Normal mode coordinate calculation

Since harmonic oscillators are considered in this theory, the bonds will never break, so it is necessary to introduce an ad hoc criterion for when a reaction occurs. Reaction is normally defined to occur when a particular bond length attains a critical value. The bond length cannot be extracted directly from a particular normal-mode coordinate, since these coordinates, typically, involve the motion of several atoms in the molecule. The bond length can, however, be calculated quite readily, by noting that the displacement of a coordinate associated with an atom of mass mr is given by Eq. (E.5) ... 

For k = 0, the Fock matrix and its derivatives with respect to the displacements of the nuclei are always block diagonal. Then one can directly apply the analytical derivative methods developed for finite systems to extended systems [69,86,87,88]. But when the displacements break the translational symmetry, the Fock matrix and its derivatives are no longer block diagonal. To solve the CPHF equations, one needs to use the symmetrized (normal mode) coordinates instead of the Cartesian coordinates of the nuclei. Efficient analytical methods have been developed to calculate the energy derivatives for k / 0 with both plane wave [89-90] and general basis functions [85]. The latter can be functions of nuclear coordinates and have linear dependence. These methods reduce the computational cost required to calculate the phonon spectrum with k 7 0 to the same as that needed for the spectrum at k = 0. 

The term M p,is the eph coupling constant, and ba is the annihilation operator of the mode a, whose frequency and normal mode coordinate are represented by Q,a and Qp, respectively. The sites for electrons i( T) coupled with phonons are restricted to the C region or a subpart of C. The focused modes should be sufficiently localized on the molecule in term of their definition. Practically, these internal modes can be calculated by means of a frozen-phonon approximation, where displaced atoms are atoms in the c region (or its subpart) denoted as a vibrational box though a check for convergence to the size of the vibrational box is necessary [90]. 

Since this transformation to normal coordinates is invertible, one can readily determine the functional dependencies of the terms in Eq. (1) using either the normal or internal coordinates. Interestingly, in our study of vibrational states of the well-known local mode molecule H20 and its deuterated analogs we found only minor differences between the results of CVPT in the internal and normal mode representations (46). The normal mode calculations, however, required significantly less computer time to run, since many terms in the Hamiltonian are constrained to zero by symmetry. For this reason we chose to use the normal mode coordinates for all subsequent studies. 

We can calculate the right-hand side of (22) using the definition of Hna (q) together with (17). In the vicinity of the intersection of interest it is appropriate to retain in (22) only the lowest-order terms in each of the Qk (k = 1, 2,.. ., 3N — 6). In addition, in order to explore the point group symmetry properties of ij/ (rel q0) and v /j(rel q0) it is desirable to choose for the Qk coordinates which display simple transformation properties under the operations of that group, such as normal mode coordinates [13,15] or, in some circumstances, symmetrized hyperspherical coordinates [12,16,17]. Such choices lead to simple point group symmetries for the / (rel q0) and (rel q0) and permit the identification of which. (q0) and q0) vanish due to symmetry [18]. 

In quantum chemical calculations, the vibrational problem is normally described in the harmonic approximation. Assuming that the vibrational problem has been solved, potential energy and each internal parameter qn can be expressed as function of Nvib normal mode coordinates Q, [1-6]... 

The matrix of dipole derivatives with regard to normal coordinates contains the derivatives of the dipole moment components with regard to each normal coordinate. The infrared intensity 1 of the normal mode is calculated according to Eq. (84)... 

I Zb ) the quantum number n that corresponds to the mode indicated has been raised to n = 1. The splitting and shifts of the monomer fundamental vibrational frequencies can be calculated by taking Eq. (6) as the perturbation and using first order perturbation theory for the ground state and the degenerate first excited state of the dimer. The normal mode coordinates Qa in Eq. (6) refer to the modes x, and Za and the... 

For linear displacements from a stationary point, separable coordinates are uniquely defined for small displacements by normal mode coordinates [62], which simultaneously diagonalize the kinetic energy to infinite order and the potential energy to second order (i.e., through quadratic terms in the potential). Thus, to the extent that one stays in a region where the quadratic expansion of the potential is trustworthy, these coordinates separate the physical motion, and they are not just an artificial mathematical imposition. Not only is the motion separable in normal coordinates, but the coordinates themselves are very convenient for calculations, since they are rectilinear. 

For the calculations on malonaldehyde we used a set of mass- and frequency scaled normal mode coordinates, obtained at the transition state depicted in Fig. 5.7b. The coordinate labeling, normal mode frequencies and physical description are outlined in Table 5.1. The coordinates were subsequently modified [78] to minimize the correlation induced by the reorganization of tlie double bonds. The shifts of the inter atomic distances only depend on the position of the proton along the transfer coordinate 21 They can be compensated by the modified coordinates qt obtained by the transformation... 

The relative merits of integrating the Cartesian or normal mode coordinates have not yet been fully explored. If the determination of the Hessian (which may be done analytically or numerically) is computationally expensive compared to the force calculation, integration of the Cartesian coordinates may be preferred (since the Hessian is not required in these algorithms). If the Hessian calculation is relatively cheap then, for a given integration accuracy, the larger step size afforded by the normal mode scheme may make this method more efficient. 

To examine the effects of anharmonicity on the decay of excited clusters, calculations were performed for hot clusters in which the alternate, truncated harmonic potential, equations (7) and (8), was employed as the atomic pair potential. While the interatomic pair potential was harmonic in form, the entire cluster was not a harmonic system since displacements along the normal mode coordinate were not subject to restoring forces which were linear functions solely of the displacements along those coordinates. The change in potential nonetheless causes a change in the density of states for the cluster. 

Miller, Handy, and Adams have recently shown how one can construct a classical Hamiltonian for a general molecular system based on the reaction path and a harmonic approximation to the potential surface about it. The coordinates of this model are the reaction coordinate and the normal mode coordinates for vibrations transverse to the reaction path these are essentially a polyatomic version of the natural collision coordinates introduced by Marcus and by Hofacker for A + BC AB 4- C reactions. One of the important practical aspects of this model is that all of the quantities necessary to define it are obtainable from a relatively modest number of db initio quantum chemistry calculations, essentially independent of the number of atoms in the system. This thus makes possible an ab initio theoretical description of the dynamics of reactions more complicated than atom-diatom reactions. 

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. 

More traditional applications of internal coordinates, notably normal mode analysis and MC calculations, are considered elsewhere in this book. In the recent literature there are excellent discussions of specific applications of internal coordinates, notably in studies of protein folding [4] and energy minimization of nucleic acids [5]. 

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... 

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