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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).
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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).
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FORTRAN
Lines of Code: 2642
Recommended Citation: D. F. McIntosh and M. R.
Peterson, QCPE 11, 342 (1977).
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