Static and dynamic polarizabilities of the non-relativistic hydrogen molecular ion

A system of charged particles described by a Hamiltonian H is in a state of energy E with wavefunction Ψ. It is perturbed by an electric multipole O^(λ)_μ of multipolarity λ and projection μ oscillating at angular frequency ω,

$$\begin{equation} V_{\rm pert} = O^{(\lambda )*}_\mu\, {\rm e}^{-{\rm i}\omega t} + O^{(\lambda )}_\mu\, {\rm e}^{{\rm i}\omega t}. \end{equation} \tag{ 1 }$$

Following the perturbation theory presented in [27] and [28], the equation of the first-order correction Ψ^{( ± )} to the wavefunction Ψ is

$$\begin{equation} (H-E\pm \omega )\Psi ^{(\pm )}(\omega ) = O^{(\lambda )}_\mu \Psi . \end{equation} \tag{ 2 }$$

The frequency-dependent multipole polarizability α_λμ(ω) for the state Ψ is given by

$$\begin{equation} \alpha _{\lambda \mu }(\omega ) = \langle \Psi ^{(+)}(\omega )+\Psi ^{(-)}(\omega )|O^{(\lambda )}_\mu |\Psi \rangle . \end{equation} \tag{ 3 }$$

For the H⁺₂ molecular ion, the three-body Hamiltonian involving the Coulomb interactions between the three particles but no spin effects reads

$$\begin{equation} H = -\frac{1}{2m_p} (\Delta _1 + \Delta _2) -\frac{1}{2} \Delta _e + \frac{1}{R}-\frac{1}{r_{e1}}-\frac{1}{r_{e2}} , \end{equation} \tag{ 4 }$$

where 1 and 2 refer to the proton coordinates, m_p is the proton mass and e refers to the electron coordinate. It depends upon the interproton distance R and the two distances r_e1 and r_e2 between the electron and the protons. Its eigenfunctions $\Psi ^{L^\pi \sigma }_M$ depend on the total orbital (or rotational) angular momentum L, its projection M, the parity π and the spatial symmetry σ of the protons. The total spin of the protons is fixed by σ = ( − 1)^S. The many bound states corresponding to the Σ_g electronic configuration have π = σ = ( − 1)^L and the three weakly bound levels corresponding to the Σ_u electronic configuration have π = −σ = ( − 1)^L. The corresponding energies are denoted as $E^{L^\pi \sigma }$. For simplicity, we shall drop the σ superscript in the following as the electric transition operators conserve the spatial proton symmetry. The polarizabilities are calculated below for the levels of the Σ_g electronic configuration.

We approximatively solve the Schrödinger equation for the hydrogen molecular ion H⁺₂ without using the Born–Oppenheimer approximation and its corrections. To this end, the Lagrange-mesh method [23–25] in perimetric coordinates [20, 21] is used as explained in [18, 19].

The perimetric system of coordinates involves the three Euler angles ψ, θ and ϕ which characterize the orientation of the molecular plane and the three combinations of interdistances:

$$\begin{eqnarray} &&x = R-r_{e2}+r_{e1}, \nonumber\\ &&y = R+r_{e2}-r_{e1}, \\ &&z = -R+r_{e2}+r_{e1}.\nonumber \end{eqnarray} \tag{ 5 }$$

The wavefunctions are expanded over a basis of angular functions as [18]

$$\begin{equation} \Psi ^{L^{\pi }}_M = \sum _{K=0}^L \mathcal{D}_{MK}^{L\pi }(\psi ,\theta ,\phi ) \Phi _K^{L^{\pi }}(x,y,z). \end{equation} \tag{ 6 }$$

The internal degrees of freedom are described by Φ^Lπ_K(x, y, z) in terms of the perimetric coordinates. The normalized angular functions $\mathcal{D}_{MK}^{L\pi }(\psi ,\theta ,\phi )$ are defined for K ⩾ 0 by

$$\begin{eqnarray} &&\fl \mathcal{D}_{MK}^{L\pi }(\psi ,\theta ,\phi ) = \frac{\sqrt{2L+1}}{4\pi }(1+\delta _{K0})^{-1/2}\nonumber\\ &&\times\, \big[D_{MK}^L(\psi ,\theta ,\phi )+ \pi (-1)^{L+K} D_{M\;-K}^L(\psi ,\theta ,\phi )\big] \end{eqnarray} \tag{ 7 }$$

