Using Simultaneous Diagonalization and Trace Minimization to Make an Efficient and Simple Multidimensional Basis for Solving the Vibrational Schrödinger Equation

In this paper we improve the product simultaneous diagonalization (SD) basis method we previously proposed [J. Chem. Phys. 122, 134101 (2005)] and applied to solve the Schrödinger equation for the motion of nuclei on a potential surface. the improved method is tested using coupled complicated Hamiltonians with as many as 16 coordinates for which we can easily find numerically exact solutions. in a basis of sorted products of one-dimensional (1D) SD functions the Hamiltonian matrix is nearly diagonal. the localization of the 1D SD functions for coordinate qc depends on a parameter we denote αc. in this paper we present a trace minimization scheme for choosing αc to nearly block diagonalize the Hamiltonian matrix. Near-block diagonality makes it possible to truncate the matrix without degrading the accuracy of the lowest energy levels. We show that in the sorted product SD basis perturbation theory works extremely well. the trace minimization scheme is general and easy to implement.