Cartesian GTOs

In the case of d-type orbitals, there are six Cartesian GTOs with pre-exponential factors of x, xy, y, xz, yz and z - Only five are linearly independent, e combi nation... 

Xhe simplification is even more dramatic for a two-electron integral, which can involve GTOs on four different centres. Formulae for integrals involving Cartesian GTOs of p, d,. .. types can be deduced from those involving s orbitals by simple differentiation. Here is the famous synopsis. 

N is a normalization factor which ensures that = 1 (but note that the are not orthogonal, i. e., 0 lor p v). a represents the orbital exponent which determines how compact (large a) or diffuse (small a) the resulting function is. L = 1 + m + n is used to classify the GTO as s-functions (L = 0), p-functions (L = 1), d-functions (L = 2), etc. Note, however, that for L > 1 the number of cartesian GTO functions exceeds the number of (27+1) physical functions of angular momentum l. For example, among the six cartesian functions with L = 2, one is spherically symmetric and is therefore not a d-type, but an s-function. Similarly the ten cartesian L = 3 functions include an unwanted set of three p-type functions. 

In the same manner as the n value constitutes an order of the WO-CETO angular part, the me 0,1 value classifies WO-CETO, and also CETO functions, into two kinds odd WO-CETO s if msl and even WO-CETO s if m=0. A similar classification may be applied over cartesian GTO fimctions, whose basis sets used in the literature are always defined as even. 

In the Cartesian scheme (Eq. (19)), there are (/+1)(/+ 2)/2 components of a given /, whereas the number of independent spherical harmonics is only 21+ 1. Usually, therefore, the Cartesian GTOs are not used individually but instead are combined linearly to give real solid harmonics (see Ref. 1). In addition, for a more compact and accurate description of the electronic structure, the GTOs (Eq. (19)) are not used individually as primitive GTOs but mostly as contracted GTOs (i.e., as fixed, linear combinations of primitive GTOs with different exponents a). 

The kinetic balance requirement in this form is quite simple to implement, but its application to Gaussian basis sets calls for some further comments. These are most easily demonstrated on Cartesian GTOs. If we use a scalar basis as described above, the main effect of the a p operator will be to differentiate the basis function. For a px GTO, we get... 

The small component basis function derived from the px GTO is no longer just a simple one-term Cartesian GTO, but a sum of two such functions albeit with the same exponential part. The part... 

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 ... 

The term spherical-harmonic is used to distinguish these orbitals from the Cartesian GTOs introduced in Section 6.6.7. Note that the GTOs of even / contain even powers of r and those of odd / contain odd powers. Thus, whereas the radial 2s and 3.s functions correspond to r exp(—ar ) and / exp(—ar ), the radial 2p and 3p functions are represented by rexp(—ar ) and r exp(—ar ). 

The three-dimensional HO functions (6.6.24) possess a rather complicated nodal structure. If we retain only the highest-order terms in the Hermite polynomials, we arrive at the following set of nodele.ss Cartesian GTOs ... 

The functional form of the AOs was discussed in Chapters 6 and 8. In nearly all calculations on polyatomic systems, the AOs are taken as fixed linear combinations of real-valued primitive Cartesian GTOs of the form [1,2]... 

In molecular calculations, we always employ full shells of GTOs - that is, the individual Cartesian components of a given shell are never used alone. As we shall see in the present chapter, the simple analytical form of the primitive Cartesian GTOs (9-1.3) allows for an efficient evaluation of the polyatomic integrals entering the standard Hamiltonian operator (2.2.18). 

As discussed in Chapter 8, the primitive Cartesian GTOs (9.1.3) are mostly used in fixed linear combinations X/i(r). A typical AO thus consists of a linear combination of primitive Cartesian GTOs of the same angular-momentum quantum number / but of different Cartesian quantum numbers i, j and k and of different exponents a. In Section 9.1.2, we shall discuss how Cartesian GTOs of the same / but different i, j and k are combined to yield the real-valued herical-harmonic GTOs next, in Section 9.1.3, we shall see how the GTOs of different exponents are combined to yield the final AOs as contracted spherical-harmonic GTOs. 

As should be apparent from our discussion so far, a large number of integrals over primitive Cartesian GTOs (9.1.3) contribute to a smaller number of integrals over contracted spherical-harmonic GTOs (9.1.13). This is especially true for the two-electron integrals (9.1.2), since the... 

We shall always work with Cartesian GTOs in the unnormalized form (9.2.1). Still, it is of some interest to consider the self-overlap of these fundamental functions. The self-overlap of the... 

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