(a) Exciting and probing a chiral vibrational wavepacket in planar COFCl. A Raman excitation (magenta arrows) creates a superposition of the two lowest out-of-plane vibrational states of the central C-atom via an electronically excited state with vibrational level |v⟩, followed by one-photon ionization (blue) populating a vibrational level |v ′ ⟩ in the parent ion's ground state. (b) Visualization of the field configuration. Electric field coils generate the static field orienting the molecules. Pump (magenta) and probe (blue) pulses are circularly polarized in a plane perpendicular to this field. (c) Rank profile of the Euler angle distribution P (ω) from Eq. (12) for different temperatures obtained with the Lebedev-Laikov method of degree L = 131. The horizontal gray line indicates the value 10 −6 used to determine the maximum rank of P (ω). (d) Relative quadrature error of PECD calculated with Eq. (8) for the process shown in (a) and (b) at 6 eV photoelectron energy. The orientation average is weighted with the Euler angle distribution from Eq. (12) at rotational temperature 5 K. The vertical gray lines indicate the number of sampling points needed to achieve degree L = 41 for a method with efficiency E = 1 and E = 2/3. The inset displays a zoom of the same data, with the horizontal gray line highlighting a relative error of 1 %. Methods are Lebedev-Laikov quadrature (L), the spherical designs from Ref. 69 (D), Gauss-Legendre product grids (GL×T), equidistant step methods with the same sampling density for β and γ (T×T) and near-uniform spherical coverings by Fibonacci spheres (F).

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... an example, we consider the pump-probe experiment proposed in Ref. 13. The process, illustrated in Fig. 3(a), comprises three-photons: a circularly polarized Raman pulse excites a chiral vibrational wavepacket in planar COFCl molecules, which is then probed by onephoton ionization with a circularly polarized femtosecond pulse. In contrast to the preceding example, the initial orientational distribution of the molecules is not uniform. ...

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... which is then probed by onephoton ionization with a circularly polarized femtosecond pulse. In contrast to the preceding example, the initial orientational distribution of the molecules is not uniform. Instead, the molecules are oriented uniaxially by a static electric field aligned with the propagation direction of the pulses, as displayed in Fig. 3(b). Consequently, the Euler angle distribution does not break the cylindrical symmetry of the light-molecule interaction and it is still possible to perform a two-angle orientation average. Although COFCl is a member of the C s symmetry group, the induced vibrational dynamics break the planar symmetry. Hence, no symmetries remain that ...

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... T is the rotational temperature of the ensemble, N (T ) is a normalization factor, E is the static electric field in the laboratory frame, d is the permanent dipole moment of the molecule in the molecular frame and R(ω) is the rotation matrix for the transformation between the two frames of reference. Figure 3(c) displays the rank profile of P (ω) for different values of T . Since P (ω) essentially describes the distribution of the rotational states of the molecule, it is not surprising that the rank profile of P (ω) also follows a Boltzmann distribution, down to the precision limit resulting from round-off errors. ...

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... a result, significantly more quadrature points are necessary to reach the same level of precision as in Sec. VI A. This is illustrated in Fig. 3(d), which shows the quadrature error of different methods as a function of the number of sampling points, n. The quadrature error reaches machine precision for methods with degree L ≥ 41, which is only slightly less than the estimated maximum rank. For spherical Gauss quadratures, which have efficiency E ≈ 1, this corresponds to about 600 ...

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... to about 600 sampling points, whereas for spherical designs and other methods with efficiency E ≈ 2/3, e.g. the Gauss product method, this amounts to more than 900 quadrature points. Methods with efficiency E ≈ 1 are the best choice for evaluating the orientation average numerically exactly. Furthermore, for larger quadrature errors (up to 0.1 %) Fig. 3(d) shows that the spherical Gauss methods outperform all other methods. In particular, the Lebedev-Laikov method exhibits the smallest error for n ≳ 100, demonstrating a slight advantage over the S 2 Gauss quadrature from Ref. 36 (not shown in the ...

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... are not necessarily the optimal choice in this regime, because their error increases significantly when their degree of exactness is not high enough. Consequently, methods with more gradually increasing rank profiles can compete with Gauss methods or can even outperform them for low number of sampling points. This is demonstrated in the inset of Fig. 3(d), which compares different methods for quadrature errors larger than 0.1 % and less then 100 sampling points. In the present example, the equidistant step method (T×T) allows to evaluate the PECD with relative error of 1 % with 72 points, and the Gauss product method (GL×T) yields a relative error of about 0.1 % with only 50 sampling ...

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... particular, Eq. (B1c) has been used to calculate the rank profiles shown in Fig. 6. To calculate the rank profile of the Euler angle distribution P (ω) shown in Fig. 3(c), we used Eq. (B2b). with zero degree (left column) and nonzero degree (right column). The number of sampling points, n, and the degree, L, for which the corresponding rank profile is shown is denoted in braces behind the method name. Methods are spherical Gauss quadrature from Ref. 35, 60, 65, 67, 80, and 81 (G), spherical designs from ...