Contracted GTOs

To overcome the primary weakness of GTO fimetions (i.e. their radial derivatives vanish at the nucleus whereas the derivatives of STOs are non-zero), it is coimnon to combine two, tliree, or more GTOs, with combination coefficients which are fixed and not treated as LCAO-MO parameters, into new functions called contracted GTOs or CGTOs. Typically, a series of tight, medium, and loose GTOs are multiplied by contraction coefficients and suimned to produce a CGTO, which approximates the proper cusp at the nuclear centre. 

Most calculations today are done by choosing an existing segmented GTO basis set. These basis sets are identihed by one of a number of notation schemes. These abbreviations are often used as the designator for the basis set in the input to ah initio computational chemistry programs. The following is a look at the notation for identifying some commonly available contracted GTO basis sets. 

The atomic unit of wavefunction is. The dashed plot is the primitive with exponent 2.227 66, the dotted plot is the primitive with exponent 0.405 771 and the full plot is the primitive with exponent 0.109 818. The idea is that each primitive describes a part of the STO. If we combine them together using the expansion coefficients from Table 9.5, we get a very close fit to the STO, except in the vicinity of the nucleus. The full curve in Figure 9.4 is the contracted GTO, the dotted curve the STO. 

Combining the full set of basis functions, known as the primitive GTOs (PGTOs), into a smaller set of functions by forming fixed linear combinations is known as basis set contraction, and the resulting functions are called contracted GTOs (CGTOs). 

There are two different ways of contracting a set of primitive GTOs to a set of contracted GTOs segmented and general contraction. Segmented contraction is the... 

Here, n corresponds to the principal quantum number, the orbital exponent is termed and Ylm are the usual spherical harmonics that describe the angular part of the function. In fact as a rule of thumb one usually needs about three times as many GTO than STO functions to achieve a certain accuracy. Unfortunately, many-center integrals such as described in equations (7-16) and (7-18) are notoriously difficult to compute with STO basis sets since no analytical techniques are available and one has to resort to numerical methods. This explains why these functions, which were used in the early days of computational quantum chemistry, do not play any role in modem wave function based quantum chemical programs. Rather, in an attempt to have the cake and eat it too, one usually employs the so-called contracted GTO basis sets, in which several primitive Gaussian functions (typically between three and six and only seldom more than ten) as in equation (7-19) are combined in a fixed linear combination to give one contracted Gaussian function (CGF),... 

When the HF basis set is minimal, this is fairly simple (there is a one-to-one correspondence in basis functions) but when it is larger, some algorithmic choices are made about how to carry out the mapping (e.g., always map to the tightest function or map based on overlap between the semiempirical STO and the large-basis contracted GTO). Thus, it is... 

In a significant paper, Bauschlicher and Schaefer213 have examined the flexibility of atomic orbitals in a molecular environment, and they have shown in calculations on diatomics involving second-row atoms (among these the 32 state of PH) that only the outermost orbitals are altered during molecular formation, and hence essentially fully contracted GTO can be used for the inner-shell orbitals. We will return to this point later. For PH the contraction procedure that was used recovered 89 % of the energy obtained with an uncontracted basis set. 

A. AH2 Molecules, inclnding H3.—The simplest of these species is H, which has been the subject of a number of recent studies. Work previous to 1970 is listed by Borkman.426 The ground state is triangular (Dsn) and it has been studied using a basis set of elliptical orbitals. Contracted GTO basis sets were used in near-HF calculations by Harrison et a/.,430 and correlated wavefunctions were computed by Salmon and Poshusta,431 and by Handy.482 In the latter calculation a new form for the correlation factor was used, and the good results suggest that it might be useful for other small molecules. 

In order to see the difference in the two approaches, below I focus on the excitation energies, AE, of the Be states that are discussed here. The nice thing about atomic spectra of this type is that there is accurate experimental information with which one can compare the results of a theoretical method (See, Tables of NIST, USA, in the WWW). Specifically, I compare the AE from the Be Fermi-sea energies for which cancellations on subtraction of total energies are expected, with those obtained from methods that have used one of the known basis sets. I consider two such publications. The first is in 1986 by Graham et al. [105] where a (9s9p5d) contracted GTO basis (61 basis functions) was used for different types of computations. I keep the full Cl (FCI) results. The second is in 2003 by Sears, Sherrill and Krylov [106], who studied aspects of "spin-flip" methods and compared them with FCI using the same basis set, which is a 6-31G. 

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). 

Table 4 SCF calculations of valence-region energies using the contracted GTO(9s5p 6s) —+ [5s3p 3s] basis...

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