A system of $N$ antisymmetric particles, moving under the influence of a fixed potential and their mutual many-particle interactions, is investigated in the ordinary Hartree-Fock scheme, having the total wave function approximated by a single Slater determinant. It is shown that all the density matrices of various orders, the wave function, and the entire physical situation depends only on a fundamental invariant $\rho (x_1, x_2)$, which is identical with the first-order density matrix. The Hartree-Fock equations are expressed in terms of this quantity.

The Hartree-Fock equations are also solved by expanding the eigenfunctions in a given complete set, and applications to the MO-LCAO theory of the electronic structure of molecules, and crystals are given. It is shown that, in this scheme, the entire physical situation depends on a charge- and bond-order matrix $R\left(\nu \mu \right)$ with respect to the ordinary atomic spin-orbitals involved. The Hartree-Fock equations for this matrix are investigated.

Finally, the ionized and excited states are investigated, and it is shown that the Hartree-Fock scheme has a high degree of physical visuality also in case of many-particle interactions. The excitation energy of the system is the difference (${{\omega }_j}^{\prime }-{\omega }_i$) between two "spin-orbital energies," being eigenvalues to the effective Hamiltonians associated with the two states under consideration.

• Received 8 July 1954

DOI:https://doi.org/10.1103/PhysRev.97.1490