Vibrational displacement coordinates

The time dependence of the displacement coordinate for a mode undergoing hannonic oscillation is given by V = V j cos2tiv /, where is the amplitude of vibration and is the vibrational frequency. Substitution into equation (Bl.2.9) witii use of Euler s half-angle fomuila yields... 

The procedure Split selects the internal displacement coordinates, q, and momenta, tt, (describing vibrations), the coordinates, r, and velocities, v, of the centers of molecular masses, angular velocities, a>, and directional unit vectors, e, of the molecules from the initial Cartesian coordinates, q, and from momenta, p. Thus, the staring values for algorithm loop are prepared. Step 1 Vibration... 

Force constant calculations are normally done in Cartesian coordinates. Suppose we have N atoms whose position vectors are Ri, R2,. .., Ra - Each of the atoms vibrates about its equilibrium position Ri g, Ri.e, , R v,e-The first step in our treatment is to define mass-weighted displacement coordinates... 

Each normal mode of vibration can be described by a normal coordinate Qi which is a linear combination of nuclear displacement coordinates of the molecule. For the symmetric stretching vibration vi of C02, the normal coordinate is of the form... 

The characters Xj for the examples in the previous section were calculated following the method described in Section 8.9, that is, on the basis of Cartesian displacement coordinates. Alternatively, it is often desirable to employ a set of internal coordinates as the basis. However, they must be well chosen so that they are sufficient to describe the vibrational degrees of freedom of the molecule and that they are linearly independent The latter condition is necessary to avoid the problem of redundancy. Even when properly chosen, the internal coordinates still do not usually transform following the symmetry of the molecule. Once again, the water molecule provides a very simple example of this problem. 

The actual calculation consists of minimizing the intramolecular potential energy, or steric energy, as a function of the nuclear coordinates. The potential-energy expressions derive from the force-field concept that features in vibrational spectroscopic analysis according to the G-F-matrix formalism [111]. The G-matrix contains as elements atomic masses suitably reduced to match the internal displacement coordinates (matrix D) in defining the vibrational kinetic energy T of a molecule ... 

The matrix to be diagonalized for finding the vibrational frequencies is the matrix product of the above G matrix for Cartesian coordinates and the corresponding F matrix for Cartesian displacement coordinates. It is noted in passing that the GF matrix is generally not symmetric, i.e. 

The second rule for isotopomer harmonic frequencies is the so-called Sum Rule which follows from Equation 3.A1.8. Equation 3.A1.8 relates the sum of the squares of all the frequencies to the sum of the diagonal matrix elements of the (FG) matrix diagonalized to obtain the frequencies. When mass weighted Cartesian displacement coordinates are used to calculate the vibration frequencies, this means that the sum of the A s (A = 4n2v12)can be found as follows (Equation3.51)... 

The electronic energy of a molecule, ion, or radical at geometries near a stable structure can be expanded in a Taylor series in powers of displacement coordinates as was done in the preceding section of this Chapter. This expansion leads to a picture of uncoupled harmonic vibrational energy levels... 

The point of changing from Cartesian displacement coordinates to normal coordinates is that it brings about a great simplification of the vibrational equation. Furthermore, we will see that the normal coordinates provide a basis for a representation of the point group to which molecule belongs. 

Thus if there are no degenerate vibrations, each normal coordinate is either unchanged or multiplied by — 1 upon application of a symmetry operation. Each Qk is a linear combination of the mass-weighted Cartesian displacement coordinates of the nuclei. If Qk is multiplied by — 1, each Cartesian displacement coordinate is multiplied by - 1, which reverses the directions of all the displacement vectors. If Qk is unchanged by a symmetry operation, then the symmetry operation sends the displacement vectors to a configuration indistinguishable from the original one. (The displacement vectors are defined relative to molecule-fixed axes, which in turn are defined relative to the nuclear positions. The effect of a symmetry... 

Just as group theory enables one to find symmetry-adapted orbitals, which simplify the solution of the MO secular equation, group theory enables one to find symmetry-adapted displacement coordinates, which simplify the solution of the vibrational secular equation. We first show that the matrices describing the transformation properties of any set of degenerate normal coordinates form an irreducible representation of the molecular point group. The proof is based on the potential-energy expression for vibration, (6.23) and (6.33) ... 

The transformed normal coordinate 0RQjVk can be expressed as some linear combination of the 3N mass-weighted displacement coordinates using (6.21), we can express 0RQjVk as a linear combination of all the normal coordinates of the molecule. If in this linear combination, the coefficient of a normal coordinate whose vibrational frequency differed... 

Symmetry Types of the Normal Modes. For this nonlinear four-atomic molecule there are 3(4) -6 = 6 genuine internal vibrations. Using a set of three Cartesian displacement coordinates on each atom, we obtain the following representation of the group C3l, ... 

The determination of these normal frequencies, and the forms of the normal vibrations, thus becomes the primary problem in correlating the structure and internal forces of the molecule with the observed vibrational spectrum. It is the complexity of this problem for large molecules which has hindered the kind of detailed solution that can be achieved with small molecules. In the general case, a solution of the equations of motion in normal coordinates is required. Let the Cartesian displacement coordinates of the N nuclei of a molecule be designated by qlt q2,... qsN. The potential energy of the oscillating system is not accurately known in the absence of a solution to the quantum mechanical problem of the electronic energies, but for small displacements it can be quite well approximated by a power series expansion in the displacements ... 

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