Variational calculations with complex Gaussian functions

The complex-valued exponential functions depending on the all inter-particles distances [1] first was used for variational calculations of the three-particle quantum-mechanical systems with Coulomb interaction between the particles. On calculations of three particle systems was shown that complex exponential basis function is especially efficient for molecular-like systems. It is necessary to underline, that calculation formulas of integrals are greatly complicated for four particle systems [2], and unknown for many particle systems. That creates grand problem for using complex exponential functions for many particle systems. The other way is to use complex-valued ``Gaussian'' functions, depending on squares of inter-particle distances. Calculations with these functions are simple by using complex numbers arithmetic. In this work we present results of testing new functions. We make a calculation of four particle systems: hydrogen and exotic positronium molecules in basis of 45 complex functions, and find energies and some other averages. For example, we obtain binding energy of positronium molecules equal to -0.513 a.u. It is shown that complex-valued ``Gaussian'' functions like complex-valued exponential functions can be significantly improve the accuracy of variational calculations of few--particle molecular systems \newline [1] T.K.Rebane and O.N.Yusupov, Zh. Eksp.Teor.Fiz . 98, 1870 (1990) \newline [2] V.S.Zotev and T.K.Rebane , Phys. Rev. A 65, N6 ( 2002)