Relative quadrature error of PECD calculated with Eq. (8) as resulting from the 2+1 REMPI process shown in Fig. 1(a) applied to randomly oriented molecules. We calculate PADs at 0.58 eV photoelectron energy, averaged over the Euler angles β and γ for α = 0. The error is shown for the maximum PECD signal, corresponding to the forward direction (θ k = 0). Methods are Lebedev-Laikov quadrature (L), spherical designs from Ref. 69 (D), equidistant step methods with the same sampling density for β and γ (T×T) and near-uniform spherical coverings by Fibonacci spheres (F). The arrows indicate the degree L for which the error of the Lebedev-Laikov method and the spherical designs reaches machine precision.

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... the superscript denotes ionization with left-(−) and right-(+) circular polarization. The relative error of the maximum PECD is shown in Fig. 2 for various quadrature methods. We observe that the equidistant product method as well as the uniform spherical coverings converge slowly compared to spherical Gauss and Chebyshev quadratures. In particular, the error of reaches machine precision for methods with L ≥ 11, corresponding to 50 sampling points for LebedevLaikov quadrature ...

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... PECD is a normalized difference of PADs, convergence for L ≥ 10 also holds. The aforementioned considerations explain the steep drop-off of the relative errors for spherical Gauss and Chebyshev methods in Fig 2. In contrast, spherical coverings and equidistant step methods lack such a feature. Overall this example demonstrates the superior performance of methods with non-zero degree if the rank of the integrand is small. ...

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... spheres provide the smallest error for 500 ≳ n ≳ 100, even outperforming the spherical designs and Gauss product methods. In contrast to our other examples, the quadrature error of methods with L = 0 shows a similar super-exponential scaling as for methods with L > 0, up to n ≈ 400, before asymptotically showing a slow expo- nential decay (cf. Figs. 2 and 4(c)). Comparison with the rank profile of P (ω) indicates that the quadrature error of the L = 0 methods is for a small number of sampling points dominated by the anisotropy originating from the weighted Euler angle ...

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... described in the perturbative regime because of the high intensity of the light. Thus, no ab initio prediction about the rank profile is possible and despite the molecules being uniformly distributed in the gas phase a high rank is to be expected. A comparison of the quadrature error for different methods is provided in Fig. 4(c). Comparison with Fig. 2 and 3(d) shows, that the SO(3) quadratures exhibit convergence behavior similar to their two-angle counterparts. This indicates that the categorization in Sec. IV provides a reliable initial estimate of a method's expected performance, regardless of the problem's ...