where D^L_MK(ψ, θ, ϕ) represents a Wigner matrix element. They have parity π and change as π( − 1)^K under permutation of the protons. Hence, Φ^Lπ_K is symmetric for ( − 1)^K = σπ and antisymmetric for ( − 1)^K = −σπ, when x and y are exchanged. When (6) is introduced in the Schrödinger equation, one obtains the system of coupled equations

$$\begin{equation} \sum _{K=0}^L \mathcal{H}_{K^{\prime }K}^{(L^{\pi })}(x,y,z) \Phi _{K}^{L^{\pi }}(x,y,z) = E \Phi _{K^{\prime }}^{L^{\pi }}(x,y,z), \end{equation} \tag{ 8 }$$

where ${\mathcal {H}}^{(L^{\pi })}_{K^{\prime }K} = \big\langle {\mathcal {D}}^{L\pi }_{MK^{\prime }}\big|H\big|{\mathcal {D}}^{L\pi }_{MK}\big\rangle$ are the matrix elements of the Hamiltonian evaluated between angular functions [17, 18].

The electric transition operators can be written in perimetric coordinates as [19]

$$\begin{eqnarray} &&\fl O^{(\lambda )}_\mu = \sum _{\kappa =0}^\lambda \frac{\mathcal {A}_\kappa ^{(\lambda )}(x,y,z)}{1+\delta _{\kappa 0}}\nonumber\\ &&\times\, \big[D^\lambda _{\mu \kappa }(\psi ,\theta ,\phi ) + (-1)^\kappa D^\lambda _{\mu -\kappa }(\psi ,\theta ,\phi )\big]. \end{eqnarray} \tag{ 9 }$$

The perimetric coefficients ${\mathcal {A}}_{\kappa }^{(\lambda )}$ are given in terms of the interprotonic distance R and the polar components of the electron coordinate (ρ, ζ) for the dipole (λ = 1) as

$$\begin{equation} \mathcal {A}_0^{(1)} = -(1+M^{-1})\zeta , \quad \mathcal {A}_1^{(1)} = -(1+M^{-1})\frac{\rho }{\sqrt{2}} \end{equation} \tag{ 10 }$$

with the total mass M = 2m_p + 1 and for the quadrupole (λ = 2) as

$$\begin{eqnarray} &&\mathcal {A}_0^{(2)} = \frac{1}{2} [R^2 - \gamma (2\zeta ^2 - \rho ^2)],\nonumber\\ &&\mathcal {A}_1^{(2)} = - \sqrt{\frac{3}{2}} \gamma \zeta \rho , \quad \mathcal {A}_2^{(2)} = - \sqrt{\frac{3}{8}} \gamma \rho ^2 \end{eqnarray} \tag{ 11 }$$

with γ = 1 − 2M⁻¹ − M⁻². See [18, 19] for the expressions of R, ρ and ζ in terms of the perimetric coordinates.

Now, one needs to solve equations (2). Their solutions depend on both M and μ for each L^πσ. For this purpose, we expand the $\Psi ^{(\pm )L^{\pi }}_{M\mu }$ functions as

$$\begin{eqnarray} &&\fl \Psi ^{(\pm )L^{\pi }}_{M\mu } = \sum _{L^{\prime }} (L\lambda M\mu |L^{\prime } M+\mu )\nonumber\\ &&\times\, \sum _{K^{\prime }=0}^{L^{\prime }} \mathcal{D}_{M+\mu K^{\prime }}^{L^{\prime }\pi ^{\prime }}(\psi ,\theta ,\phi ) \Phi _{K^{\prime }}^{(\pm )L^{\prime \pi ^{\prime }}}(x,y,z), \end{eqnarray} \tag{ 12 }$$

where $\Phi _{K}^{(\pm )L^{\prime \pi ^{\prime }}}$ also depends on L^π but this dependence is understood. The Clebsch–Gordan coefficients are introduced for later convenience. After the substitution of the functions $\Psi ^{L^{\pi }}_M$ and $\Psi ^{(\pm )L^{\pi }}_{M\mu }$ in equation (2), the dependence on M and μ disappears and one obtains the system of differential equations:

