Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　

2014 年 6 月 20 日　　

Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　Oxford

University

2014 年 6 月 20 日　　

Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　Oxford

University

2014 年 6 月 20 日　　

Adam Mickiewicz University

Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　Oxford

University

2014 年 6 月 20 日　　

Adam Mickiewicz University

University of Arizona

Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　Oxford

University

2014 年 6 月 20 日　　

Adam Mickiewicz UniversityIIMCB

University of Arizona

Accurate analytic potentials for HeH+, HeD+, HeT+,

including finite-mass, relativistic and 4th order QED

Staszek Welsh, Mariusz Puchalski, Grzegorz Lach, Wei-Cheng Tung, Ludwik Adamowicz,

Nike Dattani

　　Oxford

University京都大学（ Kyoto University)

2014 年 6 月 20 日　　

Adam Mickiewicz UniversityIIMCB

University of Arizona

Please guess !

At what number of electrons, do you think agreement between experiment and theory collapses?

1e- : HHyperfine structure142040575768(1) mHz (present best experiment)1420452 (theory – QED)

What’s missing is the effect of the nuclear structure

1e- : Mu (p+ in H is replaced by μ+) Hyperfine structure4463302780(50) Hz (experiment)4463302880(550) Hz (theory – QED)

2e- : HeHyperfine structure6739701177(16) Hz (experiment)6739699930(1700) Hz (theory, QED + nuclear structure)

Agreement possible because Hz precision, not mHz

2e- : H21975: Kolos & Wolniewicz (numerical soln to Schroedinger Eqn)

More recently:

Ev = 1 – Ev = 0 4161.16632(18) cm-1 ( experiment )4161.16612(9) ( best theory )

3e- : Li

2S

2P

3e- : Li

Experiment: 14903.632061014 +/- 0.0000005003 cm-1

Theory: 14903.631765 +/- 0.000667 cm-1

Experiment:

Theory:

3e- : Li

Energy (for lowest transition)

Radiative lifetime ?V(r) = - C3 / r3 – C6 / r6 – C8 / r8 …

Radiative lifetime : τ = ( 3ħ / 2C3 ) ( λ / 2 π )3

Oldest experimental value ? Guess !

1931 Loomis F.W. and Nusbaum R.E. Phys. Rev. 38 pg. 1447

1931 Loomis F.W. and Nusbaum R.E. Phys. Rev. 38 pg. 1447

University of Illinois Urbana-Champaign physics department:“Loomis Laboratory of Physics”

Loomis was challenged in bringing top-notch physics talent to a university in the rural Midwest. When he approached Isaac Rabi, Rabi said "I love subways and I hate cows." 

While building the department, Loomis attracted John Bardeen (2 Nobel prizes) to join the staff, and had Polykarp Kusch (1 Nobel Prize) as his graduate student.

Year Name Nobel Prize

1923 Du Vigneaud Nobel Prize in Chemistry

1929 Stanley Nobel Prize in Chemistry

1933 Kusch Nobel Prize in Physics

1947 Kilby Nobel Prize in Physics

1957 Schrieffer Nobel Prize in Physics

1969 Sharp Nobel Prize in Chemistry

???? Ben McCall

1931 Loomis F.W. and Nusbaum R.E. Phys. Rev. 38 pg. 1447

More recently: (Le Roy & Dattani)2009: C3 = 357829(8) “most accurate C3 value for any molecule ever determined,by an order of magnitude” “landmark in diatomic spectral analysis” (2011 Mitroy et al.)Theory:2009: C3 = 357810.89(7) (finite-mass corrections)2010: C3 = 357773 (relativistic corrections)2011: C3 = 357773 (third order perturbation theory)

