Harmonics representation

At large displacements from the equilibrium position, the harmonic representation of the potential energy is invalid and a more realistic model is necessary. One simple function that is often employed is the Morse potential,... 

The hyperspherical and related coordinates which have been considered in this work have served for the visualization of critical features of potential energy surfaces [91,92], crucial for the understanding of reactivity (role of the ridge [93] and the kinetic paths [94]). In [95], the PES for the O + H2 reaction was studied. A discrete hyperspherical harmonics representation is presented in [96] for proton transfer in malonaldehyde. 

In the literature three different approaches were reported based on the spherical harmonics representation of the SODFs by Wang et by... 

Simplified Harmonics Representation of the Peak Shift. When it is not necessary to find the WSODFs and the average strain and stress tensors is not of interest, one can choose a different approach that corrects only for the line shifts caused by stress. In this case, an alternative representation for 7/ with fewer parameters is possible. To accomplish this, the angles (O, fi) in Equation (120) are replaced by the direction cosines u,-. After introducing this into Equation (119) and rearranging, 7 becomes ... 

In fact. Section 4.1 is geared to the spherical harmonic representation emphasizing that the functions involved are eigenfunctions of L and L, for which the unit vectors i /f matching the eigenfunctions are appropriate. Now the functions involved are common eigenfunctions of L , H, and the parity operators, calling for fhe representation... 

The distributed multipole analysis method of Stone and co-workers is similar in concept but is based on nonredundant spherical harmonic representation of the multipoles (recall that whereas there are six second moments, only five are independent). He initially places numerous site multipoles at centers of orbital overlap. The individual monopoles are spread out along the molecular axis, and are thought to represent the distribution of charge the site dipoles are also spread out along the bond axis. This very detailed description is simplified into a three-site model, which includes a site in the F—H bond. However, the multipole expansion does not converge well, especially for the bond center site. 

Potential coefficients coefficients in the spherical harmonic representation of the gravitational potential U by the equation... 

There are likely to be fundamental limitations to the Redfield approach that will render certain types of problems intractable. For example, if the bath is highly anharmonic and strongly coupled to the system, the use of a harmonic representation of the bath may be... 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

Since the transverse buckling in a square pile of side W is 27t2/W and of a circular pile is (2.4048)2/ij2 we find for the circular pile equivalent to a square one of side 272.2 cm a radius R = 147.3 cm. It should be pointed out that this procedure is not completely unambiguous since an accurate solution of the problem requires an infinite series of Bessel functions of order 4// however, it seems reasonable that such a choice of R gives the best one harmonic representation of this sum. An estimate of the error made in neglecting the fourth harmonic is given at the end of this report. 

For example, a tetrahedron corresponding to harmonic representation associated to s and p orbital t5T)es, carrying the orbital quantum number taking the values /= 0,1, is a regular tetrahedron, describing four equivalent and equidistant states. In quantum theory, it is said that these 4 states, described by so- called hybridization spi, are forming an orthogonal basis. 

Figure 10. The coefficients g(R) in a spherical harmonics representation of orientation correlation in liquid chloroform CHCil3...

In practice the evaluation of the ERIs in the real spherical harmonic representation runs over the Cartesian representation and a transformation occurs between the Cartesian and the real spherical harmonic representation (see Table 2). This is not due to the lack of explicit formulae for ERI generation in the real spherical harmonic Gaussian representation but rather a matter of implementation. The transformation is expressed as... 

Cartesian Versus Real Spherical Harmonic Representation... 

Figure 15 Spherical harmonic representations " of the mosaic virus Ivtm viral structure. Viral RNA is included in the helix. Structure was constructed with the symmetry server library within the AVS data-flow visualization environment. (Image courtesy of Bruce Duncan, The Scripps Research Institute, La Jolla, CA)...

Figure 22 Construction of the icosahedral capsid of poliovirus using a low-order spherical harmonic representation of the viral protomers (one copy each of the proteins VPl, VP2, VP3, and VP4). " The individual boundaries of the four protein chains are texture mapped onto the surface. One pentameric assembly intermediate has been translated away from the capsid along a five-fold axis. The model was constructed, and can be manipulated interactively in real time using the symmetry server library developed in the Olson group, within the AVS data-flow visualization environment. (Image courtesy of Arthur J. Olson, The Scripps Research Institute, La Jolla, CA)...

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, ... 

Fortunately, not all regions of the potential energy surface are of equal interest or importance. For stable molecules the region around the equilibrium structure determines the rotational constants, vibrational frequencies, etc. For most molecules the energy in this region can be satisfactorily represented by a polynomial expansion in the internal coordinates. For an N atom molecule a quadratic (harmonic) representation of the surface requires only (3N-6)(3N-5) energy calculations (after the equilibrium structure has been determined). ... 

For reactive molecular systems the transition state region, i.., the region around the saddle point along the reaction path connecting reactants and products, must also be characterized. Again, for many applications a harmonic representation of the surface suffices. 

Recently, Miller, Handy, and Adams derived a "Reaction Path" Hamiltonian for a polyatomic molecule (see also the chapter by Miller in this book), They recast the internuclear Hamiltonian in terms of a reaction coordinate, which is determined by the path of steepest descents, and (3N-7) normal coordinates which describe vibrations orthogonal to the reaction path. If the harmonic representation is desired for 2M distances along the reaction coordinate, i.g., M distances in the entrance channel and M in the exit channel, the number of energy evaluations required (over and above those necessary to find the reaction path) is M(3N-5)(3N-6). 

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