Perturbation Theory

7/17/2019 Perturbation Theory

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Rayleigh-Schrdinger Perturbation Theory

Introduction

Consider some physical system for which we had already solved the Schrdinger

Euation completely but then wished to perform another calculation on the same physical system

which has been slightly modified in some way! Could this be done without solving the

Schrdinger Euation again" This would be particularly ir#some if for e$ample all we wanted to

do was to sub%ect the system &which we now feel is well described by the wave functions and

energies' to a series of small changes such as imposing a series of electric or magnetic fields of

various strengths!

Suppose the original system were described by the hamiltonianH() so that we had

a complete set of energy eigenvalues and eigenfunctions labelled by &('*

'(&

+

'(&

+

'(&

+( EH =

'(&

,

'(&

,

'(&

,( EH = &+'

'(&

-

'(&

-

'(&

-( EH =

The modification of the system is described by the addition of a term Vto the original

hamiltonian so that the new hamiltonian isH = H(.V and has solutions with unsuperscripted

symbols

+++ EH = )

,,, EH = etc!

Perturbation theory is based on the principle e$pressed in /c0aurin and Taylor

series that if a variablex is altered by a small amount then a functionf&x + ' of the variable can

be e$pressed as a power series in! Truncation of the series at the terms of different powers) ()

+) ,) defines the order of the perturbation as 1eroth) first) second order etc! So the hamiltonian

H and the general solution &i)Ei' for the perturbed system are written as

VHH += ( (2)

+


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and the Schrdinger euation for the perturbed system is

iiii EVHH =+ '& ( &'

where +++= ',&,'+&'(& iiii

and +++= ',&,'+&'(& iiii EEEE

))) '-&',&'+&

iii are the +st-order) ,nd-order) rd-order corrections to the ithstate wave

functions2)))

'-&',&'+&iii EEE are the +st-order) ,nd-order) rd-order corrections to the ith

state energies!

Substituting for i

andEibac# into the Schrdinger euation &' we have

''&&

''&&

',&,'+&'(&',&,'+&'(&

',&,'+&'(&

(

++++++=

++++

iiiiii

iii

EEE

VH

( )[ ]

+++++

=+++++'(&',&'+&'+&,'(&'+&'+&'(&'(&'(&

',&'(&',&

(

,'+&

(

'(&'(&

(

'&

'&'&

iiiiiii

iiiiii

EVEEEE

EHHVH

&3'

which on euating the same powers of gives the euations

'(&'(&'(&

( iii EH = (1eroth order &same as ens! +''+&'(&'(&'+&'+&

(

'(&

iiiiii EEHV +=+ + first order

'(&'+&'+&'(&( '&'& iiii VEEH = + first order &4'

'(&',&'+&'+&',&'(&

( '&'& iiiiii EVEEH += , second order &5'

and we could go on ! ! !

The 1eroth order euation tells us nothing new it6s %ust &+'! 7ut &4' and &5'

define the conditions of first and second order perturbation theory) which come ne$t!

+! 8irst order perturbation

&a' Energies

,


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8or this we need e! &4'! 9e #now the sets :i&('; and :Ei&('; but not the first-

order corrections li#e :i&+'; so let BiVi &F'

In other words) the first order correction to the ith energy is the e$pectation value

dV ii'(&'(&

? of V obtained using the 1eroth order wave function i&('! Gone of the other


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functions

)(ij

( are involved! So the energy corrected to first order is %ust

iViEE ii += '(&

&+('

&b' 9ave functions

Ge$t) try putting i kin &='! This provides the coefficient*

'(&'(&

ik

kEE

iVka

=

(11)

7ut this is what we need in &5' to e$press the first-order correction to the ithstate wave function!

So the wave function of the ith

state) corrected to first order) is

'(&

'&'(&'(&

'(&

k

ik ik

iiEE

iVk

=

&+,'

Hnli#e the theory leading to the first-order energyEiin en! &F') in order to e$press the wave

functionto first order the functions and energies)(ij

( and

)(ijE

( of all the other 1eroth-order

states are involved! The function in en! &=' is then normalised after calculating all thecoefficients e$pressed by en! &++'!

