Schr\"odinger equation solved for the hydrogen molecule with unprecedented accuracy

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by  Krzysztof Pachucki

Schro¨dinger equation solved for the hydrogen molecule with unprecedented accuracy
Krzysztof Pachucki∗
Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
Jacek Komasa†
Faculty of Chemistry, Adam Mickiewicz University, Umultowska 89b, 61-614 Poznan´, Poland
(Dated: January 15, 2016)
The hydrogen molecule can be used for determination of physical constants and for improved
tests of the hypothetical long range force between hadrons, which requires a sufficiently accurate
knowledge of the molecular levels. For this reason, we have undertaken a project of significant
improvements in theoretical predictions of H and perform the first step, which is the solution of
2
the nonrelativistic Schro¨dinger equation to the unprecedented accuracy of 10−12. This will inspire,
we hope, a parallel progress in the spectroscopy of the molecular hydrogen.
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1 INTRODUCTION [8], James and Coolidge [9], Kol(cid:32)os and Wolniewicz [10],
0 andmanyothershavemadetheirmarksonthehistoryof
2 research on H . Every breakthrough in the precision of
2
n The spectroscopy of simple atomic systems like hy- theoreticalpredictionshasbeenrelatedtotheprogressin
drogen [1, 2], hydrogenic ions [3], muonic hydrogen [4],
a computational techniques. Fig. 1 illustrates the progress
J muonium, positronium has been used to determine fun- made over many decades in the precision of the dissocia-
4 damental physical constants and to test the quantum tion energy D for H .
0 2
1 electrodynamic theory. Although experiments for more
complicated atomic systems like helium or lithium can

