Structure Determination in a New Class of Amorphous Cluster Compounds with Extreme Nonlinear Optical Properties

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Introduction
For the far longer time of human history, light was produced in form of burning objects, like torches and oil lamps.These illuminants emitted an electromagnetic spectrum fully comprising the part visible to humans.The invention of the light bulb was only made 150 years ago, and yet it was the first decisive breakthrough in the development of light generation.However, the underlying process was still based on hot glowing objects.Hence, it was still rather unfavorable in terms of energy efficiency, since the larger ranges of the emitted spectra were still located in the red and infrared regimes, and most of the energy dissipated as heat.After all, incandescent lamps with filaments heated up to about 3000 K have an intensity maximum which is already in the visible range, nevertheless, the infrared portion is still about 40-50%.Numerous attempts were therefore undertaken during the 20th century until today to develop new and more efficient light generation techniques.The most promising result of these efforts is probably the Light-Emitting-Diode (LED). 1) In terms of energy efficiency, this technology represents a real quantum leap in the development of light sources.Normally, such devices consist of inorganic semiconductors and their spectra are located in the near ultraviolet.][4][5] The organic LED is a similar concept based on organic semiconductors where the band gap and hence the emitted spectrum can be tuned by chemical modifications on the molecular level.][8][9] LEDs are advantageous for many daily applications and one important feature is that the radiation pattern is usually less oriented allowing their use in, e.g., displays where high viewing angles are preferred.However, for other applications a high directional emission of spectra in the visible range is often desired.In fact, the research for "Laser-like" White Light Generators (WLG) goes back to the nineteen seventies and resulted in so called Super-Continuum-Generators (SCG) where use is made of non-linear optical (NLO) effects in different materials. 10,11)he white light radiation of these SCGs is point-like, sustaining the beam characteristics of the short-pulsed high energy pumping laser, which are needed to provide the high field strengths necessary to invoke the higher order NLO effects.Common materials used here are YAG, Sapphire, CaF 2 -crystals, optical fibers and waveguide-based sources.However, the problem with such techniques is that the required experimental installations are bulky and expensive, restricting the use of these devices to pure scientific or medical applications.Recently, a group of relatively simple chemical compounds was reported to exhibit extreme nonlinear optical behavior when irradiated by an unpretentious, inexpensive continuous wave near infrared (CW-NIR) laser diode. 12,13)Also, these materials are comparatively easily and efficiently to synthesize.The development of such products into materials that can be used in applications is hence of great interest and could open up the way for a wide range of new and low-cost optical devices.They are prepared as powdered solids and emit either second-harmonics or warm white optical spectra which retain the high-directional beam characteristics of the driving laser.Their molecular building blocks are organic-inorganic hybrid-compounds consisting of cluster-like molecules with general formula [(RX) 4 Y 6 ], where X is a group-14 element, Y is a chalcogen and R represents an organic ligand attached to the group 14 atom.The inorganic constituents X and Y form an adamantane-like cage, and the organic substituents R protect the inorganic cluster cores from further reaction to more stable compounds.The resulting general shape of these clusters is displayed in Fig. 1, where the organotin sulfide cluster [(PhSn) 4 S 6 ] is shown, with phenyl ligands (Ph = -C 6 H 5 ) attached to the tin atoms of the adamantane shaped Sn-S cluster core.Variation of the ligands and=or the chalcogens or of the group 14 element allows great chemical variability of the cluster compositions.Table I outlines some of the compounds synthesized so far together with their respective optical responses.All of these materials show strong NLO-reactions either as SHG or WLG.However, while the crystalline compounds clearly react as second harmonics generators, the amorphous materials prefer to emit white spectra, indicating that disorder seems to be a mandatory requirement for WLG.The influence of morphology is exemplary demonstrated for the two systems (1) and (4) of Table I in Fig. 2. The two compounds differ only by the exchange of Sn by Si in the cluster core, but they show completely different optical and morphological properties.The Figure depicts their optical answers when being irradiated by a 980 nm laser-line (1.265 eV).While the Si-based compound replies with a distinct 2nd order harmonics at 2.53 eV, the Sn-containing material responds with a broad white spectrum ranging from about 1.5 to 3 eV.Also shown in the Figure are the respective diffraction-pattern of the two substances.The [(PhSi) 4 S 6 ] material (4) is clearly crystalline, as indicated by the Bragg peak dominated diffraction pattern, while [(PhSn) 4 S 6 ], (1), is fully amorphous and the scattered X-rays yield a structure factor SðQÞ, typical of a fully disordered system.Although both clusters are similarly shaped and differ only by one single atom type in the cluster-core, the structural morphology is vastly different, as are the optical responses.The fact that WLG is obviously never observed from crystalline cluster materials indicates that the effect is enabled by certain microscopic structural correlations or degrees of freedom that are only possible in a sufficiently disordered state.The amorphous materials (1) and (2) were clearly identified as WLGs, while (3), although also not crystalline, does not emit a white spectrum upon irradiation but clearly reacts as a SHG.We therefore explored all three cluster materials with respect to their microscopic structure to determine whether one can identify characteristic structural differences in the interatomic and intermolecular range between the amorphous SHG (3) and the two amorphous WLGs materials (1) and (2).
Here, we will give a short report on the current status of our work.

