12.6.2 Generalized SCF-MI Calculations and Additional Features

The original definition of the ALMOs used in SCF-MI calculations is based on the fragment-blocking structure of the AO-to-MO transformation matrix, i.e., for a given fragment, the associated MOs can only be expanded by the AO basis functions centered on the atoms that belong to the same fragment. Here we propose a generalized definition for SCF-MI calculations: given a set of basis vectors ($𝐆$) in which each of them is tagged to a fragment but is allowed to be spanned by any AO basis function, it defines the working basis of the SCF-MI problem. Then, within this basis, the locally projected SCF equations can be solved in a similar way, with the constraint that the MO coefficient matrix in the working basis ($𝐆$) is fragment-block-diagonal, while the MO coefficient matrix in the AO basis does not necessarily retain the blocking structure. The basis vectors in $𝐆$ can be either non-orthogonal or orthogonal between fragments. More details on the generalized SCF-MI equations are available in Ref.  556 Horn P. R., Head-Gordon M.
J. Chem. Phys.
(2015), 143, pp. 114111. Link .

This generalized SCF-MI scheme is implemented in GEN_SCFMAN (the original AO-block based scheme is available in GEN_SCFMAN as well). It is used for the variational optimization of the polarized (but CT-forbidden) intermediate state in “EDA2" (see Section 12.6.3). Another preferable feature of this generalized scheme is that the interfragment linear dependency in $𝐆$ can be properly handled. Therefore, this scheme can be used to replace the original AO-block based SCF-MI without becoming ill-defined when interfragment linear dependency occurs. In contrast, the original ALMO-EDA method that employs the AO-block based approach fails when the sum of the number of orbitals on each fragment is not equal to the number of orbitals for the super-system (the latter is determined by the total number of AO basis functions and BASIS_LIN_DEP_THRESH), which often happens when substantially large basis sets are used or when the super-system comprises a large number of fragments.

SCF-MI calculations based on the GEN_SCFMAN implementation are triggered by setting GEN_SCFMAN = TRUE and FRGM_METHOD = STOLL or GIA (the other options of FRGM_METHOD are not allowed). A subset of supported algorithms in GEN_SCFMAN are available for restricted (R) and unrestricted (U) SCF-MI, including DIIS, GDM, GDM_LS, and NEWTON_CG. While the DIIS algorithm iteratively solves for the locally-projected SCF equations, the latter two methods use the energy derivatives with respect to the on-fragment orbital rotations to minimize the energy until the gradient reaches zero. As for standard calculations using GEN_SCFMAN, internal stability analysis is also available for R- and U-SCF-MI, and one can set FD_MAT_VEC_PROD to TRUE if the analytical Hessian is not available for the employed density functional (note: for functionals containing non-local correlation, one can always use FD_MAT_VEC_PROD = FALSE).

As in the original implementation, perturbative corrections can be applied on top of the SCF-MI solution to approach the full SCF result, and this is still controlled by FRGM_LPCORR. Note that among the options introduced in Section 12.5.5, only ARS and RS are allowed here since the exact SCF calculation is actually beyond the scope of SCF-MI.

In addition, with this more general implementation users are allowed to specify some fragments to be frozen during the SCF-MI calculation, i.e., intrafragment relaxation does not occur on these fragments. This is achieved by specifying the $rem variable SCFMI_FREEZE_SS. Such a calculation can be interpreted as an active fragment being embedded in a frozen environment where the interaction between them is treated quantum mechanically.