Exponentially correlated basis sets

It is in this context that exponentially correlated basis sets (EC or ECG) become particularly interesting, since such bases behave as if they contained high metric terms, but in a very compact form. With multiple nonlinear parameters, a trial wave function may be able to represent the wave function at the cusps accurately while retaining sufficient flexibility to correctly reproduce its asymptotic behavior. 

The calculated ioi as a function of basis set and electron correlation (valence electrons only) at the experimental geometry is given in Table 11.8. As the cc-pVXZ basis sets are fairly systematic in how they are extended from one level to the next, there is some justification for extrapolating the results to the infinite basis set limit (Section 5.4.5). The HF energy is expected to have an exponential behaviour, and a functional form of the type A + 5exp(—Cn) with n = 2-6 yields an infinite basis set limit of —76.0676 a.u., in perfect agreement with the estimated HF limit of -76.0676 0.0002 a.u. ... 

Dunning has developed a series of correlation-consistent polarized valence n-zeta basis sets (denoted cc-pVnZ ) in which polarization functions are systematically added to all atoms with each increase in n. (Corresponding diffuse sets are also added for each n if the prefix aug- is included.) These sets are optimized for use in correlated calculations and are chosen to insure a smooth and rapid (exponential-like) convergence pattern with increasing n. For example, the keyword label aug-cc-pVDZ denotes a valence double-zeta set with polarization and diffuse functions on all atoms (approximately equivalent to the 6-311++G set), whereas aug-cc-pVQZ is the corresponding quadruple-zeta basis which includes (3d2flg,2pld) polarization sets. 

A variety of extrapolation algorithms have been applied to the sequences generated by the correlation-consistent cc-pVnZ basis sets [12, 51-55], Dunning and his colleagues had initially suggested fitting their calculations to an exponentially decaying function [12, 51, 52],... 

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. 

Basis sets for use in practical Hartree-Fock, density functional, Moller-Plesset and configuration interaction calculations make use of Gaussian-type functions. Gaussian functions are closely related to exponential functions, which are of the form of exact solutions to the one-electron hydrogen atom, and comprise a polynomial in the Cartesian coordinates (x, y, z) followed by an exponential in r. Several series of Gaussian basis sets now have received widespread use and are thoroughly documented. A summary of all electron basis sets available in Spartan is provided in Table 3-1. Except for STO-3G and 3 -21G, any of these basis sets can be supplemented with additional polarization functions and/or with diffuse functions. It should be noted that minimal (STO-3G) and split-valence (3-2IG) basis sets, which lack polarization functions, are unsuitable for use with correlated models, in particular density functional, configuration interaction and Moller-Plesset models. Discussion is provided in Section II. 

CO Peterson and Dunning [92] have made an extensive analysis of the role of basis sets and correlation treatments in the calculation of the molecular properties of CO. By carefully controlling the errors in the calculations, it was possible to compute properties of this small molecule to an accuracy that rivals the most sophisticated experimental studies. They made use of the correlation consistent basis sets (cc). The dissociation energy with icCAS+SDQ was computed 258.5 kcal/mol with the best method, and the experimental value is 259.6 0.1 kcal/mol. The CCSD(T) yielded 258.6 kcal/mol in excellent agreement with experiment. CASSCF, MP4, and CCSD yield results with errors bigger than 4 kcal/mol. The CBS limit was obtained by exponential extrapolation of the cc-pVDZ through cc-pV6Z for all methods. 

Calculation of the quantum dynamics of condensed-phase systems is a central goal of quantum statistical mechanics. For low-dimensional problems, one can solve the Schrodinger equation for the time-dependent wavefunction of the complete system directly, by expanding in a basis set or on a numerical grid [1,2]. However, because they retain the quantum correlations between all the system coordinates, wavefunction-based methods tend to scale exponentially with the number of degrees of freedom and hence rapidly become intractable even for medium-sized gas-phase molecules. Consequently, other approaches, most of which are in some sense approximate, must be developed. 

Recurrence formulas for the so-called Hylleraas basis (in which the parameters ui are zero) were published in 2004 by Pachucki et al. [4] that work was extended by the present author in 2009 [5] to handle full exponential correlation (general values of aU the m,- and w,). Both these sets of four-... 

Diagrammatic many-body perturbation theory calculations of the correlation energy of various diatomic molecules in their ground states using universal basis sets of even-tempered exponential-type functions. Comparison with other approaches. ... 

Because of the systematic nature of the correlation-consistent family of basis sets, it is possible to use extrapolation techniques to estimate the CBS limit. For example, the three-parameter exponential function introduced by Feller nicely describes the convergence of SCF energy to the CBS limit with... 

---