X-ray and neutron scattering
While crystal structure analysis is nowadays a standard technique in chemistry and physics, the detailed determination of structural relationships in disordered matter is still a major and demanding challenge, although diffraction studies on disordered matter are almost as old 17,18) as von Laue's first experiments of X-ray diffraction from crystal lattices in 1912. 19)Various excellent review articles on this topic exist (see, e.g., Refs.20 and 21), so we will just give a very brief outline here.In the simplest case, the total structure factor SðQÞ as the central function, is directly obtained from the pure scattering intensity IðQÞ either using X-rays or neutrons.It is the total interference function of the sample and can be interpreted as the distribution of interatomic pair distances in reciprocal space.SðQÞ is related to the pair distribution function (PDF), gðrÞ via Fourier-transform representing the same statistical relations in real space.Its values are proportional to finding an atom at position r when there is a reference atom located at the origin, averaged over the whole ensemble of atoms in the irradiated part of the sample.Average coordination numbers and interatomic distances can generally be derived from the PDF and this is in fact a straight forward task when simple systems such as monatomic glasses or liquids are being explored.However, this is no longer the case if the sample contains more than one chemical component, e.g., when complex alloys, amorphous semiconductors and metals or other technically interesting systems are being investigated.Then, the interpretation of gðrÞ or SðQÞ is no longer unambiguous, since various arrangements of different atoms may lead to similar functions.The goal of structural investigations on complicated materials using scattering techniques is therefore always to overcome this ambiguity of the total structure factor SðQÞ in order to obtain unmistakable, reliable structural information.Since SðQÞ is basically defined as a sum of scattering contributions from pairs of atoms, 20,21) it is possible to also decompose it into the sum of partial structure factors S ij ðQÞ, where i and j run over all combinations of two chemical species, 22) SðQÞ ¼ X i; j w i; j Á S i;j ðQÞ: ð1Þ For a system consisting of n chemical components, nðn þ 1Þ=2 different S ij ðQÞ exist.The partial structure factors represent the structural correlations between particles of chemical species i to particles of species j in reciprocal space and the partial pair distribution functions (PPDF) g ij ðrÞ, obtained from Fourier-transforming the S ij ðQÞ, determine the probability of finding a particle of species j at position r, if there is particle of species i at the origin.Their determination is the goal in any structure exploration on disordered materials consisting of more than one chemical component.These functions can readily be interpreted and they contain the required information to comprehend the correlations between the particles, which is the key for an understanding of the structural arrangement of atoms in disordered matter.The w ij in Eq. (1) represent the weighting factors of the individual S ij ðQÞ.They are determined by the respective concentrations and scattering lengths of each component and are defined as x i and f i in Eq. ( 2) denote the respective portion and scattering length of component i, respectively and the denominator determines the square of the concentrationaveraged scattering length of the sample [either X-ray formfactor fðQÞ or neutron scattering length b].The problem is, that the S ij ðQÞ functions of Eq. ( 1) are a priori not known and cannot directly be obtained from a single measurement of the total SðQÞ.For a direct determination of the nðn þ 1Þ=2 unknown S ij ðQÞ functions an equally number of linear equations would be needed to form a linear set of inhomogeneous equations which could then in principle be solved.However, for this, nðn þ 1Þ=2 different SðQÞ measurements of the same sample are needed, yielding nðn þ 1Þ=2 different scattering patterns, without changing the chemical composition, which is a difficult to impossible task.However, for not too complex compositions such techniques exist as, e.g., neutron diffraction with isotopic substitution (NDIS) 20) or anomalous X-ray scattering (AXS). 23,24)In the latter case, the full set of required equations is usually not obtained, and even then, the contrast between these functions is usually too small to numerically solve the resulting system of equations in a satisfactory way. 20,21)Therefore, AXS data analysis is typically combined with Reverse Monte Carlo (RMC) simulations. 25,26)However, both NDIS and AXS are sophisticated and difficult methods, and a high level of experience is required to reliably perform the associated data analyses.
Another kind of challenge exists when the elementary building blocks of an amorphous sample consist of molecules.This situation is also widely discussed in the literature, and a number of intriguing review articles exists here as well (see, e.g., Ref. 27).Again, since SðQÞ is defined as the sum of the scattering from pairs of atoms, the total structure factor, after appropriate rearrangement, can also be written as a sum of two contributions representing the intra-and intermolecular correlation respectively, and is then simply defined as, Here, f 1 ðQÞ is denoted the molecular formfactor. 27)It represents the contribution to SðQÞ that originates solely from atom-atom spacings within the molecular entities.Here, we will define it as The i; j run over all combination of atom pairs within the molecule.D m ðQÞ in Eq. ( 3) denotes the scattering from atom pairs belonging to different molecules, it hence represents the intermolecular contribution to SðQÞ.It depends on the mutual spatial orientation of the molecules and is therefore characteristic of the intermolecular interactions.If the molecular structure is known f 1 ðQÞ can be computed and D m ðQÞ is determined by the difference between SðQÞ and f 1 ðQÞ.However, the interpretation of the intermolecular contribution is a sophisticated task and a direct translation into real space is difficult.Therefore, the analysis of scattering laws obtained from molecular systems should also be supported by additional methods such as computer simulations.We will therefore briefly discuss the use of RMC simulation for the exploration of disordered molecular systems.

