Configurational function spin-free

Define the functional U(Q [v[/])=q integration over the electronic configuration space is indicated as a sub-index. The variational principle applied [6] to the (spin-free) function space v /(q) leads to the Euler-Lagrange equation ... 

Hess et al.119 utilized a Hamiltonian matrix approach to determine the spin-orbit coupling between a spin-free correlated wave function and the configuration state functions (CSFs) of the perturbing symmetries. Havriliak and Yarkony120 proposed to solve the matrix equation... 

When using a spin-free hamiltonian operator, the spin functions are introduced as mulplicative factors, yielding the spin-orbitals. There are two spin-orbitals per orbital. An electronic configuration is defined by the occupancies of the spin-orbitals. Open-shell configurations are those in which not all the orbitals are doubly occupied. 

The configurational functions of all three components of the triplet state are listed on the three lines of Equation (1.17). From top to bottom, they correspond to the occupation of the MOs 0, and 0, with two electrons with an a spin, with one electron each with a and jS spin, and with two electrons with a p spin. The z component of the total spin is equal to M5 = 1,0, and — 1, respectively. The three triplet functions are degenerate (i.e., have the same energy) in the absence of external fields and ignoring relativistic effects (i.e., with a spin-free Hamiltonian). For our purposes, it is therefore sufficient to consider only one of the components, e.g., the one corresponding to Ms = 0. 

When the one-electron operators are spin free, only the spin-singlet configuration functions contribute and hence the matrix elements to be evaluated are... 

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. 

In order to understand this, let us again consider the and S terms of the Is 2s configuration of a two-electron atom. In the framework of Slater s theory of atoms the electron density g(f) is the same for both states, and for an exact treatment we would expect (9) to differ very little between the two states. The pair density n(9i, 92) is, however, quite different for the IS and the states. If we call the two singly occupied real (orthogonal) orbitals a and b, we have (where stands for the spin-free wave function and where a (1) has the same meaning as a (fi)... 

The main restriction of the methods discussed in the main part of this chapter is the assumption of a single dominant configuration. This requirement is not met for systems with strong non-dynamical correlation effects such as biradicals (e.g. in bond-breaking situations) or transition metal compounds. In this case the reference function must be a multireference expansion. The fixed-amplitude approximation lends itself very well to a multireference extension of the formalism, as demonstrated first by Ten-no who devised an F12-based internally contracted geminal correction to the multireference MP2 (MRMP2) method. A closely related approach was considered by Torheyden and Valeev, who proposed a generalized perturbative correction to arbitrary wavefunctions and which they applied to a MRCI wavefunction. A spin-free formulation was reported recently.The approach was also used by Booth et al. to get a basis set limit estimate for their full Cl quantum Monte Carlo method, which can also be seen as an approach to tackle systems with multireference character. 

Ribonucleotide reductase differs from the other 5 -deoxyadenosyl-cobalamin requiring enzymes in a number of respects. Hydrogen is transferred from coenzyme to the C2-position of the ribose moiety without inversion of configuration. Also since lipoic acid functions in hydrogen transfer, exchange with solvent protons takes place. Furthermore, exchange between free and bound 5 -deoxyadenosylcobalamin occurs rapidly during catalysis. Evidence for a Co(I)-corrin as an intermediate for this reduction is presented in our section on electron spin resonance. 

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