10.8 Spin-component scaling approaches (SCS/SOS)

By introducing individual scaling factors for the same–spin and opposite–spin contributions to the correlation energy most second–order methods can be modified to achieve a (hopefully) better performance. SCS-MP2 has first been proposed by S. Grimme and SOS-MP2 by Y. Jung et al. (see below). The generalization of SCS and SOS to CC2 and ADC(2) for ground and excited states is described in [18]. It uses the same scaling factors as proposed for the original SCS- and SOS-MP2 approaches (see below). In the ricc2 program we have also implemented SCS and SOS variants of CIS(D) for excitation energies and of CIS(D$_\infty$) for excitation energies and gradients, which are derived from SCS-CC2 and SOS-CC2 in exactly the same manner as the unmodified methods can be derived as approximations to CC2 (see Sec. 10.2 and Ref. [277]). Please note, that the SCS-CIS(D) and SOS-CIS(D) approximations obtained in this way and implemented in ricc2 differ from the spin-component scaled SCS- and SOS-CIS(D) methods proposed by, respectively, S. Grimme and E. I. Ugorodina in [278] and Y. M. Rhee and M. Head–Gordon in [279].

A line with scaling factors has to be added in the $ricc2 data group:

$ricc2
scs  cos=1.2d0  css=0.3333d0

cos denotes the scaling factor for the opposite–spin component, css the same–spin component.

As an abbreviation
scs
can be inserted in $ricc2. In this case, the SCS parameters cos=6/5 and css=1/3 proposed S. Grimme (S. Grimme, J. Chem. Phys. 118 (2003) 9095.) are used. These parameters are also recommended in [18] for the SCS variants of CC2, CIS(D), CIS(D$_\infty$), and ADC(2) for ground and excited states.

Also, just
sos
can be used as a keyword, to switch to the SOS approach proposed by the Head-Gordon group for MP2 with scaling factors of cos=1.3 and css=0.0 (Y., Jung, R.C. Lochan, A.D. Dutoi, and M. Head-Gordon, J. Chem. Phys. 121 (2004) 9793.), which are also recommended for the SOS variants of CC2, CIS(D), CIS(D$_\infty$), and ADC(2). The Laplace-transformed algorithm for the SOS variants are activated by the additional data group $laplace:

$laplace
   conv=4

For further details on the Laplace-transformed implementation and how one can estimated whether the ${\cal O}({\cal N}^4)$-scaling Laplace-transformed or ${\cal O}({\cal N}^5)$-scaling conventional RI implementation is efficient see Sec. 9.6.

Since Version 6.6 the ${\cal O}({\cal N}^4)$-scaling Laplace-transformed implementation is available for ground and excited state gradients with CC2 and ADC(2).

Restrictions:

the spin $(S^2)$ expectation value for open-shell calculation can not be evaluated in the SCS or SOS approaches
for LT-SOS-CC2 (and the related CIS(D) and ADC(2) versions) the following further limitations apply:
- incompatible with the calculation of the $D_1$ and $D_2$ diagnostics
- second-order properties, two-photon transitions moments and induced transition moments are not implemented for the spin-component scaled variants.