^{1}, D. Van Neck

^{1}, V. Van Speybroeck

^{1}, T. Verstraelen

^{1}and M. Waroquier

^{1,a)}

### Abstract

In this paper the authors develop a method to accurately calculate localized vibrational modes for partially optimized molecular structures or for structures containing link atoms. The method avoids artificially introduced imaginary frequencies and keeps track of the invariance under global translations and rotations. Only a subblock of the Hessian matrix has to be constructed and diagonalized, leading to a serious reduction of the computational time for the frequency analysis. The mobile block Hessian approach (MBH) proposed in this work can be regarded as an extension of the partial Hessian vibrational analysis approach proposed by Head [Int. J. Quantum Chem.65, 827 (1997)]. Instead of giving the nonoptimized region of the system an infinite mass, it is allowed to move as a rigid body with respect to the optimized region of the system. The MBH approach is then extended to the case where several parts of the molecule can move as independent multiple rigid blocks in combination with single atoms. The merits of both models are extensively tested on ethanol and di--octyl-ether.

This work is supported by the Fund for Scientific Research-Flanders and the Research Board of Ghent University. This work was partly performed within the framework of the SBO-BIPOM program of the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).

I. INTRODUCTION

II. BACKGROUND AND THEORETICAL DEVELOPMENT

A. Normal modes in nonequilibrium configurations

B. Partial Hessian vibrational analysis

C. The mobile block Hessian approach

D. Discussion: PHVA versus MBH

III. APPLICATION TO THE ETHANOL MOLECULE

A. PHVA and MBH applied to the equilibrium structure

B. PHVA and MBH applied to partially optimized structures

IV. EXTENSION: MULTIPLE MOBILE BLOCKS

V. APPLICATION TO DI--OCTYL-ETHER

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Eigenvalues
- 26.0
- Normal modes
- 26.0
- Vibration analysis
- 11.0
- Frequency analyzers
- 10.0
- Ethanol
- 8.0

## Figures

Schematic representation of the basic idea behind the MBH method. The shaded blocks symbolize the parts of the molecule of which the internal geometry is kept fixed during the partial geometry optimization. In the MBH approach, they are described as rigid bodies with six degrees of freedom (translations and rotations).

Schematic representation of the basic idea behind the MBH method. The shaded blocks symbolize the parts of the molecule of which the internal geometry is kept fixed during the partial geometry optimization. In the MBH approach, they are described as rigid bodies with six degrees of freedom (translations and rotations).

Vibrational contribution to the entropy and the free enthalpy calculated with PHVA (엯) and MBH (×) frequencies are given for the different partially optimized ethanol configurations at . Benchmark values are indicated by the dashed lines. The fixed block in the MBH calculation consists of the atoms in the shaded box.

Vibrational contribution to the entropy and the free enthalpy calculated with PHVA (엯) and MBH (×) frequencies are given for the different partially optimized ethanol configurations at . Benchmark values are indicated by the dashed lines. The fixed block in the MBH calculation consists of the atoms in the shaded box.

Specification of the various configurations of di--octyl-ether with rigid bodies indicated as shaded regions. Atoms in shaded boxes are fixed at positions during the partial geometry optimization at the level.

Specification of the various configurations of di--octyl-ether with rigid bodies indicated as shaded regions. Atoms in shaded boxes are fixed at positions during the partial geometry optimization at the level.

Lowest frequencies (in ) of di--octyl-ether based on the full Cartesian Hessian belonging to the various partially optimized configurations defined in Fig. 3. Partial optimization at the level. Plot on the left displays the exact normal mode frequencies (full geometry optimization) that serve as benchmark.

Lowest frequencies (in ) of di--octyl-ether based on the full Cartesian Hessian belonging to the various partially optimized configurations defined in Fig. 3. Partial optimization at the level. Plot on the left displays the exact normal mode frequencies (full geometry optimization) that serve as benchmark.

Lowest frequencies (in ) of di--octyl-ether based on the multiple MBH model belonging to the various partially optimized configurations defined in Fig. 3. Partial geometry optimization at the level. Plot on the left displays the exact normal mode frequencies (full geometry optimization) that serve as benchmark for the other plots where two rigid bodies (defined by the configuration label) are taken into account in the frequency analysis.

Lowest frequencies (in ) of di--octyl-ether based on the multiple MBH model belonging to the various partially optimized configurations defined in Fig. 3. Partial geometry optimization at the level. Plot on the left displays the exact normal mode frequencies (full geometry optimization) that serve as benchmark for the other plots where two rigid bodies (defined by the configuration label) are taken into account in the frequency analysis.

The PHVA method implies the introduction of one block. For the multiple MBH method, two rigid blocks were used.

The PHVA method implies the introduction of one block. For the multiple MBH method, two rigid blocks were used.

Square of the overlap between the MBH normal modes and the benchmark normal mode frequencies of di--octyl-ether. The sum of the strengths is always normalized to 1 for each MBH frequency.

Square of the overlap between the MBH normal modes and the benchmark normal mode frequencies of di--octyl-ether. The sum of the strengths is always normalized to 1 for each MBH frequency.

## Tables

Normal mode frequencies (in ) of ethanol derived from the benchmark geometry, which corresponds to the geometry optimization obtained at . The rigid body is composed of the atoms in the shaded region. In the left column, translational and rotational frequencies from the full Hessian calculation are plotted before and after projection. Vibrational frequencies are not affected by this projection. The PHVA and MBH frequencies were ordered according to the maximum overlap with the benchmark modes.

Normal mode frequencies (in ) of ethanol derived from the benchmark geometry, which corresponds to the geometry optimization obtained at . The rigid body is composed of the atoms in the shaded region. In the left column, translational and rotational frequencies from the full Hessian calculation are plotted before and after projection. Vibrational frequencies are not affected by this projection. The PHVA and MBH frequencies were ordered according to the maximum overlap with the benchmark modes.

Normal mode frequencies (in ) of ethanol derived on the basis of partially optimized geometries at the level of theory. The rigid body is composed of atoms in the shaded region and its geometry is originated from a geometry optimization of the whole molecule at the low level. Benchmark frequencies are given in the left column for comparison. The PHVA and MBH frequencies were ordered according to the maximum overlap with the benchmark modes.

Normal mode frequencies (in ) of ethanol derived on the basis of partially optimized geometries at the level of theory. The rigid body is composed of atoms in the shaded region and its geometry is originated from a geometry optimization of the whole molecule at the low level. Benchmark frequencies are given in the left column for comparison. The PHVA and MBH frequencies were ordered according to the maximum overlap with the benchmark modes.

Frequencies (in ) of di--octyl-ether of various partially optimized configurations defined in Fig. 3 are compared with the benchmark frequencies of the fully optimized geometry (left column). Three approaches are used: the full Hessian calculation (Full), the PHVA method, and the MBH approach. The size of the rigid bodies is defined by the configuration label. The Full/PHVA/MBH frequencies are ordered according to the maximum overlaps with benchmark eigenmodes, which are given by the values between parentheses (in %).

Frequencies (in ) of di--octyl-ether of various partially optimized configurations defined in Fig. 3 are compared with the benchmark frequencies of the fully optimized geometry (left column). Three approaches are used: the full Hessian calculation (Full), the PHVA method, and the MBH approach. The size of the rigid bodies is defined by the configuration label. The Full/PHVA/MBH frequencies are ordered according to the maximum overlaps with benchmark eigenmodes, which are given by the values between parentheses (in %).

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