$$\begin{eqnarray} &&\fl \sum _{K=0}^{L^{\prime }} \big[{\mathcal {H}}^{(L^{\prime \pi ^{\prime }})}_{K^{\prime }K} - (E^{L^\pi }\mp \omega )\delta _{K^{\prime }K}\big]\Phi ^{(\pm )L^{\prime \pi ^{\prime }}}_{K} \nonumber \\ &&= \sum _{\kappa =0}^{\lambda }\frac{{\mathcal {A}}_{\kappa }^{(\lambda )}}{1+\delta _{\kappa 0}} \sum _{K=0}^{L} F^{LL^{\prime }\lambda }_{KK^{\prime }\kappa } \Phi ^{L^\pi }_{K}. \end{eqnarray} \tag{ 13 }$$

The coefficients $F^{LL^{\prime }\lambda }_{KK^{\prime }\kappa }$ are given by

$$\begin{eqnarray} &&\fl F_{K K^{\prime }\kappa }^{L L^{\prime }\lambda } = \frac{(2L+1)^{1/2}}{[(2L^{\prime }+1)(1+\delta _{K^{\prime }0}) (1+\delta _{K0})]^{1/2}} \nonumber \\ &&\times\, \lbrace (L \lambda K \kappa |L^{\prime } K^{\prime }) + (-1)^\kappa (L \lambda K -\kappa |L^{\prime } K^{\prime }) + \pi ^{\prime }(-1)^{L^{\prime }+K^{\prime }} \nonumber \\ &&\times\,[(L \lambda K \kappa |L^{\prime } -K^{\prime }) + (-1)^\kappa (L \lambda K -\kappa |L^{\prime } -K^{\prime }) ] \rbrace . \end{eqnarray} \tag{ 14 }$$

See the definition in [19] where a factor (LλMμ|L'M') is missing on the left-hand side of equation (21).

In order to solve the system (13), we apply the Lagrange-mesh method [23, 18] which is an approximatively variational calculation with Lagrange basis functions F^K_ijk(x, y, z) [18, 19]. The Lagrange functions have the remarkable property that they vanish at all mesh points (h_xu_i, h_yv_j, h_zw_k) of an associated mesh, but one. The u_i, v_j and w_k are the zeros of Lagrange polynomials of respective degrees N_x, N_y and N_z and, h_x, h_y and h_z are scaling factors. The variational calculation is simplified by the use of a Gauss quadrature associated with the mesh. In this method, the Φ_K(x, y, z) functions of equation (6) are expanded as [18]

$$\begin{eqnarray} &&\fl \Phi _K^{L^{\pi }}(x,y,z) = \sum _{i=1}^{N} \sum _{j=1}^{i-\delta _K} \sum _{k=1}^{N_z} C_{Kijk}^{L^{\pi }} [ 2 (1+\delta _{ij})]^{-1/2}\nonumber \\ &&\times\,\big[ F^K_{ijk}(x,y,z) + \sigma \pi (-1)^K F^K_{jik}(x,y,z) \big], \end{eqnarray} \tag{ 15 }$$

where we use the same number N of mesh points and the same scale factor h for the two perimetric coordinates x and y. Because of the symmetrization, the sum over j is limited by the value i − δ_K, where δ_K is equal to 0 when ( − 1)^K = σπ and to 1 when ( − 1)^K = −σπ. A similar expansion is used for Φ^{( ± )Lπ}_K(x, y, z) with coefficients C^{( ± )Lπ}_Kijk.

With the Gauss quadrature, equations (8) become a linear system for the coefficients C^Lπ_Kijk [18],

$$\begin{eqnarray} &&\fl \sum _{K=0}^{L} \sum _{ijk} [(1+\delta _{ij})(1+\delta _{i^{\prime }j^{\prime }})]^{-1/2}\nonumber\\ &&\times\, \big[\big\langle F^{K^{\prime }}_{i^{\prime }j^{\prime }k^{\prime }}\big|{\mathcal {H}}^{(L^{\pi })}_{K^{\prime }K}\big|F^{K}_{ijk}\big\rangle+\sigma \pi (-)^{K} \big\langle F^{K^{\prime }}_{i^{\prime }j^{\prime }k^{\prime }}\big|{\mathcal {H}}^{(L^{\pi })}_{K^{\prime }K}\big|F^{K}_{jik}\big\rangle \big]\nonumber\\ &&\times\,C_{Kijk}^{L^{\pi }} = E C_{K^{\prime }i^{\prime }j^{\prime }k^{\prime }}^{L^{\pi }}. \end{eqnarray} \tag{ 16 }$$

