Quasi-harmonic simulations

R. M. Levy, O. de la Luz Rojas, and R. A. Friesner. Quasi-harmonic method for calculating vibrational spectra from classical simulations on multidimensional anharmonic potential surfaces. J. Phys. Chem., 88 4233-4238, 1984. 

One of the advantages of the simulation of infrared and Raman spectra is the ability to separate the contribution that different conformers make to the overall spectrum. Mathieu and Grand [54] used a quasi-harmonic method using ab initio... 

Molecular dynamics (MD) simulations, the most suitable theoretical tool for the investigation of internal motions, can be used to explore both equilibrium properties and time-dependent phenomena. Based on both experimental and theoretical observations two models for the internal motion of proteins have been suggested. Within the framework of the first model internal motions arise from harmonic or quasi-harmonic vibrations that occur in a single multidimensional well on the potential energy surface [4,5,6,7]. The second model assumes that motions are a superposition of oscillations within a well and... 

With the advance of computing techniques classic LD programs have become more and more sophisticated. The PHONON program, provided from Daresbury Laboratory [69], is one such excellent example. PHONON uses the quasi-harmonic approximation and has a wide range of two body potentials embodied in the code. In addition, angular three-body bending potentials, four-body torsion potentials are also included. The program has been widely used for simulations of a variety of properties, such as dispersion curves, defects and surface phonons of crystalline and amorphous materials. 

The experimental and computational study of bacterial thioredoxin, an E. coli protein, at THz frequencies is presented. The absorption spectrum of the entire protein in water was studied numerically in the terahertz range (0.1 - 2 THz). In our work, the initial X-ray molecular structure of thioredoxin was optimized using the molecular dynamical (MD) simulations at room temperature and atmospheric pressure. The effect of a liquid content of a bacterial cell was taken into account explicitly via the simulation of water molecules using the TIP3P water model. Using atomic trajectories from the room-temperature MD simulations, thioredoxin s THz vibrational spectrum and the absorption coefficient were calculated in a quasi harmonic approximation. 

For both models, we used atomic trajectories from our room-temperature MD simulations to calculate thioredoxin s THz spectra in a quasi harmonic approximation. The absorption coefficient was calculated for different orientations of the molecule with respect to the electric field polarization. 

A normal mode calculation is based upon the assumption that the energy surface is quadratic in the vicinity of the energy minimum (the harmonic approximation). Deviations from the harmonic model can require corrections to calculated thermodynamic properties. One way to estimate anharmonic corrections is to calculate a force constant matrix using the atomic motions obtained from a molecular d)namics simulation such simulations are not restricted to movements on a harmonic energy surface. The eigenvalues and eigenvectors are then calculated for this quasi-harmonic force-constant matrix in the normal way, giving a model which implicitly incorporates the anharmonic effects. 

The idea of the Green s function/principal component analysis is closely related to the essential dynamics approach recently introduced into biomolecular simulations. Other similar works include those by Garcia, Ichiye and Karplus, Go and coworkers,and developers of the quasi-harmonic method. " The basic idea of the essential dynamics approach is to diagonalize a covariance matrix a whose elements are given by the formula... 

However, the drawback of ab initio calculations is that they usually refer to the athermal limit (T = 0 K), so that pressure but not temperature effects are included in the simulation. Although in principle the ab initio molecular dynamics approach[13] is able to overcome this limitation, at the present state of the art no temperature-dependent quantum-meehanieal simulations are feasible yet for mineral systems. Thus thermal properties have to be dealt with by methods based on empirical interatomic potential functions, containing parameters to be fitted to experimental quan-tities[14,15, 16]. The computational scheme applied here to carbonates is that based on the quasi-harmonic approximation for representing the atomic motion[17]. 

At low temperatures, if most of the anharmonic effects are due to lattice expansion, the quasi-harmonic approximation can be successfully applied. However, if the average displacement of the atoms is so large that the potential energy cannot be approximated by quadratic terms anymore, the approximation fails. In such cases, we can use a classical simulation method such as molecular dynamics to sample the phase space and calculate observables using these samples. We should note that this is strictly valid only in case of high temperatures, where Tmd Tqm-... 

Expression 11.3 provides the whole simulated spectrum, while a detailed vibrational analysis requires the unambiguous assignment of each mode contribution. Recently, a number of methods appeared in the literature aimed at the extraction of normal-mode-like analysis from ab initio dynamics [58-63]. Some of these [58-60] refer to the quasi-harmonic model introduced by Karplus [64,65] in the framework of classical molecular dynamics and individuate normal-mode directions as main components of the nuclear fluctuations in the NVE or NVT ensemble. The quasinormal model relies on the equipartition of the kinetic energy among normal modes thus problems arise when the simulation time required to obtain such a distribution is computationally too expensive, as is often the case for ab initio dynamics. Other approaches [61-63] carry out the time evolution analysis in the momenta subspace instead of the configurational space. In these approaches the basic consideration is that, at any temperature, generalized normal modes g, correspond to uncorrelated momenta such that [61]... 

Quasi-harmonic analysis is the computation of the normal modes of a molecule from atomic displacements generated by a molecular dynamics simulation. In this case, the atomic coordinate fluctuations are inversely related to the force constants, which are the second derivatives of the potential function. This formulation allows anharmonic motions, arising either from continuous diffusive motion or from transitions between wells, to be included implicitly within a harmonic representation, Brooks and co-workers " have carried out a comparison of different approaches to calculating the harmonic and quasiharmonic normal modes for the protein bovine pancreatic trypsin inhibitor (BPTI) with different force field and simulation models, Yet another approach, called essential dynamics, differs from quasi-harmonic analysis in that the atomic masses are not considered and motion is not reduced to a harmonic form, ... 

Quasi-harmonic analysis utilizes the atomic fluctuations calculated from a molecular dynamics simulation. Constmc-tion of a fluctuation matrix, which is inversely related to the... 

Normal mode frequencies that have been refined in the above fashion can also be employed to calculate thermodynamic properties. In the xfin 31 study, TSv and //vib at 300 K were calculated using (1) the original set of normal modes, (2) the set of adjusted 250 normal modes, (3) the set of adjusted 35 normal modes, and (4) the quasi-harmonic normal modes derived from the MD simulation, Both sets of adjusted normal modes were seen to be an improvement over the original set of normal modes in terms of reproducing the low frequency distribution of modes as well as the thermodynamic properties calculated from the quasi-harmonic modes derived from the MD trajectory. 

The effects of the circuit in the frequency domain were also characterized. The Fourier transform of the quasi-square waveform in Figure 8.41 was taken and the results shown in Fig. 8.44. Note that the third, fifth, seventh, and ninth harmonics are suppressed by about 40db, while the eleventh and thirteenth harmonics are about 20 dB less. The IsSpice simulation of this circuit was generated using the ICL feature of IsSpice. The format of the FOURIER command is shown below in Table 8.2. The resulting circuit characteristics in the frequency domain (Fig. 8.44) compare favorably to the resulting output from the IsSpice file (Table... 

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