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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