Fishing for positronic compounds

Dario Bressanini

QMC in the Apuan Alps QMC in the Apuan Alps VII 2012 VII 2012

TTI Vallico SottoTTI Vallico Sotto

Università dell’Insubria, Como, ITALY

Theory ahead of experiments

• e+ 1930: Dirac theory • e+ 1933: Carl Anderson experiment (cosmic

radiations)

• Ps (e+ e-) 1937 (1946): Ruark theory (coined the name)

• Ps (e+ e-) 1946: Wheeler theory (polyelectrons)

• Ps (e+ e-) 1951: Martin Deutsch experiment

• Two spin states •Singlet (para-positronium) – 0.125 ns•Triplet (ortho-positronium) – 142 ns

Annihilation via photons is inevitable…

“resistance is futile”…but a lot can happen on the way

Time scales

e+ and Ps spectroscopy• Used in

► Polymer science► Medical research► Solid state, electronics► hope to build a -ray laser

Theory ahead of experiments

• Ps- (e+ e- e-) and Ps+ (e+ e+ e-) ► inconsistent nomenclature► A- means Ae- , an added electron (as usual)► A+ means Ae+ , an added positron (but not always)► 1946, 1947: Wheeler, Hylleraas theory► 1981 Ps- seen in experiment by Mills

• HPs (p+ e+ e- e-)► 1947, Hylleraas & Ore theory► Seen in experiment 1992 Schrader

• Many calculations on atomic bound states, very few experimental results (more with molecules)

Current status for atoms

Computational techniques

• CI► slow convergence► frozen core► many atoms

• ECG-SVM► very accurate► slow optimization► 4 e-

• VMC-DMC► Compact ► VMC can be unbound► statistical error

Bressanini and Morosi: JCP Bressanini and Morosi: JCP 119119, 7037 (2003), 7037 (2003)

)()()()()ˆ1(),2,1(112 eergPsrfHPe

HPs

Polyleptons

Ps2 : e-e-e+e+

• 1946: Wheeler, unbound• 1946: Ore, unbound• 1947: Hylleraas & Ore: bound

► -0.504 a.u.

• 1947-1996: what is the energy?► Energy did not converge with time

0.001 a.u. = 0.027 eV

The energy of Ps2

•1996-1997: Energy controversy resolved

►-0.51601(1) DMC (1997, Bressanini et al.) ► agrees with Frolov & Smith, hylleraas (1996)►-0.516003788 Matyus & Reiher ECG 2012

0.001 a.u. = 0.027 eV

Ps2 : e-e-e+e+

• 2007: finally seen in experiment► Cassidy & Mills, Nature 449 195 (2007)► 60 years after theoretical prediction

• Open the possibility to study BEC of Ps• Not the end of the story…

Ps2 : e-e-e+e+

• Symmetry of Ps2 must include charge conjugation: e- e+

► problems with early calculations

• 1993: isomorph to D2h group (Kinghorn &

Poshusta)

► 0+(A1) ground state

► 0+(B2), 0+(E) excited states

• 1998: Varga, Usukura, Suzuki ECG► 1-(B2) bound L=1 state E = -0.334408 a.u.

• 2012: L=1 state detected experimentally

Pse+ Ps-

Ps+ Ps2-Ps2

Ps3Ps2+ Ps3

-

e-

Ps3+ Ps4

10 32 4

1

2

3

4

0

e-

e+

Higher systems? Psn, Ps-n

• Ps2-

► Ps2 + e- Ps2-

► Ps + Ps- Ps2-

► L=0 unbound (ECG)► What about L>0 ?

Ps2- and beyond: general

strategy• Problems with Monte Carlo:

► No starting (R) from HF/DFT

• Use a “Valence Bond-like” (R)► (R)=A[(Ps) (Ps-)] or for other fragments

• VMC and DMC unbound. Dissociation► Use a modified potential (preserving

symmetry)

► V(R) = V(e-,e+) + g (V(e-,e-) + V(e+,e+))

► (R)=A[(Ps) (Ps-) (interaction)]

► Consider the limit for g1

Ps2- total energy

• E(L=1) < E(L=0)

0 0.2 0.4 0.6 0.8 1

g

-1 .2

-1

-0.8

-0.6

-0.4

En

erg

y (a

.u.)

Ps+Ps-

Ps2+e-Ps2

- L=1

Ps2- L=0

Ps2- binding energy

• L=0 is unbound• L=1 is probably

unbound…• …but with

better nodes?

0 0.2 0.4 0.6 0.8 1

g

-0 .05

0

0.05

0.1

0.15

0.2

0.25

Bin

din

g e

ner

gy

(a.u

.)

