Hamiltonian valence-electron

A CNDO all-valence-electron HF-LCAO Hamiltonian matrix has elements... 

Hamiltonian for, 19 valence electrons in, 4. See also Valence bond model... 

Hamiltonian operator, 2,4 for many-electron systems, 27 for many valence electron molecules, 8 semi-empirical parametrization of, 18-22 for Sn2 reactions, 61-62 for solution reactions, 57, 83-86 for transition states, 92 Hammond, and linear free energy relationships, 95... 

Initially we consider a simple atom with one valence electron of energy and wave function which adsorbs on a solid in which the electrons occupy a set of continuous states Tj, with energies Ej. When the adsorbate approaches the surface we need to describe the complete system by a Hamiltonian H, including both systems and their interaction. The latter comes into play through matrix elements of the form Vai = / We assume that the solutions T j to this eigen value problem... 

In crystalline field theory, the valence electrons belong to ion A and the effect of the lattice is considered throngh the electrostahc field created by the snrronnding B ions at the A position. This electrostatic field is called the crystalline field. It is then assnmed that the valence electrons are localized in ion A and that the charge of B ions does not penetrate into the region occnpied by these valence electrons. Thns the Hamiltonian can be written as... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

We know from Chapter 1 that the probability P,f of indncing an optical transition from a state i to a state / is proportional to (1 //1), where in the matrix element Ip, and P f denote the eigenfnnctions of the ground and excited states, respectively, and H is the interaction Hamiltonian between the incoming light and the system (i.e., the valence electrons of the center). In general, we can assnme that // is a sinnsoidal... 

SRPA has been already applied for atomic nuclei and clusters, both spherical and deformed. To study dynamics of valence electrons in atomic clusters, the Konh-Sham functional [14,15]was exploited [7,8,16,17], in some cases together with pseudopotential and pseudo-Hamiltonian schemes [16]. Excellent agreement with the experimental data [18] for the dipole plasmon was obtained. Quite recently SRPA was used to demonstrate a non-trivial interplay between Landau fragmentation, deformation splitting and shape isomers in forming a profile of the dipole plasmon in deformed clusters [17]. 

Let us consider the 5s, 5p, 5d orbitals of lead and Is orbital of oxygen as the outercore and the ai, a2, os, tti, tt2 orbitals of PbO (consisting mainly of 6s, 6p orbitals of Pb and 2s, 2p orbitals of O) as valence. Although in the Cl calculations we take into account only the correlation between valence electrons, the accuracy attained in the Cl calculation of Ay is much better than in the RCC-SD calculation. The main problem with the RCC calculation was that the Fock-space RCC-SD version used there was not optimal in accounting for nondynamic correlations (see [136] for details of RCC-SD and Cl calculations of the Pb atom). Nevertheless, the potential of the RCC approach for electronic structure calculations is very high, especially in the framework of the intermediate Hamiltonian formulation [102, 131]. 

The problem of estimating crystal field parameters can be solved by considering the CFT/LFT as a special case of the effective Hamiltonian theory for one group of electrons of the whole A -electronic system in the presence of other groups of electrons. The standard CFT ignores all electrons outside the d-shell and takes into account only the symmetry of the external field and the electron-electron interaction inside the d-shell. The sequential deduction of the effective Hamiltonian for the d-shell, carried out in the work [133] is based on representation of the wave function of TMC as an antisymmetrized product of group functions of d-electrons and other (valence) electrons of a complex. This allows to express the CFT s (LFT s or AOM s) parameters through characteristics of electronic structure of the environment of the metal ion. Further we shall characterize the effective Hamiltonian of crystal field (EHCF) method and its numerical results. 

The Hamiltonian of valence electrons (39), in the so-called orthogonal representation (or in the most localized representation, neglecting orbital overlap) can be mapped on a tight-binding form Hamiltonian... 

The FE MO, HMO, and PPP methods are restricted to planar conjugated molecules (e.g., butadiene, benzene, pyridine), and make the pi-electron approximation of treating only those valence electrons that are in pi MOs (those that have eigenvalue — 1 for reflection in the molecular plane) they assume the existence of a pi-electron Hamiltonian Hm of the form... 

The EH method (developed by Wolfsberg and Helmholz and by Hoffmann) is an extension of the Hiickel method in which the pi-electron approximation is not made, but all valence electrons are treated. The method is thus applicable to nonplanar, as well as planar, molecules. The valence-electron Hamiltonian is taken as the sum of one-electron Hamiltonians //va, = 2(/ eff(0, where Hcft(i) is not explicitly defined. The valence-electron wave function is the antisymmetrized product of spin-... 

The resulting valence hamiltonian Hv includes the influence of the core on the valence electrons. In the simple case where the core function c consists of a single configuration of doubly occupied orbitals... 

Explicit orthogonality constraints can be removed by transformingthe hamiltonian so that it only acts on a specific subspace (e.g. the valence space) of the all-electron space. This can be done formally by the use of the projection operator method (see Huzinaga and Cantu18 or Kahn, Baybutt, and Truhlar20). If the core function is written as in equation (6) a projection operator may be defined for each valence electron p ... 

It is common for valence-only calculations to use a form of effective hamiltonian which is based on the eigenfunctions for atoms or ions with only one valence electron,83 This is equivalent to choosing a set of core orbitals l which satisfy... 

This Fock operator has been derived starting from the assumption of a Hartree-Fock valence function valence electrons has little influence on the core electrons, so that the many-electron valence hamiltonian may be similarly approximated as... 

The first and last terms in equation (34) consist of the valence-electron components of the all-electron hamiltonian of equation (1), and the remaining terms constitute the pseudopotential represented symbolically in equation (2). Further, if we assume that the interaction of the two cores A and B can be approximated by a point charge potential (see Kahn et al.2i for errors in this assumption),... 

Table 2 Functional forms for the one-electron operators in valence-electron pseudo-hamiltonians...

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