Understanding the physics involved in the interaction of a low energy positron with atoms and molecules is a long standing goal with both fundamental and technological implications [1,2]. The study in great detail of the phenomena of positron-matter interaction has been driven by the development of increasingly efficient positron energy moderators, which allow to obtain positron beams with appreciable fluxes and low energy spread [2]. Modern buffer-gas trap techniques allows the generation a high flow positron pulse, in the range of few eV and with energy spread of tens meV [3,4]. With such efficient mono-energetic positron beams, several state-resolved cross sections have been obtained, regarding to total, electronic excitation, ionization, and positronium formation in atomic and molecular targets [5], as well as vibrational excitation [6,7] and annihilation in molecules [4,8].

In particular the vibrational excitation of molecules by positron impact has attracted the attention of the scientific community, due to the striking manner in which a low energy positron couples to the nuclear degrees of freedom of polyatomic molecules. The physical richness of this phenomenon may involve vibrational resonances and even bound states, in which the positron is temporary trapped, before its final fate: annihilation. The theoretical models devoted to describe experimental data of vibrational excitation cross sections [9,10] and resonant annihilation spectra [11], coincide in attributing a leading role to molecular properties such as polarizability, dipole moment, ionization potential and number of π electrons [12–14].

In this work a generalization to the full three-dimensional positron-diatom configuration space, of a previously proposed two-dimensional model [15] (hereafter referred as paper I), is presented. In paper I the 0 → 1 vibrational excitation cross section of a hydrogen-like diatomic oscillator, was computed using time-dependent wave-packet propagation techniques over a two-dimensional model potential. The results suggested that the molecular polarizability of the target, as a function of the iternuclear separation, is a key property in describing the excitation of the fundamental mode of the oscillator. Here, the model potential is extended to cover the three coordinates of the positron-diatom complex and the scattering problem solved within the close-coupling formalism. The procedure is applied to the positron-H₂ system, as in paper I, but the positron-N₂ case is also studied. The present results corroborate the previous ones and suggest that the coupling of the positron to the fundamental vibrational mode of a homo-nuclear diatom can be described by a correlation-polarization potential, which involves, as the key property, the (an)isotropic molecular polarizability. The elastic cross sections for positron scattering from H₂ and N₂ are also well described below the positronium formation threshold.

In next section the model potential is described in detail. The close-coupling method is briefly summarized in Section 3. Results and discussion is presented in Section 4 and some conclusions given in Section 5.

Following paper I the potential energy surface (PES) is written as the sum of oscillator (OSC) and positron (POS) components: V R , r , θ = V o s c R + V p o s R , r , θ , (1) where R is the inter-nuclear distance and (r, θ) are the polar-Jacobi coordinates of the positron with respect to the center of mass of the target (see Figure 1 in [16]). The OSC part is represented as a harmonic oscillator potential V o s c R = 1 2 μ ω 2 R 2 , (2) with mass μ and frequency ω. The pair {μ, ω} is chosen to be, in atomic units (a.u.), {918.075, 0.020} and {12,861.925, 0.011} for H₂ and N₂, respectively.

In Eq. 1 the POS term refers to the positron-target interaction and includes the static (V _st ) and correlation-polarization (V _cp ) potentials. The former is taken to be the Hartree-Fock r-dependent spherically symmetric electrostatic potential of the diatom, clamped at the equilibrium bond distance. This term, for H₂ and N₂, is represented in the form V s t r = κ ∑ i = 1 3 s i ⁡ exp − t i r − r i , (3) where s _i , t _i and r _i are adjustable parameters and κ is a free parameter chosen ad hoc, in order to better reproduce reported total cross sections below the positronium formation threshold. It will be called the static parameter, which determines the amount of contribution of the V _st component to the potential.

The V _cp energy term, in turn, is represented using the analytical form proposed by Mitroy and Ivanov [17], V c p R , r , θ = − α R , θ 2 r 4 f ρ r , (4) where f ρ r = 1 − exp − r 6 ρ 6 , (5) is a cut-off function which damps the −1/r ⁴ term at short distances and ρ is a free parameter chosen ad hoc, in order to better reproduce reported total cross sections below the positronium formation threshold. It will be called the correlation-polarization parameter, which determines the amount of contribution of the V _cp component to the potential.

In Eq. 4, α(R, θ) represents the molecular polarizability as a function of the inter-nuclear separation (R) and the positron-diatom relative orientation angle (θ). A simple form for this quantity will be α R , θ = α 0 R + α 2 R P 2 cos ⁡ θ , (6) where α ₀(R) and α ₂(R) are the isotropic and anisotropic polarizabilities, respectively, and P ₂(cos θ) is the second-degree Legendre polynomial.

The dependence of α _⋆(R) (where ⋆ ≡ 0 or 2) on the R coordinate was represented in the form: α ⋆ R = α ⋆ 0 + γ ∑ i = 1 4 a i R i , (7) which fits, with high accuracy, the ab initio data reported by Kołos and Wolniewitz for H₂ [18] and the set of ab initio points previously reported for N₂ [19]. In Eq. 7 α _⋆(0) is the (an)isotropic polarizability at the equilibrium bond distance of the diatomic, a _i are adjustable parameters and γ is a free parameter chosen ad hoc, in order to reproduce reported vibrational excitation cross sections. It will be called the vibrational parameter, which determines the coupling strength between the positron and the target vibrational mode.

