wavefunctions are different. The integrals involve not only electron 1 on H1
and electron 2 on H2, but electron 1 on H2and electron 2 on H1.Since the in-
tegrand involves the electrons exchanging atoms, equation 13.14 is the general
form for an exchange integral,denoted by K.Exchange integrals do not have any
classical counterpart.It can be compared to a tug of war with two people on
each side: if the tuggers both change sides, the game is unchanged, right?
Classically, this is so, but quantum mechanically, it is different. The exchange
integrals Kaffect the total energy of the molecule, even something as simple as
H 2 , and their existence represents one reason why classical mechanics couldn’t
adequately describe molecules. Classical mechanics completely missed the ex-
change integral contribution to the total energy of a molecular system.
Exchange integrals appear in all systems with more than one electron. Their
contributions to the energy of the system cannot be ignored. In the case of an
excited state of the helium atom having the electron configuration 1s^12 s^1 ,Kis
about 1/10 of the Coulomb integral J, indicating that it has a substantial effect
on the predicted energy of the system.Example 13.14
What are the expected valence bond wavefunctions for lithium hydride, LiH?Solution
In an initial valence bond approximation, the 1s^2 electrons of Li would be ig-
nored, assuming that they don’t participate in the bonding because they aren’t
in the valence shell. (Their presence is accounted for by assuming some con-
stant amount of energy added to the entire diatomic system.) Therefore, we
need to consider only the single 1selectron of H and the single 2selectron in
the valence shell of Li. Of course, the proper spin functions must also be in-
cluded. Analogous to the valence bond wavefunctions for H 2 , those for LiH are
1
2
[1sH(1) 2 sLi(2) 1 sH(2) 2 sLi(1)]1
2
[(1)(2) (2)(1)]
and
1
2
[1sH(1) 2 sLi(2) 1 sH(2) 2 sLi(1)] (1)(2)1
2
[1sH(1) 2 sLi(2) 1 sH(2) 2 sLi(1)]1
2
[(1)(2)(2)(1)]
1
2
[1sH(1) 2 sLi(2) 1 sH(2) 2 sLi(1)] (1)(2)(The spatial parts of the^3 LiHwavefunctions are the same. Only the spin
parts are different.) This example uses LiH because of the correspondence be-
tween its valence bonds and those of H 2 , as it should be recognized that such
expressions, though useful, can get very complicated very quickly when the
number of valence electrons, and thus the number of valence bond wave-
functions, increases.13.11 Hybrid Orbitals
The idea of valence orbitals itself does not use much symmetry, which is the
main focus of this chapter. But the idea of hybrid orbitals does, and it also450 CHAPTER 13 Introduction to Symmetry in Quantum Mechanics
(^1) LiH
(^3)
LiH