^{1}and R. Benny Gerber

^{1,2,a)}

### Abstract

A computational study is made of the number of important anharmonic mode-mode couplings in the context of vibrational calculations for di-, tri-, and tetrapeptides. The method employed is the correlation-corrected vibrational self-consistent field (CC-VSCF) algorithm, which includes correlation effects between different vibrational modes. It is found that results of good accuracy can be obtained in calculations that include only mode-mode coupling terms, where is the number of modes. This simplification significantly accelerates CC-VSCF calculations for large molecules. A criterion based on the characteristics of the normal-mode displacements is employed to predict *a priori* unimportant coupling terms. The criterion is tested statistically using Spearman’s rank correlation coefficient. The results are illustrated by calculations for several di-, tri-, and tetrapeptides using semiempirical PM3 potential surfaces. These results are analyzed and a statistical model for error estimation is given. The decrease in the number of included coupling from to opens possibilities of anharmonic vibrational calculations for large peptides.

We wish to thank Dr. Brina Brauer for her helpful discussions. The help of Professor Mark S. Gordon and of Bosiljka Njegic is gratefully acknowledged. We thank Yair Goldberg for his assistance in the statistical analysis. This research was supported by a grant from the U.S.-Israel Binational Science Foundation (BSF No. 2004009).

I. INTRODUCTION

II. METHODOLOGY

A. The CC-VSCF method

B. The motivation for calculating only some of the normal-mode pair couplings

C. Choosing the important mode-mode coupling potentials

D. Definition of the PIC measure

E. The statistical meaning of the PIC rank

F. Torsional normal-modes

G. Computational details

III. RESULTS

A. Criterion for evaluation

B. Model for error estimation

C. The motivation for considering set of the size of

D. Comparison with optimal choice of the best pairs

E. CPU times

IV. CONCLUSIONS

### Key Topics

- Particle-in-cell method
- 33.0
- Normal modes
- 13.0
- Peptides
- 13.0
- Semi empirical calculations
- 5.0
- Wave functions
- 5.0

## Figures

Histogram of the absolute coupling potential of all the normal-mode pairs for glycine and ValGlyVal. This histogram shows that most of the normal-mode pairs have a very low coupling potential, and only a few normal-mode pairs have a high coupling potential.

Histogram of the absolute coupling potential of all the normal-mode pairs for glycine and ValGlyVal. This histogram shows that most of the normal-mode pairs have a very low coupling potential, and only a few normal-mode pairs have a high coupling potential.

An illustration of the main concept of the PIC rank. This graph includes four pairs of normal modes. The first two pairs [(A) and (B)] have a high PIC rank, while the other two pairs [(C) and (D)] have a low PIC rank. The percentiles are calculated over all the normal-mode pairs of glycine. (Pair A) is an example of two normal modes that have a very high coupling potential and, indeed, have a very high PIC rank. The PIC rank of this pair is in the 100th percentile (the highest value) and the coupling potential is , which is in the 98.5th percentile. (Pair B) is another example of two normal modes that have a very high coupling potential and, indeed, have a very high PIC rank. The PIC rank of this pair is in the 98th percentile and the coupling potential is , which is in the 99.5th percentile. (Pair C) is an example of two normal modes that have a very weak coupling potential and, indeed, have a low PIC rank. The PIC rank of this pair is in the 0.3th percentile (the lowest value) and the coupling potential is , which is in the 1st percentile. (Pair D) is another example of two normal modes that have a weak coupling potential and, indeed, have a low PIC rank. The PIC rank of this pair is in the 7th percentile and the coupling potential is , which is in the 2nd percentile.

An illustration of the main concept of the PIC rank. This graph includes four pairs of normal modes. The first two pairs [(A) and (B)] have a high PIC rank, while the other two pairs [(C) and (D)] have a low PIC rank. The percentiles are calculated over all the normal-mode pairs of glycine. (Pair A) is an example of two normal modes that have a very high coupling potential and, indeed, have a very high PIC rank. The PIC rank of this pair is in the 100th percentile (the highest value) and the coupling potential is , which is in the 98.5th percentile. (Pair B) is another example of two normal modes that have a very high coupling potential and, indeed, have a very high PIC rank. The PIC rank of this pair is in the 98th percentile and the coupling potential is , which is in the 99.5th percentile. (Pair C) is an example of two normal modes that have a very weak coupling potential and, indeed, have a low PIC rank. The PIC rank of this pair is in the 0.3th percentile (the lowest value) and the coupling potential is , which is in the 1st percentile. (Pair D) is another example of two normal modes that have a weak coupling potential and, indeed, have a low PIC rank. The PIC rank of this pair is in the 7th percentile and the coupling potential is , which is in the 2nd percentile.

