Electron Crystallography and Non-Linear OpticsI.G. VOIGT-MARTIN*Institut fur Physikalische Chemie der Universitat Mainz, D 55099 Mainz, Germany

ABSTRACT Electron crystallography can be used to obtain specific information about molecularparameters such as the polarisability, dipole moment, and hyperpolarisability. In this, work we showhow a combination of quantum mechanics and simulation methods can be used to solve severalunknown organic structures and how the calculated molecular parameters can be used to predict thecorresponding physical properties of the crystals. Microsc. Res. Tech. 46:00–00, 1999.r 1999 Wiley-Liss, Inc.

INTRODUCTIONElectron microscopic structural investigations of beam

sensitive, organic samples have reached considerablesophistication in recent years due to the application ofdirect methods (including maximum entropy statistics)and simulation techniques. In the latter case, it is evenpossible to determine atomic positions in the unit cell.However, even beyond the determination of atomicpositions, it is possible to calculate specific physicalproperties of small organic molecules from quantummechanical calculations of molecular parameters andto relate these to the crystal axes. In cases wherecrystals are too small for X-ray analysis, electroncrystallography is the only way to obtain this informa-tion.

BASIC CONCEPTS IN NON-LINEAR OPTICSThe motivation for this work has been to reach a

clearer understanding of the relationship between thestructure of organic molecular crystals and their secondorder non-linear optical (NLO) properties. The furtheraim is to find a more specifically directed route towardthe synthesis of molecules with the required moleculararchitecture. Such an approach requires close collabora-tion between specialists in organic chemistry, physicalchemistry, and physics as well as in electron microscopyand can be described by the term ‘‘crystal engineering.’’

The effect that is observed in second-order NLO isthat of frequency doubling, or second harmonic genera-tion (SHG). In our examples, an incoming beam ofinfrared light (l 5 1,047 nm) emerges as green light(l 5 523.5 nm). Practical applications are found inopto-electronic devices that process information effi-ciently and are, therefore, candidates for future commu-nication systems. Organic materials have SHG efficien-cies that are greater than those of classical inorganicmaterials like lithium niobate or potassium dihydrogenphosphate, a positive effect that is unfortunately coupledwith poorer mechanical properties. However, it is gener-ally accepted that organic compounds offer much morescope for deliberately tailoring both electronic andcrystallographic properties as well as offering the possi-bility of processing in different geometries (Desiraju,1979).

The observed physical property involved is the opti-cal hyperpolarisability x which is related to the polari-

sation P by the following expression:

P 5 e0[x(1)IJE 1 x(2)

IJKE2 1 x(3)IJKLE3 . . .], (1)

where xIJ is the linear susceptibility, xIJK the second-order susceptibility, and xIJKL the third-order suscepti-bility. The relevant term for SHG, namely xIJK, can benon-zero only if the crystal is non-centrosymmetric.

In order to ensure that a molecule crystallises in anon-centrosymmetric fashion, the molecular architec-ture is often designed by choosing suitable molecularconcepts such as chirality, but there are many otherpossibilities, as discussed in Desijaru’s book.

The classical way of determining and refining struc-ture is by X-ray diffraction, and Zyss has made majorcontributions in relating molecular properties to crystalstructure (Chemla and Zyss, 1987). However, there aremany molecules that do not crystallise easily and formcrystal platelets that are only about 10 nm thick withlateral dimensions of a few tens of nm. For suchmaterials, electron crystallography is the only solution.

In addition to the crystallographic aspects, there aremolecular criteria that need to be considered. Fororganic compounds, the molecular dipole moment µinduced by an electric field E is given by:

µi 5 µi0 1 aijE 1 bijkE2 1 gijklE3 . . . , (2)

where µi0 is the intrinsic molecular dipole moment and i,

j, k, l 5 x, y, or z. The molecular polarisability and thefirst- and second-order hyperpolarisabilities a, b, g aregiven in the molecular co-ordinate system x, y, z. Thecombined molecular and crystallographic requirementsmay be met, for example, by a molecule containing asuitable non-linear active segment plus a chiral seg-ment.

