Detailed configuration, orbital wave functions

The essential point that distinguishes between allowed and forbidden reactions is the role of the D+A configuration. If the D+A configuration is allowed by symmetry to mix into the transition state wave-function then the transition state will be stabilized and will take on character associated with that configuration. For the ethylene dimerization, D+A is precluded from mixing with DA due to their opposite symmetries. As was discussed in detail in Section 2 (p. 130), DA cannot mix with D+A" since and n orbitals are orthogonal (106). Thus for ethylene dimerization the concerted process... 

But, as described in more detail in Appendix C, this wave function, which configurationally corresponds to an a electron in the orbital and a p electron in the ( 2 orbital, is neither a singlet nor a triplet, but a 50 50 mixture of the two, and this point is emphasized by die left superscript on 4/ in Eq. (14.14). While the wave function does not represent a pure spin state, we may take advantage of the prevailing situation by noting that we may write... 

We shall not perform the somewhat elaborous calculation of the MC wave function in detail. A somewhat simpler example is the dissociation of a double bond and it is given as an exercise (exercise 2). Here we only note that the number of configuration state functions (CSF s) will increase very quickly with the number of active orbitals. In most cases we do not have to worry about the exact construction of the MC wave function that leads to correct dissociation. We simply use all CSFs that can be constructed by distributing the electrons among die active orbitals. This is the idea behind the Complete Active Space SCF (CASSCF) method. The total number of such CSFs is for N2 175 for a singlet wave function. A further reduction is obtained by imposing spatial symmetry. All these CSFs are not included in a wave... 

If we transform the MO s such that condition (5 11) is fulfilled, the resulting transition density matrix will be obtained in a mixed basis, and can subsequently be transformed to any preferred basis The generators Epq of course have to be redefined in terms of the bi-orthonormal basis, but this is a technical detail which we do not have to worry about as long as we understand the relation between (5 9) and the Slater rules. How can a transformation to a bi-orthonormal basis be carried out We assume that the two sets of MO s are expanded in the same AO basis set. We also assume that the two CASSCF wave functions have been obtained with the same number of inactive and active orbitals, that is, the same configurational space is used. Let us call the two matrices that transform the original non-orthonormal MO s [Pg.242]

The extension of the basis can improve wave functions and energies up to the Hartree-Fock limit, that is, a sufficiently extended basis can circumvent the LCAO approximation and lead to the best molecular orbitals for ground states. However, this is still in the realm of the independent-particle approximation 175>, and the use of single Slater-determinant wave functions in the study of potential surfaces implies the assumption that correlation energy remains approximately constant on that part of the surface where reaction pathways develop. In cases when this assumption cannot be accepted, extensive configuration interaction (Cl) must be included. A detailed comparison of SCF and Cl results is available for the potential energy surface for the reaction F + H2-FH+H 47 ). 

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