^{1}, M.-P. Gaigeot

^{2}, D. Borgis

^{3}and R. Vuilleumier

^{4,a)}

### Abstract

A general method for obtaining effective normal modes of a molecular system from molecular dynamics simulations is presented. The method is based on a localization criterion for the Fourier transformed velocity time-correlation functions of the effective modes. For a given choice of the localization function used, the method becomes equivalent to the principal mode analysis (PMA) based on covariance matrix diagonalization. On the other hand, a proper choice of the localization function leads to a novel method with a strong analogy with the usual normal modeanalysis of equilibrium structures, where the Hessian system at the minimum energy structure is replaced by the thermal averaged Hessian, although the Hessian itself is never actually calculated. This method does not introduce any extra numerical cost during the simulation and bears the same simplicity as PMA itself. It can thus be readily applied to *ab initio*molecular dynamics simulations. Three such examples are provided here. First we recover effective normal modes of an isolated formaldehyde molecule computed at in very good agreement with the results of a normal modeanalysis performed at its equilibrium structure. We then illustrate the applicability of the method for liquid phase studies. The effective normal modes of a water molecule in liquid water and of a uracil molecule in aqueous solution can be extracted from *ab initio*molecular dynamics simulations of these two systems at .

I. INTRODUCTION

II. THEORY

A. Fourier transformed velocity time-correlation functions

B. Localization of FTVCF

C. Effective normal modes from finite time simulations

D. Solution of the minimization problem

E. : Analogy with normal modeanalysis

F. : Comparison to PMA

G. Effective normal modes as a force-fitting procedure

1. case

2. case

H. Frame of reference

III. APPLICATIONS

A. Simulation details

B. Isolated formaldehyde molecule at

C. Water molecules in liquid water

D. Uracil in aqueous solution

IV. CONCLUSION

### Key Topics

- Normal modes
- 79.0
- Eigenvalues
- 20.0
- Molecular dynamics
- 17.0
- Infrared spectra
- 10.0
- Molecular liquids
- 8.0

## Figures

(Color) Power spectra of the effective normal modes of at in the Eckart framework rotating with the molecule (color lines) and the eigenvalues of the Hessian at (vertical line with a cross at the top).

(Color) Power spectra of the effective normal modes of at in the Eckart framework rotating with the molecule (color lines) and the eigenvalues of the Hessian at (vertical line with a cross at the top).

(Color) Effective normal modes (blue arrows) at and normal modes (red arrows) of . Modes have increasing frequency from left to right and top to bottom.

(Color) Effective normal modes (blue arrows) at and normal modes (red arrows) of . Modes have increasing frequency from left to right and top to bottom.

(Color) Power spectra of the effective normal modes in the rotating Eckart *frame* (a) and in the *laboratory* frame using the coordinate system of the Eckart frame (b) (see text). In both frames, the solid red curve corresponds to the power spectrum of the water bending mode, while the green curve and the black solid curves correspond to the power spectra of the symmetric and asymmetric stretches, respectively. In (b), the blue curves represent the power spectra of water translation modes (solid: translation perpendicular to the molecular plane, dashed: translation along the HH vector, dotted: translation along the OHO bisectrix), while the pink curves represent the power spectra of the librational modes (solid: rotation in the molecular plane, dashed and dotted: rotation around the two OH axes).

(Color) Power spectra of the effective normal modes in the rotating Eckart *frame* (a) and in the *laboratory* frame using the coordinate system of the Eckart frame (b) (see text). In both frames, the solid red curve corresponds to the power spectrum of the water bending mode, while the green curve and the black solid curves correspond to the power spectra of the symmetric and asymmetric stretches, respectively. In (b), the blue curves represent the power spectra of water translation modes (solid: translation perpendicular to the molecular plane, dashed: translation along the HH vector, dotted: translation along the OHO bisectrix), while the pink curves represent the power spectra of the librational modes (solid: rotation in the molecular plane, dashed and dotted: rotation around the two OH axes).

(Color) Effective normal modes of a water molecule in liquid water (bend, symmetric and asymmetric stretches), in the Eckart frame (blue arrows), and in the coordinate system associated to the Eckart frame (red arrows).

(Color) Effective normal modes of a water molecule in liquid water (bend, symmetric and asymmetric stretches), in the Eckart frame (blue arrows), and in the coordinate system associated to the Eckart frame (red arrows).

(Color) A ball and stick representation of the uracil molecule. Hydrogen atoms are white, carbon cyan, nitrogen are blue, and oxygen atoms are in red.

(Color) A ball and stick representation of the uracil molecule. Hydrogen atoms are white, carbon cyan, nitrogen are blue, and oxygen atoms are in red.

(Color) Power spectra of the effective normal modes of one uracil molecule in aqueous solution between 1000 and .

(Color) Power spectra of the effective normal modes of one uracil molecule in aqueous solution between 1000 and .

(Color) Effective normal modes of one uracil molecule in aqueous solution between 1000 and .

(Color) Effective normal modes of one uracil molecule in aqueous solution between 1000 and .

(Color) Effective normal modes of one uracil molecule in aqueous solution between 1400 and .

(Color) Effective normal modes of one uracil molecule in aqueous solution between 1400 and .

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