# Natural orbitals of the ground state of the two-electron harmonium atom

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## Abstract

The radial components of the natural orbitals (NOs) pertaining to the \(^1S_+\) ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength \(\omega \) at which the respective correlation factors are given by polynomials.
Together with the angular momentum *l* of the NOs, the degrees of these polynomials determine the orders of the differential equations, eigenvalues of which (arising from well-defined boundary conditions) yield the natural amplitudes. In the case of \(l=0\), analysis of these equations uncovers certain properties of the NOs whereas application of a WKB-like approximation produces asymptotic expressions for both the NOs and the corresponding natural amplitudes that hold when the latter are small negative numbers. Extensive numerical calculations reveal that these expressions remain valid for arbitrary values of \(\omega \). The approximate *s*-type NOs, which are remarkably accurate at sufficiently small radial distances and exhibit universal scaling, differ qualitatively from the eigenfunctions of the core Hamiltonian even at the \(\omega \rightarrow \infty \) limit of vanishing electron correlation.

## Keywords

Two-electron harmonium atom Natural orbitals WKB approximation Hill's asymptotics## 1 Introduction

Electronic properties of the two-electron harmonium atom are readily elucidated not only at select values of \(\omega \) but also at the weak- and strong-correlation limits. Within the former regime, which corresponds to strong confinement, the Hamiltonian (1) describes a system of two weakly coupled three-dimensional harmonic oscillators that is amenable to perturbative treatment [17, 18, 19, 20, 21, 22]. On the other hand, the strongly correlated species that ensues at small values of \(\omega \) can be regarded as a classical Wigner molecule subject to minor quantum corrections [17, 23].

In this paper, some recently derived properties of natural orbitals (NOs) and their occupation numbers pertaining to the \(^1S_+\) ground state of the two-electron harmonium atom are reported. Although this state has been the subject of numerous studies involving both rigorous mathematical analysis [1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and numerical approaches [17, 28], the attention devoted to the corresponding NOs has been limited to investigations of their asymptotic behavior at the \(\omega \rightarrow 0\) limit [23] and of their collective occupancies at various values of \(\omega \) [24, 25, 26], formulation of accurate approximations to the strongly occupied NOs at \(\omega = \frac{1}{2}\) and \(\omega = \frac{1}{10}\) [27], and finding definitive answers [28] to the questions concerning the existence of NOs with vanishing occupation numbers [17, 29, 30]. With amelioration of this unsatisfactory state of knowledge as its objective, the present work aims at uncovering universalities in NOs and their occupation numbers that persist throughout the entire range of confinement strengths.

## 2 Theory

*l*[27, 30]. In Eq. (5), \(\{\phi _{nl}(r)\} \equiv \{ \phi _{nl}(\omega ;r)\}\) (normalized according to \(\int ^{\infty }_0 [\phi _{nl}(r)]^2 \, r^2 \; \mathrm{d}r = 1\)) are the real-valued radial parts of \( \{\psi _{nlm}( \vec {r})\}\),

### 2.1 A differential equation for NOs

*q*th derivative of the respective Legendre polynomial in the definitions

*j*in the l.h.s. of Eq. (8) are polynomials of degree\(2l+2 \, [\frac{k+1}{2}]+1\) in \(r_1\) (here and in the following, [

*t*] denotes the integer part of

*t*), differentiating both sides of this equation \(2l+2 \, [\frac{k+1}{2}]+2\) times with respect to \(r_1\) produces a homogeneous differential equation for \(\chi _{knl} (r)\) that reads

*p*that simply imply vanishing of \(\chi _{knl} (r)\) and all its even-order derivatives at \(r=0\). The remaining \(l+ \, [\frac{k+1}{2}]+1\) conditions that ensue for odd

*p*are rather cumbersome. However, satisfying them is equivalent to enforcing the large-

*r*asymptotics of \(\chi _{knl}(r) \xrightarrow [r \rightarrow \infty ] \, \chi _{knl}^{>}(r)+o(1)\), where the rational function

*r*asymptotic approximants \( \{ \chi _{knl}^{>\,[q]} (r_1) \}\) defined recursively as [compare Eq. (8)]

### 2.2 The case of \(l=0\)

*r*asymptotics becomes a polynomial (note the linear dependences among its coefficients)

### 2.3 The *s*-type NOs pertaining to \(\omega =\omega _1\) and \(\omega =\omega _2\)

*r*asymptotics, namely [compare Eq. (20)]

*r*. On the other hand, integration of this eigenequation subject to the boundary conditions (17) produces

*r*as long as \(\chi _{kn0}(r) \ge 0\). As vanishing of \(\chi _{kn0}(r_0)\) at some \(r_0 > 0\) [note that \(\chi _{kn0}(0)=0\) per the boundary conditions (17)] would contradict these findings, the corresponding NO must be nodeless and the moments \(\{ \mu _{kn0,j} \}\) must be positive-valued for all

*j*.

