Physical intuitions and graphs

Abstract

The shape of the graph is not uniquely defined, depending on the ordering of graph’s levels. Nevertheless one may try to visualize and compare different approximations using corresponding graphs. In recent years development of computer programs and computer hardware has allowed to solve Schrödinger equation exactly for several molecules in rather large full model spaces. I will discuss here benchmark calculations on water in double-zeta basis (Saxe et al 1981; Harrison and Handy 1983). The 10 electrons of water distributed among 14 orbitals of double zeta basis give rise to 270 270 orbital configurations, as shown in Fig 32, corresponding to 1 002 001 singlet states of the full space. Taking into account spatial symmetry leaves 256 473 A₁ symmetry singlet states. Abelian symmetry is for the graphs representing full spaces in this orbital basis easily taken into account if the orbitals are ordered according to their symmetry species and certain vertices at the lowest levels of a given symmetry removed. Following Fig 32 graphs showing A₁, A₂, B₁, B₂ symmetry configurations for water are presented (these graphs were produced on a normal line printer and are a part of output of a computer program). The full graph is decomposed into its four symmetry versions. Such a simple symmetry adaptation is possible because there are no a₂ symmetry orbitals in the one-particle space. The graphs that easily fit on one page describe hundreds of thousands of functions. Calculations in such a large space are very expensive (4 hours of CRAY 1S time, as quoted by Harrison and Handy 1983) but allow us to study the influence of various approximations to the full space on the errors.