The Hartree-Fock Hamiltonian

The Hartree-Fock model leads to an effective one-electron Hamiltonian, called the Fockian F. The second quantized representation of the Fockian has that same form as any other one-electron operator. In the basis of orthogonalized spin-orbitals one can write  

10 Some Model Hamiltonians in Second Quantized Form 

The expression for the matrix elements F y can be derived in the second quantized formalism in an elegant manner. This derivation relies on the physical picture behind the Hartree-Fock approximation. As known, in this model the electrons interact only in an averaged manner, so correlational effects are excluded. To derive such an averaged operator, we start again with the usual Hamiltonian  

We may bring this expression into a more transparent form by appropriate interchanges of summation indices in order to have the operator string x TXv all terms. In particular, this requires a v X interchange in the first term, a v a 

It is convenient to have the Fockian in terms of spatial orbitals. The situation is simple in the closed-shell case where each orbital is required to be either doubly occupied or empty according to the restricted Hartree-Fock (RHF) method. Than, the spatial density matrix is given by Eq. (9.8), and the transcription of Eq. (10.55) can be performed in the usual manner. We find that the spatial Fock matrix is independent of spin  

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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 Hartree-Fock Hamiltonian, with elements given in (98) and (99) is written... 

A Consequence of the Instability in First-order Properties.—Suppose a first-order property which is stable to small changes in the wavefunction (though is not necessarily close to the experimental value) is calculated to, say, three decimal places does an error in the fourth matter To provide a concrete example for discussion, a method described in the next section will be anticipated, namely the finite field method for calculating electric polarizability a. In this method a perturbation term Ai—— fix(F)Fa is added to the Hartree-Fock hamiltonian and an SCF wave-function calculated as usual. For small uniform fields,... 

The majority of polarizability calculations use the FFT, perhaps primarily because it is easy to incorporate into standard SCF computer programs in the presence of a perturbation A which is a sum of one-electron operators, the Hartree-Fock SCF hamiltonian hF—h+G(R)z becomes h+A + G R), and the SCF equations are solved by any standard technique. Thus, all that is involved is to add an extra array into the Hartree-Fock hamiltonian matrix hF every iteration. The method can be extended to higher polarizabilities, and a review by Pople et al.73b gives a good introduction to the method, including a discussion of the computational errors likely to be involved. 

Thus If corr not only includes the interdependence of electron position but all other contributions that have been omitted from the Hartree-Fock Hamiltonian, e. g. relativistic effects. 

Let us briefly discuss the relationship between approaches which use basis sets and thus have a discrete single-particle spectrum and those which employ the Hartree-Fock hamiltonian, which has a continuous spectrum, directly. Consider an atom enclosed in a box of radius R, much greater than the atomic dimension. This replaces the continuous spectrum by a set of closely spaced discrete levels. The relationship between the matrix Hartree-Fock problem, which arises when basis sets of discrete functions are utilized, and the Hartree-Fock problem can be seen by letting the dimensions of the box increase to infinity. Calculations which use discrete basis sets are thus capable, in principle, of yielding exact expectation values of the hamiltonian and other operators. In using a discrete basis set, we replace integrals over the continuum which arise in the evaluation of expectation values by summations. The use of a discrete basis set may thus be regarded as a quadrature scheme. 

Fig, 13 Projected Density of States (PDOS) for the optimised (111 surface, for the a) for the Pd/Zr02, and in b) for the Pt/ZrOi interfaces when the metal is adsorbed on top of O.,. The Fermi level is marked as a dashed line, and energies are given in eV. Note the scaling of the PDOS in comparison to the total DOS. Calculations performed with the Hartree-Fock Hamiltonian. 

The true Hamiltonian operator and wave function involve the coordinates of all n electrons. The Hartree-Fock Hamiltonian operator E is a one-electron operator (i.e., it involves the coordinates of only one elec-... 

We now define a one-electron operator, the Hartree—Fock Hamiltonian Hhf-... 

The Hartree—Fock Hamiltonian is written as a sum over electrons with coordinate—spin labels s. 

From (5.17) we see that Hq is formally identical to the Hartree—Fock Hamiltonian. 

