Wavefunction Optimisation

The SCF-MI algorithm, recently extended to compute analytic gradients and second derivatives [18,41], furnishes the Hartree Fock wavefunction for the interacting molecules and also provides automatic geometry optimisation and vibrational analysis in the harmonic approximation for the supersystems. The Ml strategy has been implemented into GAMESS-US package [42]. 

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. 

In the development of the TURTLE program [3], we started by considering a multi-structure Valence Bond wavefunction and added the capability to optimise the orbitals. We tried to avoid putting restrictions on the way the wavefunction is built and to allow great flexibility in the choice of orbitals. For... 

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

When the wavefunction is expanded, using expansion parameters c, this theorem still holds if dE/dc=0, or when dc/dx=0. The first is the case for completely optimised wavefunctions and the second for wavefunctions where some, or all, of the coefficients are frozen. This can be seen when we write the derivative of E with respect to x as a sum of two terms ... 

The 7t-system is described by all five Rumer structures, which is the complete spin-space (i.e. Fig. 3 and Fig. 4). This allows a smooth transition from benzene, where the 2 Kekule structures are most important, to the highly bent Dewar benzene, where only one of the Dewar structures (Fig 4) is important. All the orbitals, doubly occupied and singly occupied are fully optimised. For each bent structure, the orbitals from the preceding less bent structure were used as initial guess. This and the choice of wavefunction ensure that an aromatic 7i-system can be identified, even when no symmetry separation exists. All orbitals were completely optimised so we have a wavefunction of the spin-coupled type. This is the type of wavefunction used by Cooper et al. [52] in their study of benzene. 

To obtain geometries, 10-orbital 10-electron complete active space self-consistent field (CASSCF) [82-84] calculations were performed with the GAMESS-UK program [6], The occupied orbital order in an SCF for flat benzene is n,2c,2n. In the bent molecule, there is no clear distinction between a- and tt-orbitals and we want to include all the tt-orbitals in the CAS-space. Thus, 10 orbitals in the active space are required. Obviously, the 5 structure VB wavefunction would have been a preferable choice to use in the geometry optimisation. However, at that time, the VB gradients were not yet available. The energies of the VBSCF at the CASSCF geometries followed the CASSCF curve closely. 

In the localised VB the orbitals are only allowed to extend over part of the molecule. For instance, for an enolate anion, the basic wavefunction would have a doubly occupied 7i-orbital, localised only on the oxygen atom and a doubly occupied 7t-orbital extending over both carbon atoms. Thus, while describing structure 2a in Fig. 14, a single determinant is still employed. Since the orbitals are completely optimised, the a-system can partly counteract the charge separation. 

We have given an account of some of the inner workings of the gradient VBSCF program TURTLE. The program is especially conceived to allow the optimisation of wavefunctions of arbitrary form. This feature is exploited in the study of resonance and delocalisation phenomena. 

J.H. van Lenthe and G.G. Balint-Kurti, VBSCF The optimisation of non-orthogonal orbitals in a general (Valence Bond) wavefunction, in 5th seminar on Computational Methods in Quantum Chemistry (Groningen, 1981). 

It follows that, if there are na and nb SCF-MI active orbitals on fragments A and B, we obtain a total of na nb optimised virtual orbitals. The spin space is described by the spin wavefunction... 

The whole procedure can be repeated n times generating - for each pair of occupied orbitals - n optimised virtual orbital pairs, whose contribution to the interaction energy is strictly decreasing up to saturation of the space, i.e. up to the full use of the SCF-MI virtual orbital space. Consistent with the employed basis set, the final MO-VB wavefunction (16) can be so improved to the desired degree of accuracy. 

The generalised multi-configuration SC (GMCSC) and optimised basis set-GMCSC (OBS-GMCSC) methods developed by Penotti can be viewed as multi-configuration extensions of the SC approach. The GMCSC wavefunction can... 

Non-variational wavefunctions, in which there is no optimisation of parameters, also exist. One approach is Moller-Plesset (MP) perturbation theory which, like Cl methods, is employed most often to improve upon a previously determined HF wavefunction. Fuller details of all these approaches may be found in the monograph by Szabo and Ostlund [40]. 

For the construction of spin eigenfunctions see, for example, Ref. [22], There are obviously many parallels to the multiconfiguration self-consistent field (MCSCF) methods of MO theory, such as the restriction to a relatively small active space describing the chemically most interesting features of the electronic structure. The core wavefunction for the inactive electrons, 4>core, may be taken from prior SCF or complete active space self-consistent field (CASSCF) calculations, or may be optimised simultaneously with the and cat. 

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