Valence only Hamiltonian

Only the most loosely bound electrons in an atom or molecule need be considered explicitly, the valence electrons being separated from a core , whose presence is simulated by using a suitable effective (valence-only) Hamiltonian - or, in practice, by assigning empirical values to the Coulomb and exchange integrals involving valence orbitals. 

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... 

Successful developments on relativistic valence-only Hamiltonians makes it more and more feasible to obtain accurate molecular spectroscopic data. These developments aim at accurately describing both relativistic and correlation effects. However, as most of the difficulties in describing relativistic effects are now overcome, highly correlated treatments remain the most difficult and challenging task, and this remains the main problem which guides the choice of physically well-foimded and relevant approximations. To reach maximal accuracy as well as efficiency, the above applications show three major approximations to the full relativistic Hamiltonian based on the separate treatment of both active/inactive electrons and physical effects, namely ... 

The present section essentially discusses the attempts to build effective valence-only Hamiltonians spanned by a valence minimal basis set, which has been the subject of several tens of papers by Freed et al. (reviewed in Ref. 78 see also Westhaus and Mukherjee ). One should notice first that the purpose of such attempts is two-fold elimination of the core electrons from the explicit treatment and reduction of the basis set for the treatment of the valence states of the molecule to a minimal valence basis set. The effective Hamiltonian should reproduce the energetic results of a calculation including the core electrons, their exclusion, polarization and correlation effects, and performed in a large basis set (for both the concentrated core distribution and the more diffuse valence cioud). 

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. 

We have so far discussed the theory and characteristics of pseudospinors and the corresponding one-particle pseudopotentials. The next stage is to examine the use of these pseudopotentials in a valence-only Hamiltonian and the properties of the energy and the SCF equations derived from this energy. The desired form of the pseudopotential Hamiltonian is one in which only the one-electron operator is modified. 

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... 

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. 

Thble3.6 Bond length Rc (A), vibrational constant coe (cm-1) and binding energy De (eV) of Eka-Au hydride (111)H without (with) counterpoise correction of the basis-set superposition error. All-electron (AE) values based on die Dirac-Coulomb-Hamiltonian (Seth and Schw-erdtfeger 2000) are compared with valence-only results obtained with energy-consistent (EC) (Dolg etal. 2001) and shape-consistent (SC) (Han and Hirao 2000) pseudopotentials (PP). The numbers 19 and 34 in parentheses denote the number of valence electrons for the Eka-Au PP. 

The indices i and j denote electrons, X and ju nuclei. is the charge of the nucleus X. For the one- and two-particle operators h and g various expressions can be inserted (e.g., relativistic, quasirelativistic or nonrelativistic all-electron or valence-only). The basic goal of quantum chemical methods is usually the approximate solution of the time-independent Schrodinger equation for a specific Hamiltonian, the system being in the state 7, i.e.. 

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],...

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