Nuclear generator coordinate method

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. 

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. 

In addition to the analysis of the topology of a conical intersection, the quadratic expansion of the Hamiltonian matrix can be used as a new practical method to generate a subspace of active coordinates for quantum dynamics calculations. The cost of quantum dynamics simulations grows quickly with the number of nuclear degrees of freedom, and quantum dynamics simulations are often performed within a subspace of active coordinates (see, e.g., [46-50]). In this section we describe a method which enables the a priori selection of these important coordinates for a photochemical reaction. Directions that reduce the adiabatic energy difference are expected to lead faster to the conical intersection seam and will be called photoactive modes . The efficiency of quantum dynamics run in the subspace of these reduced coordinates will be illustrated with the photochemistry of benzene [31,51-53]. 

Dynamics can also be used to model the mechanics and rates of reactions at a fundamental level. While a complete potential energy surface is the ideal starting point of any type of accurate dynamical computation, it can be obtained for only very simple systems. Ab initio MD can be performed by repeated calculation of forces, generation of a new geometry, and convergence of the MO coefficients for the new geometry. This iterative scheme, however, is very time consuming because accurate MO coefficients are required at each point of the simulation. The Car-Parrinello method does not optimize the electronic (i.e., MO coefficients) and nuclear coordinates separately, but simultaneously, because it could be shown that the errors in the nuclear forces and in the electronic forces cancel out. 

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