Criticality of the electron-nucleus cusp condition to local effective potential-energy theories

Abstract

Local(multiplicative) effective potential energy theories of electronic structure comprise the transformation of the Schrödinger equation for interacting fermi systems to model noninteracting fermi or bose systems whereby the equivalent density and energy are obtained. By employing the integrated form of the Kato electron-nucleus cusp condition, we prove that the effective electron -interaction potential energy of these model fermions or bosons is finite at a nucleus. The proof is general and valid for arbitrary system whether it be atomic, molecular, or solid state, and for arbitrary state and symmetry. This then provides justification for all prior work in the literature based on the assumption of finiteness of this potential energy at a nucleus. We further demonstrate the criticality of the electron-nucleus cusp condition to such theories by example of the Hydrogen molecule. We show thereby that both model system effective electron-interaction potential energies, as determined from densities derived from accurate wave functions, will be singular at the nucleus unless the wave function satisfies the electron-nucleus cusp condition.

Quantal density functional theory of the hydrogen molecule

“…The corresponding electron-interaction potential energy v ee (r) representative of these correlations as determined by the work done in the force of these fields is then symmetric about this point as also dictated by the symmetry of the molecule. The potential energy v ee (r) is also finite at each nucleus, as must be the case [23].…”

“…Furthermore, it is shown that this finiteness is a direct consequence of the satisfaction of the electron-nucleus cusp condition by the Schrodinger wave function. (As a consequence, for example, this potential energy is singular at each nucleus when determined either from Gaussian geminal [23] or configuration interaction [25] wave functions.) Hence, in order to obtain v c (0, z), we have employed our calculated results in regions other than near the nucleus, and smoothed the curve through each nucleus.…”

“…The underlying reason for the singularity, however, is that the wave function does not satisfy the electron-nucleus cusp condition exactly. In a recent paper [23], we have proved by employing the integral form of the electron-nucleus cusp condition [24], that in local effective potential energy theories and for arbitrary symmetry, the potential energy v ee (r) is finite at the nucleus. Furthermore, it is shown that this finiteness is a direct consequence of the satisfaction of the electron-nucleus cusp condition by the Schrodinger wave function.…”

On-top pair-correlation function in the homogeneous electron liquid

“…As mentioned above, the RPA gives poor description of the short-range correlations of the electrons, especially for g(r) as r → 0. In fact, the results for g ↑↓ (r) in the RPA violate the following cusp condition: [14,17,24,25,26,27] ∂g ↑↓ (r) ∂r…”

Total energy equation leading to exchange-correlation functional

“…Standing on the middle rung of Jacob's ladder, we can-not see the next rung. With more research on physical models and also constraint satisfaction [19][20][21][22][23][24][25][26][27][28], the nature of exchange-correlation functional will be gradually uncovered. However, this research can hardly help design a better exchange-correlation functional, and we believe that this is because these models or constraints can-not suggest the form of exchange-correlation functional directly.…”