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15.4 The correlation factor $\hat F_{12}$

In the F12 methods, the correlation factor $\hat F_{12}$ is approximated by a frozen expansion of Gaussian type geminals that are functions of the interelectronic distance $r_{12}$. In principle, this can be any function, but normally a Slater function

$$F_{12}(r_{12})=-\beta^{-1}\exp(-\beta r_{12})$$ (3)

is used. By default this function is approximated by an expansion of six Gaussian functions, and the exponents and coefficients are optimized to obtain the best least squares fit, using a suitable weight function. The exponent $\beta$ can be chosen using the option gem_beta=value (default gem_beta=1.0).

In addition, also linear R12-methods ($F_{12}=r_{12}$) are available (DF-MP2-R12 and DF-LMP2-R12). However, these are no longer recommended since the non-linear correlation factor yields much better accuracy, numerical stability and convergence with respect to the AO, DF and RI basis sets.