perturbation theory
DESCRIPTION
PerturbationTRANSCRIPT
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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$pression in &5'*
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'(&',&'+&'+&'(&'(&
'&
( '&'& iiiijiij
j EVEEHb +=
s before we use e! &+'
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rdorder perturbation
@ere is the energy to rdorder*
+
++= im in
)(
n
)(
i
)(
m
)(
i)i(k
)(
k
)(
i
)(ii
)EE)(EE(
iVnnVmmVi
EE
iVkkViiViEE
((((((
(
(16a)
(thorder +storder ,ndorder rdorder correction correction correction
If you loo# at how the rdorder term is an e$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
+,
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,
,,,,
, +
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
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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
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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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+=