Computational Chemistry

gi¼

ðÞ@E=@q (^1) i
ðÞ@E=@q (^2) i
...
ðÞ@E=@q (^9) i

0

B

B

B

@

1

C

C

C

A

(2.10)

and the second-derivative matrix, the force constant matrix, is

H¼

@^2 E=@q 1 q 1 @^2 E=@q 1 q 1 &&& @^2 E=@q 1 q 9

@^2 E=@q 2 q 1

...

@^2 E=@q 2 q 2 &&&

... &&&

@^2 E=@q 2 q 9

...

@^2 E=@q 9 q 1 @^2 E=@q 9 q 2 &&& @^2 E=@q 9 q 9

0 B B B B @

1 C C C C A

(2.11)

The force constant matrix is called the Hessian.^6 The Hessian is particularly
important, not only for geometry optimization, but also for the characterization of
stationary points as minima, transition states or hilltops, and for the calculation of
IR spectra (Section2.5). In the Hessian∂^2 E/∂q 1 q 2 ¼∂^2 E/∂q 2 q 1 , as is true for all
well-behaved functions, but this systematic notation is preferable: the first subscript
refers to the row and the second to the column. The geometry coordinate matrices
for the initial and optimized structures are

qi¼

qi 1

qi 2

...

qi 9

0

B

BB

@

1

C

CC

A

(2.12)

and

qo¼

qo1

qo2

...

qo9

0

BB

B

@

1

CC

C

A

(2.13)

The matrix equation for the general case can be shown to be:

qo¼qi%H%^1 gi (2.14)

which is analogous to Eq.2.9for the optimization of a diatomic molecule, which
could be written

qo¼qi%ðd^2 E=dq^2 Þ%^1 ðdE=dqÞi

(^6) Ludwig Otto Hesse, 1811–1874, German mathematician.
2.4 Geometry Optimization 29