Chapter 2 Instantaneous Normal Mode Theory for Clusters
2.2 Instantaneous normal mode analysis
2.2.4 Relation to velocity autocorrelation function
The displacements of atoms in a cluster are confined within a finite region in space and do not continue to increase with time [68]as in the bulk liquids whose mean square displacements of atoms show the asymptotic behavior of growing linearly with time so that, predicted by mean square displacements of its atoms, the diffusion coefficient of atom in a cluster should be zero. As a result, there remains an ambiguity in the study of the dynamical properties of a cluster that resorts to the mean-square displacement of atoms. Even so, the velocity autocorrelation function (VAF) also provides fruitful information of dynamical properties in a cluster, some methods were proposed to calculate the diffusion coefficient of a cluster via the VAF and will be discussed in the Chapter 7. Here, I will discuss the VAF and its power spectrum of a cluster. The definition of the VAF for a cluster is given as the following formula by Eq. (2.46) is a general form adequate to a cluster even containing different species of atoms. On the other hand, in the calculation of the VAF due to the vibrational motions of atoms in a cluster, the velocities of individual atoms should be corrected by subtracting out the parts corresponding to translational as well as the rotational motion of the whole cluster. Failing to make these corrections will result in a non-zero value of the long-time limit of C(t). For a cluster at low temperatures, the C(t) will exhibits a behavior like a under-damping oscillation as a result of atoms in the solid-like structure vibrating at a equilibrium position owing to thermal energy. As raising to high temperatures, the behavior at the beginning of C(t) will decrease to a dip and, after subsequently going through the dip, follows by a decaying tail, which is a characteristics of liquid-like behavior corresponding to the
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spikes broadened by thermal disturbance; in the liquid-like phase, the power spectrum will become smoother and, at zero frequency, give a nonzero value related to the diffusion phenomenon. These features are similar as for the bulk systems.
Now, I will give an short-time approximation for the VAF with its definition in Eq. (2.46) by considering the velocities of particles in Eq. (2.46) described under the INM coordinates , so that the associated power spectrum in Eq. (2.47) also can be described under INM approximation. I give the formal derivation in Appendix A.2. By connecting the relation between the INM coordinates and mass-weighted coordinates in Eq. (2.13) and the equation of motion of INMs in Eq. (2.21), we have
been separated into the contributions due to the stable and unstable branches. I should point out that the formula in Eq. (2.48) with both stable and unstable branches accurately predicts C(t) in the short-time regime; however, the prediction diverges badly in the long-time limit due to the imaginary-frequency term. In the stable-INM approximation [86,[87], in which the contribution due to the imaginary-frequency lobe is neglected, the C(t)is further simplified to
After making the inverse cosine transform for Eq.(2.49), the power spectral density in the stable-INM approximation is given as
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Correspondingly, in the stable-INM approximation, the C(t)is further simplified into
The power spectral density in the stable-INM approximation is given as
INM,s D( )s I( )s
, (2.55)
which can be compared with, , the power spectrum of a cluster defined in Eq. (2.47).
In general, the behaviors of C(t) and its power spectral density are very different between bulk systems in the solid and liquid phases. However, for a cluster system, not only the two quantities vary with temperature, but also the contributions of individual atom to C(t) and are very sensitive with its corresponding special position in the cluster structure at of lower temperatures. Thus a method that gives a quantitative and deeper insight is to dissect the contributions of individual atom to the VAF of the whole cluster, and I consider the normalized VAF for the j-th atom in a cluster as
(0) ( ) average over atoms in C(t) is carried out to improve the statistical accuracy since the dynamical motion for each of all atoms is assumed the same, while in the solid, in which atoms are fixed in position, and characterized by arrangements of their neighbors, Cj(t) distinct from each other depend on their corresponding local structures. This feature for cluster systems will become more explicit than bulk system due to its shell structures. From Eq. (2.56), one can examine the behavior of the VAF of individual atom and its related power spectrum,
( ) 2
0( ) cos( )
j
C t
j t dt
. (2.57)Since the Cj(t) exhibits different characteristics for atoms at different positions in the structure of the cluster. The atomic power spectrum of the solid-like phase will depend on its position in the structure.
However, in the liquid-like phase, the atoms of a cluster will mix together and are not characterized by their positions. Consequently, the atomic power spectra averaged over configurations are only distinguishable for the species with different masses.
According to the definition of Cj(t) in Eq. (2.56), there is no summation over atoms. In the INM approximation, Cj(t) can be formulated readily by inferring from Eq.(2.48) as
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where the denominator equals and the INM DOS have been separated into the stable and unstable branches as in Eq. (2.48). In the stable-INM approximation [86,[87], the Cj(t)is further simplified to
INM,s ( )
Also, at low temperatures the imaginary-frequency INMs are absent and the two quantities and are expected to be equal.
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