On factorization of molecular wavefunctions

Abstract

Recently there has been a renewed interest in the chemical physics literature of factorization of the position representation eigenfunctions {Φ} of the molecular Schrödinger equation as originally proposed by Hunter in the 1970s. The idea is to represent Φ in the form ϕχ where χ is purely a function of the nuclear coordinates, while ϕ must depend on both electron and nuclear position variables in the problem. This is a generalization of the approximate factorization originally proposed by Born and Oppenheimer, the hope being that an 'exact' representation of Φ can be achieved in this form with ϕ and χ interpretable as 'electronic' and 'nuclear' wavefunctions respectively. We offer a mathematical analysis of these proposals that identifies ambiguities stemming mainly from the singularities in the Coulomb potential energy.

Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

“…The solution to the nonlinear system of Eqs. (18) and (19) is a difficult problem [25] that lies beyond the scope of this work. Since our main goal is to learn about the topography of the atomic NAPES and its dependence upon the interparticle correlations, we employed these equations for the purposes of analysis only.…”

“…Of course, much better trial functions than (25) can be devised [27], but this level of approximation suffices for our largely qualitative analysis. Before presenting the results, we must qualitatively compare the spatial behavior of the functions (23)- (25). First, we recall that, strictly, the correlation factor in a trial function should go to a constant as r 12 → ∞ (which can occur only when r 1 → ∞ and/or r 2 → ∞), because in this limit the electrons are uncorrelated.…”

“…Thus, in regions of large r 1 and/or large r 2 the uncorrelated function (23) actually provides a much better approximation to the exact wavefunction. On the other hand, close to the nucleus the correlated functions (24) and (25) provide better approximations. Very close to the nucleus, function (25) should provide a better approximation than function (24), but at sufficiently long distance from the nucleus the situation should reverse, since the divergence of (25) is stronger.…”

“…On the other hand, close to the nucleus the correlated functions (24) and (25) provide better approximations. Very close to the nucleus, function (25) should provide a better approximation than function (24), but at sufficiently long distance from the nucleus the situation should reverse, since the divergence of (25) is stronger. Hence, we will confine our comparison of the NAPES's extracted from these functions to a region relatively close to the nucleus, which is where the electron correlation plays an important role, anyway.…”

An exact factorization perspective on quantum interferences in nonadiabatic dynamics

“…[43][44][45][46][47] The EF has been developed both in the time-independent [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64] and in the time-dependent [37][38][39][40][41][42][43][65][66][67][68] versions and analyzed under different perspectives. [44][45][46][47][69][70][71][72] When nuclear dynamics undergoes a single nonadiabatic event, we have pointed out the properties of the TDPES and related them to the, more standard, picture provided in the BO framework, i.e., BO nuclear wavefunctions evolving on multiple static potential energy surfaces (PESs). In this situation, the TDPES shows (i) a diabatic shape in the vicinity of an avoided crossing, smoothly connecting the BO PESs involved in the process, and (ii) dynamical steps bridging piecewise adiabatic shapes, far from the avoided crossing.…”