Computational studies of x-ray scattering from three-dimensionally-aligned asymmetric-top molecules

Abstract

We theoretically and numerically analyze x-ray scattering from asymmetric-top molecules three-dimensionally aligned using elliptically polarized laser light. A rigid-rotor model is assumed. The principal axes of the polarizability tensor are assumed to coincide with the principal axes of the moment of inertia tensor. Several symmetries in the Hamiltonian are identified and exploited to enhance the efficiency of solving the time-dependent Schrödinger equation for each rotational state initially populated in a thermal ensemble. Using a phase-retrieval algorithm, the feasibility of structure reconstruction from a quasiadiabatically aligned sample is illustrated for the organic molecule naphthalene. The spatial resolution achievable strongly depends on the laser parameters, the initial rotational temperature, and the x-ray pulse duration. We demonstrate that for a laser peak intensity of 5 TW/cm 2 , a laser pulse duration of 100 ps, a rotational temperature of 10 mK, and an x-ray pulse duration of 1 ps, the molecular structure may be probed at a resolution of 1Å. STEFAN PABST, PHAY J. HO, AND ROBIN SANTRA PHYSICAL REVIEW A 81, 043425 (2010)where the expansion coefficients [C(t)]The equation of motion for the expansion coefficients in the interaction picture (subscript I) readsIn the following, we assume we found [C L,I (t)] J 1 τ 1 M 1 J 0 τ 0 M 0 , the solution for the laser-only problem, i.e.,Ĥ rot +Ĥ L (t). Note that in the laser-only Hamiltonian no x-ray field is involved and we will drop the x-ray field indices in its solution.The interaction between the laser-aligned molecules and the x-ray field is taken into account by first-order perturbation theory. The solution of Eq. (7) becomesThe expectation values of interest can be calculated by043425-2 = − 1 2 2 J =0 J M=−J (−1) J +M [α pol ] [J ] L M U [J ] L −M (t), (18) | J τ M (t) has to be known to calculate the expectation values. However, | J τ M (t) can be written as a superposition of | J KM (t) [cf. Eq. (11)], and by using the same argument as for M, only | J KM (t) for K 0 are necessary to build all | J τ M (t) [cf. Eq. ( 13)]. This makes it attractive to propagate | J KM (t) rather than | J τ M (t) . Taking both symmetries together only the states | J KM (t) for 0 K,M J have to be propagated to understand the full system response, which can save up to a factor 4 in computational effort.The decoupling between even and odd K/τ states and M states also enhances efficiency and does not get destroyed by the presence of the interactionĤ L (t). Therefore, many [C L (t)] J τ M J KM remain zero throughout the alignment process. This holds for symmetric-top as good as for asymmetric-top rotors, since each |J τ M can be classified into the class E ± or O ± (cf. Sec. II A). The benefit is not just a speed-up by a factor 4; also the memory requirement to store [C L (t)] J τ M J KM , the matrixĤ L (t), and the matrices cos 2 θ lm is reduced by a factor 4. Memory size can become an issue on PCs when high J states are not negligible.

Theory of retrieving orientation-resolved molecular information using time-domain rotational coherence spectroscopy

“…It is known from the literature [4,23] and confirmed by our calculation that it is the product F ≡ I * τ , i.e., the fluence of the aligning laser pulse, that ρ(φ, θ, χ, t) depends on. This reduces the number of parameters from three to two, namely, {F, T rot }.…”

“…Field-free molecular alignment Theories of molecular alignment by an external laser pulse have been given in the literature [4][5][6]23] and there is no need to repeat them here. These theories are based on the rigid rotor approximation and tell how the molecular rotational wavepacket evolves with time in the presence of an external laser field.…”

“…For laser alignment of linear molecules, we basically use the framework of Ortigoso et al [4] (with corrections of a few minor errors and including the spin statistical weights). For laser alignment of general asymmetric-top molecules, we basically use the framework of Pabst et al [23]. The reader can refer to these references for details.…”

Gas-phase electron diffraction from laser-aligned molecules

“…A limitation of GED is that the random orientation of the molecules in the gas phase greatly reduces the information content of the diffraction patterns. It has been shown that a diffraction pattern from perfectly aligned molecules contains sufficient information to retrieve the structure directly from the diffraction pattern, without the need for comparison with a theoretical model [13,14]. It is, however, so far not possible to experimentally produce perfectly aligned molecules in the gas phase for diffraction experiments.…”

Electron gun for diffraction experiments off controlled molecules

“…Clean molecular samples can be generated by their spatial separation according to shape [13][14][15] and size [16]. Controlling the spatial orientation of the molecules leads to an enhancement of the information that can be retrieved from a diffraction pattern, as proposed theoretically [17][18][19][20] and demonstrated experimentally for x-ray [21,22] as well as for electron diffraction [23,24]. Strong alignment or orientation is generally necessary [11,20] for three-dimensional structure reconstruction [24] and can be provided in cold supersonic molecular beams by strong-field laser alignment and mixed-field orientation [25][26][27][28][29].…”