Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix D 282
D.6 Eigenvalue equations
Consider a set of linear equations as considered in the previous section which can be
writtenintheform
Ax= x
(D.49)
where
(1) A is an × -dimensional matrix,
(2) is a scalar constant, called an eigenvalue of A ( is any one of the set of
eigenvalues of A), and
(3) x is an -dimensional column vector (which can in general be complex), called
the eigenvector of A belonging to the particular eigenvalue.
Eq. D.49 is called an eigenvalue equation. It has the following property: The multiplication
of x by (and hence that of x by the matrix A) changes the length of x (by
the factor ), but not the direction.
I Eigenvalues and eigenvectors of the molecular Hamiltonian: The determination
of eigenvalues and eigenvectors of the molecular Hamiltonian H, i.e., the “energy
matrix”, or the matrix of the Hamilton operator b , is the most important problem in
spectroscopy (in general, it is the key problem!!).
I Secular equation and eigenvalues λ : Eq. D.49 can be rewritten as
(A − I) x = 0 (D.50)
In order not to obtain the trivial solutions 1 = 2 = 3 = =0, det (A − I) has
to vanish, i.e., det (A − I) =0. This is the so-called characteristic equation or
secular equation:
|A − I| =
¯
The secular equation has roots
which are called the eigenvalues.
11 − 12 1
21 22 − 2
. . .
=0 (D.51)
1 2 − ¯
{ 1 2 } (D.52)