Parameter-dependent basis functions

Systematic and theoretical studies of the dependence of bond parameters on basis functions and the optimization of the basis functions are neccessary. 

The expansion (50) of the time-dependent state vector in terms of time-independent basis functions becomes impractical for vibronic-coupling problems with more than about 7 vibrational modes, especially if one or several modes possess large coupling parameters or if the excess energy is very large. For such more demanding applications the MCTDH scheme, which is based on an expansion of the time-dependent state vector in terms of variationally determined time-dependent basis functions, has proven... 

The electronic energy W in the Bom-Oppenlieimer approxunation can be written as W= fV(q, p), where q is the vector of nuclear coordinates and the vector p contains the parameters of the electronic wavefimction. The latter are usually orbital coefficients, configuration amplitudes and occasionally nonlinear basis fiinction parameters, e.g., atomic orbital positions and exponents. The electronic coordinates have been integrated out and do not appear in W. Optimizing the electronic parameters leaves a function depending on the nuclear coordinates only, E = (q). We will assume that both W q, p) and (q) and their first derivatives are continuous fimctions of the variables q- and py... 

The gradient of the PES (force) can in principle be calculated by finite difference methods. This is, however, extremely inefficient, requiring many evaluations of the wave function. Gradient methods in quantum chemistiy are fortunately now very advanced, and analytic gradients are available for a wide variety of ab initio methods [123-127]. Note that if the wave function depends on a set of parameters X], for example, the expansion coefficients of the basis functions used to build the orbitals in molecular orbital (MO) theory. 

Parameters for elements (basis functions in ab initio methods usually derived from experimental data and empirical parameters in semi-empirical methods usually obtained from empirical data or ab initio calculations) are independent of the chemical environment. In contrast, parameters used in molecular mechanics methods often depend on the chemical environment. 

For bonded atoms, the off-diagonal terms (where i j) are taken to depend on tjje type and length of the bond joining the atoms on which the basis functions y- and Xj 0 centred. The entire integral is written as a constant, 0ij, which is not the same as the fixY in Hiickel 7r-electron theory. The are taken to be parameters, fixed by calibration against experiment. It is usual to set Pij to zero when the pair of atoms are not formally bonded. 

The wave function depends on the perturbation indirectly, via parameters in the wave function (C), and possibly also the basis functions (x). The wave function parameters may be MO coefficients (HF), state coefficients (Cl, MP, CC) or both (MCSCF). 

For this and other problems, the spread in the results depends on the initial size of the basis set. For small basis sets, the final branching ratios depend on the initial conditions—position and momentum parameters that define each basis function. Because in the TDB only 10 (out of 30) initial conditions were chosen independently, it is instructive to compare its results to the ones obtained when only 10 independent basis functions are used to represent the initial wavefunction. The purpose of this comparison is to examine the dependence of the branching ratios on the initial conditions and not to demonstrate an improvement in the results. Such an improvement is expected because the... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

The definition of the matrix in equation (60) requires some explanation The minus sign is motivated by the fact that H(x) is assumed to be an attractive potential. Division by Po is motivated by the fact that for Coulomb systems, when is so defined, it turns out to be independent of po, as we shall see below. The Sturmian secular equation (61) has several remarkable features In the first place, the kinetic energy has vanished Secondly, the roots are not energy values but values of the parameter po, which is related to the electronic energy of the system by equation (52). Finally, as we shall see below, the basis functions depend on pq, and therefore they are not known until solution... 

This rescaling reflects the idea that any increase of electronic charge at a center, as a consequence of an enrichment of the basis functions describing it, is unphysical if, lacking equipoise, atoms bonded to it suffer from poorer basis set descriptions. The parameter introduced in Eq. (8.4) is there to correct this imbalance if we follow Mayer s claim [172,173] that Mulliken s half-and-half partitioning of overlap terms between the concerned atoms should not be tampered with. It is felt that the way depends on the basis sets used for describing atoms k and I deserves attention as part of an effort aimed at letting X.rtsi approach Mulliken s limit A = 1 as closely as possible. 

In Fenske and Hall s parameter-free SCF calculations (80-84), the He1t 1-electron operator is substituted by a model 1-electron operator that has a kinetic energy and potential energy term for each atomic center in the complex. This approach assumes that the electron density may be assigned to appropriate centers. The partitioning of electron density is done through Mulliken population analyses (163) until self-consistency is obtained. The Hamiltonian elements are evaluated numerically, and the energies of the MO s depend only on the choice of basis functions and the intemuclear distance. 

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