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x - ray diffraction is a powerful method for investigating structures of molecules . x - ray crystallography has become the standard tool for identifying the structure of large molecules and proteins @xcite . recent developments in x - ray sources have opened new opportunities @xcite for imaging membrane proteins and other macromolecules that can not be crystallized @xcite . one possibility is single - molecule imaging , where one molecule at a time is probed by an intense x - ray pulse @xcite , which subjects the molecule to severe damage @xcite . a series of diffraction patterns has to be collected and classified according to the molecular orientation to get full structural information from randomly oriented molecules @xcite . new iterative phase - retrieval algorithms for noncrystalline specimens have been developed to invert the diffraction data @xcite . alternative approaches with laser - aligned molecules have been proposed , where an ensemble of molecules , rather than a single molecule , is exposed to x - ray pulses @xcite . in this approach the radiation dose for each molecule is strongly reduced and stays well below the critical value for damage @xcite . three - dimensional information on the single - molecule structure can be gained from a well - aligned ensemble by capturing many two - dimensional diffraction patterns . the ability to accumulate the signal over a large number of x - ray pulses reduces the radiation dose further . notice that alignment , rather than orientation , of the molecules is expected to be sufficient for the reconstruction of the molecular structure @xcite . nevertheless , various methods have been proposed to orient molecules @xcite by the linear stark effect @xcite , the ac stark effect @xcite , or two - color fields @xcite . much theoretical and experimental research has been done on laser - induced alignment @xcite . with nonresonant laser frequencies , an induced dipole moment can be created that couples back to the laser electric field and forces the molecule to be aligned . earlier work has focused on one - dimensional alignment of the most polarizable axis by using linearly polarized light @xcite . the demonstration of three - dimensional alignment @xcite , which requires an asymmetric - top molecule and an elliptically polarized pulse ( or two linearly polarized pulses ) opens the door for probing the three - dimensional structure . the coulomb explosion technique has been exploited to detect three - dimensional alignment @xcite , where rotational temperatures down to 1  k have been accomplished @xcite . depending on the duration of the laser pulse , @xmath1 , relative to the rotational period of the molecule , @xmath2 , the alignment dynamics can be classified into three distinct regimes . in the limits of adiabatic ( @xmath3 ) and impulsive ( @xmath4 ) alignment , the dynamics can be described analytically @xcite . impulsive alignment reveals the quantum mechanical nature of this process by showing alignment revivals after the laser pulse is over @xcite . in the adiabatic limit , the alignment dynamics follow the laser pulse shape @xcite . no analytic solution exists in the intermediate regime ( @xmath5 ) , and the time propagation of the molecular ensemble in the presence of the laser pulse has to be performed numerically @xcite . especially the quantum mechanical description of three - dimensional alignment @xcite has proved to be numerically expensive @xcite . a general theory of x - ray diffraction from laser - aligned symmetric - top molecules was developed in ref . applications to adiabatically aligned molecules may be found in refs . reference @xcite describes calculations on x - ray scattering from impulsively aligned molecules , exploiting the alignment revivals for probing field - free molecular structure . it has been shown that , for a symmetric - top molecule containing a single heavy scatterer , a holographic algorithm can successfully reconstruct the molecular structure from an x - ray scattering pattern @xcite .    in this article , we discuss x - ray scattering from an ensemble of rigid , asymmetric - top molecules aligned three - dimensionally by elliptically polarized light at finite rotational temperature . we point out the symmetries in the quantum mechanical theory of three - dimensionally aligned molecules that can be used to significantly reduce the numerical time propagation . we restrict our analysis to electronic and vibrational ground - state configurations and neglect deformations . our approach allows us to investigate x - ray diffraction from molecules in all three alignment regimes ( adiabatic , impulsive , and intermediate ) . in order to probe molecular structure of gas - phase molecules by x - ray scattering , the degree of alignment must be rather high for sufficient resolution . this favors the adiabatic alignment approach @xcite with low - temperature molecules @xcite . in sec . [ sec.theory ] , we present the theory and inherent symmetries of the hamiltonian underlying x - ray diffraction from laser - aligned molecules . section [ sec.methods ] focuses on the numerical implementation and computational efficiency , and basic ideas are presented of the phase - retrieval algorithm that is used for structure reconstruction . in sec . [ sec.results ] , the three - dimensional alignment dynamics and their impact on the reconstruction are demonstrated using the example of the organic molecule naphthalene . we conclude with a discussion of the feasibility and limitations of x - ray diffraction from laser - aligned gas - phase molecules . atomic units @xcite are employed throughout , unless otherwise noted . the hamiltonian for x - ray diffraction from laser - aligned molecules is @xcite @xmath6 where @xmath7 describes the field - free rotational motion of a molecule , @xmath8 is the hamiltonian of the free quantized x - ray fields , and @xmath9 and @xmath10 describe the interactions of a molecule with the laser and x - ray field , respectively . the electronic and vibrational states of the molecule are omitted , since we assume that the molecule remains in its electronic and vibrational ground state throughout our discussion . the x - ray field and its interaction with the molecule is described in a quantized manner . the laser field is formulated as a classical field . the eigenstates of the noninteracting system , i.e. , @xmath11 , are @xmath12 with    @xmath13    where @xmath14 are the rotational eigenstates of an asymmetric - top molecule @xcite and @xmath15 are the x - ray fock states @xcite . the density matrix of the whole system is @xmath16 where the gas - phase molecules @xcite are described by a canonical ensemble @xcite , @xmath17 is the statistical weight , and @xmath18 denotes the initial distribution of all the occupied field modes @xcite .    with including both interactions , @xmath9 and @xmath10 , the states of eq . ( [ eq.eigenstates_overall ] ) are no longer eigenstates of the system . however , each state @xmath19 can be written as @xmath20^{j_1\tau_1m_1;\{n_1\}}_{j_0\tau_0 m_0;\{n_0\ } } }      { \left|j_1\tau_1m_1;\{n_1\}\right>},\end{aligned}\ ] ] where the expansion coefficients @xmath21^{j_1\tau_1m_1;\{n_1\}}_{j_0\tau_0 m_0;\{n_0\}}}$ ] statisfy the initial condition @xmath22^{j_1\tau_1m_1;\{n_1\}}_{j_0\tau_0 m_0;\{n_0\ } } }      & = &      \delta_{j_0j_1}\delta_{\tau_0\tau_1}\delta_{m_0m_1}\delta_{\{n_0\},\{n_1\}}. \end{aligned}\ ] ]    the equation of motion for the expansion coefficients in the interaction picture ( subscript i ) reads    @xmath23^{j_2\tau_2m_2;\{n_2\}}_{j_0\tau_0 m_0;\{n_0\ } } }      & = &      \hskip-2ex \sum_{j_1\tau_1m_1,\{n_1\}}\hskip-2ex      { \left < j_2\tau_2m_2;\{n_2\}\right| }      \hat h_{l}(t ) + \hat h_{x }      { \left|j_1\tau_1m_1;\{n_1\}\right > } \\&&\times\nonumber      e^{i(e^\text{rot}_{j_2\tau_2}-e^\text{rot}_{j_1\tau_1}+e^x_{\{n_2\}}-e^x_{\{n_1\}})t }      \ { \left[c_i(t)\right]^{j_1\tau_1m_1;\{n_1\}}_{j_0\tau_0 m_0;\{n_0\}}}.\end{aligned}\ ] ]    in the following , we assume we found @xmath24^{j_1\tau_1m_1}_{j_0\tau_0 m_0}}$ ] , the solution for the laser - only problem , i.e. , @xmath25 . 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 . ( [ eq.eom1 ] ) becomes    @xmath26^{j_4\tau_4m_4;\{n_4\}}_{j_0\tau_0 m_0;\{n_0\ } } }      & = &      -i\hskip-3ex\sum_{{\genfrac{}{}{0pt}{}{j_1\tau_1m_1 , j_2\tau_2m_2}{j_3\tau_3m_3 } } } \hskip-2ex      { \left[c_{l , i}(t)\right]^{j_4\tau_4m_4}_{j_3\tau_3 m_3 } }      \int_{-\infty}^t\!\!dt'\    e^{i(e^\text{rot}_{j_2\tau_2}-e^\text{rot}_{j_1\tau_1}+e^x_{\{n_4\}}-e^x_{\{n_0\}})t ' } \\&&\times\nonumber    { \left[c^{-1}_{l , i}(t')\right]^{j_3\tau_3m_3}_{j_2\tau_2 m_2 } }    { \left < j_2\tau_2m_2;\{n_4\}\right| }      \hat h_{x }      { \left|j_1\tau_1m_1;\{n_0\}\right > }    { \left[c_{l , i}(t')\right]^{j_1\tau_1m_1}_{j_0\tau_0 m_0 } }    .\end{aligned}\ ] ]    the expectation values of interest can be calculated by @xmath27 \\\nonumber&= &    \label{eq.expval_detail }    \hskip-3ex\sum_{{\genfrac{}{}{0pt}{}{j_2\tau_2 m_2,j_1\tau_1m_1,j'\tau'm'}{\{n_2\},\{n_1\},\{n'\},\{n''\}}}}\hskip-4ex    w_{j'\tau'}\ \rho^x_{\{n'\},\{n''\ } }    { \left < j_2\tau_2 m_2;\{n_2\}\right|}\hat o{\left|j_1\tau_1m_1;\{n_1\}\right > } \\\nonumber&&\times    e^{i(e^\text{rot}_{j_2\tau_2}-e^\text{rot}_{j_1\tau_1}+e^x_{\{n_2\}}-e^x_{\{n_1\}})t }    { \left[c_i(t)\right]^{j_1\tau_1m_1;\{n_1\}}_{j'\tau ' m';\{n'\ } } }    { \left[c^*_i(t)\right]^{j_2\tau_2 m_2;\{n_2\}}_{j'\tau ' m';\{n''\}}},\end{aligned}\ ] ]    from which the alignment signals ( cf . sec . [ ssec.cos2 ] ) and the x - ray scattering probability ( cf . [ ssec.angu_prob_dist ] ) can be derived . we followed here the strategy that is laid out in ref . @xcite for x - ray diffraction from symmetric - top molecules and adapt it to the asymmetric - top case . the structures of @xmath28 and @xmath29 are the same for symmetric - top and asymmetric - top molecules . the most dramatic changes for asymmetric - top molecules happen in the laser - only system . therefore , we focus for the rest of our discussion just on the laser - only system .    