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, USAJ. Gorfinkiel and Z. Mašín from Open University, UK

C.W. McCurdy and T. Rescigno, LBNL, USA

Institute of Theoretical PhysicsFaculty of Mathematics and PhysicsCharles University in Prague

Resonant electron-molecule collisions at low energies

Vibrational (VE) and rotational excitation including elastic scattering

Electronic excitation

Dissociative electron attachment (DA)

three-body decays etc.

We study also inverse process of DA called associative detachment (AD)

),(ABe)(),(ABe ffii NABN

BA)()(ABe ABi

*ABe)(ABe AB

)(ABe)(BA fAB

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

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

Complex Kohn variational principle (Berkeley, USA)

R-matrix (UCL and OU, UK)

Schwinger multichannel variational method (CalTech, USA)

Motivation for this talk – results for e + COSMC method for electron scattering + LCP and NRM for nuclear dynamics

Fixed-nuclei eigenphase sums

Potential energy curves

Cross sections – vibrational excitation

Cross sections – vibrational excitation

Motivation for this talk – results for e + COSMC method for electron scattering + LCP and NRM for nuclear dynamics

e + CO – comparison with previous calculations

Morgan, J. Phys. B 24 (1991) 4649Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005

What is the best theoretical result and what uncertainties are there?

e + CO – comparison of potential energy curves

Morgan, J. Phys. B 24 (1991) 4649Laporta, Cassidy, Tennyson, Celiberto, PSST 21 (2012) 045005

This gives us a hint of the origin of discrepancies and uncertainties

Another example of results – e + HCl

Allan, Čížek, Horáček, Domcke, J. Phys. B 33 (2000) L209

Nonlocal resonance model by Fedor et al – Phys. Rev. A 81 (2010) 042702

HCl – structures in the VE and DA cross sections

HCl – origin of structures in the cross sections

Uncertainties in position of structures – shape and relative position of PECUncertainties in absolute values of the cross sections – details of the model used

Uncertainties in calculations of e-M collisions

1) Potential energy curves (surfaces)

2) Fixed-nuclei electron scattering

3) Model for nuclear dynamics

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

various models (SE, SEP, CAS) problem of consistency of scattering data with accurate potential

energy curves

different levels of approximation (effective range, LCP, NRM etc.) possibility of testing using two-dimensional model

Potential energy curves – e + CO systemDifferent methods with aug-cc-pVTZ basis

Potential energy curves – e + CO systemMRCI based on CASSCF(10,9) for larger basis sets

LCP model with improved potentials – e + CO systemMorse potential and adjusted width to get a good agreement with experiment

R-matrix electron scattering calculationsSchwinger 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 modelCASSCF (10,8) model – 6-311G** basis, 9-12 virtual orbitals, 13 states at R = 2.1

R-matrix electron scattering calculations – CAS modelCASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals, 13 states

R-matrix electron scattering calculations – CAS modelCASSCF (10,8) model – 6-311G** basis, 10 virtual orbitals (1 different), 13 states

R-matrix electron scattering calculations – CAS modelCASSCF (10,8) model – 6-311G** basis, 11 virtual orbitals, 9 states

Vibrational excitation cross sections – e + CO

Nuclear dynamics – local vs. nonlocal theory

Nuclear dynamics – local vs. nonlocal theorysimple two-dimensional model as a testing toolModel Hamiltonian – one nuclear (R) and one electronic (r) degree of freedom

),(2

)1(

2

1)(

2

1int22

2

02

2

rRVr

ll

dr

dRV

dR

dH

- potential energy of the neutral molecule- Morse potential

)(0 RV

l - angular momentum of the electron- p-wave (l = 1) or d-wave (l = 2)

),(int rRV - interaction potential- bound state of the electron for large R- resonance for small R Barrier for incoming electron →

shape resonance for small R

Houfek, Rescigno, McCurdy, Phys. Rev. A 73 (2006) 032721Houfek, Rescigno, McCurdy, Phys. Rev. A 77 (2008) 012710

incoming electron

vibrational m

otion

Fixed-nuclei calculations – N2-like model

22

2

2

)1()(

2

1)(

2

r

lleR

dr

dRH r

el

Electronic Hamiltonian used in fixed-nuclei calculations

Cross sections (or phase shifts) resonance position and width electron bounding energy

Exact wave function at a given energy

with the initial state

where is initial molecular vibrational state

Solution of the full 2D model

),(),(1

),(),( 0int

0 rRrRViHE

rRrR

)(Riv

)()(),( 20 krrjRrR lk

vi

elv EEEi Numerical solution using

finite elements with DVR basisand exterior complex scaling

N2-like model incoming electron

vibrational motion

NO-like model – scattered wave functions for v i = 0

Franck-Condon region

Nuclear dynamics – LCP approximation

Simple extensions of local complex potential approximation

• barrier penetration factor, nonlocal imaginary part

Vibrational excitation cross section

Dissociative attachment cross section

Local complex potential approximation

Details can be found in Trevisan et al, Phys. Rev. A 71 (2005) 052714

)(2

)()()(

2)(

2

12/1

2

2

RR

RRi

REdR

dE

ivEres

22/1

2

3

)2/(4

)( Evi

VEvv ffi kE

2

2

2

)(lim2

)( Rk

KE d

Ri

DADAvi

NO-like model

Test of LCP approximation and its extensions NO-like model

Nonlocal theoryDirect derivation by choosing a proper diabatic basis for electronic part of the problem

• discrete state

• othogonal “background” continuum states

satisfying conditions

Into which we can expand the full wave function

Using matrix elements of the electronic Hamiltonian in this basis

we finally get effective equations for nuclear motion

);( Rrk

);( Rrd

0),(

,0),(

R

Rr

R

Rr kd )();(lim rRr bdR

)2/'2/()2/)(()()(

)()(,)()(

2220'' kkkRVRHRV

RHRVRHRV

kelkkk

kelddkdeldd

),()(),()(),( RrRkdkRrRrR kkdd

)()()()',,(')()( RRVRRREFdRRRVTEii vdkdddR

)'(2/)()(')',,(12

0 RVikRVTERkdkVdRRREF dkRdk

Nonlocal theory – cross sectionsVibrational excitation and dissociative attachment cross sections

It can be shown that for a properly chosen discrete statethere is no background contribution to the DA cross section.

But the VE T-matrix consists of two terms

The resonance term is calculated within nonlocal resonance theory

The background is non-zero even for inelastic vibrational excitationand for the 2D model can calculated exactly

2

2

2

)(lim2

)( Rk

KE d

Ri

DADAvi

,)(4

)(2

2

3

ETk

E VEvv

i

VEvv fifi

)();(lim rRr bdR

)()()( ETETET bgvv

resvv

VEvv fififi

Rddkvresvv fffi

VET )(

Rvldkdkv

lkvkv

bgvv iiffiifffi

JVJVET int)(

)();()( * rJRrdrRJ lkd

ldk ii

smooth coupling (width) – works in all channels, reasonably small background

Test of nonlocal theory – NO-like model

Local vs. nonlocal theory – e + F2 modelNO-like model – works in all channels, reasonably small background

works in all channels, reasonably small background

the whole information about the dynamics is “hidden”in coupling (non-local potential)

Minimizing background – e + F2 model

ConclusionsUncertainties in fixed-nuclei calculations

Uncertainties in nuclear dynamics

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

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 frommodel calculations