The Schrödinger equation in quantum chemistry. Transl. from the 1998 Russian original (Q2715778)

The booklet is devoted to elementary methods of analyzing the many-body time-dependent Schrödinger equation. It begins with the adiabatic (or, Born-Oppenheimer) approximation to the Schrödinger equation for polyatomic molecules. It is shown that the equations of the adiabatic approximation is equivalent to an eigenvalue problem for a nonlinear operator \(H^{ad}\) and for an arbitrary test function \(\Phi\), \(\langle\phi, H^{ad}\phi\rangle\leq (\Phi,H\Phi)\); \((H\psi= E\psi)\). In the next section, a model many-body equation, e.g., NEWLINE\[NEWLINET\equiv \sum^N_{j=1}{\partial\over\partial x_j}+ \sum_{k< l} V(x_k, x_l)=0,\text{ such that }x_k<x_lNEWLINE\]NEWLINE with \(V(x,y)= V(y,x)\), is considered which has an explicit solution. The possibility of extending the method presented to the case of the many-electron Schrödinger equation is investigated. The following few sections are devoted to the investigation of geometrical properties of adiabatic potentials of diatomic and polyatomic molecules. The important result is that the ground state adiabatic potential cannot have a positive minimum. Next, the notion of an operator valued concave function is introduced. This enables one to obtain new inequalities for the eigenvalues of \(N\)-body problems. Section 11 is devoted to the mathematically rigorous proof of the Hohenberg-Kohn theorem. In the pen-ultimate section, the one-dimensional Schrödinger equation describing internal rotation is discussed to establish few interesting results. In the last section, the authors deduce a generalization of the viral theorem to the case of internal rotation in molecules.NEWLINENEWLINENEWLINEThe presentation throughout the booklet is very lucid and simple, in fact the reader's mathematical background is expected to be minimal. It is sufficient to know what a selfadjoint operator is and to be familiar with the Courant minimax principle. It is intended for students and researchers, specializing in quantum chemistry.