in the following subsections the structure of the field - free hamiltonian @xmath7 and the laser  molecule interaction matrix @xmath30 will be investigated . for this purpose it is convenient to stay in the schrdinger picture . in sec . [ ssec.eom ] we return to the equation of motion of the laser - only system and point out the symmetries of its solutions @xmath24^{j'\tau'm'}_{j\tau m}}$ ] . the measure of three - dimensional alignment is described in sec . [ ssec.cos2 ] . the theory section closes with exploiting the symmetries in the angular density distribution and the diffraction signal .      assuming that structural deformation of the molecule may be neglected , we treat the molecules as rigid rotors . the corresponding field - free hamiltonian reads @xcite @xmath31,\end{aligned}\ ] ] where @xmath32 are the rotational constants associated with the principal axes of inertia , @xmath33 are the cartesian components of the angular momentum operator in the molecular frame ; and @xmath34 are the spherical basis components . for symmetric - top ( @xmath35 ) and asymmetric - top ( @xmath36 ) rotors , the angular momentum @xmath37 and its projection on the space - fixed @xmath38-axis , @xmath39 , are conserved . the angular - momentum projection on the molecular @xmath40-axis , @xmath41 , is conserved only for symmetric - top molecules . a new quantum number @xmath42 , replacing @xmath41 , must be introduced for asymmetric - top rotors diagonalizing the field - free hamiltonian [ cf . ( [ eq.eigenenergy_hrot ] ) ] . note , the rotational energies @xmath43 are independent of the quantum number @xmath39 . hence , each energy level @xmath43 is ( @xmath44)-fold degenerate . it is possible to express the asymmetric - top eigenstates @xmath14 as a superposition of the analytically known symmetric - top eigenstates @xmath45 @xcite , @xmath46}_{k\tau } { \left|jkm\right > } , \\      \label{eq.eigenstate_symm }      { \left<\left.\phi,\theta,\chi\right|jkm\right > }      & = &      \sqrt{\frac{2j+1}{8\pi^2}}d^{*[j]}_{m , k}(\phi,\theta,\chi),\end{aligned}\ ] ] where the coefficients @xmath47}_{k\tau}$ ] are real and @xmath48 are the three euler angles connecting the space - fixed laser frame ( l ) with the principal axes of inertia in the molecular reference frame ( m ) . @xmath49}_{m , k}(\phi,\theta,\chi)$ ] is the complex conjugate of the wigner d - matrix with angular momentum @xmath37 . unfortunately , it is not possible to find a general relation between the @xmath42 and @xmath41 classification @xcite . there are two symmetries incorporated between the symmetric - top ( @xmath41 ) and asymmetric - top ( @xmath42 ) classification . the first symmetry decouples states with even @xmath41 from states with odd @xmath41 , since @xmath7 is a linear combination of @xmath50 and @xmath51 [ eq . ( [ eq.hrot ] ) ] . the state class with even @xmath41 is labeled @xmath52 , the one with odd @xmath41 is labeled @xmath53 . asymmetric eigenstates inherit this separation and are superpositions of just even or odd @xmath41 states . the invariance of @xmath7 under the substitutions @xmath54 and @xmath55 leads to the second symmetry , the wang symmetry @xcite , where asymmetric - top eigenstates decompose in symmetric and antisymmetric linear combinations , @xmath56}_{k\tau}\big[{\left|jkm\right>}+(-1)^\tau { \left|j(-k)m\right>}\big ]      , \qquad\end{aligned}\ ] ] where @xmath57 denotes the parity of @xmath42 , @xmath58 thus , overall there are four separate state classes , which are summarized in table [ tab.rotational_class ] . .[tab.rotational_class ] rotational classes of asymmetric - top rotors . [ cols= " < , < " , ]     three - dimensional alignment of naphthalene is obtained by employing elliptically polarized light . the @xmath38 and @xmath59 axes of the space - fixed frame are defined by the major and minor axes , respectively , of the ellipse that characterizes the elliptically polarized light . following ref . @xcite , we choose the ratio between the field components using @xmath60 which maximizes the three - dimensional alignment of naphthalene . perfect three - dimensional alignment with respect to the space - fixed frame is achieved , when @xmath61 . random orientation corresponds to @xmath62 for all angles . the typical distance between two neighboring carbon atoms in naphthalene is 1.4   . in order to resolve the atomic structure , the resolution must be smaller than this value . imperfect alignment limits the resolution . to build a connection between the resolution and the alignment in terms of @xmath63 , it is helpful to view the angle @xmath64 as an opening angle within which the residual motion of the atoms takes place ( cf . [ fig.naphthalene ] ) . hence , the length characterizing the smallest resolvable structure is @xmath65 where @xmath66 is the linear dimension of the molecule ( measured from the center of mass of the molecule ) . the degree of alignment needed for a resolution of 1    for the outermost carbon atom ( @xmath67   ) is @xmath68 . for a given resolution , smaller opening angles @xmath64 are required when the molecules become larger . in the adiabatic regime at sufficiently low temperature the maximum alignment for a linear rotor ( @xmath69 ) is given by @xmath70^{[2]}_0 $ ] @xcite . the