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
Adam
Mickiewicz
University
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
Adam
Mickiewicz
University
University of
Arizona
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
Adam
Mickiewicz
University
IIMCB
University of
Arizona
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
Adam
Mickiewicz
University
IIMCB
University of
Arizona
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
At what number of electrons,
do you think agreement between
experiment and theory collapses?
1e
: H
Hyperfine structure
142040575768(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 structure
4463302780(50) Hz (experiment)
4463302880(550) Hz (theory – QED)
2e
: He
Hyperfine structure
6739701177(16) Hz (experiment)
6739699930(1700) Hz (theory, QED + nuclear structure)
Agreement possible because Hz precision, not mHz
2e
: H2
1975: 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
2P
2S
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
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
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 
:H

2e : He 
: H2 
3e : Li 
+
2e : 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
Current SI units:
SI units will soon change:
25th General Conference on Weights and
Measures (18-20 November 2014)
New definition of kB , more rigorous temperature scale
Dipole polarizability (α ) for He atom
1.383759(13)
(experiment)
1.38376079(23) (theory)
Avagadro constant (known accurately)
𝑁𝐴2 (Є𝑟 + 2)
𝑘𝐵 = α
3𝑝ε0 (Є𝑟 − 1)
pressure
(held fixed)
vaccuum
permitivity
(defined)
refractive index
(measured
accurately)
New definition of kB , more rigorous temperature scale
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
Recent experiments needed +/- 0.01 cm-1 predictions 
Experiment would take several years, need better than ab initio
Experiment successful BECAUSE,
MLR’s predicted energies were much better than ab initio
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 polarizability
C7 : 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 polarizability
non-relativistic
relativistic corrections
QED 3rd order modulo Bethe ln
QED 3rd order with Bethe ln
QED 4th order, finite-mass 3rd
order
total dipole polarizability
1.383192174455(1)
-80.35(2)
30.473(1)
0.193(2)
0.49(23)
1383760.79(23)
13 digits !
for large r, we should have:
V(r) = De – C4 / r4 – C6 / r6 – C7 / r7 – C8 / r8 …
C6 : quadrupole polarizability
non-relativistic
relativistic corrections
finite-mass corrections
2.44508310433(5) 12 digits !!!
-1.750786(2) x 10-4
1.8749483(3) x 10-3
total quadrupole
polarizability
2.4467829742(4)
1e
: Mu 
:H

2e : He 
: H2 
3e : Li 
+
2e : HeH In Progress
1e
: Mu
:H
2e : He
: H2
3e : Li
5e : BeH





5e
: BeH
V(r) = - C6 / r3 – C8 / r6 – C10 / r8 …
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)
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
2e
3e
5e
5e
: Mu
: He
: H2
: Li
: BeH in progress
: LiHe