Orbitals optimisation

The aim of the present article is to present a qualitative deseription of the optimised orbitals of molecular systems i.e. of the orbitals resulting from SCF calculations or from MCSCF calculations involving a valence Cl we do not present here a new formal development (although some formalism is necessary), nor a new computational method, nor an actual calculation of an observable quantity. .. but merely the description of the orbitals. 

In the case of a valence MCSCF calculation the difference between the optimised orbitals and these atomic RHF orbitals simply represents the way in which the atoms are distorted by the molecular environment. Thus, this difference is closely related to the idea of atoms in molecules (1). However, here, the atoms are represented only at the RHF level, and the difference concerns only the orbitals, not the intra- atomic correlation. 

The starting step of the present work is a specific analysis of the solution of the Schrodinger equation for atoms (section 1). The successive steps for the application of this analysis to molecules are presented in the section 2 (description of the optimised orbitals near of the nuclei), 3 (description of the orbitals outside the molecule), and 4 (numerical test in the case of H ). The study of other molecules will be presented elsewhere. 

The Valley theorem leads to simple conditions for the optimised orbitals near the nuclei. However these conditions are not sufficient to characterize these orbitals one needs in addition to take the asymptotic form of the equations into account. 

We first consider what happens when comparing directly the optimum orbital of H2, the un-optimised orbital of (i.e. the sum of the two Is orbitals of the H atoms) and the orbitals of the H atom itself. The comparison between the values of these orbitals along the bond axis is presented on the fig. (7). 

It is seen that in the inner region (positive values of the abscissae), the atomic orbital is close neither to the optimal orbital nore to the un-optimised orbital. On the contrary, the atomic orbital is very close of the un-optimised orbital but not of the optimised one in the outer region (negative values of the abscissae). The inverse con-... 

In the case of the s wave (/ = 0) of the optimised orbital the effective energy defined in the section 2 is given here by Ce// = e- -l/R = -0.602H (R is the internuclear distance). According to the analysis of that section it is seen on the fig.(9) that the s wave of the optimum orbital obtained in the gaussian basis is actually very close to the numerical regular atomic s orbital with 6 // = -0.602 H while the s wave of the un-optimised orbital is significantly different from these two functions. 

We present in the Table 2 the ratio of the irregular solution of the hydrogen-like system with the s wave of the optimised orbital, and with the s wave of the unoptimised orbital. It is seen that the irregular numerical solution is actually much closer to be proportional to the s wave of the optimised orbital than to that of the unoptimised orbital. 

In fact, the ratio between the numerical and the optimised orbital is nearly constant (relative variation smaller than 11%) for 2< r <6 B, while the ratio with the s wave of the un-optimised orbital is multiplied by ca. 5 when r increases from 2 B to 6 B ( r=distance to the midpoint of the two nuclei). The decrease of the ratio at larger... 

In the two preceding sections (4.1 and 4.2) we have presented numerical test of the following description (resulting from the analysis of the sections 2 and 3) of the optimised orbital of ... 

We examine now a numerical test of the reciprocal of this description if a function satisfies this description, then it is the optimised orbital of. If both the deseription and the reciprocal are true we can conclude that the description is complete. 

The results of these calculations are summarised in Figs. 1,2 and 3. Fig. 1 has plots of the total energy for the three wave functions (together with the results of Kolos and Roothaan (8) for reference). The optimised orbital exponents are plotted in Fig. 2. In Fig. 3 the orbital exponents for case (iii) for the short intemuclear distance region R = 0 to R = 0.5 a.u. 

Fig. 2. The Hydrogen Molecule optimised orbital exponents for the functions of Fig. 1...

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. 

The first calculations on benzene using optimised orbitals were done by Cooper et al. [52], using their spin-coupled VB method [20]. A review [53] has appeared with an overview of their work on aromatic and anti-aromatic compounds. 

For instance it allows the complete optimisation, orbitals and geometry, of benzene (D6h symmetry), which is described by two resonating structures and of the fictional molecule cyclohexatriene (D3h symmetry), whose wave function consist of just one of the structures. A comparison of the results gives a better insight in the nature and the persistence of resonance. 

With the first of these forms, a single exchange operator, we may write down the unique equation which the optimising orbitals must satisfy in order that there be a turning point in the Hartree-Fock energy functional eqn ( 2.1) ... 

Of course, there are still many identical determinants, related to the determinant of a set of orthogonal Xi by unitary (orthonormality- preserving) transformations for the moment we will simply live with this. It will be shown later that this freedom of choice wiihtn orthonormal sets can be used to both simplify the resulting equations and define a unique set of optimising orbitals. 

---