Hartree-product Wave Functions

Let us examine the Schrodinger equation in the context of a one-electron Hamiltonian a little more carefully. When the only terms in the Hamiltonian are the one-electron kinetic energy and nuclear attraction terms, the operator is separable and may be expressed as 

Because the Hamiltonian operator defined by Eq. (4.32) is separable, its many-electron eigenfunctions can be constructed as products of one-electron eigenfunctions. That is 

A wave function of die form of Eq. (4.35) is called a Hartree-product wave function. The eigenvalue of T is readily found from proving the validity of Eq. (4.35), viz.. 

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This expression is just the one which obtains for the Hartree product wave-function. The difference between this Hartree wavefunction and the Fock wavefunction of Eq. (1) is the absence of the antisymmetrizer j4 in that equation. This means that in the Hartree wavefunction each electron can be identified with a specific molecular orbital, whereas in the Fock wavefunction all electrons make use of all orbitals. The Hartree wavefunction is of course not a proper quantum mechanical wavefunction, since it is not antisymmetric in the electrons. Moreover, for the Fock wavefunction, it is in general not possible to reduce the interorbital exchange energy to zero. But the localized molecular orbitals, as defined here, represent that set of molecular orbitals for which the energy expression comes closest to the Hartree form, i.e. they come closest to being identifiable with electrons which are not exchanged among different orbitals. 

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian That is, we wish to find orbitals that minimize (4 hp H I hp). By applying variational calculus, one can show that each such orbital i/f, is an eigenfunction of its own operator hi defined by... 

At this point it is appropriate to think about our Hartree-product wave function in more detail. Let us say we have a system of eight electrons. How shall we go about placing them into MOs In the Hiickel example above, we placed them in the lowest energy MOs first, because we wanted ground electronic states, but we also limited ourselves to two electrons per orbital. Why The answer to that question requires us to introduce something we have ignored up to this point, namely spin. 

Knowing these aspects of quantum mechanics, if we were to construct a ground-state Hartree-product wave function for a system having two electrons of the same spin, say a, we would write... 

Slater determinants have a number of interesting properties. First, note that every electron appears in every spin orbital somewhere in the expansion. This is a manifestation of the indistinguishability of quantum particles (which is violated in the Hartree-product wave functions). A more subtle feature is so-called quantum mechanical exchange. Consider the energy of interelectronic repulsion for the wave function of Eq. (4.43). We evaluate this as... 

The Hartree product wave function, equation 5.2, for helium does not comply with the anti-symmetry requirement of the Pauli Principle (42,47) that electronic wave functions must change sign on exchange of the coordinates for a pair of electrons. Fock identified this defect in the overestimation of the electron-electron repulsion term, which occurs for Hartree product wave functions, while Slater showed how to overcome this problem by writing the product wave function in the form of a determinant (6,7,42,47,64). 

The Hartree-Fodr SCF Method. The alert reader may have realized that there is something fundamentally wrong with the Hartree product wave function (11.4). Although we have paid some attention to spin and the Pauli principle by putting no more than two electrons in each spatial orbital, any approximation to the true wave... 

Using this Hartree-product wave function, the total energy of the crystal is ... 

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