Spin-Coupled calculations

The inclusion of the correlation corrections to the spin-spin coupling calculation via electron propagator is quite straightforward since at third order it can be written as... 

We have performed Spin-Coupled calculations on a series of selected carbonium ions (55). The Spin-Coupled calculations allow the study of chemical structure of die molecule, since chemical structure and connectivity are central features of VB theory. Spin-Coupled calculations for CHS+ in CSI show the system as bonded by the intuitively proposed 3c2e bond, which connects the carbon atom to two hydrogens, and three ordinary 2c2e bonds between the carbon and the other hydrogens, commonly called the tripod (Figure 3). 

Higher protonated species, such as CHe2 and CH7+3 have also been proposed (63). Spin-Coupled calculations on CH6+2 confirms the model involving 3c2e bonds (Figure 7) (53). However, these orbitals show that there are two such 3c2e bonds. 

A detailed comparison of spectral data concluded that borazine has a delocalized 7T-electron system like that of benzene (112) other workers, however, have concluded from spin-coupled calculations that borazine has litde aromatic stabilization as compared to benzene (113). 

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... 

For a system with N electrons, a spin-coupled calculation followed by a full-valence VB calculation produces a total wavefunction which is formally equivalent to a CASSCF description with all allowed distribution of N electrons in N orthogonal orbitals. Robb and co-workers [15] have shown how in principle one might transform between these two descriptions. The interpretation of the CASSCF wavefunction for benzene is not straightforward, as many of the 175 configurations make significant... 

We found that these more sophisticated spin-coupled calculations, which used larger basis sets with polarization functions on all of the atoms and which allowed the a orbitals to relax, produced a picture of bonding in the 7t-electron system of benzene which is practically identical to that described earlier. As before, we found six equivalent spin-coupled orbitals which are transformed into one another by successive C6 rotations. The overlaps between the orbitals, ordered cpa to cp6 around the ring, are reported in Table 1. In this case, the electron correlation effects incorporated in the spin-coupled model provide an energy improvement over the SCF description of 170 kJ mol - with a further lowering of 20 kJ mol -1 on including spin-coupled ionic structures. 

Spin-coupled calculations have been performed for various azobenzenes, in a manner analogous to our first series of calculations on benzene. Bond lengths and angles were taken from the volumes edited by Katritzky and Rees [17], but where we were unable to find precise C — H bond lengths we took values from related molecules. 

Spin-coupled calculations, analogous to those for the organic heterocycles, have been carried out for both molecules. Three of the spin-coupled n orbitals for borazine are related to each other by successive C3 rotations and they take the form of fairly localized, slightly distorted N(2p) functions. The other three orbitals are also transformed into one another by successive C3 operations. These are also based on N(2p) functions but clearly show significant delocalization onto neighboring boron centres. Nonetheless, most of the electron density remains on N. 

Given the concerted, almost synchronous nature of this gas-phase reaction it might seem reasonable to suppose that the electronic mechanism would resemble those for the Diels-Alder and cyclohexadiene ring-opening reactions, described above. However, our spin-coupled calculations along the IRC reveal a somewhat different picture [3]. 

An important outcome of all these spin-coupled calculations is the consistency of the descriptions. In particular, a simple and highly-visual model emerges for the behaviour of correlated n electrons in all of the aromatic molecules that we have studied. These 7t-electron systems are well described in terms of fairly localized, nonorthogonal, singly-occupied orbitals. The special stability of such systems arises in the spin-coupled model from a profoundly quantum mechanical... 

Spin-coupled calculations at the idealized Dm geometry of cyclooctatetraene reveal a description dominated by triplet coupling of pairs of electrons [12], as anticipated earlier. Expressing the total spin function in the Serber basis [29], we find that the mode made up only of triplet-coupled pairs is responsible for 75% of the total. We find that the n orbitals for this antiaromatic system (see Figure 9) adopt localized forms that resemble closely those shown in Figure 4 for benzene, rather than the antipair representation shown for cyclobutadiene in Figure 6. 

In the subsequent spin-coupled calculations [9], the < > were optimized as linear combinations of all the LMOs which correspond to X—Y bonds, plus all the virtual orbitals. This scheme is entirely equivalent to expanding the in the full basis atomic basis set, except that it maintains the orthogonality between the active orbitals and the inactive space, which consists of all the other doubly-occupied MOs. The spin function 0 was fully optimized in the full spin space. The active orbitals were thus fully optimized without constraints on their form, on the degree of localization, on the overlaps between them, or on the mode of coupling the electron spins. Nevertheless, we found for each molecule that the optimized spin-coupled orbitals consist of pairs, each clearly associated with a particular two-centre bond, and with predominantly singlet coupling of the electron spins. For example, we show in Figure 2 the pair of spin-coupled... 

Being firmly based in modem valence bond theory, the very compact descriptions of the bonding which have emerged from our many spin-coupled calculations facilitate a direct and, hopefully, very convincing description of... 

Spin-coupled calculations revealed crossing seams between potential energy surfaces arising from the I + 02(a) and I + 02 X) asymptotes. In a fully adiabatic treatment of this problem the surface crossings were avoided and somewhat difficult to locate. To facilitate interpretation of the adiabatic results, diabatic calculations were preformed with coupling between states from different asymptotes suppressed. Surface intersections were allowed and readily located using the diabatic treatment. 

Some of the most efficient algorithms currently available for spin-coupled calculations involve the expansion of the SC wavefunction in terms of Slater determinants, leading to a summation over (N-p) -dimensional cofactors for the p-particle density matrix [23]. For calculations involving the full spin space, a further saving can be achieved by the use of projected spin functions [24], An alternative strategy is provided by CASVB (see Sect. 5). 

This equation is most easily satisfied if the matrix U (P ) is diagonal. The way that this is achieved is best illustrated by an example, and we consider a spin-coupled calculation of the lowest-lying potential energy surfaces of CH In particular, we consider the perpendicular approach of C ( P) to Hj so that the symmetry point group is C2V. The coordinate system is shown in Fig. 3. We ignore the two core electrons of C" and consequently the spin-coupled wavefunctions are of the form... 

The inclusion of closed-shell configurations in spin-coupled calculations is most easily accomplished by carrying out a self-consistent field (SCF) (or small multiconfiguration SCF (MCSCF)) step first. The integrals over the basis functions are transformed to the molecular-orbital representation and input to the spin-coupled program. The use of MOs as basis functions also shows directly which of them are significant in the spin-coupled wavefunction. 

In Fig. 8a the spin-coupled and final spin-coupled VB potentials are compared for the ground state. Fig. 8b shows the associated spin-coupling coefficients. It can be seen that the spin-coupled calculation by itself yields a potential energy curve with all the essential features of the final result. The spin-coupling coefficients display much of the essential chemistry of bond formation in BH. At large distances, coefficients are C2 I and 0, which shows that the two 2s-like orbitals stemming from B are coupled to a... 

The five spin functions used in the spin-coupled calculations are the... 

Fig. 2a-h. Contour plots of (l>s p), taken from spin-coupled calculations on BH (X Z ). The nuclear separations considered here are a 2.3501 bohr b 3.0 bohr c 3.3 bohr d 3.5 bohr e 3.75 bohr f 4.0 bohr g 5.5 bohr h 10.0 bohr. Each p-space electron density has cylindrical symmetry (about the Pj-axis)... 

At convergence of a single-configuration spin-coupled calculation, the active orbitals equations based on (A-l)-electron operators from which the particular occupied orbital has been excluded (9) ... 

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