Spin-Coupled Approach

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

---

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... 

As stated in the Introduction, the GMCSC method [1][2][25] is, in its roots, a multiconfiguration generalization of the well known spin-coupled approach [3]-[6]. 

Straightforward applications of the spin-coupled approach confirm the relative energies and the preferred geometries of cyclobutadiene in these singlet and triplet states [11], without the requirement for multiconfigurational descriptions,... 

It is now easy to show that in the long range limit, where the energy curves for the (25>i -I- 1)(25b + 1) alternative spin couplings approach indefinitely closely, the ensemble-averaged coupling density derived from (41) becomes everywhere zero. The spatial density factors in (41) are in fact quite independent of coupling scheme and... 

When interpreting W in AT CASSCF wave functions, our usual choice of Fvb is motivated by the well-documented success of the spin-coupled approach, as exemplified by the various applications reviewed in Refs. [11,12]. Such a wave function takes the form... 

Taking into account that the wave function of the product is (even within spin-coupled approach) still satisfactorily approximated by one-determinant function... 

Function Qs.m an (V-electron spin function for the N active electrons. The subscripts indicate that the net spin of these electrons is S with z-component M. A characteristic feature of the spin-coupled approach now appears. Since the N valence orbitals are singly occupied, there are several distinct ways of coupling the individual spins of the electrons to each other in order to form the required overall resultant spin S. This number is denoted by and is given by ... 

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... 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

Hricovini, M., Malkina, O. L., Bizik, F., Nagy, L. T., Malkin, V. G., 1997, Calculation of NMR Chemical Shifts and Spin-Spin Coupling Constants in the Monosaccharide Methyl [1 ]) xylopyTniiosidc Using a Density Functional Theory Approach , J. Phys. Chem. A, 101, 9756. 

The everlasting need for better experimental tools and theoretical methods to explore the field of carbocation chemistry had also a significant influence on the further development of ab initio methods for the calculation of NMR shielding and indirect spin-spin coupling constants. Carbocation chemistry thus serves as a forerunner for a close integration of experimental and computational approaches in all areas of chemistry. 

---