Experiment:2011: C3 = 357557(78)2013: C3 = 357682.8(44)2013: C3 = 357835.2

1e- : Mu : H2e- : He : H2

3e- : Li2e- : HeH+

Li2

V(r) = - C3 / r3 – C6 / r6 – C8 / r8 …

Radiative lifetime of Li (2p) : τ = ( 3ħ / 2C3 ) ( λ / 2 π )3

HeH+

V(r) = - C4 / r3 – C6 / r6 – C7 / r7 …

Dipole polarizability of He : α = 2C4

Change in SI definitions

New definition of kB , more rigorous temperature scale

Current SI units: SI units will soon change:25th General Conference on Weights and Measures (18-20 November 2014)

Redefining temperature

New definition of kB , more rigorous temperature scale

Dipole polarizability () for He atom 1.383759(13) (experiment)1.38376079(23) (theory)

𝑘 𝐵=α 𝑁 𝐴23𝑝 ε 0

(Є 𝑟+2)(Є 𝑟 −1)

pressure(held fixed) vaccuum

permitivity(defined)

refractive index(measured accurately)

Avagadro constant (known accurately)

Li2

V(r) = - C3 / r3 – C6 / r6 – C8 / r8 …

Radiative lifetime of Li (2p) : τ = ( 3ħ / 2C3 ) ( λ / 2 π )3

HeH+

V(r) = - C4 / r3 – C6 / r6 – C7 / r7 …

Dipole polarizability of He : α = 2C4

Part II

Best ab initio for Li2 (6e-)

Recent experiments needed +/- 0.01 cm-1 predictions

Experiment would take several years, need better than ab initio

Alternative to ab initio : Empirical potential (MLR)

Using very little data, All energies can be predicted

very accurately

Experiment successful BECAUSE,MLR’s predicted energies were much better than ab initio

MLR (Morse / Long-Range) Potential

It’s a Morse potential,but with the correct long-range built in !!!

MLR (Morse / Long-Range) Potential

It’s a Morse potential,but we can make the long-range part correct !!!

for large r, we should have for HeH+:

V(r) = De – C4 / r4 – C6 / r6 – C7 / r7 – C8 / r8 …

So

u(r) = C4 / r4 + C6 / r6 + C7 / r7 + C8 / r8 …

V(r) = De – C4 / r4 – C6 / r6 – C7 / r7 – C8 / r8 …

C4 : dipole polarizability C6 : quadrupole polarizability, non-adiabatic dipole polarizabilityC7 : mixed dipole-dipole-quadrupole polarizability (3rd order)C8 : hyperpolarizability (4th order), octupole polarizability,

& non-adiabatic quadrupole polarizability

for large r, we should have:

V(r) = De – C4 / r4 – C6 / r6 – C7 / r7 – C8 / r8 …

C4 : dipole polarizabilitynon-relativistic 1.383192174455(1) 13 digits !relativistic corrections -80.35(2)QED 3rd order modulo Bethe lnQED 3rd order with Bethe lnQED 4th order, finite-mass 3rd order

30.473(1) 0.193(2) 0.49(23)

total dipole polarizability 1383760.79(23)

for large r, we should have:

V(r) = De – C4 / r4 – C6 / r6 – C7 / r7 – C8 / r8 …

C6 : quadrupole polarizability

non-relativistic 2.44508310433(5) 12 digits !!!relativistic corrections -1.750786(2) x 10-4

finite-mass corrections 1.8749483(3) x 10-3

total quadrupole polarizability

2.4467829742(4)

1e- : Mu : H2e- : He : H2

3e- : Li2e- : HeH+

In Progress

1e- : Mu : H2e- : He : H2

3e- : Li5e- : BeH

5e- : BeH

Most accurate empirical potential:

2006 Le Roy et al. JMS 236, 178-188

C6, C8, C10 not included couldn’t determine leading BOB term (u0 ) De had uncertainty of +/- 200cm-1

single-state fit (excited states not included)

V(r) = - C6 / r3 – C8 / r6 – C10 / r8 …

5e- : BeH C6, C8, C10 not included couldn’t determine leading BOB term (u0 ) De had uncertainty of +/- 200cm-1

single-state fit (excited states not included)

Next step!

1e- : Mu 2e- : He : H2

3e- : Li5e- : BeH5e- : LiHe

in progress

Thank you VERY MUCH !!!