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(May 16, 2021)

Level-shifting is an old technique that may facilitate SCF convergence in
systems having small HOMO/LUMO gaps.
^{
453
}
Proc. Roy. Soc. Ser. A

(1970),
320,
pp. 161.
Link
^{,}
^{
960
}
Int. J. Quantum Chem.

(1973),
7,
pp. 699.
Link
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392
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Mol. Phys.

(1974),
28,
pp. 819.
Link
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.
^{
453
}
Proc. Roy. Soc. Ser. A

(1970),
320,
pp. 161.
Link
^{,}
^{
960
}
Int. J. Quantum Chem.

(1973),
7,
pp. 699.
Link
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 4.3.1 and Section 4.5.14).

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 ${10}^{-5}$, but becomes less efficient as the convergence threshold is tightened, say, to ${10}^{-8}$.
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.

LEVEL_SHIFT

Determine whether to invoke level-shifting or not together with DIIS.

TYPE:

LOGICAL

DEFAULT:

FALSE

OPTIONS:

TURE, FALSE

RECOMMENDATION:

Use TRUE if level-shifting is necessary to accelerate SCF convergence.

GAP_TOL

HOMO/LUMO gap threshold to control whether to shift the diagonal elements of the virtual block of the Fock matrix or not.
If the HOMO/LUMO gap is less than this threshold, at a given SCF iteration, then
the diagonal elements of the virtual block of the Fock matrix are shifted. Otherwise no level-shift is applied.

TYPE:

INTEGER

DEFAULT:

300

OPTIONS:

User-defined

RECOMMENDATION:

The input number must be an integer between 0 and 9999.
The actual threshold is equal to GAP_TOL divided by 1000, in Hartree.
The default value is provided to make the level-shifting calculation run and should not be taken as optimal for any specific problem.
Trial and error may be required to find the optimal threshold.
Larger values of GAP_TOL generally lead to
level-shifting being used more frequently during the SCF convergence process.

LSHIFT

Constant shift applied to all diagonal elements of the virtual block of the Fock matrix.

TYPE:

INTEGER

DEFAULT:

200

OPTIONS:

User-defined

RECOMMENDATION:

The input number must be an integer between 0 and 9999.
The actual shift is equal to GAP_TOL divided by 1000, in Hartree.
The default value is provided to make the level-shifting calculation run and should not be taken as optimal for any specific problem.
Trial and error may be required to find the optimal threshold.
Larger level shifts make the SCF process more stable but also slow down convergence, thus requiring more SCF cycles.

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.

MAX_LS_CYCLES

The maximum number of DIIS iterations with level-shifting when SCF_ALGORITHM = LS_DIIS. See also THRESH_LS_SWITCH.

TYPE:

INTEGER

DEFAULT:

MAX_SCF_CYCLES

OPTIONS:

1
Only a single DIIS step with level-shifting, and no level-shifting for the remaining DIIS steps.
$n$
$n$ DIIS iterations with level-shifting before turning level-shifting off.

RECOMMENDATION:

None

THRESH_LS_SWITCH

The threshold for turning off level-shifting in DIIS is ${10}^{-\text{THRESH\_LS\_SWITCH}}$ when
SCF_ALGORITHM is set to LS_DIIS. See also MAX_LS_CYCLES.

TYPE:

INTEGER

DEFAULT:

4

OPTIONS:

User-defined.

RECOMMENDATION:

None

$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