Harmonic analysis potentials

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

Harmonic analysis is an alternative approach to MD. The basic assumption is that the potential energy can be approximated by a sum of quadratic terms in displacements. 

Harmonic analysis (normal modes) at given temperature and curvature gives complete time behavior of the system in the harmonic limit [1, 2, 3]. Although the harmonic model may be incomplete because of the contribution of anharmonic terms to the potential energy, it is nevertheless of considerable importance because it serves as a first approximation for which the theory is highly developed. This model is also useful in SISM which uses harmonic analysis. 

Vibrational spectroscopy has played a very important role in the development of potential functions for molecular mechanics studies of proteins. Force constants which appear in the energy expressions are heavily parameterized from infrared and Raman studies of small model compounds. One approach to the interpretation of vibrational spectra for biopolymers has been a harmonic analysis whereby spectra are fit by geometry and/or force constant changes. There are a number of reasons for developing other approaches. The consistent force field (CFF) type potentials used in computer simulations are meant to model the motions of the atoms over a large ranee of conformations and, implicitly temperatures, without reparameterization. It is also desirable to develop a formalism for interpreting vibrational spectra which takes into account the variation in the conformations of the chromophore and surroundings which occur due to thermal motions. 

The low-lying excited states of the hydrogen molecule conhned in the harmonic potential were studied using the configuration interaction method and large basis sets. Axially symmetric harmonic oscillator potentials were used. The effect of the confinement on the geometry and spectroscopic constants was analyzed. Detailed analysis of the effect of confinement on the composition of the wavefunction was performed. 

Harmonic analysis was carried out on the specimens 7 days after the impedance measurements in order to allow the specimens to settle down again. An Ono Sokki CF 910 dual channel FFT analyser was used in conjunction with a potentiostat (Thompson Mlnistat 251) to hold the specimen at its rest potential and to provide the low frequency sine wave perturbation. The second channel was used to measure the harmonic content of the resulting current. The Ono Sokki produces a dlgltially generated high purity sine wave at a chosen frequency, in this instance, 0.5 Hz. The total harmonic content of the input sine wave was less than 0.45% measured over 10 harmonics. Only the first 3 harmonics are used to calculate the corrosion current. 

Recently, Darowicki [29, 30] has presented a new mode of electrochemical impedance measurements. This method employed a short time Fourier transformation to impedance evaluation. The digital harmonic analysis of cadmium-ion reduction on mercury electrode was presented [31]. A modern concept in nonstationary electrochemical impedance spectroscopy theory and experimental approach was described [32]. The new investigation method allows determination of the dependence of complex impedance versus potential [32] and time [33]. The reduction of cadmium on DM E was chosen to present the possibility of these techniques. Figure 2 illustrates the change of impedance for the Cd(II) reduction on the hanging drop mercury electrode obtained for the scan rate 10 mV s k... 

In the above example, by changing the capacitor bank to a 500-kVAR unit, the resonance frequency is increased to 490 Hz, or the 8.2 harmonic. This frequency is potentially less troublesome. (The reader is encouraged to work out the calculations.) In addition, the transformer and the capacitor bank may also form a series resonance circuit as viewed from the power source. This condition can cause a large voltage rise on the 480-V bus with unwanted results. Prior to installing a capacitor bank, it is important to perform a harmonic analysis to ensure that resonance frequencies do not coincide with any of the characteristic harmonic frequencies of the power system. 

In planar geometry (8 = 0), where there is no electronic coupling, Vge coincides with VBA and can be labeled as 2Bj and 2A1 respectively, with the former being bound and the latter repulsive. (A third, bound, state is not coupled to the dissociative repulsive state 2A by a displacement in 8 and is considered no further.) As 8 goes away from 8 = 0, the adiabatic curves Vg e are split by the electronic coupling. In the analysis [1,2], the VB curves are fit to model potentials, assuming a harmonic wag potential which is common to both, and the coupling has the form j3 = b 8,... 

In spectroscopic analysis the first term, Vo, is usually made zero by definition. This is done by assuming that each internal coordinate, let it be a given bond distance or bond angle, is strainless at its equilibriiun value. For example, the standard sp3-sp3 carbon-carbon bond length zq is 1.53 A. The harmonic stretching potential then is represented by ... 

