Accurate S-state helium wave functions in momentum space

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by  J. Sapirstein

Version 3.0
Accurate S-state helium wave functions in momentum space
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J. Sapirstein
Department of Physics, University of Notre Dame, Notre Dame, IN 46556
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Abstract

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High accuracy helium wave functions based on exponentials with random coefficients are trans-
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formed into momentum space. The utility of the wave functions is demonstrated through cal-
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culation of the expectation value of various operators needed to evaluate relativistic and QED
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[email protected]
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I. INTRODUCTION
Inrecent yearsabasissetconsistingofrandomexponentialshasbeenusedwithincreasing
frequency to carry out calculations in helium [1], [2], [3]. For singlet and triplet S-states it
can be written as
φ(~r ,~r ) = C [e−αir1−βir2−γir12 e−αir2−βir1−γir12], (1)
1 2 i
X ±
i
where α , β , and γ are parameters that are chosen randomly in certain ranges, and the
i i i
spin wavefunctions are understood. Through careful choice of those ranges Korobov [1] has
been able to obtain a ground state energy of
E = 2.903 724 377 034 119 598 311 159 a.u.. (2)
−
Whileeven higher accuracies arepossible with basissets thatincorporateknown nonanalytic
behaviors of the wave function [4], the simple form of the above wave function makes the
evaluationofhigherordercorrectionscomingfromrelativisticandQEDcorrectionsrelatively
straightforward. Inthisnotewewillpresent calculationsbasedonthemomentumspaceform
of Eq. 1, where the wave function is defined through
d3r d3r
φ(p~ ,~p ) = 1 2e−ip~1·~r1e−ip~2·~r2φ(~r ,~r ). (3)
1 2 Z (2π)3 1 2
The Fourier transform is evaluated by first noting that
∂3 e−αr1−βr2−γr12
e−αr1−βr2−γr12 =
−∂α∂β∂γ r r r
1 2 12
∂3 d3q eiq~1·~r1 d3q eiq~·~r12 d3q eiq~2·~r2
1 2
= . (4)
−∂α∂β∂γ Z 2π2 ~q 2 +α2 Z 2π2~q2 +γ2 Z 2π2 ~q 2 +β2
1 2
To simplify the following discussion, we Fourier transform only the above expression, and
define it as φ (p~ ,p~ ), with the generalization to Eq. 1 being clear. We see that
i 1 2
1 ∂3 1 1 1
φ (p~ ,~p ) = d3q . (5)
i 1 2 −π3∂α∂β∂γ Z (p~ ~q)2 +α2~q2 +γ2(p~ +~q)2 +β2
1 2
−
The integral over d3q has been carried out analytically in Ref. [5], and leads to
2 ∂3 θ
φ (p~ ,~p ) = , (6)
i 1 2
−π∂α∂β∂γ √x
2
where
y = γ[ p~ +p~ 2 +(α+β)2]+β(γ2 +α2 +p~ 2)+α(γ2 +β2 +p~ 2)
2 1 1 2
| |
x = [ p~ +p~ 2 +(α+β)2][p~ 2 +(α+γ)2][p~ 2 +(β +γ)2] y2
2 1 1 2
| | −
√x
θ = arctan( ). (7)
y
While this equation for φ (p~ ,~p ) seems quite compact, the action of the three derivatives
i 1 2
leads to a considerably more complicated expression. It simplifies if we introduce the auxil-
iary parameters
dx dx dx
X = ,X = ,X =
1 2 3
dα dβ dγ
dy dy dy
Y = ,Y = ,Y =
1 2 3
dα dβ dγ
d2x d2x d2x
X = ,X = ,X =
23 12 13
dβdγ dαdβ dαdγ
T = X X +X X +X X ,T = X +X +X ,T = Y X +Y X +Y X
1 1 23 2 13 3 12 2 1 2 3 3 1 23 2 13 3 12
T = Y +Y +Y ,T = X Y Y +X Y Y +X Y Y
4 1 2 3 5 1 2 3 2 1 3 3 1 2
T = Y X X +Y X X +Y X X . (8)
6 1 2 3 2 1 3 3 1 2
In terms of these we find
′′
2 3θ y (T +2yT ) T
2 4 3
φ (p~ ,~p ) = [ [2xT 5X X X ]+ +
i 1 2 −π 8x7/2 1 − 1 2 3 D2 D2
4yT 2(x 3y2)Y Y Y (40xy3 +33x2y +15y5)X X X
5 1 2 3 1 2 3
+ − +
− D3 D3 8x3D3
yT (3y2+5x) 2T 2
1 1
], (9)
− 4x2D2 − D3 − D
where D = x+y2 and y′′ = 2(α+β+γ). We note the following symmetry properties of the
momentum space wave function. Because the basic functions that form the wave function,
x and y, are invariant under the simultaneous replacement ~p ~p and α β, for singlet
1 2
↔ ↔
states φ(p~ ,~p ) = φ(p~ ,~p ) and for triplet states φ(p~ ,~p ) = φ(p~ ,p~ ). Both symmetries
1 2 2 1 1 2 2 1
−
also hold when the magnitudes of p~ and ~p are switched.