Profile of the mean errors according to the different subject type. The regression lines are as follows: For the glycines (glycine and diglycine to tetraglycine), with ; for the valines (only 2 values valine and divaline), with ; and for the alanines (alanine, dialanine, and trialanine), with .

Profile of the mean errors according to the different subject type. The regression lines are as follows: For the glycines (glycine and diglycine to tetraglycine), with ; for the valines (only 2 values valine and divaline), with ; and for the alanines (alanine, dialanine, and trialanine), with .

The distribution of the absolute difference between the PIC CC-VSCF and the full CC-VSCF. The distribution line is taken from exponential distribution with , which is the mean value of the errors of ValGlyVal.

The distribution of the absolute difference between the PIC CC-VSCF and the full CC-VSCF. The distribution line is taken from exponential distribution with , which is the mean value of the errors of ValGlyVal.

A plot which describe a comparison between the observed values and the expected normal values. If the observed variable matches the normal distribution, the points cluster around a straight line.

A plot which describe a comparison between the observed values and the expected normal values. If the observed variable matches the normal distribution, the points cluster around a straight line.

The average error of the CC-VSCF while using only the normal-mode pairs with the highest PIC rank.

The average error of the CC-VSCF while using only the normal-mode pairs with the highest PIC rank.

The average error of the CC-VSCF with (1) PIC , (2) normal-mode with the strongest coupling, and (3) normal-mode with the strongest coupling. In the first case, the regression line is with , in the second case, with , and in the third case, with .

The average error of the CC-VSCF with (1) PIC , (2) normal-mode with the strongest coupling, and (3) normal-mode with the strongest coupling. In the first case, the regression line is with , in the second case, with , and in the third case, with .

The CPU time of the PIC version (the CPU time of CC-VSCF method when calculating only the normal-mode pairs with the highest PIC rank) and the CPU time of the full version (the CPU time of CC-VSCF method when calculating all the normal-mode pairs). The linear line is for the original version with , and for the improved version the line is with

The CPU time of the PIC version (the CPU time of CC-VSCF method when calculating only the normal-mode pairs with the highest PIC rank) and the CPU time of the full version (the CPU time of CC-VSCF method when calculating all the normal-mode pairs). The linear line is for the original version with , and for the improved version the line is with

## Tables

Spearman’s rank correlation coefficient for several peptides. The correlation is between the PIC rank and the coupled potential for the group of the entire normal-mode pairs for each subject. All the correlations were found to be significant with -value smaller than 0.0005.

Spearman’s rank correlation coefficient for several peptides. The correlation is between the PIC rank and the coupled potential for the group of the entire normal-mode pairs for each subject. All the correlations were found to be significant with -value smaller than 0.0005.

The accuracy of CC-VSCF method when using only the normal-mode pairs with the highest PIC rank.

The accuracy of CC-VSCF method when using only the normal-mode pairs with the highest PIC rank.

The accuracy of CC-VSCF method when using only the normal-mode pairs with the highest coupled potential (optimal choose).

The accuracy of CC-VSCF method when using only the normal-mode pairs with the highest coupled potential (optimal choose).

The percentage of the normal-modes pairs with highest potential among the normal-mode pairs with the highest PIC rank. Let be a group that contained the normal-mode pairs with the highest coupling potential among the normal-modes pairs. . Let be a group that contains the normal-modes pairs, chosen, using the PIC criteria. .

The percentage of the normal-modes pairs with highest potential among the normal-mode pairs with the highest PIC rank. Let be a group that contained the normal-mode pairs with the highest coupling potential among the normal-modes pairs. . Let be a group that contains the normal-modes pairs, chosen, using the PIC criteria. .

The CPU time of the PIC version (the CPU time of CC-VSCF method when calculating only the normal-mode pairs with the highest PIC rank) and the CPU time of the full version (the CPU time of CC-VSCF method when calculating all the normal-mode pairs).

The CPU time of the PIC version (the CPU time of CC-VSCF method when calculating only the normal-mode pairs with the highest PIC rank) and the CPU time of the full version (the CPU time of CC-VSCF method when calculating all the normal-mode pairs).

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