Page 121 - Bruno Linder Elementary Physical Chemistry

August 18, 2010 11:37 9in x 6in b985-ch10 Elementary Physical Chemistry

106 Elementary Physical Chemistry

Neither of these functions represent correctly the H 2 molecule. What is
missing are electron spin contributions. Denoting the spins as α (spin up)
and β (spin down), we can construct four types of spin functions, namely
[α 1 α 2 ], [β 1β 2 ], [α 1 β 2 + β 1 α 2], [α 1β 2 − β 1α 2] (10.1e)

where the subscripts refer to electrons 1 and 2. Note that of the four spin
functions [ ], the first three are symmetric with respect to exchange of the
electrons (do not change sign); the fourth spin function is anti-symmetric.
There is an important rule that stipulates that for a wave-function to
be proper, the function must change sign upon interchange of two electrons.
Accordingly, the first three spin functions of Eq. (10.1a), being symmetric,
can only combine with the orbital function Ψ* given in (10.1a), which is
anti-symmetric. The fourth spin function, which is anti-symmetric, can only
combine with the symmetric function Ψ of (10.1c). Thus, there is one Ψ
function and there are 3 Ψ* functions. The state, characterized by Ψ, is
called singlet state and the state characterized by Ψ* is called triplet state.

Note: A linear combination (singlet or triplet) is the only formula
which gives a correct description of combing atoms. It establishes that
you cannot say whether electron 1 is on A and electron 2 on B or vice
versa. In quantum mechanics, identical particles are indistinguishable,
in contrast to classical mechanics where they are not. In general, a
bond between A and B is

Ψ(A − B) = ψ A(1)ψ B(2) + ψ A(2)ψ B (1) (10.2)

Example 10.1.
(a) Describe the valence bond between two N atoms.
Solution
2
2
First note that the electron configuration of the N atom is 1s 2s 2p 1 x
1
1
2p 2p . The 1s electrons are so deeply embedded that they are unlikely
z
y
to take part in the bond formation and are ignored. The 2s electrons are
also more deeply embedded than the p electrons and will also be ignored in
bond formation. Only 2p electrons will be considered to participate in bond
formation. It is common practice to take the Z-axis to be the internuclear
axis. The 2pz orbitals of the two atoms point towards each other, while