7.11. SCF Stability Analysis

The SCF stability analysis evaluates the electronic Hessian (with respect to orbital rotations) at the point indicated by the SCF solution to determine the lowest eigenvalues of the Hessian. If one or more negative eigenvalues are found, the SCF solution corresponds to a saddle point and not a true local minimum in the space considered in the analysis. A typical case are stretched bonds of diatomics, where the symmetry of the initial guess leads to a restricted solution instead of the often preferred unrestricted one. Several spaces are theoretically possible[780]. However, ORCA limits itself to the analysis RHF/RKS in the space of UHF/UKS or UHF/UKS in the space of UHF/UKS. As such, it is on the available for the SCF parts of DFT and HF.[80] We mention passing, that a stability analysis is also available for the CASSCF type wave function and is described elswhere in more detail (Section Detecting CASSCF Instabilities). In the following, HF is used to indicate both HF and KS. Consider the following input (unless indicated otherwise, default values are shown):

! BHLYP def2-SVP NORI

%scf 
 guess hcore # for illustrative purposes only 
 HFTyp UHF # default based on spin multiplicity
 STABPerform true # default false
 STABRestartUHFifUnstable true # restart the UHF-SCF if unstable 
 STABNRoots 3                  # number of eigenpairs sought
 STABMaxDim 3                  # Davidson expansion space = MaxDim * NRoots
 STABMaxIter 100               # maximum number of Davidson iterations
 STABNGuess 4096               # size of initial guess matrix: 4096 x 4096
 STABDTol 0.0001               # convergence criterion from iteration to iteration
 STABRTol 0.0001               # convergence criterion max residual norm
 STABlambda +0.5               # mixing parameter 
 STABORBWIN -1, -1, -1, -1,  -1, -1, -1, -1 # defines the donor / acceptor spaces
	                       # 4 parameters for RHF
                               # 8 paramters for UHF (4 alpha, 4 beta)
                               # orbital window, -1 refers to automatic determination
 STABEWIN -5.0, 5.0            # lower and upper cutoff in Eh for automatic freezing 
 #---------------------------------------------------------------------------------
 # alternative specification using a sub-block: 
 stab
  NRoots 3
  MaxDim 3 # etc.
 end
end

* xyz 0 1 
h 0.0 0.0 0.0
h 0.0 0.0 1.4
*

The determination of the electronic Hessian is structurally comparable to the TDHF/CIS/TDDFT procedure. Thus, many options are very similar and the user is encouraged to read the section on TDDFT (Section Excited States via RPA, CIS, TD-DFT and SF-TDA) to clarify some of the options given here. Since one is usually only interested in the qualitative determination “stable or not?”, three roots should be sufficient to find the lowest eigenvalue. By the same philosophy, StabMaxDim, StabMaxIter, StabNGuess and the convergence criteria were chosen. The parameter StabLambda refers to the \(\lambda\) of equation 37 of reference [780], which determines the mixing of the original SCF solution and the new orbitals to yield a new guess. Choosing this value is not trivial, since positive and negative values can lead to different new solutions (at least in principle). The convergence of the ensuing SCF depends on it, as well, since all SCF procedures require a sufficiently good guess to converge in a decent number of iterations (or even at all).

The orbital window and the energy window can be specified. Note that the StabEWIN will be overridden by the appropriate StabORBWIN values. The automatic determination is also influenced by the %method FrozenCore settings. Tests have shown that significant curtailing of the actual orbital window can drastically influence the results to the point of qualitative failure.

Current limitations on the method are:

  • Only single-point-like calculations are supported. For geometry optimizations etc., one must use the guess MORead feature Choice of Initial Guess and Restart of SCF Calculations to employ the guess obtained here. Likewise, one must extract a geometry and run a separate calculation if one is interested in the SCF stability.

  • As for TDDFT, NORI, RIJONX, and RIJCOSX are supported. RI-JK is not supported.

  • Other, more advanced features like finite-temperature calculations and relativistic calculations (beside ECPs) are not possible at this time.

Overall, the user is cautioned against using the stability analysis blindly without critically evaluating the result in terms of energy difference and by investigating the orbitals (by the printout or by plotting). Its usefulness cannot be denied, but it is certainly not black-box.

An SCF stability analysis with default settings can be requested via STABILITY, SCFSTABILITY, SCFSTAB or STAB on the simple input line.