Molecular RMC simulation
In standard RMC simulations, computer generated hardsphere atoms of identical composition as a real sample are iteratively moved inside a simulation box with periodicboundary conditions and a scattering law SðQÞ calc is computed after each step until it matches the experimental SðQÞ.Individual partial pair correlation functions are then obtained from the virtual ensemble which gives further insight into the specific microscopic arrangements of the real material. 28)The problem of RMC however is, that the simulated SðQÞ for a multi component system will not be unique, 29) as was already mentioned above.Therefore, strong simulation constraints are required to enforce unambiguous simulation results.Beneath hard constraints like minimum atomic distances for the different atomic sizes, so called cutoff-limits, additional scattering laws of the same sample as, e.g., additional X-ray and neutron scattering data may be used as further constraints, e.g., from AXS 20,21,30) or from NDIS. 20,31,32)The situation is however different for molecular substances, where complete well-defined molecular units are moved in a single step instead of single atoms, and the molecular structure is preserved throughout the simulation.This is a strong constraint, and a molecular RMC (m-RMC) simulation carried out to match a total experimental SðQÞ will therefore provide a reliable simulation output.Such approaches were already performed in the past, they were however limited to small or symmetrically simple, rigid molecules ignoring internal degrees of freedom. 33,34)The reason for this limitation is, that the implementation of m-RMC in common simulation software packages is difficult, and there are a number of problems to overcome.First, a reasonable molecular structure must be assumed beforehand.Internal degrees of freedom like rotations about bonds must be considered as well as possible molecular deformations in the condensed amorphous phase so as to only enable the simulation to change structural properties that are well reflected in the available data.A purely rigid molecular entity might not necessarily provide a meaningful structural model for a complex molecular system. 35)The uncertainty of atomic positions due to thermal vibrations is also problematic.Typically, this is considered by a Debye-Waller approach, which, however, cannot account for the real vibrational volume and may lead to unrealistically high free volumes. 36)t is also questionable whether an artificial isotropic broadening of the atom positions is realistic, or whether one should rather expect ellipsoid-like spatial distributions in the molecular case.
][39] In our m-RMC simulation we have implemented the additional option to slightly and independently move intramolecular atoms in order to simulate corresponding molecular distortions or anisotropic vibrations.

EXAFS and XANES
A complementary view of the short-range structural features can be gained from X-ray spectroscopy methods performed in the vicinity of an X-ray absorption edge.Typically, two techniques are distinguished based on the energetic region that is probed: the first is X-Ray Absorption Near Edge Structure (XANES), probing the energy range "near the edge", conventionally within about 50 eV of the absorption edge.Beyond this region, Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy can extend to more than 1000 eV above the edge.From a XANES experiment, also called Near Edge X-ray Absorption Fine Structure, we can obtain information on the local electronic properties as well as on structural features, i.e., chemical bonds as well as long-or intermediate-range interactions.The interpretation of a XANES spectrum can be rather challenging, however, and is effectively assisted by computer simulations, using, e.g., FEFF 40,41) or FDMNES. 41)EXAFS spectroscopy, on the other hand, is based on the oscillatory modulation of the X-ray absorption coefficient as a function of X-ray photon energy beyond an X-ray absorption edge, 42) and can provide one-dimensionally averaged, elementspecific structural information, such as atomic distances and vibrations.Both methods are well developed and meanwhile belong to standard characterization techniques for investigations of atomic-scale properties.They are extremely versatile, and can be applied to ordered as well as disordered states of matter.Phenomenologically, the XAFS spectrum χ is defined as the normalized, oscillatory part of the X-ray absorption above a given absorption edge.The oscillations arise from an interference effect: the outgoing photoelectron that is excited from the incoming X-ray, typically from a lowlying K-or L-shell, will be scattered from neighboring atoms, and the amplitude of all the reflected electron waves at the position of the absorbing atom add either constructively or destructively to the outgoing photoelectron wave.The photoelectron wavevector is conventionally denoted k, in contrast to Q in X-ray scattering, and can be written as where E 0 is the threshold energy of the absorption edge, and m e is the mass of the electron.With this parameter, we can define the XAFS spectrum ðkÞ as a sum of contributions from all neighboring atoms, or more generally, from all possible scattering paths j. 42) ðkÞ ¼ Here, R j is the absorber-scatterer distance, S 2 0 is the so-called amplitude reduction factor (related to many-body effects), N j is the coordination number of j, f j ðkÞ is the backscattering amplitude from each neighboring atom, ðkÞ is the mean-free path of the photoelectron, j is the variance in the absorberscatterer distance (commonly interpreted as Debye-Waller factor), and ðkÞ is the total phase shift experienced by the photoelectron.The factor due to the mean free path ðkÞ of the photoelectron (typically 5 to 30 Å) is largely responsible for the relatively short range probed by EXAFS, which is generally within the first or second coordination shell in amorphous compounds, and in the order of magnitude of 10 Å in crystals.It should also be noted that single scattering paths are dominating in the low R j region, but multiple scattering effects quickly become important beyond the first coordination shell of an atom and lead to a veritable multitude of scattering paths j.It is readily seen that the structural information is contained in the parameters R j , N j , and j .These can be fitted directly with available software like the DEMETER program package, 43) but can also be determined by RMC modelling.The other parameters fðkÞ, ðkÞ, and ðkÞ are nowadays typically calculated from programs like FEFF 44) and given as input into the structure refinement process.For the interpretation of EXAFS data, the Fourier transform magnitudes of the EXAFS spectra, jFTð ðkÞÞj, are helpful.Similar to pair correlation functions obtained from Fourier transformed SðQÞs in X-ray scattering, these functions provide a handy illustration of the 1-D average real-space environment around the absorber atom.It is important to keep in mind, however, that these functions are not pair correlations in the strict sense.Most obviously, the EXAFS phase shift j ðkÞ typically leads to a shift of the real-space signal by some 0.1 Å.Additionally, the oscillatory nature of the (complex valued) ðkÞ functions frequently leads to signals with several peaks in jFTð ðkÞj but belonging to the same scattering path.For further reading and more detailed information on these techniques, we refer to the vast existing literature, e.g., regarding XANES 45) or EXAFS. 42,44,45)