1 2
There are various uses for the wave function of helium in momentum space, notably
application to scattering calculations. While most work has been carried out with Fourier
transforms of Hartree-Fock wave functions, an approach that includes correlation more com-
pletely is given in Ref. [6]. In that work, a Hylleraas basis set was Fourier transformed and
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applied to the calculation of a number of helium properties, notably the Compton profile.
While the wave function used here gives a more accurate energy (odd powers of r were not
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included in the basis set of Ref. [6]), we do not find appreciably different answers for any
of the quantities calculated there. In particular, we find the same small difference between
using fully correlated wave functions and Hartree-Fock wave functions for the Compton
profile. Rather than pursuing this line of research we instead now discuss applications of
the momentum space wave function to the calculation of higher order relativistic and QED
corrections to the energies of the 11S , 21S , and 23S states of neutral helium.
0 0 1
These corrections are obtained from a set of operators O , with associated energy shifts
i
E = φ O φ . The first set of operators was derived by Breit [7], and describe corrections of
i i
h | | i
order α2 a.u.. The equation he used in the derivation has certain difficulties connected with
negative energy states [8], but later treatments using the Bethe-Salpeter equation [9], [10],
[11] treat negative energy states consistently and allow the systematic treatment of higher
order corrections.
Most recent calculations, however, use the technique of effective field theory [12] to derive
the operators. We note in particular the compact rederivation of the Douglas-Kroll [11]
results for contributions to the fine structure of helium in order α4 a.u. of Ref. [13], and the
derivation of contributions to the energy of the ground state to the same order in Ref. [14].
The idea of effective field theory is to compare free-particle scattering amplitudes in QED
to an effective nonrelativistic theory, with operators added to the Schr¨odinger Hamiltonian
that account perturbatively for the difference of the amplitudes. Once the operators have
been determined from considering free-particle scattering, they are used as perturbations in
standard bound-state Rayleigh-Schr¨odinger perturbation theory to calculate energy shifts.
In this approach it is natural to work in momentum space, and the operators are then
Fourier transformed to coordinate space for numerical evaluation. Here, however, because
we have formed wave functions in momentum space, we avoid this step, and work exclusively
in momentum space. This has the advantage of simplicity, but the disadvantage of being
less accurate than coordinate space. We now give a brief rederivation of the Breit operators
using effective field theory, and illustrate their numerical evaluation in momentum space.
The connection of the scattering amplitude of two electrons with momentum p , p to
1 2
4
scatter into states with momentum p , p to an energy shift is
3 4
1
E = d3p d3p d3p d3p φ∗(p~ ,~p )O (p~ ,~p ;~p ,~p )φ(p~ ,~p ). (10)
i (2π)6 Z 1 2 3 4 3 4 i 3 4 1 2 1 2
We work in the center of mass frame, with initial nuclear momentum p~ p~ , final nuclear
1 2
− −
momentum p~ p~ . If only electron-electron interactions (ee) are considered, momentum
3 4
− −
conservation allows us to write
O ee(p~ ,~p ;~p ,~p ) = (2π)3δ3(p~ +p~ p~ p~ )M ee(p~ ;~p ,p~ ) (11)
i 3 4 1 2 3 4 1 2 i 3 1 2
− −
with the associated energy shift
1
E ee = d3p d3p d3p φ∗(p~ ,~p +p~ ~p )M ee(p~ ;p~ ,p~ )φ(p~ ,~p ). (12)
i (2π)3 Z 1 2 3 3 1 2 − 3 i 3 1 2 1 2
Ifinsteadweconsider diagramsinwhichoneoftheelectrons, takentobeelectron1,interacts
with the nucleus (eN), and electron 2 does not participate, we can write
O eN(p~ ,~p ;p~ ,p~ ) = (2π)3δ3(p~ p~ )M eN(p~ ,p~ ) (13)
i 3 4 1 2 4 2 i 3 1
−
with the energy shift
1
E eN = d3p d3p d3p φ∗(p~ ,~p )M eN(p~ ,p~ )φ(p~ ,~p ). (14)
i (2π)3 Z 1 2 3 3 2 i 3 1 1 2
Diagrams with all three particles interacting have no delta functions, and have to be evalu-
atedwiththe12-dimensionalintegralofEqn. 10. Inallcaseswenotethat3oftheintegration
variables can be carried out trivially, and that the adaptive Monte-Carlo program VEGAS
[15] can be used to numerically evaluate the integrals, though with far less accuracy than
available from coordinate space techniques. This loss of accuracy is due entirely to the fact
that the multidimensional integrals have to be carried out numerically: the wave functions
themselves are quite accurate. In the calculations presented here we use 600 basis functions,
and the energy eigenvalues are accurate to more than 14 digits for the ground state and 11
for the excited S states.
