Coordinate systems normal modes

In the second equality we have expanded the coordinate deviation <5x in normal modes coordinates, and expressed the latter using raising and lowering operators. The coefficients are defined accordingly and are assumed known. They contain the parameter a, the coefficients of the normal mode expansion and the transformation to raising/lowering operator representation. Note that the inverse square root of the volume Q of the overall system enters in the expansion of a local position coordinate in normal modes scales, hence the coefficients scale like... 

We next solve the secular equation F — I = 0 to obtain the eigenvalues and eigenvectors o the matrix F. This step is usually performed using matrix diagonalisation, as outlined ii Section 1.10.3. If the Hessian is defined in terms of Cartesian coordinates then six of thes( eigenvalues will be zero as they correspond to translational and rotational motion of th( entire system. The frequency of each normal mode is then calculated from the eigenvalue using the relationship ... 

A vibrations calculation is the first step of a vibrational analysis. It involves the time consuming step of evaluating the Hessian matrix (the second derivatives of the energy with respect to atomic Cartesian coordinates) and diagonalizing it to determine normal modes and harmonic frequencies. For the SCFmethods the Hessian matrix is evaluated by finite difference of analytic gradients, so the time required quickly grows with system size. 

The situation simplifies when V Q) is a parabola, since the mean position of the particle now behaves as a classical coordinate. For the parabolic barrier (1.5) the total system consisting of particle and bath is represented by a multidimensional harmonic potential, and all one should do is diagonalize it. On doing so, one finds a single unstable mode with imaginary frequency iA and a spectrum of normal modes orthogonal to this coordinate. The quantity A is the renormalized parabolic barrier frequency which replaces in a. multidimensional theory. In order to calculate... 

We can now search the normal modes for each system, looking for ones which exhibii these types of displacements. In each case, there is only one mode with any significant displacements in the required coordinates. 

The Rouse model, as given by the system of Eq, (21), describes the dynamics of a connected body displaying local interactions. In the Zimm model, on the other hand, the interactions among the segments are delocalized due to the inclusion of long range hydrodynamic effects. For this reason, the solution of the system of coupled equations and its transformation into normal mode coordinates are much more laborious than with the Rouse model. In order to uncouple the system of matrix equations, Zimm replaced S2U by its average over the equilibrium distribution function ... 

As before, we make the fundamental assumption of TST that the reaction is determined by the dynamics in a small neighborhood of the saddle, and we accordingly expand the Hamiltonian around the saddle point to lowest order. For the system Hamiltonian, we obtain the second-order Hamiltonian of Eq. (2), which takes the form of Eq. (7) in the complexified normal-mode coordinates, Eq. (6). In the external Hamiltonian, we can disregard terms that are independent of p and q because they have no influence on the dynamics. The leading time-dependent terms will then be of the first order. Using complexified coordinates, we obtain the approximate Hamiltonian... 

To render the KP theory feasible for many-body systems with N particles, we make the approximation of independent instantaneous normal mode (INM) coordinates [qx° 3N for a given configuration xo 3W [12, 13], Hence the multidimensional V effectively reduces to 3N one-dimensional potentials along each normal mode coordinate. Note that INM are naturally decoupled through the 2nd order Taylor expansion. The INM approximation has also been used elsewhere. This approximation is particularly suited for the KP theory because of the exponential decaying property of the Gaussian convolution integrals in Eq. (4-26). The total effective centroid potential for N nuclei can be simplified as ... 

Fig. 20. A resonance state for the H + HD system at Ec = 1.2 eV at a total angular momentum of J = 25. In (a) the wavefunction is shown in the H + HD Jacobi coordinates for the collinear subspace. In (b) the wavefunction has been sliced perpendicular to the minimum energy path and is plotted in symmetric stretch and bend normal mode coordinates.

Molecular mechanics calculations are an attempt to understand the physical properties of molecular systems based upon an assumed knowledge of the way in which the energy of such systems varies as a function of the coordinates of the component atoms. While this term is most closely associated with the conformational energy analyses of small organic molecules pioneered by Allinger (1), in their more general applications molecular mechanics calculations include energy minimization studies, normal mode calculations, molecular dynamics (MD) and Monte Carlo simulations, reaction path analysis, and a number of related techniques (2). Molecular mechanics... 