Experiments and Computations
For the phenyl-tin-sulfur and cyclopentadienyl-tin-sulfur clusters, (1) and (2) in Table I, respectively, X ray scattering experiments were performed in transmission geometry at beamline P02.1 46) of the PETRA III storage ring of the Deutsches Elektronen-SYnchrotron (DESY) using a primary energy of 59.87 keV.The samples were confined in X-ray capillaries of 1 mm outer diameter and scattering data were collected using a flat pixel detector with pixel size of 200 Â 200 µm 2 .The sample-detector distance was set to 240.18 mm.Raw data were processed using the software DAWN. 47)An in-house Bruker D5000 diffractometer, equipped with a Goebel mirror to optimize the primary beam from a Mo tube, was used to study a freshly prepared sample of the amorphous SHG material (3), also contained in a 1 mm X-ray capillary for transmission measurements.An attempt to also obtain a decent SðQÞ for (3) from the PETRA III synchrotron source failed due to crystallization of the sample.All data were corrected for background-and air-scattering, self-absorption, polarization and Compton contribution and were then normalized to SðQÞ.The X-ray SðQÞs of the inorganic cluster materials are dominated by scattering from the heavier atoms of the Sn-S cluster cores, whereas the organic ligands contribute only with small weighting factors.Therefore, an additional neutron scattering experiment was performed on (2) where a degree of deuteration of about 90% was accomplished to suppress the incoherent scattering from the hydrogen atoms.For neutrons, the scattering length of the D and C atoms are of similar magnitude as those for Sn and S. The experiment was performed on the D4C diffractometer at the Institute Laue-Langevin in Grenoble=France using an incident wavelength of 0.4957(1) Å. Patterns were measured for the sample in a vanadium can of 6.05 mm inner diameter and 0.15 mm wall thickness, the empty can, the empty instrument and a vanadium rod of 6.08 mm diameter for normalization purposes.The data were corrected for background scattering, attenuation, multiple scattering and inelastic scattering as is outlined in, e.g., Ref. 48.EXAFS spectra were obtained at the sulfur K-and L 3 -edges and at the K-edge of the tin atoms.The Sn K-edge data were obtained at beamline P65, located at PETRA III (DESY) and the sulfur K-and tin L 3 -edge data were measured at beamline BL-11 of the Hiroshima Synchrotron HiSOR in Japan.At BL-11, the measurements were carried out at room temperature and the sample was directly measured, sandwiched between two polypropylene foils.At P65 additional measurements at 34 K were carried out and the samples were mixed with graphite and pressed to pellets.The scans were performed in transmission mode with absorption lengths ¼ 2:0 (P65, DESY) and ¼ 1:5 (BL11, HiSOR).The data were analyzed using the DEMETER software package (Athena and Artemis). 43)EXAFS spectra were normalized and background was calculated using the AUTOBK algorithm.An m-RMC-analysis on (1) was performed, for which we have manipulated an existing RMC simulation code based on the RMC_POT++ program, 49) which already offers the option to group atoms as a rigid molecule.We have generated a code enabling molecular translation and rotation, and also free rotation of the phenyl groups. 50)This code was then further optimized in terms of computational-and simulation speed. 51)uring the simulation 216 molecules of the phenyl-based molecule (1) were moved in a cubic simulation box with 55.836 Å edge length.Two sets of simulations were carried out.In the first set, the molecules consisted of rigid copies of the DFT-calculated phenyl-cluster (1).In the second simulation, the sulfur and tin atoms were allowed to slightly vary their coordinates within the cluster-core.The motion of chemically bonded atom pairs was thereby restricted within certain limits.The Sn-S bond, however, was left more or less unconstrained between 2.05 and 2.65 Å, since the weighting factors for both, X-ray and EXAFS data, are large for this correlation, providing sufficient information density for a reliable simulation.In contrast, the variability of the Sn-C bond was strongly constrained between 2.05 and 2.25 Å to compensate for the smaller weighting factor and thus the smaller information density in the data.The C atoms were not allowed to move intramolecularly.The attached Sn atoms thus always remained close to their original coordinates, which inevitably ensured an intact overall molecular structure during the simulation.