### 2.4 Approximate solutions of Eq. (16)

*r*approximation (30) is asymptotically exact at the \(\lambda _{kn0} \rightarrow 0\) limit and is expected to be accurate for

*r*satisfying the inequality \(\exp (\omega _k \, r^2) \ll \beta _{kn0}^4\).

In principle, an approximation valid throughout the entire range of radial distances could be obtained by stitching the above result with the large-*r* asymptotics given by Eq. (20). However, the resulting quantization of the natural amplitudes turns out to be unduly sensitive to the choice of the stitching point, rendering such an approach impractical. Instead, one can either set \(A_{kn0}\) to zero or determine it by matching the radial location of the outermost node of the approximant (30) with that of its exact counterpart. When employed in conjunction with exact natural amplitudes, either of those choices produced NOs of remarkable accuracy (see the next section of this paper).

*r*asymptotics of the approximant (30) is not compatible with the polynomial (20) unless the latter is set to zero. In that case, one obtains

*n*follows the asymptotics of \((-1)^{n+1} \, \sqrt{2} \, \exp [-\pi (n+\frac{1}{4})]\). In light of these facts, one expects the quantity \(\Delta _{kn0}\) in the expression

*n*only weakly.

*n*radial nodes). Although the approximants (35) are rather inaccurate, for a given \(\omega _k\) they form a set of orthonormal functions whose simplicity facilitates evaluation of asymptotic expressions for matrix elements of one-electron operators. In particular, one obtains

## 3 Numerical verification of theoretical predictions

*m*that read, respectively, \(\frac{127-48 \pi +36 \ln 2}{72 \, \pi } \; \omega ^{-1} \approx 5.11440 \cdot 10^{-3} \; \, \omega ^{-1} \) and \(\frac{-2053+720 \pi -300 \ln 2}{1800 \, \pi } \; \omega ^{-1} \approx 1.77291 \cdot 10^{-4} \; \omega ^{-1} \), in agreement with the previously published estimates [17]. The numerical data of the present study that are quoted below have been calculated with 2000-digit arithmetic available within the algebraic manipulation software [33] from matrices truncated at

*M*= 1000.

*s*-type NOs are ordered according to decreasing absolute magnitudes of the corresponding negative-valued natural amplitudes, with \(n=1\) assigned to first such NO, etc.). Inspection of Figs. 1 and 2 reveals remarkable accuracy of the reduced radial parts [given by \(\exp (-\frac{3\,\omega }{8} \, r^2) \, \chi _{n0}(\omega ;r)\) or, equivalently, \(\exp (\frac{\omega }{8} \, r^2) \, r \, \phi _{n0}(\omega ;r)\)] of the 10th, 20th, and 30th NOs computed from the approximants (30) at both \(\omega =\frac{1}{2}\) and \(\omega =\frac{1}{10}\). The excellent agreement at sufficiently small radial distances is unaffected by the choice of \(A_{kn0}\) being either set to zero or determined by matching the radial location of the outermost node of the approximant with that of its exact counterpart. On the other hand, as expected, neither choice gives rise to correct asymptotics at \(r \rightarrow \infty \) [which is, however, faithfully reproduced by \(\exp (-\frac{3\,\omega }{8} \, r^2) \, \chi ^{>}_{kn0}(r)\)]. These features of the approximants are found to carry over not only to other values of

*k*, such as in the case of \(\omega =\omega _3\) (Fig. 3a), but also to arbitrary magnitudes of the confinement strength such as \(\omega =1\) (Fig. 3b) and even the \(\omega \rightarrow \infty \) limit of vanishing correlation [Fig. 3c, note that \( \bar{\chi }_{n0}(\bar{r}) = \lim \limits _{\omega \rightarrow \infty } \, \omega ^{-1/4} \, \chi _{n0}(\omega ;\omega ^{-1/2} \, \bar{r}) \)], uncovering approximate universal scaling of the

*s*-type NOs.

