bei48482_FM

270 Chapter Eight

(a)

(b)

Total

electron

energy

Proton a Proton b

R

0 r

V

Electron

Figure 8.4(a) Potential energy of an electron in the electric field of two nearby protons. The total

energy of a ground-state electron in the hydrogen atom is indicated. (b) Two nearby protons corre-

spond quantum-mechanically to a pair of boxes separated by a barrier.

shared by two protons is less confined than one belonging to a single proton, which

means that it has less kinetic energy. The total energy of the electron in H 2 is there-

fore less than that of the electron in HH. Provided the magnitude of the proton-

proton repulsion in H 2 is not too great, then, H 2 ought to be stable.

8.3 THE H 2 MOLECULAR ION

Bonding requires a symmetric wave function

What we would like to know is the wave function of the electron in H 2 , since from

we can calculate the energy of the system as a function of the separation Rof the

protons. If E(R) has a minimum, we will know that a bond can exist, and we can also

determine the bond energy and the equilibrium spacing of the protons.

Solving Schrödinger’s equation for is a long and complicated procedure. An in-

tuitive approach that brings out the physics of the situation is more appropriate here.

Let us begin by trying to predict what is when R, the distance between the protons,

is large compared with a 0 , the radius of the smallest Bohr orbit in the hydrogen atom.

In this event near each proton must closely resemble the 1swave function of the

hydrogen atom, as pictured in Fig. 8.5. The 1swave function around proton ais called

aand that around proton bis called b.

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