For a molecule of fixed size, increasing the number of basis functions
*per atom*, $n$, leads to $\mathcal{O}({n}^{4})$ growth in the number of significant
four-center two-electron integrals, since the number of non-negligible product
charge distributions, $|\mu \nu \u27e9$, grows as $\mathcal{O}({n}^{2})$. As a result, the use
of large (high-quality) basis expansions is computationally costly. Perhaps
the most practical way around this “basis set quality” bottleneck is the use
of auxiliary basis expansions.^{Feyereisen:1993, Dunlap:2000, Jung:2005}
The ability to use auxiliary basis sets to accelerate a variety of electron
correlation methods, including both energies and analytical gradients, is a
major feature of Q-Chem.

The auxiliary basis $\{|K\u27e9\}$ is used to approximate products of Gaussian basis functions:

$$|\mu \nu \u27e9\approx |\stackrel{~}{\mu \nu}\u27e9=\sum _{K}|K\u27e9{C}_{\mu \nu}^{K}$$ | (6.19) |

Auxiliary basis expansions were introduced long ago, and are now widely recognized as an effective and powerful approach, which is sometimes synonymously called resolution of the identity (RI) or density fitting (DF). When using auxiliary basis expansions, the rate of growth of computational cost of large-scale electronic structure calculations with $n$ is reduced to approximately ${n}^{3}$.

If $n$ is fixed and molecule size increases, auxiliary basis expansions reduce the pre-factor associated with the computation, while not altering the scaling. The important point is that the pre-factor can be reduced by 5 or 10 times or more. Such large speedups are possible because the number of auxiliary functions required to obtain reasonable accuracy, $X$, has been shown to be only about 3 or 4 times larger than $N$.

The auxiliary basis expansion coefficients, $\mathbf{C}$, are determined by minimizing the deviation between the fitted distribution and the actual distribution, $\u27e8\mu \nu -\stackrel{~}{\mu \nu}|\mu \nu -\stackrel{~}{\mu \nu}\u27e9$, which leads to the following set of linear equations:

$$\sum _{L}\u27e8K|L\u27e9{C}_{\mu \nu}^{L}=\u27e8K|\mu \nu \u27e9$$ | (6.20) |

Evidently solution of the fit equations requires only two- and three-center integrals, and as a result the (four-center) two-electron integrals can be approximated as the following optimal expression for a given choice of auxiliary basis set:

$$\u27e8\mu \nu |\lambda \sigma \u27e9\approx \u27e8\stackrel{~}{\mu \nu}|\stackrel{~}{\lambda \sigma}\u27e9=\sum _{K,L}{C}_{\mu \nu}^{L}\u27e8L|K\u27e9{C}_{\lambda \sigma}^{K}$$ | (6.21) |

In the limit where the auxiliary basis is complete (*i.e.* all products of
AOs are included), the fitting procedure described above will be exact.
However, the auxiliary basis is invariably incomplete (as mentioned above,
$X\approx 3N)$ because this is essential for obtaining increased computational
efficiency.

Standardized auxiliary basis sets have been developed by the Karlsruhe group
for second-order perturbation (MP2)
calculations of the correlation energy.^{Weigend:1998, Weigend:2002b}
Using these basis sets, absolute errors in the correlation energy are small
(*e.g.*, below 60 $\mu $Hartree per atom), and errors in relative energies are smaller still
At the same time, speedups of 3–30$\times $ are realized.
This development has made the routine use of
auxiliary basis sets for electron correlation calculations possible.

Correlation calculations that can take advantage of auxiliary basis expansions
are described in the remainder of this section (MP2, and MP2-like methods) and
in Section 6.16 (simplified active space coupled cluster methods
such as PP, PP(2), IP, RP). These methods automatically employ auxiliary basis
expansions when a valid choice of auxiliary basis set is supplied using the
AUX_BASIS_CORR or
AUX_BASIS keyword which is used in the same way as the BASIS
keyword.
The PURECART *$rem* is no longer needed here, even if using
a auxiliary basis that does not have a predefined value. There
is a built-in automatic procedure that provides the effect
of the PURECART *$rem* in these cases by default.