Exploring Unorthodox Dimensions for Two-Electron Atoms - The

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Exploring Unorthodox Dimensions for Two-Electron Atoms Published as part of The Journal of Physical Chemistry virtual special issue “Veronica Vaida Festschrift”. Dudley R. Herschbach,*,†,‡ John G. Loeser,§ and Wilton L. Virgo† †

Institute for Quantum Science and Engineering, Texas A&M University, College Station, Texas 77843, United States Department of Chemistry and Chemical Biology, Harvard University, Cambridge, Massachusetts 02138, United States § Department of Chemistry, Oregon State University, Corvallis, Oregon 97331, United States ‡

ABSTRACT: Melding quantum and classical mechanics is an abiding quest of physical chemists who strive for heuristic insights and useful tools. We present a surprisingly simple and accurate treatment of ground-state two-electron atoms. It makes use of only the dimensional dependence of a hydrogen atom, together with the exactly known first-order perturbation value of the electron−electron interaction, both quintessentially quantum, and the D → ∞ limit, entirely classical. The result is an analytic formula for D-dimensional two-electron atoms with Z ≥ 2. For D = 3 helium, it gives accuracy better than 2 millihartrees, which is better than current density functional theory. A kindred explicit formula for correlation energy exploits interpolation between D → ∞ and D = 1 or 2; when added to the Hartree−Fock energy, it improves accuracy for D = 3 helium to better than 0.1 millihartrees.

T

dependence. Hence, D = 3 helium can be obtained to good approximation from the exactly known D → ∞ limit by just adding the difference between the first-order terms. The resulting error is reduced to 0.059% (1.7 millihartrees). Making use as well of the D = 1 limit lowers the error to 0.021%, and combining with the HF approximation reduces the error much further, to 0.0012% (0.035 millihartrees). Such dimensional maneuvers may find applications to other properties.

wo-electron systems continue to bring out fresh perspectives, including recent work on correlation energy functionals,1 transformative reduced density matrix methods,2 lower bounds,3 and entanglement.4,5 Here we reconsider a dimensional scaling approach to electronic structure applied long ago to helium.6−10 The original approach was prompted by a perturbation technique used to treat quantum chromodynamics.11 That approach aimed to develop a dimensional perturbation theory, starting with the D → ∞ limit and adding terms in powers of δ = 1/D. It was arduous as the δ-series proved to be explosively asymptotic, yet terms up to 30th order were subdued by summation techniques that attained very high accuracy (one part in 109) for D = 3 helium.12 Another approach, at its outset merely empirical and heuristic, found that linear interpolation in δ between D → ∞ and D = 1 gave good accuracy (better than current density functional theory) with very modest effort.8,13 To test both the perturbation and interpolation methods, state-of-the-art Hylleraas−Pekeris variational computations were performed for S states of twoelectron atoms14 as well as for the Hartree−Fock (HF) approximation.15 These covered a wide range of dimensions 1 ≤ D ≤ ∞ and nuclear charge 1 ≤ Z ≤ 6, with high accuracy (one part in 108). The dimensional scaling version that we present here is rudimentary, yet surprisingly accurate. In conventional quantum mechanics treating D = 3 helium, textbooks begin by evaluating the electron−electron interaction, 1/r12, by firstorder perturbation theory; the result is in error by 5.29% (0.15 hartree). For the D → ∞ limit, the corresponding first-order result is in error by 5.55%. In dimensional scaling, however, the first-order term actually provides most of the dimension © XXXX American Chemical Society

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BASIC THEORY For a hydrogenic atom in D-dimensional space, the total ground-state energy (in hartree atomic units) is −1/2(Z/β)2, where β = 1/2(D − 1). For a two-electron atom, the total energy except for electron repulsion is −(Z/β)2; thus, the energy becomes singular as D → 1 and vanishes as D → ∞. As originally proposed by Herrick and Stillinger,6 we adopt scaled units, whereby ED = (Z/β)2εD and the reduced energy εD remains finite in both limits. For heliogenic atoms, exact expressions have been obtained7,16 for both the total and HF energy in the D → ∞ limit ε∞ = −1 −

5 2 1 λ + [λ 4 + λ(128 + λ 2)3/2 ] 32 2048

ε∞HF = −1 + 2−1/2λ −

λ2 8

(1a)

(1b)

