It has been well established from spectroscopic measurements that the total mole- cular energy of a diatomic molecule is the sum of an electronic, a vibrational, and a rotational energy:
Eðn;v;Jị ẳenỵ vỵ1 2
honỵJðJỵ1ịBn;v ð12-15ị Heren,v, andJ are the electronic, vibrational, and rotational quantum numbers, respectively; on is the vibrational frequency corresponding to the electronic state n; andBn;vis its rotational constant. We will derive the above relation by solving the molecular Schro¨dinger equation
ẵHnucðRị ỵHelðr;Rịðr;Rị ẳEðr;Rị ð12-16ị while making use of the Born-Oppenheimer approximation.
We assume that the electronic eigenstate of the molecule is described by Eq. (12-12) so that we may substitute
ðr;Rị ẳfnðX;Y;ZịFnðr;Rị ð12-17ị into the Schro¨dinger equation (12-16). The result is
ẵHnuclðRị ỵenðRịfnðX;Y;Zị ẳE fnðX;Y;Zị ð12-18ị since we may divide the Schro¨dinger equation by the electronic eigenfunction Fnðr;R).
This equation may be solved by transforming the nuclear coordinates (X;Y;Z) into polar coordinates (R;y;f),
XẳRsinycosf Y ẳRsinycosf ZẳRcosy
ð12-19ị
and by introducing the nuclear angular momentum operators
Lxẳh i Y q
qZZ q qY
etc: ð12-20ị
analogous to Eq. (7-6). This allows us to write the nuclear Hamiltonian Hnucl of Eq. (12-9) in the form
Hnuclẳ h2 2m
q2 qR2þ2
R q qR
þ L2
2mR2þZaZbe2
R ð12-21ị
Here we have made use of Eqs. (3-51) and (7-31).
The eigenvalues and eigenfunctions of the angular momentum operatorL2 were derived in Chapters 6 and 7; the result was described in Eq. (7-30), and we may write it as
L2Jðy;fị ẳh2JðJỵ1ịJðy;jị ð12-22ị The eigenfunctionsfnof the Schro¨dinger equation (12-18) may therefore be repre- sented as
fnðX;Y;Zị ẳgn;JðRịJðy;jị ð12-23ị
NUCLEAR MOTION OF DIATOMIC MOLECULES 165
Substitution into Eq. (12-18) leads to the following equation:
ẵHnuclðRị ỵenðRịfnðX;Y;Zị ẳ h2 2m
q2 qR2þ2
R q qR
gðRịJðy;jị þ L2
2mR2 gðRịJðy;jị þ ZaZbe2
R ỵenðRị
gðRịJðy;jị
ẳEgðRịJðy;jị ð12-24ị
By making use of Eq. (12-22) this may be simplified to
h2 2m
q2 qR2þ2
R q qR
gðRị ỵUnðRịgðRị ẳEgðRị ð12-25ị
with
UnðRị ẳenðRị ỵZaZbe2
R ỵh2JðJỵ1ị
2mR2 ð12-26ị
We introduce a final small adjustment by substituting gðRị ẳcðRị
R ð12-27ị
into Eq. (12-25), which leads to
h 2m
q2c
qR2ỵUnðRịcðRị ẳEcðRị ð12-28ị This is the customary form of the Schro¨dinger equation representing the vibrational motion of a diatomic molecule.
In order to solve the vibrational Schro¨dinger equation (12-28), it is necessary to have a general understanding of the behavior of the functionsUn (R), which are called the molecular potential curves. We first consider the molecular ground state, and we have sketched a typical potential curveU1(R) of a stable diatomic molecule in Figure 12-2. The potential function has a minimum
U1ðR1ị ẳe1ðR1ị ỵZaZbe2 R1
ỵh2JðJỵ1ị
2mR21 ð12-29ị at its equilibrium nuclear distanceR1. Strictly speaking, the potential function and the position of its minimum also depend on the value of its rotational quantum
numberJ, but this constitutes only a very small correction and it is usually ignored.
The potential function tends to infinity whenRapproaches zero due to the Coulomb repulsion of the two nuclei. For very large values ofRthe potential curve asymp- totically approaches a constant value
U1ð1ị ẳU1ðR1ị ỵD ð12-30ị whereDis the dissociation energy of the molecule.
The solution of the Schro¨dinger equation (12-28) for the electronic ground state nẳ1 may now be derived by expanding the potential functionU1ðRịas a power series in terms of its coordinate around its minimum RẳR1,
U1ðRị ẳU1ðR1ị ỵ1
2k q2þ1
6k3 q3ỵ ð12-31ị where
kẳ q2U1ðRị qR2
R1
kn ẳ qnU1ðRị qRn
R1
qẳRR1 ð12-32ị U2(R)
U1(R)
5 4
3 2
1
Figure 12-2 Potential curves of a diatomic molecule. The curves are derived from a calculation on the hydrogen molecular ion.
NUCLEAR MOTION OF DIATOMIC MOLECULES 167
The customary approximation consists of terminating the power series after its sec- ond term, which leads to
h2 2m
d2 dq2þ1
2k q2
cẳec ð12-33ị
where
eẳEU1ðR1ị ð12-34ị We note that this equation (12-33) is identical to the Schro¨dinger equation (6-50) of the harmonic oscillator, which we discussed in Section 6.IV. Its eigenvaluesev
are given by
evẳ ðvỵ1=2ịho vẳ1;2;3;etc: ð12-35ị oẳ
ffiffiffi k m s
ð12-36ị
according to Eqs. (6-62) and (6-65). The corresponding eigenfunctions cvðqịare described by Eqs. (6-63) and (6-66). Finally, by combining Eqs. (12-35), (12-34), and (12-29), we find that the molecular energiesEð1;v;Jịare given by
Eð1;v;jị ẳe1ðR1ị ỵZaZbe2 R1
ỵ ðvỵ1=2ịhoỵJðJỵ1ịh2
2mR21 ð12-37ị This is consistent with the empirical result of Eq. (12-15) if we define the rotational constant as
B1ẳ h2
2mR21 ð12-38ị
Since the reduced mass of a diatomic molecule is known, the equilibrium ground state internuclear distance is easily derived from experimental results of the rotational constant B. It may be instructive to present some numerical results, and we have listed the rotational constantsBand ground state internuclear distances R1for some diatomic molecules in Table 12-1. The customary unit for the rotational constantB, cm1, refers to the wave number s, which is the inverse of the wave- length of the corresponding transition:
Eẳhnẳhcs ð12-39ị
It may be seen that the rotational constantsB1depend primarily on the reduced masses of the nuclei, so that their values for the hydrogen molecule and the various
hydrides are significantly larger than their values for other diatomics. The values of R1depend partially on the size of the atoms but also on the strength of the bond; this may explain the rather large values for both Cl2and Li2.
In Table 12-1 we also list the values of the vibrational frequenciesn, again in terms of cm1. The latter depend on the values of the force constantsk, which are related to the strength of the chemical bonds and, to a lesser extent, the nuclear masses. This explains the large value for HF, which has a very strong bond, and the small values for LiH and Li2, whose chemical bonds are much weaker.
Finally, we comment briefly on the potential curves corresponding to the various electronic eigenstates of the molecules. These may be divided into two different types, both sketched in Figure 12-2. The first type, denoted byU1ðRị, has a mini- mum at an internuclear distanceR1. The second type, denoted byU2ðRị, does not have a minimum. Excitation from the ground state to this electronic eigenstate therefore leads to molecular dissociation.