Physical Chemistry Third Edition

920 22 Translational, Rotational, and Vibrational States of Atoms and Molecules

Appendix E and was applied to the hydrogen atom in Chapter 17. We relabel the

center-of-mass term as the translational term,Ĥtr:

ĤnucĤc+ĤrelĤtr+Ĥrel (22.2-5)

̂Htr−h ̄

2

2 M

∇c^2 +Vext(xc,yc,zc) (22.2-6)

Ĥrel−h ̄

2

2 μ

∇^2 +V(r) (22.2-7)

where we now drop the subscript AB on the internuclear distance. The sum of the

masses of the two nuclei is denoted byMand their reduced mass is denoted byμ:

MmA+mB (22.2-8)

μ

mAmB

mA+mB

(22.2-9)

We neglect the mass of the electrons compared with the nuclei and considerMto

the mass of the entire molecule. The operator∇c^2 is the Laplacian operator for the

center-of-mass coordinates and the operator∇is the Laplacian operator for the relative

coordinates. Thexcoordinate of the center of mass is

xc

mAxA+mBxB

M

(22.2-10)

and the relativexcoordinate is

xxB−xA (22.2-11)

Theyandzcoordinates are similarly defined.

Since the Hamiltonian operator in Eq. (22.2-5) contains the two terms in Eq. (22.2-6)

and Eq. (22.2-7) it can be solved by the separation of variables, yielding one Schrödinger

equation for̂Htrand one Schrödinger equation for̂Hrel. We assume that the molecule

is contained in a rectangular box. The Schrödinger equation for̂Htris like that of a

structureless particle in the box, and the particle-in-a-box energy wave functions and

the particle-in-a-box energy eigenvalues from Chapter 15 can be used. We now focus

on the Schrödinger equation for̂Hrel, which can be solved by a second separation of

variables in the same way as was done for the hydrogen atom.

The relative HamiltonianĤrelgives the same Schrödinger equation as in Eq. (17.2-2)

since that equation applies to any two-body system in whichV depends only onr.

We now omit the subscript AB onr. This equation is solved by the trial function of

Eq. (17.2-3):

ψrel(r,θ,φ)R(r)Y(θ,φ) (22.2-12)

TheYfactor represents the same spherical harmonic functions as in Chapter 17, which

are eigenfunctions of̂L^2 , the operator for the square of the angular momentum. Sub-

stitution ofψrelR(r)Y(θ,φ) into the time-independent Schrödinger equation and

division byR(r)Y(θ,φ) gives Eq. (17.2-5):

−

1

R

d

dr

(

r^2

dR

dr

)

+

2 μr^2

h ̄^2

(V−Erel)+

1

h ̄^2

1

Y

̂L^2 Y 0 (22.2-13)

To solve this equation for the nuclear motion of a diatomic molecule, we must have a

representation of the potential energy functionV(r).