Normal-mode displacements

The frequency and normal mode displacement output. Once you know the sort of displacement to expect, you can determine which normal mode corresponds to it and its associated frequency. 

Note that the frequency calculation produces many more frequencies than those listed here. We ve matched calculated frequenices to experimental frequencies using symmetry types and analyzing the normal mode displacements. The agreement with experiment is generally good, and follows what might be expected of Hartree-Fock theory in the ground state. ... 

The b2, b 1 and a2 blocks are formed in a similar manner. The eigenvalues of each of these blocks provide the squares of the harmonic vibrational frequencies, the eigenvectors provide the normal mode displacements as linear combinations of the symmetry adapted... 

Figure 1 Left Enol-keto tautomerism in salicylaldimine (SA) and normal mode displacements for skeleton modes 1 4 and 1/30. Middle H/D diabatic potential energy curves Ua(Qu) for mode i/u (lowest states ground state, bolding and stretching fundamental, first bolding overtone arrows indicate laser excitation). Right two-dimensional (Qj4,Q3o) cuts through the adiabatic PES (obtained upon diagonalizing the field-free part of Eq. (1)) which has dominantly H/D stretching character but includes state and mode couplings (contours from 0 to 7400 cm-1).

Now in the strongly overdamped limit (8 -> 0) and for times longer than t = 4m/f(4 — 8), the first term of Eq. (1.3) dominates and the normal mode displacement decays in time according to ... 

The normal mode displacements are sketched in Figure 9.4. The notation u, v for the degenerate pair of H symmetry and , //, ( for the T2 triplet is standard. Actually, these projections had already been done in Section 6.4, but this example has been worked in full here to illustrate the projection operator method of finding normal modes. 

Fie 2- Normal mode displacement vectors for the target modes as calculated in harmonic approximation of the AT-(H20) =i,2 PES using DFT/B3LYP with a 6-31++G(d,p) basis set. 

The phase relationships describing the internal vibrations of a molecule can disappear in larger systems and the normal mode displacements, which in theory are generalised over the whole molecule, become localised on one part or another. The forces carrying the phase information are too weak and it is lost. What can be true in large molecular systems is certainly found in lattices. 

FIGURE 1.1.1 Sketch of the potential energy surfaces (in the monomer coordinate representation) related to electron transfer, showing the vertical transitions, the normal mode displacement AQ, and the relaxation energies A,/ and Xp. 

The matrix L 2) connects the 3n-6 n is the number of atoms in the [nonlinear] molecule) normal coordinates with the set of 3n mass-weighted Cartesian coordinates, (2, the vectors and correspond to the stationary points on the adiabatic potential surfaces of states 1 and 2, respectively. Then, the normal mode displacements Ag 2) are obtained by projecting the displacements = q,<°> - q2<° onto the normal-mode vectors. Finally, substituting the calculated quantities into Equations 1.1.10, 1.1.13, and 1.1.14 provides the total relaxation energy. 

State calculations. With the extensions provided, the method can be applied to the full Watson Hamiltonian [51] for the vibrational problem. The efficiency of the method depends greatly on the nature of the anharmonic potential that represents couphng between different vibrational modes. In favorable cases, the latter can be represented as a low-order polynomial in the normal-mode displacements. When this is not the case, the computational effort increases rapidly. The Cl-VSCF is expected to scale as or worse with the number N of vibrational modes. The most favorable situation is obtained when only pairs of normal modes are coupled in the terms of the polynomial representation of the potential. The VSCF-Cl method was implemented in MULTIMODE [47,52], a code for anharmonic vibrational spectra that has been used extensively. MULTIMODE has been successfully applied to relatively large molecules such as benzene [53]. Applications to much larger systems could be difficult in view of the unfavorable scalability trend mentioned above. 

The example given is particularly simple because the postulated set of collision-induced displacements coincides with that of mode 16b. That this is so could have been seen by comparison of Figs. 30a and 1 and needs no analysis. In other cases even simple atomic displacement patterns generate superpositions of normal mode displacements, and these must be decomposed systematically. A collision that leads to the displacement pattern shown in Fig. 30b generates excitation of B2g, 2, E g and /lju modes, which are modes 5, 17, 10, and 11 of Fig. 1. TTie analysis suggests that the amplitudes of 5 and 17 are much greater than those of 10 and 11, and that... 

We proceed by expanding the diabatic excited-state PE functions and coupling elements in terms of normal-mode displacements ... 

Fig. 6.4. Ground- and excited-state potential energy curves illustrating possible dissociation pathways. For a diatomic molecule r is the internuclear distance in a polyatomic molecule r is a normal mode displacement. Vibrational separations are greatly exaggerated. The vertical arrows correspond to Franck-Condon allowed transitions. Where possible, molecular examples are indicated, (a) Repulsive excited state—immediate dissociation, (b) Crossing of bound and repulsive excited states—intersystem crossing leads to predissociation. (c) Metastable excited state—tunneling leads to dissociation, (d) Bound state— excitation energy greater than dissociation limit.

Figure 6.4 may be transcribed for polyatomic molecules. A similar diagram exists for each vibrational mode where a normal mode displacement corresponds to the interparticle distance. The situation is complicated further because the vibrational energy is rapidly partitioned among many... 

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