read “nu tilde”;n, “nu bar”, has been used less frequently.c¼velocity of light,k¼
force constant for the vibration,m¼reduced mass of the molecule¼(mAmB)/(mA+
mB);mAandmBare the masses of A and B.
The force constantkof a vibrational mode is a measure of the “stiffness” of the
molecule toward that vibrational mode – the harder it is to stretch or bend the
molecule in the manner of that mode, the bigger is that force constant (for a
diatomic moleculeksimply corresponds to the stiffness of the one bond). The
fact that the frequency of a vibrational mode is related to the force constant for the
mode suggests that it might be possible to calculate the normal-mode frequencies of
a molecule, that is, the directions and frequencies of the atomic motions, from its
force constant matrix (its Hessian). This is indeed possible:matrix diagonalization
of the Hessian gives the directional characteristics (which way the atoms are
moving), and the force constants themselves, for the vibrations. Matrix diagonali-
zation (Section 4.3.3) is a process in which a square matrixAis decomposed into
three square matrices,P,D, andP%^1 :A¼PDP%^1 .Dis a diagonal matrix: as withk
in Eq.2.17all its off-diagonal elements are zero.Pis a premultiplying matrix and
P%^1 is the inverse ofP. When matrix algebra is applied to physical problems, the
diagonal row elements ofDare the magnitudes of some physical quantity, and each
column ofPis a set of coordinates which give a direction associated with that
physical quantity. These ideas are made more concrete in the discussion accom-
panying Eq.2.17, which shows the diagonalization of the Hessian matrix for a
triatomic molecule, e.g. H 2 O.
H¼
@^2 E=@q 1 q 1 @^2 E=@q 1 q 2 &&& @^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 90
B
B
B
@
1 C C C A ¼
q 11 q 12 &&& q 19
q 21 q 22 &&& q 29
...q 91 q 92 &&& q 990 B B B B @
1 C C C C A
k 1 0 &&& 0
0 k 2 &&& 0
...00 &&& k 90 B B B B @
1 C C C C A
P%^1
Pk(2.17)
Equation 2.17 is of the formA¼PDP%^1. The 9(9 Hessian for a triatomic
molecule (three Cartesian coordinates for each atom) is decomposed by diagona-
lization into aPmatrix whose columns are “direction vectors” for the vibrations
whose force constants are given by thekmatrix. Actually, columns 1, 2 and 3 ofP
and the correspondingk 1 ,k 2 andk 3 ofkrefer totranslationalmotion of the
molecule (motion of the whole molecule from one place to another in space);
these three “force constants” are nearly zero. Columns 4, 5 and 6 ofPand the
correspondingk 4 ,k 5 andk 6 ofkrefer to rotational motion about the three principal
32 2 The Concept of the Potential Energy Surface