Hartree-Fock Critical Nuclear Charge in Two-Electron Atoms
HHartree–Fock Critical Nuclear Charge in Two-Electron Atoms
Hugh G. A. Burton a) Physical and Theoretical Chemistry Laboratory, Department of Chemistry, University of Oxford, South Parks Road, Oxford, OX1 3QZ,U.K. (Dated: 8 January 2021)
Electron correlation e ff ects play a key role in stabilising two-electron atoms near the critical nuclear charge, representingthe smallest charge required to bind two electrons. However, deciphering the importance of these e ff ects relies on fullyunderstanding the uncorrelated Hartree–Fock description. Here, we investigate the properties of the ground state wavefunction in the small nuclear charge limit using various symmetry-restricted Hartree–Fock formalisms. We identify thenuclear charge where spin-symmetry breaking occurs to give an unrestricted wave function that predicts the ionisation ofan electron. We also discover another critical nuclear charge where the closed-shell electron density detaches from thenucleus, and identify the importance of fractional spin errors and static correlation in this limit. I. INTRODUCTION
How much positive charge is required to bind two elec-trons to a nucleus? This simple question has been subjectto intense research and debate ever since the early 1930s.
High-precision calculations have only recently convergedon a critical nuclear charge for binding two electrons of Z c = .
911 028 224 077 255 73(4).
For Z > Z c , the two-electron atom ( Z e e) is bound and stable, with an energy lowerthan the ionised system ( Z e + e). However, for Z < Z c , theenergy of the bound atom becomes higher than the ionisedsystem, causing an electron to spontaneously detach from thenucleus. As a result, the critical charge marks the threshold forstability in the three-body problem and can be interpretedas a quantum phase transition. While the critical nuclear charge is fascinating in its ownright, the two-electron atom also provides an essential modelfor understanding the performance of electronic structure ap-proximations. It is the simplest chemical system where elec-tron correlation is present, which is thought to be essentialin binding the two electrons near Z c . In particular, com-paring the closed-shell HF energy to the exact energy of theionised system shows that HF theory fails to predict a stabletwo-electron atom with Z c < Z < .
031 177 528, includingH – . However, interpreting exactly how correlation influ-ences the stability of the two-electron atom is made di ffi cultby an incomplete understanding of the HF approximation forsmall Z .In many ways, placing artificial restrictions on the wavefunction makes the HF description more complicated to in-terpret than the exact result. For example, the restricted HF(RHF) formalism can only predict doubly-occupied orbitals, and one might ask if comparing the RHF energy to the exactone-electron energy is a fair way to identify the RHF criticalcharge. Alternatively, the unrestricted HF (UHF) approachallows the spin-up and spin-down electrons to occupy di ff er-ent spatial orbitals, providing a qualitatively correct modelfor the dissociation of a single bound electron in H – at theexpense of broken spin-symmetry. The onset of HF sym-metry breaking is marked by instability thresholds in the orbital a) Electronic mail: [email protected]
Hessian, which have also been interpreted as critical chargesin closed-shell atoms. These sudden qualitative changes inthe HF wave function can also be probed using the average ra-dial electronic positions, providing an alternative indicator forelectron ionisation that does not rely on energetic comparisonswith the exact result. However, to the best of our knowledge,the exact nuclear charge for UHF symmetry breaking, and thequalitative properties of HF wave functions near this point,remain unknown.Previous studies on the two-electron atom using HF theoryhave primarily focussed on the large Z , or “high-density”, limit(see e.g., Ref. 20). In this limit, the closed-shell RHF wavefunction provides a good approximation to the exact result,creating a model for understanding dynamic correlation. Alternatively, the small- Z “low-density” limit, where staticcorrelation becomes significant, remains far less explored. Theprimary challenges of small Z include the presence of HFsymmetry-breaking and convergence issues that occur withdi ff use basis functions. One recent HF study has been unableto reliably converge the RHF approximation for Z < . hindering attempts to understand HF theory for smaller Z .Consequently, the small- Z limit also provides a model forunderstanding how to predict strong static correlation, whichremains a major computational challenge.In this contribution, we investigate the properties of theRHF and UHF ground-state wave functions in the small Z limit. We use numerical Laguerre-based HF calculations tocompute the exact location of the UHF symmetry-breakingthreshold. By investigating the average radial positions inthe RHF and UHF wave functions, we assess how each HFformalism predicts electron detachment near the critical charge.Our results suggest that the UHF symmetry-breaking thresholdrepresents the onset of ionisation and forms a branch pointsingularity in the complex Z plane. Alternatively, RHF theorypredicts a closed-shell critical point where half the electrondensity becomes ionised, leading to strong static correlationfor small Z . a r X i v : . [ phy s i c s . c h e m - ph ] J a n II. TWO-ELECTRON ATOMIC HAMILTONIAN
The Z -scaled Hamiltonian for the two-electron atom with aninfinite nuclear mass is H = − (cid:16) ∇ + ∇ (cid:17) − ρ − ρ + Z ρ , (1)where ρ i = r i / Z is the scaled distance of electron i from thenucleus, ρ = | ρ − ρ | is the scaled inter-electronic distance,and the unscaled distances have atomic units a . Nuclearcharges are given in atomic units e . The exact wave function isdefined by the time-independent Schrödinger equation H Ψ ( x , x ) = ˜ E Ψ ( x , x ) (2)with the spin-spatial coordinate x i = ( ρ i , σ i ) and the scaledenergy ˜ E = E / Z . The electron-electron repulsion can beconsidered as a perturbation to the independent-particle modelwith the coupling strength λ = / Z , giving the power seriesexpansions ˜ E ( λ ) = (cid:80) ∞ k = ˜ E ( k ) λ k and Ψ ( λ ) = (cid:80) ∞ k = Ψ ( k ) λ k . Thecritical nuclear charge Z c can then be identified from the radiusof convergence of these series, defined by the distance ofthe closest singularity to the origin in the complex λ plane. Both E ( λ ) and | Ψ ( λ ) | have complicated singularities on thepositive real axis at λ c = / Z c , which have been interpreted asa quantum phase transition in the complete-basis-set limit. The HF wave function is a single Slater determinant Ψ HF ( x , x ) built from the antisymmetrised product of theoccupied spin-orbitals ψ i ( x ). These orbitals are self-consistenteigenfunctions of the one-electron Fock operator ˆ f ( x ), withthe corresponding eigenvalues defining orbital energies. The Z -scaled Fock operator isˆ f ( x ) = ˆ h ( x ) + Z (cid:88) i = (cid:104) ˆ J i ( x ) − ˆ K i ( x ) (cid:105) , (3)with the one-electron Hamiltonianˆ h ( x ) = − ∇ − ρ , (4)and the Coulomb and exchange operators denoted as ˆ J i ( x ) andˆ K i ( x ) respectively (see Ref. 17). The total HF energy is˜ E HF = (cid:88) i = ( h i + f i ) , (5)with the matrix elements h i = (cid:104) ψ i | ˆ h | ψ i (cid:105) and f i = (cid:104) ψ i | ˆ f | ψ i (cid:105) .The self-consistent two-electron component of the Fock op-erator can be considered as a perturbation with the couplingstrength λ = / Z . For large Z ( λ → As Z becomes smaller and λ grows, the self-consistent repulsionbecomes increasingly dominant over the one-electron nuclearattraction. Eventually, it becomes energetically favourable fora pair of lower-energy UHF solutions to emerge where eitherthe spin-up or spin-down electron becomes detached from the nucleus. This phenomenon is analogous to the Coulson–Fischer point in stretched H , where the spin-up and spin-downelectrons localise on opposite atoms, and is closely related toWigner crystallisation. By analytically continuing an equiv-alent two-electron coupling parameter to complex values, wehave recently shown that the UHF wave functions form a non-Hermitian square-root branch point at the symmetry-breakingthreshold.
