{'fraction_non_alphanumeric': 0.04946376340591485, 'fraction_numerical': 0.015924601884952875, 'mean_word_length': 4.220902612826603, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Two-particle coalescence conditions revisited

The notion of the n-th order local energy, generated by the n-th power of the Hamiltonian, has been introduced. The n-th order two-particle coalescence conditions have been derived from the requirements that the n-th order local energy at the coalescence point is non-singular and equal to the n-th power of the Hamiltonian eigenvalue. The first condition leads to energy-independent constraints. The second one is state-specific. The analysis has been done using a radial, one-dimensional, model Hamiltonian. The model is valid in the asymptotic region of r ~ 0. The coalescence conditions set the relations between the expansion coefficients of the radial wave function into a power series with respect to r.


I. INTRODUCTION
Two-particle Hamiltonians, since nearly a century, have been used as a playground for testing suitability of a variety of methods and models to the description of properties of simple quantum systems. The simplest ones, the hydrogen-like atom and the spherical harmonic oscillator not only served as a test of quantum mechanics, but also as a basis for the development of analytical methods of solving the Hamiltonian eigenvalue problem and for the studies on the properties of its spectrum. By a proper change of coordinates, a two-particle Hamiltonian can be expressed as a sum relative motion. Consequently, the resulting twoparticle eigenvalue problem can be separated to two independent one-particle problems: one describing the free motion of the centre of mass and the second one, describing the relative motion of the two particles. In general, an external potential prevents the separability. An exception is the parabolic confinement. Two-particle Schrödinger equations, independent of the form of the interaction potential, are separable also in parabolic external potentials.
The interaction potential V in the equation describing the relative motion depends only on the interparticle distance r 12 ≡ r. Therefore, the Hamiltonian commutes with the angular momentum operators and, after the elimination of the angular part, its eigenvalue problem transforms to an infinite set of eigenvalue equations H Φ νλ (r) = E νλ Φ νλ (r) (1) of one-dimensional radial Hamiltonians where λ = 0, 1, 2, . . . is the angular momentum quantum number and µ is the reduced mass. 1 In the case of two identical fermions the wave function is symmetric (a singlet pair) if λ is even and antisymmetric (triplet) if λ is odd.
The point r = 0 corresponds to the coalescence of the two particles. The information about the behaviour of the exact wave functions at this point is important both for the understanding of general properties of many-particle systems and for the construction of variational trial functions. Therefore, the subject attracted much interest -see, e.g. [1][2][3][4][5][6][7][8][9][10][11][12][13][14] 1 It is convenient to use the radial Hamiltonian in the self-conjugate form which does not contain the first-order derivative. and references therein. Hamiltonian (2) offers a simple and easy to treat model. Though this model describes a "bare" pair of particles, under certain assumptions it can be generalized so that, after some modifications, it can be also applied to studies on the coalescence conditions in N -particle systems. In particular, if the distance r between two coalescing particles is much smaller than the distance to any of the remaining N − 2 particles, then one may expect that the influence of these particles on the properties of the coalescing pair can be described by a two-particle effective potential, parametrically dependent on the coordinates of the other particles. After an approximate separation of the centre of mass of the two particles and a spherical averaging, we end up with a radial equation describing the pair of particles in the vicinity of the coalescence point. From here one can derive constraints on the asymptotic form of the exact wave functions at the limit of r = 0. The best known of these constraints, Kato's cusp condition [1], can be derived from the requirement that in the case of two Coulomb-interacting particles the local energy is non-singular at r = 0. Higherorder coalescence constraints have been obtained using some other universal properties of the exact wave functions in the vicinity of r = 0 [2][3][4][5]. In a similar way the effects of the electron-electron coalescence on the electron density can be investigated. The earliest works on this subject were published already half a century ago [6], but the links with the structure of the first-order density matrix and of the natural orbitals have been discovered very recently [7][8][9]. A detailed analysis of the wave function coalescence constraints, referred to as general coalescence conditions for the exact wave functions has been given by Kurokawa et al. [10][11][12].
A sensitive tool for the exploration of the behaviour of Φ νλ (r) at r = 0 is the local energy.
Let Ψ νλ (r) be a trial function which for specific values of parameters, and for r << 1, is equal to Φ νλ (r). We define the n-th order local energy as If Ψ νλ (r) = Φ νλ (r), i.e. it is the exact eigenfunction of H, then E (n) νλ (r) = E n νλ . In this paper we derive the general two-particle coalescence conditions, as the ones of Kurokawa et al. [10][11][12], from the properties of the local energies at r = 0. 2  separable, model in which the radial part of the interaction is described by Hamiltonian (2).
The constraints are derived using the information about the behaviour of the wave function at r = 0. Therefore, the results are valid for both discrete and continuous spectra.
For eigenfunctions of Hamiltonian (2) the local energies of all orders have to be nonsingular at the coalescence point. This property implies that the wave function has to compensate r = 0 singularities generated by the Hamiltonian. The constraints imposed by the enforcement of this property are, for a given λ, energy-independent, i.e. they are common to all wave functions Ψ νλ (r) which belong to the space spanned by the eigenfunctions of the radial Hamiltonian (2). In the case of Coulomb-interacting particles and n = 1 this constraint leads to Kato's cusp condition [1].
If at the coalescence point Ψ νλ (r) behaves as an eigenfunction of H corresponding to the eigenvalue E νλ then This property is, by definition, energy-dependent. Therefore the constraints imposed by its enforcement are state-specific.
In the next section general coalescence conditions are derived and in Section III an example of application is given. A graphical method of deriving explicit form of the energyindependent coalescence conditions is presented in the Appendix. Atomic units are used in this paper.

