where Q is the amplitude and 8* is the phase constant of the motion, and the frequency of a normal vibration is
In order to obtain a reasonably accurate set of force constants, an approximate set of force constants is used to calculate the fundamental frequencies. If there are large
discrepancies between the calculated and observed frequencies, the force constants are adjusted and the calculations repeated until a satisfactory correlation is obtained
(Nakamoto 1986). When agreement between the calculated and observed frequencies is reached, the force constants are considered to be a representation of the potential energy of the system.
One of the difficulties encountered with this type of calculation is that the number of force constants of a molecule is generally larger than the number of
frequencies, resulting in solutions which are not unique. For small molecules with high symmetry the calculations are more satisfactory as a number of the force constants are identical. (Califano 1976).
A generalised valence force (GVF) field consisting of bond stretching and bending force constants and the interaction force constants between each coordinate may be used, however the number of interaction constants is often too large to obtain reliable results. The simpler Urey-Bradley (UB) force field was introduced by Shimanouchi (1949) consisting of stretching, bending and repulsive force constants. The repulsive force constants represent the forces between non-bonded atoms. The method for calculation of UB constants is described by Overend and Scherer (1960). Fewer force constants are used for this method and since they relate specifically to stretches, bends and non-bonded interactions between two atoms should be easier to transfer to similar molecules. The general form of the potential field is given by
2V =X [2K i-riAri + Kj(Ar;)2]
+ 2 j 2Hijri2Aaij + Hjj(rjAaij)2] i<j
+ X l ^ q i j A q i j + Fij(A qij)2] i<j
where K, K ' are the stretching force constants, H, H ' are the bending force constants and F, F are the repulsive force constants between the non-bonded atoms. Ar, Act, and Aq are the changes in the bond lengths, bond angles and non-bonded atom separations respectively, and i and j represent the atoms involved in the vibration.
The advantage of using the UB force field is that the final force constants can be directly related to the internal coordinates, and when there is little internal torsion in the molecule, they are often transferable between similar molecules (Shimanouchi 1963). One disadvantage o f this method is that redundancies may occur in the coordinates as the non-bonded distances must be included. Since the coordinates are not independent, the linear terms in the potential energy equation may not be zero (Califano 1976). The relationship between the molecular parameters is
q ij =
r?
+Tf -
2 r i r j C o s ( X i j (Califano 1976)Using this relationship the redundant coordinates may be removed from the potential energy equation. The linear terms then become zero and F ' is introduced into the quadratic terms.
(rj-rjcosogj) (rjsincxij)
SlJ qjj tij -■ q.. L (Overend & Scherer 1960) The relationship between F and F has been established for the short distances between two non-bonded atoms such that F ' = -0.1F (Cahfano 1976). In all these calculations it is assumed that the repulsive forces between two atoms across 3 bonds is negligible (Califano 1976).
The programs NORCORD and OVER carry out a normal coordinate analysis using the UB force field method described by Overend and Scherer (1960). The sequence of calculations and perturbation cycle is given in the flowchart shown in figure 1.6.
Input to the program NORCORD consists of the Cartesian coordinates, the internal coordinates, the symmetry blocks of the molecule and the U matrix. The U matrix gives the magnitude of the contribution of each internal coordinate (columns) for each symmetry coordinate (rows). The internal coordinates include all the stretches and bends and the symmetry coordinates are determined using the appropriate character table, depending on the symmetry group of the molecule. AH redundancies are removed during the calculations, a G matrix o f kinetic energy data is computed, transformed into symmetry coordinates and saved on disk in a form ready for use by the program OVER. The internal and symmetry coordinates used for each symmetry group are given in chapter five, and the detailed input to NORCORD for each molecule is given in appendix B.
where
NORCORD
OVER
adjust force constants calculate F matrix solve IGF - E XI = 0 calculate frequencies and observed frequencies Input force constants, Z matrix Input cartesian coordinates
internal coordinates, symmetry block and U matrix
Transform coordinates and calculate G matrix
end
Figure 1.6. A block diagram of the force constant refinement.
The program OVER requires a set of approximate force constants to calculate the fundamental vibrational frequencies and the potential energy distribution. The exact values of the force constants within the molecules being studied were not available, so values found in the literature for C-H, C-F, C-Cl and C-C bonds (Bucker & Nielsen
1963, Naito et al. 1955) have been used to give an approximate set o f force constants for the initial calculations. The input data also includes the observed frequencies from an infrared spectrum of the molecule, the kinetic energy and symmetry coordinate
information from NORCORD, and the
Z
matrix. TheZ
matrix transforms the force constants into the F matrix in the required coordinates, such that the secular equationIGF
-E X \
= 0 can be solved for allX
and hence the normal frequencies calculated (Overend & Scherer 1960). The coefficients o fZ
are calculated using the table ofrelationships between the force constants and Fy and F{- in the publication by Overend and Scherer (1960). The method used for constructing the input to OVER is detailed in chapter five and the final force constants and calculated frequencies are given in chapter six. The detailed input for each molecule is given in appendix B. A copy of the
programs adapted for use on the VAX computer was available here at the Australian National University. Errors were found in the programs when using bond lengths correct to only two decimal places or attempting to refine five or more force constants. Considerable time was spent amending the programs to facilitate their use, correcting the errors, and adapting them to run on the SUN/UNIX system. The transferability of force constants between different CFCs and different HFCs has been investigated in this work, and the results shown in chapter six. Attempts to fit the calculated
frequencies to the observed frequencies for individual molecules have also been made, and the resulting trends in the force constants over each set o f molecules is given in chapter six.
1.6.2 Ab initio calculations.
The ab initio method of computing model chemical structures and molecular
properties uses the laws o f quantum mechanics , the fundamental constants c, m, e, and h, (the speed of light, the masses and charges o f electrons and nuclei, and Plank's constant respectively), and a set of mathematical approximations to calculate the solutions of the Schrödinger equations for the system (Foresman and Frisch 1993). The time-independent Schrödinger equation for the energy o f a wavefunction 'F can be written as,
H*F = (Atkins 1986.)
the Hamiltonian operator,
h
-
s v2 *
vfor a moving particle of mass m, where,