Remarkably, following a complex-valued con-tour around this point leads to the interconversion of the degen-erate solutions, and allows a ground-state wave function to besmoothly evolved into an excited-state wave function. III. COMPUTATIONAL DETAILS
In the present work, we follow Ref. 13 and express thespatial component φ p ( r ) of the HF spin-orbital ψ p ( x ) usingthe spherically-symmetric Laguerre-based functions χ µ ( r ) = exp (cid:18) − Ar (cid:19) L (1) µ ( Ar ) , (6)giving φ p ( r ) = ∞ (cid:88) µ = χ µ ( r ) C µ ·· p . (7)Here we employ the nonorthogonal tensor notation of Head–Gordon et al. The non-linear parameter A controls the spatialextent of the basis functions and is optimised alongside thecoe ffi cients C µ ·· p . In practice, this expansion is truncated at afinite basis set of size n . To avoid previous issues with iterativesolutions to the HF equations for small Z , we optimise the C µ ·· p coe ffi cients for a fixed A value using the quasi-Newton Geo-metric Direct Minimisation (GDM) algorithm. The optimal A value is then identified through another quasi-Newton minimi-sation with the orbital coe ffi cients re-optimised on each step.All calculations were performed in a developmental versionof Q-C hem , and analytic expressions for the Laguerre-basedintegrals are provided in the Supporting Information. IV. RESULTSA. Spin-Symmetry Breaking Critical Point
First, we identify the critical nuclear charge for HFsymmetry-breaking Z UHFc using a bisection method to locatethe point where the lowest orbital Hessian eigenvalue of theRHF solution vanishes. The convergence of Z UHFc with re-spect to the basis set size is shown in Table I, giving a bestestimate of Z UHFc = .
057 660 253 46(1). This value is con-verged for n ≥
24, for which converged RHF energies for Heand H – are also obtained as E RHF (He) = − .
861 679 995 612 23(1) E RHF (H – ) = − .
487 929 734 370 84(1) , in agreement with the best variational benchmarks up to 10decimal places. We believe that this is the first numer-ically precise estimate of a symmetry-breaking threshold inthe complete-basis-set HF limit, and therefore defines a newtype of benchmark value within electronic structure theory. Asexpected, we find Z UHFc >
1, and thus our result is consistentwith previous observations of UHF symmetry breaking in thehydride anion.
TABLE I: Convergence of the UHF symmetry-breakingthreshold Z UHFc and the associated energy E UHF ( Z UHFc ) withrespect to basis set size. Best estimates and quoted errorscorrespond to the mean and standard deviation of theconverged values n ≥
24 respectively. n Z
UHFc / e E UHF ( Z UHFc ) / E h
10 1 .
057 651 800 057 − .
570 335 516 8712 1 .
057 658 412 462 − .
570 345 373 2414 1 .
057 659 966 054 − .
570 347 687 1216 1 .
057 660 213 291 − .
570 348 055 2218 1 .
057 660 248 206 − .
570 348 107 1920 1 .
057 660 252 818 − .
570 348 114 0522 1 .
057 660 253 391 − .
570 348 114 9124 1 .
057 660 253 461 − .
570 348 115 0126 1 .
057 660 253 464 − .
570 348 115 0128 1 .
057 660 253 462 − .
570 348 115 0130 1 .
057 660 253 464 − .
570 348 115 0232 1 .
057 660 253 439 − .
570 348 114 9834 1 .
057 660 253 473 − .
570 348 115 0336 1 .
057 660 253 458 − .
570 348 115 0138 1 .
057 660 253 477 − .
570 348 115 0340 1 .
057 660 253 466 − .
570 348 115 02Best 1 .
057 660 253 46(1) − .
