Accurate analytic potentials for BeH, BeD & BeT Nike Dattani & Staszek Welsh Oxford University & Kyoto University　　 2014年 6月 20日

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 Used 1000 digits for maple calculation

MLR (Morse / Long-Range) Potential It’s a Morse potential, but with the correct long-range built in !!!

MLR (Morse / Long-Range) Potential 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 … It’s a Morse potential, but we can make the long-range part correct !!!

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 1.383192174455(1) 13 digits ! relativistic corrections -80.35(2) QED 3rd order modulo Bethe ln QED 3rd order with Bethe ln QED 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  : H  2e- : He : H2 3e- : Li 2e- : HeH+ In Progress Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

1e- : Mu  : H  2e- : He : H2 3e- : Li 5e- : BeH Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

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) Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

5e- : BeH    Next step!  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! Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

1e- : Mu 2e- : He : H2 3e- : Li 5e- : BeH in progress 5e- : LiHe Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

Experimental energy gap is hyperfine structure according to table 4 of: http://arxiv.org/pdf/hep-ph/0509010.pdf He (from Table VIII of http://www.fuw.edu.pl/~krp/papers/hfs_theory_v2.pdf): Experimental: 6 739 701.177(16) Theoretical: 6 739 699.93 Fourth order means: alpha^4

Thank you  VERY MUCH !!!