Received: June 22, 2017 Revised: July 19, 2017 Published: July 31, 2017 A

DOI: 10.1021/acs.jpca.7b06148 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A where λ = 1/Z. Corresponding results17,18 for the D → 1 limit are ε1 = −1 +

⎡3 2 ⎤ 2 ⎢ 1 5 ⎥ 3 λ −⎢ − − ⎥λ + ⎢⎣ ⎥λ + ... ⎦ ⎣ 2 8 3π 6π 128 ⎦ (2a)

ε1HF = −1 +

λ λ2 − 2 12

(2b)

A closed formula for ε1 has not yet been found, but a highly accurate value is available from the first 20 terms in the 1/Z series.19 When expressed in a 1/Z perturbation series, the reduced energy is given by εD = −1 + εD(1)λ + εD(2)λ 2 + ...

(3)

where the first-order coefficient is6 ⎛D 1⎞ ⎛ 1⎞ ⎛D⎞ εD(1) = Γ⎜ + ⎟Γ⎜D + ⎟ /Γ⎜ ⎟Γ(D + 1) ⎝2 ⎠ ⎝ 2 2⎠ ⎝ 2 ⎠

(4)

It represents the expectation value, ⟨1/r12⟩, of the electron− electron repulsion evaluated with the zeroth-order hydrogenic wave function, exp(−r1 − r2). Accordingly, εD(1) is universal and applies for the HF approximation as well. It is readily obtained from a recursion relation8 ⎫ ⎧⎡⎛ 1 ⎞⎛ 3 ⎞⎤ εD + 2(1) = ⎨⎢⎜D + ⎟⎜D + ⎟⎥ /[D(D + 2)]⎬εD(1) 2 ⎠⎝ 2 ⎠⎦ ⎭ ⎩⎣⎝

Figure 1. Contributions to the energies of neutral two-electron atoms as a function of δ = 1/D: (a) “exact” (i.e., highly accurate) and HF reduced energies, εD and εDHF; (b) same, but with the first-order interelectron repulsion energy deducted, as in eq 6; (c) reduced correlation energy, ΔεD = εD − εDHF. Data from refs14 and 15.

(5)

with ε1(1) = 1/2 and ε2(1) = 3π/16; thus, εD(1) = 5/8, 105π/512, 21/32, and 2−1/2 respectively, for D = 3, 4, 5, and ∞. Figure 1 shows in panel (a) the reduced energies εD and εDHF obtained for helium from nearly exact (accurate to eight digits) variational calculations.14,15 Panel (b) shows that subtracting εD(1)λ removes most of the dimensional dependence. Moreover, what is left is nearly linear in δ = 1/D, both for the total energy and HF approximation. Consequently, as seen in panel (c), the correlation energy is also nearly linear. Energies Derived from the D → ∞ Limit. The residual of the total energy after deduction of the first-order term, (εD − εD(1)λ), is nearly independent of D; hence εD can be approximated by εD = ε∞ + (εD(1) − ε∞(1))λ

yet linger for D = 3 but rapidly dwindle away for the D = 1 and D → ∞ limits. Energies Derived from Linear Interpolation. As seen in Figure 1, deducting the first-order perturbation term rendered the dependence on δ = 1/D nearly linear and nearly flat for εD − εD(1)λ but slanted for the corresponding HF approximation and hence also for the correlation energy. Interpolation between D → ∞ and D = 1 was previously found to work well.8,13 We now find that results for εD and εDHF are improved by incorporating εD(1). Also, we consider interpolation between D → ∞ and D = 2, both because the D = 1 limit has much less scope and because conventional quantum computations for D = 2 are less onerous than those for D = 3. We discuss separately the correlation energy because in it the first-order perturbation terms cancel out. The 1 to ∞ interpolation provides

(6)

This provides an analytic formula, comprised of exactly known functions of D: ε∞ (eq 1a) and the first-order coefficients εD(1) (eq 4). Table 1 gives results obtained for D = 1, 2, and 3 and Z = 2, 3, and 6; upper rows display highly accurate values (boldface) from ref 14, and lower ones show εD from eq 6. Percentage errors indicate the remaining dimension dependence, lacking in eq 6. In Table 2, that dependence is exhibited for helium via partial sums of the λ-perturbation series εD = −1 +

∑K εD(k)λk

εD = δε1 + (1 − δ)ε∞ + [εD(1) − δε1(1) − (1 − δ)ε∞(1)]λ (8)

and for D = 3 gives 1 2 ε3 = ε1 + ε∞ − 0.0130712λ 3 3

(7)

