The generator coordinate method

Wheeler and collaborators [3], in the context of nuclear physics, showed at that time that the limit in the variational procedure potential itself was not reached. Indeed, the Rayleigh-Ritz (RR) variational scheme teaches us how to obtain the best value for a parameter in a trial function, i.e., exponents of Slater (STO) or Gaussian (GTO) type orbital, Roothaan or linear combination of atomic orbitals (LCAO) expansion coefficients and Cl coefficients. Instead, the generator coordinate method (GCM) introduces the Hill-Wheeler (HW) equation, an integral transform algorithm capable, in principle, to find the best functional form for a given trial function. We present the GCM and the HW equation in Section 2. 

On the other hand, as early as 1968, Somorjai [4] had introduced an integral transform method, closely related to GCM, for atomic and molecular systems. An extensive review of this method may be found in Ref. [5]. In this context, accurate correlated functions for two- and three-electron atomic systems were obtained by Thakkar and Smith [6]. Nonetheless, Somorjai used the integration limits of the integral transform as variational parameters. While this was a very innovative option, to some extent it masked the full potential of the HW equation. 

It is also worth mentioning that in 1976 a self-consistent GCM was introduced [7] as an improvement over the harmonic approximation in quantum lattice dynamics applications show an interesting parallelism with the self-consistent random phase approximation. 

No doubt, the price to pay for such a powerful tool is high. In Section 3 we show the nontrivial analjrtical work that was needed to extract from the HW equation the exact Is function and energy for the hydrogen atom from the deliberately wrong Gaussian trial function [8]. 

not surprisingly, applications to nontrivial cases had to rely on either approximations (in nuclear physics, preference was given to the Gaussian overlap approximation [9]) or numerical solutions (see Section 5). 

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Van Leuven, P., and Lathouwers, L. (1980), The Generator Coordinate Method in Molecular Physics, in Quantum Dynamics of Molecules, R. G. Woolley (Ed.), Plenum, New York. 

We present a new mapping procedure from fermion onto boson spaces. The procedure is based on the Generator Coordinate Method. We show an application of this technique to systems of nucleons moving in a single j-orbit. The extension of this microscopic investigation to many j-orbit systems is also examined. 

The procedure we are going to expose is based on the Generator Coordinate Method (GCM). The GCM looks for solutions of the Schrodinger equation of the form... 

Mito, Y. and Kamimura, M. (1976). The generator coordinate method for composite-particle scattering based on the Kohn-Hulth6n variational principle,... 

It has also been pointed out that there is an interpretation to our theory based upon the generator coordinate approach [44 6]. A close scrutiny of Lathouwers et al. paper shows that the present approach can be embedded into the methodology of the generator coordinate method (GCM), as it was pointed out by one referee. The postulate PI ensures the existence of a set of fixed nuclei so that the GCM, initially thought for... 

The Generator Coordinate Method for Atomic and Molecular Systems Revision and Further Developments... 

The generator coordinate method (GCM), as initially formulated in nuclear physics, is briefly described. Emphasis is then given to mathematical aspects and applications to atomic systems. The hydrogen atom Schrodinger equation with a Gaussian trial function is used as a model for former and new analytical, formal and numerical derivations. The discretization technique for the solution of the Hill-Wheeler equation is presented and the generator coordinate Hartree-Fock method and its applications for atoms, molecules, natural orbitals and universal basis sets are reviewed. A connection between the GCM and density functional theory is commented and some initial applications are presented. 

Osvaldo and I certainly became friends. We later met many times, in Uppsala, Norway, Florida (where 1 had also the privilege to meet Professor Slater 1 believe he and a few others - Hartree, Fock, Bom, Oppenheimer - set the foundations in the 1930s of what we now call quanmm chemistry). I remember specially when, in Rio de Janeiro, during the 6th Escola Latinoamericana de Quimica Teorica in 1988, Osvaldo had very stimulating words on our first results with the generator coordinate method, which I review below. 

Reaction coordinate. Another technique for generating an initial guess is to use the reaction coordinate method (also called coordinate driving ). A parameter is chosen that, when varied, carries the reactant into the product. The parameter may be a bond distance or bond angle, or may involve a parameter including a dummy atom. 

Molecular level computer simulations based on molecular dynamics and Monte Carlo methods have become widely used techniques in the study and modeling of aqueous systems. These simulations of water involve a few hundred to a few thousand water molecules at liquid density. Because one can form statistical mechanical averages with arbitrary precision from the generated coordinates, it is possible to calculate an exact answer. The value of a given simulation depends on the potential functions contained in the Hamiltonian for the model. The potential describing the interaction between water molecules is thus an essential component of all molecular level models of aqueous systems. 

Jorge FE, de Castro EVR, Da SUva ABF. A universal Gaussian basis set for atoms cerium through lawrencium generated with the generator coordinate Hartree-Fock method. J Comp Chem. 1997 18(13) 1565-9. 

A descriptor for the 3D arrangement of atoms in a molceulc can be derived in a similar manner. The Cartesian coordinates of the atoms in a molecule can be calculated by semi-empirical quantum mechanical or molecular mechanics (force field) methods, For larger data sets, fast 3D structure generators are available that combine data- and rule-driven methods to calculate Cartesian coordinates from the connection table of a molecule (e.g., CORINA [10]). 

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