Antisymmetric wave function, definition

In order to leam more about the nature of the intermolecular forces we will start with partitioning of the total molecular energy, AE, into individual contri butions, which are as close as possible to those we defined in intermolecular perturbation theory. Attempts to split AE into suitable parts were undertaken independently by several groups 83-85>. The most detailed scheme of energy partitioning within the framework of MO theory was proposed by Morokuma 85> and his definitions are discussed here ). This analysis starts from antisymmetrized wave functions of the isolated molecules, a and 3, as well as from the complete Hamiltonian of the interacting complex AB. Four different approximative wave functions are used to describe the whole system ... 

The notion of electrons in orbitals consists essentially of ascribing four distinct quantum numbers to each electron in a many-electron atom. It can be shown that this notion is strictly inconsistent with quantum mechanics (7). Definite quantum numbers for individual electrons do not have any meaning in the framework of quantum mechanics. The erroneous view stems from the original formulation of the Pauli principle in 1925, which stated that no two electrons could share the same four quantum numbers (8), This version of the principle was superseded by a new formulation that avoids any reference to individual quantum numbers for separate electrons. The new version due to the independent work of Heisenberg and Dirac in 1926 states that the wave function of a many-electron atom must be antisymmetrical with respect to the interchange of any two particles (9,10). 

The definitions are here given under the assumption that the wave function XP is either antisymmetric or symmetric for a trial function without symmetry property, one has to replace the binomial factor NCV before the integrand by a factor l/p and sum over the N(N—l). . . (N—p+l) possible integrals which are obtained by placing the fixed coordinates x, x 2,. . ., x P in various ways in the N places of the first factor W and the fixed coordinates xv x2,. . xv similarly in the second factor W. By using Eq. II.8 we then obtain... 

Tlie exact definition is slightly more complicated, since the wave function has to be properly antisymmetrized and projected onto the actual basis, but for illustration the bove form is sufficient. Such R12 wave functions may then be used in connection with the Cl, MBPT or CC methods described above. Consider for example a TT calcnlation... 

By definition, the Hamiltonian of a system of identical particles is invariant under the interchange of all the coordinates of any two particles. The wave function describing the system must be either symmetric or antisymmetric under this interchange. If the particles have integer spin, the wavefunction is symmetric and the particles are called bosons if they have half-integer spins, the wavefunction is antisymmetric and the particles are fermions. Our discussion will be restricted to electrons, which are fermions. 

Pauli exclusion principle follows mathematically from definition of wave function for a system of identical particles - it can be either symmetric or antisymmetric (depending on particles spin). 

This remark is important because almost all the calculations thus far performed to get molecular interaction energies have been based on the UF proeedure, which still remains the basic starting approach for all the ab initio ealeulations. The HF procedure gives the best definition of the molecular wave function in terms of a single antisymmetrized product of molecular orbitals (MO). To improve the HF description, one has to introduce in the caleu-lations other antisymmetrized products obtained from the basic one by replacing one or more MOs with others (replacement of occupied MOs with virtual MOs). This is a proce-... 

The wave function is antisymmetric with respect to the exchange of the coordinates of any two electrons, and, therefore p is symmetric with respect to such an exchange. Hence, the definition of p is independent of the label of the electron we do not integrate over. According to this definition. 

There is a class of configurations that we can add to equation (25) that do not change the target definition asymptotically but which do allow the target to polarize when the scattered electron interacts with it. Those are single excitations from the HF target wave function antisymmetrized with the Gaussian orbitals in equation (5). 

---