wehaveto

take

into

accountonlytheterm

ei

tbecause

wearedealingwithab

sorption

(theother

term

eitleadsto

fast

oscillations,wou

ldberelevantwhen

wewantto

treat

stim

ulatedem

issionandcompute

M21).

=

M12

= 2 H

1=

eA0

2m

e2|p

|1(3.195)

2|p

|1=

2 i

1d3r=

ime

2|r

|1(3.196)

=

M12

=i 2

eA02|r

|1=

E0 22|er

|1(3.197)

Because

eristhedipoleop

erator,M

12iscalled

dipole-m

atrix-elementan

dtheab

ove

approxim

ationdipole

approxim

ation.

=

|M12|2

=e2 4|2

|E0r

|1|2

(3.198)

ifE

0=

E0ex,

=

M2 12

=e2E

2 0

4|2

|x|1

|2(3.199)

Butthesystem

issymetric=

(x;y;z)areidentical

=

|2|x

|1|2

=1 3|2

|r|1

|2(3.200)

=

M2 12

=e2E

2 0

12|2

|r|1

|2(3.201)

1 2 0E

2 0:averaged

energydensity

forplanewaveatthefrequency

=

u(

)=

1 2 0E

2 0(

)forspectralenergydensity,an

dwith

(

)=

(E

)dE d

=(E

)(3.202)

=

(E

)=

2u(

)

0E

2 0

(3.203)

Therefore,withW

1

2=

B12u(

)wehave

B12

=e2

3 02|2

|r|1

|2(3.204)

A12

=e23

3 0c3|2

|r|1

|2(3.205)

Indipoleapproxim

ation:

|2|r

|1|2=

| 2r1d3r|2

=0

=

transitiondipole

allowed

(3.206)

if|2

|r|1

|2=

0=

transitiondipole

forbidden(3.207)

Oscillatorstrengh(see

chapter2)isproportionnalto

|2|r

|1|2 .

80

3.4

Molecu

les

Source:

Fox,Chapter8.

3.4.1

H2molecu

le

Letsstart

withtheeasiestmolecule

H2.

Weexpectedthatthetw

oprotonssharethetw

oelectronsin

thegroundstate

(low

estenergy).

Typicalforcovalent

bond,i.e.

co-valence,sharingvalence,sharing

outere

.

Moleculesareusuallytreatedin

Born-O

ppenheimer

approxim

ation,whichmeans:

weseparate

electronic

andnuclearmotion,justied

bythehugedierence

inmasses.

For

ethenuclei

arenotmoving,wecantreatrelative

positionsof

nuclei

asparametersandsolveelectronic

problem

En(R

),withR

-positionof

nuclei.

Nuclei

seepotentialcreatedbye

,wesolveproblem

fornuclei

insecondstep.

Problem

forthetw

oe

(electron1an

d2)

(x

1,y

1,z

1,x

2,y

2,z

2)

(3.208)

H=

2

2me(

2 1+2 2

)

e2

4 0

( 1 r 1l+

1 r 1r+

1 r 2l+

1 r 2r

1

|r 1r 2|)

(3.209)

e.g.r 1

lisdistance

betweenelectron1andleftnucleus.

Complicated6-dim

ensionalproblem! 81

Approxim

ationap

proach

tondgroundstate:ifthetw

onuclei

are

farapart,we

havetw

ohydrogen

atoms:

l(r)

=100(|r

r lp|)

(3.210)

r(r)

=100(|r

r r

p|)

(3.211)

withr lpandr r

pare

positionof

leftan

drightprotonsrespectively.

(r

1,r

2)

=l(r 1)

r(r

2)

(3.212)

Electron1is

aroundleft

protonandelectron2is

aroundrightproton,electronsare

independent.

Whatabouttheprobabilitydensity?Problem,wehave6dim

ensions!

Probability

density

tondeither

e:

(r)

=

|(r,r

2)|2

d3r 2

+

|(r

1,r)|2

d3r 1

(3.213)

with,e.g., |

(r,r

2)|2

d3r 2

isprobabilityto

ndelectron2,

whateverelectron1is

doing!

For

(r

1,r

2)=

l(r 1)

r(r

2),theprobabilitydensity

lookslike:

Ofcourse,

exchangingtheelectronsisphysicallyequivalent:

(r

1,r

2)

=r(r

1)

l(r 2)

(3.214)

Note

thatr(r

1)

r(r

2)or

l(r 1)

l(r 2)arenotrelevantto

ndagroundstate

(2e

aroundoneproton).

82

Stateswheretheelectronsare

nolonger

assigned

toonespecicproton(shared

electronstates)

canbefoundaslinearcombination:

(r

1,r

2)

=al(r 1)

r(r

2)+b

r(r

1)

l(r 2)

(3.215)

Norm

alization(

randlare

realandnorm

alized,aandbare

real):

(r 1,r

2)

(r1,r

2)d

3r 1d3r 2

(3.216)

=a2+b2

+2ab

l|r2

! =1

(3.217)

a bcanbechosenfreely.

Compute

expectationvalueforenergy H

forallratio

a bandproton-protondis-

tances(variationalapproach,possible

because

weknow

100(r)=

e

r/a0

a3 0

).Integrals

haveto

beevaluatednumerically.

Approxim

ationof

ground-state:thecongurationwendwiththelowestenergy.

Itturnsou

t:groundstate

a=

b

E=

H=

2ER

13.6eV

3.2eV

exactvalue4.52eV

(3.218)

Distance

betweenprotons:

0.87A(0.74A

)

83

Symmetricstate

s:sw

appingr 1

andr 2

changes

nothing.

Importantother

possibility-antisymmetricstate

a:sw

appingr 1

andr 2

changes

sign.

Generic

phenomena:lettw

osystem

swithidenticaleigenfunctionsinteract:

withoutinteraction:degeneracy

ofsym/antisymmetricstate,

withinteraction:degeneracy

getslifted

Esym0.

Compatible

withBloch:acanbechosenarbitrarily!

V0=

0

n(x)=

Ansin[k(x

na)]+B

ncos[k(x

na)]

for(n

1)a