Potential harmonic expansion

Transition state theory is very often used in its harmonic approximation. The harmonic approximation is applicable under the normal assumptions of transition state theory, but further demands that the potential energy surface is smooth enough for a harmonic expansion of the potential energy to make sense. Since the harmonic expansion is performed in the initial state and in a first-order saddle point on the... 

By performing a normal mode analysis in the initial state and in the saddle point, it is then possible to obtain the harmonic expansion of the potential in the reactant region ... 

The L (R) matrix is chosen such that the matrix L (R)TF L (R) = Q (R) is diagonal with elements iv. The eigenvectors of F are arranged as columns in the L (R) matrix. It should also be noticed that V B sol(S (R), R) can no longer be written as a sum of an intramolecular (gas-phase potential) and an intermolecular part as in Eq. (10.18), because the harmonic expansion of the potential around the saddle point is based on the total potential energy surface and not just on the intramolecular part. By combining Eqs (10.19), (10.21), and (10.23) we see that the absolute position coordinates of the atoms in the activated complex around the saddle point of the total potential energy surface can be written as... 

It has been found useful to represent the interaction potential for a dimer of homonuclear diatomic molecules [4,5,46,58] as a spherical harmonic expansion, separating radial and angular dependencies. The radial coefficients include different types of contributions to the interaction potential (electrostatic, dispersion, repulsion due to overlap, induction, spin-spin coupling). For the three dimers of atmospheric relevance, we provided compact expansions, where the angular dependence is represented by spherical harmonics and truncating the series to a small number of physically motivated terms. The number of terms in the series are six for the N2-O2 systems, corresponding to the number of configurations of the dimer (for N2-N2 and O2-O2 this number of terms is reduced to five and four, respectively). 

Figure 10. The potential F in the presence of lich the motion of the Brownian particle takes place. The right well is (xmsideted. The dashed curve expresses the harnuniic e q>anaon of the potential V around x a. The r ions 0 < x < a and a.< x

Figure 11.18 Predictions of the tumbling parameter A as a function of reduced concentration C/C2 from the Smoluchowski equation for hard rods with the Onsager potential. The exact result from the spherical-harmonic expansion is shown, compared to approximate results from an analytic formula and from the perturbation expansion of Kuzuu and Doi. The open circles (O) are estimates from the periods of shear stress oscillations in transient shearing flows for PEG solutions (see Walker et al. 1995), and the closed circle ( ) is from a direct conoscopic measurement of Muller et al. (1994).

Spherical harmonic functions are important in many problems in Chemistry and Physics. Spherical harmonic functions are central in discussions of rotation, motion in a central potential, multipole expansion, cluster bonding, spherical wave expansions and many more topics. The calculation of symmetrized powers of representations give a way of obtaining the... 

V. Aquilanti, G. Grossi, and A. Lagani, On hyperspherical mapping and harmonic expansions for potential energy surfaces. J. Chem. Phys., 76 1587-1588, 1982. 

Much of the theoretical work in this field has focused on the use of empirical potentials for the two minima, and good results have been obtained (23,24). This can be expected, because the minima corresponding to the different spin states are generally rather close, and they are well approximated by harmonic expansion of the surfaces around the minima. However, ab initio computations can be expected to contribute positively to this field if they are able to characterize the crossing of the PESs in a less ad hoc manner. 

The first one, on the binding of an electron by a confined polar atom, is a challenging alternative to [4,5]. The second one is an alternative in the choice of boundary to [43] for the molecular hydrogen close to a confining plane. Section 5.4 emphasizes the importance of using the complete harmonic expansions, outside and inside the sources, of the electrostatic potential of atoms and molecules. In particular, the dipole fields in spherical, prolate and oblate spheroidal coordinates define other alternatives to [4,5]. 

In Section 3 reference was made to the superintegrability of the Laplace equation in parallel to that of the Schrodinger equation for the hydrogen atom. This subsection goes back to that common feature of the respective harmonic functions and atomic wave functions in their common systems of coordinates. This is particularly important in the investigation of two or more electron systems, for which the respective harmonic expansions of the Coulomb inter-electronic potential becomes the important tool for physically and computationally tractable calculations. 

As a specific experience, we can point out that our works on the helium atom confined by paraboloids [46,47] were preceded by variational calculations for the hydrogen atom in the respective confinement situations [49] and by the construction of the paraboloidal harmonic expansion of the Coulomb potential [50]. The last reference also includes the corresponding expansions in prolate and oblate spheroidal coordinates. 

As said before, in practice the integral equations for moleeular liquids are almost always solved using spherieal harmonic expansions. This is beeause the basic form [8.76] of the OZ relation eontains too many variables to be handled effieiently. In addition, harmonic expansions are neeessarily truncated after a finite number of terms. The validity of the trun-eations rests on the rate of eonvergence of the harmonic series that depends in turn on the degree of anisotropy in the intermolecular potential. 

