# 6.6.4 Spin-Biased MP2 Methods (SCS-MP2, SOS-MP2, and MOS-MP2)

The accuracy of MP2 calculations can be significantly improved by semi-empirically scaling the opposite-spin (OS) and same-spin (SS) correlation components with separate scaling factors, as shown by Grimme.328 Scaling with 1.2 and 0.33 (or OS and SS components) defines the SCS-MP2 method, but other parameterizations are desirable for systems involving intermolecular interactions, as in the SCS-MI-MP2 method, which uses 0.40 and 1.29 (for OS and SS components).226

Results of similar quality for thermochemistry can be obtained by only retaining and scaling the opposite spin correlation (by 1.3), as was recently demonstrated.442 Furthermore, the SOS-MP2 energy can be evaluated using the RI approximation together with a Laplace transform technique, in effort that scales only with the 4th power of molecular size. Efficient algorithms for the energy442 and the analytical gradient589 of this method are available since Q-Chem v. 3.0, and offer advantages in speed over MP2 for larger molecules, as well as statistically significant improvements in accuracy.

However, we note that the SOS-MP2 method does systematically underestimate long-range dispersion (for which the appropriate scaling factor is 2 rather than 1.3) but this can be accounted for by making the scaling factor distance-dependent, which is done in the modified opposite spin variant (MOS-MP2) that has recently been proposed and tested.587 The MOS-MP2 energy and analytical gradient are also available in Q-Chem 3.0 at a cost that is essentially identical with SOS-MP2. Timings show that the 4th-order implementation of SOS-MP2 and MOS-MP2 yields substantial speedups over RI-MP2 for molecules in the 40 heavy atom regime and larger. It is also possible to customize the scale factors for particular applications, such as weak interactions, if required.

A fourth order scaling SOS-MP2/MOS-MP2 energy calculation can be invoked by setting the CORRELATION keyword to either SOSMP2 or MOSMP2. MOS-MP2 further requires the specification of the $rem variable OMEGA, which tunes the level of attenuation of the MOS operator:587  $g_{\omega}(r_{12})=\frac{1}{r_{12}}+c_{\mathrm{MOS}}\frac{\mathrm{erf}\left({% \omega r_{12}}\right)}{r_{12}}$ (6.22) The recommended OMEGA value is $\omega=0.6$ bohr${}^{-1}$.587 The fast algorithm makes use of auxiliary basis expansions and therefore, the keyword AUX_BASIS should be set consistently with the user’s choice of BASIS. Fourth-order scaling analytical gradient for both SOS-MP2 and MOS-MP2 are also available and is automatically invoked when JOBTYPE is set to OPT or FORCE. The minimum memory requirement is 3$X^{2}$, where $X$ = the number of auxiliary basis functions, for both energy and analytical gradient evaluations. Disk space requirement for closed shell calculations is $\sim 2OVX$ for energy evaluation and $\sim 4OVX$ for analytical gradient evaluation. Summary of key$rem variables to be specified:

CORRELATION RIMP2 SOSMP2 MOSMP2 sp (default) single point energy evaluation opt geometry optimization with analytical gradient force evaluation with analytical gradient user’s choice (standard or user-defined: GENERAL or MIXED) corresponding auxiliary basis (standard or user-defined: AUX_GENERAL or AUX_MIXED no default $n$; use $\omega=n/1000$. The recommended value is $n=600$ ($\omega=0.6$ bohr${}^{-1}$) Optional Optional Turns on spin-component scaling with SCS-MP2(1), SOS-MP2(2), and arbitrary SCS-MP2(3)

Example 6.7  Example of SCS-MP2 geometry optimization

$molecule 0 1 C H 1 1.0986 H 1 1.0986 2 109.5 H 1 1.0986 2 109.5 3 120.0 0 H 1 1.0986 2 109.5 3 -120.0 0$end

$rem JOBTYPE opt EXCHANGE hf CORRELATION rimp2 BASIS aug-cc-pvdz AUX_BASIS rimp2-aug-cc-pvdz BASIS2 racc-pvdz Optional Secondary basis THRESH 12 SCF_CONVERGENCE 8 MAX_SUB_FILE_NUM 128 SCS 1 Turn on spin-component scaling DUAL_BASIS_ENERGY true Optional dual-basis approximation N_FROZEN_CORE fc SYMMETRY false SYM_IGNORE true$end


Example 6.8  Example of SCS-MI-MP2 energy calculation

$molecule 0 1 C 0.000000 -0.000140 1.859161 H -0.888551 0.513060 1.494685 H 0.888551 0.513060 1.494685 H 0.000000 -1.026339 1.494868 H 0.000000 0.000089 2.948284 C 0.000000 0.000140 -1.859161 H 0.000000 -0.000089 -2.948284 H -0.888551 -0.513060 -1.494685 H 0.888551 -0.513060 -1.494685 H 0.000000 1.026339 -1.494868$end

$rem EXCHANGE hf CORRELATION rimp2 BASIS aug-cc-pvtz AUX_BASIS rimp2-aug-cc-pvtz BASIS2 racc-pvtz Optional Secondary basis THRESH 12 SCF_CONVERGENCE 8 MAX_SUB_FILE_NUM 128 SCS 3 Spin-component scale arbitrarily SOS_FACTOR 0400000 Specify OS parameter SSS_FACTOR 1290000 Specify SS parameter DUAL_BASIS_ENERGY true Optional dual-basis approximation N_FROZEN_CORE fc SYMMETRY false SYM_IGNORE true$end


Example 6.9  Example of SOS-MP2 geometry optimization

$molecule 0 3 C1 H1 C1 1.07726 H2 C1 1.07726 H1 131.60824$end

$rem JOBTYPE opt METHOD sosmp2 BASIS cc-pvdz AUX_BASIS rimp2-cc-pvdz UNRESTRICTED true SYMMETRY false$end


Example 6.10  Example of MOS-MP2 energy evaluation with frozen core approximation

$molecule 0 1 Cl Cl 1 2.05$end

$rem JOBTYPE sp METHOD mosmp2 OMEGA 600 BASIS cc-pVTZ AUX_BASIS rimp2-cc-pVTZ N_FROZEN_CORE fc THRESH 12 SCF_CONVERGENCE 8$end