We show that the exact wave function for two electrons, interacting through a
Coulomb potential but constrained to remain on the surface of a
$\mathcal{D}$-sphere ($\mathcal{D} \ge 1$), is a polynomial in the
interelectronic distance $u$ for a countably infinite set of values of the
radius $R$. A selection of these radii, and the associated energies, are
reported for ground and excited states on the singlet and triplet manifolds. We
conclude that the $\mathcal{D}=3$ model bears the greatest similarity to normal
physical systems.Comment: 4 pages, 0 figur

A. D. Polyanin

A. G. Ushveridze

Peter M. W. Gill

Pierre-François Loos

R. G. Parr

Physical Review Letters

English

arXiv

We show that the exact wave function for two electrons, interacting through a Coulomb potential but constrained to remain on the surface of a D-sphere (D≥1), is a polynomial in the interelectronic distance u for a countably infinite set of values of th

Loos, Pierre-Francois

Gill, Peter

The Australian National University

Two Electrons on a Hypersphere: A Quasiexactly Solvable ModelPierre-François Loos and Peter M.W. Gill*Research School of Chemistry, Australian National University, Canberra, Australian Capital Territory 0200, Australia(Received 5 July 2009; published 18 September 2009)We show that the exact wave function for two electrons, interacting through a Coulomb potential butconstrained to remain on the surface of a D-sphere (D  1), is a polynomial in the interelectronicdistance u for a countably infinite set of values of the radius R. A selection of these radii and the associatedenergies are reported for ground and excited states on the singlet and triplet manifolds. We conclude thatthe D ¼ 3 model bears the greatest similarity to normal physical systems.DOI: 10.1103/PhysRevLett.103.123008 PACS numbers: 31.15.ac, 31.15.ve, 31.15.vjQuantum mechanical models for which it is possible tosolve explicitly for a finite portion of the energy spectrumare said to be quasiexactly solvable [1]. They have ongoingvalue and are useful both for illuminating more compli-cated systems and for testing and developing theoreticalapproaches, such as density-functional theory (DFT) [2–4]and explicitly correlated methods [5–8]. One of the mostfamous two-body models is the Hooke’s law atom, whichconsists of a pair of electrons repelling Coulombically buttrapped in a harmonic external potential with force con-stant k. This system was first considered nearly 50 yearsago by Kestner and Sinanoglu [9], solved analytically in1989 for one particular k value [10], and later for a count-ably infinite set of k values [11].A related system consists of two electrons trapped on thesurface of a sphere of radius R. This has been used by Berryand collaborators [12–15] to understand both weakly andstrongly correlated systems and to suggest an ‘‘alternat-ing’’ version of Hund’s rule [16]. Seidl utilized this systemto develop new correlation functionals [17] within theadiabatic connection in DFT [18]. We will use the term‘‘spherium’’ to describe this system.In recent work [19], we examined various schemes anddescribed a method for obtaining near-exact estimates ofthe 1S ground state energy of spherium for any given R.Because the corresponding Hartree-Fock (HF) energies arealso known exactly [19], this is now one of the mostcomplete theoretical models for understanding electroncorrelation effects.In this Letter, we consider D-spherium, the general-ization in which the two electrons are trapped on aD-sphere of radius R. We adopt the convention that aD-sphere is the surface of a ðDþ 1Þ-dimensional ball.