Second-derivative coupling matrix systems

The same relations (11) and (12) hold for the Gibbs free energy in the (N, p,T) ensemble. Equation (11) is also valid for a quanmm mechanical system. Note that for a linear coupling scheme such as Eq. (10), the first term on the right of Eq. (12) is zero the matrix of second derivatives can then be shown to be definite negative, so that the free energy is a concave function of the Xi. 

As a consequence, field methods, which consist of computing the energy or dipole moment of the system for external electric field of different amplitudes and then evaluating their first, second derivatives with respect to the field amplitude numerically, cannot be applied. Similarly, procedures such as the coupled-perturbed Hartree-Fock (CPHF) or time-dependent Hartree-Fock (TDHF) approaches which determine the first-order response of the density matrix with respect to the perturbation cannot be applied due to the breakdown of periodicity. 

Normal modes are useful because they correspond to collective motions of the atoms in a coupled system that can be individually excited. The three normal modes of water are schematically illustrated in Figure 5.15 a non-linear molecule with N atoms has 3N — 6 normal modes. The frequencies of the normal modes together with the displacements of the individual atoms may be calculated from a molecular mechanics force field or from the wavefunction using the Hessian matrix of second derivatives Of course, if we... 

Thus, magnetic field-like perturbations yield much easier response (or coupled perturbed ) equations in which flie contributions from any local potential vanish, hr fact, in the absence of HF exchange the A-matrix becomes diagonal and the linear equation system is trivially solved. This then leads to a sum-over-orbital -like equation for the second derivative that resembles in some way a sum-over-states equation. One should, however, carefully distinguish the sum-over-states picture from linear response or analytic derivative techniques since they have a very different origin. For electric field-like perturbations or magnetic field-like perturba-... 

Investigation of the reaction valley in the harmonic approximation At each path point, the orthogonal directions to the RP are described by a quadratic (harmonic) approximation of the potential V R), which implies the calculation of the second derivatives of Vf/f) with regard to the internal coordinates. A coupling of translational and vibrational motions of the reaction complex can be described, which is the basis for a more quantitative investigation of reaction mechanism and reaction dynamics. Calculations can be done for most of the reaction systems considered by approach (2). Of course, a routine, inexpensive calculation of the matrix of second derivatives of V(R) is desirable. 

A second more extensive extension would be the development of programs to solve a number of coupled second order PDEs - for example 3 PDEs in three physical variables. These could also be time dependent. Such coupled PDEs occur frequently in physical problems. While extending the previous code for this case is relatively straightforward in principle, this is not a trivial task. Only some considerations for coupled equations will be considered here for N coupled equations. If one allows in the most general case, all ranges of derivatives to be expressed in the coupled equations, then one has N variables at each node and each defining PDE has N variables. Thus the number of node equations is increased by the factor N and the number of possible non-zero elements per row is increased by the factor N, giving a value of NXN = as the increased factor for the possible number of non-zero matrix elements. For the case of 3 coupled variables this is a factor of 9. Thus the computational time for eoupled systems of equations can increase very fast. 

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