, M.C. Zammit 1,2 , J.S. Savage & I. Bray 1Deuring et al. Jones Subramanian and Kumar Nickel et al....

Electron scattering from molecular hydrogen

D.V. Fursa1, M.C. Zammit1,2, J.S. Savage1 & I. Bray1

1Curtin University, Perth, Australia 2Los Alamos National Laboratory, Los Alamos, USA

Acknowledgements

Australian Research Council

Curtin University

Los Alamos National Laboratory

Pawsey Supercomputing Centre

Method: H2 molecule

Born-Oppenheimer approximation

Fixed-nuclei approximation, R = fixed

Solve for the electronic wave function

Adiabatic-nuclei approximation: perform FN calculations at a number of R, study nuclear motion

• spherical coordinate code: the origin at the midpoint

• spheroidal coordinate code: the best approach for diatomic molecules (nuclear motion)

fn,l (2lr)µ(2lr)l+1e-lrLn2l+2(2lr)

modeling of infinite number

of bound and continuum states

with a finite number of pseudostates

N

n

tnpntpi F1

)()()(),( xxxx A• N-state multi-channel expansion

• The Schrödinger equation is converted to an

integral LS equation for the T-matrix

• Cross sections

Tfi(k f ,k i) =V fi(k f ,k i) + d3kòn=1

N

åV fn (k f ,k)Tni(k,k i)

E + i0 -en - k 2 /2

Qfi(R)µ Tfi(R)2

Target Hamiltonian HT is diagonalized in a Sturmian (Laguerre) basis

Electron scattering from H2

• Is a benchmark system with a range of applications

• Accurate measurements exist for some major processes

• Theory had not done sufficiently well:

Previous largest CC calculations use 9-states RM and SMC or 41-states RMPS

• There is room for improvement…

Fixed-nuclei calculations at R = 1.448 a.u.

Size of projectile partial wave expansion Lmax ABS to speed up p.w. convergence

Size of expansion in total M, P, S

Size of close-coupling expansion

• number of states in close-coupling expansion

• size of underlying one-electron (Sturmian) basis

o maximum value of orbital angular momentum lmax

o number of functions Nl for given l

• Two-electron configuration interaction: choice of configurations, (1ss,nl), (nl,n’l), n, n’ < 2

Largest calculations:

- 491-state, Nl =17-l, lmax = 3, projectile p.w. Lmax = 8

Convergence is established with:

- 427-state, Nl =15-l, lmax = 3, projectile p.w. Lmax = 8

- 259-state, , Nl =15-l, lmax = 2, projectile p.w. Lmax = 6

- 92-state: only negative energy (bound) states from 491 state model

- 9-state: first 9 states of H2

Convergence & accuracy studies

Accuracy of the target structure Two-electron energies (a.u.) at R=1.4 a.u.

State Present Ref.

X 1Sg -1.162 -1.174a

b 3Su -0.770 -0.784b

a 3Sg -0.710 -0.714c

c 3Pu -0.701 -0.707d

B 1Su -0.697 -0.706e

E,F 1Sg -0.687 -0.692e

C 1Pu -0.683 -0.689f

e 3Su -0.683 -0.689g

h 3Sg -0.640 -0.644h

B’ 1Su -0.625 -0.629i

D 1Pu -0.621 -0.624f

B” 1Su -0.600 -0.602e

D’ 1Pu -0.598 -0.600h

aBranchett and Tennyson J. Phys. B 25, 2017 (1992)

OOS (length) at R=1.4 a.u.

transition Present Ref.a

X 1Sg B 1S

u 0.2769 0.2995

X 1Sg C 1Pu 0.3368 0.3508

X 1Sg B’ 1S

u 0.0578 0.0603

X 1Sg D 1P

u 0.0832 0.0913

X 1Sg B” 1S

u 0.0221 0.0353

X 1Sg D’ 1P

u 0.0344 0.0534

aKolos etal, J. Chem. Phys. 84 (1986) 3278 bKolos & Wolniewicz, J. Chem. Phys. 43 (1965) 2429 cKolos & Wolniewicz, J. Chem. Phys. 48 (1968) 3672

dKolos & Rychlewski, J. Mol. Spectrosc. 66 (1977) 428 eLiu & Hagstrom, Phys. Rev. A 48 (1993) 1966 fWolniewicz & Dressler, J. Chem. Phys. 143 (1988) 386 gKolos & Rychlewski, J. Mol. Spectrosc. 13 (1990) 237 hSharp, At. Data Nucl. Data Tables 2 (1971) 119

iWolniewicz, Chemical Physics Letters 31 (1975) 248

Static dipole polarizability (a.u.) at R=1.4 a.u.

