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(June 30, 2021)

The relaxed constraint algorithm (RCA) is an ingenious and simple means of minimizing the SCF energy that is particularly effective in cases where the initial guess is poor. The latter is true, for example, when employing a user-specified basis (when the “core” or GWH guess must be employed) or when near-degeneracy effects imply that the initial guess will likely occupy the wrong orbitals relative to the desired converged solution.

Briefly, RCA begins with the SCF problem as a constrained minimization of the
energy as a function of the density matrix,
$E(\mathbf{P})$.
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Int. J. Quantum Chem.

(2000),
79,
pp. 82.
Link
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J. Chem. Phys.

(2001),
114,
pp. 10616.
Link
The constraint is that the density
matrix be idempotent, $\mathbf{P}\cdot \mathbf{P}=\mathbf{P}$, which basically forces the
occupation numbers to be either zero or one. The fundamental realization of
RCA is that this constraint can be relaxed to allow sub-idempotent density
matrices, $\mathbf{P}\cdot \mathbf{P}\le \mathbf{P}$. This condition forces the occupation
numbers to be between zero and one. Physically, we expect that any state with
fractional occupations can lower its energy by moving electrons from higher
energy orbitals to lower ones. Thus, if we solve for the minimum of $E(\mathbf{P})$
subject to the relaxed sub-idempotent constraint, we expect that the ultimate
solution will nonetheless be idempotent.

In fact, for Hartree-Fock this can be rigorously proven. For density functional theory, it is possible that the minimum will have fractional occupation numbers but these occupations have a physical interpretation in terms of ensemble DFT. The reason the relaxed constraint is easier to deal with is that it is easy to prove that a linear combination of sub-idempotent matrices is also sub-idempotent as long as the linear coefficients are between zero and one. By exploiting this property, convergence can be accelerated in a way that guarantees the energy will go down at every step.

The implementation of RCA in Q-Chem closely follows the “Energy DIIS”
implementation of the RCA algorithm.
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J. Chem. Phys.

(2002),
116,
pp. 8255.
Link
Here, the current
density matrix is written as a linear combination of the previous density
matrices:

$$\mathbf{P}(x)=\sum _{i}{x}_{i}{\mathbf{P}}_{i}$$ | (4.39) |

To a very good approximation (exact for Hartree-Fock) the energy for $\mathbf{P}(x)$ can be written as a quadratic function of x:

$$\mathbf{E}(x)=\sum _{i}{E}_{i}{x}_{i}+\frac{1}{2}\sum _{i}{x}_{i}({\mathbf{P}}_{i}-{\mathbf{P}}_{j})\cdot ({\mathbf{F}}_{i}-{\mathbf{F}}_{j}){x}_{j}$$ | (4.40) |

At each iteration, $x$ is chosen to minimize $\mathbf{E}(x)$ subject to the constraint that all of the ${x}_{i}$ are between zero and one. The Fock matrix for $\mathbf{P}(x)$ is further written as a linear combination of the previous Fock matrices,

$$\mathbf{F}(x)=\sum _{i}{x}_{i}{\mathbf{F}}_{i}+\delta {\mathbf{F}}_{xc}(x)$$ | (4.41) |

where $\delta {\mathbf{F}}_{xc}(x)$ denotes a (usually quite small) change in the exchange-correlation part that is computed once $x$ has been determined. We note that this extrapolation is very similar to that used by DIIS. However, this procedure is guaranteed to reduce the energy $\mathbf{E}(x)$ at every iteration, unlike DIIS.

In practice, the RCA approach is ideally suited to difficult convergence
situations because it is immune to the erratic orbital swapping that can occur
in DIIS. On the other hand, RCA appears to perform relatively poorly near
convergence, requiring a relatively large number of steps to improve the
precision of a good approximate solution. It is thus advantageous in many
cases to run RCA for the initial steps and then switch to DIIS either after
some specified number of iterations or after some target convergence threshold
has been reached. Finally, note that by its nature RCA considers the energy as
a function of the density matrix. As a result, it cannot be applied to
restricted open shell calculations which are explicitly orbital-based. Note:
RCA interacts poorly with INCDFT, so INCDFT is disabled by default when an RCA
or RCA_DIIS calculation is requested. To enable INCDFT with such a
calculation, set INCDFT = 2 in the *$rem* section. RCA may also have
poor interactions with incremental Fock builds; if RCA fails to converge,
setting INCFOCK = FALSE
may improve convergence in some cases.

Job-control variables for RCA are listed below, along with an example of its use.

RCA_PRINT

Controls the output from RCA SCF optimizations.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
No print out
1
RCA summary information
2
Level 1 plus RCA coefficients
3
Level 2 plus RCA iteration details

RECOMMENDATION:

None

MAX_RCA_CYCLES

The maximum number of RCA iterations before switching to DIIS when
SCF_ALGORITHM is RCA_DIIS.

TYPE:

INTEGER

DEFAULT:

50

OPTIONS:

N
N RCA iterations before switching to DIIS

RECOMMENDATION:

None

THRESH_RCA_SWITCH

The threshold for switching between RCA and DIIS when SCF_ALGORITHM is RCA_DIIS.

TYPE:

INTEGER

DEFAULT:

3

OPTIONS:

N
Algorithm changes from RCA to DIIS when Error is less than ${10}^{-N}$.

RECOMMENDATION:

None

$molecule 0 2 H 1.004123 -0.180454 0.000000 O -0.246002 0.596152 0.000000 O -1.312366 -0.230256 0.000000 $end $rem UNRESTRICTED true METHOD hf BASIS cc-pVDZ SCF_GUESS gwh SCF_ALGORITHM RCA_DIIS THRESH 9 $end