II. COALESCENCE CONSTRAINTS
If we assume a Coulomb-like behaviour of V (r) at r = 0, expand it to a power series about this point, and retain the first q + 2 terms of the expansion then we get where, in the case of a parabolic confinement, α 2 contains a contribution from the external potential. For Coulombic systems and for r << 1, the term corresponding to p = −1 is dominant and determines the physical character of the potential. If α −1 > 0 then the potential is repulsive (describing, for example, the interaction between two electrons); if α −1 < 0, it can describe an attractive electron-nucleus interaction. Formally, the potential parameters are unrestricted. If α −1 = 0, then the potential loses its Coulombic character but, nevertheless, the higher-order coalescence conditions impose wave function constraints dependent on α p with p > 0 (cf. Subsection II A 2). 3 The higher-order terms in Eq. (5) improve the analytic representation of the potential in the vicinity of the coalescence point, but have no physical meaning for large values of r. For example, a term α q r q , α q > 0, q ≥ 1 prevents any kind of dissociation of the two particles.
For r ∼ 0 the radial functions describing the coalescing pair, can be expressed as where ν refers to the energy eigenvalue and prefactor r λ+1 compensates the singularity at r → 0 generated by the centrifugal potential λ(λ+1)/r 2 [16]. As the normalisation condition we set ψ νλ (0) = 1. Since Ψ νλ (r) is analytic [15] it can be expanded to a power series of r.
We assume that ψ νλ (r) has an asymptotic expansion of order s about r = 0: (to simplify notation, indices ν and λ in c i are omitted). For r << 1, Eqs. (6) and (7), with properly defined expansion coefficients, give a correct representation of the eigenfunctions of H at the vicinity of the coalescence point. Note that Ψ νλ (r) provides an asymptotic form of the corresponding eigenfunction but has no physical meaning outside of the coalescence region. First, the model is physically meaningful only for small r. Second, Ψ νλ (r), as defined in Eq. (6), is not square-integrable in the range r ∈ (0, ∞).
Our aim is to derive the conditions limiting the values of the coefficients of expansion (7) from two requirements defining the behaviour of E (n) νλ (r) at r = 0. The first group of these conditions, referred to as the energy-independent constraints, is a consequence of the requirement that E (n) νλ (r) is non-singular at r = 0. The second group, the energy-dependent constraints, follows from the condition given by Eq. (4). Since we are interested in the properties of the local energies at r = 0, the behaviour of the wave functions outside of the asymptotic region is irrelevant for this discussion.
Since lim 0 , the first-order local energy is asymptotically, at r ∼ 0, equal to where, according to Eq. (14), As it follows from Eq. (16), E −1 = 0. Therefore, the first order coalescence constraint reads If c and the procedure described by Eqs. (9) -(18) may be repeated with superscripts (0) and (1) replaced, respectively, by (1) and (2). where with c and where c (n) From here we have Corollary: The coalescence constraints are independent of the free parameter, α 0 , in the potential.
According to Theorem 1, α −1 c (n−2) 0 Since E νλ can be included to α 0 , the Corollary implies that the coalescence conditions derived from the requirement that the n-th order local energies are non-singular at r = 0 are, as expected, independent of the energy eigenvalues.
The coefficient c where A graphical method of deriving t mi coefficients is presented in the Appendix. From the first-order (n = 1) constraint one can get the well known cusp condition generated by the Coulomb singularity: This condition defines the coefficient of the linear term of the expansion of the radial function.
In the case of two-electron coalescence it is equivalent to Kato's cusp condition [1]. For n = 2 we get, additionally, the second-order constraint: where c 1 can be eliminated using Eq. (27). In energy-independent constraints the odd coefficients, c 1 , c 3 , c 5 , . . ., can be expressed as linear functions of the even ones, c 0 , c 2 , c 4 , . . .: 4 c 4 .
· · · · · · · · · · · · where D (1) In general, The asymptotic expansion of the wave function [Eq. (9)] becomes where we set s = 2n − 1. Alternatively, we can arrange the expansion according to the even coefficients: where Then, where A (j) p is a j p -term combination of p-fold products of A i . In particular, A p is given in the Appendix. For n = 4 the constraints for the pure Coulomb potential read 1 = α 1 c 0 + A 2 c 3 . As it was shown above, c We assume that the energy-independent constraints are fulfilled. Consequently, the local energies of all orders from 1 to n are non-singular at r = 0. The energy-dependent constraints result from the application of condition (4). It is convenient to include α 0 -the free parameter in the potential -to the eigenvalue. We definẽ where, to simplify notation, indices ν and λ in are omitted. 4 We assume that E Therefore, formally, the replacement of H byH and E (n) (0) byẼ (n) (0) is equivalent to setting In particular, where f i are linear combinations of products of the potential parameters α p , p = −1, 1, 2, 3, 4 and A p , p = 2, 3, 4, 5, 6. Using relation (43), one can replace the eigenvalue parameter in the energy-dependent constraints (42) by c 2 .