a

CCC 6.43 4.64 5.23

Ref.a 6.38 4.58 5.18

aKolos and L. Wolniewicz, J. Chem. Phys. 46, 1426 (1967)

Accuracy due to the structure is about 9%

Electron scattering from H2

The best estimate of CCC cross sections is produced for energies from 0.1 to 300 eV for

• total cross section

• total ionization cross section

• elastic scattering (ICS, DCS)

• excitation cross sections (ICS, DCS) for

- b 3Su+ , a 3Sg

+, c 3Pu , e 3Su

+, h 3Sg+

- B 1Sg+, C 1Pu, E,F 1Sg

+, B’ 1Sg+, D 1Pu, B” 1Sg

+, D’ 1Pu

- more transitions are available on request

e--H2 recommended cross sections of

Yoon et al. J. Phys. Chem. Ref. Data

37(2008)913

are all derived from experimental data

e--H2 total cross section

1

10

100

0.1 1 10 100

Cross Section (units of a0

2)

Incident Energy (eV)

CCC

Ferch et al.

van Wingerden et al.

Hoffman et al.

Deuring et al.

Jones

Subramanian and Kumar

Nickel et al.

Zhou et al.

All major scattering processes

are accounted for correctly:

- elastic scattering

- excitations

- ionization

Zammit, Savage, Fursa & Bray, Phys. Rev. Lett. 16 (2016) 233201

e--H2 total ionization cross section

Zammit, Savage, Fursa & Bray, Phys. Rev. Lett. 16 (2016) 233201

0

1

2

3

4

50 100 150 200 250 300

Cro

ss s

ection (

un

its o

f a

02)

Incident energy (eV)

CCC(491)

CCC(427)

CCC(259) 0

1

2

3

4

50 100 150 200 250 300


Rapp and Englander-Golden

Krishnakumar and Srivastava

Straub et al.

Lindsay and Mangan

CCC

RMPS

TDCC

recommended

recommended cross sections are due to Yoon et al.

J.Phys.Chem.Ref.Data 37(2008)913

e--H2 excitation ICS & DCS: b 3Su+

Zammit, Savage, Fursa & Bray,

submitted to PRA

0

0.5

1

1.5

2

2.5

3

10 20 30 40 50 60

Cro

ss s

ection (

un

its o

f a

02)


b 3Su

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

0

0.5

1

1.5

2

2.5

3

10 20 30 40 50 60


Khakoo and Segura

Khakoo et al.

Nishimura and Danjo

CCC

SMC

RM

RMPS

DW

recommended

SMC, RM ~ 9-state



e--H2 excitation ICS & DCS:

b 3Su+


submitted to PRA

0

0.5

1

1.5

2

2.5

3

10 20 30 40 50 60

Cro

ss s

ection (

un

its o

f a

02)


b 3Su

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

0

0.5

1

1.5

2

2.5

3

10 20 30 40 50 60


Khakoo and Segura

Khakoo et al.

Nishimura and Danjo

CCC

SMC

RM

RMPS

DW

recommended

SMC, RM ~ 9-state



0

10

20

30

40

15 eV

CCC(491)

CCC(427)

CC(92)

CC(9)

0

10

20

30

40

CCC(491)

SMC

RM

DW

0

10

20

30

40

Diffe

ren

tia

l cro

ss s

ectio

n (

un

its o

f 1

0-2

a0

2/s

r)

13 eV

0

10

20

30

40

Khakoo and Segura

Nishimura and Danjo

Hall and Andric

0

10

20

30

40

0 30 60 90 120 150 180

12 eV

0

10

20

30

40

0 30 60 90 120 150 180

Scattering angle (degree)

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50 60

Cro

ss s

ection (

un

its o

f a

02)


a 3Sg

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

0

0.2

0.4

0.6

0.8

1

10 20 30 40 50 60


Wrkich et al.