polarizability @xmath71^{[2]}_0 $ ] is approximately proportional to @xmath72 , and @xmath73 scales approximately as @xmath74 @xcite . the smallest resolvable dimension @xmath75 is thus proportional to @xmath76 and might be expected to decrease with increasing molecular size . note , however , that the temperature required to suppress thermal effects ( @xmath77 ) also decreases as a function of the molecular size . additionally , in order to remain in the adiabatic regime , the laser pulse duration must increase with increasing molecular size . in practice this means the laser intensity will decrease . for large molecules it is more realistic to consider the high temperature limit ( @xmath78 ) , where the degree of alignment is a competitive interplay between rotational temperature and coupling strength @xmath79 ; more precisely , @xmath80 @xcite . by using the same scaling arguments , we find @xmath81{ktr}$ ] . the expected resolution is now reversed and increases with molecular size and rotational temperature . consequently , larger molecules can be resolved less precisely . mk ( solid ) and @xmath82  k ( dotted ) . the pump laser shown in gray has a peak intensity of @xmath83  tw/@xmath0 , a width of @xmath84  ps , and an ellipticity of @xmath85 . the upper panel pictures the alignment dynamics on the time scale of the pulse duration . the lower panel is a close - up view of the region highlighted in the upper graph . at @xmath86  ps , nonadiabatic oscillation enhances the alignment for all axes at @xmath87  mk and is ideal for a 1-ps x - ray probe pulse . ]    in fig . [ fig.cos2_t10mk ] , the alignment dynamics of naphthalene at 10  mk and 1  k , respectively , are shown . we assumed a gaussian laser pulse with a peak intensity of @xmath83  tw/@xmath0 and a pulse duration of @xmath84  ps ( fwhm ) . at a rotational temperature of @xmath87  mk , almost all naphthalene molecules are in the rotational ground state . the rotational period of naphthalene at such a low temperature , @xmath88  ps , is of the order of @xmath84  ps , suggesting that the alignment dynamics are quasi - adiabatic @xcite . this is consistent with the observation that the overall alignment follows the laser pulse shape and , in addition , clear nonadiabatic features ( fast oscillations ) are visible . one can make use of the nonadiabatic behavior by probing the molecules with an x - ray pulse of 1 ps duration , which is fast enough to resolve the nonadiabatic oscillations . for instance , at @xmath86  ps the alignment is transiently enhanced in all three dimensions ( cf . [ fig.cos2_t10mk ] ) . at @xmath82 k , the rotational period of naphthalene is small in comparison to @xmath1 , so the alignment dynamics are adiabatic and fast oscillations are significantly suppressed . as a consequence of the increased thermal motion , the maximum degree of alignment at @xmath82  k is clearly reduced . the diffraction signal @xmath89 [ cf . eq . ( [ eq.diff_signal ] ) ] collected over the x - ray pulse duration reflects a pulse - averaged , effective alignment @xmath90 where @xmath91 is the normalized x - ray flux . we assume a gaussian temporal envelope for the x - ray pulse , with a full width at half maximum of @xmath92 . in fig . [ fig.cos2_eff ] , @xmath93 is shown as a function of @xmath92 . we may conclude from fig . [ fig.cos2_eff ] that the effective alignment of the body - fixed axes decreases rapidly when the x - ray pulse duration is longer than the laser pulse duration . an enhancement in the effective alignment is visible for @xmath94  ps at @xmath87  mk , where nonadiabatic oscillations are not suppressed and can be resolved by the x - ray pulse .    , for @xmath87  mk ( solid ) and @xmath82  k ( dotted ) . the pump laser has a peak intensity of @xmath83  tw/@xmath0 , a width of @xmath84  ps , and an ellipticity of @xmath85 . the x - ray pulses are centered at @xmath95  ps . ]    when @xmath96 , the influence of @xmath92 is rather weak and the degree of alignment is mainly affected by the rotational temperature , @xmath97 . figure [ fig.cos2_temp ] shows the effective alignment of the molecular axes as a function of @xmath97 for @xmath98  ps and @xmath99  ps . below 0.25 k , the best aligned axis is the body - fixed @xmath100 axis in the space - fixed @xmath101 direction for @xmath102  ps , despite the fact that no laser field is applied in this direction . the strong alignment of the @xmath100 axis translates to well - aligned naphthalene molecules in the polarization plane of the laser ( @xmath103 plane ) . within the plane , the body - fixed @xmath40 axis is always more strongly aligned in the @xmath38 direction than the @xmath104 axis is aligned in the @xmath59 direction . ( recall that @xmath105 and @xmath106 . ) for higher temperatures , the body - fixed @xmath40 axis is the most strongly aligned axis , since the alignment of the @xmath100 axis is affected by the more rapidly decreasing alignment of the @xmath104 axis . ps ( solid ) and @xmath99  ps ( dotted ) . the pump laser has a peak intensity of @xmath83  tw/@xmath0 , a width of @xmath84  ps , and an ellipticity of @xmath85 . the x - ray pulses are