Orbital optimisation

We will describe the main features of our program and give examples of the use of the code for studying the aromaticity in various molecules. 

Alternatively, they may be fixed from the outset. To the structures, weights can be attributed, which add up to one, using a formula given by Chirgwin and Coulson [7]. 

The orbital optimisation is based on the Generalised Brillouin Theorem [8] as extended to non-orthogonal wavefunctions [9,10]  

The excitation operator does not have to adhere to the unitary condition, as is the case for orthogonal orbitals. Each Brillouin matrix element (Eq. (4)) represents the stationary condition for the mixing of orbitals iffj and iffj according to Wi - Vi + Vj- The wavefunction consisting of Vo and all singly excited states 

For orthogonal orbitals this procedure is often called the SuperCI method. 

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The interest of IlJ in the present context is that it provides a good test for the present orbital optimisation theory because one knows the exact solution. 

Thus we will use the result of calculations of the wave function of expanded in a gaussian basis to provide numerical tests of the qualitative discussion on the orbital optimisation theory presented in the above sections 2 and 3. 

The accuracy of the results obtained here using gaussian bases - and the usefulness of the numerical tests based on these results - can be seen from the values given in the Table 1. It is seen that the dissociation energy De obtained in the largest basis used here is excellent (error equal to 0.01 eV). On the other hand, the error on the value obtained using the minimum basis is as high as 1.35 eV (or 48% in relative value). This proves, if need be, the importance of the orbital optimisation studied in the present article. 

It is also useful to note that the major part (77%) of the effect of the orbital optimisation is obtained in the intermediate basis where no polarisation orbital is used. 

A special aspect of this description appears if one starts the orbital optimisation process with orbitals obtained by linear combinations ofRHF orbitals of the isolated atoms (LCAO approximation s.str.). Let Pn.opt and be the starting and final orbitals of such a calculation. Then the difference between c n.opi and Papt in the vicinity of each atom merely consists in a distortion of the atomic orbitals of each atom. This distortion just compensates the contribution of the orbitals of the other atoms to Pn.ctpt in order to restore the proportionality between the partial waves of ipopi and the appropriate atomic orbital. 

Clearly, several aspects of the orbital optimisation remain to be clarified. Firstly a numerical test using a system more complex than Ilj should be made. What happens to 7T orbitals or strongly hybridized orbitals should be also examined. It would be also interesting to explain how the optimisation - as described here - is related to an energy lowering, as well as the practical use of the present description in actual calculations, etc. .. These different aspects will be examined in forthcoming publications. 

The Ab Initio Valence Bond program TURTLE has been under development for about 12 years and is now becoming useful for the non-specialist computational chemist as is exemplified by its incorporation in the GAMESS-UK program. We describe here the principles of the matrix evaluation and orbital optimisation algorithms and the extensions required to use the Valence Bond wavefunctions in analytical (nuclear) gradient calculations. For the applications, the emphasis is on the selective use of restrictions on the orbitals in the Valence Bond wavefunctions, to investigate chemical concepts, in particular resonance in aromatic systems. 

The usual convergence acceleration/stabilisation tools may be employed in this orbital optimisation. For instance, we have implemented level shifting and DIIS [11]. 

Therefore, the dependence on the coefficients does not enter the gradient expression not for fixed orbitals, which is the classical Valence Bond approach and not for optimised orbitals, irrespective of whether they are completely optimised or if they are restricted to extend only over the atomic orbitals of one atom. If the wavefimction used in the orbital optimisation differs, additional work is required. This would apply to a multi-reference singles and doubles VB (cf. [20,21]). Then we would require a yet unimplemented coupled-VBSCF procedure. Note that the option to fix the orbitals is not available in orthogonal (MO) methods, due to the orthonormality restriction. 

There is still freedom in the choice of atomic orbitals used in Eq. (50). For instance, one can use fixed atomic orbitals, which eliminates the (sometimes costly) orbital optimisation. One can also use fully optimised, potentially delocalised orbitals in the spin-coupled / Coulson-Fischer sense. Finally, one can use real atomic orbitals by limiting each orbital to its own atom. This often gives a clearer physical picture of chemical bonding. It generates for instance optimal hybrids [9,10]. 

The combination of an approximate form for the hessian with a second-order perturbation expression for the energy results in an overall orbital optimisation strategy that scales extremely favourably both with the number of active electrons and with the number of basis functions. In this way, the current upper limit to the applicability of the method is fixed by the determination of the SC occupied orbitals, ° the number of active electrons that can be treated is therefore currently about 12-14. Obviously, the use of an alternative formalism for the reference configuration, e.g. SCF or GVB-SOPP, could allow the basic method to be extended to larger systems. 

The basis sets used in the CEPA calculations were large. For example with H2HF, the F atom basis was a (10s,6p) contracted to [6s,4p] augmented with extra diffuse s and p orbitals, and 3d and 4f orbitals optimised to produce the dipole and quadrupole polarisabilities of HF. The results reported here for ArHCN did not include f orbitals in the basis set, although the sensitivity of the results to these functions is currently being examined. The CEPA potential for ArOH was calculated by Esposti and Wemer. In all cases, basis set superposition errors were accounted for. 

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