Primitive GTO

Another family of basis sets, commonly referred to as the Pople basis sets, are indicated by the notation 6—31G. This notation means that each core orbital is described by a single contraction of six GTO primitives and each valence shell orbital is described by two contractions, one with three primitives and the other with one primitive. These basis sets are very popular, particularly for organic molecules. Other Pople basis sets in this set are 3—21G, 4—31G, 4—22G, 6-21G, 6-31IG, and 7-41G. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

The functions put into the determinant do not need to be individual GTO functions, called Gaussian primitives. They can be a weighted sum of basis functions on the same atom or different atoms. Sums of functions on the same atom are often used to make the calculation run faster, as discussed in Chapter 10. Sums of basis functions on different atoms are used to give the orbital a particular symmetry. For example, a water molecule with symmetry will have orbitals that transform as A, A2, B, B2, which are the irreducible representations of the C2t point group. The resulting orbitals that use functions from multiple atoms are called molecular orbitals. This is done to make the calculation run much faster. Any overlap integral over orbitals of different symmetry does not need to be computed because it is zero by symmetry. 

The next step was to represent each Slater atomic orbital as a fixed linear combination of Gaussian orbitals so a Slater-type orbital with exponent f is written as a sum of GTOs with exponents a, q 2, and so on. For example, in the case of three primitive GTOs we might write... 

Thus, to give an STO-3G fit to a Is orbital with exponent = 1 we have to take the three primitive Is GTOs with exponents cti =0.109818, aj — 0.405 771 and 03 = 2.227 66. Their individual variations with distance is shown in Figure 9.3. 

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. 

To remind you, for an atomic STO with exponent f different from 1, the rf s remain the same but the exponents of the primitives have to be multiplied by For a carbon atom, the Is STO is represented as a combination of four primitive GTOs, whilst the 2s and 2p STO are represented by two basis functions, one consisting of three primitives and the other of one. 

The primitive GTOs with exponents 18050.0 through 0.2558 are Is type, and the remainder are 2p type. The two most diffuse s functions (those with exponents 0.7736 and 0.2558) are the main components of the 2s STO, and they are allowed to vary freely in molecular calculations. The Is primitive with exponent 2.077 turns out to make substantial contributions to both the atomic Is and 2s orbitals, so that one is also left free. The remaining seven distinct primitive Is GTOs describe the atomic Is orbital, and a careful examination of the ratios of their... 

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),... 

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