Displacement coordinates

Figure Al.2.6. Anliannonic stretch nonnal modes of a synnnetric triatoniic. The plot is similar to figure Al.2.5. except the nonnal modes are now anliannonic and can be curvilinear hi the bond displacement coordinates r, r. The antisyimnetric stretch is curved, but the synnnetric stretch is linear because of synnnetry.

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... 

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

Let us consider tire energy expanded tlirough second order in two sets of displacement coordinates Ax and Aq. The two coordinate systems are related by... 

One starts with the Hamiltonian for a molecule H r, R) written out in terms of the electronic coordinates (r) and the nuclear displacement coordinates (R, this being a vector whose dimensionality is three times the number of nuclei) and containing the interaction potential V(r, R). Then, following the BO scheme, one can write the combined wave function [ (r, R) as a sum of an infinite number of terms... 

Here, we discuss the motion of a system of three identical nuclei in the vicinity of the D3/, configuration. The conventional coordinates for the in-plane motion are employed, as shown in Figure 5. The noraial coordinates Qx, Qy, Qz), the plane polar coordinates (p,(p,z), and the Cartesian displacement coordinates (xi,yhZi of the three nuclei (t = 1,2,3) are related by [20,94]... 

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... 

The procedure Merge transforms the internal displacement coordinates and momenta, the coordinates and velocities of centers of masses, and directional unit vectors of the molecules back to the Cartesian coordinates and momenta. Evolve with Hr = Hr(q) means only a shift of all momenta for a corresponding impulse of force (SISM requires only one force evaluation per integration step). 

Defining mass-weighted Cartesian displacement coordinates r/ . 

Mass weighting the generalized displacement coordinates qi and qj retains the form of Eq. (5-37) even when the actual masses are not unit masses. 

More generally, it is possible to combine sets of Cartesian displacement coordinates qk into so-called symmetry adapted coordinates Qrj, where the index F labels the irreducible representation and j labels the particular combination of that symmetry. These symmetry adapted coordinates can be formed by applying the point group projection operators to the individual Cartesian displacement coordinates. 

To illustrate, again consider the H2O molecule in the coordinate system described above. The 3N = 9 mass weighted Cartesian displacement coordinates (Xl, Yl, Zl, Xq, Yq, Zq, Xr, Yr, Zr) can be symmetry adapted by applying the following four projection operators ... 

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... 

It is convenient to define a set of 3N mass-weighted Cartesian displacement coordinates q, q2,..., q N such that the first three q s are the components of Qi, the fourth, fifth and sixth q s are the components of Q2, and so on. The kinetic energy T can therefore be written... 

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 stability of the resulting P(VI) is emphasized by displacement coordinate experiment as shown in Scheme 11 [59]. 

With the use of the Cartesian displacement coordinates defined in Fig. 3 a basis vector is of the form... 

To illustrate the application of Eq. (37), consider the ammonia molecule with the system of 12 Cartesian displacement coordinates given by Eq. (19) as the basis. The reducible representation for the identity operation then corresponds to the unit matrix of order 12, whose character is obviously equal to 12. The symmetry operation A = Cj of Eq. (18) is represented by the matrix of Eq. (20) whore character is equal to zero. Hie same result is of course obtained for die operation , as it belongs to the same class. For the class 3av the character is equal to two, as exemplified by the matrices given by Eqs. (21) and (22) for the operations C and Z), respectively. The representation of the operation F is analogous to D (problem 12). 

It i usually convenient to work with a set of internal displacement coordinates, 5, as they have chemical significance. In the limit of small amplitudes of atomic displacements, the two sets of coordinates are linearly related. Thus,... 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

The operations shown in Fig. 2. can be easily recast in matrix language. For completeness, the displacement coordinates of the oxygen atom (numbered )... 

Fig, 3 Cartesian displacement coordinates for the ammonia molecule, The Z(CO a is perpendicular to the plane of the paper (which is not a plane of symmetry)... 

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. 

In the spectra of alkyl cobinamides two peaks have been observed at 3.89 and 4.42 which were assigned to the protons of a water molecule coordinated at the lower axial site (130). To confirm this assignment, it was found that addition of cyanide to methyl cobinamide, which displaces coordinated water, caused the peaks to disappear. Likewise, addition of excess D2O caused disappearance of the peaks through either ligand exchange or proton-deuteron exchange. 

Randall and Alberly (136) have studied the binding of various ligands to aquocobalamin using stopped flow techniques. This work suffers from the fact that it is not clear if the added ligand is displacing coordinated water or coordinated benzimidazole. One might be led to believe that the reaction studied in this work is in fact displacement of benzimidazole because the kinetics are at least inconsistent with a mechanism in which unimolecular dissociation of coordinated water is the rate limiting step. 

Decarbonylation of the acyl is likely to be metal-assisted (Ru11) giving rise to a Ru carbonyl, which is subsequently decarbony-lated by nucleophilic attack by nBu3P. This phosphine can displace coordinated carbonyl, as exemplified by reaction 3 ... 

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 ... 

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