Excitation functions minima

The excitation function together with the transition-state wavefunctions for FNO are shown on the left-hand side of Figure 10.15. This figure underlines very clearly the success of the simple picture in interpreting the quantum mechanically calculated final state distributions. The reflection principle is obvious in the light of Section 6.3 and needs no further explanation. Each maximum and each minimum in the distribution has its counterpart in the transition state wavefunction. 

Following the customary terminology, we will call inactive holes the inactive occupied orbitals, doubly filled in every model CSF. The inactive particles will refer to aU the orbitals unoccupied in every CSF. Orbitals which are occupied in some (singly or doubly) but unoccupied in others are the active orbitals. In our spin-free form, the labels are for orbitals only, and not for spin orbitals. From the mode of definition, no active orbital can be doubly occupied in every model CSF. We want to express the cluster operator T, inducing excitations to the virtual functions, in terms of excitations of minimum excitation rank, and at the same time wish to represent them in a manifestly spin-free form. To accomplish this, we take as the vacuum—for excitations out of 4> — the largest closed-shell portion of it, For each such vacuum, we redefine the holes and particles, respectively, as ones which are doubly occupied and unoccupied in < 0 a-The holes are denoted by the labels. .., etc. and the particle orbitals are denoted as a, etc. The particle orbitals are totally unoccupied in any or are necessarily... 

The value of n = (2.25 0.20) MeV - extracted for the excitation energy of the ground state in the second minimum of Pu - is in good agreement with the fission isomer energy obtained from the well-known method of extrapolated excitation functions of various experiments (see, e.g., Wagemans 1991 and references therein). Thus having proven the reliability of this method, an excellent tool has been obtained to address the question of the depth of the (hyperdeformed) third minimum of the potential surface. This tool will be discussed in O Sect. 5.5. 

Figure 13 shows the cross section for insertion events whose minimum energy was nearly that of the potential energy surface minimum. The cross section for abstraction includes only those reactions whose minimum energy was nearly that of products. Owing to the use of these more restrictive definitions, the excitation functions for insertion and abstraction do not necessarily sum to the overall excitation function but any discrepancy is always quite small. 

The self-consistent field function for atoms with 2 to 36 electrons are computed with a minimum basis set of Slater-type orbitals. The orbital exponents of the atomic orbitals are optimized so as to ensure the energy minimum. The analysis of the optimized orbital exponents allows us to obtain simple and accurate rules for the 1 s, 2s, 3s, 4s, 2p, 3p, 4p and 3d electronic screening constants. These rules are compared with those proposed by Slater and reveal the need for the screening due to the outside electrons. The analysis of the screening constants (and orbital exponents) is extended to the excited states of the ground state configuration and the positive ions. 

How are the additional determinants beyond the HF constructed With N electrons and M basis functions, solution of the Roothaan-Hall equations for the RHF case will yield N/2 occupied MOs and M — N/2 unoccupied (virtual) MOs. Except for a minimum basis, there will always be more virtual than occupied MOs. A Slater detemfinant is determined by N/2 spatial MOs multiplied by two spin functions to yield N spinorbitals. By replacing MOs which are occupied in the HF determinant by MOs which are unoccupied, a whole series of determinants may be generated. These can be denoted according to how many occupied HF MOs have been replaced by unoccupied MOs, i.e. Slater determinants which are singly, doubly, triply, quadruply etc. excited relative to the HF determinant, up to a maximum of N excited electrons. These... 

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... 

The dissociation problem is solved in the case of a full Cl wave function. As seen from eq. (4.19), the ionic term can be made to disappear by setting ai = —no- The full Cl wave function generates the lowest possible energy (within the limitations of the chosen basis set) at all distances, with the optimum weights of the HF and doubly excited determinants determined by the variational principle. In the general case of a polyatomic molecule and a large basis set, correct dissociation of all bonds can be achieved if the Cl wave function contains all determinants generated by a full Cl in the valence orbital space. The latter corresponds to a full Cl if a minimum basis is employed, but is much smaller than a full Cl if an extended basis is used. 

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