^{1,2}

^{3}

^{1}

^{2}

^{3}

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 ^{1}Σ_{g} and ^{3}Σ_{u} states of H_{2} are calculated and compared with the accurate variational energies of Kolos and Wolniewitz.

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 [

Nonrelativistic quantum theory fails us, however, even for two-electron problems. According to experimental observation, the electrons have intrinsic angular momentum comprising two-spin-1/2 states, such that the total wave function must be antisymmetric on electron exchange, which is satisfied either by a product of the even spatial wave function adduced above with a spin state which is odd on electron exchange (singlet state) or by a product of the adduced odd spatial wave function with a spin state which is even on electron exchange (triplet state). The generalization is of course the Slater determinantal wave function of spin orbitals which guarantees the correct antisymmetric permutation symmetry for

Readers should recognize that an electron’s spin state and its spatial correlation with another electron are strongly related, as suggested by our observation that a successful simulation of correlation is also accompanied by the correct exchange symmetry. Notice, however, that such a relationship between electron spin and the electron-electron Coulomb interaction appears to be grossly at odds with our intuitive understanding of quantum chemistry, likely due to the absence of a particle-trajectory picture in the standard methodologies. Bohm formulated a quantum dynamical approach in which the phase-amplitude solution of Schroedinger’s equation has a formal relationship to classical hydrodynamics [

We therefore seek a quantum trajectory approach for real electrons in which the spin of an individual electron is correctly accounted for, which means that we must look to Dirac’s rather than to Schroedinger’s equation. One of the desiderata of Dirac’s program [

should be inferred from the equation of motion for the relativistic electron,

whereupon [

One more step is needed to show the relationship of electron spin and electron exchange correlation, namely, the evaluation of the electron-electron Coulomb interaction using quantum trajectories. Considering the case of two electrons, one of which is described by (

Pauli’s exclusion principle is obeyed if (

Quantum trajectories in the y direction at _{2}. Solid: spin-up electron. Dotted: spin-down electron. The trajectories are calculated from (

Quantum eigentrajectories in the _{2} correlate with increasing time in the formation of an antibonding state. Upper: spin-up electron. Lower: spin-up electron. The eigentrajectories are calculated from (_{2}.

Spectrum at _{2}. Solid: spin-up electron. Dotted: spin-down electron.

The envelop of trajectory amplitudes shown in Figure

Quantum eigentrajectories in the _{2}.

Expectation values of the Hamiltonian and their time averages for

Time average of the Hamiltonian expectation values for the

The time-independent solution of (

The trajectories are quantum mechanical since they can be calculated either using Dirac’s equation in the regime of relativistic velocities or Schrodinger’s equation in the regime of nonrelativistic velocities. The quantum trajectories are spin dependent since they depend explicitly on Pauli’s vector. Our result, for the first time, gives a relativistic correction which is of order

We believe that this unforeseen Dirac correction of order

In the nonrelativistic regime of electron velocity, (

Written out explicitly for up (upper sign) or down (lower sign) spin states the nonrelativistic current is

The time-dependent Schroedinger’s equation [(_{2} using an algorithm described previously [^{3}. The accuracy of the calculation could be improved by mesh refinement of the uniform mesh or by use of an adaptive mesh (which we do not have at our disposal); however, we find that for

The time-dependent wave function for each electron is evolved, and its spectrum of eigenvalues and eigenfunctions is obtained using the methods described in the pioneering paper of Feit et al. [

Readers should be reminded that the evolution of the wave function by solving (

Figure

Figure

Wave functions along _{2}. Solid: spin-up electron. Dotted: spin-down electron. Convergence is shown for the solid curves at 0.75

Wave functions along _{2}. Solid: spin-up electron. Dotted: spin-down electron. The dotted curve which is symmetric about the two proton positions is calculated for the interelectronic interaction artificially set equal to zero, demonstrating that the correct dissociation limit into two hydrogen atoms, as shown by the solid and dotted curves asymmetric about the two protons, depends critically on the interelectronic interaction.

Figure

Figure

Notice that the trajectories are separated into a ground-state trajectory centered about one nucleus and an excited-state trajectory moving at large distances from the two centers of attraction. Such a state is known in quantum chemistry as antibonding.

Figure

Finally Figure

One cannot extract more than three significant figures from the calculations at the present level of accuracy, but we are confident that the results can be systematically improved for accuracy by further refinements in the grid meshes and especially by using improved, energy-conserving numerical methods to integrate the quantum trajectories in order to avoid run away, the underestimation of the interelectronic potential energy, and as a consequence the overestimation of the binding energy. These problems notwithstanding, we believe that the quantum-trajectory theory (a) establishes the relationship between the spin of an individual electron and the Fermi-Dirac statistical behavior of an ensemble of electrons and (b) achieves the pair-wise correlation of electrons by evaluating Coulomb’s law as an instantaneous interaction in the time rather than as a quantum stationary-state mean or exchange interaction.

The author is grateful to T. Scott Carman for his support of this work. This work was performed under the auspices of the Lawrence Livermore National Security, LLC (LLNS), under Contract no. DE-AC52-07NA273.