Optical potential in electron-molecule scattering

Roman Čurík

• Some history or “Who on Earth can follow this?”• Construction of the optical potential or “Who needed that molecule anyway?”• Static and Polarization or “Is this good for anything?”• What changes with Pauli principle? or “Someone should really paint those electrons with different colors”

History

• 1956 – W.B.Riesenfeld and K.M. Watson summarized Perturbation expansions for energy of many-particle systems.

Different choices of O leads to different perturbation methods:

- Brillouin-Wigner- Rayleigh-Schrödinger- Tanaka-Fukuda- Feenberg

MOVOHE

M ][1

10

• 1958 – M.H. Mittleman and K.M. Watson applied Feenberg perturbation method for electron-atom scattering with exchange effects neglected

• 1958 – Feshbach method

• 1959 – B.A. Lippmann, M.H.Mittleman and K.M. Watson included Pauli’s exclusion principle into electron-atom scattering formalism

• 197X – C.J.Joachain and simplified Feenberg method without exchange

N+1 electron scattering

r1r2

rNr

molecule

• Nuclear mass >> mass of electrons• Vibrations and rotations of molecule not

considered• Electronically elastic scattering

scattered electron

N electronsM nuclei

• Formulation in CMS

Total hamiltonian H is split into free part H0 and a “perturbation” V:

N

i i

M

A A

A

M

A

N

ji ji

N

i Ai

AM

M

ZV

ZH

HkH

VHH

11

1 1

221

0

0

||

1

||

||

1

||

rrRr

rrRr

)...1()...1(0 NwNH nnn

Initial and final free states:

',0.

2/3

,0.

2/3

)2(

1

)2(

1

Si

f

Si

i

f

i

e

e

rk

rk

fff

iii

EH

EH

0

0

20

20

2

12

1

ff

ii

kwE

kwE

Exact solution can be provided by N+1 electron

Lippmann-Schwinger equation

Function describes both the elastic and inelastic scattering. For example it can be expanded in diabatic expansion over states of the target as:

)(

0

)( 1

ViHEi

i

)(

nnn NNN )1()..1()1...1()(

Since we don’t have any rearrangement and target remains in the same electronic state, can we reduce the size of the problem and formulate scattering equations as for scattering of a single electron by some single-electron optical potential?

where corresponds to the elastic (coherent) part of the total wave function .

)(

0

)( 1

copti

ic ViHE

)(c)(

We define projection operator onto ground states:

Elastic part can be obtained by projection

Full N+1 L-S equations for are

00000 t

tt

)(0

)( c

)(

iHEEG

TEG

VEG

iii

ii

0

)(0

)(0

)(

)()(0

)(

1)(

)(

)(

By applying from the left we get

where the elastic part of the T operator can be defined as follows,

0

iii TEG )()(000

)(0

0)(

0)(

000 , GGii

i

T

ic

C

TG 00)(

0)(

00000000 CC TTTT

We have projected L-S equation

Definition of optical potential

gives desired form

or

iCic TG )(0

)(

iCcopt TV )(

)()(0

)( coptic VG

CoptoptC TGVVT )(0

Thus the optical potential Vopt is defined as an operator which, through the Lippmann-Schwinger equation leads to the exact transition matrix TC corresponding to the elastic scattering of the incident particle by the molecule.

Finally we project above equation onto the ground state of molecule via

and the single-electron equation is obtained:

)(00

)(0

)( 000 coptic VG

)(2/ˆ

1)2()( )(

2,.2/3)(

,,,,rV

ikEer

tit

i

tiopt

i

i

kk

rk

Where is a single-particle optical potential obtained

by

The definition of Vopt does not imply that optical potential

is an Hermitian operator. In fact, hermitian Vopt would lead to TC such that is unitary.

In fact after applying of machinery of optical theorem

it can be shown that

optV

00 optopt VV

TiSC 21

)()(3

Im)2(

2 ioptii

inel Vk

Optical potential and the many-body problem

Following the method of Watson et al we introduce an operator F such that

Thus, in contrast with Π0, the new operator F reconstructs the full many-body wave function from its elastic scattering part.

In order to connect it with many-body problem we start

from the full N+1 electron L-S equation:

.)()(

)(0

)(

c

c

F

and apply Π0 from left

thus the optical potential , which does not act on the internal coordinates of the target, is given by

)()(0

)( VGi

,)(0

)(0

)(

)()(000

)(0

C

V

iC

i

opt

FVG

VG

optV

00 FVVopt

In order to determine we must therefore find the operator F.

To carry out this we just play with above equations

with extracted from above we get

optV

)()(0

)( VGi

)()(0

)( CiC FVGF

)()(0

)(0

)(0

)()( CCCC FVGFVGF

i

or finally

This is an exact L-S equation for F, which can be solved in few ways:– 2 body scattering matrices lead to Watson equations– Perturbation Born series in powers of the interaction

V, namely

Once again, single particle optical potential was

FVGF ]1[1 0)(

0

...]1[]1[]1[1 0)(

00)(

00)(

0 VGVGVGF

00 FVVopt

so the optical potential is given in perturbation series as

...0]1[]1[0

0]1[000

0)(

00)(

0

0)(

0

VGVGV

VGVVVopt

Optical potential for the molecules

First order term leads in so-called Static-Exchange Approximation with exchange part still missing because

Pauli exclusion effects have been neglected so far

with

00)1( VV

N

i i

M

A A

AZV11 ||

1

|| rrRr

0||

10

|| 11

)1(

N

i i

M

A A

AZVrrRr

HF approximation

Let’s take first term:

)(...)1()(!