Results and Discussion
Total structure factors SðQÞ, as obtained from X-ray scattering on (1), (2), and (3) are displayed in Fig. 3 as the lower three curves by the symbols.The neutron-based SðQÞ is shown as the upper curve, also displayed by symbols.Corresponding molecular formfactors f 1 ðQÞ were calculated according to Eq. ( 4) and are displayed by the lines.Structural data based on DFT calculations 13,14) were used for the computations.They confirm that the molecular structure is well reflected by in the experimental data for the amorphous solids, and that the basic structural motifs of heteroadamantane clusters can basically be confirmed for all of the materials.Special positions of the organic ligands which can freely rotate around the Sn-C axis in the free single molecules, were not considered in these calculations.They were used as suggested by the DFT calculations for minimum free energy.However, the difference to the real situation in the amorphous solid seems not to be particularly important for the X-ray based SðQÞs, due to the smaller weighting factors of the C-C, C-H, and H-H correlations.SðQÞs of the amorphous samples and f 1 ðQÞs therefore show clear similarities because both are mainly determined by the scattering from the molecular Sn-S core and not from the organic ligands.The most striking difference is a pronounced peak around 0.6 Å −1 in the experimental data, which is not present in f 1 ðQÞ.In this Q-range, the latter increases monotonically with vanishing momentum transfer and intersects the SðQÞ axis at a value given by the number of atoms in the molecule.The experimental SðQÞ peak, located around 0.6 Å −1 in all of the X-ray SðQÞs thus represents a distinct intermolecular structural correlation.For the neutronbased SðQÞ of (2) however, the situation is clearly different.The coherent scattering lengths of C and D are now comparable with that of Sn and even more as twice as large as that of S. Accordingly, the weighting factors of the C-C, C-D, D-D, and C-Sn correlations are larger than the Sn-S correlation.Hence, the cluster core does no longer dominate the scattering pattern.Therefore, the respective rotational orientation of the organic ligands plays an important role for the shape of f 1 ðQÞ.The deviation from the real situation in the amorphous solid, which is unknown, is therefore now reflected by the difference between the structure factor and the molecular formfactor, especially in the lower Q-range.
Here, an intense peak appears around 0.9 Å −1 in f 1 ðQÞ which now however, is clearly intramolecular.This peak is also seen in SðQÞ, however here, it is shifted to slightly larger Q at about 1.1 Å −1 .
Figure 4 shows the corresponding PDFs as obtained by Fourier-transformation of the experimental SðQÞs in Fig. 3. Five different interatomic distances exist within the Sn-S cluster core, as is illustrated by the inset in Fig. 4. The respective values as obtained from the DFT-calculated molecules are marked by vertical dashed lines.Also given as grey dashed lines are the values for the C-H, C-C, and the Sn-C chemical bonds. 13,14)It should however be noted, that the PDFs are subject to non-negligible truncation errors in the lower r-region, which we did not further address in our data reduction.The PDFs were set to zero below r-values of about 1 Å which is close to the smallest physically reasonable value of the C-H bond.The truncation error influence is indicated by the dashed part of gðrÞ in the lower r-range.For the X-ray PDFs the oscillations influence the pattern up to about 2 Å where the contribution to gðrÞ is weak since this is the range of the negligible C-H and C-C correlations.For the neutron gðrÞ of (2) these correlations have the largest weighting factors and these bonds are therefore prominently represented in the PDF.However, these correlations will not be further discussed in this report.For r-values above about 2 Å, gðrÞ is determined mainly by correlations with heavy Sn atoms, so that the pattern there is determined by intense peaks that are only weakly influenced by the truncation problem.The Sn-C bond in (2) is longer than in the other two compounds.It is represented by a peak in the neutron-based gðrÞ at 2.28 Å.For the X-ray-based PDFs this signal is slightly shifted to higher r (2.35 Å) due to the larger weighting of the neighboring Sn-S correlation there.A further strong peak appears in all X-ray gðrÞs at around 3.8 Å comprising the Sn£Sn and S£S spacings.The Sn-Sn correlation possesses the dominating weighting for X-rays.However, its position at the high r side of this peak indicates that the Sn£Sn spacing may be smaller in the amorphous solids than in the calculated single molecules.For the WLGs (1) and (2) this shift seems to be slightly larger than for the SHG (3).Each Sn atom of the cluster core is also surrounded by three further S atoms which are located equally spaced around 4.6 Å in the DFTcalculated molecules (Sn£S long ).For the amorphous SHGmaterial (3) a broad peak exists ranging from 4.3-5.1 Å and is centered around the DFT value.It is also visible in the gðrÞs of the WLGs (1) and ( 2), but there it seems to be slightly shifted towards larger distances.Similarly, each sulfur is surrounded by four equally spaced sulfur neighbors (S£S) and one which sits further away at about 5.7 Å (S£S long ).Broad peaks are here visible for the SHG material (3) and the two WLG materials (1) and (2), again shifted to slightly larger r values.The PDFs in Fig. 4 hence suggest that the lengths of the chemical bonds in the cluster core (Sn-  2), and (3), from bottom to top, respectively.The symbols of the upper data set represent the neutron SðQÞ from a deuterated sample of (2).Also shown as lines are the respective molecular formfactors as obtained from Eq. ( 4).For the neutron data of (2) a deuteration degree of 90% was estimated.C and Sn-S) of the explored amorphous materials are similar to the respective values of the DFT-calculated clusters.However, nearest-neighbor distances spanning less than two bonds, like the Sn£Sn and S£S spacings, appear to be smaller than in their DFT-computed counterpart, while more distant spacings, spanning more than two bonds (Sn£S long and S£S long ), are longer than calculated.This suggests that the cluster nuclei in the amorphous materials may be subject to some distortions and they appear to be slightly larger for the WLG materials than for the SHG material.This would define a clear difference between the two material classes.However, it is neither possible to make quantitative nor sustainable statements about such small effects only on basis of the total PDFs.Therefore, we have additionally carried out EXAFS studies on the amorphous WLG (1) and the amorphous SHG (3) to obtain more detailed structural information. 52)The corresponding data as obtained at the sulfur K-edge are displayed in Fig. 5 together with their real space Fourier-transforms.
The data were fitted using structural details from the DFT calculated molecules 13,14) as starting parameters.Both spectra can be fitted reasonably well (solid lines in Fig. 5).However, it is obvious that the fits for the SHG material (3) are considerably better than for the WLG material (1).This is a further hint that the molecular structure of the WLG-cluster experiences stronger modifications when being transferred into a dense amorphous phase as the SHG-cluster.The fit quality for the phenyl cluster (1) can however be improved by including an additional S-S or S-Sn fitting path at intermediate distance.This is shown in Fig. 5 by the dotted lines in the spectra for the phenyl cluster (1).Such an additional fitting path is indication that an additional atom is situated nearby either due to a distortion of the molecular structure or due to an additional intermolecular correlation.However, it must be stated, that the inclusion of this additional fitting path leads to a considerable dependence among the fitting parameters.Hence, additional constraints were needed to increase reliability of this procedure.To obtain further sustainable indications for the structural situation of the Sn-S cluster core, we have performed m-RMC simulations on the phenyl based WLG material (1)   where X-ray diffraction and EXAFS data simultaneously served as constraints. 53)X-ray and Sn K-edge EXAFS data were both obtained from the PETRA III facility at DESY in Hamburg.They are depicted as symbols in Figs.6(a) and 6(b), respectively.SðQÞ and EXAFS of a virtual ensemble of 216 molecules were iteratively computed during the simulation and should ideally converge towards the experimental curves.The thinner lines in Fig. 6 represent the outcome of the rigid-simulation attempt, where inflexible molecules of the DFT-calculated type were employed as molecular entities.It already reflects the essential details of the scattering pattern, but significant deviations still exist.In particular, a phase shift is visible, between the experimental and the simulated oscillations, which is especially pronounced between 8 and 12 Å −1 .Such a phase shift indicates that the real molecular structure in the amorphous solid deviates in shape and size from the DFT-calculated prediction.The differences between experiment and simulation are even more evident in the EXAFS data, where the simulation differs substantially from experiment also indicating that distances in the phenyl cluster (1) of the amorphous phase differ from the theoretical prediction.The result of the dynamic-simulation, where tin and sulfur atoms were allowed to vary their positions is depicted by the thicker lines in the two graphs.The atomic degrees of freedom unavoidably led to a variation of the molecular structure, but also to a rapid convergence of the simulations towards the experimental curves.The agreement with experiment is now excellent.Just small differences still remain at lower momentum transfer, possibly owed to the uncertainty of the exact carbon positions

