Explicitly correlated Gaussian functions

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 ... 

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

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

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. 

In a non-adiabatic calculation, the spatial part of the ground state A-particle wave function eqn.(19) can be expanded in terms of the following explicitly correlated gaussian functions ... 

A variety of functional forms has been used for several very small systems. These include the molecule H2, the ion H3, and the dimer He-He for which Hylleraas functions. Singer polymals, and explicitly correlated Gaussian functions of very high accuracies have been used in QMC of all types. The optimization of these functions has usually been carried out by means of the technique of minimizing the variance in local energies described by Conroy the 1960s. In fact, it has only rarely been done in any other way. 

W. Cencek and J. Rychlewski,/. Chem. Phys., 98, 1252 (1993). Many-Electron Explicitly Correlated Gaussian Functions. 1. General Theory and Test Results. 

In this Table we also find a second form of exponentially correlated function s]. Such functions have been referred to as explicitly correlated Gaussians (ECG). They take the form,... 

As mentioned above, the centerpiece of our methodology is use of explicitly correlated gaussian basis functions which we will now write in a more general form,... 

As we have suggested recently [68] the technique involving separation of the CM motion and representation of the wave function in terms of explicitly correlated gaussians is not only limited to non- adiabatic systems with coulombic interactions, but can also also extended to study assembles of particles interacting with different types of two- and multi-body potentials. In particular, with this approach one can calculate the vibration-rotation structure of molecules and clusters. In all these cases the wave function will be expanded as symmetry projected linear combinations of the explicitly correlated fa of eqn.(29) multiplied by an angular term, Y M. 

D.B. Kinghorn, Explicitly Correlated Gaussian Basis Functions Derivation and Implementation of Matrix Elements and Gradient Formulas Using Matrix Differential Calculus, Ph.D. dissertation, Washington State University, 1995. 

Kinghom, D.B. and Adamowicz, L. 1997. Electron Affinity of Hydrogen, Deuterium, and Tritium A Nonadiabatic Variational Calculation Using Explicitly Correlated Gaussian Basis Functions. Journal of Chemical Physics 106 4589-4595. 

Until very recently, the use of rii-dependent wave functions has been confined to two-, three-, and four-electron systems. For those systems, it is possible to perform very accurate variational calculations with explicitly correlated wave functions. See, for example, the computations on two-electron systems discussed in Section 2 or the variational calculations in a basis of Gaussian geminals presented in Section 5.2,1. 

Recently, impressive calculations using Hylleraas wave functions have been done for the H2 molecule by the Hylleraas method [44,63], the Iterative Complement Iteration (ICI) [36], and Explicitly Correlated Gaussian (ECG) [12] methods. Few molecules have yet been calculated using Hylleraas-type wave functions HeH+ and some other species [72] using the Hylleraas method, the helium dimer He2 interaction energy [46] and the ground state of the BH molecule [7], both using the ECG method. 

Unfortunately, extending Hylleraas s approach to systems containing three or more electrons leads to very cumbersome mathematics. More practical approaches, known as explicitly correlated methods, are classified into two categories. The first group of approaches uses Boys Gaussian-type geminal (GTG) functions with the explicit dependence on the interelectronic coordinate built into the exponent [95]... 

For LiH and LiD, 244-term non-BO wave functions were variationaUy optimized. The initial guess for the LiH non-BO wave function was built by multiplying a 244-term BO wave function expanded in a basis of explicitly correlated functions by Gaussians for the H nucleus centered at and around (in all three dimensions) a point separated from the origin by the equilibrium distance of 3.015 bohr along the direction of the electric field. Thus the centers... 

The family of variational methods with explicitly correlated functions includes the Hylleraas method, the Hyller-aas Cl method, the James-Coolidge and the KcAos-Wolniewicz approaches, as well as a method with exponentially correlated Gaussians. The method of explicitly correlated functions is very successful for two-, three-, and four-electron systems. For larger systems, due to the excessive number of complicated integrals, variational calculations are not yet feasible. 

All explicitly correlated calculations were performed at the CCSD(F12) level of theory, as implemented in the TurbomOLE program [58, 69]. The Slater-type correlation factor was used with the exponent 7 = 1.0 aQ. It was approximated by a linear combination of six Gaussian functions with linear and nonlinear coefficients taken from Ref. [44]. The CCSD(F12) electronic energies were computed in an all-electron calculation with the d-aug-cc-pwCV5Z basis set [97]. For all cases we used full CCSD(F12) model (see Subsection 4.9 for the discussion about models implemented in Turbomole), the open-shell species were computed with a UHF reference wave function. The explicitly correlated contributions to the relative quantities are collected in Tables 10 and 11 under the label F12 . 

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