Single-electron Hamiltonian

Hatree-Fock Approximation and Single Electron Hamiltonian... 

Inorderto study the implications ofthe periodic structure oflattices on the electronic structure of the corresponding solids we consider a single electron Hamiltonian of the form... 

In this Equation 4.7, the charge density of the more electronegative atom is used. The single-electron Hamiltonian can be expressed as... 

According to the approach of Beattie and Landsberg [31], the intrinsic Auger lifetime is calculated by perturbation method. The perturbation operator of Auger mechanism, i.e., of Coulomb interaction of two electrons, is calculated by subtracting Hartree-Fock single-electron Hamiltonian from the complete Hamiltonian of the system (and actually the simplest Hamiltonian that still sees the Coulomb interaction of the Auger process). It has the form of a screened Coulomb potential... 

The corresponding fiinctions i-, Xj etc. then define what are known as the normal coordinates of vibration, and the Hamiltonian can be written in tenns of these in precisely the fonn given by equation (AT 1.69). witli the caveat that each tenn refers not to the coordinates of a single particle, but rather to independent coordinates that involve the collective motion of many particles. An additional distinction is that treatment of the vibrational problem does not involve the complications of antisymmetry associated with identical fennions and the Pauli exclusion prmciple. Products of the nonnal coordinate fiinctions neveitlieless describe all vibrational states of the molecule (both ground and excited) in very much the same way that the product states of single-electron fiinctions describe the electronic states, although it must be emphasized that one model is based on independent motion and the other on collective motion, which are qualitatively very different. Neither model faithfully represents reality, but each serves as an extremely usefiil conceptual model and a basis for more accurate calculations. 

Given a real electronic Hamiltonian, with single-valued adiabatic eigenstates of the form n) = and x ), the matrix elements of A become... 

Under the single-electron approximation, Hamiltonian (9-6) becomes... 

For the hydrogen atom, and for the hydrogen-like ions such as He, Li, ..., with a single electron in the field of a nucleus with charge +Ze, the hamiltonian (the quantum mechanical form of the energy) is given by... 

For large molecules, very many terms contribute to the electronic Hamiltonian. To simplify the notation, I am going to collect together all those terms that depend explicitly on the coordinates of a single electron and write them as... 

An important property of the electron Hamiltonian (Eq. (3.3)) is that for arbitrary hopping amplitudes the spectrum of the single-electrons slates is symmetric with respect to c=0 if is the electron amplitude on site n of an eigenstate with energy c, then the state with amplitudes —)"< > is also an eigenstate, with energy -c. In particular, in the uniformly dimerized stale, the gap between the empty conduction and the completely filled valence bands ranges from -A, to A(). 

The Hamiltonian for a single electron in orbit around a fixed nucleus of charge Z is... 

Since our Hamiltonian involves a sum of hett(i), which are only functions of the coordinates and momenta of a single electron, we can use separation of variables and reduce the problem to m identical one-electron problems... 

The Hamiltonian (3.4) is a function of the usual spatial coordinates x, y, z or r, 0, (j)). Electrons possess the intrinsic property of spin, however, which is to be thought of as a property in an independent, or orthogonal, space (spin space). Spin is actually a consequence of the theory of relativity but we shall merely graft on the property in an ad hoc fashion. The spin, s, of an electron (don t confuse with s orbitals ) takes the value 1/2 only. The z component of spin, m, takes (25 + 1) values of ms, ranging 5, 5-l,...-s. Thus for the single electron, = +1/2 or -1/2, also labelled a or p, or indicated by t or i. 

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. 

In practice, each CSF is a Slater determinant of molecular orbitals, which are divided into three types inactive (doubly occupied), virtual (unoccupied), and active (variable occupancy). The active orbitals are used to build up the various CSFs, and so introduce flexibility into the wave function by including configurations that can describe different situations. Approximate electronic-state wave functions are then provided by the eigenfunctions of the electronic Hamiltonian in the CSF basis. This contrasts to standard HF theory in which only a single determinant is used, without active orbitals. The use of CSFs, gives the MCSCF wave function a structure that can be interpreted using chemical pictures of electronic configurations [229]. An interpretation in terms of valence bond structures has also been developed, which is very useful for description of a chemical process (see the appendix in [230] and references cited therein). 

Figure 3. Two electrons in three orbitals system. Configuration (a) is the reference configuration. Single electron excitations with spin-flip produce configurations (bf(g). Two-electron excitations with a single spin-flip produce configurations (h)-(j). Note that non-spin-flipping excitations or excitations that flip the spin of two electrons produce M = l configurations, which do not interact through the Hamiltonian with the final M =0 states, and thus are not...

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. 

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