Configuration state functions definition

Most of the formalism to be developed in the coming sections of these lecture notes will be independent of the specific definition of the configurational basis, in which we expand the wave function. We therefore do not have to be very explicit about the exact nature of the basis states hn>. They can be either Slater determinants or spin-adapted Configuration State Functions (CSF s). For a long time it was assumed that CSF s were to be preferred for MCSCF calculations, since it gives a much shorter Cl expansion. Efficient methods like GUGA had also been developed for the solution of the Cl problem. Recent... 

Not all of these determinants are eigenfunctions of the total squared spin operator S, but one may construct linear combinations of determinants that are [287,288]. Then, the complexity of the problem can be reduced by definition of these so-called configuration state functions (CSF),... 

The purpose of this contribution is to give an overview of the results which center around the atomic density function and the recovery of the periodicity. Since all the calculations are based on atomic density functions, it is appropriate to revisit the construction of these densities in some depth. First a workable definition of the density function is established in the framework of the multi-configuration Hartree-Fock method (MCHF) [2] and the spherical harmonic content of the density function is discussed. A spherical density function is established in a natural way, by using spherical tensor operators. The proposed expression can be evaluated for any multi-configuration state function corresponding to an atom in a particular well-defined state and a recently developed extension of the MCHF code [3] is used for that purpose. Three illustrative examples are given. In the next section relativistic density functions for the relativistic Dirac-Hartree-Fock method [4] are defined. The latter will be used for a thorough analysis of the influence of relativistic effects on electron density functions later on in this paper. 

The first satisfactory definition of entropy, which is quite recent, is that of Kittel (1989) entropy is the natural logarithm of the quantum states accessible to a system. As we will see, this definition is easily understood in light of Boltzmann s relation between configurational entropy and permutability. The definition is clearly nonoperative (because the number of quantum states accessible to a system cannot be calculated). Nevertheless, the entropy of a phase may be experimentally measured with good precision (with a calorimeter, for instance), and we do not need any operative definition. Kittel s definition has the merit to having put an end to all sorts of nebulous definitions that confused causes with effects. The fundamental P-V-T relation between state functions in a closed system is represented by the exact differential (cf appendix 2)... 

With regard to the former, one would like to include as many important configurations as possible. Unfortunately, the definition of an important configuration is often debatable. One popular remedy is the full-valence complete active space SCF (CASSCF) approach in which configurations arising from all excitations from valence-occupied to valence-virtual orbitals are chosen. [29] Since this is equivalent to performing a full Cl within the valence space, the full-valence CASSCF method is limited to small systems. Nevertheless, the CASSCF approach using a well-chosen (often chemically motivated) subspace of the valence orbitals has been shown to yield a much improved depiction of the wave function at all points on a potential surface. Furthermore, the choice of an active space can be adjusted to describe excited state wave functions. 

The determinantal functions must be linearly independent and eigenfunctions of the spin operators S2 and Sz, and preferably they belong to a specified row of a specified irreducible representation of the symmetry group of the molecule [10, 11]. Definite spin states can be obtained by applying a spin projection operator to the spin-orbital product defining a configuration [12]. Suppose d>0 to be the solution of the Hartree-Fock equation. From functions of the same symmetry as d>0 one can build a wave function d>,... 

The preceding discussion has demonstrated how the specificity inherent in the cyclic transition state may be used to fix structural elements in Cope products, establishing double bonds and other functional groups in definite relationships in a variety of acyclic and cyclic skeletons. This final section treats the remarkable consequences of cyclic transition states in stereochemical control, including alkene configuration and both relative and al olute configuration at stereogenic centers. ... 

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