Charge cloud Gaussian

As a next step we have to take care of the electrostatic potential contribution of the Gaussian charge clouds themselves. 

In a first step we associate with each d-function a diffuse cloud of point charges qj of opposite sign located at r = rj - n. It is convenient (yet not crucial) to represent these clouds by Gaussians that is... 

The coefficients, in Eq. 10 are chosen so as to minimize the difference between the atomic orbitals given by Eqs. 9 and 10. Once chosen, these coefficients are fixed in all subsequent ab initio calculations that is, is completely specified by Eq. 10. The size of the exponent a in the Gaussian determines how close to the nucleus the electron charge is or conversely how diffuse the electron charge is. For higher-level basis sets, two sets of valence atomic orbitals are used (these are called split-valence basis sets). One set lies close to the nucleus and mimics closely the true-valence atomic orbitals. The other set is more diffuse (smaller a) and enables the molecular orbital to respond to electron cloud deformation away from the nucleus due to chemical bond formation (see Table 1). 

The derivation of the value 27/16 is a common textbook example s of the variational principle. Since HeH has a net positive charge, we expect, however, that the electron cloud is considerably contracted relative to that for the free atoms He and H. For illustrative purposes we will use the standard STO-3G value of C2 1 24 for H and let the standard value for He be Ci = 1.6875 x 1.24 = 2.0925 which, like the value for H, is also a factor of 1.24 larger than the best atom value. Figure 3.6 shows our coordinate system and basis functions for the HeH calculation. As described previously, each of the STO-3G basis functions of Fig. 3.6 or Eqs. (3.249) and (3.250) is the sum of the three primitive Gaussians. 

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