Gaussian primitives

Schmidt M W and Ruedenberg K 1979 Effective convergence to compiete orbitai bases and to the atomic Hartree-Fock iimit through systematic sequences of Gaussian primitives J. Chem. Phys. 71 3951-62... 

We find that there are two Gaussian primitives and one unpaired electron from the output... 

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. 

Minimal basis sets use fixed-size atomic-type orbitals. The STO-3G basis set is a minimal basis set (although it is not the smallest possible basis set). It uses three gaussian primitives per basis function, which accounts for the 3G in its name. STO stands for Slater-type orbitals, and the STO-3G basis set approximates Slater orbitals with gaussian functions. ... 

Figure 4-7 The SCF energy of the neon atom converges exponentially with the number of Gaussian primitive basis functions.

IG and 6-3IG. These are commonly used split-valence plus polarization basis sets. These basis sets contain inner-shell functions, written as a linear combination of six Gaussians, and two valence shells, represented by three and one Gaussian primitives, respectively (noted as 6-3IG). When a set of six d-type Gaussian primitives is added to each heavy atom and a single set of Gaussian p-type functions to each hydrogen atom, this is noted as and... 

Bardo, R. D. and Ruedenberg, K., Even-tempered atomic orbitals. VI. Optimal orbital exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and oxygen in molecules,. 1. Chem. Phys. 60, 918-931 (1974). 

The method that is used in most of the work described in this chapter is the distributed multipole analysis (DMA) of Stone,which is implemented in the CADPAC ab initio suite. DMA is based on the density matrix p,y of the ab initio wavefunction of the molecule, expressed in terms of the Gaussian primitives q that comprise the atomic orbital basis set ... 

E Matrix elements of the truncated harmonic potential F Matrix elements in Gaussian primitive basis functions... 

Gaussian primitives, however, yields closed forms that are remarkably... 

F Matrix elements in Gaussian primitive basis functions... 

An expansion of the Morse potential, for example, in a set of Gaussian functions is given by eq (C4) in Appendix C. Matrix elements of the Morse potential in terms of the Gaussian primitive basis functions are therefore simply three center overlap integrals [49], These matrix elements can be evaluated for each term in the sum and then converted to the final expression in a straightforward manner. 

For each of the three Gaussian primitives in Table 1.4, construct their projections on the radial mesh in cells B 9 to D 29. For example, the gl) entry in cell B9 is... 

Figure 1.15a Comparisons of the variations of the Reeves sto-ng ls) basis sets with exact variation for the hydrogen ls) radial function. Note, the substantial disagreement for the simple single Gaussian function and the increasing agreement with the numbers of Gaussian primitives used.

Figure 1.15c Comparisons of the variations of the Reeves sto-ng ls) basis sets with exact variation for the hydrogen ls) radial function. Note, the substantial disagreement for the simple single Gaussian function and the increasing agreement with the numbers of Gaussian primitives used. But, note, too. that the sto-2g 2p) set does not return a good fit to the variation with radial distance of the exact hydrogen 2p> function. The radial distance, in this example, is assumed to coincide with distance along the s-axis.

These Gaussian sets are very different to the linear combinations determined by Reeves, in that these basis sets were determined to match the equivalent Slater functions [see Section 2.6 page 70]. Like the Reeves sets, though, because of the need, at the time, to calculate integrals as simply as possible, the Gaussian sets in the table for the 2s hydrogen orbital are linear combinations of 1 s Gaussian primitives. If you read the Hehre, Stewart and Pople paper (33) you will see that their normalization criterion is over all the coordinates. Thus, in comparisons with Slater functions, we need to be careful to apply a consistent normalization condition. In this discussion, the choice is to normalize only over the radial coordinate and so to apply equation 1.9, with equation 1.15 multiplied by (1/47t) /2. 

Enter the appropriate formulae for the Gaussian primitives in the active cells of columns. 

Fill in the basic parameters for the Gaussian primitives in cells C 3 to E 4 and, then, the primitive quantum number in cell F 3, with the Slater exponent for the Is orbital in boron in cell G 3. Since the Slater function is to be included in the chart displaying the variations with radial distance, repeat these entries in cells G 7 and G 8. 

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