The relationship between microscopic and macro-scopic parameters is given by:

xIJK(22v;v1,v2) 5 N/V[fI(v)fJ(v)fK(v)SScosuIi

3 cosuJjcosuKkbijk(22v;v1;v2)],(3)

Contract grant sponsor: Deutsche Forschungsgemeinschaft.*Correspondence to: I.G. Voigt-Martin, Institut fur Physikalische Chemie der

Universitat Mainz, Jakob Welder Weg 11, D 55099 Mainz, Germany.E-mail: voightmar@mail.uni-mainz.de

Received 14 October 1999; accepted in revised form 28 January 1999

MICROSCOPY RESEARCH AND TECHNIQUE 46:178–201 (1999)

r 1999 WILEY-LISS, INC.

where V is the unit cell volume, N is the number ofmolecules per unit cell, fI(v) are local field factors atfrequency v for the I-direction in the crystal, etc., andthe uIi are the rotation angles relating microscopic andmacroscopic axes. The local field factors fI(v) depend onthe linear polarisability term aIi, which is related to therefractive indices of the crystal. The macroscopic suscep-tibility coefficients dIJK, which are actually measured inan experiment, are directly related to xIJK and to thedirection of the incoming and outgoing beams withrespect to the crystal axes. They depend on the crystalsymmetry.

In addition, a number of other factors such as non-critical phase matching, infrared and visible transpar-ency, ease of synthesis, thermal stability, and mechani-cal strength must be considered.

Our approach to electron crystallography, which in-cludes quantum mechanical and molecular packingcalculations as well as the maximum entropy (ME)approach, offers the advantage of giving deeper insightinto the molecular mechanisms that give rise to theobserved physical properties, as will become apparentin the following.

OUTLINE OF EXPERIMENTAL PROCEDUREThe basic consideration is therefore to relate the

following:

Molecular architecture

<

Characterisation of molecular parameters

<

Structure and relation of molecularproperties to crystal axes

<

Physical properties

The molecular and crystallographic parameters in-volved and their relationship are schematically shownin Figure 1.

Since b and x are third-rank tensors with 27 compo-nents, this diagram is, of course, highly simplified andapplicable only to linear molecules in which only diago-nal components of the hyperpolarisability have signifi-cant values and can be represented as a vector.

Our experimental procedure can be summarised inthe following 14 steps:

1. A molecule with a suitable chemical architecture issynthesised according to the principles outlinedabove.

2. We establish whether the molecule has an NLOeffect in solution by Electric Field Induced SecondHarmonic generation (EFISH) and Hyper RayleighScattering (HRS) methods and measure, µ, a, b(Voigt-Martin et al., 1997a).

3. The conformation of the molecule in the gas phaseis calculated by semi-empirical methods usingMOPAC (Stewart, 1989), or ab initio quantummechanical calculations with GAUSSIAN or TUR-BOMOL (Richards and Cooper, 1983).

4. Design crystal to produce non-centosymmetry.

5. Screen the crystals for NLO effect (Loos-Wilde-nauer et al., 1995).

6. Obtain electron diffraction patterns in at least 8different projections (Voigt-Martin et al., 1995a,b,1997b; Yakimanski et al., 1997).

7. Routine check of d-values with X-ray powder diffrac-tion (Voigt-Martin et al., 1995a,b, 1997b; Yakiman-ski et al., 1997).

8. Quantify electron diffraction intensities (Kolb andKothe, 1997).

9. Simulation of diffraction patterns and packingenergy calculations (Voigt-Martin et al., 1995a,b,1997b; Yakimanski et al., 1997).

10. High resolution imaging and image restoration(Voigt-Martin et al., 1995a,b, 1997b; Yakimanski etal., 1997).

11. Simulation of images (Voigt-Martin et al., 1995a,b;Yakimanski et al., 1997).

12. Check for dynamical scattering (Voigt-Martin etal., 1995a,b, 1997b; Yakimanski et al., 1997).

13. Ab initio structure determination using ME ap-proach (Voigt-Martin et al., 1995a,b, 1997b; Yaki-manski et al., 1997).

14. Relate molecular a, µ, b to crystal co-ordinates andcalculate macroscopic optical susceptibility (Voigt-Martin et al., 1995a,b, 1997b; Yakimanski et al.,1997).

Fig. 1. Schematic diagram showing relationship between molecu-lar parameters and macroscopic properties depending on mutualpacking.