,! Second order perturbation

To second order the wave function iand the energyEi of the ithstate are

i> i&('. i

&+'. i&,'

and Ei>Ei&('.Ei

&+'.Ei&,'

s was done to e$press the +st order corrections i&+'to the wave functions in &=') we also write the

,nd order corrections i&,'to the functions as a linear combination

of the 1erothorder set:i&(';*

='&

'(&',&

ij

jji b

&+'

and substitute this e$pression in &5'*

3


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'(&',&'+&'+&'(&'(&

'&

( '&'& iiiijiij

j EVEEHb +=

s before we use e! &+'


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rdorder perturbation

@ere is the energy to rdorder*

+

++= im in

)(

n

)(

i

)(

m

)(

i)i(k

)(

k

)(

i

)(ii

)EE)(EE(

iVnnVmmVi

EE

iVkkViiViEE

((((((

(

(16a)

(thorder +storder ,ndorder rdorder correction correction correction

If you loo# at how the rdorder term is an e$tension of the ,ndorder term you can guess how to

write any higher order terms! The thorder correction to the energy would be

=im in !inimii!

iEEEEEE

iV!nVmmViE

'&''&& '(&'(&'(&'(&'(&'(&'&

(17)

Some points concerning Rayleigh-Schrdinger Pertubation Theory

+! lthough we stopped at second order) provided you were persistent enough there would be no

restriction to proceeding to as high an order of perturbation as you wished) using the euations

developed in the earlier part of this account! The energy interval LEikin the denominators of the

correction terms &+() &+5' or &+=' show that successively higher order perturbations ma#e

successively smaller contributions!

,! ll the rth-order corrections to the wave functions) i&r') and energies i&r') involve matri$

elements BkVi of the perturbation operator V as in &+4' in the numerator and energy differences

i&r'k&r'in the denominator) with the sole e$ception of i&+'in &F'! Physically this means that the

procedure consists of mixing in functions into i&(') particularly from the set of high-energy

unoccupied states k&('!

! 7ecause of the latter point) RS perturbation theory cannot be used if the state k&('to be mi$ed

with i&('is energetically degenerate to this state!

5


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3! 8rom the perturbation corrections li#e those in ens! &+4' and &+5' the mi$ing in of higher-

order states ma#es the denominator negative! The effect is therefore to"tabi#izethe lower-energy

states!

4! Suppose that we were investigating the states of a molecule Athat was influenced by

another molecule Bat a distance$from it! Then the perturbation operator Vwould consist of

those terms describing the coulomb attractions and repulsions of the particles of Awith those of

B! The basis set of functions would be the complete set :i&('; of functions for both molecules!

The e$pression for the perturbed energy states would start off li#e &+5' but would e$plore the

various orders until the higher-order terms become negligible! 9hen the successive terms i&+')

i&,') i&') ! ! ! are e$amined they are found to be of the forms$+) $,) $3) $5) $D) ! ! ! which can

be interpreted as the mutual interactions of the net charge) dipoles) uadrupoles) octupoles) !!!

&both permanent and induced' that are created on the two molecules! This result is sometimes

interpreted in terms of the non-bonded 0ondon or van der 9aals forces arising from the

fluctuating electronic charges on the molecules A andB! 9hile this description is a useful one)

you don H( + V

=


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9e shall treat the electronic repulsion term as a perturbation Vof the (thorder

@amiltonianH() so

+,(

,

3 r

eV

=

&+D'

Zeroth order

Ignoring the electron-repulsion term &+D' we e$pressH(as a sum of the two remaining

parts N one for each of the electrons &+' and &,'

H(>H(&+' .H(&,'

where H(&+' > +(

,,

,

3'+&

, r

%e

m

and H(&,' > ,(

,,

,

3',&

, r

%e

m

These componentsH(&+' andH(&,' are each @amiltonians for @e.which is a Mhydrogen-li#e &('&+'

H(&,'&('&,'> &('&,'

Remember N we #now &('and e$actly! The ground state is described by the hydrogenic +"atomic

orbitals

&('>'Oe$p& (+ a%r' &+F'

>( ) ,(

3,

3,

me%

&energy of Mthe @-li#e atom &('&+'&('&,'

Kperating on it withH(&+),' we have

H(&+),'&('&+),' > H(&+' .H(&,'Q&('&+'&('&,'

>H(&+' &('&+'&('&,' .H(&,' &('&+'&('&,'

D


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In the first term on the right hand sideH(&+' operates on &('&+' giving &('&+' and leaving &('&,'

unchanged! Similarly in the second termH(&,' operates on &('&,' giving &('&,' and leaving &('&+'

unchanged) so we have

H(&+),'&('&+),' >&('&+'&('&,' . &('&+'&('&,'

> ,&('&+'&('&,'

So the eigenvalue euation

H(&+),'&('&+),' > ,&('&+),'

tells us that the @e atom (thorder energy is ,) i!e! twice the energy of the @e.ion! This is 43!3

e so that the (thorder energy is

E&('> +(D!D e

This is physically meaningful because each of the two electrons is described as if it were

in a @e.atom) whose energy is ! The term +,(

,

3 r

e

that accounts for their mutual repulsion has

been omitted to formH(! 7ut the energy is far too negative* the fact that the actual ground-state

electronic energy of @e is =F!( e &the sum of the first two ioni1ation energies' shows that it is

essential to include the interelectronic repulsion term in the @amiltonian!