The eigenvalues of the Hamiltonian matrix are denoted as $E^{L^\pi }_v$, where v is the level of vibrational excitation. The corresponding wavefunctions are available analytically when the coefficients C^Lπ_Kijk are introduced in (15) and (6). In our previous work [19], the spectra of the four lowest vibrational, v = 0 to 3, and 41 rotational states, L = 0 to 40, were obtained as well as analytic approximations for the corresponding wavefunctions that are employed here.

A similar treatment of the differential system (13) leads to the inhomogeneous algebraic system of equations:

$$\begin{eqnarray} &&\fl \sum _{K=0}^{L^{\prime }}\sum _{ijk} \big\lbrace [(1+\delta _{ij})(1+\delta _{i^{\prime }j^{\prime }})]^{-1/2} \big[\big\langle F^{K^{\prime }}_{i^{\prime }j^{\prime }k^{\prime }}\big|{\mathcal {H}}^{(L^{\prime \pi ^{\prime }})}_{K^{\prime }K}\big|F^{K}_{ijk}\big\rangle \nonumber \\ &&+\,\sigma \pi (-)^{K} \big\langle F^{K^{\prime }}_{i^{\prime }j^{\prime }k^{\prime }}\big|{\mathcal {H}}^{(L^{\prime \pi ^{\prime }})}_{K^{\prime }K}\big|F^{K}_{jik}\big\rangle \big]\nonumber\\ &&-\,(E^{L^\pi } \mp \omega ) \delta _{K^{\prime }K}\delta _{i^{\prime }i}\delta _{j^{\prime }j}\delta _{k^{\prime }k}\big\rbrace C_{Kijk}^{(\pm )L^{\prime \pi ^{\prime }}} \nonumber \\ &&= \sum _{\kappa =0}^\lambda (1+\delta _{\kappa 0})^{-1/2} \mathcal {A}_{\kappa }^{(\lambda )}(h_x u_{i^{\prime }}, h_y v_{j^{\prime }}, h_z w_{k^{\prime }}) \sum _{K=0}^{L} F_{KK^{\prime }\kappa }^{LL^{\prime }\lambda } C_{Ki^{\prime }j^{\prime }k^{\prime }}^{L^{\pi }}.\nonumber\\ \end{eqnarray} \tag{ 17 }$$

The coefficients $F_{KK^{\prime }\kappa }^{LL^{\prime }\lambda }$ are for the dipole transition operator (λ = 1):

$$\begin{eqnarray} &&\fl F_{KK^{\prime }0}^{LL^{\prime }1} = (L1K0|L^{\prime }K^{\prime }), \nonumber \\ &&\fl F_{KK^{\prime }1}^{LL^{\prime }1} = (L1K1|L^{\prime }K^{\prime })(1+\delta _{K0})^{1/2}\nonumber\\ &&-\,(L1K-1|L^{\prime }K^{\prime })(1+\delta _{K^{\prime }0})^{1/2}, \end{eqnarray} \tag{ 18 }$$

and for the quadrupole transitions operator (λ = 2):

$$\begin{eqnarray} &&\fl F_{KK^{\prime }0}^{LL^{\prime }2} = (L2K0|L^{\prime }K^{\prime }), \nonumber \\ &&\fl F_{KK^{\prime }1}^{LL^{\prime }2} = (L2K1|L^{\prime }K^{\prime }) (1+\delta _{K 0})^{1/2}\nonumber\\ &&-\, (L2K-1|L^{\prime }K^{\prime })(1+\delta _{K^{\prime } 0})^{1/2}, \nonumber \\ &&\fl F_{KK^{\prime }2}^{LL^{\prime }2} = (L2K2|L^{\prime }K^{\prime }) (1+\delta _{K 0})^{1/2}\nonumber\\ &&+\, (L2K-2|L^{\prime }K^{\prime })(1+\delta _{K^{\prime } 0})^{1/2} - (L22-1|L^{\prime }1) \delta _{K1} \delta _{K^{\prime }1}.\nonumber\\ \end{eqnarray} \tag{ 19 }$$