(9)

The 2 to ∞ interpolation reads

where the summation runs from k = 1 to K. Highly accurate values of the perturbation coefficients, εD(k), were obtained in ref 6 for D = 1 and 3 and ref 8 for D → ∞. Deducting the higher-order partial sums in sequence makes evident that the first-order (k = 1) contribution is indeed major, the secondorder (k = 2) term is minor, and higher-orders contribute little

εD = 2δε2 + (1 − 2δ)ε∞ + [εD(1) − 2δε2(1) − (1 − 2δ)ε∞(1)]λ

(10)

and for D = 3 gives B

DOI: 10.1021/acs.jpca.7b06148 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A Table 1. Energies Derived from the D → ∞ Limita Z

D=1

D=2

D=3

D→∞

2

−0.788 843 2 −0.787 996 −0.850 883 6 −0.850 381 −0.921 123 0 −0.920 968 0.11/0.056/0.016

−0.743 738 887 −0.743 471 −0.820 841 089 −0.820 699 −0.906 170 701 −0.906 123 0.036/0.017/0.0053

−0.725 931 094 −0.725 496 −0.808 879 268 −0.808 715 −0.900 173 517 −0.900 135 0.060/0.020/0.0043

−0.684 442 285

3 6 % errors

−0.781 345 851 −0.886 450 405

a

Energies in hartree units. Highly accurate numbers from ref 14 are in boldface to compare with those obtained from eq 6. Percentage errors are listed at the bottom of each column, in the order Z = 2/3/6.

Some further aspects of correlation energy appear in the λperturbation series

Table 2. Removal of Dimension Dependence by Partial Sumsa K

D=1

D=3

D→∞

1 2 3 4 5

−1.038 848 −0.998 150 −0.999 898 −1.000 001 −1.000 002

−1.038 431 −0.999 014 −1.000 102 −1.000 046 −1.000 014

−1.037 996 −0.998 933 −0.999 969 −0.999 999 −1.000 000

CE = Z2ΔεD = ΔεD(2) + ΔεD(3)λ + ΔεD(4)λ 2 + ...

Table 5 and Figure 2 demonstrate that the efficacy of D interpolation holds up for two-electron cations. The interpolated D = 3 correlation energy for Z ≥ 3 is appreciably less accurate than that for Z = 2, and the error is of opposite sign, but Z → ∞ remains accurate within 0.7 millihartree or better. In a monumental paper,20 Davidson and co-workers fitted to data for Z = 2−144 a four-term expression for the D = 3 correlation energy; in millihartree units

Entries are from εD − ∑K εD(k)λk, for Z = 2, where the summation runs from k = 1 to K (see eq 7). Input data are from refs 6 and 8. Values of εD are in Tables 1 and 3; values of εD(1) from eq 4 are in Table 3. a

ε3 =

2 1 ε2 + ε∞ − 0.003401342λ 3 3

(12)

CE = − 46.670914 + 9.880741λ − 0.852948λ 2 (11)

− 0.79598λ 3

(13)

This expression reproduces their “exact” data for Z = 2−20 within 2 microhartree units but implies that the correlation energy at the Z → ∞ limit is −46.6709 rather than −46.663, an accurate value directly computed.21 By fixing the leading term at −46.663 while fitting the Z = 2−20 data by least-squares, we obtained

Table 3 displays (in boldface) highly accurate reduced energies, εD, obtained from variational calculations14 and exact first-order coefficients, εD(1), for helium with the pertinent dimensions. The third row shows εD − εD(1)λ, exhibiting again that the D dependence is slight, as seen in Figure 1b. These boldface data serve as references in comparing the approximate energies derived from eqs 6, 8, and 10, designated, respectively, as flat (0−∞) and the interpolation versions (1−∞) and (2−∞). The percentage errors show that interpolation does better than the ultrasimple flat scheme but only by about factors of 2 or 3 and therefore remains near 1 millihartree for D = 3 helium. Correlation Energies. As exhibited in Figure 1c, the dimensional dependence of the correlation energy, ΔεD = εD − εDHF, invites interpolation. Equations 8−11 serve for ΔεD, but the first-order terms cancel out and the input quantities become Δε∞, Δε1, or Δε2. Table 4, with format like Table 3, displays input and results. Remarkably, both the simple (1−∞) and (2−∞) interpolations deliver correlation energies with D dependence that mesh with εDHF to render εD with error reduced more than 10-fold.

