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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.
                         1
              M  1            [ V  O] M
                     E  H0  O

  Different choices of O leads to different perturbation
  methods:
  - Brillouin-Wigner
  - Rayleigh-Schrödinger
  - Tanaka-Fukuda
  - Feenberg
• 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
• Formulation in CMS
                                                N electrons
                                                M nuclei
                               r     rN
                                           r2
          scattered electron          r1

                                   molecule


• Nuclear mass >> mass of electrons
• Vibrations and rotations of molecule not
  considered
• Electronically elastic scattering
Total hamiltonian H is split into free part H0 and a
“perturbation” V:
          H  H0 V
          H 0  21 k 2  H M
                      M   N                N
                                ZA                  1
          HM                        
                 A 1 i 1 | ri  R A |  i  j | ri  r j |
                  M             N
                      ZA                1
         V                 
             A 1 | r  R A |  i 1 | r  ri |




          H 0  n (1... N )  wn  n (1... N )
Initial and final free states:
                       1
               i              eik i .r 0  S ,
                    (2 )3 / 2
                        1
               f               e f 0  S , '
                                  ik .r

                     (2 )3 / 2

                         H 0i  Eii
                         H 0 f  E f  f
                                  1 2
                        Ei  w0  ki
                                  2
                                  1 2
                        E f  w0  k f
                                   2
Exact solution can be provided by N+1 electron
Lippmann-Schwinger equation

                                   1
                ()
                        i                V  ()
                              Ei  H 0  i

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

         (  ) (1... N  1)    
                                 n
                                      n   (1.. N ) n ( N  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?

                               1
            ()
                    i                Vopt c(  )
                          Ei  H 0  i
             c




where c( ) corresponds to the elastic (coherent) part of
the total wave function  (.)
We define projection operator onto ground states:
               0   0 t 0 t  0 0
                      t

Elastic part can be obtained by projection
                           c( )  0 ( )

Full N+1 L-S equations for  ( ) are

               (  )  i  G0  ) ( Ei ) V  (  )
                              (



               (  )  i  G0  ) ( Ei ) T i
                              (



               ()              1
             G       (E) 
                           E  H 0  i
               0
By applying  0 from the left we get

         0  (  )   0 i   0 G0 ) ( Ei ) T i
                                    (




             0   i  i   , 0 G0)  G0) 0
                                   (      (
                                                       

            c(  )  i  G0  )  0 T  0 i
                            (
                                    
                                   
                                     TC


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

      TC  0 T 0  0 0 T 0 0  0 TC 0
We have projected L-S equation

               c(  )  i  G0 ) TC i
                               (



Definition of optical potential

                    Vopt c( )  TC i

gives desired form

              c(  )  i  G0 ) Vopt c(  )
                              (



or

                   TC  Vopt  Vopt G0) TC
                                     (
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

             0 c(  )  0 i  0 G0 )  0 Vopt  0 c(  )
                                   (



and the single-electron equation is obtained:
                                                                        1
   ()
                (r )  (2 )   3 / 2
                                        e   i k i .r
                                                        ,                        Vopt  ki,), t (r )
                                                                                            (
    k i , , t                                           t           ˆ
                                                                 Ei  k 2 / 2  i
Where Vopt is a single-particle optical potential obtained
by
                   Vopt  0 Vopt 0

The definition of Vopt does not imply that optical potential
is an Hermitian operator. In fact, hermitian Vopt would
lead to TC such that SC 1 2i T is unitary.

In fact after applying of machinery of optical theorem
it can be shown that
                         (2 )3
             inel   2        Im  i(  ) Vopt  i(  )
                           ki
           Optical potential and
         the many-body problem
Following the method of Watson et al we introduce an
operator F such that
                     c(  )   0  (  )
                      (  )  F c(  ) .

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:
                 (  )  i  G0 ) V  (  )
                                (



and apply Π0 from left

           0  (  )   0 i   0 G0  ) V  (  )
                                      (



           C )  i  G0  )  0 V F C ) ,
            (            (              (
                               
                                    Vopt


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

                       Vopt  0 V F 0
In order to determine Vopt we must therefore find the
operator F.
To carry out this we just play with above equations

              (  )  i  G0 ) V  (  )
                             (




          F C )  i  G0 ) V F C )
             (            (         (




with i extracted from above we get

 F C )  C )  G0 )  0 V F C )  G0 ) V F C )
    (       (       (             (       (         (
or finally
                    F  1  G0  ) [1   0 ] V F
                             (



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

 F  1  G0  ) [1   0 ] V  G0  ) [1   0 ] V G0  ) [1   0 ] V  ...
          (                     (                   (



 Once again, single particle optical potential was

                          Vopt  0 V F 0
so the optical potential is given in perturbation series as

Vopt  0 V 0  0 V G0  ) [1   0 ] V 0
                    (


               0 V G0  ) [1   0 ] V G0  ) [1   0 ] V 0  ...
                     (                   (
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

                          V (1)  0 V 0
with
                          M              N
                               ZA                1
                  V                 
                      A 1 | r  R A |  i 1 | r  ri |

                      M                N
                             ZA                1
        V   (1)
                                   0             0
                    A 1 | r  R A |  i 1 | r  ri |
                        HF approximation
                                    1
               1.. N 0                 ( P) P g1 (1) ... g N ( N )
                                    N! p

Let’s take first term:

           1
     0            0 
       | r  r1 |
    1                                             1
       d1...dN  ( p) P g1 (1)...g * ( N )
                            *
                                                          ( p' ) P' g1 (1)...g N ( N )
                                              | r  r1 | p '
                                      N
    N!          p

  1    N
             g i* (r1 ) g i (r1 )

  N
        dr1 | r  r |
      i 1               1
Thus the first order optical potential provides the static
(and exchange) potential generated by nuclei and
fixed bound state wave function of the molecule:
                       M
                              ZA               ( x)
         V   (1)
                                     dx
                     A 1 | r  R A |        |rx|

with HF density
                               N
                    ( x)     g i* (x) g i (x)
                              i 1
            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 larger
  scattering angles that are ruled by electrons with
  small impact parameters.

