Correlated gaussians

NON-BORN-OPPENHEIMER VARIATIONAL CALCULATIONS OE ATOMS AND MOLECULES WITH EXPLICITLY CORRELATED GAUSSIAN BASIS EUNCTIONS... 

Born-Oppenheimer Integrals over Correlated Gaussians... 

VII. The Use of Shifted Gaussians in Non-BO Calculations on Polyatomic Molecules A. Bom-Oppenheimer Calculations in a Basis of Explicitly Correlated Gaussians... 

The symmetry requirements and the need to very effectively describe the correlation effects have been the main motivations that have turned our attention to explicitly correlated Gaussian functions as the choice for the basis set in the atomic and molecular non-BO calculations. These functions have been used previously in Born-Oppenheimer calculations to describe the electron correlation in molecular systems using the perturbation theory approach [35 2], While in those calculations, Gaussian pair functions (geminals), each dependent only on a single interelectron distance in the exponential factor, exp( pr ), were used, in the non-BO calculations each basis function needs to depend on distances between aU pairs of particles forming the system. 

In our atomic calculations, the -type explicitly correlated Gaussian functions have the following form ... 

The above function is a one-center correlated Gaussian with exponential coefficients forming the symmetric matrix A]. <1) are rotationally invariant functions as required by the symmetry of the problem—that is, invariant with respect to any orthogonal transformation. To show the invariance, let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3-space) that is applied to rotate the r vector in the 3-D space. Prove the invariance ... 

The w-particle one-center correlated Gaussians, (j), can also be expressed in the more conventional form used in the electronic structure calculations as... 

In this form, the w-particle correlated Gaussian is a product of n orbital Gaussians centered at the origin of the coordinate system and n n — l)/2 Gaussian pair functions (geminals). 

To describe bound stationary states of the system, the cji s have to be square-normalizable functions. The square-integrability of these functions may be achieved using the following general form of an n-particle correlated Gaussian with the negative exponential of a positive definite quadratic form in 3n variables ... 

Electron Affinity of Hydrogen, Deuterium, and Tritium Atoms Obtained with 300 Explicitly Correlated Gaussian Functions ... 

The general form of an n-pseudoparticle correlated Gaussian function is given by... 

Functions (46) have been succesfully used in numerous quantum-mechanical variational calculations of atomic and exotic systems where there is, at most, one particle (nuclei), which is substantially heavier than other constituents. However, as is well known, simple correlated Gaussian functions centered at the origin cannot provide a satisfactory convergence rate for nearly adiabatic systems, such as molecules, containing two or more heavy particles. In the diatomic case, which we we will mainly be concerned with in this section, one may introduce in basis functions (46) additional factors of powers of the intemuclear distance. Such factors shift the peaks of Gaussians to some distance from the origin. This allows us to adequately describe the localization of nuclei around their equilibrium position. 

The evaluation of matrix elements for exphcitly correlated Gaussians (46) and (49) can be done in a very elegant and relatively simple way using matrix differential calculus. A systematic description of this very powerful mathematical tool is given in the book by Magnus and Neudecker [105]. The use of matrix differential calculus allows one to obtain compact expressions for matrix elements in the matrix form, which is very suitable for numerical computations [116,118] and perhaps facilitates a new theoretical insight. The present section is written in the spirit of Refs. 116 and 118, following most of the notation conventions therein. Thus, the reader can look for information about some basic ideas presented in these references if needed. 

It is well known that the convergence of variational expansions in terms of correlated Gaussians, both the simple ones and those with premultipliers, strongly depends on how one selects the nonlinear parameters in the Gaussian... 

It can be shown that the basis of spherical explicitly correlated Gaussian functions with floating centers (FSECG) form a complete set. These functions... 

Here we will present the formulae needed for calculating the reduced one-particle density matrices from the floating correlated Gaussians used in this work. The first-order density matrix for wave function T (ri,r2,..., r ) for particle 1 is defined as... 

The results for the non-BO diagonal polarizability are shown in Table XIII. Our best—and, as it seems, well-converged—value of a, 29.57 a.u., calculated with a 244-term wave function, is slightly larger than the previously obtained corrected electronic values, 28.93 and 28.26 a.u. [88,91]. It is believed that the non-BO correction to the polarizability will be positive and on the order of less than 1 a.u. [92], but it is not possible to say if the difference between the value obtained in this work and the previous values for polarizability are due to this effect or to other effects, such as the basis set incompleteness in the BO calculations. An effective way of testing this would be to perform BO calculations of the electronic and vibrational components of polarizability using an extended, well-optimized set of explicitly correlated Gaussian functions. This type of calculation is outside of our current research interests and is quite expensive. It may become a possibility in the future. As such, we would like the polarizability value of 29.57 a.u. obtained in this work to serve as a standard for non-BO polarizability of LiH. 

A. Born-Oppenheimer Calculations in a Basis of Explicitly Correlated Gaussians... 

A new upper bound for the BO energy of the ground state of H J was recently obtained by Cencek et al. [48] using explicitly correlated Gaussians. Below we... 

All values are calculated for an optimized 50-term explicitly correlated Gaussian basis set and are in atomic units. 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

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