Internal coordinate hessian target

1. Implementation of internal coordinate hessian target

Goal: replacement of vibrational frequencies fitting to internal coordinate hessian fitting.
Potential issue with the vibrational frequencies fitting: direct fit of MM sorted sequence of vibrational frequencies to QM sorted sequence of frequencies doesn’t guarantee the modes are correctly matched.

Target type	Description

Target type	Description
Vibrational frequencies target	compute QM and MM vibrational frequencies using normal mode analysis and sort from lowest to highest to yield sorted sequences respectively. Objective function contribution: Sum of squared differences of QM and MM frequencies, scaled by a factor of d=200cm-1
Internal coordinate hessian target	Compute MM cartesian hessian and convert into primitive redundant internal coordinates.

Targets: 10 targets from 2nd Gen. Roche set. (Same hessian data used to generate hessian and vibrational frequencies targets respectively);
Initial objective function(X2) set to 10 for each target;
Fitting result
- MAE, Error of overlap matrix measured and plotted after permuting MM vibrational modes to maximize overlap with QM vibrational modes;

	MAE : mean absolute error of MM vib. freq. to QM vib. freq. after rearrangement	Error of overlap matrix :\|\| M_overlap- I \|\|, Error of overlap matrix between QM and MM normal modes to identity matrix

	MAE : mean absolute error of MM vib. freq. to QM vib. freq. after rearrangement	Error of overlap matrix :\|\| M_overlap- I \|\|, Error of overlap matrix between QM and MM normal modes to identity matrix
Initial guess	199.03	6.09
Vib. freq. targets	166.35 (16.4 %)	6.07 (0.3 %)
Hessian targets	127.10 (36.1 %)	5.28 (13.3 %)

(Values in the table are averages of the 10 targets, values in parentheses are percent decrease.)

target #004
- Initial guess
- Fitted to vibrational frequencies targets
- Fitted to hessian targets (w/o normalization)

(1) normalization scheme

hess_normalize_type	Description

hess_normalize_type	Description
0	No normalization. Same weight on each hessian element.
1	weight on each element= 1/ sum of squared QM Hessian elements for each element type
2	weight1 + additional factor of 100 for diagonal elements