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