CE = − 46.663 + 9.7344λ − 0.19297λ 2 − 1.59867λ 3 (14)

This reproduces the data even more closely, with errors mostly less than 0.5 microhartrees. The leading term in the correlation energy, ΔεD(2) in eq 12 is solely a function of D and represents the correlation energy in the Z → ∞ limit. Loos and Gill have evaluated ΔεD(2) over the range of D = 2−8 for helium and three other two-electron systems.21 They found that in the large-D limit, ΔεD(2) has the same value for helium and the other systems, dubbed spherium, ballium, and hookium. Moreover, they proved that the limit is invariant for two opposite-spin electrons subject to any radial confining potential in a D-dimensional space.22 The D → ∞

Table 3. Energies from Dimensional Interpolations for Z = 2a quantitya εD εD(1) εD − εD(1)λ εD(0−∞) εD(1−∞) εD(2−∞) % errors

D=1 −0.788 848 2 0.5 −1.038 848 2 −0.787 996 −0.788 530 0.108/−/0.040

D=2

D=3

D→∞

−0.743 738 887 0.589 048 622 −1.038 263 320 −0.743 471 −0.743 898

−0.725 931 094 0.625 −1.038 431 094 −0.725 496 −0.725 780 −0.725 674 0.060/0.021/0.035

−0.684 442 285 0.707 106 781 −1.037 995 676

0.036/0.021/−

a As in Table 1, energies are in hartree units, and highly accurate reference or exact reference quantities are in boldface. Percentage errors are shown at the bottom of the columns, in the order of eqs 6/8/10: (0−∞)/(1−∞)/(2−∞).

C

DOI: 10.1021/acs.jpca.7b06148 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A Table 4. Interpolated Correlation Energies for Z = 2a quantity

D=1

D=2

D=3

D→∞

εDHF ΔεD = εD − εDHF ΔεD(1−∞) ΔεD(2−∞) εDHF + ΔεD εD(1−∞) εD(2−∞) % errors

−0.770 833 333 −0.018 014 9 − −0.017 971 124 −0.788 848 1 − −0.788 804 457 0.0055

−0.731 380 487 −0.012 358 4 −0.012 380 288 − −0.743 738 887 −0.743 683 367 − 0.0075

−0.715 419 999 −0.010 511 095 −0.010 502 084 −0.010 487 492 −0.725 931 094 −0.725 922 083 −0.725 907 491 0.0012/0.0032

−0.677 696 609 −0.006 745 676 − − −0.684 442 285 − −

a

As in Tables 1 and 3, energies are in hartree units, and highly accurate reference quantities from ref 14 and 15 are in boldface. Percentage errors are shown for the εD obtained by adding to εDHF the interpolated correlation energy (either 1−∞ or 2−∞).

Table 5. Quality of D Interpolation of the D = 3 Correlation Energya Z

exact CE

interpolated CE

error

2 3 5 10 20 ∞

−42.044 −43.498 −44.737 −45.693 −46.177 −46.663

−42.007 −43.827 −45.247 −46.293 −46.808 −47.320

−0.037 0.31 0.51 0.60 0.63 0.66

when brought together improve the result nearly a 100-fold. It also makes more cogent the previous use of interpolation including D = 1 that further improves accuracy. The second lesson might be viewed as hybridizing distinct species, D = 3 quantum and D → ∞ classical. In the version of dimensional generalization used here, all vectors acquire D Cartesian coordinates. The Laplacian and Jacobian are Ddependent, but the form of the Coulombic potential energy is not altered6 (except for the D = 1 limit). As D → ∞, classical mechanics sets in because in D-scaled space the electron wave functions shrink to δ functions. The reduced ground-state energy ε∞ is obtained from the minimum of an effective potential comprised of centrifugal and Coulombic contributions.8,10 The minimum specifies, in D-scaled coordinates, a rigid electron configuration. Among consequent heuristic insights, we note three. At D → ∞, helium has a symmetrical bent structure with an angle of θ = 95.3° between the two electron−nucleus vectors, resulting from competition between centrifugal (favoring 90°) versus Coulombic (favoring 180°) interactions. For D = 3 helium, the electron distribution is broad, but when the Jacobian weighting is properly included, it likewise is bent with an angle peaking near 95° (contrary to the usual presumption of 180°). Despite the classical character of the D → ∞ limit, the uncertainty principle is lurking still. As D → ∞, the uncertainty product of the D-scaled momentum spread and location spread declines gently but remains finite.8 Recently, the D → ∞ limit has been linked with the venerable Bohr model and provided useful accuracy for potential curves of H2 and other small molecules.24,25 Exploiting the D dependence of the first-order perturbation term together with D → ∞ should enhance previous dimensional scaling treatments of two-electron excited states.26 Particularly amenable would be certain doubly excited states that stem from interdimensional degeneracies.27 The first-order ⟨1/r12⟩ terms, obtained from wave functions of D-dimensional hydrogen atoms, enjoy use of an exact isomorphism6,10 relating ΨD to Ψ3 via the principal quantum number n → n + (1/2)(D − 3) and the orbital angular momentum l → l + (1/2)(D − 3). At present, we consider that developing this project may lead to connecting dimensional scaling with the extraordinary twoelectron density matrix approach of Mazziotti,2,3 which he has established as the basic variable in many-electron quantum chemistry. Another inviting prospect is examining dimensional dependence of entanglement.4,5