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due to their small weighting.The difference between thin and thick curves clearly shows that the rigid DFT-calculated molecule is in fact not a suitable model to reproduce the experimental data of the amorphous solid.The virtual molecular ensemble of the dynamic m-RMC simulation allows to extract partial PDFs for the Sn-Sn, Sn-S, and the S-S correlations.In turn, this allows direct access to the structural properties of the Sn-S cluster core in the amorphous solid.They are shown in Fig. 7. Dashed lines represent the partial PDFs of the rigid simulation, while the black solid lines give the PPDFs of the dynamic approach.Clear differences between the two data sets exist.For the rigid molecule, the PPDFs are entirely determined by the expected intramolecular distances of the core atoms which are marked by arrows for the Sn-S and S-S correlations.Distinct intermolecular correlations cannot be identified.E.g., for the Sn-Sn correlation, one only observes the atomic distance inside the Sn tetrahedron.Apart from that, the PPDF grows within the statistical error continuously towards one with increasing r, representing a pure stochastic probability of encountering intermolecular Sn atoms without any distinct correlation.The PPDFs of the dynamic simulation show a completely different picture: The Sn£Sn distance is slightly smaller than found in the DFT-calculation and the peak is slightly asymmetric, already indicating that the Sn 4 frame is subject to some distortion.Also, between about 5.This curve defines the number of neighboring atoms hidden under the PPDF peaks.In the rigid cluster core, every sulfur atom is surrounded by four identical neighbors located at 4.01 Å as is indicated by a dashed vertical line.In the amorphous solid these neighbors split into two groups, each containing two atoms, as indicated by the PRDF integral.One group is represented by a larger higher narrower peak at about 3.68 Å and the second by a smaller but broader peak at about 4.18 Å.Each sulfur atom of the rigid cluster also has a further well-defined, but more distant single sulfur neighbor at 5.72 Å, again indicated by a vertical dashed line.In the amorphous solid, this correlation is only visible as a broad peak situated between 5 and 6 Å with maximum slightly shifted to smaller distance.
The Sn atom of the rigid model cluster is chemically bonded to 3 sulfur atoms each at 2.44 Å but also by three identical neighbors, two bonds away, located 4.63 Å apart (Sn£S long ) as shown in the lower graph of Fig. 8(b) by the vertical lines.However, in the amorphous solid the latter correlation splits into three components situated between 3.5 and 5.5 Å, each containing one single atom.Again, the center of this group is shifted to values below the original rigid cluster value.The ensemble averages displayed in Figs.7  and 8 clearly demonstrate, that the cluster cores in the amorphous [(PhSn 4 )S 6 ] solid (1) are considerably deformed as compared to their counterparts computed with DFT methods.It is also obvious that it is the sulfur which is mainly responsible for these structural variations.Analyzing the atom-atom correlation between different molecules gives a similarly interesting picture.This is visualized by the upper graphs of Fig. 8.A pronounced correlation peak ranging from about 3 to 4.4 Å and centered at about 3.7 Å is found for the intermolecular S-S correlation which is depicted in Fig. 7(a).The corresponding PRDF-integral reveals one  atom, indicating that every cluster core of the RMC ensemble is on average surrounded by one S-atom of another cluster core on this length scale.It should be noted that such distances are well inside the range of intramolecular S£S spacings.The intermolecular Sn-Sn correlation in the upper plot of Fig. 8(b) shows a similar increase in the RPDF integral, but involves only about half an atom on a comparable length scale, indicating that it is the sulfur that imposes the dominant relationships between the molecules in the amorphous WLG solid.
In order to get more information on the relative spatial arrangement of different clusters from the dynamic computer ensemble, we have computed molecular centers of mass (MCMs) for each of the molecules and calculated the corresponding PDF of these centers, which is shown in Fig. 9. Distinct peaks are visible in the molecular PDF indicating that the molecular arrangement comprises clear intermolecular correlations.A first peak is centered at 7.5 Å and ranges from about 5.5 to 8 Å.This coincides exactly with values from a quantum chemical approach where dimers were considered as a minimal model for the amorphous system. 16)The integral over the corresponding RDF (solid line in Fig. 9, right scale) shows that each molecule is on average surrounded by two neighbors (dashed lines indicate the integration area).Another maximum appears between 10 and 14 Å, indicating that the molecules may arrange into chains.The inset in Fig. 9 illustrates the distribution of MCMs inside the simulation box.Bonds between molecules with distances between 5.5 and 8 Å were drawn to visualize the mutual spatial correlations of the clusters.In fact, it can be seen that chain-like structures of different lengths, partly branched, are indeed formed.A closer analysis shows, that about 20-25% of all clusters are bound in dimers with spacings between 6-7.5 Å. 53) The mutual arrangement of the phenyl clusters bound in such chain like structures is an alternating staggered configuration, where the ligands of one cluster are located in the voids between those of another cluster.As an example, Fig. 10 shows the arrangement of molecules in an arbitrarily chosen four atomic linear chain.Similar mutual molecular arrangements have already been predicted by the quantum chemical calculations. 16)Such an arrangement is also found in the crystalline compound (4) containing molecular building blocks very similar to those studied in our m-RMC exploration, where just Sn has been replaced by Si. 16) We thus find that the molecules are relatively strongly correlated and form superstructures of linear and partly branched chains.Thereby, their molecular centers of mass approach distances close to those predicted by theory for dimers. 16)Here, the spacings are, however, slightly larger which can be attributed to the fact that the interaction in the solid is shared by more than one neighbor.Moreover, the individual molecules are clearly distorted as compared to the respective DFT-calculated structures and these distortions result mainly from position changes of the sulfur atoms.It is noticeable that the overall difference between intra-and intermolecular S-S distances seems to disappear.Apparently, the sulfur atoms try to distribute themselves evenly over the sample, as far as this is possible despite their inclusion into the molecular adamantane framework.To illustrate this, some sulfur-sulfur distances are given for the four-molecule chain shown in Fig. 10 where intramolecular and intermolecular distances are respectively depicted.

