Physical Chemistry Third Edition

20.1 The Born–Oppenheimer Approximation and the Hydrogen Molecule Ion 825

We can omit the constantVnnfrom the electronic Hamiltonian and later add that

constant to the energy eigenvalues (see Problem 15.6). The Born–Oppenheimer Hamil-

tonian is now

ĤBOĤel+Vnn (20.1-3)

whereĤelis the electronic Hamiltonian:

̂Hel−h ̄

2

2 m

∇^2 +

e^2

4 πε 0

(

−

1

rA

−

1

rB

)

(20.1-4)

The electronic Schrödinger equation is

̂HelψelEelψel (20.1-5)

whereEelis the electronic energy eigenvalue andψelis a one-electron wave function

that we call amolecular orbital, since it involves motion of the electron around the

entire molecule. The Born–Oppenheimer energy is

EBOEel+Vnn (20.1-6)

The electronic Schrödinger equation for the H+ 2 ion can be solved by transforming to a

coordinate system that is calledconfocal polar elliptical coordinates. One coordinate

isξ(rA+rB)/rAB, the second coordinate isη(rA−rB)/rAB, and the third coor-

dinate is the angleφ, the same angle as in spherical polar coordinates. The solutions to

the electronic Schrödinger equation are products of three factors:

ψelL(ξ)M(η)Φ(φ) (20.1-7)

The factorΦ(φ) is the same factorΦthat occurs in hydrogen-like atomic orbitals. As

in that case, we can choose either the complexΦfunctionsΦmor the realΦfunctions

ΦmxandΦmy. The other factors are more complicated and we do not display the formu-

las representing them.^2 These molecular orbitals are calledexact Born–Oppenheimer

molecular orbitals. They are not exact solutions to the complete Schrödinger equation,

but they contain no approximations other than the Born–Oppenheimer approximation.

Figure 20.3 shows the Born–Oppenheimer energy of H+ 2 as a function ofrABfor

the two states of lowest energy. We denote the molecular orbital for the ground state by

ψ 1 since it is customary to number orbitals from the lowest energy to the highest. The

ground-state energy has a minimum atrAB 1. 06 × 10 −^10 m106 pm 1 .06 Å.

We denote this equilibrium value ofrABbyre. For large values ofrABthe energy

approaches a constant value, which we set equal to zero. The difference in energy

between this constant value and the value of the energy atrABreis denoted by

Deand is called thedissociation energyof the molecule. The dissociation energy of

H+ 2 is equal to 2.8 eV. This dissociation energy is large enough that the molecule is

chemically bonded in the ground state. We say that the molecule has a bond order of

1 /2, corresponding to one shared electron. We denote the molecular orbital for the first

excited state byψ 2. This state has an energy that decreases smoothly asrABincreases. If

the molecule makes a transition to the first excited state it will then dissociate, forming

a hydrogen atom and a H+ion (a bare nucleus).

Figure 20.4 shows sketches of the orbital regions forψ 1 andψ 2. Both of these orbitals

contain the factorΦ 0 and are independent ofφ. We say that they arecylindrically

(^2) D. R. Bates, K. Ledsham, and A. L. Stewart,Phil. Trans. Roy. Soc.,A246, 215 (1963).