Valence-Only Model Hamiltonian

In ECP theory an effective model Hamiltonian approximation for Jifnp is seeked, which only acts on the electronic states formed by the valence electrons  

The subscripts c and v denote core and valence, respectively, hy and gv stand for effective one- and two-electron operators, Vcc represents the repulsion between all cores and nuclei of the system, and Vcpp is a core polarization poten- 

Scalar-quasirelativistic and quasirelativistic ECPs use a formally nonrelativistic model Hamiltonian 

Relativistic contributions result only from the parametrization of the ECP kcv which describes the interaction of a valence electron with all nuclei and cores present in the system. The molecular ECP is assumed to be a superposition of 

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This valence model Hamiltonian contains high-order many-electron operators due to the presence of products of projection operators and would be clearly much too complicated for practical calculations. Moreover, no computational savings would result compared to a standard all-electron treatment, since the derivation given so far merely corresponds to a rewriting of the Fock equation for a valence orbital in a quite complicated form (eqns. 41 and 43). Actual reductions of the computational effort can be achieved only at the price of approximations, i.e., by the actual elimination of the core electron system and the simulation of its influence on the valence electrons by introducing a simplified valence-only model Hamiltonian containing the pseudopotential If higher than two-electron operators in eqn. 45 are omitted this corresponds to the formal substitutions ... 

Figure 21. Valence spinors of 53I from average level multi-configuration calculations using the AE Dirac-Coulomb-Hamiltonian (solid lines) and a PP valence-only model Hamiltonian (dashed lines) [193],...

Pseudopotentials, effective core potentials and model potentials By means of the introduction of pseudopotentials the many-electron system is divided in a core-electron system implicitly included by the pseudopotentials and a valence-electron system explicitly treated in the quantum-chemical calculations. The all-electron Hamiltonian is replaced by the following valence-only model Hamiltonian for Hy valence electrons ... 

In ECP theory an effective model Hamiltonian only acting on the explicitly treated valence electrons is searched. There are several choices for the formulation of such a valence-only model Hamiltonian, i.e., four-, two-, or one-component approaches and explicit or implicit relativistic treatment [20], Nonrelativistic, scalar-relativistic, and quasirelativistic ECPs use a formally nonrelativistic valence-only model Hamiltonian implicitly including relativistic effects [19]... 

The pseudopotential approximation was originally introduced by Hellmann already in 1935 for a semiempirical treatment of the valence electron of potassium [25], However, it took until 1959 for Phillips and Kleinman from the solid state community to provide a rigorous theoretical foundation of PPs for single valence electron systems [26]. Another decade later in 1968 Weeks and Rice extended this method to many valence electron systems [27,28], Although the modern PPs do not have much in common with the PPs developed in 1959 and 1968, respectively, these theories prove that one can get the same answer as from an AE calculation by using a suitable effective valence-only model Hamiltonian and pseudovalence orbitals with a simplified nodal structure [19],... 

At least five approximations have to be made in order to work with a computationally practical valence-only model Hamiltonian as in equation 6.1. To compensate for the errors... 

Fourth, the valence-only model Hamiltonian given in equation 6.1 assumes a classical point charge Coulomb repulsion between cores/nuclei. However, for very large cores this approximation might be too crude and mutually penetrating or overlapping cores need to be accounted for by additional corrections [19]. 

A rigorous way to construct a valence-only model with ab initio methods is to make use of the effective Hamiltonian theory presented in Chap. 1. First, we define the basis of the model space as I, ba, aa, and use the matrix of Eq. 5.4 to... 

We overview our valence bond (VB) approach to the ir-electron Pariser-Parr-Pople (PPP) model Hamiltonians referred to sis the PPP-VB method. It is based on the concept of overlap enhanced atomic orbitals (OEAOs) that characterizes modern ab initio VB methods and employs the techniques afforded by the Clifford algebra unitary group approach (CAUGA) to carry out actual computations. We present a sample of previous results, sis well sis some new ones, to illustrate the ability of the PPP-VB method to provide a highly correlated description of the ir-electron PPP model systems, while relying on conceptusilly very simple wave functions that involve only a few covalent structures. 

Modern relativistic effective core potentials provide a useful tool for accurate quantum chemical investigations of heavy atom systems. If sufficiently small cores are used to minimize frozen-core and other errors, they are able to compete in accuracy with the more rigorous all-electron approaches and still are, at the same time, economically more attractive. Successful developments in the field of valence-only Hamiltonians turned relativistic effects into a smaller problem than electron correlation in practical calculations. Both the model potential and the pseudopotential variant have advantages and disadvantages, and the answer to the question which approach to follow may be a matter of personal taste. Highly accurate correlated all-electron calculations are becoming... 

Both steps, introduction of the pseudopotential and simplification of the Hamiltonian, may formally be carried out in any order, resulting in different expressions for the valence-only Hamiltonian (Hafner and Schwarz 1979, Pyper 1980a,b, 1981, Pyper and Marketos 1981). The derivation of pseudopotentials to be used within the Dirac-Hartree-Fock scheme has also been attempted (Ishikawa and Malli 1981, 1983, Dolg 1995). We note that the derivation of valence-only Hamiltonians used in pseudopotential theory is neither unique nor may be carried out without a multitude of approximations. Therefore, in practice a proper adjustment of free parameters in a suitably chosen model Hamiltonian seems to be much more important than a lengthy theoretical derivation. 

The measurements were carried out using polarized-light from synchrotron radiation. The angle-resolved UPS spectra were recorded for specific directions of photon incidence, photon polarization, and electron exit, chosen in order to resolve the momentum dependence of the 7t-electron energy bands which could be observed in this experiment. Details are available elsewhere63. The UPS results are analysed not only with the help of the valence effective Hamiltonian (VEH) method, but also with the help of new quantum-chemical calculations based upon the excitation model method64. The full VEH band structure is shown in Fig. 7.32. 

Much work has been done in this field by Zeiri and Shapiro, and subsequently by others. For example, semi-empirical potential surfaces have been reported for the alkali atom (M = Li, Na, K, Rb) and halogen molecule (XY = F2,Cl2,Br2,l2) reactions . In this model, the three valence orbitals were represented by Slater-type orbitals centred on M, X and Y. Three VB structures were used a covalent structure based on the spatial configuration MXY and two ionic structures based on M X Y and M XY . The full Hamiltonian was written as a sum of diatomic and atomic terms, but, contrary to the DIM formalism, only the ground-state potentials were required. The approach of Zeiri and Shapiro is very cheap and can be applied to heavy systems. 

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