We begin by evaluating the relativistic mass increase (RMI) operator, which we treat as
an ee diagram. The contribution to the scattering amplitude of this operator is
p 4 +p 4
M ee(p~ ;~p ,~p ) = (2π)3 1 2 δ(p~ p~ ), (15)
RMI 3 1 2 − 8m3 3 − 1
5
which gives the energy shift
1
E ee = d3p d3p φ∗(p~ ,p~ )(p4 +p4)φ(p~ ,~p ). (16)
RMI −8m3 Z 1 2 1 2 1 2 1 2
The result is tabulated in the first row of Table 1.
We next turn to corrections to Coulomb scattering between the electrons. In this case
the nonrelativistic scattering operator is
4πα
M ee(p~ ;~p ,~p ) = . (17)
C 3 1 2 p~ ~p 2
3 1
| − |
This corresponds to the coordinate space potential α/ ~r ~r . To calculate relativistic
1 2
| − |
corrections, we use Dirac spinors to describe scattering and work in Coulomb gauge. We
introduce the notation DC to refer to the scattering amplitude with exchange of a Coulomb
photon, and DT for the scattering amplitude with exchange of a transverse photon. The
DC scattering amplitude can then be Taylor expanded in powers of p/m, with the leading
corrections given by
p~ p~ 2 p~ p~ 2
M ee(p~ ;~p ,~p ) = M ee(p~ ;~p ,~p )[1 | 3 − 1| | 4 − 2| +
DC 3 1 2 C 3 1 2
− 8m2 − 8m2
i~σ (p~ p~ ) i~σ (p~ ~p )
1 3 1 2 4 2
· × + · × ], (18)
4m2 4m2
with the understanding that ~p = ~p + ~p ~p . The first two correction terms are Darwin
4 1 2 3
−
terms, and sum to πα/m2δ3(~r ~r ) in coordinate space. The last two are spin-orbit op-
2 1
−
erators, which do not contribute to the S-states considered here. We tabulate the Darwin
terms in the second row of Table 1.
Considering now Coulomb scattering between an electron and the nucleus, which we take
to have charge Z although only Z = 2 will be considered here, the nonrelativistic limit is
given by
4παZ
M eN(p~ ,~p ) = (19)
C 3 1
− p~ p~ 2
3 1
| − |
with associated energy shift
1 4παZ
E eN = d3p d3p d3p φ∗(p~ ,~p ) φ(p~ ,~p ). (20)
C −(2π)3 Z 1 2 3 3 2 p~ p~ 2 1 2
3 1
| − |
Relativistic corrections are now obtained by introducing a Dirac spinor only for the electron
(thenucleus is treatedherein theinfinite masslimit), andwe find forexchange of aCoulomb
photon
p~ p~ 2 i~σ (p~ p~ )
M eN(p~ ,~p ) = M eN(p~ ,~p )[1 | 3 − 1| + 1 · 3 × 1 ] (21)
DC 3 1 C 3 1 − 8m2 4m2
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Again only the Darwin term contribute for S-states, but now corresponds to πZα/m2δ3(~r ),
1
which we tabulate in the third row of Table 1.
