mass weighting. By diagonalizing the force constant matrix according to e
L
{
F
x e
L ¼
Λ, the normal mode eigenvectors and eigenvalues are obtained.
Usually, the normal mode vectors e l μ are renormalized according to L ¼
e
L M
R
À Á 1=2 , where the elements of the mass matrix M
R are given by m
R
μ ¼
e l
{
μ
e l μ
À1
and represent the reduced mass of mode μ. Equation 3 can be written in
different ways. For example, without mass weighting as shown in Eq. (4):
F
x L ¼ MLΛ
ð4Þ
One obtains L
{ F
x L ¼ K and L
{ ML ¼ M
R , which define the diagonal normal force
constant matrix K and the reduced mass matrix M
R , respectively.
One can express the molecular geometry in terms of internal coordinates q n rather
than Cartesian coordinates x n , and by this, the Wilson equation adopts a new form:
[86]
F
q e
D ¼ G
À1 e
DΛ
ð5Þ
where e
D collects the normal mode vectors e d μ (μ ¼ 1, . . ., N vib ) column-wise, and
matrix G ¼ BM
À1 B
{ (Wilson G-matrix) gives the kinetic energy in terms of internal
coordinates [86]. The eigenvector matrix e
D has the property to diagonalize F
q and to
give e
D
{
F
q e
D ¼ Λ. If one does not mass weight the matrix D and works with
F
q D ¼ G
À1 DΛ, diagonalization leads to D
{ F
q D ¼ K.
Properties of a Local Mode The local mode vector a n associated with the internal
coordinate q n , which leads the nth local mode, is given by [224]:
a n ¼
K
À1 d
{
n
d n K
À1 d
{
n
ð6Þ
where the local mode is expressed in terms of normal coordinates Q μ . K is the
diagonal normal mode force constant matrix (see above) and d n is a row vector of the
matrix D. The local mode force constant k
a
n of mode n (superscript a denotes an
adiabatically relaxed, i.e., local mode) is obtained via Eq. (7):
k
a
n ¼ a
{
n Ka n ¼ d n K
À1 d
{
n
À
Á À1
ð7Þ
Local mode force constants, contrary to normal mode force constants, have the
advantage of being independent of the choice of the coordinates used to describe the
molecule in question [76, 224]. In recent work, Zou and co-workers proved that the
compliance constants Γ nn of Decius [117] are simply the reciprocal of the local mode
force constants: k
a
n ¼ 1=Γ nn [85, 226, 243].
238
E. Kraka and M. Freindorf
L
{
F
x e
L ¼
Λ, the normal mode eigenvectors and eigenvalues are obtained.
Usually, the normal mode vectors e l μ are renormalized according to L ¼
e
L M
R
À Á 1=2 , where the elements of the mass matrix M
R are given by m
R
μ ¼
e l
{
μ
e l μ
À1
and represent the reduced mass of mode μ. Equation 3 can be written in
different ways. For example, without mass weighting as shown in Eq. (4):
F
x L ¼ MLΛ
ð4Þ
One obtains L
{ F
x L ¼ K and L
{ ML ¼ M
R , which define the diagonal normal force
constant matrix K and the reduced mass matrix M
R , respectively.
One can express the molecular geometry in terms of internal coordinates q n rather
than Cartesian coordinates x n , and by this, the Wilson equation adopts a new form:
[86]
F
q e
D ¼ G
À1 e
DΛ
ð5Þ
where e
D collects the normal mode vectors e d μ (μ ¼ 1, . . ., N vib ) column-wise, and
matrix G ¼ BM
À1 B
{ (Wilson G-matrix) gives the kinetic energy in terms of internal
coordinates [86]. The eigenvector matrix e
D has the property to diagonalize F
q and to
give e
D
{
F
q e
D ¼ Λ. If one does not mass weight the matrix D and works with
F
q D ¼ G
À1 DΛ, diagonalization leads to D
{ F
q D ¼ K.
Properties of a Local Mode The local mode vector a n associated with the internal
coordinate q n , which leads the nth local mode, is given by [224]:
a n ¼
K
À1 d
{
n
d n K
À1 d
{
n
ð6Þ
where the local mode is expressed in terms of normal coordinates Q μ . K is the
diagonal normal mode force constant matrix (see above) and d n is a row vector of the
matrix D. The local mode force constant k
a
n of mode n (superscript a denotes an
adiabatically relaxed, i.e., local mode) is obtained via Eq. (7):
k
a
n ¼ a
{
n Ka n ¼ d n K
À1 d
{
n
À
Á À1
ð7Þ
Local mode force constants, contrary to normal mode force constants, have the
advantage of being independent of the choice of the coordinates used to describe the
molecule in question [76, 224]. In recent work, Zou and co-workers proved that the
compliance constants Γ nn of Decius [117] are simply the reciprocal of the local mode
force constants: k
a
n ¼ 1=Γ nn [85, 226, 243].
238
E. Kraka and M. Freindorf