centered at @xmath95  ps . ]    as may be seen in fig . [ fig.cos2_eff ] , the effective alignment @xmath107 of all molecular axes is at least 0.96 for 10  mk naphthalene molecules probed by x - ray pulses shorter than @xmath108  ps . hence , the smallest resolvable dimension limited by residual pendular motion is @xmath109   . only the rotational ground state is initially occupied , and the minimum intensity required to accomplish 1    resolution is @xmath83  tw/@xmath0 . dissociation and ionization play only a minor role at this intensity , but become important at higher intensities @xcite . with an effective alignment of @xmath110 at 1  k and @xmath99  ps , the smallest resolvable dimension @xmath75 is 1.73   , which is larger than the distance between neighboring carbon atoms .      in fig . [ fig.diffpic_front ] , diffraction patterns @xmath89 of three - dimensionally aligned naphthalene are shown for perfect alignment as well as for imperfect alignment at different temperatures and x - ray pulse widths . the laser parameters are the same as for fig . [ fig.cos2_t10mk ] . the laser polarization plane is assumed to be the @xmath103 plane so that the molecules are aligned as illustrated in fig . [ fig.naphthalene ] . the diffraction patterns in fig . [ fig.diffpic_front ] are two - dimensional planar slices through the three - dimensional @xmath111 space , taken at @xmath112 . ( due to the curvature of the ewald sphere @xcite , experimental scattering patterns do not correspond to exactly planar slices through @xmath111 space , but this is of no consequence here . ) the signal strength falls rapidly for high @xmath113 . therefore , to highlight the structure at high momentum transfers , the scattering intensities are shown on a logarithmic scale . the effective alignment in fig . [ fig.diffpic_front ] decreases clockwise , from @xmath114 for perfect alignment to @xmath110 for naphthalene molecules at 1  k probed by a 100-ps x - ray pulse . by increasing the x - ray pulse width from 1 to 100  ps , the effective alignment at @xmath87  mk decreases from @xmath115 to @xmath116 [ cf . [ fig.diffpic_front](b ) and ( d ) ] , and is significantly smaller than the impact of temperature rise from 10  mk to 1  k [ cf . [ fig.diffpic_front](c ) and ( d ) ] .     of naphthalene . ( a ) perfect alignment . ( b ) @xmath87  mk , @xmath98  ps , x - ray pulse centered at @xmath86  ps ( cf . [ fig.cos2_t10mk ] ) . ( c ) @xmath82  k , @xmath117  ps , x - ray pulse centered at @xmath95  ps . ( d ) @xmath87  mk , @xmath99  ps , x - ray pulse centered at @xmath95  ps . for the alignment pulses in ( b ) , ( c ) , and ( d ) , we assumed @xmath83  tw/@xmath0 and @xmath84  ps . ] although the basic features of the x - ray scattering pattern for perfectly aligned naphthalene ( fig . [ fig.diffpic_front]a ) are preserved in the scattering patterns for laser - aligned naphthalene [ figs . [ fig.diffpic_front](b ) , ( c ) , and ( d ) ] , the contours are washed out and the contrast between maxima and minima is less pronounced with decreasing alignment . incoherent averaging for laser - aligned naphthalene renders the diffraction patterns more cylindrically symmetric with respect to the @xmath101 direction , which limits the accessible structural information particularly at high momentum transfer . side views of naphthalene in @xmath111 space are shown in fig . [ fig.diffpic_side ] , where fig . [ fig.diffpic_side](a ) depicts the long side ( @xmath118 ) and fig . [ fig.diffpic_side](b ) the short side ( @xmath119 ) of naphthalene . both display strong similarities to multi - slit diffraction patterns , consistent with the planar structure of naphthalene being well aligned in the @xmath103 plane .    . ( b ) diffraction signal of naphthalene for @xmath119 . other parameters are the same as for fig . [ fig.diffpic_front](b ) . ] the achievable real - space resolution depends on two aspects . on the one hand , the smallest resolvable dimension @xmath75 is a function of the degree of alignment [ eq . ( [ eq.align_res ] ) ] . on the other hand , the pixel size of the real - space structure reconstruction is determined by the maximum momentum transfer @xmath120 in the diffraction signal @xmath89 . here , @xmath121 @xmath122 , corresponding to a pixel size of 0.5   . the approximate range in momentum space within which the assumption of coherent scattering holds may be defined by @xmath123 . in fig . [ fig.diffpic_highq ] , the diffraction signals of fig . [ fig.diffpic_front ] are shown up to @xmath124  @xmath122 . the respective ranges defined by @xmath125 are highlighted . it is not possible to increase the resolution of the real - space structure reconstruction by choosing @xmath120 much greater than @xmath125 . in fact , numerical tests have indicated that data beyond @xmath125 can lead to poor convergent structures . , except @xmath124  @xmath122 . the momentum transfer @xmath125 ( see the text ) is highlighted by a solid green circle in ( b)(d ) . ]    for 12-kev x - ray photons , a real - space pixel size of 0.5    requires the detection of photons scattered up to @xmath126 with respect to the x - ray