Optimal Displacement Increment for Numerical Frequencies (SFP)

This single-figure presentation (SFP) was submitted to the 2016 Virtual Winterschool on Computational Chemistry (http://winterschool.cc -- registration required). 

1H-pyrrolo[3,2-h]quinoline [Gorski, 2012] was optimized in ORCA v3.0.3 [Neese, 2012; http://orcaforum.cec.mpg.de] using RPBE [Perdew, 1992 and 1996] with the def2-TZVP basis sets [Weigend, 1998], and the def2-TZVP/J auxiliary bases [Weigend, 2006] for the RI approximation [Vahtras, 1992]. The nuclear Hessian, normal modes, and harmonic vibrational frequencies were computed using analytical (ANFREQ) and numerical (NUMFREQ) methodologies.  The numerical Hessians were computed with nuclear (Cartesian) displacement increments ranging from 0.0001 Bohr to 0.1 Bohr.  The geometry optimization was conducted using the parameters of the TIGHTOPT simple input keyword; KS-SCF and CP-SCF calculations used VERYTIGHTSCF thresholds.

An analysis was conducted of the quality of the numerical Hessians obtained, using the analytical Hessian as reference. The form of MAD[Delta] was chosen to consider both parallel and anti-parallel normal mode vectors as identical, and to accommodate any numerical glitches where the magnitude of a dot product might exceed unity. Where modes were found to be out of sequence, the appropriate elements of the list of vibrational frequencies were swapped to match.

A clear minimum was found in the deviation of both normal modes and vibrational frequencies from the analytical Hessian reference. The region where the best-matching numerical Hessians were found spanned displacement increments between 0.005 and 0.03 Bohr.

Future work will examine other molecular systems and theoretical methods in an effort to identify robust guidelines/approaches for selection of the displacement increment, particularly for cases where the analytical Hessian is unavailable.

The dataset with which the above analysis was performed is available at doi:10.5281/zenodo.44767.