Cartesian Gaussian-type orbitals

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 . 

Canonical ensemble 60 Cartesian components 4 Cartesian Gaussian-type orbitals 161 CASSCF (Complete Active Space Self Consistent Field) model 205 cc-pVDZ (Correlation-consistent Basis Sets) 175, 201 Centrifugal effects 276 Charge element 15 Choice of origin 297... 

The nonorthogonal basis functions Xx( )are referred to as atomic orbitals (AOs) and are often taken to be Cartesian Gaussian-type orbitals (GTOs) of the (unnormalized) form ... 

Cartesian Gaussian-type orbitals (GTOs) characterized by the... 

The use of Cartesian Gaussian-type orbitals (GTOs) in ab initio work may come as a surprise to anyone who recalls the functional form of the hydrogen atom orbitals, hydrogen exp( - Ir). Cartesian GTO s have the form ... 

Gaussian type orbitals can be written in terms of polar or cartesian coordinates... 

Boys (1950) proposed an alternative to the use of STOs. All that is required for there to be an analytical solution of the general four-index integral formed from such functions is that the radial decay of the STOs be changed from e to e. That is, the AO-like functions are chosen to have the form of a Gaussian function. The general functional form of a normalized Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is... 

We use an alternative to this method, that enables a fast and accurate evaluation of the two-center integrals. Analytical integration is possible when linear combinations of Gaussian Type Orbitals are used to describe the atomic states [1,9, 10]. Imperfect behavior of such gaussian functions at large distances does not affect the results, since the two-center matrix-elements (7) have an exponential decay for increasing intemuclear distances. For example, for integrals and expressed in cartesian coordinates, one has to evaluate expressions such as... 

Kaijser and Smith [17] have presented analytic forms for many of the Slater-type orbitals and to Gaussian-type orbitals in both spherical harmonic and Cartesian form. The total momentum-space electron density p(p) is given by... 

Gaussian-type orbitals (GTOs), usually represented in their Cartesian form, have been proposed (i +j + k = I determines the type of the orbital (s, p, etc.)). 

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. 

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. 

If the group is rotational or helical and ij> is not 5-type, then the />, on each site become linear combinations of basis functions related by the rotation matrix of the appropriate angular momentum and the appropriate rotational or helical step angle [27]. It is traditional to use Cartesian-Gaussian orbital basis sets in quantum-chemical calculations [28], but solid-spherical-harmonic Gaussians [29] are best for symmetry adaption and matrix element evaluation. Including an extra factor of (-)M in the definition of the solid spherical harmonics [30]... 

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