propagation axis . the azimuthally averaged differential x - ray scattering cross section per naphthalene molecule for @xmath127 @xmath122 is @xmath128 . since the largest distance between carbon atoms in naphthalene is @xmath129   , the area of a pixel in momentum space must not exceed @xmath130  @xmath131 . at a scattering angle of @xmath126 , this corresponds to a solid angle @xmath132 . for a molecular beam width of 1  mm and an x - ray focus area of @xmath133  @xmath134m@xmath135 , it has been demonstrated that the number of molecules in the interaction volume can be as large as @xmath136 @xcite . hence , requiring a minimum of five scattered photons per pixel @xcite , the estimated acquisition time for one diffraction pattern is around 70  seconds with an x - ray fluence of @xmath137 photons / pulse / mm@xmath135 and a repetition rate of 1  khz . we applied the phase - retrieval algorithm described in sec . [ ssec.phase-retrieval ] to the naphthalene diffraction signals shown in fig . [ fig.diffpic_front ] . the maximum momentum transfer of @xmath138@xmath122 in fig . [ fig.diffpic_front ] corresponds to a pixel size of 0.5   , which is of the order of the structure we want to resolve in naphthalene . as a consequence , each pixel encodes a lot of structure information . calculations have shown that the reconstruction routine becomes sensitive to the set of initial random phases . therefore , we follow the spirit of ref . @xcite and average over reconstructions obtained for 100 different sets of initial phases to define a quality criterion , which we apply subsequently to all 100 reconstructions to select the most meaningful results and average over these selected reconstructions . our quality criterion is defined as follows : ( 1 ) all pixels of the unfiltered averaged result that have at least 20% of the maximum electron density are selected to build a density core region . ( 2 ) the quality criterion uses the density core region and selects only the reconstructions that have at least 50% of their total electron density within this density core region . the filtered averages for the x - ray scattering patterns of fig . [ fig.diffpic_front ] are displayed in fig . [ fig.reconstruct_front ] . in our calculations , we employed a grid spacing in @xmath111 space of 0.16  @xmath122 , corresponding to a maximum object size 8 times larger than the size of naphthalene .    . the pixel size is 0.5   . ( a ) perfect alignment . ( b ) @xmath87  mk , @xmath98  ps , x - ray pulse centered at @xmath86  ps . ( c ) @xmath82  k , @xmath99  ps , x - ray pulse centered at @xmath95  ps . ( d ) @xmath87  mk , @xmath99  ps , x - ray pulse centered at @xmath95  ps . ] figure [ fig.reconstruct_front ] illustrates that with better effective alignment in the diffraction pattern more structural information can be reconstructed . in the perfect alignment case ( cf . [ fig.reconstruct_front](a ) , the positions of the atoms can be resolved . a comparison with the perfect alignment reconstruction in fig . [ fig.diff_res_perfect](b ) , where the pixel size is 0.125     and no averaging over several initial random phases was performed , illustrates that the reconstruction of the carbon positions is more difficult when the pixel size is comparable with the size of the atoms . incoherent averaging for imperfectly aligned molecules affects the effective resolution further , since the assumption of coherent scattering limits the usable diffraction signal to @xmath139 . as a consequence , @xmath120 should be of the order of @xmath125 and , therefore , the pixel size in the reconstruction is limited by the degree of effective alignment . all results displayed in fig . [ fig.reconstruct_front ] recover the two - ring structure of naphthalene , indicating an effective resolution below 2   . especially the two carbon atoms connecting both carbon rings are distinctly visible . figure [ fig.reconstruct_front](b ) and [ fig.reconstruct_front](d ) show the reconstruction for naphthalene at 10  mk with @xmath140ps and @xmath141ps , respectively . both have a similar effective alignment and reconstruction . in both cases , the electron density peaks on the two - ring structure indicate the positions of all carbon atoms and are consistent with the theoretically expected effective resolutions , which are below the typical distance between neighboring carbon atoms ( 1.4   ) . the predicted effective resolution at 1  k ( @xmath142 ) is sufficient to resolve the overall structure of naphthalene , which can be seen in fig . [ fig.reconstruct_front](c ) . for the central region of naphthalene , the effective resolution improves and makes it possible to identify the two central carbon atoms at 1  k. we have theoretically studied the theory of x - ray diffraction from asymmetric - top molecules that are three - dimensionally aligned by elliptically polarized light and have discussed a phase - retrieval method in order to reconstruct molecular structure from the x - ray scattering pattern . the interaction of the gas phase molecules with the laser and x - ray fields was studied in the density matrix formalism . we have assumed rigid