Expansion, coefficient configurational

The next step might be to perform a configuration interaction calculation, in order to get a more accurate representation of the excited states. We touched on this for dihydrogen in an earlier chapter. To do this, we take linear combinations of the 10 states given above, and solve a 10 x 10 matrix eigenvalue problem to find the expansion coefficients. The diagonal elements of the Hamiltonian matrix are given above (equation 8.7), and it turns out that there is a simplification. 

This is perhaps the easiest method to understand. It is based on the variational principle (Appendix B), analogous to the HF method. The trial wave function is written as a linear combination of determinants with the expansion coefficients determined by requiring that the energy should be a minimum (or at least stationary), a procedure known as Configuration Interaction (Cl). The MOs used for building the excited Slater determinants are taken from a Hartree-Fock calculation and held fixed. Subscripts S, D, T etc. indicate determinants which are singly, doubly, triply etc. excited relative to the... 

On the experimental side, one may expect most progress from thermodynamic measurements designed to elucidate the non-configurational aspects of solution. The determination of the change in heat capacity and the change in thermal expansion coefficient, both as a function of temperature, will aid in the distinction between changes in the harmonic and the anharmonic characteristics of the vibrations. Measurement of the variation of heat capacity and of compressibility with pressure of both pure metals and their solutions should give some information on the... 

Core matrix elements, H, will be specified with individual methods. Indices k and m refer to closed and open shells, respectively c and y have their usual meaning of expansion coefficients and repulsion integrals, respectively. Numerical values of constants f, a, and b depend on the electronic configuration under study e.g., for a system having an unpaired electron in a nondegenerate... 

The Multi-Configuration Self-Consistent Field (MCSCF) method includes configurations created by excitations of electrons within an active space. Both the coefficients ca of the expansion in terms of CSFs and the expansion coefficients of the... 

Within the SCF-CI method a fixed set of molecular orbitals is used. This means that during the calculation (leading to slow convergence) the individual molecular orbitals remain unchanged. A method where the linear expansion coefficients and the LCAO coefficients are optimized simultaneously is the multi-configuration SCF (MCSCF). 

An alternative to the Co-rich perovskites is the Sr-doped LaFe03 which has a lower thermal expansion coefficient and a superior chemical compatibility with doped Ce02 electrolyte. LaFe03 is expected to be more stable than Ni- and Co-based perovskites because the Fe3+ ion has the stable electronic configuration 3d5. It is, therefore, expected that compositions in the system (La,Sr)(Co,Fe)03 will have desirable properties for intermediate temperature SOFC cathode applications. 

Table 2. Configuration interaction expansion coefficients for the wavefunctions...

The matrix elements of co can be calculated from the expansion coefficient of the ab initio configuration interaction method. The Cl wavefunction with single and double excitations can be written... 

Equation (2.18) is a linear variation function. (The summation indices prevent double-counting of excited configurations.) The expansion coefficients cq, c, c%, and so on are varied to minimize the variational integral. o) is a better approximation than l o)- In principle, if the basis were complete. Cl would provide an exact solution. Here we use a truncated expansion retaining only determinants D that differ from I Tq) by at most two spin orbitals this is a singly-doubly excited Cl (SDCI). 

It will represent a two-shell multi-configuration wave function in which the coefficients of expansion in configurations are the conventional Clebsch-Gordan coefficients. 

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