(Thus, for example, the Berry system is a 2-spherium.) Weshow that the Schrödinger equation for the 1S and the 3Pstates can be solved exactly for a countably infinite set of Rvalues and that the resulting wave functions are polyno-mials in the interelectronic distance u ¼ jr1  r2j. Otherspin and angular momentum states can be addressed in thesame way using the ansatz derived by Breit [20].The electronic Hamiltonian, in atomic units, isĤ ¼ r212r222þ 1u; (1)and, because each electron moves on a D-sphere, it isnatural to adopt hyperspherical coordinates [21,22].For 1S states, it can be then shown [19] that the wavefunction SðuÞ satisfies the Schrödinger equationu24R2 1d2Sdu2þð2D 1Þu4R2D 1udSduþ Su¼ ES:(2)By introducing the dimensionless variable x ¼ u=2R, thisbecomes a Heun equation [23] with singular points at x ¼1; 0;þ1. Based on our previous work [19] and the knownsolutions of the Heun equation [24], we seek wave func-tions of the formSðuÞ ¼ X1k¼0skuk; (3)and substitution into (2) yields the recurrence relationskþ2 ¼skþ1 þ ½kðkþ 2D 2Þ 14R2  Eskðkþ 2ÞðkþDÞ (4)with the starting valuesfs0; s1g ¼ f0; 1g D ¼ 1;f1; 1=ðD 1Þg D  2: (5)Thus, the Kato cusp conditions [25] areSð0Þ ¼ 0; S00ð0ÞS0ð0Þ ¼ 1 (6)for electrons on a circle (D ¼ 1) andS0ð0ÞSð0Þ ¼1D 1 (7)in higher dimensions. We note that the ‘‘normal’’ Katovalue of 1=2 arises forD ¼ 3, suggesting that this may thePRL 103, 123008 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending18 SEPTEMBER 20090031-9007=09=103(12)=123008(4) 123008-1  2009 The American Physical Societymost appropriate model for atomic or molecular systems.We will return to this point below.The wave function (3) reduces to the polynomialSn;mðuÞ ¼Xnk¼0skuk (8)(where m is the number of roots between 0 and 2R) if, andonly if, snþ1 ¼ snþ2 ¼ 0. Thus, the energy En;m is a root ofthe polynomial equation snþ1 ¼ 0 [where degsnþ1 ¼bðnþ 1Þ=2c], and the corresponding radius Rn;m is foundfrom (4) which yieldsR2n;mEn;m ¼ n2n2þD 1: (9)Sn;mðuÞ is the exact wave function of the mth excited stateof 1S symmetry for the radius Rn;m.If we write the 3P state wave function as [20]3 ¼ ðcos1  cos2ÞTðuÞ; (10)where 1 and 2 are the Dth hyperspherical angles of thetwo electrons [21,22], the symmetric part satisfies theSchrödinger equationu24R2 1d2Tdu2þð2Dþ 1Þu4R2Dþ 1udTduþ Tu¼ ET;(11)and the antisymmetric part provides an additional kineticenergy contribution D=ð2R2Þ.Substituting the power series expansionTðuÞ ¼ X1k¼0tkuk (12)into (11) yields the recurrence relationtkþ2 ¼tkþ1 þ ½kðkþ 2DÞ 14R2  Etkðkþ 2ÞðkþDþ 2Þ (13)with the starting valuesft0; t1g ¼ f1; 1=ðDþ 1Þg (14)yielding the cusp conditionT0ð0ÞTð0Þ ¼1Dþ 1 : (15)The wave function (12) reduces to the polynomialTn;mðuÞ ¼Xnk¼0tkuk (16)when the energy En;m is a root of tnþ1 ¼ 0, and thecorresponding radius Rn;m is found from (13) which yieldsR2n;mEn;m ¼ n2n2þD: (17)Tn;mðuÞ is the exact wave function of the mth excited stateof 3P symmetry for the radius Rn;m.It is