      (1)
• V   is Hermitian and therefore no electronically
  inelastic processes can be described by this term.
      (Correlation -) Polarization

     Vopt  0 V 0  0 V G0) [1 0 ] V 0  ...
                         (



We notice that
                           1  0    
                                      n0
                                            n n

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

                                0V n nV 0
      V   ( 2)
                     E  kˆ
                     n0
                                2
                                    / 2  ( wn  w0 )  i
                            i
Adiabatic approximation
                                                     ˆ
                                                    k2
assumes that the change of kinetic energy Ei 
                                                     2
may be neglected comparing to excitation
energies wn-w0, then
                        0V n nV 0
             Vad  
               ( 2)
                                      .
                    n0   w0  wn
Adiabatic approximation is local (non-local properties
have been removed with kinetic energy operator
neglected in denominator) and real. So
                                           (
                                         Vad2 ) again does
not account for the removal of particles from elastic
channel above excitation threshold.
                    M                N
                       ZA                    1
     0V n  0                n  0               n
               A1 | r  R A |      i 1 | r  ri |
            
                         0

Angular expansion of the Coulomb operator
                     
            1            rl           1 1
                    l 1 Pl (r
                               ˆ.ri )   2 r.ri  ...
                                 ˆ          ˆ
        | r  ri | l 0 r             r r

gives approximate expression
                               N
                        1
                0 V n  2 0 r. ri n
                            ˆ
                       r      i 1
The adiabatic approximation to second order optical
potential then becomes
                                   
                 V  ( 2)
                   ad                 4
                                            ,
                                   2r
where
                            N                   N
                      r. 0  ri n n  r j 0 .r
                      ˆ                      ˆ
           2             i 1                j 1

                n0                 wn  w0
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.
Approximation of the average excitation energy

                                                                 1
rV   ( 2)
            r'    r
                   n0
                               0V n
                                              ˆ
                                         Ei  k 2 / 2  ( wn  w0 )  i
                                                                                    n V 0 r'


Introducing complete set of plane waves we obtain:

                                                         0V n nV 0
 rV     ( 2)
               r'       dk
                       n0
                                     rk
                                           Ei  k / 2  ( wn  w0 )  i
                                                         2
                                                                                      k r'

                         1                                            0V n nV 0
                   
                       (2 )   3     dk e
                                   n0
                                              i k .( r r ')

                                                               Ei  k 2 / 2  ( wn  w0 )  i
       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)
            k n l  k(rl ) n (r1...r0 (l )... N )
                                             r

Our scattering problem can be defined via solutions of
full N+1 electron Schrödinger equation

                 ( E  H ) (0...N )  0 ,

where  is antisymmetric in all pairs of electrons.
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

    rl l k n l  (electronl outgoing waves)
        

                    1 for l  0
              l  
                    1 for l  0

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

                 lim  k n            kn 
                 r0           0 0
                         k ,n
Final expressions for the optical potential for
undistinguishable particles have very similar form

    Vopt  0 V 0  0 V G0) [1 0 ] V 0  ...
                        (




This time V is modified interaction potential:
              M               N
                    ZA                 1
      V                                  (1  P0i ) ,
          A 1 | r0  R A |  i 1 | r0  ri |


where P0 i swaps 0-th and i-th electrons.
Thus the additional exchange term has in coordinate
representation form (HF orbitals assumed):
                                    g i (r ) g i* (r ' )
                                    N
                 r Vex r '   
                               i 1     | r  r'|

Exchange part is non-local and short-range interaction
as can be seen from its effect on wave function
                                               N
                                                             (r ' ) g i* (r ' )
r Vex    dr ' r Vex r '  (r ' )    g i (r )  dr '
                                              i 1              | r  r'|

Second order using HF approximation leads to the sum
of 2 terms, called polarization and correlation
contributions as follows:
                                                                 1
rV   ( 2)
            r'    r
                   n0
                               0V n
                                              ˆ
                                         Ei  k 2 / 2  ( wn  w0 )  i
                                                                                    n V 0 r'


Introducing complete set of plane waves we obtain:

                                                         0V n nV 0
 rV     ( 2)
               r'       dk
                       n0
                                     rk
                                           Ei  k / 2  ( wn  w0 )  i
                                                         2
                                                                                      k r'

                         1                                            0V n nV 0
                   
                       (2 )   3     dk e
                                   n0
                                              i k .( r r ')

                                                               Ei  k 2 / 2  ( wn  w0 )  i
           r V ( 2) r'  VLR (r, r' , Ei )  VSR (r, r' , Ei )



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

				
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