13.7 The Second-Generation ALMO-EDA Method

13.7.1 Generalized SCFMI Calculations and Additional Features

The original definition of the ALMOs used in SCFMI 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 SCFMI 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 SCFMI 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 SCFMI equations are available in ref. 390.

This generalized SCFMI 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 13.7.2). 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 SCFMI 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.

SCFMI 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) SCFMI, 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-SCFMI, and one can set FD_MAT_VEC_PROD to TRUE if the analytical Hessian is not available for the employed density functional.

As in the original implementation, perturbative corrections can be applied on top of the SCFMI solution to approach the full SCF result, and this is still controlled by FRGM_LPCORR. Note that among the options introduced in Section 13.6, only ARS and RS are allowed here since the exact SCF calculation is actually beyond the scope of SCFMI.

In addition, with this more general implementation users are allowed to specify some fragments to be frozen during the SCFMI 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.

SCFMI_MODE
       Determine whether generalized SCFMI is used and also the property of the working basis.
TYPE:
       INTEGER
DEFAULT:
       0 (“1" is used by basic “EDA2" calculations).
OPTIONS:
       0 AO-block based SCFMI (the original definition of ALMOs). 1 Generalized SCFMI with basis vectors that are non-orthogonal between fragments. 2 Generalized SCFMI with basis vectors that are orthogonal between fragments.
RECOMMENDATION:
       None

SCFMI_FREEZE_SS
       Keep the first several fragments unrelaxed in an SCFMI calculation.
TYPE:
       INTEGER
DEFAULT:
       0 (all fragments are active)
OPTIONS:
       n Freeze the first n fragments.
RECOMMENDATION:
       None

Example 13.12  Generalized SCFMI calculation for the water dimer with single Roothaan-step perturbative correction. For this specific case, the result is identical to that given by AO-block based SCFMI (SCFMI_MODE = 0).

$molecule
0 1
--
0 1
O  -1.551007  -0.114520   0.000000
H  -1.934259   0.762503   0.000000
H  -0.599677   0.040712   0.000000
--
0 1
O   1.350625   0.111469   0.000000
H   1.680398  -0.373741  -0.758561
H   1.680398  -0.373741   0.758561
$end

$rem
   JOBTYPE          sp
   METHOD           b3lyp
   GEN_SCFMAN       true
   BASIS            6-31+G(d)
   GEN_SCFMAN       true
   SCF_CONVERGENCE  8
   THRESH           14
   SYMMETRY         false
   SYM_IGNORE       true
   FRGM_METHOD      stoll
   FRGM_LPCORR      rs
   SCFMI_MODE       1 !gen scfmi (non-orthogonal)
$end