Multiconfigurational wavefunctions

Once the requisite one- and two-electron integrals are available in the MO basis, the multiconfigurational wavefunction and energy calculation can begin. Each of these methods has its own approach to describing tlie configurations d),. j included m the calculation and how the C,.] amplitudes and the total energy E are to be... 

I. Optimization of the Energy for a Multiconfiguration Wavefunction A. The Energy Expression... 

Once the requisite one- and two-electron integrals are available in the molecular orbital basis, the multiconfigurational wavefunction and energy calculation can begin. 

F. Grein and T. C. Chang, Chem. Phys. Lett., 12,44 (1971), Multiconfiguration Wavefunctions Obtained by Application of the Generalized Brillouin Theorem. 

In both cases, the extra pair of configurations entered the multiconfiguration wavefunction with a weight that was comparable to that assumed in it by the fully symmetric configuration. Here and in what follows, a configuration s weight is defined as its Chirgwin-Coulson occupation number [3 4] [2] (for alternative definitions, see e.g. Refs. [35] and [36]). 

In the last few years, the polarizable continuum model for the study of solvation has been extended to consider multideterminantal wavefunctions. Such novel techniques allow the study of the most important solvent effects on chemical reactions. In this context, the valence bond theory provides a way to analyze such effects through the transcription of the, generally, complicated multiconfigurational wavefunctions into sums of few selected classical structures, which are, in fact, more useful to understand the electron distribution rearrangement along a reaction path. In this chapter, the valence bond analysis of CASSCF wavefunctions calculated for chemical reactions in solution is discussed in details. By way of example, the results for some basic chemical processes are also reported. 

Methods based on the use of determinants, particularly those of Balint-Kurti and co-workers [37], share the advantages of simplicity and flexibility and in spite of a certain lack of mathematical elegance they also readily admit orbital optimization and the use of multiconfiguration wavefunctions. 

It is important to dispense with the received wisdom that MO theory is in some sense more fundamental than VB approaches. On the other hand, it is certainly not our intention to argue that the MO description is somehow wrong . In the particular case of benzene, we quantify to what extent the conventional MO and VB models can be considered reliable approximate representations of a particular type of multiconfigurational wavefunction that is more sophisticated than those obtained from either approach. We conclude that we should not have any serious qualms about switching between the MO and VB representations, according to the nature of the particular problem being addressed. 

Here then is the crux of the computational difficulty. The reactant, (3Z)-3-hexene-l,5-diyne, is well described by a single-configuration reference wavefunction. The product, p-benzyne, is likely to have appreciable diradical character and necessitates a multiconfiguration wavefunction. The transition state will exist somewhere in between. The choice of computational method suited to describe all three structures equally well is nontrivial, and in the next section we discuss the various approaches employed and results obtained by a number of research groups. 

With the two radical centers farther apart in 42 than in 41, it is reasonable to expect the meta isomer to express greater biradical character than the ortho isomer. Therefore, a multiconfiguration wavefunction will be necessary to adequately describe 41. The two configurations that doubly occupy either the radical bonding orbital (llaj) or antibonding orbital (7 2)... 

Again, a multiconfiguration wavefunction will be necessary to describe the expected large biradical character of p-benzyne 43. This wavefunction will be dominated by the two configurations that define the bonding and antibonding interactions between the radical centers. 

The next controversy concerning the benzynes is the stracture of m-benzyne. Does it exist as the monocyclic biradical 42 or as the bicyclic closed-shell species 44 Answering this question with a computational approach will take some care. While the biradical character of 42 is small, a multiconfiguration wavefunction (Eq. (5.3)) is likely to be necessary for adequate description of its electronic structure. On the other hand, 44 is a closed-shell species and its electronic configuration can be expressed by a single Slater determinant made from the molecular orbitals shown in Figure 5.12 ... 

Though not discussed above, in all the studies mentioned the trial wavef unctions included pair correlation functions. J j. as prescribed by Reynolds et al. ( ). Moskowitz et al. (48.49) have shown that the product of a relatively simple multiconfiguration wavefunction with pair correlation functions can provide a rather accurate approximation to the exact wavefunction. In our calculations and in those of Hammond et al. (59) the many-electron local potential, has been obtained by allowing the REP to... 

There are many variations on the procedure sketched above. The simplest MO wavefunctions are Hartree-Fock (HF) wavefunctions in which a single determinant appears in the expansion in equation 6 and the MO coefficients are the only variable parameters. More elaborate types are Cl wavefunctions in which only the Cl coefficients are allowed to vary. The values of the MO coefficients are fixed, having been taken, usually, from a previous HF calculation. The most general types are multiconfigurational wavefunctions, in which both sets of coefficients are treated variationally. 

The doubly excited configurations in Eq. (121) are obtained by applying two-electron excitation operators to the complete reference function 4 o- If Tq is a multiconfiguration wavefunction we can write... 

Here stands for a Hartree-Fock or multiconfiguration wavefunction for atom A in state i for a single configuration, it is identical to the function of... 

The second notion is concerned with aspects of the issue of the a priori identification of ND and D correlations and the choice of the state-specific set of zero-order orbitals and multiconfigurational wavefunctions in terms of which this identification is assumed and implemented. In this context, 1 use examples from published results and from new computations. 

Photoabsorption transition probabilities and cross-sections for the two categories contain different satellite peaks due to the presence (or absence) of different zero-order Fermi-sea SACs in initial and final sfafes, for example, Ref. [101]. This is in accordance wifh FOTOS, where, as explained in Section 4, the essential features of the transition probability and the related phenomena are explained by using in the zero-order transition matrix element the Fermi-sea multiconfigurational wavefunctions of the initial and the final states of the transition [26b, 45]. 

The introduction in the early 1970s of the concept and the methodology of the Fermi-sea as the zero-order orbital set for the construction of the state-specific multiconfigurational wavefunction played on the themes... 

The sum of determinants in Eq. (8) can accommodate multiconfiguration wavefunctions, which are especially important in systems with a neardegeneracy features. Perhaps the simplest manifestation of this is in the... 

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