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QCMP012. 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
Converted by Timothy J. O'Leary, Department
of Health and Human Services, Bethesda,
Maryland 20205
This set of three programs allows the user to
analyze a general vibrational problem in
terms of the method of Wilson, Decius and
Cross1. 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.
Schactschneider2 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 coordinate).
Twoimportantdifferencesbetween
Schactschneider'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 for 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. Schactschneider, 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 77 (Microsoft FORTRAN)
Lines of Code: 2650
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