Wave function multiconfiguration

The time dependence of the molecular wave function is carried by the wave function parameters, which assume the role of dynamical variables [19,20]. Therefore the choice of parameterization of the wave functions for electronic and nuclear degrees of freedom becomes important. Parameter sets that exhibit continuity and nonredundancy are sought and in this connection the theory of generalized coherent states has proven useful [21]. Typical parameters include molecular orbital coefficients, expansion coefficients of a multiconfigurational wave function, and average nuclear positions and momenta. We write... 

The methods used to describe the electronic structure of actinide compounds must, therefore, be relativistic and must also have the capability to describe complex electronic structures. Such methods will be described in the next section. The main characteristic of successful quantum calculations for such systems is the use of multiconfigurational wave functions that include relativistic effects. These methods have been applied for a large number of molecular systems containing transition metals or actinides, and we shall give several examples from recent studies of such systems. 

The main terms of the multiconfigurational wave function were found... 

The difficulties are mainly caused by two problems (1) the fact that even a qualitatively correct description of excited states often requires multiconfigurational wave functions and (2) that dynamic electron correlation effects in excited states are often significantly greater than in the electronic ground states of molecules, and may also vary greatly between different excited states. For explanations of the concepts invoked in this section, see Section 3.2.3 of Chapter 22 in this volume. Therefore, an accurate modeling of electronic spectra requires methods that account for both effects simultaneously. 

The major difficulty in wave function based calculations is that, starting from an independent-particle model, correlation between electrons of opposite spin must somehow be introduced into T. Inclusion of this type of electron correlation is essential if energies are to be computed with any degree of accuracy. How, through the use of multiconfigurational wave functions, correlation between electrons of opposite spin is incorporated into is the subject of Section 3.2.3. 

An analysis in terms of VB structures (see exercise 3) shows that this configurational mixing corresponds to approximately 40% diradical character in the wave function for ozone. The RHF wave function, on the other hand, contains only 12% of the diradical VB structure (the result was obtained using Hiickel values for the coefficients of the orbitals (2 11)). It is clear from these considerations that a correct treatment of the electronic structure for the ozone molecule must be based on a multiconfigurational wave function. 

It is clear that we need a multiconfigurational wave function in order to describe this process properly. 

Clearly a quantum chemical calculation of the energy surface for this reaction would have to be based on a multiconfigurational wave function, with four active orbitals, the k orbitals of the two ethene molecules, and four active electrons. However, a complication appears cyclobutane is a quadratic molecule with all four carbon-carbon bonds equal. Our wave function does not have this property. The CC bonds between atoms A and B are treated using... 

Similar iterative schemes were used to determine the MO s for multiconfigurational wave functions, in the early implementations. Fock-like operators were constructed and diagonalized iteratively. The convergence problems with these methods are, however, even more severe in the MCSCF case, and modem methods are not based on this approach. The electronic energy is instead considered to be a function of the variational parameters of the wave function - the Cl coefficients and the molecular orbital coefficients. Second order (or approximate second order) iterative methods are then used to find a stationary point on the energy surface. 

The standard prescription in this situation would be to apply the ab initio methods and to attempt to take into account missing correlations in their framework. This is, however, possible only for the systems of very modest size due to the M5 -=-M7 scalability of the correlated ab initio methods already mentioned. Using DFT methods in the situation when explicit correlations are necessary may be completely wrong, due to the structural deficiency of this class of methods, which precludes any treatment of nontrivial parts of correlations, requiring multiconfigurational wave function. 

Malcolm NOJ, McDouall JJW (1996) Combining multiconfigurational wave functions with density functional estimates of dynamic electron correlation, J Phys Chem, 100 10131-10134... 

We shall finish with a small example that illustrates the difference between the multiconfigurational wave function approach described in this chapter and the commonly used density functional (DFT) theory. It concerns the quadratic cyclobutadiene system, which is a transition state between the two equivalent rectangular forms of the molecule (cf. Fig. 5-3). 

Ab initio MO methods based on HF or small multiconfigurational wave-functions have been the method of choice, up to the present, for studies of organic systems and other molecules with light nuclei. The properties of stable species on the PES are often reproduced very well by calculations with just HF wavefunctions. Studies of reactions usually require the more sophisticated and expensive techniques, such as Cl or MP perturbation theory, that take into account the effects of the correlation between the electrons that is omitted from the HF approximation. The additional energy lowering computed with these methods with respect to that obtained with an HF calculation is called the correlation energy. A detailed and up-to-date discussion of the accuracy of state-of-the-art MO methods when applied to a variety of problems may be found in the book by Hehre et al. 

Shepard, R., Discussion of some multiconfiguration wave function optimization methods , 183rd ACS National Meeting, March 1982. 

The starting point for the introduction of the SOC and SSC interactions is a calculation of matrix elements over multiconfigurational wave functions... 

Since the numerical evaluation of these coupling terms on the basis of multiconfigurational wave functions is very demanding and is difficult to apply to polyatomic molecules the kinetic couplings have been approximate to asymmetric Lorentzian functions [104]... 

A similar effect can be found in atoms where the ns shell is filled but the np shell is empty or only partially filled. The symmetry then allows the multiconfigurational wave function ... 

We could of course have given many more examples where it is necessary to use a multiconfigurational approach to the wave function. A few more will be given at the end of the chapter, but we now turn to the question of how to construct and evaluate multiconfigurational wave functions. 

As long as a satisfactory multireference coupled-cluster theory is missing, there are various options for states that need a zeroth-order multiconfiguration wave function. One possibility is to start from an MC-SCF calculation and to improve this by selected Cl. Since the MC-SCF part is basically extensive, while the Cl part is not, and since one can hardly go beyond external double excitations, one tends to include as many configurations in the MC-SCF part as possible. However, MC-SCF is usually of CAS (complete active space) [154] type, e.g. like full Cl, which restricts the possible size of the active space. Such multireference Cl scheme have been very popular for describing excited states, reaction barriers, dissociation processes etc. 

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