First order

8rom e! &F' the first order correction to the energy isE+"&+'> B+"V+"

',&'+&',)+&+

?',)+&3

'(&

+,

'(&

(

,'+&

dd

r

eE =

Substituting for &('from &+F' and performing the integration leads to

E&+'> 3!( e!

This brings the energy of helium to first order to

E>E&('.E&+'

> +(D!D . 3!( > =3!D e e$ptl! value =F!( eQ

F


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Second order

In order to form the matri$ elements dViVk ik

'(&?'(& reuired for substitution ine! &+4' we need all the (th-order functions for &(') i!e!

))) '(&

,

'(&

,

'(&

+ +!""

and all the corresponding (thorder energies

))) '(&

,

'(&

,

'(&

+ +!"" EEE

7ut these are #nown e$actly since they are the solutions of a hydrogen-li#e atom! Evaluation of

E&,'from es! &+4' and &+D' gives 3! e) so the energy of helium to second order is

E>E&('.E&+'.E&,'

> +(D!D . 3!( 3! > =F!+ e e$ptl! value =F!( eQ

Higher orders

The mi$ing of the higher order states'(&

k into the ground state'(&

+" should also include

the continuum) i!e! states with energies greater than 1ero &which is the ma$imum energy obtained

from the 7ohr formula n >,,

(,

3,

, n

me%

'! The development of'thorder perturbation theory is

tedious but routine) as is the numerical calculation of all the reuired matri$ elements!

Calculations have been performed up to +thorder giving

E> ,!F(=,3 hartree &atomic units of energy'

> =F!(+5+ e

&E$ptl! value =F!( e'

+(


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,! Star# effect +* The shifting of the torsional energy levels of an K@ group by an

electric field!

Consider a part K@ of a molecule in which the K@ group rotates around the K bond!

K@ has an electric dipole moment whose component in the plane perpendicular to the K bond&rotation a$is' is ! &If (is the dipole moment along K-@ then > (cos ' 9hen an electric field F

is applied in this plane &i!e! perpendicular to the rotation a$is') rotates &in thexyplane' through

a1imuthal angles monitored as so that successively comesinto and out of alignment with F.

The coupling of the dipole with the field produces an energy

V&' > F> cos

which is an energy perturbation term to the hamiltonianH(and is the component of the K@

bond dipole moment perpendicular to the rotation a$is! The total hamiltonian is

@ >H&. V!

In the absence of an electric field &F > (' the solution of the Schrdinger Euation describing the

internal rotation)

is the familiar one of a particle confined to a circle*

where)(=m )+ ), ! ! !

9e shall e$plore the perturbation of the torsional energy levels to second order by calculating the

matri$ elements of that are reuired in the e$pression from en! &+4'*

'(&'(&

,

,,'(&(

,

E

d

d

)H ==

im

m e,+'(&

=

mEm

,

,,'(& =

++


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++= '&

'(&6

'(&

66,,

6, mm mmm

EE

mVmmVmmVm

mE

&,('

7oth first- and second-order perturbation terms on the right of this euation reuire the matri$

elements Vmm*where Vmm*> BmVm* where m> mfor the first-order and m m for the second-

order terms! 9e shall first evaluate the general matri$ element Vmm*!

Substituting for

V> cos > U&ei. ei'

and the 1eroth order functions

m

ime,

+

the matri$ element Vmm*

6mVmbecomes

Vmm*

deeee

dee imiiimimim 6,

(

6,

('&

3cos

,

+ +==

3

=

+ +

,

(

,

(

'+6&'+6& dede mmimmi

7oth integrals are of the form

de in

,

( where nis an integer! Gow such an integral is 1ero

unless n > () in which case it is , &see footnote+'! Then as m = m + either of the two integrals

contributes , and we have

Vmm ,+

6 = &,('

This is our reuired non-1ero matri$ element of perturbation V! Gote that as Vmm> ( there is no

first order correction to the energy!

9e can now calculate the second order perturbation term of the energy!

1If n ()

[ ] [ ] [ ] (+(++,sin,cos++ ,(,

(

,

(

=+=+== inninineinde inin

Ktherwise &n > (')

,,

(

=d

+,


13/17

,

,,,,

, +

mEm

=

!

,

,,

(,(

E

=

Rotational level m > ( of the rotor is therefore lowered by a uantity proportional

to the suare of the field intensity and of the dipole moment of the K@ bond! The ,-fold

degeneracies of the levels are not lifted! &7ut they are when higher-order perturbation

theory is appliedJ'! Recall that is the component of the K@ bond dipole moment

perpendicular to the rotation a$is) and we should write

9hat if the field were applied in a direction other than perpendicular to the

torsional a$is" Then V> F would have components VzandVxfrom the couplings along

and perpendicular to the this a$is! The shift fromVxwould be of the same form as the

one we %ust calculated &but smaller because Fhas a smaller component in the circle

around which the dipole moment is rotating') and Vzwould becos where is the

&constant' angle between and F!

