Consider an s -type Gaussian centered on nucleus A and one on nucleus B ; we are considering real functions, which is what basis functions normally are: g A ¼ a A e ± a A j r ± r A j 2 ; g B ¼ a B e ± a B j r ± r B j 2 ð 5 : 153 Þ 5.3 Basis Sets 233
where j r ± r A j 2 ¼ ð x ± x A Þ 2 þ ð y ± y A Þ 2 þ ð z ± z A Þ 2 and j r ± r B j 2 ¼ ð x ± x B Þ 2 þ ð y ± y B Þ 2 þ ð z ± z B Þ 2 ð 5 : 154 Þ with the nuclear and electron positions in Cartesian coordinates (if these were not s - type functions, the preexponential factor would contain one or more cartesian variables to give the function – the “orbital” – nonspherical shape). It is not hard to show that g A g B ¼ a C e ± a C j r ± r C j 2 ¼ g C ð 5 : 155 Þ The product of g A and g B is the Gaussian g C , centered at r C . Now consider the general electron-repulsion integral ð rs j tu Þ ¼ Z Z f ² r ð 1 Þ f s ð 1 Þ f ² t ð 2 Þ f u ð 2 Þ r 12 dv 1 dv 2 ð 5 : 156 ¼ 5 : 73 Þ If each basis function f were a single, real Gaussian, then from Eq. 5.155 this would reduce to ð v = w Þ ¼ Z Z f v ð 1 Þ f w ð 2 Þ r 12 dv 1 dv 2 ð 5 : 157 Þ i.e. three- and four-center two-electron integrals with four basis functions would immediately simplify to tractable two-center integrals with two functions. Actually, things are a little more complicated. A single Gaussian is a poor approximation to the nearly ideal description of an atomic wavefunction that a Slater function provides. Figure 5.12 shows that a Gaussian (designated STO-1G) is rounded near r ¼ 0 while a Slater function has a cusp there (zero slope vs a finite slope at r ¼ 0); the Gaussian also decays somewhat faster than the Slater function at large r . The solution to the problem of this poor functional behaviour is to use several Gaussians to approximate a Slater function. In Fig. 5.12 a single Gaussian and a linear combination of three Gaussians have been used to approximate the Slater function shown; the nomenclature STO-1G and STO-3G mean “Slater-type orbital (approxi- mated by) one Gaussian” and “Slater-type orbital (approximated by) three Gaus- sians”, respectively. The Slater function shown is one suitable for a hydrogen atom in a molecule ( z ¼ 1.24 [ 31 ]) and the Gaussians are the best fit to this Slater function. STO-1G functions were used in our illustrative Hartree–Fock calculation on HHe + (Section 184.108.40.206.5), and the STO-3G function is the smallest basis function used in standard ab initio calculations by commercial programs. Three Gaussians are a good speed versus accuracy compromise between two and four or more [ 31 ]. The STO-3G basis function in Fig. 5.12 is a contracted Gaussian consisting of three primitive Gaussians each of which has a contraction coefficient (0.4446, 234 5 Ab initio Calculations
0.5353 and 0.1543). Typically, an ab initio basis function consists of a set of primitive Gaussians bundled together with a set of contraction coefficients. Now consider the two-electron integral ( rs | tu ) (Eq. 5.156 ¼ 5.73 ). Suppose each basis function is an STO-3G contracted Gaussian, i.e.
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