The Augmented Roothaan-Hall method has been developed by T. Helgaker and coworkers and is extensively discussed in Ref. [271]. The basic idea of the method is that the density matrix is optimized directly to minimize total energy. At each step, the new density matrix is parametrized in terms of matrix exponent:
Pnew = exp(-X) Pold exp(X),
here, X is an anti-symmetric step matrix subject to the following conditions:
X = argmin{E(P(X))} - X minimizes the energy
|X| < h - length of X is smaller than or equal to some trust radius
The optimal X is found using a Conjugate Gradient method, possibly with pre-conditioning. The trust radius is updated based on how well the energy change is predicted.
ARHOPTIONS Input
It is possible to specify ARH options without turning on the method itself unconditionally. The ARHOPTIONS subkey has the same arguments as the ARH subkey (see below). It should be used in combination with the ALLOW ARH keyword. The ARH procedure will then be invoked automatically if SCF has trouble converging.
SCF
ARHOPTIONS ....
...
End
SYMMETRY NOSYM
ALLOW ARH
ARH Input
The ARH procedure is invoked using an ARH keyword in the SCF input block. This subkey ARH can be used in addition to the other subkeys of the block key SCF.
SCF
ARH {CONV=conv} {ITER=iter} {NSAVED=nsaved} {START=start}
{FINAL} ...
...
End
SYMMETRY NOSYM
All parameters in the ARH keyword are optional. The following arguments determine the main parameters of the ARH procedure.
CONV=conv
ARH convergence criterion. When the RMS gradient and its maximum components are both lower than the criterion, the ARH procedure will be considered converged. The default value is 10-4.
ITER=iter
Maximum number of ARH iteration to perform. Please note that in difficult cases a huge number of iterations may be required for complete SCF convergence. The default value is 500.
FINAL
Determines whether SCF is continued after ARH has completed. If this option is set, one Fock matrix diagonalization will be performed to get orbitals and the SCF procedure will be halted. By default this option is OFF.
START=start
Sets the SCF cycle number on which the ARH method is invoked. The default value is 2. Using a larger value may provide a better starting guess for the ARH minimization.
NSAVED=nsaved
Sets the number of saved density and Fock matrices used for augmentation of the electronic Hessian. The default value is 8. A larger nsaved value should be used in difficult cases when the number of orbitals very close to the Fermi level is large.
The default minimization method is Untransformed Pre-conditioned Conjugate Gradient. The following two parameters may be used to change this.
NOPRECOND
Disables pre-conditioning during the CG minimization. This option should not be used if atoms heavier than the second-row elements are present.
TRANSPCG
Specifying this option will enable the use of the Transformed Pre-conditioned CG method, which may result in better SCF convergence in some cases.
At each SCF step, the procedures begins by performing usual CG minimization keeping track of the total step length. If at some micro-iteration the step length exceeds the trust radius, the procedure switches to trust-radius optimization in the reduced space, which, in turn, is halted as soon as the level-shift parameter mu has converged. The final step is then calculated as a Newton step in the reduced space of all the trial vectors generated during CG minimization. The following options may be used to modify this behavior.
NOSWITCHING
Setting this option turns OFF the switching from the normal CG to a trust-radius minimization in reduced space. Using this option helps to reduce the total number of SCF cycles is some cases.
SHIFTED
Setting this option will turn ON the trust-radius optimization from the first micro- iteration.
CGITER=cgiter
Sets the maximum number of micro-iterations.
The next two options determine the trust radius.
TRUSTR=trustr
Initial value for the trust radius. Default: 0.5
MAXTRUSTR=maxtrustr
The maximum trust radius value. This is set to 0.5 by default and should never be changed.
ARH Notes and Recommendations
Restriction: The method currently works for symmetry NOSYM calculations only. The NOSYM requirement comes from the fact that during direct optimization of the density matrix it may have a symmetry lower than that of the molecule.
The method requires the total energy to be calculated at each step, which makes it much more expensive compared to the standard SCF procedure that does not need or use the energy. Therefore, the method should only be used when the standard SCF procedure fails. Another complication caused by the use of the total energy is that somewhat higher integration accuracy may be required to get stable SCF convergence, and that the method may not be applicable in all cases. It is also recommended to use the ADDDIFFUSEFIT keyword to increase accuracy of the total energy and, thus, improve convergence. Please refer to the TOTALENERGY keyword for more information.
