CHAP. 12: BASIC TERMS OF CHEMICAL PHYSICS [CONTENTS] 422
whereV(~v) is the total potential energy of a system of N molecules, andViare the energies
of the individual molecules. For each molecule we then separate the terms corresponding to
different types of motion:
- translation,
- rotation,
- vibration,
- movement of electrons around the nuclei,
Vi=Vtrans+Vrot+Vvib+Vel. (12.21)
Thus we also separate the wavefunctions and the corresponding energy eigenvalues of these
motions from one another. The solution of the Schr ̈odinger equation for basic types of particles
motion is presented in the following sections.
12.2.3 Translation
The simplest type of molecular motion is a translation inside a rectangular box—parallelepiped
of the sidesa,bandc. The Schr ̈odinger equation has the form
−
h^2
8 π^2 m(
∂^2 Ψ
∂x^2+
∂^2 Ψ
∂y^2+
∂^2 Ψ
∂z^2)
+Vtrans(x, y, z)Ψ =EtransΨ, (12.22)whereVtrans(x, y, z) is the potential energy of a particle given by the relation
Vtrans(x, y, z) ={
0 , inside box
∞, outside box.(12.23)
The translational energy of the particle is then given by the eigenvalues
Etrans=h^2
8 m(
n^2 x
a^2+
n^2 y
b^2+
n^2 z
c^2)
, (12.24)wherenx, ny, andnzare translational quantum numbers (nx,ny,nz= 1, 2 , 3.. .). The wave-
function is uniquely determined by the triplet (nx, ny, nz).