BioPHYSICAL chemistry

(13.7)

E=E 1 +E 2

Each of the two equations is dependent upon only a set of coordinates,
allowing the wavefunctions to be solved explicitly. The resulting solutions
will not be demonstrated here but the similarity of the equations to those
for the hydrogen atom are consistent with the new solutions, behaving as
modified forms of the hydrogen atom wavefunction due to the potential
of the second proton included.
In solving this problem, there is always an ambiguity in identifying which
electron is which. When the two atoms are far apart, the electrons are each
localized around one nucleus. When the two atoms come within bonding
distance, then the electrons can be found on either of the nuclei. These two
electrons have identical properties and there is no identifiable property
that would indicate that the two electrons were switched. Therefore, we
must consider the two electrons to be identical. To reflect this property,
the allowed wavefunctions are written as the linear combination of two
solutions, with the second simply reflecting the possible switch:

ψ(r 1 ,r 2 ) =ψA(r 1 )ψB(r 2 ) ±ψB(r 1 )ψA(r 2 ) (13.8)

These wavefunctions are substituted into Schrödinger’s equation with
one of the terms previously neglected: the interaction between the
nuclei. The general dependence of this interaction on
the separation between the nuclei, rAB, arises from two
contributions, the bonding and electrostatic interactions
(Figure 13.2). As discussed for the multi-electron atoms,
the two electrons will tend to be paired together, with
opposite spin direction rather than having the same spin
direction. Thus, the pairing of the electrons naturally
results in a bond.
The bonding properties are a result of contributions
of two factors. When two uncharged atoms are brought
close together, their electrons interact. Random variations
in the distributions of the electrons of one atom give
rise to a transient dipole in a neighboring atom. The
fluctuation-induced interaction becomes more negative
as the separation decreases and results in a weak inter-
action, termed a London dispersion interaction, that has
an approximate dependence of 1/r^6 (Figure 13.2).

−∇ − +

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Z^2

2

2

2

2

(^24022)

11

m

r

e

rr

ψ

πε

() ψ(

AB

rrEr 222 )()= ψ

−∇ − +

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Z^2

1

2

1

2

(^24) 01 1

11

m

r

e

rr

ψ

πε

() ψ(

AB

rrEr 111 )()= ψ

CHAPTER 13 CHEMICAL BONDS AND PROTEIN INTERACTIONS 273

0

Minimum

energy

To t a l

rAB

Potential energy

Contribution of

repulsive electrostatic

interaction

between nuclei α

Attractive

London

dispersion

contribution α

Equilibrium

bond

distance

1

r^12

1

r^6

Figure 13.2

Potential energy for

two atoms separated

by a distance r, with

both the attractive

and repulsive

contributions shown.