BioPHYSICAL chemistry

CHAPTER 13 CHEMICAL BONDS AND PROTEIN INTERACTIONS 271

(13.1)

where p 1 and p 2 are the momenta of elec-
trons 1 and 2, respectively. For four particles,
there are a total of six interactions between
individual particles. Each of the two nuclei
can interact with the two electrons with a
potential that decreases inversely with dis-
tance since the two charges have opposite
signs. For example, the A nucleus and electron 1 have an electrostatic
potential with a distance dependence of −1/rA1. The other two electrostatic
interactions are between electrons 1 and 2, with a distance dependence
of +1/r 12 , and two nuclei, with a distance dependence of +1/rAB. In these
latter two cases the sign of the interaction is positive as the two charges
have the same sign. The total electrostatic potential for these four charges
can be written as:

(13.2)

For two charges, q 1 and q 2 , separated by a distance r 12

Now that the potential for the molecule has been established, we can write
Schrödinger’s equation. The wavefunction will be dependent upon six
coordinates, the three that describe the first electron and the three that
describe the second electron. For simplicity, this dependence of the wave-
function will be denoted as ψ(r 1 ,r 2 ). Substituting the appropriate operators
for the classical kinetic and potential energies yields Schrödinger’s equation
for the hydrogen molecule:

=Eψ(r 1 ,r 2 ) (13.3)

−∇+∇ − +

Z^2

1

2

2

2

12

2

(^240)

11

m

rr

e

rr

()(,)ψ

πε A1 A 2 2B1B212AB

++−− (,

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

1111

rrrrψr^1 rr^2 )

Vr

qq

r

()=^12

4 πε 012

Potential energy
A1 A2 B1

=−−−

e

rrr

2

(^40)

111

πε

−++

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

111

rrrB2 12 AB

Kinetic energy==( ) =

1

2

1

22

22

2

m

m

m

p

m

99

Kinetic energy=+

p

m

p

eem

1

2

2

2

22

e 1 e 2

HA HB

rAB

rA 2 rB^1

rA 1 rB 2

r 12





Figure 13.1

The potential of a

hydrogen model is

modeled as arising

from electrostatic

interactions involving

the two electrons,

e 1 and e 2 , and two

nuclei, HAand HB.