Uncertainties in calculations of
low-energy resonant electron collisions with
diatomic molecules
Karel Houfek
in collaboration with
J. Horáček, M. Čížek and M. Formánek from Prague
V. McKoy and C. Winstead from CalTech, USA
J. Gorfinkiel and Z. Mašín from Open University, UK
C.W. McCurdy and T. Rescigno, LBNL, USA
Institute of Theoretical Physics
Faculty of Mathematics and Physics
Charles University in Prague
Resonant electron-molecule collisions at low energies
Vibrational (VE) and rotational excitation including elastic scattering





e  AB ( i , N i )  ( AB )  e  AB ( f , N f )
Electronic excitation

e  AB  ( AB )  e  AB
*
Dissociative electron attachment (DA)


e  AB ( i )  ( AB )  A  B

three-body decays etc.
We study also inverse process of DA called associative detachment (AD)



A  B  ( AB )  e  AB ( f )
Theoretical description of electron-molecule collisions
First fixed-nuclei calculations provide
•
potential energy curves (surfaces) of neutral molecule (standard)
and molecular ion where electron is bound (more difficult)
•
fixed-nuclei scattering data (eigenphase sums, cross sections) –
several methods available

Complex Kohn variational principle (Berkeley, USA)

R-matrix (UCL and OU, UK)

Schwinger multichannel variational method (CalTech, USA)
from which a model for nuclear dynamics is constructed within some approximation
•
local complex potential approximation – simplest to use, first choice
•
nonlocal, complex, and energy-dependent potential – universal, but difficult
•
R-matrix approach of Schneider et al – applied only to N2 and CO
Motivation for this talk – results for e + CO
SMC method for electron scattering + LCP and NRM for nuclear dynamics
Fixed-nuclei eigenphase sums
Potential energy curves
Cross sections – vibrational excitation
Motivation for this talk – results for e + CO
SMC method for electron scattering + LCP and NRM for nuclear dynamics
Cross sections – vibrational excitation
e + CO – comparison with previous calculations
What is the best theoretical result and what uncertainties are there?
Morgan, J. Phys. B 24 (1991) 4649
Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005
e + CO – comparison of potential energy curves
This gives us a hint of the origin of discrepancies and uncertainties
Morgan, J. Phys. B 24 (1991) 4649
Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005
Another example of results – e + HCl
Allan, Čížek, Horáček, Domcke, J. Phys. B 33 (2000) L209
HCl – structures in the VE and DA cross sections
Nonlocal resonance model by Fedor et al – Phys. Rev. A 81 (2010) 042702
HCl – origin of structures in the cross sections
Uncertainties in position of structures – shape and relative position of PEC
Uncertainties in absolute values of the cross sections – details of the model used
Uncertainties in calculations of e-M collisions
1) Potential energy curves (surfaces)

various methods (HF, CASSCF, CI, CC)

absolute energies are not important

relative shape and position of curves (surfaces) is crucial, problem
of size consistency (N and N+1 electrons)

known difficulties to obtain correct electron affinities
2) Fixed-nuclei electron scattering

various models (SE, SEP, CAS)

problem of consistency of scattering data with accurate potential
energy curves
3) Model for nuclear dynamics

different levels of approximation (effective range, LCP, NRM etc.)

possibility of testing using two-dimensional model
Potential energy curves – e + CO system
Different methods with aug-cc-pVTZ basis
Potential energy curves – e + CO system
MRCI based on CASSCF(10,9) for larger basis sets
LCP model with improved potentials – e + CO system
Morse potential and adjusted width to get a good agreement with experiment
R-matrix electron scattering calculations
Schwinger multichannel variational method cannot describe the target beyond the
Hartree-Fock approximation –> impossible to have a better description of the
target consistent with electron scattering calculations
Possible with UK R-matrix polyatomic codes – several different scattering models
available
•
SE – static exchange – target at HF level
•
SEP – static exchange plus polarization – target at HF level
•
CAS – close-coupling model (only a few excited target states included) based
on complete active space (CASSCF) calculations of the target
R-matrix electron scattering calculations – CAS model
CASSCF (10,8) model – 6-311G** basis, 9-12 virtual orbitals, 13 states at R = 2.1
R-matrix electron scattering calculations – CAS model
CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals, 13 states
R-matrix electron scattering calculations – CAS model
CASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals (1 different), 13 states
R-matrix electron scattering calculations – CAS model
CASSCF (10,8) model – 6-311G** basis, 11 virtual orbitals, 9 states
Nuclear dynamics – local vs. nonlocal theory
Vibrational excitation cross sections – e + CO
Nuclear dynamics – local vs. nonlocal theory
simple two-dimensional model as a testing tool
Model Hamiltonian – one nuclear (R) and one electronic (r) degree of freedom
H 
1
d
2
2  dR
2
 V0 ( R ) 
1 d
2 dr
2
2

l ( l  1)
2r
2
 V int ( R , r )
V0 (R)
- potential energy of the neutral molecule
- Morse potential
l
- angular momentum of the electron
- p-wave (l = 1) or d-wave (l = 2)
V int ( R , r ) - interaction potential
- bound state of the electron for large R
- resonance for small R
Houfek, Rescigno, McCurdy, Phys. Rev. A 73 (2006) 032721
Houfek, Rescigno, McCurdy, Phys. Rev. A 77 (2008) 012710
Barrier for incoming electron →
shape resonance for small R
Fixed-nuclei calculations – N2-like model
Electronic Hamiltonian used in fixed-nuclei calculations
H el ( R )  
1 d
2 dr
2
 r
2
  (R) e
2

l ( l  1)
2r
Cross sections (or phase shifts)
2
resonance position and width electron bounding energy
Solution of the full 2D model
Exact wave function at a given energy E  E v  E el
i

