bei48482_FM

274 Chapter Eight

energy of H 2 is the 13.6 eV of the hydrogen atom plus the 2.65-eV bond energy,

or 16.3 eV in all.

In the case of the antisymmetric state, the analysis proceeds in the same way except

that the electron energy EAwhen R0 is that of the 2pstate of He. This energy is

proportional to Z^2 n^2. With Z2 and n2, EAis just equal to the 13.6 eV of the

ground-state hydrogen atom. Since EAS13.6 eV also as RS, we might think that

the electron energy is constant, but actually there is a small dip at intermediate dis-

tances. However, the dip is not nearly enough to yield a minimum in the total energy

curve for the antisymmetric state, as shown in Fig. 8.7, and so in this state no bond

is formed.

8.4 THE HYDROGEN MOLECULE

The spins of the electrons must be antiparallel

The H 2 molecule has two electrons instead of the single electron of H 2 . According to

the exclusion principle, both electrons can share the same orbital(that is, be described

by the same wave function nlml) provided their spins are antiparallel.

With two electrons to contribute to the bond, H 2 ought to be more stable than

H 2 —at first glance, twice as stable, with a bond energy of 5.3 eV compared with

Figure 8.7Electron, proton repulsion, and total energies in H 2 +as a function of nuclear separation R

for the symmetric and antisymmetric states. The antisymmetric state has no minimum in its total

energy.

Up = Proton potential energy

ES = Electron energy (symmetric state)

EStotal = H 2 + energy (symmetric state)

EA = Electron energy (antisymmetric state)

EAtotal = H 2 + energy (antisymmetric state)

30

20

10

0

• 10


• 13.6


• 16.3


• 20


• 30


• 40


• 50

Energy, eV

1.06 × 10 –^10 m

Nuclear separation R, nm

Total energy of isolated

hydrogen atom

0.2 0.3 0.4

Bond energy = 2.65 eV

EA

Up

EStotal

ES

0.1

EAtotal

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