C. The lowest-order constraints
For the reader's convenience, several first coefficients of the expansion of ψ (0) νλ are given:

III. EXAMPLE
We consider two Coulomb-interacting particles in a parabolic confinement, i.e. we set α −1 = 0, α 2 = 0 and α i = 0 if i = −1, 2. The radial Schrödinger equation (1) reads where subscript ν has been replaced by the corresponding energy . In the case of two electrons (µ = 1/2, α −1 = 1) the interaction is repulsive. In the case of two µ = 1/2 particles with opposite charges (electron-positron pair) α −1 = −1 -the interaction is attractive. The spectrum of the confined system (α 2 > 0) in both cases is purely discrete. In the unconfined systems (α 2 = 0) the positive energy spectrum is continuous and the continuum spreads from 0 to ∞. In the case of electron-positron pair discrete states with < 0 also appear.
As it results from Eq. (46), the transformation r → −r is equivalent to the replacement According to Eqs. (45) the coefficients in the asymptotic expansion (7) are equal to · · · · · · · · · .
It is convenient to split expansion (7) to two parts: the first one (F α −1 ) one describing the interaction of unconfined particles and the second one (∆ α 2 ), describing the effect of confinement: The expression for α −1 = −1 can be obtained by the substitution r → −r. The parabolic confinement does not affect c 1 , c 2 and c 3 . Therefore, up to the cubic term, the asymptotic expansion (7) of ψ λ is the same whether or not there is a parabolic confinement.
In Fig. 1 (48) including only the free-particle term F 1 (dashed lines), and also the confinement contribution ∆ α 2 (dash-dotted lines). Surprisingly, the free-particle wave functions corresponding (with term ∆ α 2 neglected). 5 Explicit expressions for the continuous spectrum wave functions can be found, e.g. in the monograph by Bethe and Salpeter [18]. The expansion given by Eq. (48) with ∆ α2 (r) = 0 is the same as the one obtained from the expansion of the exact eigenfunctions.

IV. FINAL REMARKS
We introduced the notion of the local energy of the n-th order (3)  α 2 = 1/4), differences between the wave functions with confinement effects included and neglected only become noticeable for r > 1. ACKNOWLEDGEMENT We thank Dr. Heinz-Jürgen Flad (Technische Universität München) for useful discussions.

DISCLOSURE STATEMENT
No potential conflict of interest was reported by the authors.

Equation c
(2) is represented in Fig. 4 -the graph has been obtained by connecting three segments of Fig. 3 (for n = 2) into one diagram.
In general, expression (26) for c  The uppermost part (the first five rows) of the most general graph (all α p = 0) is shown in Fig 5. Contributions from α −1 and from A j are present in all orders. Contributions from α 1 start from the second order. From the third order up, we have also contributions from α 2 and α 3 . In the next order contributions from α 4 and α 5 appear. And so on -each next order activates two more terms of the expansion of V (r).
The graph corresponding to the pure Coulomb potential, i.e. to the case of α p = 0 if p = −1, is given in Fig. 6. It is isomorphic with the Pascal triangle. There are j p paths one can reach node c