*r*by the approximate reduced radial parts of the NOs displayed in Figs. 1, 2, and 3 are much larger than those of the actual natural orbitals. Second, as the number of nodes increases with

*n*, the spacing between them becomes proportionally reduced, leaving the radial extents of the NOs barely changed. This observation explains the large-

*n*asymptotic behavior of the expectation values of both the kinetic energy operator [Eq. (36)] and the operator describing the interaction with the confining potential [Eq. (37)]. The asymptotic constancy of \(v_{n0}(\omega )\) with respect to

*n*, which reflects the almost constant radial extents of the NOs, is consistent with the quadratic dependence of \(t_{n0}(\omega )\) on

*n*that is reminiscent of that known for a particle in one-dimensional box. These asymptotics are expected to persist for all values of \(\omega \), including the limit of \(\omega \rightarrow \infty \) at which the interelectron interaction becomes vanishingly small in comparison with the other components of the Hamiltonian (1). When juxtaposed against the linear dependences on

*n*exhibited by the kinetic and potential energy components pertaining to the

*s*-type wavefunctions of the three-dimensional harmonic oscillator (which become more and more diffuse upon successive excitations), these asymptotics vividly illustrate the fundamental difference between the natural orbitals and the eigenfunctions of the one-electron Hamiltonians that are left in \(\hat{H}\) upon removal of the interelectron interaction term.

The natural amplitudes \(\{ \lambda _{kn0} \}\) computed for \(k=1\), \(k=2\), \(k=3\), and \(k=4\) are found to follow the asymptotics (34) as attested by both the smallness and the weak dependence on *n* of the expression \(\big ( -\frac{\pi \, \omega _k^2 \, \lambda _{kn0}}{8 \, C_{k1}} \big )^{-1/4}-n\) (Fig. 4a). In general, the rate at which this asymptotics is approached appears to diminish with increasing *k*. Quite unexpectedly, the natural amplitudes \(\{ \lambda _{nl}(\omega ) \}\) computed at arbitrary values of \(\omega \) and even for \(l \ne 0\) turn out to exhibit the same large-*n* behavior (which is reflected in the values of \(\big [ -\frac{\pi \, \omega ^2 \, \lambda _{nl}(\omega )}{4 \, \Psi (\omega ;\vec 0,\vec 0)} \big ]^{-1/4}-n\) plotted against *n* in Fig. 4b). The analogous plots of the quantities \(\big [ -\frac{\sqrt{3} \; (E_k-2\;h_{kn0})}{\pi \, \omega _k} \big ]^{1/2} -n\) (Fig. 5a) and \(\big ( \frac{\sqrt{3} \; \omega _k \, \eta _{kn0}}{8 \, C_{k1}} \big )^{-1/2}-n\) (Fig. 5b) confirm the predicted asymptotics (36), (37), and (40).

## 4 Conclusions

The radial components of the natural orbitals (NOs) pertaining to the \(^1S_+\) ground state of the two-electron harmonium atom are found to satisfy homogeneous differential equations at the values of the confinement strength \(\omega \) at which the respective correlation factors are given by polynomials. Together with the angular momentum *l* of the NOs, the degrees of these polynomials determine the orders of the differential equations, eigenvalues of which (arising from well-defined boundary conditions) yield the natural amplitudes. In the case of \(l=0\), analysis of these equations uncovers certain properties of the NOs whereas application of a WKB-like approximation produces asymptotic expressions for both the NOs and the corresponding natural amplitudes that hold when the latter are small negative numbers. Extensive numerical calculations reveal that these expressions remain valid for arbitrary values of \(\omega \). The approximate *s*-type NOs, which are remarkably accurate at sufficiently small radial distances and exhibit universal scaling, differ qualitatively from the eigenfunctions of the core Hamiltonian even at the \(\omega \rightarrow \infty \) limit of vanishing electron correlation.

*n*th negative-valued natural amplitude and \(n^{4}\) tends to a constant at the \(n \rightarrow \infty \) limit implies the asymptotic \(n^{-8}\) decay of the occupation numbers \( \{\nu _{nl}(\omega ) \} \) of the NOs. In fact, the expression

*n*, appears to be universal, i.e., to hold for arbitrary \(\omega \) and

*l*. Interestingly, naive summation of this expression over

*n*yields collective occupancies that exhibit the same proportionality constant of \(\frac{|\Psi (\omega ;\vec {0},\vec {0})|^2}{\omega ^4} \) as those that follow from the large-

*l*asymptotics of Hill [34].

## Notes

### Acknowledgements

The research described in this publication has been funded by the National Science Center (Poland) under Grant 2016/21/B/ST4/00597. Support from MPI PKS Dresden is also acknowledged.

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