All the parameters involved in the present model potentials, for H₂ and N₂, are collected in tables 1 and 2. Figures 1, 2, show the isotropic and anisotropic molecular polarizabilities as functions of R, along with the curves from Eq. 7 for γ = 1 (thick lines) and for those values of γ (thin lines) for which the computed 0 → 1 vibrational excitation cross sections show good agreement with previous reports (see Section 4.2). Contour plots of the PESs for the positron around the diatoms, clamped at the equilibrium bond distances, are shown in Figures 3, 4. The salient feature of the PES for H₂ correspond to a minimum for geometries perpendicular to the molecular axis, suggesting that the positron tends to surrounds the hydrogen target favouring axial configurations. A different situations is observed for the positron-N₂ complex, with the minimum potential energy located on both side of the nitrogen atoms.

The close-coupling formalism for non-reactive collisions has been extensively revisited in the literature [20–22]. Here we present the main ideas behind the method, for the particular case of a structureless particle (the positron) scattering from a vibrating diatomic target.

Assuming a complete, discrete, orthonormal basis of the internal states of the system, the total wave function, in the space fixed reference frame, can be written in the form Ψ J M j l v = ∑ j ′ l ′ ν ′ ψ j ′ l ′ ν ′ J j l ν r r Y j ′ l ′ J M r ̂ , R ̂ χ ν ′ R , (8) where χ _ν′(R) represents vibrational wavefunctions of the target and Y j ′ l ′ J M is a basis for the total angular momentum (J) and projection (M), obtained by coupling the target (j) and the projectile (l) angular momentum, via the Clebsch-Gordan coefficients. Note that j does not include the angular momentum due to the movement of the electrons of the target. After substitution of Eq. 8 in the time-independent Schrödinger equation, the following set of coupled equations, for a given J and M, is obtained d 2 d r 2 + k j ′ ν ′ 2 − l ′ l ′ + 1 r 2 ψ j ′ l ′ ν ′ J j l ν = 2 m ℏ 2 ∑ j ″ l ″ ν ″ ⟨ j ′ l ′ ν ′ J | V | j ″ l ″ ν ″ J ⟩ ψ j ″ l ″ ν ″ J j l ν , (9) where m is the reduced mass of the positron-diatom system, V is the projectile-target interaction potential and k j ′ ν ′ 2 is the wavenumber of the j′ν′-th channel.

Eq. 9 gives an exact solution for the quantum scattering problem, within the space expanded by Eq. 8, with the boundary condition ψ(r = 0) = 0.

As usually done, the interaction potential is represented in a basis of Legendre polynomials V R , r , θ = ∑ λ c λ R , r P λ cos ⁡ θ , (10) leading in Eq. 9 to the matrix elements of the potential ⟨ j ′ l ′ ν ′ J | V | j ″ l ″ ν ″ J ⟩ = ∑ λ c λ ν ′ ν ″ r f λ j ′ l ′ j ″ l ″ J , (11) where c λ ν ′ ν ″ r = ⟨ χ ν ′ R | c λ r , R | χ ν ″ R ⟩ , (12) and f _λ (j′l′j ^″ l ^″ J) are the Percival-Seaton coefficients [21,22].

The set of Eq. 9 is integrated using the Manolopoulos diabatic modified log-derivative method [23], as implemented in the MOLSCAT code [24], with r ranging from 0.01 to 100 a.u, with a fixed step of 0.01 a.u. The diatom was treated as a harmonic oscillator. The expansion (10) runs through the Legendre polynomials of even degree up to λ _max = 12, the vibrational quantum number includes ν = 0, 1, 2, 3 and the total angular momentum ranges from 0 to J _max = 12. Note that convergence is reached for these maximum values of λ, ν and J.

In this work a model potential is proposed, which describes the elastic and vibrational excitation cross sections of H₂ and N₂ by impact of a low energy positron. The potential energy surface depends on the three coordinates involved in a particle-diatom interaction, generalizing to the full configuration space a previous two-dimensional model. The coupling of the positron to the vibrational modes of the target is included in an attractive correlation polarization energy term, properly damped at short range, which involves the target (an)isotropic polarizability as a function of the vibrational coordinate. The dependence on the relative positron-target orientation is accounted for by the anisotropic component of the polarizability. A static spherically symmetric energy term describes the repulsive short range region. In addition to a set of adjustable parameters used to obtain an analytical representation of accurate ab initio points, the potential involves three free parameters, which serve to calibrate the static, correlation-polarization and vibrational parts of the potential, so as to reproduce reported cross sections.

The elastic and 0 → 1 vibrational excitation cross sections, computed using the close-coupling method, show good agreement with previous experimental and theoretical reports. This is achieved for specific values of the free parameters of the model potential. The good description of the here reported 0 → 1 vibrational excitation cross sections for H₂ and N₂, suggests a mechanism for the coupling of the positron to the vibration of the diatoms. In this picture the positron responds instantly to a correlation-polarization potential whose attractiveness oscillates following the vibrational mode of the target. This dependence of the potential on the vibrational coordinate is included through the variation of the molecular polarizability with the internuclear separation. This simple model can be arguably extended to a multimode situation where resonant positron annihilation is mediated by positron binding to a non-polar poliatomic molecule.