Conclusions
Two amorphous Sn-S based WLG materials with organic phenyl and cyclopentadienyl ligands, together with another amorphous Sn-S based SHG material containing naphthyl attached to the tin atoms, were explored using X-ray and neutron scattering, EXAFS and molecular RMC simulation to identify microscopic structural differences between the two material classes.Overall, the results indicate that the adamantane-like molecular building blocks in the WLG materials are subject to deformations that are not observed that way in the SHG material.Molecular RMC simulations performed for the phenyl-based WHG material suggest that  these distortions are mainly resulting from positional variations of the sulfur atoms.Moreover, it appears that the sulfur atoms in the amorphous WLG materials attempt to distribute as evenly as possible over the amorphous solid, and are only hindered by their molecular integration.It seems that more S£S short spacings are needed to reproduce the X-ray and EXAFS patterns than are actually present in the cluster cores.Therefore, intermolecular S£S spacings are employed, which, however, is only possible by appropriate deformation.In the real solid, this process may be driven by strong corecore interactions which were identified previously by theory. 16)These deformations also explain the high degree of isotropic interactions between the clusters reported there, which elsewhere was made responsible for the tendency to suppress crystallization. 53)The nearly uniform distribution of sulfur atoms resembles a close-meshed net, and since no chemical bonds exist between the sulfur atoms one may identify this as a vibrational network.Such a network would offer the option to generate a corresponding phonon-based atom dynamic which could then offer a wide range of k values to enable a broad receptivity for infrared radiation.Since such a scenario is exclusively restricted to disordered materials, it would explain why only amorphous Sn-S based materials show the WLG effect.

