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In computational quantum chemistry the STO-3G basis is a popular minimal choice for basic work – the parameters to an expansion of primitive Gaussian functions are commonly deduced by a nonlinear least squares fit that maximises the overlap with a given Slater-type orbital (STO) (e.g. W. J. Hehre, R. F. Stewart and J. A. Pople, J. Chem. Phys. 1969, 51, 2657).

For example, for the simplest STO,

$$ \phi_\mathrm{1s}(\zeta_1, \mathbf{r}) = (\zeta_1^3/\pi)^{1/2}\exp(-\zeta_1 \mathbf{r})\tag{1} $$

the STO-3G basis function is

$$ \phi_\mathrm{1s}'(\mathbf{r}) = \sum_k^{N=3} c_k\,g_\mathrm{1s}(\alpha_k,\mathbf{r})\tag{2} $$

where

$$ g_\mathrm{1s}(\alpha_k, \mathbf{r}) = (2\alpha_k/\pi)^{3/4}\exp(-\alpha_k \mathbf{r}^2). \tag{3} $$

and the coefficients $c_k$ and $\alpha_k$ that maximise $\langle \phi_\mathrm{1s}'(\mathbf{r}) | \phi_\mathrm{1s}(\zeta_1, \mathbf{r}) \rangle$ are sought.

Why is $\langle \phi_\mathrm{1s}'(\mathbf{r}) | \phi_\mathrm{1s}(\zeta_1, \mathbf{r}) \rangle$ the target instead the goal being to minimize $\langle E \rangle = \langle \phi_\mathrm{1s}'(\mathbf{r}) | \hat{H} | \phi_\mathrm{1s}'(\mathbf{r}) \rangle$? Would the coefficients corresponding to the latter condition not be a better representation of the STO?

I did the fits myself and it seems so (I get within 0.60% of the true H 1s orbital energy ($-0.5 E_h$) compared with 1.02%)

Minimizing $\langle E \rangle$:

minimize

Maximizing overlap:

maximising overlap

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    $\begingroup$ I'm guessing here, because I'm not sure where to look for proof. However, it seems to me that these basis sets were never intended to be of production quality. On the other hand, you can pretty easily do the fit on the overlap (almost) by hand, but for the least energy fit you need to solve for the expectation values first. Computing power was incredibly expensive in the early days and why bother with a first order approximation when you already know that you'll make worse later on. $\endgroup$ Jul 14 at 18:41
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    $\begingroup$ One minor quibble with @Martin-マーチン comment. I think these were intended as “proof of principle” basis sets - that several Gaussian could be “about as good” as Slater but much faster. And for sure it would have been easier to convince people by high overlap with the STO, as well as the much easier task to fit. I don’t know if they were intended as “production” or not, but they were widely used for a while. $\endgroup$ Jul 17 at 14:16
  • $\begingroup$ @BuckThorn I’m happy for you to migrate it if you want! $\endgroup$
    – issy
    Jul 18 at 8:58
  • $\begingroup$ Another problem is that basis sets are often parameterized using atomic data, but are often intended to be used on molecules. It may be that parameterizing to minimize the energy for the bare atoms gives worse results for molecules than parameterizing based on the atomic overlap. I don't know if that's the case or not. I'm just saying the basis sets really shouldn't be designed to maximize performance for individual atoms but rather for molecules. $\endgroup$
    – jheindel
    Jul 19 at 23:20

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The STO-$n$G basis sets were one of the first Gaussian basis sets. They were proposed in J. Chem. Phys. 51, 2657 (1969). The idea was merely to formulate Gaussian basis versions of a pre-existing minimal Slater-type orbital (STO) basis set that had been optimized for molecules.

The biggest issue with the STO-$n$G basis sets is that they are minimal basis sets that are known to be unreliable for chemistry. Another big issue is that the naming is misleading: STO-$\infty$G would converge to the original minimal STO basis. Both of these features originate from the obsolete nature of the basis set.

Loads of advances in basis set design have been taken since the days of Pople, Hehre and coworkers. For instance, Davidson and Feller wrote almost 40 years ago

The first molecular calculations used a minimum set of STO's, with exponents chosen by Slater’s rules. Later, exponents were chosen to minimize the SCF energy of the atom.

in Chem. Rev. 86, 681 (1986). Modern basis sets are indeed typically designed by minimization of energy for atoms and/or molecules, and come in systematic families ranging from small, split-valence basis sets for qualitative calculations to large polarized multiple-$\zeta$ basis sets that afford quantitative accuracy. Several recent reviews of basis sets can be found in the literature; see e.g. Hill, Int. J. Quantum Chem. 113, 21 (2012) and Jensen, WIREs CMS 3, 273 (2013).

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