eigenvalues are obtained by the variational method. Lower bounds are much more difficult to obtain, and there is no method that gives accurate lower bounds even for rather simple systems at present.

For accurate upper bounds of eigenvalues we need to approximate true eigenfunctions by some explicit functions. To obtain such functions we approximate functions by antisymmetrized products of one-electron functions. The one-electron functions are obtained as solutions to the Hartree-Fock equation. The Hartree-Fock equation can not be solved exactly even for small numbers of electrons and nuclei. A standard way to solve the equation by numerical calculations is the self-consistent field (SCF) method. I am also interested in how to obtain a particular solution to the equation by SCF method.

In order to study large molecules we need to consider local structures of the molecules. Real molecules are not disorderly clusters. They are composed connecting many orderly local structures like chains and have specific three-dimensional structures, which is an underlying idea in organic chemistry. The idea of local structures has not been formulated as a mathematically rigorous notion yet.