Khakoo and Trajmar

Ajelo and Shemansky

CCC

RM

SMC

DW

recommended

0

1

2

60 eV

CCC(491)

CCC(427)

CC(92)

CC(9)

0

1

2

Khakoo and Trajmar

Wrkich et al.

CCC(491)

SMC

RM

DW

0

5

10

Diffe

ren

tial cro

ss s

ectio

n (

un

its o

f 1

0-2

a0

2/s

r)

30 eV

0

5

10

0

5

10

15

20 eV

0

5

10

15

0

5

10

15

0 30 60 90 120 150 180

17.5 eV

0

5

10

15

0 30 60 90 120 150 180


e--H2 excitation ICS & DCS: a 3Sg+


submitted to PRA

SMC, RM ~ 9-state



0

0.2

0.4

0.6

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


E,F 1Sg

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

Born

0

0.2

0.4

0.6

200 300 10 100


Wrkich et al.

recommended Liu et al.

CCC

RM

SMC

DW

e--H2 excitation ICS & DCS: E,F 1Sg+


submitted to PRA

SMC, RM ~ 9-state



0

0.2

0.4

0.6

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


E,F 1Sg

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

Born

0

0.2

0.4

0.6

200 300 10 100


Wrkich et al.


CCC

RM

SMC

DW

0

5

10

15

30 eV

CCC(491)

CCC(427)

CC(92)

CC(9)

0

5

10

15

Wrkich et al.

CCC(491)

0

5

10

15

20

Diffe

ren

tial cro

ss s

ectio

n (

un

its o

f 1

0-2

a0

2/s

r)

20 eV

0

5

10

15

20

RM * 0.1

0

5

10

15

20

0 30 60 90 120 150 180

17.5 eV

0

5

10

15

20

0 30 60 90 120 150 180



E,F 1Sg+


submitted to PRA

SMC, RM ~ 9-state



0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


B 1Su

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

Born

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100


Srivastava and Jensen

Khakoo and Trajmar

Kato et al.

Wrkich et al.


CCC

RM

SMC

DW

e--H2 excitation ICS & DCS: B 1Sg+


submitted to PRA

SMC, RM ~ 9-state


J.Phys.Chem.Ref.Data 37(2008)913 Oscillator Strength

Accurate theory 0.274 Phys. Rev. A 60, 1226 (1999)

Fixed-nuclei CCC 0.288

Kato (measured) 0.241 ± 0.048 Phys. Rev. A 77, 062708 (2008)

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


B 1Su

+

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

Born

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100


Srivastava and Jensen

Khakoo and Trajmar

Kato et al.

Wrkich et al.


CCC

RM

SMC

DW

0.1

1

10

100

30 eV

CCC(491)

CCC(427)

CC(92)

CC(9)

0.1

1

10

100

Khakoo and Trajmar

Wrkich et al.

CCC(491)

SMC

RM

DW

0.1

1

10

100

Diffe

ren

tia

l cro

ss s

ectio

n (

un

its o

f 1

0-2

a0

2/s

r)

20 eV

0.1

1

10

100

0.1

1

10

100

0 30 60 90 120 150 180

17.5 eV

0.1

1

10

100

0 30 60 90 120 150 180



B 1Sg+


submitted to PRA

SMC, RM ~ 9-state



0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


C 1Pu

CCC(491)

CCC(427)

CCC(259)

CC(92)

CC(9)

Born

0

0.3

0.6

0.9

1.2

1.5

1.8

2.1

200 300 10 100


Khakoo and Trajmar

Kato et al.

Wrkich et al.