illuminating to begin by examining the simplest 1Sand 3P polynomial solutions. Except in the D ¼ 1 case,the first 1S solution hasR1;0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2D 1Þð2D 2Þ8s; E1;0 ¼ 1D 1 ; (18)and the first 3P solution hasR1;0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Dþ 1Þð2Dþ 2Þ8s; E1;0 ¼ 1Dþ 1 : (19)These are tabulated for D ¼ 1; 2; 3; 4, together with theassociated wave functions, in Table I.In the D ¼ 1 case (i.e., two electrons on a circle), thefirst singlet and triplet solutions have E2;0 ¼ 2=3 andE1;0 ¼ 1=2, respectively, for the same value of the radius(ffiffiffi6p=2  1:2247). The corresponding wave functions arerelated by S2;0 ¼ uT1;0. Unlike T1;0, the singlet wave func-tion S2;0 vanishes at u ¼ 0 and exhibits a second-ordercusp condition, as shown in (6).For the 2-spherium (D ¼ 2 case), we know from ourprevious work [19] that the HF energy of the lowest 1Sstate is EHF ¼ 1=R. It follows that the exact correlationenergy for R ¼ ffiffiffi3p =2 is Ecorr ¼ 1 2= ffiffiffi3p  0:1547,which is much larger than the limiting correlation energiesof the heliumlike ions (0:0467) [26] or Hooke’s lawatoms (0:0497) [27]. This confirms our view that electroncorrelation on the surface of a sphere is qualitatively differ-ent from that in three-dimensional physical space.The 3-spherium (D ¼ 3 case), in contrast, possesses thesame singlet and triplet cusp conditions—Eqs. (7) and (15)—as those for electrons moving in three-dimensionalphysical space. Indeed, the wave functions in Table IS1;0ðuÞ ¼ 1þ u=2 ðR ¼ffiffiffiffiffiffiffiffi5=2pÞ; (20)T1;0ðuÞ ¼ 1þ u=4 ðR ¼ffiffiffi7p Þ (21)have precisely the form of the ansatz used in Kutzelnigg’sTABLE I. Radius R, energy E, and wave function SðuÞ or TðuÞof the first 1S and 3P polynomial solutions for two electrons on aD-sphere.State D 2R E SðuÞ or TðuÞ1S 1ffiffiffi6p2=3 uð1þ u=2Þ2ffiffiffi3p1 1þ u3ffiffiffiffiffi10p1=2 1þ u=24ffiffiffiffiffi21p1=3 1þ u=33P 1ffiffiffi6p1=2 1þ u=22ffiffiffiffiffi15p1=3 1þ u=33ffiffiffiffiffi28p1=4 1þ u=44ffiffiffiffiffi45p1=5 1þ u=5PRL 103, 123008 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending18 SEPTEMBER 2009123008-2increasingly popular R12 methods [5,6]. Moreover, it canbe shown [28] that, as R ! 0, the correlation energy Ecorrapproaches 0:0476, which nestles between the corre-sponding values for the heliumlike ions (0:0467) [26]and the Hooke’s law atom (0:0497) [27]. Again, thissuggests that the D ¼ 3 model (‘‘electrons on a hyper-sphere’’) bears more similarity to common physical sys-tems than the D ¼ 2 model (‘‘electrons on a sphere’’).Numerical values of the energies and radii, for polyno-mial wave functions in D ¼ 1; 2; 3, are reported inTables II (for 1S states) and III (for 3P states).For fixed D, the radii increase with n but decrease withm, and the energies behave in exactly the opposite way. AsR (or, equivalently, n) increases, the electrons tend tolocalize on opposite sides of the sphere, a phenomenonknown as Wigner crystallization [29], which has also beenTABLE II. Radii Rn;m and energies En;m for1S states of two electrons on a D-sphere (D ¼ 1; 2; 3).D ¼ 1 D ¼ 2 D ¼ 3n=m 0 1 2 3 0 1 2 3 0 1 2 3Radius1 0.8660 1.58112 1.2247 2.6458 4.06203 3.3912 5.4312 1.4150 7.5154 2.24044 6.5439 1.9178 9.2211 3.7379 11.961 