Level-shifting is an old technique that may facilitate SCF convergence in systems having small HOMO/LUMO gaps.468, 992, 404 If the gap is small, a simple Fock matrix diagonalization (Roothaan step) may alter the energetic ordering of the molecular orbitals, so that after re-populating the electrons according to the aufbau principle, the overall effect is a discontinuous switch in the electron configuration, and the SCF process fails to converge. To remedy this fluctuating SCF behavior, one can shift the diagonal elements of the virtual block of the Fock matrix (“level-shifting”) to increase the calculated HOMO/LUMO gap before diagonalization. With proper level-shifting, the energetic ordering of the molecular orbitals is preserved during diagonalization and thus the shapes of the orbitals are changed in a continuous way at each SCF cycle, leading to a stable iterative process. Using perturbation theory, one can show that a proper level shift guarantees that the total energy is lowered after Fock matrix diagonalization.468, 992 It is important to note, however, that SCF solutions obtained via level-shifting are not necessarily stable ground states. To check the stability of the converged electronic state, we recommend invoking the keywords STABILITY_ANALYSIS or INTERNAL_STABILITY (see Section 4.3.3 and Section 4.5.15).
In cases where DIIS or some other SCF algorithm converges quickly, application of a level shift usually slows down convergence. In addition, experiences show that level-shifting can converge difficult SCF cases to moderate thresholds such as , but becomes less efficient as the convergence threshold is tightened, say, to . To obtain tightly-converged solutions, level-shifting should be combined with DIIS or another convergence algorithm. For difficult SCF cases, invoking level-shifting in the early SCF iterations and then turning it off later (in favor of DIIS, say) is often the best strategy. As such, Q-Chem offers a hybrid SCF algorithm that combines level-shifting with DIIS. This is invoked by setting SCF_ALGORITHM = LS_DIIS. Level-shifting can also be used on its own (without DIIS), by means of the following $rem variables.
Note: If either GAP_TOL or LSHIFT or both are explicitly specified, then LEVEL_SHIFT is automatically set to TRUE. Setting LEVEL_SHIFT = FALSE disables any values of GAP_TOL and LSHIFT. Invoking the LS_DIIS algorithm also disables any setting of LEVEL_SHIFT.
$molecule 0 1 U -0.7734808 -0.8815596 -0.8853446 O -1.3090665 -2.1863261 -2.7399692 O -1.6134743 1.0032462 -1.9673881 O -0.2537507 0.4215612 0.9749395 O 0.0643962 -2.7662217 0.1985884 O -2.4384926 -1.2003830 -0.1700214 O 0.8915310 -0.5627363 -1.6006679 H -0.5266214 -2.1731728 -3.3131242 H -2.5519330 1.0551966 -1.7276040 H 0.6520521 0.7395638 0.8360306 H -0.6807322 -3.1486006 0.6879451 $end $rem METHOD = B3LYP BASIS = LANL2DZ ECP = fit-LANL2DZ MAX_SCF_CYCLES = 200 THRESH = 10 LEVEL_SHIFT = TRUE GAP_TOL = 200 LSHIFT = 200 SCF_CONVERGENCE = 5 $end
$molecule 6 2 O 0.7515076 1.2954050 1.0605230 O 0.7506760 -1.2982554 1.0594277 Co 0.0016554 -0.0007951 0.0017848 O -1.4949030 -0.0008880 1.0616115 O 1.4981395 -0.0006773 -1.0578583 O -0.7482665 -1.2970503 -1.0566523 O -0.7473745 1.2963844 -1.0559284 $end $rem METHOD = B3LYP BASIS = 6-31G SCF_ALGORITHM = LS_DIIS MAX_SCF_CYCLES = 200 GUI = 2 GAP_TOL = 100 LSHIFT = 200 SCF_CONVERGENCE = 8 STABILITY_ANALYSIS = TRUE $end