Recent ab initio calculations have attempted to probe the fundamental source of the reversal of H/D preference in ionic as compared to neutral systems, using water as a test base. A harmonic analysis of the potential energy surface of the water dimer, computed with a 6-31G basis set, indicates that the preference for D in the bridging site can be explained in a manner similar to that described earlier for HF - HF. The frequency of the bending motion of the bridging atom is sensitive to its mass this effect leads to a lower vibrational energy of some 0.2 kcal/mol when the heavier D undergoes this motion. The computations indicated that electron correlation has little effect upon this conclusion, even its quantitative aspects. While the treatment was purely harmonic in nature, other calculations have indicated that anharmonicity effects yield very little distinction between one isotopomer and the next. 

The following sections develop three subjects the classical approximations for the strain/stress in isotropic polycrystals, isotropic polycrystals under hydrostatic pressure and the spherical harmonic analysis to determine the average strain/stress tensors and the intergranular strain/stress in textured samples of any crystal and sample symmetry. Most of the expressions that are obtained for the peak shift have the potential to be implemented in the Rietveld routine, but only a few have been implemented already. 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

Secondly, we have calculated from the same potential the equilibrium structure of each of the Van der Waals dimers and estimated the barriers to internal rotation of the monomers. We have also estimated the frequencies of the Van der Waals vibrations by means of a harmonic analysis. These Van der Waals vibrations are not directly observed in the photodissociation spectra, but since they affect the monomer orientations and the distance between the monomers, they influence the positions and the widths of the observed dimer peaks. 

Fig. 6.1. Laser cooling towards the zero-point energy of the motion using three lasers. The quantum number riy characterizes the quantized motion of a single Hg+ ion in the harmonic effective potential of the confining rf quadrupole trap (eigenfrequency jjjl K — 2.96 MHz, huj = 12 neV). Using side band laser transitions Any = —1 transitions can be pumped. Analysis of side band resolved spectra reveals that the system can be cooled so far that it occupies the ground state ny = 0 for 95% of the time, corresponding to a temperature of 47 fiK. As soon as the cooling laser is switched off, the ion motion is heated with a rate of (dny/dt) = 6/s. After 2 s it reaches a mean value of (riy) = 12 corresponding to T 1.7 mK.

Harmonic analysis (intrusive). This technique is related to EIS in that an alternating potential perturbation is applied to one sensor element in a three-element probe, with a resultant current response. Not only the primary frequency but higher-order harmonic oscillations are analyzed in this technique. Theory has been formulated whereby all kinetic parameters (including the Tafel slopes) can be calculated explicitly. No other technique offers this facility. At present, the technique remains largely unproven and rooted in the laboratory domain. 

The application of normal mode analysis to macromolecules such as proteins and nucleic acids has only recently become more common. Normal modes can be calculated either using harmonic analysis, where the second derivative matrix of the potential energy is calculated for a minimized structure, or using quasi-harmonic analysis, where the matrix of correlations of atomic displacements is calculated from a molecular dynamics (MD) trajectory. At temperatures below about 200 K, protein dynamics are primarily harmonic. Above this temperature there is appreciable non-harmonic motion which can be studied using quasi-elastic scattering techniques. There is evidence that such anharmonic motions are also important for protein function and quasi-harmonic analysis allows them to be incorporated implicitly to some extent within a harmonic model. 

Currently, there are several computational methods for the calculation of the normal modes of a molecule. Harmonic analysis is the direct computation of normal modes by finding the eigenvalues and eigenvectors, which are the solutions to the secular equation of motion involving the second derivatives of the potential function and mass weighted coordinates. It is essential that the calculation be started from a fully energy minimized conformation for the resulting normal mode frequencies to be accurate. 

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, ... 

The foregoing harmonic analysis of vital rhythms by Kamiya (40) is in itself commendable research work. But I wish to carry the conclusions still further. There is present in all protoplasm the potential capacity to pulsate rhythmically when nature needs to use this capacity to its fullest extent, as in heart muscle, or occasionally, as in intestine muscle, it is available. 

The harmonic potential is a good starting place for a discussion of vibrating molecules, but analysis of the vibrational spectrum shows that real diatomic... 

The sequence of levels shown in Figure 2 closely resembles the level diagram found by Mayer and Jensen by analysis of observed nuclear properties, with the help of the calculated level sequences for harmonic-oscillator and square-well potential func-... 

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