5.33205 10.693 4.7071 14.012 7.0848 1.9256 17.404 9.3775 2.85546 15.841 8.4583 2.5522 19.804 11.448 4.7683 23.846 14.410 6.53507 21.989 13.199 5.9404 26.597 16.817 8.6593 2.4123 31.287 20.439 11.158 3.44158 29.136 18.936 10.277 3.1515 34.389 23.190 13.583 5.7566 39.728 27.466 16.768 7.6903Energy1 1.0000 0.50002 0.6667 0.2857 0.18183 0.1957 0.1271 1.8729 0.0930 1.04594 0.0934 1.0875 0.0706 0.4294 0.0559 0.28145 0.0547 0.2821 0.0446 0.1743 2.3597 0.0371 0.1279 1.37986 0.0359 0.1258 1.3817 0.0306 0.0916 0.5278 0.0264 0.0722 0.35127 0.0253 0.0703 0.3471 0.0223 0.0557 0.2100 2.7065 0.0197 0.0461 0.1546 1.62538 0.0188 0.0446 0.1515 1.6110 0.0169 0.0372 0.1084 0.6035 0.0152 0.0318 0.0854 0.4058TABLE III. Radii Rn;m and energies En;m for3P states of two electrons on a D-sphere (D ¼ 1; 2; 3).D ¼ 1 D ¼ 2 D ¼ 3n=m 0 1 2 3 0 1 2 3 0 1 2 3Radius1 1.2247 1.9365 2.64582 3.3912 4.7958 6.20483 6.5439 1.9178 8.6227 2.6738 10.718 3.41114 10.693 4.7071 13.435 6.2041 16.205 7.67485 15.841 8.4583 2.5522 19.241 10.665 3.3588 22.678 12.852 4.12856 21.989 13.199 5.9404 26.043 16.094 7.5340 30.142 18.979 9.07017 29.136 18.936 10.277 3.1515 33.842 22.505 12.615 4.0095 38.600 26.077 14.897 4.81308 37.283 25.671 15.599 7.1177 42.640 29.907 18.650 8.8083 48.054 34.155 21.654 10.411Energy1 0.5000 0.3333 0.25002 0.1739 0.1304 0.10393 0.0876 1.0196 0.0706 0.7343 0.0588 0.58014 0.0525 0.2708 0.0443 0.2078 0.0381 0.16985 0.0349 0.1223 1.3433 0.0304 0.0989 0.9972 0.0267 0.0832 0.80676 0.0248 0.0689 0.3401 0.0221 0.0579 0.2643 0.0198 0.0500 0.21887 0.0186 0.0439 0.1491 1.5858 0.0168 0.0380 0.1210 1.1974 0.0153 0.0335 0.1025 0.98218 0.0144 0.0303 0.0822 0.3948 0.0132 0.0268 0.0690 0.3093 0.0121 0.0240 0.0597 0.2583PRL 103, 123008 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending18 SEPTEMBER 2009123008-3observed in other systems [11,30]. As a result, for large R,the ground state energies of both the singlet and the tripletstate approach 1=ð2RÞ. Analogous behavior is observedwhen D ! 1 [31,32].In conclusion, we have shown that the system of twoelectrons, interacting via a Coulomb potential but con-strained to remain on a D-sphere, can be solved exactlyfor an infinite set of values of the radius R. We find that the3-spherium (D ¼ 3 model), wherein the electrons areconfined to a three-dimensional surface of a four-dimensional ball, has greater similarity to normal physicalsystems than the more familiar D ¼ 2 case.We believe that our results will be useful in the futuredevelopment of correlation functionals within density-functional theory [33], intracule functional theory [34–39], and explicitly correlated methods [5–8]. 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Chem. Phys. 10, 3447 (2008).[39] J. K. Pearson, D. L. Crittenden, and P.M.W. Gill, J. Chem.Phys. 130, 164110 (2009).[40] T.M. Henderson, K. Runge, and R. J. Bartlett, Chem.Phys. Lett. 337, 138 (2001).PRL 103, 123008 (2009) P HY S I CA L R EV I EW LE T T E R Sweek ending18 SEPTEMBER 2009123008-4

Two Electrons on a Hypersphere: A Quasiexactly Solvable Model

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Two electrons on a hypersphere: a quasi-exactly solvable model

Two electrons on a hypersphere: a quasi-exactly solvable model

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