Wavefunction, spin-coupled

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...

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... 

For the benzenoid aromatic molecules considered here, we wish to include electron correlation for the N electrons of the 7t-eleetron system but not for the 2n a electrons. This is achieved most readily if we first carry out a standard Hartree-Fock MO theory calculation, yielding a set of orthogonal self-consistent field (SCF) molecular orbitals tp(. The spin-coupled wavefunction may then be written... 

Spin-coupled calculations were then performed for the six n electrons, with the thirty-six electrons of the ct framework accommodated in doubly-occupied MOs. The form of the spin-coupled wavefunction... 

The spin-coupled wavefunction provides an improvement over the SCF energy of 199 kJ mol-1 (0.0758 hartree), which is considerable. This arises from the effects of electron correlation in the jr-electron system. The- distortion of the spin-coupled orbitals for benzene occurs because of the small amount of ionic character needed to describe the C — C n bonds. Ionic structures in spin-coupled theory are those in which one or more of the orbitals is allowed to be doubly occupied. Allowing for the different numbers of allowed spin functions, and including the spin-coupled configuration, a total of 175 VB structures can be generated in this way. A... 

The difference in energy between one of the Kekule-type structures (RJ and the full spin-coupled wavefunction, with all five Rumer structures, can reasonably be termed the resonance energy. In this way, we obtain a value of 88 kJ mol-1. If we allow also for the spin-coupled ionic structures, this value is increased to 106 kJ mol"l. 

As described in other Chapters in this book, the single-configuration spin-coupled wavefunction takes the form [5]... 

The ab initio spin-coupled wavefunction for an TV-electron system is based on a single product of singly-occupied orbitals, In general, these orbitals are... 

Just as in correlated descriptions based on MO theory, it is neither practical nor desirable to include electron correlation for all of the electrons in large systems. In common with other strategies, the orbital space is partitioned into inactive , active and unoccupied or virtual subspaces. Electron correlation is incorporated only for the active space, which corresponds to that part of the electronic structure which interests us most. A convenient representation of a general spin-coupled wavefunction takes the form ... 

The spin-coupled-like component of the CASSCF wavefunction, the SC(f.c.) wavefunction and the fully-variational spin-coupled wavefunction, which are all exceedingly similar to one another, vindicate the familiar description of benzene in terms of two Kekule and three para-bonded structures. A crucial difference,... 

This is not quite the end of this particular story. In the very special case of electron spins which are strictly triplet coupled, the spin-coupled wavefunction is not altered by taking the sums and differences of the corresponding orbitals (a+c), (a-c), (b+d) and (b-d), neglecting normalization. The term antipairs has been coined for such an alternative mode of description. It turns out that this orbital representation already corresponds to the required B1/t symmetry at the square-planar geometry, so that there are now more free parameters in the spin space the relevant spins need no longer be exactly triplet coupled. We find that... 

The spin-coupled wavefunctions for the systems considered here take the form... 

The spin-coupled wavefunction is based on a single configuration of N singly occupied nonorthogonal orbitals ... 

The U matrices play an important role in the theory of spin-coupled wavefunctions as will be seen. These matrices are always real, and since they are group representation matrices, they are also orthogonal ... 

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