Level-shift function approach

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

The coupling of a finite cluster with bulk metal material is treated through a Green function s method. First, the density of states (DOS) of the bulk contact is calculated as indicated above. Next, the influence of the DOS of the bulk contact on the broadening and shifting of the discrete energy levels of the molecular orbitals (MO) of the cluster is accounted for via our DFT-Green function approach, as will be explained below. This yields the total DOS of the cluster as affected by the continuum. 

The local-density functional approach was used to compare the band structures of the sW-trans conformation of unsubstituted polysilane with a 4/1 helical conformation and with an dll-trans conformation of dimethyl-substituted poly silane. In line with previous theoretical studies, the electronic wave functions in the vicinity of the Fermi level are primarily silicon-back-bone states, with the major effect of methyl substitution being a decrease in the gap. The predicted absorption spectra for the dll-trans conformations of unsubstituted and dimethyl-substituted polysilane are similar for nearthreshold absorption. Given this similarity, we believe that the shift in energy and strong anisotropy of threshold absorption that we predict for the two extremes of the dll-trans conformation and the dll-gauche model will also occur in alkyl-substituted systems, which are currently under investigation. 

Aq becomes as5miptotically a g/ g, i.e., the steepest descent formula with a step length 1/a. The augmented Hessian method is closely related to eigenvector (mode) following, discussed in section B3.5.5 2. The main difference between rational function and trust radius optimizations is that, in the latter, the level shift is applied only if the calculated step exceeds a threshold, while in the former it is imposed smoothly and is automatically reduced to zero as convergence is approached. 

The EF algorithm [57] is based on the work of Ceijan and Miller [69] and, in particular, Simons and coworkers [70, 71]. It is closely related to the augmented Hessian (rational function) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to adding a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

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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). 

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