Valence-only effective Hamiltonians

The DFT/MRCI approach reproduces excitation energies and other spin-independent properties of experimentally known electronic states of pyranthione and dithio-succinimide excellently. As far as phosphorescence lifetimes of dithiosuccinimide are concerned, calculations have not yet been completed. For the T] state of pyranthione we find that phosphorescence and nonradiative decay via intersystem-crossing to the So state are concurrent processes occurring at approximately equal rates in the range of 104 s-1, in good accord with experimental data. The Ti - So radiative transition borrows its intensity from two sources  

The computed spin-orbit splitting in the Ti state of D = —18 cm-1 is mainly due to interaction with the close by T2 state. A rapid depletion of the S1 state via intersystem crossing to the Ti state can be mediated by the T2 state if spin relaxation within the triplet levels is fast. 

A further reduction of the computational effort in investigations of electronic structure can be achieved by the restriction of the actual quantum chemical calculations to the valence electron system and the implicit inclusion of the influence of the chemically inert atomic cores by means of suitable parametrized effective (core) potentials (ECPs) and, if necessary, effective core polarization potentials (CPPs). Initiated by the pioneering work of Hellmann and Gombas around 1935, the ECP approach developed into two successful branches, i.e. the model potential (MP) and the pseudopotential (PP) techniques. Whereas the former method attempts to maintain the correct radial nodal structure of the atomic valence orbitals, the latter is formally based on the so-called pseudo-orbital transformation and uses valence orbitals with a simplified radial nodal structure, i.e. pseudovalence orbitals. Besides the computational savings due to the elimination of the core electrons, the main interest in standard ECP techniques results from the fact that they offer an efficient and accurate, albeit approximate, way of including implicitly, i.e. via parametrization of the ECPs, the major relativistic effects in formally nonrelativistic valence-only calculations. A number of reviews on ECPs has been published and the reader is referred to them for details (Bala-subramanian 1998 Bardsley 1974 Chelikowsky and Cohen 1992 Christiansen et 

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  

The subscripts c and v denote core and valence, respectively. hy and gy are effective one- and two-electron operators, VCc represents the repulsion between all cores and nuclei of the system, and Vccp denotes the CPPs. The total number of electrons in the neutral system n and the number of valence electrons nv are related by the charges of the nuclei Z and the corresponding core charges Q  

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Basis Set Through Effective Hamiltonian Approaches Construction of Valence-only Effective Hamiltonians... 

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],... 

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... 

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. 

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. 

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