Cartesian gaussian

Cartesian Gaussian-type orbitals (GTOs) Jfa.i.f( ( characterized by the quantum numbers a, b and c, which detail the angular shape and direction of the orbital, and the exponent a which governs the radial size . 

McMurchie L E and Davidson E R 1978 One-and two-electron integrals over Cartesian Gaussian functions J. Comp. Phys. 26 218-31 Gill P M W 1994 Molecular integrals over Gaussian basis functions Adv. Quantum Chem. 25 141-205... 

Ihiil arc added lo a basis lo form a star basis arc a simple set of un COM traded 3d primitive Ganssiaus (five Hermiic Gaussians) in S rO-NG basis sets bill Cartesian Gaussian s in th e split-valen ce basis sets. 

Thus, in a two-electron integral of the form p,v a), the product < (1)< (1) (where 0 and may be on different centres) can be replaced by a single Gaussian function that is centred at the appropriate point C. For Cartesian Gaussian functions the calculation is more complicated than for the example we have stated above, due to the presence of the Cartesian functions, but even so, efficient methods for performing the integrals have been devised. 

The presence of a single polarization function (either a full set of the six Cartesian Gaussians dxx, d z, dyy, dyz and dzz, or five spherical harmonic ones) on each first row atom in a molecule is denoted by the addition of a. Thus, STO/3G means the STO/3G basis set with a set of six Cartesian Gaussians per heavy atom. A second star as in STO/3G implies the presence of 2p polarization functions on each hydrogen atom. Details of these polarization functions are usually stored internally within the software package. 

Sometimes it turns out that we need to include a number of polarization functions, not just one of each type. The notation 4-31G(3d, 2p) indicates a standard 4-31G basis set augmented with three d-type primitive Cartesian Gaussians per centre and two p-type primitives on every hydrogen atom. Again, details of the... 

It does not make a significant difference that in practice one uses cartesian Gaussians rather than Gaussians with explicit inclusion of spherical harmonics. One... 

Details of the extended triple zeta basis set used can be found in previous papers [7,8]. It contains 86 cartesian Gaussian functions with several d- and f-type polarisation functions and s,p diffuse functions. All cartesian components of the d- and f-type polarization functions were used. Cl wave functions were obtained with the MELDF suite of programs [9]. Second order perturbation theory was employed to select the most energetically double excitations, since these are typically too numerous to otherwise handle. All single excitations, which are known to be important for describing certain one-electron properties, were automatically included. Excitations were permitted among all electrons and the full range of virtuals. 

Appendix A Cartesian Gaussian Spinors and Basis Functions... 

Appendix B Expansion of Cartesian Gaussian Basis Functions Using Spherical Harmonics... 

The procedure for evaluating the matrix elements of the P,T-odd interaction operator H,/ with Cartesian Gaussian spinors is a bit complicated and different... 

In terms of Cartesian Gaussian spinors, the basis functions can be defined as a linear combination of the following Gaussian spinors [57] ... 

APPENDIX B EXPANSION OF CARTESIAN GAUSSIAN BASIS FUNCTIONS USING SPHERICAL HARMONICS... 

Canonical ensemble, multiparticle collision dynamics, single-particle friction and diffusion, 116-118 Cartesian Gaussian spinors ... 

McMurchie LE, Davidson ER (1978) One- and two-electron integrals over cartesian gaussian functions. J Comp Phys 26 218... 

HF molecule we use the Dunning s contraction [As2pl2s [24] of Huzinaga s 9s5p As) [25] primitive set, augmented with a d polarization function with exponent 1.6 for F, and a set of p functions for H with exponent 0.75. We run different calculations for the equilibrium distance Re = 1.733 bohr), 1.5 Re, 2.0 Re and 3.0 Re. In this case, Cartesian Gaussians with 6fil-functions are used. Comparison is made with nearFCI (nFCI, up to sextuple excitation, i.e. CISDTQQS) [26]. 

The Hy-CI function used for molecular systems is based on the MO theory, in which molecular orbitals are many-center linear combinations of one-center Cartesian Gaussians. These combinations are the solutions of Hartree-Fock equations. An alternative way is to employ directly in Cl and Hylleraas-CI expansions simple one-center basis functions instead of producing first the molecular orbitals. This is a subject of the valence bond theory (VB). This type of approach, called Hy-CIVB, has been proposed by Cencek et al. (Cencek et.al. 1991). In the full-CI or full-Hy-CI limit (all possible CSF-s generated from the given one-center basis set), MO and VB wave functions become identical each term in a MO-expansion is simply a linear combination of all terms from a VB-expansion. Due to the non-orthogonality of one-center functions the mathematical formalism of the VB theory for many-electron systems is rather cumbersome. However, for two-electron systems this drawback is not important and, moreover, the VB function seems in this case more natural. 

In the above equation, is the antisymmetrizer working on both the space—and spin coordinates, gXrj )are primitive Cartesian Gaussian functions Eq. (23) and... 

The problem of evaluating matrix elements of the interelectron repulsion part of the potential between many-electron molecular Sturmian basis functions has the degree of difficulty which is familiar in quantum chemistry. It is not more difficult than usual, but neither is it less difficult. Both in the present method and in the usual SCF-CI approach, the calculations refer to exponential-type orbitals, but for the purpose of calculating many-center Coulomb and exchange integrals, it is convenient to expand the ETO s in terms of a Cartesian Gaussian basis set. Work to implement this procedure is in progress in our laboratory. 

Unfortunately, the exponential radial dependence of the hydrogenic functions makes the evaluation of the necessary integrals exceedingly difficult and time consuming for general computation, and so another set of functions is now universally adopted. These are Cartesian Gaussian functions centered on nuclei. Thus, gj( 1) is a function centered on atom I ... 

This analysis is typical of the approach to electron repulsion integrals. Use of cartesian gaussian functions gives rise to a more general basic integral... 

---