Fig. 1 .
Fig. 1. (Color online) Representation of the [(PhSn) 4 S 6 ] cluster (1), one of the amorphous white light generators.Organic phenyl ligands are attached to the Sn atoms of the tins-sulfur cluster core.

Fig. 2 .
Fig. 2. (Color online) NLO-responses from (a) the crystalline and (b) the amorphous materials, (1) and (4), respectively (top).The driving excitation is visible at 1.265 eV in each spectrum.The 2nd-harmonic of (a) is clearly seen at 2.53 eV, while (b) depicts a broad WLG.The respective X-ray patterns are also shown below indicating that the SHG-material is clearly crystalline (left) while the WLG material (right) shows the typical structure factor SðQÞ of an amorphous solid.

Fig. 3 .
Fig. 3. (Color online) Symbols represent structure factors SðQÞ as obtained from X-ray scattering for molecules (1), (2), and (3), from bottom to top, respectively.The symbols of the upper data set represent the neutron SðQÞ from a deuterated sample of (2).Also shown as lines are the respective molecular formfactors as obtained from Eq. (4).For the neutron data of (2) a deuteration degree of 90% was estimated.

Fig. 4 .
Fig. 4. (Color online) Total pair distribution functions gðrÞ as obtained by Fourier transforming the SðQÞs of Fig. 3 for molecules (1), (2), and (3), respectively.The upper curve represents the neutron PDF of (2).Dashed vertical lines indicate atom-atom spacings in the DFT calculated molecules. 13,14)Dashed parts of gðrÞ curves indicate the influence of truncation errors from Fourier transformation.

Fig. 5 .
Fig. 5. (Color online) Experimental EXAFS data (symbols) for the amorphous materials (1) and (3), respectively (left-hand side), and corresponding Fourier transforms to R-space (right-hand side).Solid lines represent fits to the data using the DFT-calculated models.Dotted lines in spectra for (1) denote an extended fit using an additional scattering path to fit the data.

Fig. 6 .
Fig. 6. (Color online) (a) Comparison between experimental (symbols) and dynamic m-RMC-simulated SðQÞ (thicker line) of (1).Thinner line represents simulation employing rigid molecular entities based on DFTcalculation. 13)(b) Comparison between experimental Sn K-edge EXAFS data (symbols) and m-RMC-simulation.Thinner line represents fit of rigid DFTcalculated clusters, thicker line shows the result of the dynamic simulation.
5 and 6.2 Å, clear indication exists for intermolecular correlations.PPDFs containing sulfur correlations deviate even more significantly from the rigid molecule expectations.Except for the direct Sn-S bond, all other S£S and Sn£S distances seem to split into different spacings indicating strong deformation of the cluster core.We therefore further separated the PPDFs into inter-and intramolecular contributions which are shown in Fig. 8.The intramolecular contributions of the S-S and the Sn-S correlations are shown in the lower graphs of Figs.8(a) and 8(b) which respectively represent the S-S and the Sn-S relations.The intermolecular PDFs are displayed in the upper graphs.The solid lines, belonging to the right-hand scales, represent the integrals over the respective partial radial distribution function (PRDF) 4r 2 ngðrÞ, with n being the neighboring atom density.

Fig. 7 .
Fig. 7. (Color online) Partial PDFs g SnSn ðrÞ, g SnS ðrÞ, and g SS ðrÞ, as obtained from the dynamic RMC-simulation (solid black lines).Same functions are also shown, obtained from the rigid RMC-simulation (dashed lines).

Fig. 8 .
Fig. 8. (Color online) g SnS ðrÞ and g SS ðrÞ of Fig. 7 further split into intraand intermolecular contributions.Dashed vertical lines represent values for the DFT-calculated cluster core. 13)Solid lines give the integral values over the radial distribution functions, determining the number on neighbors as function of r.

Fig. 9 .
Fig. 9. (Color online) Molecular PDF, gðrÞ, as obtained from the MCMs of the dynamic simulation ensemble.Full line represents again the integral over the RDF to give the number of surrounding molecules.The inset shows MCMs from the dynamic simulation with bonds for spacings between 5.5 and 8 Å.

Fig. 10 .
Fig. 10.(Color online) Fourfold linear chain, arbitrarily chosen out of the dynamic m-RMC simulation ensemble.H-atoms have been omitted to retain clarity.Molecules in chains prefer a staggered mutual configuration with respect to the organic ligands.MCMs are displayed by grey bullets connected by dashed lines.Also shown are some intramolecular and intermolecular distances, indicating similarity between intra-and intermolecular sulfursulfur spacings.