Molecular integration

Flelgaker T and Taylor P R 1995 Gaussian basis sets and molecular integrals Modem Electronic Structure Theory yo 2, ed D R Yarkony (Singapore World Scientific) section 5.4, pp 725-856... 

Dupuis M, Rys J and King H F 1976 Evaluation of molecular integrals over Gaussian basis functions J. Chem. Phys. 65 111-16... 

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... 

Although we include adsorption here following the chapter on mass transfer, we should be clear that it is a very specific process in its fullest fundamental meaning. Adsorption is the process by which molecules in the fluid phase in contact with a solid move to the solid surface and interact with it. Once at the solid surface these molecules may be reversible or irreversible adsorbed, that is, they may come back off the surface to the fluid phase with their full molecular integrity intact, or they may be so strongly bound that the rate of removal is for all purposes close enough to zero to be considered zero. 

To some other experts the meaning of the term ab initio is rather clear cut. Their response is that "ab initio" simply means that all atomic/molecular integrals are computed analytically, without recourse to empirical parametrization. They insist that it does not mean that the method is exact nor that the basis set contraction coefficients were obtained without recourse to parametrization. Yet others point out that even the integrals need not be evaluated exactly for a method to be called ab initio, given that, for instance, Gaussian employs several asymptotic and other cutoffs to approximate integral evaluation. 

There seems to be a certain complementarity between the degree of difficulty in evaluating HKL for various one-electron sets y)k and the order of the secular equation needed to obtain a certain accuracy in the result. The work carried out in getting extensive tables of molecular integrals has also been of essential value for facilitating the calculation of the matrix elements HKL. 

Cannes, L.Z., Martinez del Rio, C., Koch, P. (1998). Natural abundance variations in stable isotopes and their potential uses in animal physiological ecology. Comparative Biochemistry and Physiology - Part A Molecular Integrative Physiology, Vol. 119, No. 3, pp. 725-737. (http //dx.doi.org/10.1016/S1095-6433(98)01016-2)... 

Helgaker, T., Taylor, P. R., 1995, Gaussian Basis Sets and Molecular Integrals in Modem Electronic Structure Theory, Part II, Yarkony, D. R., (ed.), World Scientific, Singapore. 

Now any decision to break from these simple symmetry ideas must not be taken lightly since, as we shall see, this step automatically excludes the use of a whole class of semi-empirical methods for the evaluation of molecular integrals. 

The second of these four consequences has proved to be the most unfortunate. Even when a set of parameters has been consciously optimised within the MO model (and there can be no objection of principle to the conscious use of the MO framework as a numerical interpolation device), the temptation to improve on the MO results has proved irresistable. We can therefore And Cl and VB calculations using molecular integrals which have been constrained by the invariance requirement to be meaningful only in the MO framework. 

It means, for example, that atomic data can only rarely be used as a substitute for molecular integrals since the atom-in-molecule orbitals are not the same as the separate atom orbitals — worse, they are no longer equivalent among themselves. An atomic self-repulsion integral (0j0, 0j0j) is different if 0j is the lone-pair hybrid of NH3 or the bond-pair hybrid as the Gillespie-Nyholm rules suggest. 

It must be stressed that the use of GHOs in no way depends on this latter Gaussian expansion procedure since all molecular integrals reduce to standard STO forms. It has merely proved expedient to make use of the expansion method in calculations reported earlier. This ab initio technique provides the raw data with which to establish the patterns of behaviour of the GHOs. We can now address ourselves to one of our stated aims the development of approximation methods for a quantitative theory of valence. 

The gradients of the molecular integrals with respect to the nonlinear variational parameters (i.e., the exponential parameters Ak and the orbital centers Sk) were derived using the methods of matrix differential calculus as introduced by Kinghom [116]. It was shown there that the energy gradient with respect to all nonlinear variational parameters can be written as... 

B. O. Roos and P.E.M. Siegbahn The Direct Configuration Interaction Method from Molecular Integrals.,... 

B. O. Roos and P.E.M. Siegbahn The Direct Configuration Interaction Method from Molecular Integrals., inH. F. S. III(ed.) Methods of Electronic Stmcture Theory., Plenum, New York, p. 277 (1977). 

---