570 348 115 01(2)
Figure 1 (top panel) compares the Z -scaled RHF energy(red) and the symmetry-broken UHF energy (blue dashed) asfunctions of Z − with n =
26. We also consider the exact one-electron energy (black) that corresponds to the ionised atom,the exact two-electron energy (grey dashed; reproduced fromRef. 8), and a fractional spin RHF calculation (dashed orange)with half a spin-up and half spin-down electron (see Sec. IV C).The UHF symmetry-breaking threshold Z UHFc (black dot) oc-curs below the exact one-electron energy, and thus Z UHFc isgreater than the HF critical nuclear charge previously identi-fied using energetic arguments. This suggests that the RHFapproximation is already an inadequate representation of theexact wave function before it becomes degenerate with the one-electron atom. Beyond this point, the RHF energy continuesto increase, while the UHF energy rapidly flattens towards theexact one-electron result. There is therefore a small relaxationregion during which the UHF approximation approaches aqualitative representation of the one-electron atom.Radial electron position expectation values (cid:104) r (cid:105) provide fur-ther insights into the properties of the two-electron atom closeto electron detachment. The exact wave function yieldsan “inner” and “outer” electron, with repulsive interactionspushing the inner electron closer to the nucleus than in thecorresponding hydrogenic system. For Z > Z UHFc , the RHFradial electron position closely matches the averaged exact h r i / a − . − . − . − . − . − . − . . ˜ E / ( Z − E h ) One ElectronRHFUHF (elec. 1) Two ElectronFractional RHFUHF (elec. 2) . . . . . . Z − / e − − . . . . H e ss . E i g e n v a l u e FIG. 1: Z -scaled energy (top) and average radial position (cid:104) r (cid:105) (middle) using various HF formalisms and exact one- andtwo-electron results. Exact two-electron data are reproducedfrom Ref. 8. The lowest two Hessian eigenvalues for the RHFsolution (bottom) show the onset of UHF symmetry breakingand a persistent zero eigenvalue for small Z .two-electron result (grey dashed). However, the RHF resultstarts to deviate from the two-electron value as electron cor-relation e ff ects become significant for Z < Z UHFc . In contrast,the additional flexibility of the UHF wave function correctlypredicts the separation of an inner and outer electron. Thisionisation occurs almost immediately for Z < Z UHFc , as indi-cated by a sudden increase in (cid:104) r (cid:105) for the dissociating electron(Fig. 1: middle panel), while the bound electron tends towardsthe exact one-electron result. Comparing the radial distributionfunctions P ( r ) = r | ψ ( r ) | for each electronic orbital at Z = However, at Z =
1, the outer electron has essentiallyionised from the atom in the UHF approximation, but remainsclosely bound to the nucleus in the fully-correlated description.The UHF wave function therefore rapidly approximates theionised atom for Z < Z UHFc , and Z UHFc can be interpreted as acritical charge for a stable two-electron atom. This approxi-mation essentially overlocalises the electron density between Z c < Z < Z UHFc (including H – ), as previously observed fortwo-electrons on concentric spheres, and fails to capture thecorrelation required to describe the exact critical charge.Figure 1 also indicates that the dissociation of the outer elec- r / a . . . . . . r | ψ ( r ) | RHFFractional RHF UHF ( α )UHF ( β ) One Electron
100 200 4000 . . FIG. 2: Radial distribution functions for di ff erent HF orbitalscompared to the exact one-electron wave function at Z = Fur-thermore, there is small region where the average radial posi-tion of the inner electron tends towards the one-electron result.It is known that the exact two-electron system exhibits a shaperesonance as the nuclear charge goes through the critical point,with the outer electron remaining at a finite distance from thenucleus.
One might therefore interpret the region where theUHF electron positions tend towards the one-electron result asan approximation of this resonant stability regime.For Z < Z c , the exact wave function is an equal combina-tion of two configurations where either the spin-up or spin-down electron remains bound to the nucleus. In contrast, thesingle-determinant nature of the UHF wave function meansthat only one of these configurations can be represented: theUHF orbitals are “pinned” to one resonance form. Theremust therefore be a wave function singularity at Z UHFc wherethe UHF approximation branches into a form with either thespin-up or spin-down electron remaining bound. The math-ematical structure of this point can be revealed by followinga continuous pathway around Z UHFc in the complex Z plane.When Z is analytically continued to complex values, the Fockoperator becomes non-Hermitian and we must consider theholomorphic HF approach. In the remainder of this Sec-tion, we fix the non-linear A parameter to its value at Z UHFc asthe non-Hermitian energy is complex-valued and cannot bevariationally optimised.Figure 3 shows the real component of (cid:104) r (cid:105) for the (initial)inner electron along a pathway which spirals in towards Z UHFc ,parametrised as Z ( ξ ) = Z UHFc − (cid:32) . − . ξ π (cid:33) exp(i ξ ) . (8)Remarkably, after one complete rotation ( ξ = π ), the innerand outer electrons have swapped, indicating that the degen-erate UHF solutions have been interconverted. A second fullrotation is required to return the states to their original forms.The two degenerate UHF wave functions are therefore con-nected as a square-root branch point in the complex- Z plane,in agreement with our previous observations in analyticallysolvable models. Furthermore, the branch point behaves as a quasi-exceptional point, where the two solutions becomeidentical but remain normalised (see Ref. 27), providing thefirst example of this type of non-Hermitian HF degeneracy inthe complete-basis-set limit.FIG. 3: Average radial position (cid:104) r (cid:105) of the inner electron alonga spiral contour in the complex Z plane converging on Z UHFc using n =
