BioPHYSICAL chemistry

idea of this approach is to assume that the second term is much smaller

than the first term. This means that we can write for the wavefunctions:

ψ=ψ 0 +ψ 1 (12.44)

E=E 0 +E 1

Again, the second term is much smaller than the first. Substitution of these

terms into Schrödinger’s equation yields:

(12.45)

This equation simplifies because ψ 0 is the solution to the 0 terms and we

can write that:

(12.46)

The actual solution to this problem would take some time to solve, but

for our purposes the important aspect is that there is a well-defined pro-

cedure to perform the calculation. For biological systems, however, the

calculation is difficult because they are very complex; simple molecules

need to be compared in isolated situations (such as a vacuum).

Helium atom

Let us apply this approach to the simplest atom that has

more than one electron, helium. In this case there are two

electrons and one nucleus. As was true for the hydrogen

atom, the potential is established by the electrostatic

interactions among the charges (Figure 12.14). For helium,

the atomic number must be included to account for the

nuclear charge. There are a total of three interactions:

two are between the nucleus and each electron, and one

is between the two electrons. Since there are now two

electrons, there are two sets of coordinates that must

be accounted for in the equation. With this in mind,

the classical expression for energy conservation and

Schrödinger’s equation is written as:

(12.47)

−∇+∇ − +−

Z^2

1

2

2

2

12

2

24 m 01 2

rr

eZ

r

Z

r

()()ψ

πε

() ()

1

12

r rr^12 E rr^12

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

ψψ=

p

m

p

m

eZ

r

Z

rr

1

2

2

22

(^224) 01 2 12

1

+− +−

⎡

⎣

⎢

⎢

⎤

πε ⎦⎦

⎥

⎥

=E

ψψ 10 =∑a

−∇ + ++ + =

Z^22

2 m ()()()(ψψ01 0101 0VVψψ E++E10 1)(ψψ)

262 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Electron 1
 1

r 1

r 12

r 2

Electron 2

 1

Nucleus

 2

Figure 12.14The potential for the
helium atom is modeled as consisting
of electrostatic interactions that are
dependent on the relative distances
between the charges.