Vibrational normal coordinates

The coordinate axes are usually orthonormal, but this is not a requirement, since they can be orthogonalized by the methods in Section 16.4. 

The potential energy is approximated by a second-order Taylor expansion around the stationary geometry xq. 

The energy for the expansion point, V(xo), may be chosen as zero, and the first derivative is zero since xq is a stationary point. 

Here F is a 3A atom x 3iVatom (force constant) matrix containing the second derivatives of the energy with respect to the coordinates. 

39) is first transformed to mass-dependent coordinates by a G matrix containing the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, which is in line with the Born-Oppenheimer approximation that the electrons follow the nuclei). 

A unitary transformation is then introduced whieh diagonalizes the FG matrix, yielding eigenvalues s, and eigenvectors q,. The kinetic energy operator is still diagonal in these eoordinates. 

In the q-eoordinate system, the vibrational normal coordinates, the 3A-dimensional Sehrddinger equation can be separated into 3N one-dimensional Sehrddinger equations. 

In the q-coordinate system the vibrational normal coordinates, the 3A -dimensional 

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There are similar equations for the other operations of the point group. f Therefore, Ql forms a basis for the I 1 representation and is a vibrational normal coordinate, Q% and Qs together form a basis for the T representation and they are also vibrational normal coordinates, the rest correspond to X = 0 and are translational or rotational normal coordinates, Qt and Q6 belong to r, Q9 to r 1, Q7 to P 1 and and Q0 to r. ... 

For an unsymmetrical linear molecule there are N—1 non-degenerate vibrational normal coordinates belonging to the symmetry species 2+ (see Table 7-8.1) and they are of the longitudinal type and N—2 pairs of vibrational normal coordinates (doubly degenerate), each pair belonging to the II symmetry species and they are of the transverse type. 

Here fabs/em(v) is the transition intensity. The vibrational reorganization energy is defined in terms of force constants, k, and displacements, of the vibrational normal coordinates Qa as Xy = in... 

We expand the inverse moment of the inertia matrix as a power series in the vibrational normal coordinates ... 

Derivative Techniques 240 10.4 Lagrangian Techniques 242 10.5 Coupled Perturbed Hartree-Fock 244 10.6 Electric Field Perturbation 247 10.7 Magnetic Field Perturbation 248 10.7.1 External Magnetic Field 248 13.1 Vibrational Normal Coordinates 312 13.2 Energy of a Slater Determinant 314 13.3 Energy of a Cl Wave Function 315 Reference 315 14 Optimization Techniques 316... 

In equation 27 is the distance vector of the equilibrium internuclear distance between the atoms k and i. S and S are the vector components associated with the matrix that transforms normal coordinate displacements to internal coordinate displacements (along and perpendicular to bond directions). The sum is over the atoms and transformation vectors associated with the yth vibrational normal coordinate. 

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