One-electron Hamiltonians

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 ... 

I h e preceding discussion mean s that tli e Matrix etjuatiori s already described are correct, except that the Fuck matrix, F. replaces the effective one-electron Hamiltonian matrix, and th at K depends on th e solution C ... 

In an ab initio method, all the integrals over atom ic orbital basis function s are com puted and the bock in atrix of th e SCK com puta-tiori is formed (equation (6 1) on page 225) from the in tegrals. Th e Kock matrix divides inui two parts the one-electron Hamiltonian matrix, H, and the two-electron matrix, G, with the matrix ele-m en ts... 

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 principal semi-empirical schemes usually involve one of two approaches. The first uses an effective one-electron Hamiltonian, where the Hamiltonian matrix elements are given empirical or semi-empirical values to try to correlate the results of calculations with experiment, but no specified and clear mathematical derivation of the explicit form of this one-electron Hamiltonian is available beyond that given above. The extended Hiickel calculations are of this type. 

For this simple Hamiltonian, which involves the sum of one-electron Hamiltonians, we can use a wave function of the form... 

The matrix H 3 can be obtained fiom the general ex)x ession for the superoperator Hamiltonian matrix H33 recently derived (126). For the non-diagonal one-electron Hamiltonian it can be written as... 

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. 

The one-electron Hamiltonian operator h = h +v, with kinetic energy operator, generates a complete spectrum of orbitals according to the Schrodinger equation... 

However, there also exists a third possibility. By using a famous relation due to Dirac, the relativistic effects can be (in a nonunique way) divided into spin-independent and spin-dependent terms. The former are collectively called scalar relativistic effects and the latter are subsumed under the name spin-orbit coupling (SOC). The scalar relativistic effects can be straightforwardly included in the one-electron Hamiltonian operator h. Unless the investigated elements are very heavy, this recovers the major part of the distortion of the orbitals due to relativity. The SOC terms may be treated in a second step by perturbation theory. This is the preferred way of approaching molecular properties and only breaks down in the presence of very heavy elements or near degeneracy of the investigated electronic state. 

In contrast to the useful conceptual framework provided by the approximate approach just described, the results of more detailed molecular orbital calculations have on the whole been rather disappointing. Thus, although some semi-empirical SCF treatments were attempted, most of the earlier MO calculations for metallocene systems (18, 161, 162, 163, 164,165) suffered from such deficiencies as the neglect of the a-framework, or the use of various one-electron Hamiltonians, for example the various Wolfsberg-Helmholz techniques. Of late, Drago and his coworkers have carried out further Extended Htickel type computations for a wide range of both metallocene and bis-arene species (153, 154), and similar... 

The Fock operator F for an atomic or molecular system is a modified one-electron Hamiltonian (cf. Eq. (1.9b)),... 

If one has determined the operator 7 by a method which does not simultaneously determine the CMO s, then Eq. (26) can be looked upon as a one-electron Schroedinger equation to be solved for the CMO s. In this sense, the Fock operator can be thought of as an effective one-electron hamiltonian. Thus, a one-electron variational problem can be set up namely, we require... 

The very simplest theoretical approach, with linear electron-phonon coupling, is in terms of a two-center (a,b) one-electron Hamiltonian (27), with just one harmonic mode, u>, associated with each center. This is (in second quantized notation, with H = 1) ... 

If we fix the cores and assume that each electron moves in the average potential generated by the cores and other electrons, this is the so-called Hartree or one-electron approximation. For this model we arrive at a relatively simple expression for the Hamiltonian for determining the electronic band structure and wave functions. The one-electron Hamiltonian is... 

The indices are all defined in terms of the Hiickel molecular orbital method. This has been described on many occasions, and need not be discussed in detail here, but a brief statement of the basic equations is a necessary foundation for later sections. The method utilizes a one-electron model in which each tt electron moves in a effective field due partly to the a-bonded framework and partly to its averaged interaction with the other tt electrons. This corresponds conceptually to the Hartree-Fock approach (Section IX) but at this level no attempt is made to define more precisely the one-electron Hamiltonian h which contains the effective field. Instead, each 7r-type molecular orbital (MO) is approxi-... 

If 3C is the effective one-electron Hamiltonian operator for the chain, if/(r) satisfies the equation... 

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