BioPHYSICAL chemistry

Multiply by ψ0*and integrate:

(12.57)

The operators used in Schrödinger’s equation are called Hermetian oper-

ators and have the property:

(12.58)

and

(12.59)

Resulting in the expression for the first-order energy correction:

(12.60)

For the n 1 =1, n 2 =1 state this is:

(12.61)

dτ 1 =r 12 sinθ 1 dr 1 dθ 1 dφ 1 (12.62)

dτ 2 =r 22 sinθ 2 dr 2 dθ 2 dφ 2

The term r 12 must be rewritten in spherical coordinates, where the >or

<signs denote the larger or smaller values of r 1 or r 2 :

(12.63)

Experimentally, a value of −79.01 eV is determined for the n 1 =1 and

n 2 =1 state of helium. Substitution of these values into E^0 gives −108.8 eV.

The first-order term has a value of +34.0 eV, giving −74.8 eV, and using

up to the third order gives −78.9 eV.

Spin–orbital coupling

The electron spin influences the energies of the electronic states because

associated with the spin is a magnetic dipole moment. Likewise, the orbital

angular momentum, for the states with lgreater than 0, will possess

a dipole moment due to the angular momentum. The interaction of

the spin magnetic moment with the magnetic field arising from orbital

14

rl 12 211 11

r

r

YY

ml

l

l l

m

l

m

l

= ll()

∑∑ +

<

>

+

π

θφ*(()θφ 22

E

Z

a

(^12) eeZr a Zr a
6
0
6
2
12
(^110) / (^22)
,

= /

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟∫∫−−

π

00

2

012

4 12

e 1

πε r

ττ

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟dd

EH^1010 =∫ψψτ* d

∫∫ψψτ^010 EEdd==1 0 0ψψτE1**

∫∫ ∫ψψτψψ τψψ τ^001 HH Eddd===^100 ()^100 ()∫ψψψτ001E d

∫∫ ∫ψψτψψτ ψψτψ^001 HE Hdd d−=^001 −+^010 ∫∫^0 E^10 ψτd

264 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY