Physical Chemistry , 1st ed.

overall antisymmetric wavefunction. The following summarizes the properties,

which can easily be verified:

spatial

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] symmetric

spatial

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] antisymmetric

(1)(2) symmetric

(1)(2) symmetric

1

2

[(1)(2) (2)(1)] symmetric

1

2

[(1)(2) (2)(1)] antisymmetric

The symmetric spatial function is combined with the antisymmetric spin func-

tion to make a single antisymmetric total wavefunction for H 2. Since the en-

ergy of the molecule depends on the spatialwavefunction, this combination

represents a state that has a degeneracy of 1 and so is called a singletstate. The

antisymmetric spatial function can be combined with three symmetric spin

functions to make three additional and individual antisymmetric total wave-

functions for H 2. Again, since the total energy depends almost entirely on the

spatial part of the wavefunction, these three different wavefunctions have the

same energy, and so these represent an energy level for H 2 that has a degener-

acy of 3. It is called a tripletstate. The complete wavefunctions are

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] 

1

2

[(1)(2) (2)(1)]

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] (1)(2)

(13.12)

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] 

1

2

[(1)(2) (2)(1)]

1

2

[1sH1(1)  1 sH2(2)  1 sH1(2)  1 sH2(1)] (1)(2)

Which state has the lowest energy and represents the ground state? As might

be expected, the sum of two negative numbers (recall that the energy of the

hydrogen atom itself is negative, due to the attraction of the proton and elec-

tron) is lower, or more negative, than the difference of two negative numbers

(where the subtraction of a negative yields, ultimately, an addition process).

There is also experimental evidence that the lowest-energy state of H 2 is a sin-

glet state. Therefore the singlet wavefunction in equation 13.12 is the approx-

imation of the valence bond theory for the ground state of H 2. The triplet state

is, by definition, an excited state.

In evaluating the energy of the wavefunctions in equation 13.12, one can set

up a perturbational or variational treatment to come up with some expression

for the energy of the molecule. Although we won’t do this in its entirety, we

will illustrate some parts of it that introduce some unique features of molecu-

lar quantum mechanics. Assume that the energy of the ground state is deter-

mined solely by the spatial part of the wavefunction. (As mentioned above, this

448 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics

(^3) 
H 2 ,valence
(^1) H
2 ,valence
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