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QED calculations of H2 molecule
Vladimir A. Yerokhin St. Petersburg State Polytechnical University
in collaboration with

Krzysztof Pachucki Warsaw University

FFK 2013, Pulkovo, St. Petersburg, October 7-11


Do we need QED for H2?


Dissociation energy of H2
Source: Faraday Discuss. 150, 51 (2011)

An order of magnitude in precision every 14 years

QED :

-0.1964 (9)

5, 3039 (2009)

Theory: J. Chem. Theor. Comput.


H2: accurate test of QED in molecules
Measurement of the fundamental ground tone vibration (rotationless vibrational splitting ) in H2, HD, and D2 [Amsterdam, W. Ubachs group, 2013]

Accurate test of QED in chemically bound systems

QED (H2): -0.02145(9) cm-1

100 times larger than the experimental error

Theory: K. Pachucki, J. Komasa 2013


Spectroscopy of H2:bounds on fifth forces

Search for an additional interaction between particles at a given length scale <-> search for deviation from the Newtonian law of gravity. · Gravitational experiments: 10-2 m < < 106 km · Casimir force experiments: 10-7 m < < 10-4 m · Spectroscopy of H2: chemical bond distance length scale
PRL 110, 193601 (2013): H2 constraint G < 6 в 10-8 eV·A for >> R = 0.74 A


QED: from atoms to simple molecules

Atomic hydrogen

Atomic helium

Molecular hydrogen

r1A

r1A r2A r12

r1A r2A r12 r2A r2B R

e-,1

r12

e-,2

r1A p,A

r2A R

r1B

r2B p,B


Energy levels of light molecules: QED theory
Perturbative expansion in two parameters 1/137 is the fine-structure constant = m/M is the electron-to-nucleus mass ratio 10-3 for H
2

"usual" chemistry


Nonrelativistic theory
Schroedinger equation:


Adiabatic approximation

Born-Oppenheimer energy:


Adiabatic approximation

Born-Oppenheimer energy:

Adiabatic correction:


Beyond the adiabatic approximation
[K. Pachucki and J. Komasa, J. Chem. Phys. 130 164113 (2009)]

Radial nonadiabatic Schroedinger equation:

Effective R-dependent nuclear masses:


Effective nuclear masses in H2


Relativistic correction ( 2)
Expectation value of the Breit-Pauli Hamiltonian on the nonrelativistic wave function:


Leading QED correction ( 3)
Expectation values on non-local and singular operators need to be calculated Bethe logarithm:

Regularized 1/r

3 12

operator:


Higher-order QED correction ( 4)

· · ·
·

An open challenge. Very difficult to calculate, known only for H, He and H2+. Involves even more singular operators. Choice of the basis for representation of the wave function is important.


Wave function
Schroedinger equation is solved by variational principle. Wave function is represented by a linear combination of basis functions

Choice of the basis functions:

· ·

Explicitly correlated basis set. Includes explicitly all interparticle distances r 1A r2A r12 r2A r2B Cusp condition fulfilled ?


Explicitly correlated basis sets
Explicitly correlated Gaussians (ECG)

Positive: fast, stable, all integrals are calculated analytically Negative: cusp condition is not fulfilled => slow convergence, thousands of nonlinear parameters to be optimized

Explicitly correlated exponentials

Positive: cusp condition is fulfilled => potentially much more powerful than ECG. The most general two-center two-electron basis. Negative: how to calculate integrals with such functions??? Haven't been used so far due to overwhelming technical difficulties.


Master two-center integral
A problem: integral to be calculated analytically:

Previously: expansions in r, spherical-wave expansions of r12 ...

K. Pachucki, Phys. Rev. A 86, 052514 (2012):

where and i,j are polynomials Integral over t remains to be evaluated numerically.


Other two-center integrals

Integrals with additional positive powers of r's can be calculated from f(r) and its derivatives over r by recurrence relations. Integrals with additional negative powers of r's can be calculated by numerically integrating f(r) over the nonlinear parameters:

Difficulties: numerical instabilities


First calculations with exponential basis
K. Pachucki and V. A. Yerokhin, Phys. Rev. A 87, 062508 (2013):


Conclusion
Theory of H2 molecule reached a very high level of sophistication. Excellent agreement between theory and experiment. Modern experiments require calculations of higher-order QED Corrections in H2.

Towards calculations of m6 corrections: Proof-of-principle calculations with exponential two-center two-electron basis set are reported. Basis set yields a very compact representation of the wave Function, but is difficult to work with.


Theory versus Experiment: current status
Dissociation energy in H2 and D2

Theory: Pachucki, Komasa J. Chem. Phys. 130, 164113 (2009)