If the electric field were applied!ara##e#to the rotational a$is) would still go

from ( to , but as it rotated) the K@ bond dipole would ma#e a constant angle with the

field! The energy shift would then be simply Vsin where sin is the angle made by the

bond dipole with the torsional a$is &and the electric field'!

'(&

+

'(&

(

'(&

+

'(&

(

',&

(

(++((++(

+

=

EE

VV

EE

VVE

+


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! Star# effect ,* Wegenerate perturbation theory!

Energy splittings in the @ atom

9e perturb the (thorder hamiltonian @(of the @ atom by adding to it a term

V = ez

Sincezis antisymmetric &or odd' any diagonal matri$ element Vmmis 1ero i!e!

(? =

+

dzz mm

!

8or this reason there is no Star# effect to +storder PT!

In order to go to higher order we must form off-diagonal matri$ elements Vmn! These

elements will couple the +" ground state'(&

+" to higher states such as'(&

," )'(&

, ++!

)'(&

, (!

)'(&

+, !

which will result in a"iftedenergy level of what was the ," ground state) but since this state is

non-degenerate there will be no Star#"!#itting!

9e therefore ma#e the problem more interesting by replacing the ground state by the n>

, state spanned by the four functions ,") ,!.+) ,!() ,!-+! 9e should li#e to #now to what e$tent

this 3-fold degeneracy is removed by the Star# effect!

Xeroth order functions

The four basis functions of the n> , shell of the @ atom) and their energies &all eual)

En>,' are #nown e$actly! The functions can be written as products of functions involving

spherical coordinates r)- in the form R&r' &' &') and the only factor of these which will concern

us will be &' which is of the form eimwhere m> ( for ,s and +) () + respectively for ,!++) ,!() ,!+!

9ritten this way these functions are obviously eigenfunctions of the angular momentum operator

lz>

i

with eigenvalues m !

" &"' > "ei& lz &"' > ( &"'

!.+ &!.+' > !ei lz &!++' > .+ &!.+'

!( &!(' > ,!ei& lz &!(' > ( &!('

!-+ &!+' > !ei lz &!+' > + &!+'

+3


15/17

where "and !are the parts of the ,"and ,!K functions without the &' factor!

The reason that these functions are eigenfunctions of lzis because this operator commutes with

the hamiltonian! 7ut aslzalso commutes with the perturbation term V = ez) matri$ elements Vmm*

formed from functions functions mand m*corresponding to different angular momenta m and

6m are 1ero)

i!e! if Bil1j > ( then BiVj > ( also!

This is because) as lzis a hermitian operator its eigenfunctions are orthogonal if they belong to

different eigenvalues) i!e! matri$ elements Bmlzm > (! The only non-1ero elements of Vare

for basis functions corresponding to the same eigenvalue of lz. @owever) for the same reason as

in the application of perturbation theory of a Star# electric field to a molecular torsion) diagonal

matri$ elements ofVare 1ero because the perturbation V = ez) is not totally symmetric) i!e!

BmVm > ( for all m!

So we have the following!

(a) ll diagonal elements of Vare 1ero

(b)Gon-1ero elements of Vmust be from pairs of functions corresponding to the same

eigenvalue of lz.

This leaves only one pair of functions that forms a non-1ero element with V* that between ,"and

,!&! Since their degeneracy does not allow them to be used in perturbation theory we shall need

to calculate the eigenvalues of the energy matri$ of Vspanned by the four basis functions

&") !++) !() p+'

ll the elements of Vwill be 1ero e$cept two) those formed by &,"and ,!(') and by &,!(and ,"'!

The 1eroth order energyE&('is the energy of the ," and ,!orbitals &all four have the same energy)

because for all hydrogen-li#e atoms with one electron) a subshell with principal uantum number

nis n,-fold degenerate! The perturbation energy matri$ Vis therefore

V>

((((

((((

(((

(((

(

(

"V!

!V"

+4


16/17

The complete hamiltonian matri$) H> H0. Vis therefore

H >

'(&,

'(&

,

'(&

,

'(&

,

(((

(((

(((

(((

E

E

E

E

.

((((

((((

(((

(((

>

'(&

,

'(&

,

'(&

,

'(&

,

(((

(((

((

((

E

E

E

E

where'(&

,E is the energy of the n> , shell &recall that for a hydrogen atom'(&

,

'(&

, !" EE = ' and the

perturbation matri$ element Bijis assigned to a parameter v

i!e! z!z"e ,,=

&+F'

that is proportional to the electric field! His Mbloc#-diagonal


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Perturbation Theory