 (R, r)   (R, r) 
0
1
E  H  i
V int ( R , r )  ( R , r )
0
with the initial state
 ( R , r )   vi ( R )
0
2k

rj l ( kr )
where  v ( R ) is initial molecular vibrational state
i
N2-like model
incoming electron
Numerical solution using
finite elements with DVR basis
and exterior complex scaling
NO-like model – scattered wave functions for vi = 0
Franck-Condon region
Nuclear dynamics – LCP approximation
Local complex potential approximation
2


1 d
i
 (R) 
E 


E
(
R
)


(
R
)

(
R
)



res
2

 E
2

2
2

dR




1/ 2
 v (R)
i
Simple extensions of local complex potential approximation
• barrier penetration factor, nonlocal imaginary part
Details can be found in Trevisan et al, Phys. Rev. A 71 (2005) 052714
Vibrational excitation cross section

Dissociative attachment cross section

VE
vi  v f
(E ) 
4
3
2
ki
2 K DA
 v (  / 2 )
1/ 2
f
2
NO-like model
DA
vi
(E ) 
ki 
2
lim  d ( R )
R
2
E
2
Test of LCP approximation and its extensions
NO-like model
Nonlocal theory
Direct derivation by choosing a proper diabatic basis for electronic part of the problem
•
discrete state  d ( r ; R )
•

othogonal “background” continuum states  k ( r ; R )
satisfying conditions
 d (r , R )
R
 k (r , R )
 0,
R
lim  d ( r ; R )   b ( r )
0
R
Into which we can expand the full wave function
 ( R , r )   d ( R ) d ( r , R ) 
 kdk 
k
( R ) k ( r , R )
Using matrix elements of the electronic Hamiltonian in this basis

V d ( R )   d H el ( R )  d ,
V dk ( R )   d H el ( R )  k

V kk ' ( R )   k H el ( R )  k '  (V 0 ( R )  k / 2 )  ( k / 2  k ' / 2 )
2
2
2
we finally get effective equations for nuclear motion
E  TR
 V d ( R )  d ( R ) 
F (E, R, R') 
 dR ' F ( E , R , R ' ) 

d
( R )  V dk i ( R )  v i ( R )
 dR '  kdkV dk ( R ) E  T R  V 0 ( R )  k / 2  i 
2

1
V dk ( R ' )
Nonlocal theory – cross sections
Vibrational excitation and dissociative attachment cross sections

VE
vi  v f
(E ) 
4
2
ki
3
T
VE
vi  v f
2 K DA
2
2

(E ) ,
DA
vi
(E ) 
ki 
2
lim  d ( R )
2
R
lim  d ( r ; R )   b ( r )
It can be shown that for a properly chosen discrete state
R
there is no background contribution to the DA cross section.
But the VE T-matrix consists of two terms
T v i  v f (E )  T v i  v f (E )  T v i  v f (E )
VE
res
bg
The resonance term is calculated within nonlocal resonance theory
Tv i  v f (E )   v f V dk f  d
res
R
The background is non-zero even for inelastic vibrational excitation
and for the 2D model can calculated exactly

Tv i  v f (E )   v f  k f V int  v i J k i   v f V dk f J dk i  v i
bg
J dk i ( R ) 
l
l
 dr  d ( r ; R ) J k i ( r )
*
l
l
R
Test of nonlocal theory – NO-like model
smooth coupling (width) – works in all channels, reasonably small background
Local vs. nonlocal theory – e + F2 model
NO-like model – works in all channels,
reasonably small background
works in all channels,
reasonably small background
Minimizing background – e + F2 model
the whole information about the dynamics is “hidden”
in coupling (non-local potential)
Conclusions
Uncertainties in fixed-nuclei calculations

shape of potential energy curves (surfaces), relative positions of
potentials for neutral molecule and molecular negative ion
– advanced quantum chemistry methods (MRCI, CCSD(T) etc) are
necessary, basis sets limit, no fitting to Morse or similar analytical
potential –> comparison with available experimental data (electron
affinities, spectroscopic constants etc.)

problem of consistency of scattering data with accurate potential
energy curves
– it is necessary to go beyond HF description of the target,
adjusting parameters of electron scattering calculations to get
correct electronic energies where electron is bound –> estimating
errors by comparison of several scattering models
Uncertainties in nuclear dynamics

nonlocal theory necessary in many cases, simple extensions of
local complex potential approximation can sometimes improved the
results, but it strongly depends on the system

unknown background contribution –> uncertainties estimates from
model calculations