12.5.1 Locally-Projected SCF

Constrained locally-projected SCF is an efficient method for removing the SCF diagonalization bottleneck in calculations for systems of weakly interacting components such as molecular clusters and molecular complexes. ⁶⁴⁶ Khaliullin R. Z., Head-Gordon M., Bell A. T.
J. Chem. Phys.
(2006), 124, pp. 204105. Link ^{, 560} Horn P. R. et al.
J. Chem. Phys.
(2013), 138, pp. 134119. Link The method is based on the equations of the locally-projected SCF for molecular interactions (SCF-MI). ¹²²⁶ Stoll H., Wagenblast G., Preuss H.
Theor. Chem. Acc.
(1980), 57, pp. 169. Link ^{, 409} Gianinetti E., Raimondi M., Tornaghi E.
Int. J. Quantum Chem.
(1996), 60, pp. 157. Link ^{, 910} Nagata T. et al.
J. Chem. Phys.
(2001), 115, pp. 3553. Link ^{, 646} Khaliullin R. Z., Head-Gordon M., Bell A. T.
J. Chem. Phys.
(2006), 124, pp. 204105. Link ^{, 560} Horn P. R. et al.
J. Chem. Phys.
(2013), 138, pp. 134119. Link In the SCF-MI method, the occupied molecular orbitals on a fragment can be expanded only in terms of the atomic orbitals of the same fragment. Such constraints produce non-orthogonal MOs that are localized on fragments and are called absolutely-localized molecular orbitals (ALMOs). The ALMO approximation excludes charge-transfer from one fragment to another. It also prevents electrons on one fragment from borrowing the atomic orbitals of other fragments to compensate for incompleteness of their own AOs and, therefore, removes the BSSE from the interfragment binding energies. The locally-projected SCF methods perform an iterative minimization of the SCF energy with respect to the ALMOs coefficients. The convergence of the algorithm is accelerated with the locally-projected modification of the DIIS extrapolation method. ⁶⁴⁶ Khaliullin R. Z., Head-Gordon M., Bell A. T.
J. Chem. Phys.
(2006), 124, pp. 204105. Link

The ALMO approximation significantly reduces the number of variational degrees of freedom of the wave function. The computational advantage of the locally-projected SCF methods over the conventional SCF method grows with both basis set size and number of fragments. Although still cubic scaling, SCF-MI effectively removes the diagonalization step as a bottleneck in these calculations, because it contains such a small prefactor. In the current implementation, the SCF-MI methods do not speed up the evaluation of the Fock matrix and, therefore, do not perform significantly better than the conventional SCF in the calculations dominated by the Fock build.

Two locally-projected schemes are implemented. One is based on the locally-projected equations of Stoll et al., ¹²²⁶ Stoll H., Wagenblast G., Preuss H.
Theor. Chem. Acc.
(1980), 57, pp. 169. Link the other uses the locally-projected equations of Gianinetti et al.. ⁴⁰⁹ Gianinetti E., Raimondi M., Tornaghi E.
Int. J. Quantum Chem.
(1996), 60, pp. 157. Link These methods have comparable performance. The Stoll iteration is only slightly faster than the Gianinetti iteration but the Stoll equations might be a little bit harder to converge. The Stoll equations also produce ALMOs that are orthogonal within a fragment. The type of the locally-projected SCF calculations is requested by specifying either STOLL or GIA for the FRGM_METHOD keyword.