With the expansions of the wavefunction $\Psi ^{L^{\pi }}_M$ and of the first order correction $\Psi ^{(\pm )L^{\pi }}_{M\mu }$ due to an electric perturbation of the hydrogen molecular ion, the polarizability (3) can be directly calculated as

$$\begin{eqnarray} &&\fl \alpha _{\lambda \mu }^{(LM)}(\omega ) = \sum _{L^{\prime }=|L-\lambda |}^{L+\lambda } \frac{2L+1}{2L^{\prime }+1} (L\lambda M\mu |L^{\prime }M+\mu )^2 \alpha ^{L L^{\prime }}_{\lambda }(\omega ),\nonumber\\ \end{eqnarray} \tag{ 20 }$$

where

$$\begin{eqnarray} &&\fl \alpha ^{(L L^{\prime })}_{\lambda }(\omega ) = \frac{2L^{\prime }+1}{2L+1} \sum _{\kappa =0}^\lambda \frac{1}{1+\delta _{\kappa 0}} \sum _{K^{\prime }=0}^{L^{\prime }} \sum _{K=0}^{L} F_{KK^{\prime }\kappa }^{LL^{\prime }\lambda } \nonumber \\ &&\times\, \big[ \big\langle \phi ^{(+)L^{\prime \pi ^{\prime }}}_{K^{\prime }}\big| \mathcal{A}^{(\lambda )}_\kappa\big| \phi ^{L^\pi }_{K} \big\rangle + \big\langle \phi ^{(-)L^{\prime \pi ^{\prime }}}_{K^{\prime }} \big| \mathcal{A}^{(\lambda )}_\kappa \big| \phi ^{L^\pi }_{K} \big\rangle \big]. \end{eqnarray} \tag{ 21 }$$

The perimetric matrix elements $\big\langle \phi ^{(\pm )L^{\prime \pi ^{\prime }}}_{K^{\prime }} \big| \mathcal{A}^{(\lambda )}_\kappa \big| \phi ^{L^\pi }_{K} \big\rangle$ are calculated from (10) and (11) by integration over the perimetric coordinates with the volume element (x + y)(y + z)(z + x) dx dy dz. In practice, the sums over K and K' are truncated at K_max. The α^(LM)_λμ components satisfy the symmetry relations α^{(L − M)}_{λ − μ} = α_λμ^(LM).

The remaining calculation of the matrix elements is particularly simple within the Lagrange-mesh method. Because the Lagrange functions vanish at all mesh points but one, the matrix elements in (21) are simply obtained with the Gauss quadrature as

$$\begin{eqnarray} &&\fl \big\langle \phi ^{(\pm )L^{\prime \pi ^{\prime }}}_{K^{\prime }} \big| \mathcal{A}^{(\lambda )}_\kappa \big| \phi ^{L^\pi }_{K} \big\rangle \approx \sum _{i=1}^N \sum _{j=1}^{i-\delta _K} \sum _{k=1}^{N_z} C_{K^{\prime }ijk}^{(\pm )L^{\prime \pi ^{\prime }}}\nonumber\\ &&\times\, C_{Kijk}^{L^\pi } \mathcal {A_\kappa ^{(\lambda )}}(h u_i,h v_j,h_z w_k) \end{eqnarray} \tag{ 22 }$$

where $\delta _K = \max (\delta _{K_i},\delta _{K_f})$.

Due to the orthogonality relations of the Clebsch–Gordan coefficients, the polarizabilities averaged over M,

$$\begin{equation} \alpha ^{(L)}_{\lambda }(\omega ) = \frac{1}{2L+1} \sum _{M=-L}^L \alpha ^{(LM)}_{\lambda \mu }(\omega ), \end{equation} \tag{ 23 }$$

do not depend on μ and can be easily calculated with

$$\begin{equation} \alpha ^{(L)}_{\lambda }(\omega ) = \frac{1}{2\lambda +1} \sum _{L^{\prime }=|L-\lambda |}^{L+\lambda } \alpha ^{(LL^{\prime })}_\lambda (\omega ). \end{equation} \tag{ 24 }$$

Then, any μ-component of the polarizability (20) and the average polarizability (23) can be obtained from the 2λ + 1 coefficients $\alpha ^{(LL^{\prime })}_\lambda (\omega )$.