rotor molecules . symmetries in the time - dependent hamiltonian and in its solutions as well as in the angular density distribution were exploited and used to improve numerical efficiency . in combination with further computational aspects , a significant enhancement in numerical efficiency has been achieved . a feasibility study of x - ray diffraction from three - dimensionally laser - aligned molecules has been performed using the organic molecule naphthalene . we have linked the degree of alignment to an effective resolution and have studied the impact of x - ray pulse duration and temperature on the diffraction patterns . a phase - retrieval reconstruction was performed on diffraction patterns taken for different effective alignments . the enhancement of incoherent averaging due to imperfect alignment destroys structural information and hinders the recovery of detailed atomic configurations within the molecule . the reconstruction of naphthalene confirms our discussion that the degree of alignment is a good indicator for the achievable resolution in the reconstruction . to decode structures on an atomic length scale , high degree of alignment in all three molecular axes must be achieved . molecules have to be cooled down to a few kelvin or even sub - kelvin temperatures so rotational motion is sufficiently reduced . experiments have shown that it is feasible to cool molecules to 1  k @xcite . to accomplish rotational temperatures well below 1 k is a great challenge and would open the opportunity to image large gas - phase molecules at atomic resolution . the problem of incoherent averaging over a finite range of different orientations has to be further addressed , especially in the case where molecular symmetries do not coincide with the symmetries of the diffraction pattern imposed by alignment .    with the ability to detect molecular structure , x - ray scattering from gas phase molecules can be used to study torsion effects and laser - induced deformations that are expected to occur during alignment in the presence of intense laser pulses @xcite . by systematically varying the delay time between pump ( laser ) and probe ( x - ray ) pulse , it is possible to follow molecular motion on an ultrafast time scale . of particular interest is the study of physical and chemical processes in the presence of intense laser fields , which simultaneously provide the required alignment for imaging of reactions with atomic resolution in space and time . we thank cassandra hunt for helpful comments on the manuscript . this work was supported by the office of basic energy sciences , u.s . department of energy under contract no . de - ac02 - 06ch11357 . in the density matrix , statistical weights , @xmath143 , define the relative number of particles in an ensemble that are in a given quantum mechanical state for a well - defined temperature @xmath97 . the nuclear spin statistical weights @xmath144 are based on symmetry arguments and represent the number of the allowed nuclear spin states for a given rotational state @xmath14 . the overall symmetry @xmath145 of the total molecular wave function is independent of rotational or nuclear spin states and determined by the spin statistic theorem @xcite . however , @xmath145 is also a direct product of symmetries of the different quantum states @xcite , @xmath146 where @xmath147 is the rotational state symmetry and @xmath148 is the nuclear spin state symmetry . molecules are in their electronic and vibrational ground states and their contributions can be omitted in eq . ( [ eq.mol_symm ] ) . the representation of symmetries depends on the symmetry group of the molecule . the molecular symmetry group of asymmetric - top molecules without an inversion center is @xmath149 or lower @xcite . the rotational symmetry group of an asymmetric - top rotor is always @xmath150 ( isomorphic to @xmath149 ) @xcite . if there is no common symmetry class between @xmath147 and @xmath148 , @xmath151 . in other words , molecular symmetries come only into play when molecular rotations are identical to particle exchanges . all symmetry representations are written as linear combinations of irreducible representations ( irrep ) of @xmath150 in table [ tab.character_table ] . in addition the character table for the nuclear spin and rotational states are shown . the four molecular rotations and particle exchanges defining @xmath150 are : @xmath152 . cccccp1excc & & & + & @xmath52 & @xmath153 & @xmath154 & @xmath155 & & @xmath37 even & @xmath37 odd    ' '' ''     + @xmath145 & 1 & 1 & 1 & 1 & &    ' '' '' + @xmath156 & 1 & 1 & @xmath157 & @xmath157 & & @xmath158 & @xmath159 + @xmath160 & 1 & 1 & @xmath161 & @xmath161 & & @xmath159 & @xmath158 + @xmath162 & 1 & -1 & @xmath157 & @xmath161 & & @xmath163 & @xmath164 + @xmath165 & 1 & -1 & @xmath161 & @xmath157 & & @xmath164 & @xmath163 + @xmath148 & @xmath166 & @xmath167 & @xmath167 & @xmath167 & &    ' '' ''    the characters of the overall wave function @xmath168 can be derived from the pauli principle @xcite and correspond to the overall sign changes induced by the molecular rotations . first , we assume all carbon atoms in