26. On each rotation, the UHF wave functiontransitions between the two degenerate solutions.
B. Closed-Shell Critical Point
We now consider the fate of the RHF ground state as Z con-tinues to decrease below Z UHFc . Intuitively, one might expectthat the doubly-occupied RHF orbitals would be unable to de-scribe the open-shell atom with an ionised electron. Indeed,King et al. have observed a smooth and finite (cid:104) r (cid:105) value for theRHF wave function as low as Z = .
85, with erratic conver-gence for lower nuclear charges. A similar nuclear charge Z = .
84 was identified in Ref. 19 as a singlet instability thresh-old, where the orbital Hessian contains a zero eigenvalue withrespect to symmetry-pure orbital rotations. These observationssuggest that the RHF approximation somehow breaks down at Z ≈ .
84, but we are not aware of any detailed insight into thisbehaviour.By using the gradient-based GDM algorithm, we have ac-curately converged the RHF ground state for all nuclear chargesand can now firmly establish its properties in the small- Z limit.Remarkably, we find a sudden increase in (cid:104) r (cid:105) at Z RHFc = . Z (Fig. 1: bottom panel). Zero Hessian eigenvalues generallyindicate a broken continuous symmetry in the wave function,such as a global spin-rotation, and define the so-called“Goldstone” manifold of degenerate states. In this instance,the new zero-eigenvalue Hessian mode corresponds to a spin-symmetry-breaking orbital rotation that also leads to an “inner”and “outer” electron. Since the energy is constant along thismode, this additional zero Hessian eigenvalue suggests thatthe RHF approximation has become unstable with respect toelectron detachment. Consequently, the sudden increase in (cid:104) r (cid:105) at Z RHFc qualitatively represents a closed-shell critical nuclearcharge at Z RHFc = . Z RHFc , we consider the cumulative radial distribution functionof the doubly-occupied RHF orbital N RHF ( r ) = (cid:90) π (cid:90) π (cid:90) r | ψ RHF ( r (cid:48) ) | r (cid:48) sin θ d r (cid:48) d θ d φ, (9)as shown in Fig. 4. The single-step structure at Z > Z RHFc andis consistent with a single peak in the radial distribution func-tion (see e.g. Fig. 2), indicating that the electrons are closelybound to the nucleus. For Z < Z RHFc , this cumulative density ��� � �� ��� �������������������
FIG. 4: Cumulative radial distribution function for the RHFwave function using n =
26. For Z < .
82, this functionadopts a double-step structure corresponding to an inner andouter peak in the radial electron density. At Z = .
41, eachpeak contains half the electron density (dashed line). adopts a double-step structure corresponding to a radial den-sity peak close to the nucleus, and another representing anunbound electron. The magnitude of the second step continuesto grow for smaller Z as the outer peak becomes increasinglyunbound until, at Z = .
41, the inner and outer peaks bothcontain exactly one electron. For smaller Z , all the electrondensity becomes unbound. Remarkably, the RHF wave func-tion for 0 . < Z < Z RHFc is therefore providing a closed-shellrepresentation of the open-shell atom by delocalising the elec-tron density over the bound and unbound radial “sites”. Thisdelocalisation allows the RHF wave function to provide a qual-itatively correct representation of the exact one-body density,but fails to capture any two-body correlation between the boundand unbound electrons.
C. Fractional Spin Error
Although the RHF radial density for Z < Z RHFc appears tobe approximating the exact result, the RHF energy remainsconsistently above the one-electron hydrogenic energy. Theclosed-shell nature of the RHF orbitals means that the inner andouter radial density peaks both contain half a spin-up electronand half a spin-down electron bound to the nucleus. As aresult, the RHF electron distribution for small Z tends towardsa description of the one-electron atom that also contains halfa spin-up and half a spin-down electron. We have confirmedthis limiting behaviour by computing the RHF energy witha half-occupied orbital, also known as the “spin-unpolarised”atom with fractional spins. As expected, this half-occupiedRHF solution becomes degenerate with the two-electron RHFenergy at small Z (Fig 1: top panel).Remarkably, even though a one-electron atom always hasa bound ground state, we find that the fractional spin RHFwave function predicts an additional critical nuclear chargeat Z fracc = .
