Concise Physical Chemistry

c20 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

THE HYDROGEN MOLECULE ION 319

e–

rB

R

rA

FIGURE 20.1 The hydrogen molecule ion, H+ 2.

where|
〉is the state vector for theentire molecule. The inner product 〈
|
〉

is 1, so

E=〈
|Hˆ|
〉

In chemical applications the state vector is usually written as the equivalent wave

function:

E=

∫

∣

∣Hˆ

∣

∣
dτ

If the state vector or wave function is exact, the energy will be exact.

20.2 THE HYDROGEN MOLECULE ION

A stepping stone toward full-scale molecular structure and energy calculations is the

hydrogen molecule ion H+ 2. Conversion of the problem to elliptic coordinates and

subsequent solution by numerical methods has been carried out, so we know the

answer before we start. The energy of the bound state of H+ 2 is about 268 kJ mol−^1.

Unfortunately, this numerical method cannot be extended to larger molecules or ions,

so we shall use the known result to help us to develop of an approximate method,

which can then be applied to molecular systems large enough to be important in

chemistry.

The geometry of the H+ 2 problem is given in Fig. 20.1. Assume that the nuclei are

stationary^1 at a distanceRfrom one another. This gives us a problem of one electron

in the field of positive nuclei A and B over distancesrAandrB. The Schrodinger ̈

equation is similar to that of the hydrogen atom, except that there are two centers of

positive charge rather than just one. For any selected value ofR,

[

−^12 ∇^2 −

1

rA

−

1

rB

]

=Eelectronic

(^1) This is the Born–Oppenheimer approximation.