In addition, serious basic theoretical problems are encountered with the application of a harmonic expansion of the potential in terms of angular displacements to the calculation of the librational frequencies of a-Na (Section IJC.2). It is expected that quantum mechanical calculations now in progress (Jacobi and Schnepp, 1971) will clarify this problem. Further experimental work also is required to settle the question of the dependence of the intensity of the 60 cm Raman line reported by Anderson and co-workers on sample preparation. Only such additional work can contribute further to the assignments of the q = 0 librational frequencies. 

As we describe the dynamics of a reactive chemical process, we do not use a harmonic expansion approach for the potential energy surface (PES), but instead we rely on ab initio calculations. The PES are determined for the relevant region of the coordinate space spanned by our selected coordinates, fully optimized with respect to the remaining internal coordinates. Subsequently, the data points are projected onto the reduced set of reactive coordinates, which can for the dynamical calculations be further restricted to (r, Lp). 

The model using spherical harmonics expansions for the RF potential can be derived from Eq. (26) by introducing spherical boundary conditions. The procedure has already been outlined by this author [6] and will not be repeated here. Semi-continuum models. In this type of approach, the first solvation shell is represented in the supermolecule and, consequently, enters into the quantum chemical description. Basically, the radius of the sphere embedded in the continuum dielectric is much larger than for the desolvated solute. This model has been used in several occasions, e.g. solvated electron, electron transfer in solution. 

The BH and ST potentials are expressed as truncated spherical harmonic expansions. This type of representation must be used cautiously since it is known that such expansions are slowly convergent. 

In the small-curvature tunneling approximation, k(T) requires, in addition to some of the information detailed above, the curvature components Cm(,s) of the curvature of the reaction path, where each curvature component measures the projection of the curvature vector on a particular generalized normal mode direction m. Calculation of Kl-CT(r) or kPOMT(7 requires, in addition, values of the Bom-Oppenheimer potential V in the reaction swath, typically at points where it cannot be computed from the available harmonic expansion around the MEP. 

Even more crucial in this discussion, the calculation of IR spectra with MD is related only to the time-dependent dipole moment of the molecular system, requiring neither any harmonic expansion of the transition dipole moment nor the knowledge of normal modes, in contrast to harmonic calculations. Therefore, if the dipole moments and their fluctuations are accurately calculated along the trajectory, the resulting IR spectmm should be reliable. The vibrations therefore do not rely directly on the curvature of the potential energy surface at the minima... 

As a consequence of the harmonic expansion of the valley potential, just the kinetic part, but not the potential energy part of the RPH contains the coupling terms between different vibrational modes. Hence, the RPH is an adiabatic Hamiltonian and a consequence of the adiabaticity of the RPH is that the frequencies ru (r) of generalized normal modes with the same symmetry must not cross. 

In HTST, a harmonic expansion of the PES is invoked both in the IS and in the saddle point separating the IS and the FS. The HTST is therefore appUcable under the same general assumptions as mentioned for TST but further demands that the PES is smooth enough for a local harmonic expansion of the PES to be reasonable. This means that it is necessary that the potential is reasonably well represented by its second-order Taylor expansion around these two expansion configurations. The general idea is that the partition functions in Equation (4.20) can be evaluated analytically for the harmonic expansion of the PES around the expansion points. This leads to very simple expressions for the rate constants and gives reasonable rate constants for... 

As noted above the theoretical studies of reactions (R1)-(R3) discussed here concentrate on the characterization of the molecular potential energy surfaces in the reactants, transition state, and products regions. In all cases the energies have been represented by harmonic expansions in a suitable set of internal coordinates. While our emphasis will be on the energetics of the reactions, we... 

It is clear that nonconfigurational factors are of great importance in the formation of solid and liquid metal solutions. Leaving aside the problem of magnetic contributions, the vibrational contributions are not understood in such a way that they may be embodied in a statistical treatment of metallic solutions. It would be helpful to have measurements both of ACP and A a. (where a is the thermal expansion coefficient) for the solution process as a function of temperature in order to have an idea of the relative importance of changes in the harmonic and the anharmonic terms in the potential energy of the lattice. 

The model of non-correlated potential fluctuations is of special interest. First, it can be solved analytically, second, the assumption that subsequent values of orienting field are non-correlated is less constrained from the physical point of view. The theory allows for consideration of a rather general orienting field. When the spherical shape of the cell is distorted and its symmetry becomes axial, the anisotropic potential is characterized by the only given axis e. However, all the spherical harmonics built on this vector contribute to its expansion, not only the term of lowest order... 

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