Effective Hamiltonian valence-only

Continuum effective Hamiltonian needs a definition of the electronic charge distribution pMe. All quantum methods giving this quantity can be used, whereas other methods must be suitably modified. Quantum methods are not limited to those based on a canonical molecular orbital formulation. Valence Bond (VB) and related methods may be employed. The interpretation of reaction mechanisms in the gas phase greatly benefits by the shift from one description to another (e.g. from MO to VB). The same techniques can be applied to continuum effective Hamiltonians. We only mention this point here, which would deserve a more detailed discussion. 

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. 

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. 

The Wolfsberg60-Helmholz61-Hoffmann62 extended Huckel theory considers only the valence electrons, for which the effective Hamiltonian Hvai is the sum of one-electron Hamiltonians which are not specified explicitly. 

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. 

The generality and importance of the above results cannot be overemphasized. The wavefunction for the valence electrons may be optimized by variation of y alone, using the effective Hamiltonian in (47) with appropriate orthonormality constraints. In practice this means that, for a function built up from orbitals r of the valence space, it is only necessary to replace matrix elements < r h a > by... 

In ECP theory an effective Hamiltonian approximation for the all-electron no-pair Hamiltonian Hnp is derived which (formally) only acts on the electronic states formed by nv valence electrons in the field of N frozen closed-shell atomic-like cores ... 

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... 

Obviously, the sfss technique is not bounded to be applied only in AIMP calculations or in other valence-only calculations, but it can be used with any relativistic Hamiltonian which can be separated in spin-free and spin-dependent parts [48]. Being a very simple procedure, it is an effective means for the inclusion of dynamic correlation and size consistency in spin-orbit Cl calculations with any choice of Cl basis, such as determinants, double-group adapted configuration state functions, or spin-free Cl functions. In the latter case [46], the technique reduces to changing the diagonal elements of the spin-orbit Cl matrix. 

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 ... 

In principle, it should also be possible to add a semi-loced potential to the non-relativistic all-electron Hamiltonian to eirrive at a quasi-rela-tivistic all-electron method. One such suggestion has been made by Delley [76], but the resulting method has only been tested for valence properties, which could also have been obtained by valence-only methods. Effective core potential methods have the advantage of a reduced computational effort (compared to all-electron methods) and are a valuable tool as long as one is aware of the limited domain of valence-only methods. Properties for which density variations in the atomic core are important should not be calculated this way. Examples are the electric field gradient at the nucleus or the nuclear magnetic shielding. 

Algebraic manipulations then yield an effective Hamiltonian which acts only on the valence space and nevertheless produces the exact energies for all valence states ... 

If applied to the reference state normal order enables us immediately to recognize those terms which survive in the computation of the vacuum amplitudes. The same applies for any model function and, hence, for real multidimensional model spaces, if a proper normal-order sequence is defined for all the particle-hole creation and annihilation operators from the four classes of orbitals (i)-(iv) in Subsection 3.4. In addition to the specification of a proper set of indices for the physical operators, such as the effective Hamiltonian or any other one- or two-particle operator, however, the definition and classification of the model-space functions now plays a crucial role. In order to deal properly with the model-spaces of open-shell systems, an unique set of indices is required, in particular, for identifying the operator strings of the model-space functions (a)< and d )p, respectively. Apart from the particle and hole states (with regard to the many-electron vacuum), we therefore need a clear and simple distinction between different classes of creation and annihilation operators. For this reason, it is convenient for the derivation of open-shell expansions to specify a (so-called) extended normal-order sequence. Six different types of orbitals have to be distinguished hereby in order to reflect not only the classification of the core, core-valence,... orbitals, following our discussion in Subsection 3.4, but also the range of summation which is associated with these orbitals. While some of the indices refer a class of orbitals as a whole, others are just used to indicate a particular core-valence or valence orbital, respectively. 

The derivation of the MCP formalism was detailed in the work of Hdjer and Chung [59] and one of the present authors (TZ) gave a comprehensive discussion of their derivation in the second chapter of his Ph.D. thesis [60], Interested readers should refer to those two works for a thorough exposition of this method while here, in the interest of brevity, we provide only the necessary information about MCP and focus on its level-shift operator, whose physical meaning needs more clarification. Following the same philosophy as the one behind the ECP, the MCP is naturally derived from all-electron atomic calculations at the Hartree-Fock level, and the final expression of the effective hamiltonian for the closed-shell valence electrons is... 

D. Effective Hamiltonians Spanned by Neutral-only Valence-bond Determinants Magnetic (Heisenberg) Hamiltonians and their Possible... 

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