h] be as accurate as for hydrogen, the precision of theo- 104
p retical predictions, at the moment, is not sufficient to 103
- determine physical constants, such as the fine structure 102
m
constant α, the Rydberg constant (Ry), or the absolute -1m 101 [2]
he valueofthenuclearchargeradius. Incontrast,thehydro- /c0 100
c gen molecule, thanks to its simplicity, has already been δD 10-1 [4]
[6]
s. usedforthemostaccuratedeterminationofthedeuteron 10-2 [8]
c magnetic moment from NMR measurements [5], and for
i thestudiesoftheunknownhypotheticalfifthforceatthe 10-3 [10]
s 10-4
y atomicscale[6]. Theprecisionachievedrecentlyfortran- 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010
h sition frequencies, of an order of 10−4cm−1, has been
Year
p
verified by a series of measurements, which resulted in
[
strong bounds on the fifth force. We will argue in this FIG. 1. The accuracy of theoretical predictions of dissocia-
1 work, that it is possible to achieve 10−6cm−1 accuracy tion energy D of H versus time, with the linear fit on the
0 2
v for energy levels of the hydrogen molecule, which not logarithmic scale.
5
only will improve tests of quantum electrodynamics the-
8
ory and will put stronger bounds on the fifth force, but
6 Not always the results of calculations have been in
3 also will allow a resolution of the proton charge radius agreement with measured values, which has questioned
0 puzzle,whichstandsasaviolationoftheStandardModel
the validity of the theoretical approach. For example,
1. of fundamental interactions [7]. in 1964 Kol(cid:32)os and Wolniewicz [11] solved variationally
0
Theimprovementintheoreticalpredictionsforthehy- the nonadiabatic Schr¨odinger equation for the hydrogen
6
1 drogen molecule can be achieved by the calculation of molecule. Thecalculateddissociationenergyappearedto
: the yet unknown higher order α4Ry quantum electrody- be higher than the measured one [12], which was in con-
v
namicscorrectionandbyamoreaccuratesolutionofthe tradictionwiththevariationalprinciple. Fiveyearslater,
i
X nonrelativistic Schr¨odinger equation. This second im- HerzbergmeasuredD0 againwiththeaccuracyincreased
r provement is performed in this work, while the calcula- to a few tenths of reciprocal centimeter and obtained a
a tion of the QED effects is in progress. The solution of valuehigherthanpreviouslyby5cm−1inagreementwith
the Schr¨odinger equation for the hydrogen molecule has the Kol(cid:32)os and Wolniewicz predictions.
beenpursuedsincealmostthebeginningofthequantum Further progress in theoretical predictions was related
mechanicstheory. Overtime,therehavebeenmanycon- to the calculations of the leading relativistic corrections
tributionstothedevelopmentofmethodswithincreased and to approximate QED effects [13–17]. Later on, due
precision of theoretical predictions. Heitler and London to rapid development of computer power, the methods
2
basedonexponentiallycorrelatedGaussian(ECG)func- the recursions, can be expressed in the following form
tionshavebeendevelopedbothintheBorn-Oppenheimer
approximation [18] and in the direct nonadiabatic ap- (cid:90) dV e−trABe−u(r1A+r1B)e−w(r2A+r2B)
proach [19, 20]. Very recently, we have introduced a (4π)3 r r r r r r
AB 12 1A 1B 2A 2B
nonadiabatic perturbation theory (NAPT), which al- (cid:16) (cid:17) (cid:16) (cid:17)
(cid:34) ln 2u ln 2w
lowed the accuracy of about 10−3cm−1 to be achieved 1 t+u+w t+u+w
= − −
foralltherovibrationallevelsofH andisotopomers[18]. 4uw t−u+w t+u−w
2
However, the complexity of NAPT in the higher order of (cid:16) (cid:17) (cid:16) (cid:17)
ln 2uw ln 2(u+w) (cid:35)
electron-nucleus mass ratio [21] makes further improve- (u+w)(t+u+w) t+u+w
+ + . (4)
ments in accuracy quite complicated. For this reason, t+u+w t−u−w
we propose another approach to the direct solution of
the Schr¨odinger equation by the use of the special in- Theintegralswithadditionalpositivepowersofinterpar-
tegration technique for explicitly correlated exponential ticle distances are obtained by straightforward algebraic
functionsandpresentitsfirstresultsinthisLetter. Over recursionrelations,whichneverthelessaretwolongtobe
fifty years after Kol(cid:32)os and Wolniewicz, we approach the written explicitly here. All of them can be derived from
problem of solving the four-body Schr¨odinger equation, a single fourth-order differential equation [25], which is
with the precision aim for D0 at the level of 10−7cm−1. satisfied by the general four-body integral, and are ex-
pressed in terms of logarithmic and rational functions.
Details of this recursion method will be described else-
THEORY where.
The main purpose of this work is to solve accurately
the stationary Schr¨odinger equation HˆΨ=EΨ for a di- NUMERICAL RESULTS
atomic molecule with the nuclei of charge Z and Z
A B
and finite masses M and M Results of our calculations for H are presented in
A B 2
Tab. I in the form a sequence of energies resulting from
1 1 1 1
Hˆ =− ∇2 − ∇2 − ∇2− ∇2 increasing length (K) of expansion (2). The selection of
2MA A 2MB B 2me 1 2me 2 K was made on the basis of the saturation of consecu-
+ ZAZB + 1 − ZA − ZA − ZB − ZB , (1) tive ’shells’ limited by (cid:80)5k=1ki ≤ Ω with k0 fixed at 30.
r r r r r r The observed regular convergence, obeying the inverse
AB 12 1A 2A 1B 2B
power low, permits a firm extrapolation to the complete
using the variational approach. The trial wave function
basissetaswellasanestimationoftheuncertainty. The
final value agrees well with the previous estimation of
K
Ψ((cid:126)r ,(cid:126)r ,R(cid:126) ,R(cid:126) )=(cid:88)c Sˆψ ((cid:126)r ,(cid:126)r ,R(cid:126) ,R(cid:126) ) (2) −1.16402503084(6) a.u. obtained by Bubin et al. [20]
1 2 A B k {k} 1 2 A B
but has a significantly smaller uncertainty.
k=1
is expanded in properly symmetrized (Sˆ), four-particle
TABLE I. Convergence of the Schr¨odinger equation eigen-
basis of exponential functions
valueE(ina.u.) andofthecorrespondingdissociationenergy
D (in cm−1) for H with the size of the basis set.
0 2
ψ =exp[−αr −β(ζ +ζ )]rk0 rk1ηk2ηk3ζk4ζk5,
{k} AB 1 2 AB 12 1 2 1 2
Ω K E D
(3) 0
10 36642 −1.1640250308214 36118.79773257
where ζ =r +r and η =r −r are coordinates 11 53599 −1.1640250308709 36118.79774343
i iA iB i iA iB
12 76601 −1.1640250308804 36118.79774552
closely related to the prolate spheroidal coordinates of
13 106764 −1.1640250308825 36118.79774597
i-th electron, r are interparticle distances, α and β are
ij ∞ ∞ −1.164025030884(1) 36118.7977463(2)
nonlinearvariationalparameters,andk arenon-negative
i
integers collectively denoted as {k}. For its resemblance
to the electronic James-Coolidge function, we call this Further increase in the accuracy of eigenvalue of the
basis function the nonadiabatic James-Coolidge (naJC) four-bodySchr¨odingerequationisfeasible,buttheprob-
function. lem we face is the lack of the parallel code in multi-
Application of the naJC function for evaluation of the precision arithmetics for the LDLT matrix decomposi-
matrix elements leads to a certain class of integrals. Ef- tion with pivoting, which results in a long computation
ficient evaluation of these integrals has become feasible time. However, current uncertainties in the electron-
since the discovery of the analytic formulas [22, 23] and proton (proton-deuteron) mass ratio and in the Rydberg
the corresponding recursive relations [24, 25]. For exam- constantaremuchmoresignificantthanthoseduetonu-
ple, the master integral, which is the starting point for merical uncertainties. For example, the CODATA 2014
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[26]electron-protonmassratiohasarelativeuncertainty ACKNOWLEDGMENT
of 9.5·10−11, which affects the eigenvalue of H at the
2
level of 4.3·10−12 a.u. and the corresponding dissoci-
ation energy at 8.5·10−7cm−1. Similarly, the current
This research was supported by the National Sci-
uncertainty in the Rydberg constant affects the conver-
enceCenter(Poland)GrantsNo. 2012/04/A/ST2/00105
sionofD valuefroma.u. toreciprocalcentimetersatthe
0
level of 2.1·10−7cm−1. This indicates, that one cannot (K.P.)and2014/13/B/ST4/04598(J.K.),aswellasbya
computing grant from the Poznan Supercomputing and
exclude the possibility, to determine the electron-proton
Networking Center, and by PL-Grid Infrastructure.
mass ratio from future high precision studies of H .
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∗ [email protected]
† [email protected]
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