10..1 1 NggPP

NN N

p

||

)()(d

1

)()...1(')'(||

1)()...1()(d1...dN

!

1

0||

10

1

11*

11

'1

1

**1

1

rr

rrr

rr

rr

iiN

i

pN

pN

gg

N

NggPpNggPpN

Thus the first order optical potential provides the static (and exchange) potential generated by nuclei and

fixed bound state wave function of the molecule:

with HF density

||

)(d

||1

)1(

xr

xx

Rr

M

A A

AZV

N

iii gg

1

* )()()( xxx

How good is Static (-Exchange) Approximation?

• Static (-Exchange) approximation leads to correct interaction at very small distances from nuclei.

• Therefore one can expect results improving with higher collision energies ( > 10 eV) and for largerscattering angles that are ruled by electrons withsmall impact parameters.

• is Hermitian and therefore no electronically inelastic processes can be described by this term.

)1(V

(Correlation -) Polarization

...0]1[000 0)(

0 VGVVVopt

We notice that

where n runs over all intermediate states of the target except the ground state. Then

0

01n

nn

0 0

2

)2(

)(2/ˆ00

n ni iwwkE

VnnVV

Adiabatic approximationassumes that the change of kinetic energy

may be neglected comparing to excitation

energies wn-w0, then

Adiabatic approximation is local (non-local properties have been removed with kinetic energy operator neglected in denominator) and real. So again does

not account for the removal of particles from elastic channel above excitation threshold.

2

ˆ2kEi

.00

0 0

)2(

n n

ad ww

VnnVV

)2(adV

Angular expansion of the Coulomb operator

gives approximate expression

nnZ

nVN

i i

M

A A

A

1

0

1 ||

10

||00

rrRr

....ˆ11

)ˆ.ˆ(||

12

01

iil

ll

l

i rrP

r

rrrrr

rr

nr

nVN

ii

1

2.ˆ0

10 rr

The adiabatic approximation to second order optical potential then becomes

where

is the dipole polarizability of the molecule. Thus we see

if the orbital relaxation caused by strongest second order of interaction V is allowed, rise of a long-range potential

behaved as r-4 can be noticed.

,2 4

)2(

rVad

0 0

11

ˆ.00.ˆ

2n n

N

jj

N

ii

ww

nn rrrr

Approximation of the average excitation energy

Introducing complete set of plane waves we obtain:

'0)(2/ˆ

10'

0 02

)2( rrrr

n ni

VniwwkE

nVV

0 02

)'.(3

0 02

)2(

)(2/

00d

)2(

1

')(2/

00d'

n ni

i

n ni

iwwkE

VnnVe

iwwkE

VnnVV

rrkk

rkkrkrr

The effect of Pauli principle

We define asymptotic states of l-th electron being in continuum as (0-th coordinate stands for scattered particle now)

Our scattering problem can be defined via solutions of

full N+1 electron Schrödinger equation

where is antisymmetric in all pairs of electrons.

))...(...()( 01 Nnllln rrrrkk

,0)...0()( NHE

A boundary condition must be added to fix uniquely.

The physics of the problem dictates the boundary condition: As rl approaches infinity, for l arbitrary,

approaches the asymptotic form

For evaluation of cross section we need to calculate the flux of the scattered electrons “0” only.

waves)outgoing (electron lnll

rl k

0for1

0for1

l

ll

That is, all the N+1 particles enter the problem symmetrically. Each of them at infinity carries the same ingoing and the same outgoing flux. Hence the total flux is N+1 times the flux of one particle. Since only the out/in ration of fluxes appear in cross section it is sufficient to calculate the flux of particle 0 alone. Or, we may regard particle 0 as distinguishable in obtaining scattering cross section from .

The flux of “0” electrons scattered is calculated from

nr

nn,

000

limk

kk

Final expressions for the optical potential for undistinguishable particles have very similar form

This time V is modified interaction potential:

where swaps 0-th and i-th electrons.

Thus the additional exchange term has in coordinate representation form (HF orbitals assumed):

...0]1[000 0)(

0 VGVVVopt

,)1(||

1

|| 10

01 0

N

ii

i

M

A A

A PZ

VrrRr

iP0

Exchange part is non-local and short-range interaction as can be seen from its effect on wave function

Second order using HF approximation leads to the sum

of 2 terms, called polarization and correlation contributions as follows:

N

i

iiex

ggV

1

*

|'|

)'()('

rr

rrrr

N

i

iiexex

ggVV

1

*

|'|

)'()'('d)()'(''d

rr

rrrrrrrrr

Introducing complete set of plane waves we obtain:

'0)(2/ˆ

10'

0 02

)2( rrrr

n ni

VniwwkE

nVV

0 02

)'.(3

0 02

)2(

)(2/

00d

)2(

1

')(2/

00d'

n ni

i

n ni

iwwkE

VnnVe

iwwkE

VnnVV

rrkk

rkkrkrr

• V(1) 0 excitations (only ground state)• V(2) single excitations• V(3) double excitations• V(N) …..

),',(),',(')2(iSRiLR EVEVV rrrrrr