The Hartree Hamiltonian

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. 

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Equation (1.10) represents the Hartree Hamiltonian and Eq. (1.8) has to be solved by iteration, in the sense that a guessed trial wave function 1) is introduced in Eq. (1.10) and the Schrodinger equation Eq. (1.8) solved. The resulting wave function is again introduced in Eq. (1.10) and Eq. (1.8) is again solved until self-consistency is achieved. 

The Hartree approximation is usefid as an illustrative tool, but it is not a very accurate approximation. A significant deficiency of the Hartree wavefiinction is that it does not reflect the anti-synnnetric nature of the electrons as required by the Pauli principle [7], Moreover, the Hartree equation is difficult to solve. The Hamiltonian is orbitally dependent because the siumnation in equation Al.3.11 does not include the th orbital. This means that if there are M electrons, then M Hamiltonians must be considered and equation A1.3.11 solved for each orbital. 

While the equations of the Hartree-Fock approach can he rigorously derived, we present them post hoc and give a physical description of the approximations leading to them. The Hartree-Fock method introduces an effective one-electron Hamiltonian. as in equation (47) on page 194 ... 

Ihe Fock operator is an effective one-electron Hamiltonian for the electron in the poly-tiectronic system. However, written in this form of Equation (2.130), the Hartree-Fock... 

The Hartree-Fock determinant and all of the substituted determinants are eigenfunctions of Hg these are the solutions to the part of the divided Hamiltonian for which we have a solution. Thus ... 

Adding E and E yields the Hartree-Fock energy (since Hg+V is the full Hamiltonian) ... 

Note that the factor of 1/2 has disappeared from the energy expression this is because the G matrix itself depends on P, which has to be taken into account. We write SSg in terms of the Hartree—Fock Hamiltonian matrix h, where... 

This shows that, when we have found the correct electron density matrix and correctly calculated the Hartree-Fock Hamiltonian matrix from it, the two matrices will satisfy the condition given. (When two matrices A and B are such that AB = BA, we say that they commute.) This doesn t help us to actually find the electron density, but it gives us a condition for the minimum. 

The MPn method treats the correlation part of the Hamiltonian as a perturbation on the Hartree-Fock part, and truncates the perturbation expansion at some order, typically n = 4. MP4 theory incorporates the effect of single, double, triple and quadruple substitutions. The method is size-consistent but not variational. It is commonly believed that the series MPl, MP2, MP3,. .. converges very slowly. 

In a recent paper Ostrovsky has criticized my claiming that electrons cannot strictly have quantum numbers assigned to them in a many-electron system (Ostrovsky, 2001). His point is that the Hartree-Fock procedure assigns all the quantum numbers to all the electrons because of the permutation procedure. However this procedure still fails to overcome the basic fact that quantum numbers for individual electrons such as t in a many-electron system fail to commute with the Hamiltonian of the system. As aresult the assignment is approximate. In reality only the atom as a whole can be said to have associated quantum numbers, whereas individual electrons cannot. 

The correlation energy for a certain state with respect to a specified Hamiltonian is the difference between the exact eigenvalue of the Hamiltonian and its expectation value in the Hartree-Fock approximation for the state under consideration. 

In this review, we have mainly studied the correlation energy connected with the standard unrelativistic Hamiltonian (Eq. II.4). This Hamiltonian may, of course, be refined to include relativistic effects, nuclear motion, etc., which leads not only to improvements in the Hartree-Fock scheme, but also to new correlation effects. The relativistic correlation and the correlation connected with the nuclear motion are probably rather small but may one day become significant. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

In the early sixties, it was shown by Roothaan [ 1 ] and Lowdin [2] that the symmetry adapted solution of the Hartree-Fock equations (i.e. belonging to an irreducible representation of the symmetry group of the Hamiltonian) corresponds to a specific extreme value of the total energy. A basic fact is to know whether this value is associated with the global minimum or a local minimum, maximum or even a saddle point of the energy. Thus, in principle, there may be some symmetry breaking solutions whose energy is lower than that of a symmetry adapted solution. 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

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