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18.4 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
\begin{displaymath}
F_{12}(r_{12})=-\beta^{-1}\exp(-\beta r_{12})
\end{displaymath} (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).

It is also possible to use geminals with different exponents for core-core and core-valence calculation [see Mol. Phys. 109, 407 (2011)] In this case the exponents are specified as

gem_beta= $[\beta_1,\beta_2]$
gem_beta= $[\beta_1,\beta_2,\beta_3]$

The smallest $\beta$ value is used for valence correlation, the second-smallest for core-valence and the largest for core-core pairs. If only 2 values are given, core-core and core-valence pairs are treated with the same exponent. Note that the core directive is usually needed to include correlation of inner-shell orbitals.

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.



Next: 18.5 Basis sets Up: 18 Explicitly correlated methods Previous: 18.3.3 The fixed amplitude   Contents   PDF

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