matrix, Eq. (2.25), for a given set of ICs followed by using Eq. (2.24). By grouping
the non-redundant ICs according to
s ¼ s
1
1 ; s
1
2 ; . . .; s
1
k 1
; s
2
1 ; s
2
2 ; . . .; s
2
k 2
; . . .; s
N typ
1 ; s
N typ
2 ; . . .; s
N typ
k N typ
;
ð2:46Þ
where the superscript indicates the type number and the relation
X N typ
I¼1
k I ¼ K;
ð2:47Þ
is satisfied, the structures of the B
−1 and F matrices are as those presented in
Fig. 2.5.
Then, for a given set of optimal SFs f
opt
¼ f
opt
1 ; f
opt
2 ; . . .; f
opt
N typ
, called SQM SFs
or simply FF SFs, the diagonal FCs are multiplied by the corresponding factor, and
off-diagonal ones—by the geometric mean of the corresponding SFs, i.e.,
F
scal
l I m J
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
f
opt
I f
opt
J
q
F lm
ð2:48Þ
where additional indices in l I and m J are used to indicate that coordinates s l and s m
belong to types I and J, respectively. Obviously, this means that off-diagonal FCs in
a diagonal block b II corresponding to given type I (see Fig. 2.5) are multiplied by
the same factor as diagonal ones.
Fig. 2.5 Structures of the B
−1 and F matrices obtained by grouping internal coordinates into types
74
O. Bąk and P. Borowski