Spin-coupled Wavefunctions

Another approach is spin-coupled valence bond theory, which divides the electrons into two sets core electrons, which are described by doubly occupied orthogonal orbitals, and active electrons, which occupy singly occupied non-orthogonal orbitals. Both types of orbital are expressed in the usual way as a linear combination of basis functions. The overall wavefunction is completed by two spin fimctions one that describes the coupling of the spins of the core electrons and one that deals with the active electrons. The choice of spin function for these active electrons is a key component of the theory [Gerratt ef al. 1997]. One of the distinctive features of this theory is that a considerable amount of chemically significant electronic correlation is incorporated into the wavefunction, giving an accuracy comparable to CASSCF. An additional benefit is that the orbitals tend to be... 

T orbital for benzene obtained from spin-coupled valence bond theory. (Figure redrawn from Gerratt ], D L oer, P B Karadakov and M Raimondi 1997. Modem valence bond theory. Chemical Society Reviews 87 100.) figure also shows the two Kekule and three Dewar benzene forms which contribute to the overall wavefunction Kekuleform contributes approximately 40.5% and each Dewar form approximately 6.4%. 

Spin-coupled theory has been used to smdy the changes that occur in the electronic wavefunction as a system moves along the intrinsic reaction coordinate for the case of the conrotatory and disrotatory pathways in the electrocyclization of cyclobutene to c/x-butadiene. Against intuitive expectations, conrotatory opening of cyclobutenes was found to be promoted by pressure. Ab initio MO and density functional calculations have indicated that the ring opening of the cyclobutene... 

The CASVB strategy [1-9] uses a very efficient algorithm for the transformation of CASSCF [10] structure spaces, for the interpretation of CASSCF wavefunctions, and for the fully-variational optimization of VB wavefunctions. Important features for the quality of the final description include the unbiased optimization of both the VB orbitals and the mode of spin coupling, and also flexibility in the choice of the form of wavefunction. 

A natural starting point for modem valence bond applications lies in the optimization of the spin-coupled wavefunction [28], which consists of a single covalent configuration of N singly occupied orbitals ... 

Such a spin-coupled wavefunction is optimized with respect to the core wavefunction (if applicable), as well as to the nonorthogonal valence bond orbitals,... 

For the constmction of spin eigenfunctions see, for example. Ref. [36]. The spin-coupled wavefunction may be extended by adding further configurations, in which case we may speak of a multiconfigurational spin-coupled (MCSC) description. In the... 

We have outlined in the previous Section an efficient solution to the problem posed in (1). For the second point, one quickly realizes that the space 0 0 must take a full Cl (or, CASSCF) form, if no particular restrictions are to be placed on the valence bond orbitals. For the single-configuration spin-coupled wavefunction, there is for this reason an important link to N, N, A) CASSCF wavefunctions. 

Fig. 1. Key values of opemtion counts for the nonorthogonal treatment ofN-electron systems with S = 0. The operation countfor is appropriate to a spin-coupled wavefunction, while the other quantities relate...

One of the most useful types of constraint is the restriction of the spin coupling to just a single mode. Many molecular systems are described rather well by the perfect pairing mode of spin coupling, for example. A useful alternative, especially when this is not the case, is to base the structure coefficients on the CASSCF wavefunction in the VB orbital basis ... 

We focus in this Section on particular aspects relating to the direct interpretation of valence bond wavefunctions. Important features of a description in terms of modem valence bond concepts include the orbital shapes (including their overlap integrals) and estimates of the relative importance of the different stmctures (and modes of spin coupling) in the VB wavefunction. We address here the particular question of defining nonorthogonal weights, as well as certain aspects of spin correlation analysis. 

The combination of modem valence bond theory, in its spin-coupled (SC) form, and intrinsic reaction coordinate calculations utilizing a complete-active-space self-consistent field (CASSCF) wavefunction, is demonstrated to provide quantitative and yet very easy-to-visualize models for the electronic mechanisms of three gas-phase six-electron pericyclic reactions, namely the Diels-Alder reaction between butadiene and ethene, the 1,3-dipolar cycloaddition of fulminic acid to ethyne, and the disrotatory electrocyclic ringopening of cyclohexadiene. 

A complete set of spin eigenfunctions, e.g. oo i l = 1, 2,. .., 5) in the case of a six-electron singlet, can be constructed by means of one of several available algorithms. The most commonly used ones are those due to Kotani, Rumer and Serber [13]. Once the set of optimized values of the coefficients detining a spin-coupling pattern is available [see in EQ- (2)], it can be transformed easily [14] to a different spin basis, or to a modified set reflecting a change to the order in which the active orbitals appear in the SC wavefunction [see Eq. (1)]. 

Figure 6. Composition of the active space spin-coupling pattern / oo in Eq. (2)]from the SC wavefunction for the disrotatory ring-opening of cyclohexadiene along the CASSCF(6,6) IRC, expressed in terms of Chirgwin-Coulson weights Pq [seeEq. (4)] in the Rumer basis [seeEq. (3)].

For quantum chemistry, first-row transition metal complexes are perhaps the most difficult systems to treat. First, complex open-shell states and spin couplings are much more difficult to deal with than closed-shell main group compounds. Second, the Hartree—Fock method, which underlies all accurate treatments in wavefunction-based theories, is a very poor starting point and is plagued by multiple instabilities that all represent different chemical resonance structures. On the other hand, density functional theory (DFT) often provides reasonably good structures and energies at an affordable computational cost. Properties, in particular magnetic properties, derived from DFT are often of somewhat more limited accuracy but are still useful for the interpretation of experimental data. 

The purpose of this review is to discuss the main conclusions for the electronic structure of benzenoid aromatic molecules of an approach which is much more general than either MO theory or classical VB theory. In particular, we describe some of the clear theoretical evidence which shows that the n electrons in such molecules are described well in terms of localized, non-orthogonal, singly-occupied orbitals. The characteristic properties of molecules such as benzene arise from a profoundly quantum mechanical phenomenon, namely the mode of coupling of the spins of the n electrons. This simple picture is furnished by spin-coupled theory, which incorporates from the start the most significant effects of electron correlation, but which retains a simple, clear-cut visuality. The spin-coupled representation of these systems is, to all intents and purposes, unaltered by the inclusion of additional electron correlation into the wavefunction. 

We start with a description of the spin-coupled wavefunction for the general case in which electron correlation is included for all of the electrons. We return later to the question of a-n separation. In the spin-coupled approach to molecular electronic structure, an A-electron system is described by N orbitals, all of which are allowed to be distinct and non-orthogonal. One consequence of the non-orthogonality of these singly-occupied orbitals is that there is usually more than one way of coupling together the spins of the individual electrons so as to achieve the required overall... 

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