Abstract
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 c0 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.
Highlights
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
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 electronelectron 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]
Summary
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. 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 electronelectron 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]. The standard methodologies [3], for all their success, simulate many-electron quantum states in an ad hoc manner, based on experimental observation, but tell us nothing fundamental about the physical basis of how the spin state of an individual electron is related to the Pauli exclusion principle and the observed Fermi-Dirac statistical behavior of an aggregate of electrons
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