Force-constant matrix terms

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

The transformation T we adopt is induced by the wave function normalization condition which, in terms of the weights, reads w + W3 = 1. From (3.5), it is apparent that if T sends the vvm set into a new set wm with ivi = vvi + iv3 = 1 as one of its elements, then both the first row and the first column of the transformed polarization component of the solvent force constant matrix K, "/ = T. Kp°r. T (T = T) are zero, since the derivatives of wi are zero. Given the normalization condition and the orthogonality requirement — with the latter conserving the original gauge of the solvent coordinates framework — one can calculate T for any number of diabatic states [42], The transformation for the two state case is... 

The model of a reacting molecular crystal proposed by Luty and Eckhardt [315] is centered on the description of the collective response of the crystal to a local strain expressed by means of an elastic stress tensor. The local strain of mechanical origin is, for our purposes, produced by the pressure or by the chemical transformation of a molecule at site n. The mechanical perturbation field couples to the internal and external (translational and rotational) coordinates Q n) generating a non local response. The dynamical variable Q can include any set of coordinates of interest for the process under consideration. In the model the system Hamiltonian includes a single molecule term, the coupling between the molecular variables at different sites through a force constants matrix W, and a third term that takes into account the coupling to the dynamical variables of the operator of the local stress. In the linear approximation, the response of the system is expressed by a response function X to a local field that can be approximated by a mean field V ... 

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] ... 

The percentage may be greater or less than 100, because of contributions of ofl-diagonal terms in the force-constant matrix. b Torsional vibration. 

For a non-linear molecule of N atoms, there are 3N — 6 (3N — 5 if the molecule is linear) internal vibrational coordinates. If we wish to include the off-diagonal force constants then there are (3N — 6) diagonal and (3N — 6)2 — (3N — 6) off-diagonal terms. Only half of the off-diagonal force constants are unique, since (for example) kco.cs must equal kcs.co- In other words, the force constant matrix has to be symmetric. This gives 1/2(3JV — 6)(3N - 5) independent force constants, a number that usually far exceeds the available experimental vibration frequencies. A complete determination of all force constants requires analysis of the spectra of many isotopically substituted molecules. Many of the off-diagonal terms turn out to be very small, and spectroscopists have developed systematic simplifications to the force field in order to make as many of the off-diagonal terms as possible, vanish. 

The vibrations may be described by different sets of basis coordinates. To start with, there are the changes of the 3n Cartesian coordinates X of the molecule. Chemists favor descriptions of the motions and the force constants in terms of bond lengths and bond angles. These are known as internal coordinates R. The equivalent internal coordinates of a molecule which possesses a certain symmetry, may change either in-phase or out-of-phase. The simultaneously occurring relative changes of the bond parameters of equivalent bonds are described by symmetry coordinates S. Normal coordinates Q describe motions as linear combinations of any set of basis coordinates. Different coordinate. systems can be transformed into each other by matrix multiplication. For further details, see Sec. 5.2. 

The in-1 vibrational frequencies, C0 (s), are obtained from normal-mode analyses at points along the reaction path via diagonalization of a projected force constant matrix that removes the translational, rotational, and reaction coordinate motions. The B coefficients are defined in terms of the normal mode coefficients, with those in the denominator of the last term determining the reaction path curvature, while those in the numerator are related to the non-adiabatic coupling of different vibrational states. A generalization to non-zero total angular momentum is available [59]. 

This may be further simplified by noting that for every term in the sum we will have a partner term —R , and also by using the fact that the force constant matrix is the same for these partner terms. As a result, we have... 

The conclusion of this analysis is that we can obtain the dynamical matrix by performing lattice sums involving only those parts of the force constant matrix that do not involve the self-terms (i.e. K q). We note that the analysis we have made until now has been performed in the somewhat sterile setting of a pair potential description of the total energy. On the other hand, the use of more complex energy functionals does not introduce any new conceptual features. 

Because of the term 5K, when we evaluate dco(q,a)/dV, as demanded by eqn (5.72) in our deduction of the thermal expansion coefficient, it will no longer vanish as it would if we were only to keep K. The phonon frequencies have acquired a volume dependence by virtue of the quasiharmonic approximation which amounts to a volume-dependent renormalization of the force constant matrix. 

Here V(%i,... is the potential for the reactive coordinates when the substrate coordinates are either frozen at some reference geometry or relaxed to minimize V(%i,..., Xjv ). Since in general we are not dealing with a minimum energy configuration, forces j]j(x) appear in Eq. (4.1), which act on the normal mode coordinates Q . Finally, the last term in Eq. (4.1) contains the force constant matrix K (x), which is diagonal at that configuration for which the normal modes have... 

---