Photoionization cross section

As an example of the numerical treatment of the electronic continuum for a two-electron molecule, consider a calculation of the photoionization spectrum of HeH⁺. The system is chosen to demonstrate the applicability of the present approach also to heteronuclear systems. The purpose of this section is to demonstrate how the box discretization leads to the ability to calculate transitions into the electronic continuum of states. For the sake of simplicity, the investigation is limited to the case of a parallel orientation of the molecular axis with respect to the field as well as to a consideration of only single internuclear distance R= 1.45a0.

The partial (X¹Σ → ¹Σ) photoionization cross section (for single electron ejection) is given by

σ(E) = 4π²(E−E₀) 3c

X

µ

|D_µ(E)|² (3.1)

where E0 is the ground-state energy and µ specifies all possible degenerate final states with energy E. For the calculation of the electronic dipole transition moment

D_µ(E) =hΨ_Eµ|D|Ψˆ ₀i (3.2)

with the dipole operator ˆD in length gauge the ground and final state wave functions Ψ₀ and Ψ_Eµare required. The latter has to be properly energy normalized. Since the box-discretized calculation yields wave functions with bound-state normalization, it is neces-sary to perform a renormalization [47]. In the present case of the two-electron diatomic problem, a further complication arises from the occurrence of a multiply degenerate elec-tronic continuum even between the first and second ionization thresholds [84, 85]. In [84]

5 10 15 20 25 30

Photoelectron energy (eV)

10^-3 10^-2 10^-1

Dipole transition moment (a.u)

Figure 3.1.: Electronic dipole transition moments between the ground state of HeH⁺ and

1Σ continuum states (at R = 1.45a0). Every point represents the transition to a single discretized continuum state. The shown results were obtained using different box sizes in order to achieve a denser spectrum. In this way an almost continuous spectrum is yielded.

(Published in [61].)

a one-center approach was used and renormalization was performed by characterizing the states, i. e. attributing to every state a channel indexµ, according to their leading l quantum number at the asymptotic limit. Within every channel the renormalization factors were obtained from the (partial) density of states. A detailed investigation re-veals, however, that in some cases it is possible that two degenerate states are both simultaneously dominated by two differentl quantum numbers. In order to be able to handle also this situation the following procedure was developed.

If the photoelectron is far away from the remaining ion left in its 1σ ground state, an energy-normalized molecular wave function with singlet spin symmetry behaves asymp-totically as

Ψ_Eµ(1,2)→ 1

√

2(ψ1σ(1)ψµ(2) +ψµ(1)ψ1σ(2)). (3.3) Since for large distances Rξ/2 → r and η → cosθ, the one-electron continuum wave

functionψ_µ with energy =E−_1σ is asymptotically given by ψµ(ξ, η, φ)→ 1

√ 2πk

2

rsin{kr+ 2

kln 2kr+δµ()}Ω_µ(θ, ) exp(imφ) (3.4) due to the standing-wave boundary conditions imposed by the box. In Eq. (3.4) the wavenumber k = √

2, the phase shift δµ(), and the angular functions Ωµ(θ, ) are introduced. The latter functions fulfill the normalization condition

2π Z π

0

Ω²_µ(θ, ) sinθdθ= 1. (3.5)

Matching the calculated bound-state-normalized molecular wave function to the form given in Eqs. (3.3) and (3.4) gives the desired normalization factor. In the atomic case the angular part is simply given by a single spherical harmonic, Ω_µ(θ, ) exp(imφ) = Ylm(θ, φ). Different channels µare then uniquely characterized by the angular quantum number l. In a diatomic molecule, l is no longer a good quantum number, and thus Ω_µ(θ, ) exp(imφ) is analyzed in terms of linear combinations of spherical harmonics with a common value of m,

Ω_µ(θ, ) exp(imφ) =X

l

C_µl()Y_lm(θ, φ). (3.6)

Note, the coefficientsC_µl() are continuous functions with respect to energy .

Using box discretization and thus imposing the boundary condition that the wave func-tions are zero atξmaxleads to a selection of the continuum solutions obtained for a given box size. If the box is sufficiently large, the continuum wave functions have reached their correct asymptotic behavior at ξmax. In this case, they are satisfying the equation sin(kr_max+²_kln 2kr_max+δ_µ()) = 0 withr_max=Rξ_max/2 and thus Eq. (3.4) reduces to

ψ_µ(ξ, η, φ)→ 1

√ 2πk

2

r sin{k(r−r_max) + 2

kln(r/r_max)}Ω_µ(θ, ) exp(imφ) (3.7) The normalization can now be performed without knowledge of δµ(). The correctly renormalized electronic transition dipole momentsD_µ(E) for different but still unknown channels µ are given in Fig. 3.1. In the calculation the following basis set parameters were used. Inξ direction ˜n_ξ= 300 and k_ξ= 8 with a knot sequence {ξ_i^b}that is slightly denser for small values ofξ. This is done using a geometric progression withgξ = 1.1 for the first 30 intervals and a linear knot sequence afterwards. A linear knot sequence{η_i^b} with ˜n_η = 30, k_η = 6 was used inη direction. The configuration set includes about 3000

5 10 15 20 25 30

Photoelectron energy (eV)

0 0.5 1 1.5

Cross section (Mb)

Figure 3.2.: Partial (X¹Σ→¹Σ) photoionization cross-section of HeH⁺ in the fixed-nuclei approximation at R = 1.45a₀. The result obtained in the present method (solid line) is compared to the result given in [86] (red dashed line). The blue arrow indicates the position of the second (almost invisible) resonance. (Published in [61].)

configurations with both electrons in low energy orbitals to describe electron correlation.

In addition, five configuration series are used to describe the continuum. In those one electron occupies one of the 1σ−5σ orbitals and the other is put into one of the possible 1500 orbitals withNη 64.

Since the density of states obtained with a single box is insufficient for both the deter-mination of the coefficientsC_µl() and a resolution of narrow resonant structures in the dipole moments, a variation of the box sizeξmaxin between 500 and 510 with step size 1 was performed. In Fig. 3.1, 5 differentµchannels are easily recognized. The occurrence of the five channels is due to the limitation of the calculation to 06N_η 64. The knowl-edge of the expansion coefficients allows the attribution of every state (and its dipole moment) to one channel. For a given channel µ the continuous dipole function D_µ(E) is obtained from its values at discrete energies via interpolation. Using Eq. (3.1) the photoionization cross section can be calculated. In Fig. 3.2 the result is compared to the calculation in [86] where explicitly correlated basis functions and the complex-scaling

method were used. The agreement is very good, even on the absolute scale. The main difference occurs at the first resonance at about 16 eV where a small shift is observed.

The probable reason for the difference is the more accurate inclusion of correlation in [86]

that leads to a lower energy for the doubly excited state responsible for the resonant structure in the spectrum. As was discussed in [86] and is confirmed by the present calculations, the second resonance is practically invisible due to the contributions of different channels canceling each other.