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342. General Vibrational Analysis Programs Utilizing
the Wilson GF Matrix Method for a General Unsymmetrized
Molecule
by Douglas F. McIntosh and Michael R. Peterson, Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 1A1 This set of three programs allows the user to analyze a general vibrational problem in terms of the method of Wilson, Decius and Cross.1 The programs allow for a general solution with a complete set of internal coordinates or, if desired, a less complete and more restricted basis set. Previous methods of solving the general vibrational equation GFL = Ll where G and F are the familiar matrices of the Wilson method, L is the eigenvector matrix and l is the diagonal eigenvalue matrix, have been hampered by the requirement that the basis vectors (internal coordinates) be orthonormalized. This is generally accomplished by applying both symmetry projection operators and a Schmidt orthogonalization process. In the methodology of the present programs, the basis vectors need not be orthogonal. This allows for the direct solution of the vibrational problem without recourse to the use of Symmetry Coordinates, although provision has been made for their use if desired. The programs will generate the required 3N-6 (or 3N-5, for linear molecules) eigenvalues expected for a complete analysis from 3N-6 (3N-5) to 3N basis vectors. The normal difficulties inherent in redundant coordinates or modes present no problems. Program 1 computes the Wilson B matrix which will be used in all the remaining programs. The BMAT program is a modification of the original GMAT program of J. H. Schachtschneider2 and allows for the calculation of six different types of internal coordinates, namely: (1) Bond Stretch, (2) Valence Angle Bend, (3) Out-of-Plane Wag, (4) Torsion, (5) Linear Bend (defining 2 internal coordinates), and (6) Linear Bend (defining 1 internal coodinate).Two important differences between Schachtschneider's original GMAT program and BMAT is the inclusion of R. L. Hilderbrandt's method of normalization of the torsional coordinate3 and the use of new formulae for the Out-of-Plane Wag. Program 2 is a dual-purpose program allowing the user to compute either the complete set of eigenvalues for a series of isotopically related molecules or the series of matrices important to the interpretation of a vibrational problem. These matrices include (1) the L matrix, (2) the Potential Energy Distribution matrix, (3) the Atom Displacement matrix, and (4) the Root Mean Square Amplitudes of Vibration between Atom Pairs (bonded and nonbonded), Cartesian Displacements from the equilibrium geometry and Internal Coordinates. The first part of program 2 also allows the user to manipulate the F matrix with individual calculations, to obtain an approximate fit of experimental data. Program 3 utilizes the SIMPLEX optimization algorithm of Nelder and Mead4 to refine the guessed force constants via a non-linear least-squares analysis between calculated and observed frequencies (eigenvalues). __________ References: 1. E. Bright Wilson, Jr., J. C. Decius and Paul Cross, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra (New York: McGraw-Hill Book Co., 1955). 2. J. H. Schachtschneider, Reports 231/64 and 57/65, Shell Development Co., West Hollow Research Center, P.O. Box 1380, Houston, Texas 77001. 3. R. L. Hilderbrandt, J. Mol. Spectroscopy, 44, 599 (1972). 4. J. A. Nelder and R. Mead, Computer Journal, 7, 1809 (1965). _________ FORTRAN Lines of Code: 2642 Recommended Citation: D. F. McIntosh and M. R. Peterson, QCPE 11, 342 (1977). |