Projected force constant matrix

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] ... 

A normal mode representation of the Hamiltonian for the reduced system involves the diagonalization of the projected force constant matrix Ffmai, which... 

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]. 

Here the Q are the generalized normal coordinates and the to are the associated harmonic frequencies. They are obtained at each point on the path by diagonalizing the force constant matrix for which the reaction path direction as well as directions corresponding to rotations and translations have been projected out. The projected force constant matrix has seven zero eigenvalues corresponding to overall rotations, translations, and the reaction path direction. It also has 3N-7 nonzero eigenvalues corresponding to vibrations transverse to the path. 

The curvature coupling elements are thus simply off-diagonal matrix elements of the unprojected force constant matrix in the basis of eigenvectors of the projected force constant matrix. The classical notion that a trajectory will overshoot the path and climb the wall if the path curves on the way down the hill is a reflection of this curvature coupling. Climbing the wall in a transverse direction is tantamount to exchanging energy between the reaction path and the transverse vibration. 

In the simplest case, we can use a normal mode analysis for the fast q vibrations. Since the x mode is potentially a high amplitude degree of freedom, we should employ instantaneous normal modes. In this treatment, the frequencies and normal mode coordinates are obtained by diagonalizing the projected force constant matrix ... 

Figure 6.4 Normal mode frequencies of HOOH versus torsional angle computed from the projected force constant matrix. The OH stretch frequencies are shown in the upper panel, the OOH bending frequencies in middle panel and the 00 stretch frequency in the lower panel.

The first one is the assumption of a harmonic reaction valley, which can be fully described with the help of the mass-weighted projected force constant matrix K(r) evaluated at each path point of interest. 

If there are real frequencies of the same magnitude as the rotational frequencies , mixing may occur and result in inaccurate values for the true vibrations. For this reason the translational and rotational degrees of freedom are nonnally removed from the force constant matrix before diagonalization. This may be accomplished by projecting the modes out. Consider for example tire following (normalized) vector describing a translation in the x-direction. 

It should be noted that the force constant matrix can be calculated at any geometry, but the transformation to nonnal coordinates is only valid at a stationary point, i.e. where the first derivative is zero. At a non-stationary geometry, a set of 3A—7 generalized frequencies may be defined by removing the gradient direction from the force constant matrix (for example by projection techniques, eq. (13.17)) before transformation to normal coordinates. 

For this work we use D2Q9 and periodic boundary conditions in the inflow and outflow plane and non-slip boundary (bounce-back) conditions on the walls and the porous matrix. Bounce-back conditions were used whenever the fluid hit a node of the porous matrix. Our porous media is represented by blocks that are projections in the plane of actual three dimensional geometries Stability is improved by considering the porous matrix as made out of these blocks and makes the code less noisy as well. To initialize the lattice, a constant body force (F) is used and acts during the simulations, which physically corresponds to a constant pressure gradient. In this work we focus only on externally applied pressure namely we deal with pressure-driven flows. 

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