Ps2- L=0

Ps2- L=1

Ps3 and Ps3-

0.2 0.4 0.6 0.8 1

g

-0.2

0

0.2

0.4

0.6

Bin

din

g e

ner

gy

(a.u

.)

Ps3

Ps3-

• Preliminary results

• Ps3 and Ps3-

unbound so far• Not explored yet

all excited states

Z=3 e+ Li

• 1976: Hylleraas CI, e+Li unbound (Clary)

• 1996: DMC, e+Li unbound (Yoshida & Miyako)

• 1997: ECG, e+Li bound (Ryzhikh & Mitroy, Strasburger & Choinacki)

• Li+ + Ps → e+Li BE=0.0025 a.u. (first neutral atom to bind e+)

• 1999: DMC, e+Li bound (Mella, Morosi & Bressanini)

e+Li

Li+2Li+3 LiLi+

10 32 4

1

2

0

e-

e+

Li-

5 6

Z=3 e+ Li

• 1996: DMC, LiPs unbound, BE = -0.011(4) a.u. (Harju, Barbiellini & Nieminen)

• 1997: DMC, LiPs bound, BE = 0.028(5) a.u. (Yoshida & Miyako)• 1998: DMC, LiPs bound, BE = 0.0096(8) (Bressanini, Mella &

Morosi)• 1998: ECG-SVM, LiPs bound BE = 0.01051 (Ryzhikh & Mitroy)

• Li + Ps → LiPs BE=0.01237 a.u.

e+Li

Li+2Li+3 LiLi+

10 32 4

1

2

0

e-

e+

LiPs

Li-

5 6

Z=3 e+ Li

• Li+ + Ps2 → Li+Ps2 • SVM-FC BE = 0.009 a.u. • DMC BE = 0.012 a.u. (preliminary)• +Z=3, 4e- 2e+

e+Li

Li+2Li+3 LiLi+

10 32 4

1

2

0

e-

e+

LiPs

Li+Ps2

Li-

5

Li-Ps

LiPs2 Li-Ps2

6

e+ Z=1

HPs-HPs

HPs2e+HPs

H

10 32 4

1

2

0

e-

e+

H-Ps2

5

H+ H-

e+H

H+Ps3 HPs33 H-Ps3

• +Z=1, 4e- 2e+ H- + Ps2 → H-Ps2 Ps- + HPs → H-Ps2 • ECG-SVM (Varga not converged) BE = 0.004 a.u. • DMC BE = 0.006 a.u. (preliminary)• What about Z=2 HePs2 ?

H -Ps2

• (H-)(Ps2) unbound

• (Ps-)(HPs) unbound

• c+ (1-c)

bound

0 0.2 0.4 0.6 0.8 1c mixing

-1 .060

-1.056

-1.052

-1.048

-1.044

-1.040

E (

a.u

.)

H - + Ps2

HPs + Ps-

H -Ps2

Dissociation

Fishing for positronic compounds

Which atom?

How many e-?

How many e+?

Which state?

e+ Z=2

• He (1s2 1S) does not bind e+

• He (1s2s 3S) binds e+ (very weakly)• He- (1s2s2p 4Po) and He- (2p3 4So) do not

bind e+

e+He

He+Ps

He+

10 32 4

1

2

0

e-

e+

HePs2

5

He+2 He He-

HePs He-Ps

He-Ps2

Excitedstate

LiPs

1s

2s

2p

ePpsLi 32221

1s+

2s+

2p+

oPpe 22

x

oSLiPs 4

UNBOUND

Stable with respect to dissociation into Li(2Po) + Ps(2Po) Ethr = -7.472656532

Preliminary DMC: BE = 0.47 mH, SVM = 0.21 mH, CI-FC = 0.02 mH

Molecules

2 4 6 8 10

R (a.u.)

-8 .2

-8

-7.8

-7.6

-7.4

En

erg

y (a

.u.) LiH

e+LiH

H2 em+ en

-

• H2Ps H2 + Ps H + HPs

• H2Ps- H2 + Ps- H- + HPs

H2Pse+H2

H2Ps+ H2Ps2-

H2+ H2

-

10 32 4

1

2

0

e-

e+

H2Ps-

H2Ps2

5

H2

Unbound

H2Ps-

2 4 6 8 10R (a.u.)

-1.5

-1.4

-1.3

-1.2

-1.1

En

erg

y (a

.u.) H2 + Ps -

H- + HPs

H 2Ps -

Thank youStill a lot of work to do