BioPHYSICAL chemistry

CHAPTER 12 THE HYDROGEN ATOM 263

This is first solved at zeroth order; that is, neglecting the interactions
between electrons. These solutions are denoted by the superscript zero:

(12.48)

Assume that the wavefunction can be separated as the product of two
contributions:

ψ(r 1 r 2 ) =ψ(r 1 )ψ(r 2 ) (12.49)

This yields two separate equations:

(12.50)

(12.51)

where

E 10 +E 20 =E^0 (12.52)

The first-order correction for energy is determined by modifying
Schrödinger’s equation to include the interaction. First, rewrite
Schrödinger’s equation using:

(12.53)

(H^0 +H^1 )(ψ^0 +ψ^1 ) =(E^0 +E^1 )(ψ^0 +ψ^1 ) (12.54)

Ignoring the two second-order terms yields:

H^1 ψ^0 +H^0 ψ^1 =E^1 ψ^0 +E^0 ψ^1 (12.55)

H^0 ψ^1 −E^0 ψ^1 =−H^1 ψ^0 +E^1 ψ^0 (12.56)

H

e

r

1

2

(^4012)

1

=+

⎡

⎣

⎢

⎢

⎤

⎦

⎥

πε ⎥

H

m

eZ

r

Z

r

0

2

1

2

2

2

2

(^24) 01 2

=− ()∇ +∇ − +

⎡

⎣

⎢

⎢

Z

πε

⎤⎤

⎦

⎥

⎥

−∇ − =

Z^2

2

20

2

2

02

0

22

00

24 m r^2

eZ

r

ψ rE r

πε

() ψψ() ())

−∇ − =

Z^2

1

20

1

2

01

0

11

00

24 m r^1

eZ

r

ψ rE r

πε

() ψψ() ())

−∇+∇ − +

Z^2 ⎡

1

2

2

20

12

2

(^24) 01 2

11

m

rr

e

rr

()()ψ

πε ⎣⎣

⎢

⎢

⎤

⎦

⎥

⎥

ψψ^0 ()rr 12 =E^00 ()rr 12

p

i

→∇ →rr

Z