The effect of transverse photon exchange between the electrons is simplified if we neglect
retardation, which enters in order α3 a.u., and in this approximation we have
πα πα πα
M ee(p~ ;p~ ,~p ) = (p~ +p~ ) (p~ +p~ ) (~σ ~σ ~σ qˆ~σ qˆ)
DT 3 1 2 q4 − m2q2 1 3 · 2 4 − m2 1 · 2 − 1 · 2 ·
2πiα 2πiα
~σ (p~ ~p )+ ~σ (p~ p~ ), (22)
−m2q2 1 · 2 × 4 m2q2 2 · 3 × 1
where q = p~ p~ . The first two terms, referred to as orbit-orbit terms, or as E(2), are
3 1
| − |
usually evaluatedbyFouriertransforminginto coordinatespace. Withthepresent approach,
however, they are quite easily treated, and the result presented in the fourth row of Table
1. The third term is another delta function, and the last terms again vanish for the S-state
considered here. The effect of transverse photon exchange for eN scattering vanishes in the
infinite nuclear mass limit used in this work.
Theoperatorsconsidered so farinthisnotehave been studied formanydecades, andhave
all been evaluated with much higher accuracy than presented here [16]. The utility of the
present approach lies in the fact that operators that enter in higher order, generally derived
in momentum space, are both fairly complicated when Fourier transformed to coordinate
space, and in addition need to be evaluated with less accuracy than the terms treated above.
We illustrate this point with relativistic corrections that contribute in order mα(Zα)6 to
S-states. A complete set of operators for triplet states has been derived by Pachucki [17]
usinganeffective fieldtheoryapproach, andweconsider herethecorrectionstoone-Coulomb
photon exchange, Eq. 20 of that paper,
4πα 1 4
V = [q4 (p~ ~p ) (p~ p~ )(~σ ~σ )+
1 q2 64m4 − 3 3 × 1 · 4 × 2 1 · 2
5
[(p2 p2)2 +(p2 p2)2]+3q2(p2 +p2 +p2 +p2)]. (23)
2 4 − 2 3 − 1 1 3 2 4
This expression corresponds to the next term in the p/m expansion in Eq. 18. While Ref.
[17] treats triplet states, this particular result is also valid for the singlet case. We note
that the last term vanishes for triplet states because of the symmetry φ(p~ ,~p ) = φ(p~ ,p~ )
1 2 2 1
−
mentioned earlier, as also noted in Ref. [17]. The resulting energy shift of the 23S state is
1
Eee = 0.0062(2)mα6. (24)
−
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While again not of high accuracy, we note the extremely simple nature of the coding, which
is almost identical to the program that evaluates the Darwin term. This contrasts with the
more complicated coordinate space calculation, where numerous derivatives must be applied
towavefunctions,leadingtoamuchlengthierexpression. Aswiththemα4 corrections, much
higher accuracy is available from working in coordinate space, with -0.006 344 7 mα6 the
known result [18]. However, we note the momentum space accuracy corresponds to 3.7 kHz,
to be compared with the experimental accuracy [19] of 60 kHz.
While theformula forV given above isvalid forsinglet states, it gives a linearly divergent
1
result in that case. It is quite simple, however, to regulate this divergence in momentum
space, where onesimply imposes the cutoff p~ < Λ. An applicationof this momentum space
i
| |
regulator to the case of ground state positronium hyperfine splitting can be found in [20]. In
Table II we show results for the expectation value of V for the ground state of helium with
1
different cutoffs Λ, with the linear dependence on Λ clearly visible. When combined with
other linearly divergent terms in a complete calculation a Λ independent result will obtain
in the limit of large Λ. By improving the accuracy found in this part of the calculation
this procedure can be used to check the results of Ref. [3] without explicitly canceling the
divergences: work on this problem is in progress.
II. CONCLUSIONS
We have presented the formula for the momentum space form of a powerful basis set for
helium. While it has the potential for proving useful for scattering calculations on helium,
we have concentrated on evaluating expectation values of operators that give relativistic and
QED corrections to energy levels. Because these operators are derived in momentum space,
this allows one to work entirely in momentum space. The next step in this research is the
extension to states with nonvanishing angular momentum. The most important application
we have in mind is to the fine structure of helium P states, where recent high-precision mea-
surements by Hessels and collaborators [21] have the potential of allowing a determination
of the fine structure constant α to a precision of 4 ppb. Unfortunately, the present state of
theory is unclear, where the most complete calculation by Drake [22], while consistent with
the fine structure interval ν measured in Ref. [21], is inconsistent with measurements of
01
the interval ν [23], [24]. This inconsistency has also been noted in Ref. [2]. It is possible
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that the relative simplicity of the method developed here can shed light on this situation.
Acknowledgments
ThisworkwassupportedinpartbyNSFGrantNo.PHY-0097641. Conversations withK.
Pachucki and S. Morrison are gratefully acknowledged, with particular thanks to the former
for providing details of his 23S calculation and helpful comments on the manuscript.
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