Energies in millihartees. CE = Z2Δε3; “exact” values are from ref 20, and those for Z → ∞ are from ref 21. Interpolated results are from (1−∞) and for Z → ∞ obtained using Z2Δε∞ = −31.25 and Z2Δε1 = −79.460 from eqs 1 and 2.

a

Figure 2. Correlation energies, Z2ΔεD, for two-electron atoms as a function of λ = 1/Z, with Z = 2−20. Curves display dimensional limits D = 1 and D → ∞ and the interpolated D = 3 results from Δε3 = 1/ 3Δε1 + 2/3Δε∞ ; points are from highly accurate D = 3 values given in ref 20.

limit of ΔεD(2) is given directly23 by eq 1 as Δε∞(2) = −(5/32) + (1/8) = −(1/32); hence, in unscaled units Δε∞(2)/β 2 = −(1/32)/β 2 = −(1/8)/(D − 1)2 ≈ −(1/8)δ 2 + ...

(15)

where the rightmost form is that given by Loos and Gill.

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COMMENTS This paper offers two chief lessons, applicable beyond twoelectron atoms. The first might be considered philosophical. As an approximation, it defies the adage “two wrongs don’t make a right”. In eq 6 and in Tables 1−3, a pair of sibling first-order calculations, at D = 3 and D → ∞, both inadequate in accuracy,

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AUTHOR INFORMATION

ORCID

Dudley R. Herschbach: 0000-0003-3225-0648 D

DOI: 10.1021/acs.jpca.7b06148 J. Phys. Chem. A XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry A Notes

(22) Loos, P.-F.; Gill, P. M. W. Invariance of the Correlation Energy at High Density and Large Dimension in Two-Electron Systems. Phys. Rev. Lett. 2010, 105, 113001. (23) Virgo, W. L.; Herschbach, D. R. Comment on Dimension Dependence of Helium. Chem. Phys. Lett. 2015, 634, 179−180. (24) Svidzinsky, A.; Chen, G.; Chin, S.; Kim, M.; Ma, D.; Murawski, R.; Sergeev, A.; Scully, M.; Herschbach, D. Bohr Model and Dimensional Scaling Analysis of Atoms and Molecules. Int. Rev. Phys. Chem. 2008, 27, 665−723 and work cited therein.. (25) Svidzinsky, A.; Scully, M. O.; Herschbach, D. R. Bohr’s Molecular Model: A Century Later. Phys. Today 2014, 66 (January), 33−39; 67 (August), 8. (26) Goodson, D. Z.; Watson, D. K. Dimensional Perturbation Theory for Excited States of Two-electron Atoms. Phys. Rev. A: At., Mol., Opt. Phys. 1993, 48, 2668−2678 and work cited therein.. (27) Goodson, D. Z.; Watson, D. K.; Loeser, J. G.; Herschbach, D. R. Energies of Doubly Excited Two-Electron Atoms from Interdimensional Degeneracies. Phys. Rev. A: At., Mol., Opt. Phys. 1991, 44, 97− 102 and work cited therein..

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are grateful to Marlan Scully for his enthusiastic encouragement of this work and for support provided by the Institute for Quantum Science and Engineering at Texas A&M University.

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REFERENCES

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DOI: 10.1021/acs.jpca.7b06148 J. Phys. Chem. A XXXX, XXX, XXX−XXX