After solving the linear system (17) for the coefficients C^{( ± )Lπ}_Kijk, the approximation to the first correction $\Psi ^{(\pm )L^{\pi }}_{M\mu }$ are obtained. For convenience, we use the same wavefunctions Ψ^Lπ_M for the free hydrogen molecular ion as in [19]. The same numbers of mesh points are employed: N = N_x = N_y = 40 for the x and y coordinates and N_z = 20 for the z coordinate. The scale parameters are h = h_x = h_y = 0.14 and h_z = 0.4. For a given K value, the total number of basis states is then 16 400 or 15 600 depending on the parity of K. The size of the sparse Hamiltonian matrix is larger by about a factor (K_max + 1) when K is limited by K ⩽ K_max ⩽ L. For L > 2, like in [18], calculations are performed with K_max = 2. The conventional value for the proton mass: m_p = 1836.152701 au is used.

Table 1 shows a convergence test of the energies and static polarizabilities of the first four vibrational levels v = 0 to 3 for L = 0 as a function of the numbers of mesh points. The nonlinear scale parameters are h_x = h_y = 0.14 and h_z = 0.4. The results of table 1 show that the convergence with N_z is reached faster than with N, i.e. already with N_z = 20. For the slower N convergence, one observes that adding five points leads to an exponential improvement by a factor of about 30 for both energies and polarizabilities. By extrapolation, the absolute accuracy of the dipole polarizability α⁽⁰⁾₁ with (40, 20) is about 10⁻¹⁰ for the (0⁺, 0) state and decreases to 10⁻⁸ for (0⁺, 1), 10⁻⁷ for (0⁺, 2) and 10⁻⁶ for (0⁺, 3). We have checked for L = 0 and L = 30 that the v = 0 polarizabilities do not change by more than 2 × 10⁻⁹ when h_x = h_y vary from 0.12 to 0.16 and h_z varies from 0.35 to 0.60. Note that the Hamiltonian matrices completely change when the scale parameters are modified. The chosen values, where the accuracy is optimal over the whole rotational bands, lie within this stability plateau.

Table 1. Convergence of the energies E^{0 +}_v and dipole polarizabilities α⁽⁰⁾_{1, v} as a function of the numbers of mesh points N and N_z for the ground state (0⁺, 0) and the three lowest vibrational levels (0⁺, 1), (0⁺, 2) and (0⁺, 3). The most accurate theoretical and experimental results are also presented.

N	Nz	$E^{0^+}_0$	α(0)1, 0	$E^{0^+}_1$	α(0)1, 1
20	20	−0.597 138 545	3.168 934	−0.587 144 6	3.9035
25	20	−0.597 139 051 2	3.168 732 36	−0.587 155 355	3.897 787
30	20	−0.597 139 062 838	3.168 725 999	−0.587 155 670 45	3.897 569 78
35	20	−0.597 139 063 116 8	3.168 725 807 04	−0.587 155 679 003	3.897 563 54
40	20	−0.597 139 063 123 2	3.168 725 802 80	−0.587 155 679 206	3.897 563 36
35	30	−0.597 139 063 116 8	3.168 725 807 03	−0.587 155 679 003	3.897 563 54
40	30	−0.597 139 063 123 2	3.168 725 802 79	−0.587 155 679 206	3.897 563 36
50	25	−0.597 139 063 123 4	3.168 725 802 65	−0.587 155 679 212	3.897 563 35
[9]			3.168 7256
[10]			3.168 726
[14]			3.168 725 803		3.897 563 360
[15]			3.168 725 802 67
[3] (exp.)			3.167 96(15)
N	Nz	$E^{0^+}_2$	α(0)1, 2	$E^{0^+}_3$	α(0)1, 3
20	20	−0.577 660	4.8621	−0.568 50	6.27
25	20	−0.577 748 20	4.824 013	−0.568 884	6.030
30	20	−0.577 751 788	4.821 5997	−0.568 907 47	6.010 59
35	20	−0.577 751 901 0	4.821 504 47	−0.568 908 459	6.009 379
40	20	−0.577 751 904 4	4.821 500 47	−0.568 908 497	6.009 329
35	30	−0.577 751 901 0	4.821 504 47	−0.568 908 459	6.009 379
40	30	−0.577 751 904 4	4.821 500 47	−0.568 908 497	6.009 329
50	25	−0.577 751 904 5	4.821 500 36	−0.568 908 498	6.009 327
[14]			4.821 500 365		6.009 327 479

The accuracy of the present matrix elements as well as the accuracies of other observables in [18] and of the transition probabilities in [19] indicate that the present wavefunctions have relative accuracies better than 10⁻⁷ for v = 0 to 2 over most of the various rotational bands and are a little less good for v = 3.

