Set up the force equations
Write in matrix form and look for harmonic solutions
To solve this matrix for , set the determinant equal to 0. Consider a general determinant equation of the form
Matrix (5) is of this form when we define
and
Expanding (6) by minors gives
(8)
Identify the coefficients as follows
Plugging in (7) gives
Since for our problem, (9) simplifies to
One solution is therefore degenerate with
. This mode corresponds to a translation of the molecule.
The other two solutions are given via the quadratic equation
where and are given by equations (10) and (11).
Spring
© 1996-2007 Eric W. Weisstein