Ho is the Hamiltonian chosen to be best suited for modelling the target states. If a single determinant is a sufficiently-accurate model for j) then the definition (7.22) is self-consistent if Hq is the Hartree—Fock Hamiltonian. However, the self-consistent potential is not the same for all target states j). The one-electron potential is discussed in chapter 5. 

The Hartree-Fock SCF orbitals ipx considered so far have an energy associated with them. This means that they diagonalize the Hartree-Fock hamiltonian matrix (cf. page 21) ... 

The case of atoms provides further examples. Using the Z-dependent perturbation theory, it is possible to compare the results obtained either from Hartree-Fock orbitals or from simple hydrogenic functions (79). Except for the lowest orders, the results for hydrogenic (non-SCF) functions are better. There has been some discussion about this result, because in the case of He (80), it makes some difference whether one takes the Hartree or the Hartree-Fock Hamiltonian as a starting point (both having the same zeroth-order wave function). Nevertheless, it can be said that the extra effort in performing first a Hartree-Fock calculation may not be worth while, at least if one intends to make a perturbation calculation subsequently. 

It has been mentioned in the introduction that many authors [4] believed that the model-Hamiltonian Hg = OH would give a better basis for the semi-empirical quantum theory than the derivations starting from e.g. the Hartree-Fock Hamiltonian. One has previously had the dilemma that the parameters in the semi-empirical approach determined fi om selected experiments were usually rather different from those calculated by means of the ab-initio methods. This applied e.g. to Slater s F- and G-integrals in the theory of atomic spectra, to Hiickel s parameters a and P in the theory of conjugated systems, or to the y parameter in the Pariser-Parr-Pople scheme. Careful studies by Karl Freed and his group [9] in Chicago have shown that the discrepancy between the two sets of parameters disappears, if one bases the semi-... 

Recently, Hoyland and Goodman and I Haya have pointed out that the Hartree-Fock Hamiltonian matrix elements for the ionized states should be different from those calculated for the ground state. This implies that Koopmans theorem should be modified to get rid of the deficiency mentioned above. 

E is the energy of the /th molecular orbital, and is a matrix element of the Hartree-Fock Hamiltonian F, which is given by ... 

The Hartree-Fock Hamiltonian operator can be obtained by the procedure in which the total energy of the system is varied to be minimum by an infinitesimal change of each molecular orbital. See Ref. 10. 

The optimum orbitals have an interesting physical interpretation. Obviously, by construction, they are the components of the best possible single determinant and, as such, they are the solution of a single-particle Schrddinger-like equation, so that an examination of the terms in the Hartree Fock Hamiltonian will tell us a lot about their interpretation. What we might call the parent Hamiltonian of the HF Hamiltonian — the one used in the single-determinant variational method — induces the appearance of the one-particle kinetic energy and nuclear attraction terms in the HF Hamiltonian the difference between the parent and the HF lies in the way in which electron repulsion is represented. 

As we noted above, in fact calculations on stable molecules usually do generate canonical Hartree-Fock orbitals which have the symmetry of the molecular framework. It seems likely that this is a contingent, numerical fact rather than one of principle. If the one-electron terms in the Hartree-Fock Hamiltonian are so large as to dominate the form of the molecular orbitals, these MOs will, presumably, take up a distribution which optimises the energy due to the dominant terms in the Hamiltonian, and the electron-repulsion terms involving the potentially symmetry-breaking effects are too weak to chauige this situation. In molecules, radicals or (particularly) anions which have very weakly bound electrons, one would expect to see the effect of broken symmetry. 

The one-electron Hamiltonian term in the Hartree-Fock Hamiltonian depends on the nuclear coordinates,... 

The true Hamiltonian operator and wave function involve the coordinates of all n electrons. The Hartree-Fock Hamiltonian operator F is a one-electron operator (that is, it involves the coordinates of only one electron), and (13.148) is a one-electron differential equation. This has been indicated in (13.148) by writing F and as functions of the coordinates of electron 1 of course, the coordinates of any electron could have been used. The operator F is peculiar in that it depends on its own eigenfunctions [see Eqs. (13.149) to (13.152)], which are not known initially. Hence the Hartree-Fock equations must be solved by an iterative process. 

Here we have assumed that p can be held fixed at its Hartree-Fock value, given by the minimum of the Hartree-Fock hamiltonian, Eq.(ll). 

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