Normal coordinate theory

It is customary in applying normal coordinate theory to use the experimentally observed fundamental frequencies as the basis of the calculation of the quadratic force constants. From a strict viewpoint this is not justifiable inasmuch as the observed fundamental frequencies do not have the same values as they would if the anharmonic terms were zero. In order to calculate the quadratic force constants accurately, it is necessary to use the so-called mechanical frequencies of vibration, which are the frequencies which the molecule would exhibit if the anharmonic terms in V were all zero. When an empirical formula of the type given in (3) has been obtained, the mechanical frequencies can be calculated from the relations... 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

With the aid of a normal coordinate analysis involving different isotopomers a linear structure of the Pd-Si-0 molecule is deduced. The results of ab initio MP2 calculations (Tab. 4) confirm the experimentally obtained IR spectra and their interpretation. The Pd-C bond in PdCO is similar to the Pd-Si bond in PdSiO which means, that the donor bond is strengthened by x acceptor components. This conclusion is in line with the high value of the Pd-Si force constant (exp. f(PdSi) = 2.69, f(SiO) = 8.92 mdyn/A) as well as with the energy of PdSiO (Pd + SiO —> PdSiO + 182 kJ/mol for comparison Pd + CO —> PdCO + 162 kJ/mol, MP2 level of theory). 

There is a small complication in that the frequency to is different for the reduced and oxidized states so that one has to take an average frequency. Marcus has suggested taking u>av = 2woxu>red/(wox + a>reci)-When several inner-sphere modes are reorganized, one simply sums over the various contributions. The matter becomes complicated if the complex is severely distorted during the reaction, and the two states have different normal coordinates. While the theory can be suitably modified to account for this case, the mathematics are cumbersome. 

The harmonic vibrational infrared spectrum of the four bases, as well as of the two base pairs, has been calculated at the DFT and DFT-solvent level of theory by finite differentiation of the forces acting on atoms along the normal coordinates. In the following, we briefly analize the results, which... 

There is an alternative and very direct way to generalize the Rouse-Zimm model for non-Gaussian chains. This approach takes advantage of the expression given by the original theory for the chain elastic potential energy in terms of normal coordinates ... 

In this approach, the diffusion constant, Di, is related to the corresponding characteristic time, x, describing the distortions of the normal coordinate, Westlund et al. (85) used the framework of the general slow-motion theory to incorporate the classical vibrational dynamics of the ZFS tensor, governed by the Smoluchowski equation with a harmonic oscillator potential. They introduced an appropriate Liouville superoperator ... 

Wexler D, Zink JI, Reber C (1994) Spectroscopic Manifestations of Potential Surface Coupling Along Normal Coordinates in Transition Metal Complexes. 171 173-204 Wiest O, Houk KN (1996) Density Functional Theory Calculations of Pericyclic Reaction Transition Structures. 183 1 -24 Willett P,see Artymiuk PJ (1995) 174 73-104... 

For polyatomic molecules, (7.2) and (7.6) give the wave numbers and transition moment for an electronic transition. Besides the S selection rule (7.7), there are electronic selection rules stating between which electronic symmetry species transitions are allowed these are derived using group theory (Section 9.11). Equation (7.13) applies to polyatomics, except that Pei is now a function of the 3N —6 (or 3Af —5) normal coordinates Qr... 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

The normal coordinates are certain linear combinations of the 3N displacement coordinates of the nuclei from their equilibrium positions the correct linear combinations are found from the secular equation (6.17) and the associated equations (6.18) and (6.19). We now apply group theory to the problem. 

Pi orbitals, 68, 310, 317 Pi-star orbital, 309, 310 Planarity, 224-225 Planck, M., 122 Plane of symmetry, 53 Plane-polarized wave, 114 Plane polar normal coordinates, 272 Point groups, 53-56,388-389 character tables of, 458-462 see also Group theory... 

M. Quack In answer to the question by Prof. Kellman on exact analytical treatments of anharmonic resonance Hamiltonians, I might point out that to the best of my knowledge no fully satisfactory result beyond perturbation theory is known. Interesting efforts concern very recent perturbation theories by Sibert and co-workers and by Duncan and co-workers as well as by ourselves using as starting point internal coordinate Hamiltonians, normal coordinate Hamiltonians, and perhaps best, Fermi modes [1]. Of course, Michael Kellman himself has contributed substantial work on this question. Although all the available analytical results are still rather rough approximations, one can always... 

In this connection, a procedure is of interest, which has been followed in the treatment of dilute polymer solutions. In this theory a convective coordinate system is used which moves with the chain molecule under consideration. In this way, the indicated difficulty is overcome when a certain simplified model is used for the chain molecule. The set of coordinates of this model, as related to the convective coordinate system, is then transformed into a set of normal coordinates which behave independently. For a more detailed treatment and discussion of this matter the reader is referred to Chapters 3 and 5 of this review. 

Marcus attempted to calculate the minimum energy reaction coordinate or reaction trajectory needed for electron transfer to occur. The reaction coordinate includes contributions from all of the trapping vibrations of the system including the solvent and is not simply the normal coordinate illustrated in Figure 1. In general, the reaction coordinate is a complex function of the coordinates of the series of normal modes that are involved in electron trapping. In this approach to the theory of electron transfer the rate constant for outer-sphere electron transfer is given by equation (18). 

We shall now briefly review the Kubo-Oxtoby theory of vibrational line-shape. The starting point for most theories of vibrational dephasing is the stochastic theory of lineshape first developed by Kubo [131]. This theory gives a simple expression for the broadened isotropic Raman line shape (/(< )) in terms of the Fourier transform of the normal coordinate time correlation function by... 

In the theory of small vibrations it is shown that, by a linear transformation of the displacements xj yj Zj to a set of normal coordinates Qk, the kinetic energy and the potential energy may be transformed simultaneously into diagonal form so that... 

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