naphthalene are @xmath169c with nuclear spin 0 . by remembering only odd permutations of half - integer particles ( fermions ) change the sign of the overall wave function @xcite , we find all group operation leave the overall sign for naphthalene unchanged , since all hydrogen permutations are even . hence , @xmath170=1,\ \forall p\in v$ ] and @xmath171 . the rotational symmetry classes , in which each rotational state can be classified ( cf . sec . [ ssec.hrot ] ) , coincide with the irreps of @xmath150 @xcite . however , whether or not @xmath37 is even or odd defines which irrep corresponds to which symmetry class ( cf . table [ tab.character_table ] ) . the sum of characters over all possible nuclear spin states @xmath172 $ ] for each operation @xmath173 can be derived by the formula @xcite @xmath174    & = &    \prod_{g_p}\ , ( 2s_{g_p}+1)\ \text{sgn}(p_{g_p})^{2s_{g_p}},\end{aligned}\ ] ] where the atom group @xmath175 is defined by the minimum number of identical atoms in the molecule such that after the atom permutation @xmath173 each atom in @xmath175 is replaced by an atom of @xmath175 . @xmath176 is the sign of the permutation @xmath173 within in the atom group @xmath175 , which is @xmath177 or @xmath178 depending whether the permutation is even or odd in @xmath175 . the smallest atom group consist of one atom , which does not change position under the operation @xmath173 . @xmath179 is the nuclear spin of the atoms in @xmath175 .    imposing eq . ( [ eq.mol_symm ] ) , nuclear spin and rotational symmetries are linked together . ( for naphthalene : @xmath180 ) . the multiplication of symmetry representations is done by multiplying their characters . the nuclear spin statistical weights @xmath144 are the number of irreps in @xmath148 such that the product with a given rotational class is @xmath145 . as an example : when @xmath181 and j is even , then @xmath182 and only the product @xmath183 @xcite fulfills eq . ( [ eq.mol_symm ] ) . since there are 60 @xmath163 irreps in @xmath148 , the nuclear spin statistical weight is 60 . for all rotational states of naphthalene , the nuclear spin statistical weights are @xmath184 the squares of the matrix elements of the rotation matrix @xmath186 , i.e. @xmath185 , give a complete picture of the three - dimensional alignment . knowing four @xmath185 is sufficient to describe all , where at least two axes of each reference frame have to be involved in the four @xmath185 . our choice is @xmath187 , since it is one of the least computationally expensive choices . the @xmath185-matrices are diagonal in @xmath39 or @xmath41 when @xmath188 or @xmath189 , respectively . only the matrix @xmath190 is nondiagonal in @xmath41 and @xmath39 . each @xmath185 can be written in terms of @xmath191}_{mk}(\phi,\theta,\chi)$ ] @xcite , where the matrix elements of the wigner d - functions are given in refs . the matrix elements of the four matrices that we have chosen are :    @xmath192    \bigg ) , \\ { \left < jkm\right|}\cos^2\theta_{za}{\left|j'k'm'\right > }    & = &    \frac{1}{3 } \delta_{jj'}\delta_{kk'}\delta_{mm ' }    -    \sqrt{\frac{1}{6}}\sqrt{\frac{2j+1}{2j'+1}}{\left<\left.jm;20\right|j'm'\right > } \\\nonumber & & \times    \bigg (          \sqrt{\frac{2}{3}}{\left<\left.j , k;2,0\right|j',k'\right > }      -        \big[{\left<\left.j , k;2,2\right|j',k'\right>}+{\left<\left.j , k;2,-2\right|j',k'\right>}\big ]      \bigg ) , \\ { \left < jkm\right|}\cos^2\theta_{xa}{\left|j'k'm'\right > }    & = &    \frac{1}{3 } \delta_{jj'}\delta_{kk'}\delta_{mm ' }    +    \frac{1}{4}\sqrt{\frac{2j+1}{2j'+1 } } \\\nonumber & & \times      \bigg (          \sqrt{\frac{2}{3}}{\left<\left.j , m;2,0\right|j',m'\right > }        -        \big[{\left<\left.j , m;2,2\right|j',m'\right>}+{\left<\left.j , m;2,-2\right|j',m'\right>}\big ]      \bigg ) \\\nonumber & & \times      \bigg (        \sqrt{\frac{2}{3}}{\left<\left.j , k;2,0\right|j',k'\right > }        -            \big[{\left<\left.j , k;2,2\right|j',k'\right>}+{\left<\left.j , k;2,-2\right|j',k'\right>}\big ]      \bigg ) . \ ] ]                                  m. r. howells , t. beetz , h. n. chapman , c. cui , j. m. holton , c. j. jacobsen , j. kirz , e. lima , s. marchesini , h. miao , d. sayre , d. a. shapiro , j. c. h. spence , and d. starodub , journal of electron spectroscopy and related phenomena * 170 * , 4 - 12 ( 2009 ) .                                                                                                                                                                        h. n. chapman , a. barty , s. marchesini , a. noy , s. p. hau - riege , c. cui , m. r. howells , r. rosen , h. he , j. c. h. spence , u. weierstall , t. beetz , c. jacobsen , and d. shapiro , j. opt . a * 23 * , 1179 ( 2006 ) .
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 schrdinger equation for each rotational state initially populated in a thermal ensemble . using a phase - retrieval algorithm , 
 the feasibility of structure reconstruction from a quasi - adiabatically - 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/@xmath0 , 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 1  .