The main difference between the two approaches is that PGH consider the dynamics in the normal modes coordinate system. At any value of the damping, if the particle reaches the parabolic barrier with positive momentum i n the unstable mode p, it will immediately cross it. The same is not true when considering the dynamics in the system coordinate for which the motion is not separable even in the barrier region, as done by Mel nikov and Meshkov. In PGH theory the... 

This semiclassical turnover theory differs significantly from the semiclassical turnover theory suggested by Mel nikov, who considered the motion along the system coordinate, and quantized the original bath modes and did not consider the bath of stable normal modes. In addition, Mel nikov considered only Ohmic friction. The turnover theory was tested by Topaler and Makri, who compared it to exact quantum mechanical computations for a double well potential. Remarkably, the results of the semiclassical turnover theory were in quantitative agreement with the quantum mechanical results. 

While Eq. (9.49) has a well-defined potential energy function, it is quite difficult to solve in the indicated coordinates. However, by a clever transfonnation into a unique set of mass-dependent spatial coordinates q, it is possible to separate the 3 Ai-dirncnsional Eq. (9.49) into 3N one-dimensional Schrodinger equations. These equations are identical to Eq. (9.46) in form, but have force constants and reduced masses that are defined by the action of the transformation process on the original coordinates. Each component of q corresponding to a molecular vibration is referred to as a normal mode for the system, and with each component there is an associated set of harmonic oscillator wave functions and eigenvalues that can be written entirely in terms of square roots of the force constants found in the Hessian matrix and the atomic masses. 

Figure 15.3 Secondary KIEs are associated with normal modes other than the reaction coordinate, one of which is shown in this diagram. The heavy and light vibrational frequencies both change on going from the reactant (R) to the TS structure ( ) because in this example the mode is tighter in the TS structure, the difference between the heavy and light ZPVEs increases, and this causes the potential energy of activation to be larger for the light isotopomer than the heavy one (an example of an inverse secondary KIE). In a real many-atom system there are potentially a large number of modes that will contribute to the secondary KIE. some in a normal fashion and some in an inverse fashion...

Sidebar 10.3 outlines the useful analogy to normal-mode analysis of molecular vibrations, where the null modes correspond to overall translations or rotations of the coordinate system that lead to spurious alterations of coordinate values, but no real internal changes of interatomic distances. For this reason, the internal metric M( of (10.29) is the starting point for analyzing intrinsic state-related (as opposed to size-related) aspects of a given physical system of interest. 

A normal-mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix, which in turn generates a reduced-dimension potential-energy surface in terms of the mass-weighted coordinates of the reaction path [64] ... 

In order to show that this procedure leads to acceptable results, reference is briefly made to the normal coordinate transformation mentioned at the end of Section 2.2. By this transformation the set of coordinates of junction points is transformed into a set of normal coordinates. These coordinates describe the normal modes of motion of the model chain. It can be proved that the lowest modes, in which large parts of the chain move simultaneously, are virtually uninfluenced by the chosen length of the subchains. This statement remains valid even when the subchains are chosen so short that their end-to-end distances no longer display a Gaussian distribution in a stationary system [cf. a proof given in the appendix of a paper by Ham (75)]. As a consequence, the first (longest or terminal) relaxation time and some of the following relaxation times will be quite insensitive for the details of the chain... 

In this equation is the internal friction factor of thej-th normal mode and Qjj1 is the inverse transformation matrix of Zimm. In other words, Cerf assumed that one can ascribe a separate internal friction factor to every normal mode. This assumption is critisized by Budtov and Gotlib (183) as, in this way, the elements of the internal friction matrix in the laboratory coordinate system x, y, z, viz. 

There is some similarity between Ferry s treatment of concentrated systems (14), (123) [eq. (4.4)] and Cerf s just mentioned approach. In both cases the normal coordinate transformation is assumed to be possible along the lines given for infinitely dilute solutions of kinetically perfectly flexible chains (Rouse, Zimm). Only afterwards, different external (Ferry) or internal (Cerf) friction factors are ascribed to the various normal modes. 

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). 

---