Biomedical Engineering Reference
In-Depth Information
of 0's along the diagonal. In our case,
GG
−
1
will have
p
1's along the diagonal and 3
N
p
zeros. There aremany algorithms for calculating such a generalized inverse. The zerosmight
actually be tiny numbers, depending on the accuracy of the numerical algorithm used.
It can be shown that the gradient vector referred to the independent variables is related
to the Coordinate gradient vector by
−
grad
U
=
G
−
1
Bu
(grad
U
)
C
(5.11)
Similar considerations apply to the Hessian, and formulae are available in the literature.
5.12 Redundant Internal Coordinates
All coordinate systems are equal in principle, but I have stressed above that:
•
they should ideally give transferable force constants from molecule to molecule;
•
they should ideally be capable of being represented as harmonic terms in
U
, i.e. cubic
and higher corrections should not be needed.
These requirements can be best satisfied by local internal valence coordinates such as bond
lengths, bond angles and dihedral angles. The expression 'local' in this context means that
the coordinates should extend to only a few atoms.
Pulay
et al.
(1979) and Schlegel and co-workers (see Peng
et al.
1994, 1996) both
investigated the use of redundant internal coordinates for gradient optimizations. Pulay
et al.
defined an internal coordinate system similar to that used by vibrational spectroscop-
ists. It minimizes the number of redundancies by using local symmetry coordinates about
each atom and special coordinates for ring deformations, ring fusions, etc. Schlegel and
co-workers used a simpler set of internal coordinates composed of all bond lengths, valence
angles and dihedral angles.
The mathematical considerations outlined above also apply here.
Packages like Gaussian 03 offer a choice between Z-matrix and Cartesian coordinates
for input of the molecular geometry. Gaussian 03 then routinely uses redundant internal
coordinates for the optimizations rather than the Z-matrix.
References
Baker, J. and Hehre, W.J. (1991)
J. Comput. Chem.
,
12
, 606.
Fletcher, R and Powell, M.J.D. (1963)
Comput. J.
,
6
, 163.
Fletcher, R and Reeves, C.M. (1964)
Comput. J.
,
7
, 149.
Hendrickson, J.B. (1961)
J. Am. Chem. Soc.
,
83
, 4537.
Peng, C and Schlegel, H.B. (1994)
Israel J. Chem.
,
33
, 449.
Peng, C., Ayala, P.Y. and Schlegel, H.B. (1996)
J. Comput. Chem.
,
17
, 49.
Polak, E. and Ribière, G. (1969)
Rev. Fr. Inform. Rech. Operation
,
16-R1
, 35.
Pulay, P., Fogarasi, G., Pang, F. and Boggs, J.E. (1979)
J. Am. Chem. Soc.
,
101
, 2550.
Westheimer, F.H. (1956) in
Steric Effects in Organic Chemistry
(ed. M.S. Newman), John Wiley &
Sons, Inc., New York.
Wiberg, K. (1965)
J. Am. Chem. Soc.
,
87
, 1070.