A comparison with the best theoretical results is also presented in table 1. The results of Yan et al [15] for the ground state and of Hilico et al [14] for the vibrationally excited states are more accurate than ours. When the number of mesh points are increased to N = 50, N_z = 25 our results are in agreement with those presented by Hilico et al [14]. The ground-state polarizability has been investigated in several papers [9, 10, 13–15] and all of them are in agreement in at least seven significant digits. However, a difference with the experimental result [3] arises at the fifth significant digit. Relativistic corrections only slightly reduce the problem [13]. The origin of this disagreement remains unclear but does not arise from inaccuracies in the numerical resolution of the three-body Schrödinger equation.

Table 2 presents an extensive list of the components α^{(LL − 1)}₁, α^(LL)₁ and α^{(LL + 1)}₁ of the dipole polarizabilities for the full rotational band (L = 0 to 39) corresponding to the lowest vibrational level v = 0. According to (20), any M-component can be calculated with multiplications by Clebsch–Gordan coefficients. The average dipole polarizability is presented in the last column. The error is at most of a few units on the last displayed digit.

The static dipole polarizabilities α^(LM)₁₀ and their averages α^(L)₁ are presented in table 3 for the first and second rotational levels L = 1, 2 and the four lowest vibrational levels v = 0 − 3. Our results for the (1⁻, v) levels agree with the results obtained by Moss [11] and by Moss and Valenzano [12]. The (1⁻, 0) theoretical result falls within the experimental error bar [4].

The dynamic electric dipole polarizability of the (0⁺, 0) ground state is presented in table 4 for some values of the frequency ω up to about the dissociation threshold. The calculation is performed under the same conditions as for ω = 0 in table 2. The accuracy is about 10⁻⁹. The two partial components α^{( ± )}₁ = 〈Ψ^{( ± )}(ω)|O⁽¹⁾_μ|Ψ〉 of equation (3) are also displayed. One observes that α^{( − )}₁ becomes progressively larger than α^{( + )}₁.

The (0⁺, 0) dynamic polarizability is also displayed in figure 1 as a function of the frequency ω. It is a monotonically increasing function. No resonant frequencies appear because electric dipole transitions are forbidden. In fact, a first resonance should correspond to the weakly bound (1⁻, 0) level of the Σ_u excited electronic configuration at ω ≈ 0.097 395 556 but it is not visible because the transition probability is too small.

Zoom In Zoom Out Reset image size

Static electric quadrupole polarizabilities are obtained from (20) for λ = 2 with the same unperturbed wavefunctions as for λ = 1. Table 5 presents the results for the three vibrational levels v = 0 − 3 of the lowest rotational state L = 0. For the ground state (0⁺, 0) the absolute accuracy, which was investigated by varying the number of mesh points, is about 10⁻⁶. A previous more accurate result obtained by Zhang and Yan [29] agrees in nine significant digits. For higher vibrational quantum numbers, the accuracy decreases by one digit when v increases by one unit.

Table 5. Static quadrupole polarizabilities α⁽⁰⁾₂ for the four lowest L = 0 vibrational levels of the hydrogen molecular ion. The results are obtained with N = 40, N_z = 20 and h = 0.14, h_z = 0.4.

v	$E^{0^+}_v$	α(0)2, v
0	−0.597 139 063 123	1371.895 156
[29]		1371.895 160 758
1	−0.587 155 679 21	1884.814 26
2	−0.577 751 904 5	2556.2505
3	−0.568 908 498	3432.596

