Just like geometry optimization, there is no practical way to be 100% sure that you have the global minimum of SCF solutions.
But there are checks you can do to make sure that the SCF solution you got is a reasonable minimum. One of them is checking the electronic hessian at the SCF solution and determining the lowest eigenvalues. If one or more negative eigenvalues are found then the solution is a saddle point in the wavefunction space. Read Nike's answer on how to do this with Gaussian and CFOUR.
There are options to do this in other softwares as well, however, not all features might be available.
In Orca, you can request a stability analysis from
! B3LYP def2-SVP NoRI SP
# only single point calculations supported
STABPerform true # turns on SCF stability analysis
STABRestartUHFUnstable true # restart the UHF-SCF calculation if unstable
STABNRoots 3 # (default 3) number of lowest eigenvalues sought
STABlambda +0.5 # (default +0.5) mixing parameter for new UHF-SCF calculation
*xyz 0 1
H 0.0 0.0 0.0
H 0.0 0.0 1.4
STABlambda parameter determines how much the old SCF and new orbitals for the new guess. Note that the stability calculation can be done for both RHF/RKS and UHF/UKS references, but restarting the calculation if the solution is unstable (
STABRestartUHFUnstable) is only allowed for UHF/UKS references.
Refer to the Orca manual for other available options.
GAMESS can also do stability analysis, but only for RHF references. The keyword
UHFCHK=.TRUE. has to be added to
! B3LYP/aug-cc-PVDZ on H2
$BASIS GBASIS=ACCD $END
$CONTRL SCFTYP=RHF RUNTYP=ENERGY ISPHER=1
DFTTYP=B3LYP UHFCHK=.TRUE. $END
H 1.0 -7.56825 2.73003 0.00000
H 1.0 -6.86039 2.74400 0.00000
This checks pairs of orbitals for RHF->UHF instabilities. By default, it only checks HOMO-1, HOMO, LUMO, LUMO+1 and LUMO+2; but more orbitals can be checked by using the
NLUMO keywords in the same section.
You can also turn off any type of symmetry usage by adding the keyword
$CONTRL, and see if that changes the SCF solution.
Keep in mind though, that none of these methods are fool-proof and they should not be used as black-boxes. It is always good to check the orbital populations, energies, spin densities etc. to see if that makes chemical sense.
Also note that these codes generally perform calculations with Gaussian type basis sets, and mainly for small molecules. In your question, you used the word "phonon" which makes me think that you might be considering periodic systems with plane wave basis sets. I do not have any experience with those but I believe that the same principles should apply.