Boundary condition determined wave functions for the ground states of one- and two-electron homonuclear molecules

Abstract

Simple analytical wave functions satisfying appropriate boundary conditions are constructed for the ground states of one-and two-electron homonuclear molecules. Both the asymptotic condition when one electron is far away and the cusp condition when the electron coalesces with a nucleus are satisfied by the proposed wave function. For H 2 ϩ , the resulting wave function is almost identical to the Guillemin-Zener wave function which is known to give very good energies. For the two electron systems H 2 and He 2 ϩϩ , the additional electron-electron cusp condition is rigorously accounted for by a simple analytic correlation function which has the correct behavior not only for r 12 →0 and r 12 →ϱ but also for R→0 and R→ϱ, where r 12 is the interelectronic distance and R, the internuclear distance. Energies obtained from these simple wave functions agree within 2 ϫ10 Ϫ3 a.u. with the results of the most sophisticated variational calculations for all R and for all systems studied. This demonstrates that rather simple physical considerations can be used to derive very accurate wave functions for simple molecules thereby avoiding laborious numerical variational calculations.

Simple model potential and model wave functions for (H–alkali)+ and (alkali–alkali)+ ions

“…4 In the development of these wave functions, local properties of the wave functions play a significant role. [5][6][7][8] Alkali atoms are in some ways similar to the hydrogen atom. They have one electron outside the closed shell core, and many of their properties are determined primarily by the wave function of the valence electron.…”

Simple correlated wave-function for excitons in 0D, quasi-1D and quasi-2D quantum dots

“…[16] To assure a high-quality wave-function it is particularly important that the wave-function satisfy the cusp conditions, [17,18] representing the behavior of the exact wave function at the coalescence of two particles. It is also important to take into account the asymptotic conditions, [19] which represent the behavior when one of the particles goes to infinity. Bertini et al [20] have employed explicitly correlated trial wave-functions for ground and excited states of Be and Be − fulfilling cusp and asymptotic conditions by means a Padé factor exp[(ar+br 2 )/(1+cr)] for the electron-nucleus part and a Jastrow factor exp[ar/(1+cr)] for the inter-electronic part.…”

Simple model potential and model wave functions for (Na, K, Rb, Cs)–H molecules

“…This behavior has been found to play an important role 6,15 in the development of wave functions for two-electron systems.…”

“…As such it has been analyzed from many different approaches within the Born-Oppenheimer approximation, with simple Heitler-London wave function, 1 with molecular orbitals, 2 with Hartree-Fock approximation, 3 and to a high level of accuracy by using a variational approach 4 with a large number of basis states. Recently, the molecular wave functions have been analyzed in terms of some local properties 5,6 which the exact wave functions possess. This has been very helpful in developing simple wave functions which are quite accurate and which provide significant physical insight into their structure.…”