One-electron Hamiltonian effective

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 ... 

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. 

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. 

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... 

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

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

In Eqs. (3) and (4), H stands for the effective one-electron Hamiltonian the integration is over the whole space. In the simple method, a number of simplifying assumptions about the values of the integrals ftjk, j, and Sjk are introduced in the case of molecules containing no heteroatoms these are known as the Hiickel approximations. It seems useful to use this designation also for molecules with heteroatoms, and in the present review this method will be referred to as the Hiickel molecular orbital (HMO) method according to Streitwieser s suggestion.4... 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

The tight-binding band structure calculations were based upon the effective one-electron Hamiltonian of the extended Huckel method. [5] The off-diagonal matrix elements of the Hamiltonian were calculated acording to the modified Wolfsberg-Helmholtz formula. All valence electrons were explicitly taken into account in the calculations and the basis set consisted of double- Slater-type orbitals for C, O and S and a single- Slater-type orbitals for H. The exponents, contraction coefficients and atomic parameters were taken from previous work [6],... 

C is an effective one-electron Hamiltonian representing the kinetic energy, the field of the nuclei, and the smoothed-out distribution of the other electrons. 

At the lowest level of sophistication of quantum treatments, the tight-binding method and the semi-empirical HF method reduce the complexity of the interacting electron system to the diagonalization of an effective one-electron Hamiltonian matrix, whose elements contain empirical parameters. The electronic wave functions are expanded on a minimal basis set of atomic or Slater orbitals centered on the atoms and usually restricted to valence orbitals. The matrix elements are self-consistently determined or not, depending upon the method. 

There are several ways to define ionic charges. When the eigen-states of the effective one-electron Hamiltonian are expanded on a localised basis set, the probability of presence of the electrons contains a site contribution (square modulus of the projection of the wave function on the sites) and a bond term, related to the overlap of the basis functions. In standard Mulliken population analysis [16], each bond contribution is equally shared between the atom pair. The charges then depend sensitively upon the choice of the basis set and it is meaningless to compare absolute values obtained with different methods. Only charge variations within a given method bear significance. 

In the application of DHF to multipole polarizabilities, the first task is assembling the derivative Fock operators because in SCF the Fock operator is the effective (one-electron) Hamiltonian for the system. The parameters of differentiation are the elements of the expansion of the electrical potential, Vy, Fj, F, V y,.. ., . From Eqn. (28), one may see that the... 

Even though the spin orbitals obtained from (2.23) in general do not have the full symmetry of the Hamiltonian, they may have some symmetry properties. In order to study these Fukutome considered the transformation properties of solutions of (2.24) with respect to spin rotations and time reversal. Whatever spatial symmetry the system under consideration has, its Hamiltonian always commutes with these operators. As we will see, the effective one-electron Hamiltonian (2.25) in general only commutes with some of them, since it depends on these solutions themselves via the Fock-Dirac matrix. 

Thus in general the elements of the group G = S x T do not commute with the effective one-electron Hamiltonian (2.25). Some of these operations may, however, commute with 9tefr and in such a case they form a subgroup of G. That subgroup characterizes the GHF solution under study in the sense that the corresponding Fock-Dirac matrix is invariant under the elements g of the invariance group of the GHF solution,... 

Hiickel theory and it was introduced in order to describe the n electrons in planar conjugated molecules, a and / are the Coulomb and resonance integrals, respectively, of some effective one-electron Hamiltonian operator 65>. 

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