CCC

RM

SMC

DW

e--H2 excitation ICS & DCS: C 1Pu


submitted to PRA

SMC, RM ~ 9-state



Oscillator Strength

Accurate theory 0.351 Phys. Rev. A 60, 1226 (1999)

Fixed-nuclei CCC 0.342

Kato (measured) 0.226± 0.045 Phys. Rev. A 77, 062708 (2008)

0.1

1

10

100

30 eV

CCC(491)

CCC(427)

CC(92)

CC(9)

0.1

1

10

100

Khakoo and Trajmar

Wrkich et al.

CCC(491)

RM

SMC

DW

0.1

1

10

100

Diffe

ren

tia

l cro

ss s

ectio

n (

un

its o

f 1

0-2

a0

2/s

r)

20 eV

0.1

1

10

100

0.1

1

10

100

0 30 60 90 120 150 180

17.5 eV

0.1

1

10

100

0 30 60 90 120 150 180


e--H2 elastic DCS at 1, 6, and 30 eV

1

10

0 30 60 90 120 150 180


2)

Scattering Angle (deg.)

1 eV X 1Sg

+

CCC

VCC

Muse et al.

Furst et al.

Linder and Schmidt

Brunger et al.

0.1

1

10

0 30 60 90 120 150 180


2)


30 eV X 1Sg

+

CCC

Muse et al.

Shyn and Sharp

Nishimura et al.

Khakoo and Trajmar

Srivastava et al.

1

10

0 30 60 90 120 150 180


2)


6 eV X 1Sg

+

CCC

VCC

Muse et al.

Furst et al.

Nishimura et al.

Linder and Schmidt

• First large-scale close-coupling calculations for H2

• First comprehensive theoretical dataset of e-H2 excitation cross sections

• Uncertainties:

structure: < 9%

close-coupling: < 5%

total: < 10%, better than 5% for most transitions

• Future work

- scattering from H2 excited states: electronic and vibrational

- vibrational and rotational excitations of H2

- isotopologues D2, T2, HD, HT, DT

- scattering from HeH+, He2+

- hydrates (LiH, OH, …) and other diatomics (O2, N2, …)

Conclusions

• All cross sections are available at access-free databases

LXcat: http://fr.lxcat.net/home/

Conclusions

ALADDIN: https://www-amdis.iaea.org/infoaladdin.php

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


B¢ 1Su

+

CCC

CCC(491)

CCC(427)

CCC(259)

CC(92)

DW

Born

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

200 300 10 100


D 1Pu

Liu et al.

e--H2 excitation ICS: B’ 1Sg+, D 1Pu, B” 1Sg

+, D’ 1Pu


submitted to PRA

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

200 300 10 100

Cro

ss s

ection (

un

its o

f a

02)


B² 1Su

+

CCC

CCC(491)

CCC(427)

CCC(259)

CC(92)

Born

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

200 300 10 100


D¢ 1Pu

Glass-Maujean et al.

Accuracy of the target structure

Two-electron energies (a.u.) at R=1.448 a.u.

State Present Present

accurate

X 1Sg -1.161 -1.173

b 3Su -0.782 -0.797

a 3Sg -0.715 -0.719

c 3Pu -0.707 -0.713

B 1Su -0.704 -0.713

E,F 1Sg -0.693 -0.698

C 1Pu -0.690 -0.695

e 3Su -0.647 -0.651

h 3Sg -0.634 -0.637

d 3Pu -0.63234 -0.635

B’ 1Su -0.631 -0.635

D 1Pu -0.628 -0.630

B” 1Su -0.606 -0.609

D’ 1Pu -0.604 -0.607

OOS (length) at R=1.448 a.u.

transition Present Present

accurate

X 1Sg B 1S

u 0.2881 0.3129

X 1Sg C 1Pu 0.3420 0.3636

X 1Sg B’ 1S

u 0.0593 0.0580

X 1Sg D 1P

u 0.0843 0.0852

X 1Sg B” 1S

u 0.0225 0.0210

X 1Sg D’ 1P

u 0.0349 0.0336

Accuracy due to the structure is

about 9% for the B 1Su OOS

and better than 9% for other transitions

Present accurate = large CI in spheroidal coordinates

, M.C. Zammit 1,2 , J.S. Savage & I. Bray 1Deuring et al. Jones Subramanian and Kumar Nickel et al....

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