Electronic Hamiltonian Operator

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. 

H being the effective 1-electron Hamiltonian operator and Ek the energy of the kth state. Inserting (1.1) in (1.2) leads to... 

We follow closely previous expositions of the theory (4, 5) and include only the particular features needed for our present discussion. Let us imagine a mixed-valence system composed of two subunits, A and B, which are associated with formal oxidation states M and N, respectively. We designate the corresponding electronic Hamiltonian operators H and H, and if the... 

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. 

We ve derived a complete many-electron Hamiltonian operator. Of course, the Schrodinger equation involving it is intractable, so let s consider a simpler problem, involving the one-electron hamiltonian... 

The electronic Hamiltonian operator He may be derived from the classical energy expression by replacing all momenta p, by the derivative operator, pt => —ihV(i) = —ih(8/8ri), where the first i is the square root of —1. Thus,... 

There are n spatial orbitals ij/ since we are considering a system of 2n electrons and each orbital holds two electrons. The 1 in parentheses on each orbital emphasizes that each of these n equations is a one-electron equation, dealing with the same electron (we could have used a 2 or a 3, etc.), i.e. the Fock operator (Eq. 5.36) is a one-electron operator, unlike the general electronic Hamiltonian operator of Eq. 5.15, which is a multi-electron operator (a 2n electron operator for our specific case). The Fock operator acts on a total of n spatial orbitals, the ij/1, Jj2,, i// in Eq. 5.35. 

For a given set of nuclear coordinates R and a stationary electronic wavefunction CFel) normalised to 1 (i.e., (vPel f el) = 1 holds true), the electronic energy can be calculated in the Bom-Oppenheimer approximation as the expectation value of the electronic Hamiltonian operator as follows ... 

The best energy attainable within this approximation is the so-called Hartree-Fock (HF) energy. The difference between this energy and the exact eigenvalue for the electronic hamiltonian operator is denoted as correlation energy. Several schemes have been proposed in order to improve this situation. For the systems... 

The energy of such a determinantal wavefunction, ( R), is obtained by the quantum mechanical averaging of the electronic Hamiltonian operator given in equation (6)... 

It is well known, that the full electrostatic Hamiltonian operator may be written, in terms of an electronic Hamiltonian operator and a nuclear kinetic operator ... 

The Group Theory for Non-Rigid Molecules considers isoenergetic isomers, and the interconversion motions between them. Because of the discernability between the identical nuclei, each isomer possesses a different electronic Hamiltonian operator in (3), with different eigenfunctions, but the same eigenvalue. In contrast, a non-rigid molecule has an unique effective nuclear Hamiltonian operator (7). 

In order to define these two subgroups Altmann wrote down the electronic Hamiltonian operator of the molecule under study in terms of the coordinates, r,, and masses of the nuclei, m,-. 

The second mapping relation acts on the electronic Hamiltonian operator. This quantity can be rewritten in the diabatic basis as... 

Finally, hc i) is the one-electron Hamiltonian operator for electron i in the average field produced by the Nc core electrons,... 

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