Figure 2 presents the dynamic electric quadrupole polarizability α⁽⁰⁾₂ for the ground state (0⁺, 0) as a function of the angular frequency ω. Below ω = 0.03, four resonant frequencies are visible in the figure. They correspond to the first four quadrupole transitions from the ground state: (0⁺, 0) → (2⁺, v) for v = 0 − 3. The theoretical frequencies $E^{2^+}_v - E^{0^+}_0$ are about 0.000 793 86, 0.010 735 43, 0.020 098 83 and 0.028 903 07 [19] in increasing order of the final vibrational quantum number v (see table 1 of [19]). Over the considered range, the dynamical polarizability can be approximated by the first four terms of its expansion,

$$\begin{eqnarray} &&\fl \alpha ^{(0)}_2 \approx \frac{1}{5} \sum _{v=0}^3 \left[ \frac{|\mbox{$\langle $}\Psi ^{2^+}_v||O^{(\lambda )}||\Psi ^{0^+} \mbox{$\rangle $}|^2}{E^{2^+}_v - E^{0^+}_0 + \omega } + \frac{|\mbox{$\langle $}\Psi ^{2^+}_v||O^{(\lambda )}||\Psi ^{0^+} \mbox{$\rangle $}|^2}{E^{2^+}_v - E^{0^+}_0 - \omega } \right].\nonumber\\ \end{eqnarray} \tag{ 25 }$$

The approximation consists in neglecting higher bound states and the continuum. The corresponding squared reduced matrix elements of the (2⁺, v) → (0⁺, 0) transitions can be deduced from tables 3 and 4 of [19] as 2.705 894, 0.098 499, 0.000 836 and 0.000 022. These fast decreasing numbers indicate that the jumps at the resonant frequencies become progressively narrower which makes them hardly visible in the figure for v = 2 and 3.

Zoom In Zoom Out Reset image size

For ω = 0, approximation (25) is smaller than the result of table 5 by 4.793. Strikingly, over the ω range of figure 2, this difference remains comprised between 4.79 and 4.80. An extrapolation to v > 3 indicates that, except at resonances, the contribution of higher excited bound states is negligible. We thus interpret the difference between the exact result and approximation (25) as the contribution of the continuum.

If we consider the hydrogen molecular ion H⁺₂ immersed in a Debye plasma, then a screening effect appears and the Coulomb potential of the Hamiltonian (4) has to be replaced by [26]

$$\begin{equation} V=\frac{{\rm e}^{-R/D}}{R}-\frac{{\rm e}^{-r_{e1}/D}}{r_{e1}} - \frac{{\rm e}^{-r_{e2}/D}}{r_{e2}}, \end{equation} \tag{ 26 }$$

where D is the temperature- and density-dependent Debye length. Using the Lagrange-mesh technique is as easy as for the free case since one only has to replace numerical values of the Coulomb potentials appearing in (4) by numerical values of potential (26) in the code. Table 6 presents the energies E and the static dipole polarizabilities α^(L)₁ for the four lowest vibrational levels of L = 0 and 1 calculated with the Debye length D = 1. After optimizing the Lagrange-mesh parameters, we use the same number of points for each coordinate N_x = N_y = N_z = 30 and the nonlinear parameters h_x = h_y = 0.1 and h_z = 0.9. Both the energies and the polarizabilities significantly increase with respect to tables 1 and 3. The (0⁺, 0) ground state result can be compared with the one computed by Kar and Ho [26].

Table 6. Energies, components and averages of the dipole polarizabilities of the molecular ion H⁺₂ placed in a hot plasma for L = 0 and 1 and v = 0 − 3. The Debye length is D = 1. The results are for N = N_z = 30 and h = 0.1, h_z = 0.9.

v	$E^{0^+}_0$			α(0 1)1	α(0)1
0	−0.045 224 024 1544			93.479 429 41	31.159 809 80
[26]	−0.045 223				31.19
1	−0.039 889 095 0941			136.460 0271	45.486 675 70
2	−0.035 215 772 9611			195.744 4170	65.248 139 03
3	−0.031 147 0986			277.0149	92.3383
v	$E^{1^-}_0$	α(1 0)1	α(1 1)1	α(1 2)1	α(1)1
0	−0.045 039 701 165 7	14.762 177 88	24.894 288 19	54.402 520 66	31.352 995 58
1	−0.039 724 411 284 0	22.545 997 09	34.869 374 93	79.931 687 71	45.782 353 24
2	−0.035 069 356 9614	33.742 628	47.946 493	115.379 733	65.689 6180
3	−0.031 017 615	49.798	64.826	164.334	92.9862