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 [834, 835]. The method is based on the equations of the locally-projected SCF for molecular interactions (SCF MI) [861, 862, 863, 834, 835]. 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 [834].
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. [861], the other utilizes the locally-projected equations of Gianinetti et al. [862] 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.
Example 12.302 Locally-projected SCF method of Stoll
$molecule
0 1
--
-1 1
B 0.068635 0.164710 0.123580
F -1.197609 0.568437 -0.412655
F 0.139421 -1.260255 -0.022586
F 1.118151 0.800969 -0.486494
F 0.017532 0.431309 1.531508
--
+1 1
N -2.132381 -1.230625 1.436633
H -1.523820 -1.918931 0.977471
H -2.381590 -0.543695 0.713005
H -1.541511 -0.726505 2.109346
H -2.948798 -1.657993 1.873482
$end
$rem
METHOD BP86
BASIS 6-31(+,+)G(d,p)
FRGM_METHOD STOLL
$end
$rem_frgm
SCF_CONVERGENCE 2
THRESH 5
$end
Locally-projected SCF cannot quantitatively reproduce the full SCF intermolecular interaction energies for systems with significant charge-transfer between the fragments (e.g., hydrogen bonding energies in water clusters). Good accuracy in the intermolecular binding energies can be achieved if the locally-projected SCF MI iteration scheme is combined with a charge-transfer perturbative correction [834]. To account for charge-transfer, one diagonalization of the full Fock matrix is performed after the locally-projected SCF equations are converged and the final energy is calculated as infinite-order perturbative correction to the locally-projected SCF energy. This procedure is known as single Roothaan-step (RS) correction [834, 282, 864]. It is performed if FRGM_LPCORR is set to RS. To speed up evaluation of the charge-transfer correction, second-order perturbative correction to the energy can be evaluated by solving the linearized single-excitation amplitude equations. This algorithm is called the approximate Roothaan-step correction and can be requested by setting FRGM_LPCORR to ARS.
Both ARS and RS corrected energies are very close to the full SCF energy for systems of weakly interacting fragments but are less computationally expensive than the full SCF calculations. To test the accuracy of the ARS and RS methods, the full SCF calculation can be done in the same job with the perturbative correction by setting FRGM_LPCORR to RS_EXACT_SCF or to ARS_EXACT_SCF. It is also possible to evaluate only the full SCF correction by setting FRGM_LPCORR to EXACT_SCF.
The iterative solution of the linear single-excitation amplitude equations in the ARS method is controlled by a set of NVO keywords described below.
Restrictions. Only single point HF and DFT energies can be evaluated with the locally-projected methods. Geometry optimization can be performed using numerical gradients. Wave function correlation methods (MP2, CC, etc..) are not implemented for the absolutely-localized molecular orbitals. SCF_ALGORITHM cannot be set to anything but DIIS, however, all SCF convergence algorithms can be used on isolated fragments (set SCF_ALGORITHM in the $rem_frgm section).
Example 12.303 Comparison between the RS corrected energies and the conventional SCF energies can be made by calculating both energies in a single run.
$molecule
0 1
--
0 1
O -1.56875 0.11876 0.00000
H -1.90909 -0.78106 0.00000
H -0.60363 0.02937 0.00000
--
0 1
O 1.33393 -0.05433 0.00000
H 1.77383 0.32710 -0.76814
H 1.77383 0.32710 0.76814
$end
$rem
METHOD HF
BASIS AUG-CC-PVTZ
FRGM_METHOD GIA
FRGM_LPCORR RS_EXACT_SCF
$end
$rem_frgm
SCF_CONVERGENCE 2
THRESH 5
$end
For some systems good accuracy for the intermolecular interaction energies can be achieved without converging SCF MI calculations and applying either the RS or ARS charge-transfer correction directly to the FRAGMO initial guess. Set FRGM_METHOD to NOSCF_RS or NOSCF_ARS to request the single Roothaan correction or approximate Roothaan correction, respectively. To get a somewhat better energy estimate set FRGM_METHOD to NOSCF_DRS and NOSCF_RS_FOCK. In the case of NOSCF_RS_FOCK, the same steps as in the NOSCF_RS method are performed followed by one more Fock build and calculation of the proper SCF energy. In the case of the double Roothaan-step correction, NOSCF_DRS, the same steps as in NOSCF_RS_FOCK are performed followed by one more diagonalization. The final energy in the NOSCF_DRS method is evaluated as a perturbative correction, similar to the single Roothaan-step correction.
Charge-transfer corrections applied directly to the FRAGMO guess are included in Q-Chem to test accuracy and performance of the locally-projected SCF methods. However, for some systems they give a reasonable estimate of the binding energies at a cost of one (or two) SCF step(s).
Evaluation of the basis-set superposition error (BSSE) is automated in Q-Chem. To calculate BSSE-corrected binding energies, specify fragments in the $molecule section and set JOBTYPE to BSSE. The BSSE jobs are not limited to the SCF energies and can be evaluated for multi-fragment systems at any level of theory. Q-Chem separates the system into fragments as specified in the $molecule section and performs a series of jobs on (a) each fragment, (b) each fragment with the remaining atoms in the system replaced by the ghost atoms, and (c) on the entire system. Q-Chem saves all calculated energies and prints out the uncorrected and the BSSE corrected binding energies. The $rem_frgm section can be used to control calculations on fragments, however, make sure that the fragments and the entire system are treated equally. It means that all numerical methods and convergence thresholds that affect the final energies (such as SCF_CONVERGENCE, THRESH, PURECART, XC_GRID) should be the same for the fragments and for the entire system. Avoid using $rem_frgm in the BSSE jobs unless absolutely necessary.
Important. It is recommended to include PURECART keyword in all BSSE jobs. GENERAL basis cannot be used for the BSSE calculations in the current implementation. Use MIXED basis instead.
Example 12.304 Evaluation of the BSSE corrected intermolecular interaction energy
$molecule
0 1
--
0 1
O -0.089523 0.063946 0.086866
H 0.864783 0.058339 0.103755
H -0.329829 0.979459 0.078369
--
0 1
O 2.632273 -0.313504 -0.750376
H 3.268182 -0.937310 -0.431464
H 2.184198 -0.753305 -1.469059
--
0 1
O 0.475471 -1.428200 -2.307836
H -0.011373 -0.970411 -1.626285
H 0.151826 -2.317118 -2.289289
$end
$rem
JOBTYPE BSSE
METHOD MP2
BASIS 6-31(+,+)G(d,p)
$end