|
576. General Vibrational Analysis System
by Douglas F. McIntosh and Michael R. Peterson,
Department of Chemistry, Lash Miller Chemical
Laboratories, University of Toronto, 80 St. George
St., Toronto, Ontario, Canada M5S 1A1
This package of 4 FORTRAN programs has been designed to
allow the user a simple and reliable method for
generating a complete vibrational analysis using the GF
matrix method of Wilson, Decius and Cross (1). The
analysis may be done using either an unsymmetrized or a
symmetrized basis set. This new set is meant to be a
replacement for our original publication, "General
Vibrational Analysis Programs Utilizing The Wilson GF
Matrix Method for a General Unsymmetrical Molecule"
(QCPE 342).
The Wilson methodology requires the definition of a
basis set which will be used to construct the normal
coordinates. These programs allow for the definition
of unsymmetrized or symmetrized basis sets, referred to
as Internal and Symmetry Coordinates, respectively.
These are related to Cartesian Coordinates via the B
and U Matrices through the following relationships:
R = BX
S = UR = UBX = BSymX
where R, S and X represent column vectors of the
Internal, Symmetry and Cartesian Coordinates,
respectively. The BSym Matrix, which is the product of
the U and B Matrices, is referred to as the "Symmetry
Adapted B Matrix". Using B or BSym we may determine
the Inverse Kinetic Energy Matrices, G and GSym:
G = BM1Bt
GSym = (BSym)M-1(BSym)t = UMB-1BtUt
where the superscript "t" indicates "transpose" and the
M-1 is a diagonal matrix which contains the inverses
of the masses of the atoms (included 3 times each to
account for motions along the x, y and z directions).
Using the G or GSym Matrix, the Vibrational Secular
Equation may be formulated in either an unsymmetrized
or a symmetrized fashion:
GFL = Ll
GSymFSymLSym = LSym l
The problem reduced to the determination of the
eigenvector (L or LSym ) and the eigenvalue (l)
matrices.
The four programs included in this package are:
(1) UMAT (Vibrational Analysis Program 1)
This is a multi-functional program which is intended to
"set-up" the entire analysis. It may be used to:
(i) Predict the symbolic form of the Potential Energy
(IFF) Matrix
(ii) Calculate Orthonormal Symmetry Coordinates
(Symmetry Adapted Linear Combinations of Internal
Coordinates)
(iii) Output the Symmetry Adapted B Matrix (BSym) in a
format suitable for input to Programs 2 and 3
(iv) Symmetrize F and IFF to yield the FSym and IFSYM
Matrices, the latter being the block-diagonalized,
symbolic form of the Potential Energy Matrix
Included with the UMAT program are a number of data
files which contain the necessary information for a
large number of symmetry point groups:
Non-Axial Groups: Ci and Cs
Axial Groups: Cn and S2n(n = 1 to 8)
Cnh and Cnv (n = 2 to 8)
Dihedral Groups: Dn, Dnh and Dnd
(n = 2 to 8)
Cubic Groups: T, Th, Td, O and Oh
Icosahedral Groups: I and Ih
The Linear Groups, C v and D h, may be conveniently
handled by running either C2v or C4v for the former and
D2h or D4h for the latter (see documentation for
further details).
(2) BMAT (Vibrational Analysis Program 1A)
This is a modification of the GMAT program of J. H.
Schachtschneider (8). BMAT generates an unsymmetrized
B Matrix and outputs it in a format suitable for input
to Programs 2 and 3. Like UMAT, it allows for the
calculation of the following types of Internal
Coordinates: (i) Bond Stretch, (ii) Valence Angle
Bend, (iii) Out-of-Plane Wag, (iv) Torsion and (v)
Linear Bend. Two major innovations have been included
in BMAT, namely, Hilderbrandt's normalization of the
Torsional Coordinate (9) and the new formulae of
McIntosh, Michaelian and Peterson for the Out-of-Plane
Wag (10). BMAT is included as a subroutine in the UMAT
program, which is intended as its replacement. It is
included in the package for those users who prefer to
run unsymmetrized basis sets (i.e., internal
coordinates) and because of its utility in producing
the Bd Matrix (for Root-Mean-Square Amplitude
calculations).
(3) FTRY-ATOM-RMSA-INTY (Vibrational Analysis Program
2)
Program 2 is the "heart" of the package. It is also
multi-functional and may be run in one of 4 modes:
(i) The FTRY Option will produce the complete set of
frequencies for a series of isotopically related
molecules. This option may be repeatedly run to
manually refine the molecule's General Quadratic Force
Field and obtain a better fit between the calculated
and observed frequencies.
(ii) The ATOM Option will yield a Normal Coordinate
Analysis for the first molecule, only, of this series
(included are the (a) Eigenvector (L), (b) Potential
Energy Distribution (POT), (c) Atomic Displacements
(AA) and the (d) Root-Mean-Square Amplitude (äR, äX,
äd) Matrices).
(iii) The RMSA Option allows the user an alternate
method of calculating the R.M.S. Amplitudes of
Vibration.
(iv) The INTY Option generates the Infrared Intensities
of the frequencies of the series of isotopically
related molecules. This is based on a point charge
model and yields only approximate values. All
intensities are ratioed against the largest value
(which is arbitrarily assigned a value of 10.00).
(4) FFIT (Vibrational Analysis Program 3)
Program 3 utilizes the SIMPLEX optimization algorithm
of Nelder and Mead (7) to refine the best-guessed force
constants via a non-linear least-squares analysis
between the calculated and observed frequencies.
FORTRAN 77 (VAX)
Lines of Code: 6309
| 
