San José State University

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Thayer Watkins
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 The Hartree-Fock Method for Finding Self-Consistent Field Wave Functions for Multi-electron Atoms

In 1926 Erwin Schrödinger published his work that provided physicists with a simple way to formulate the quantum dynamics of the electrons in an atom. The Schrödinger Equation is easy to formulate but in all but the simplest cases impossible to solve analytically and not easy to solve numerically. The procedure first involves declaring the Hamiltonian for the system. For example, the Hamiltonian for a helium atom is

#### H = K1 + K2 + V1 + V2 + V12

where K1 and 2 are the kinetic energies of the two electrons. V1 and V2 are the potential energies of the two electrons in the field of the two protons in the nucleus. V12 is the potential energy due to the two electrons with respect to each other.

If r1 and r2 are the distances of the two electrons from the nucleus and v1 and v2 are their velocities then the Hamiltonian reduces to

#### H = ½mv1² + ½mv2² −2k/r1² −2k/r1² + k/(|r1−r2|)²

where r1 and r2 are the position vectors of the two electrons with respect to nucleus. The symbol k represents a constant that is the product of the constant for the electrostatic force and the square of the charge of an electron.

In the late 1920's Douglas Hartree began trying to find ways to simplify the numerical solution. He formulated the concept of the Self-Consistent Field. In this method the effect on a single electron of the rest of the electrons is assumed to reduce to a central field which is added the central field established by the nucleus of the atom. Starting with an approximation of this central field the wave function of the single electron can be found. This wave function is the used to determine the central field of the other electrons. This procedure is applied iteratively until the solutions converge, or at least do not change by a significant amount.

Hartree's procedure involved computing the eigenvectors and their eigenvalues of the discrete version of the self-consistent field version of the Schrödinger equation. The eigenvectors corresponded to the wavefunctions for the electron and the eigenvalues to the negative of their energies. Hartree compared the computed eigenvalues had a good correspondences with the energies of X-rays required to knock the various electrons out of their orbitals; i.e., their ionization energies.

Shortly after Hartree published his method in 1928, J.A. Gaunt found that the eigenvalue for an electron could be determined to a close approximation by computing the energy of the atom with the electron and the energy of the ion in which the electron is missing.

Hartree assumed that the wave function of a multi-electron atom was the product of the wave functions of the individual electrons. John Slater in 1929 found that theoretically and empirically it was better to take the multi-electron wave function as being the determinant formed from individual electron wave function, which came to be known as the Slater determinant of the system. The Slater determinant automatically produced multi-electron wave function that are anti-symmetric.

Vladimir Fock modified Hartree's method so as to obtain on each step wave functions that satisfied the theoretical requirement of a solution to the n electron problem.

Tjalling Koopmans published in 1934 a further refinement. He found that in general the spin-orbitals can be be chosen in a way such that a matrix of interaction energies is diagonal and thus the eigenvalues are simply equal the diagonal elements. See Koopmans' Theorem for more on this.

In the 1920's there was a competition between the wave mechanics of Werner Heisenberg based on infinite matrices and the quantum mechanics of Erwin Schrödinger based on partial differential equations. Schrödinger showed that the two formulation are equivalent and over time the theory was couched in terms of Schrödinger's formulation but when anyone does numerical computation in effect they are utilizing Heisenberg's wave mechanics with the infinite matrices truncated. For understanding the computations and the approximation involved in solving a physical system a matrix formulation has definite advantages. For that reason a matrix version of a model for a helium atom is given in Matrix Model of Helium.