Quantum-Dynamical Theory of Electron Exchange Correlation

The relationship between the spin of an individual electron and Fermi-Dirac statistics (FDS), which is obeyed by electrons in the aggregate, is elucidated. The relationship depends on the use of spin-dependent quantum trajectories (SDQT) to evaluate Coulomb’s law between any two electrons as an instantaneous interaction in space and time rather than as a quantum-mean interaction in the form of screening and exchange potentials. Hence FDS depends in an ab initio sense on the inference of SDQT from Dirac’s equation, which provides for relativistic Lorentz invariance and a permanent magnetic moment (or spin) in the electron’s equation of motion. Schroedinger’s time-dependent equation can be used to evaluate the SDQT in the nonrelativistic regime of electron velocity. Remarkably FDS is a relativistic property of an ensemble of electron, even though it is of order in the nonrelativistic limit, in agreement with experimental observation. Finally it is shown that covalent versus separated-atoms limits can be characterized by the SDQT. As an example of the use of SDQT in a canonical structure problem, the energies of the 1Σg and 3Σu states of H2 are calculated and compared with the accurate variational energies of Kolos and Wolniewitz. 1. Introduction One may consider that quantum chemistry is dominated by theoretical and computational efforts to achieve an accurate description of electron exchange correlation, evolving such workhorse methodologies as Hartree-Fock-Configuration Interaction, Density Functional Theory, and numerous variations on the theme of nonrelativistic quantum mechanics applied to problems of chemical interest. But yet, owing to historical happenstance, more heat than light has been generated concerning the fundamental physical understanding of exchange-correlation. Even in early calculations in which correlation was built into the wave function it was recognized that the concept of exchange tended to lose meaning in a calculation in which correlation was treated to high accuracy [1]. As another example it was shown that a high-order perturbation calculation in which the electron-electron interaction is treated as the perturbation is able to achieve order by order the correct permutation symmetry of the wave function starting in zeroth order with a simple unsymmetrized product of orbitals [2]. This conundrum can be easily understood for the two-electron, one-nucleus problem by a simple change from nucleus centered to Jacobi coordinates in which one vector connects the two electrons, and the other connects the center of mass of the two