41, where the (half) electrons suddenly becomeunbound (Fig 1: middle panel). This critical charge matchesthe point where half the electron density has ionised from thenucleus in the conventional RHF approach (dashed line inFig. 4). It is well-known that RHF with fractional spins failsto predict the correct energy for one-electron atoms, despitethe fact that HF theory should be exact in this limit, and causesthe static correlation error that leads to the RHF breakdownfor stretched H . We therefore conclude that this staticcorrelation also creates an artificial critical nuclear charge inone-electron atoms at Z fracc = .
41, and is responsible for thefailure of conventional RHF in the small Z limit. Identifyingsimilar artificial critical charges using a density functionalapproximation would almost certainly provide new insightsinto the failures of such methods for anionic energies andelectron a ffi nities. V. CONCLUDING REMARKS
In summary, we have used average radial electronic positionsto understand where HF theory predicts electron detachmentin the two-electron atom, providing alternative critical nuclearcharges in the RHF and UHF formalisms. For UHF theory,this critical charge corresponds to a spin-symmetry-breakingthreshold Z UHFc = .
057 660 253 46(1) where one electronsuddenly ionises from the nucleus. In contrast, at the RHFcritical charge Z RHFc = .
82, a secondary peak appears in theradial distribution function at large distances from the nucleus.These results provide a broader perspective on electron corre-lation in the small- Z limit. For example, the RHF (cid:104) r (cid:105) valuestarts to deviate from the exact two-electron result at the UHFsymmetry-breaking threshold, suggesting that Z UHFc marks theonset of static correlation. This static correlation is furthersupported by the existence of degenerate UHF solutions repre-senting the dominant configurations in the exact wave function.Since the UHF radial distribution functions are qualitativelyincorrect for Z c < Z < Z UHFc , this static correlation must be es-sential for binding the two-electron atom near Z c . Furthermore,the breakdown of the half-occupied one-electron RHF result at Z fracc = .
41 indicates that fractional spin errors occur for small Z , reinforcing the importance of static correlation.For Z < Z c , the exact wave function contains two dominantresonance forms with the spin-up or spin-down electron ionisedfrom the nucleus. As the electrons are indistinguishable, theone-electron density is delocalised between the bound andunbound “sites”, and this is reflected in the RHF wave function.However, instantaneous electron-electron correlations ensurethat, when one electron is bound to the nucleus, the otherelectron becomes unbound. UHF theory provides a snapshotof these correlations, with one electron permanently boundto the nucleus, but it cannot describe the resonance betweenthe two sites. Alternatively, when each orbital can contain aspin-up and spin-down component in generalised HF (GHF),the symmetry-broken UHF solutions form a continuum ofGHF solutions parameterised by a global spin rotation. Theresonance between the two sites is therefore represented by thiscontinuum, and could be computed using a (nonorthogonal)linear combination of stationary wave functions.
Finally, the degenerate UHF wave functions form a square-root branch point in the complex Z plane at Z UHFc . Followinga continuous complex path around this point interconvertsthe two degenerate solutions and swaps the dissociated elec-tron, while a second rotation returns the solutions to theiroriginal forms. We have previously observed this behaviourin analytic models, but our current results suggest that thisphenomenon extends to the complete-basis-set limit.
Fur-thermore, Ref. 27 shows that these complex branch points canallow a ground-state wave function to be smoothly “morphed”into an excited-state wave function by following a continuouscomplex contour. The two-electron atom therefore provides anew model for understanding these complex connections nearthe complete-basis-set limit, and we intend to continue thisinvestigation in the future.
SUPPORTING INFORMATION
See the supporting information for analytic derivations ofthe Laguerre-based one- and two-electron integrals.
ACKNOWLEDGEMENTS
I gratefully thank New College, Oxford for funding throughthe Astor Junior Research Fellowship. I also thank Hazel Coxfor providing the exact two-electron numerical data from Ref. 8,and Pierre-François Loos for countless inspiring conversationsand critical comments on this manuscript.
DATA AVAILABILITY
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