179EC AND NLO

EXPERIMENTAL METHODS AND RESULTSChoice of Suitable Chemical Architecture

For the investigation described here, molecules withthe following chemical architecture were synthesizedbecause of their delocalized p-systems:

Screening Molecules for NLO EffectHyper Rayleigh Scattering (HRS). The HRS tech-

nique (Willets et al., 1992) was further developed byClays and Persoons (1991) as a method of determiningthe molecular second order hyperpolarisability. By col-lecting the frequency doubled light scattered perpen-dicular to the propagation direction of an intense laserbeam in an isotropic liquid, one obtains informationabout the second-order polarisability of the solute. Ifthe experiment is performed under different polarisa-tion conditions of the fundamental laser and the col-lected light, it is possible to determine five independentterms containing products of b components. We havedescribed details about these calculations on DMACB,CNBA, and NPHU in the specialised literature (Voigt-Martin et al., 1995a,b, 1997a,b; Yakimanski et al.,1997) and have combined the results in a teachingcourse on crystallography (Voigt-Martin and Kolb, 1997).If only one b component is significant (as is the case inmany linear molecules with delocalised p-systems), themolecular fixed frame may be chosen such that thetensor components that finally emerge after some calcu-lation are bxxx, byyy or bzzz. These components are theneasy to relate to those calculated by MOPAC.

Electric Field Induced Second Harmonic Gen-eration (EFISH). The EFISH technique (Levine andBethea, 1975) utilises a static electric field E0 to inducean effective second order susceptibility, x(2)(22v;v,v;E0) 5 3x(3)(22v;v,v;0) · E0, in a liquid solution.

The evaluation of concentration-dependent EFISHmeasurements yields partial molar third-order polaris-abilities. Finally, with the ground state dipole obtainedfrom permittivity measurements, it is possible to evalu-ate the vector parts of the second order polarisability.Thus, the value of b that is obtained is its projection onµ. We have reported detailed results on DMABC else-where (Voigt-Martin et al., 1997b).

Generation of a Molecular Model andDetermination of Gas Phase Dipole Moment m

and Hyperpolarisability b

From quantum mechanical calculations, we can onlyobtain the gas phase conformation of the molecule. Ithas been shown that crystallisation generally affectsonly the torsional angles of the molecule (Fillipini andGavezotti, 1993; Gavezotti, 1991), therefore the gasphase conformation is sufficiently accurate to use as astarting conformation for the packing energy calcula-tions which will be performed subsequently.

However, because the crystal field causes these adjust-ments in molecular conformation, it is generally notessential to invoke ab initio quantum mechanical calcu-lations, unless more accurate values of b are required,as is indeed sometimes the case (Voigt-Martin et al.,1998a). Generally, the semi-empirical AM1 and PM3values calculated by MOPAC are sufficiently accurateto initialize the crystallography programs. Since second-order polarizabilities are frequently defined using differ-ing conventions, it must be noted that the semi-empirical calculations use the finite field technique(Kurtz et al., 1990). These methods have been param-etrized for gas phase properties such as ground stategeometries, dipoles µ, and heats of formation, but notfor the second-order polarizability b. Generally, thesecond-order polarizability values obtained by thesemethods are intermediate between those obtained byself-consistent field (SCF) and second-order perturba-tion (MP2) calculations.

The minimum-energy gas-phase conformations of themolecules at zero Kelvin discussed here were calculatedby using the PM3 method incorporated in the programpackage MOPAC. Most molecules have several mini-mum energy conformations. To begin simulation, theconformation which can best be fitted into the experi-mentally determined unit cell (see Electron Diffraction)will be chosen initially. For this initial conformation,the dipole moment, the linear polarisability, and thehyperpolarisability tensor components are calculated.

When comparing the values for b with the results ofthe spectroscopic measurements, great care has to betaken: The finite field method incorporated in MOPACgives values for the static second order polarizabilityb(0;0,0). The optical values b(22v;v,v) obtained fromthe experiments are enhanced with respect to the staticones by dispersion and by the influence of the reactionfield in solution (Clays et al., 1993; Terhune et al., 1965;Wolff et al., 1997). In general, the calculated values aretherefore expected to be significantly lower than theones obtained experimentally. In order to obtain fre-

180 I.G. VOIGT-MARTIN

quency-dependent values, it is necessary to resort to abinitio quantum mechanical calculations using pro-grams such as TURBOMOL or GAUSSIAN. Such calcu-lations are in progress in this group but have not beenpublished yet.

The results of our semi-empirical quantum mechani-cal calculations on DMACB, CNBA, and NPHU areindicated in Figure 2, together with the values of µand b.

For these molecules only the diagonal components ofthe hyperpolarisability tensor have significant values,so that b can be represented as a vector. The results aresummarised below:

DMABC

µ 5 14.5 ? 10230 Cm b 5 12 ? 10250 Cm3/v2

CNBA

µ 5 14.54 ? 10230 Cm b 5 1.33 ? 10250 Cm3/v2

NPHU

µ 5 24.78 ? 10230 Cm b 5 4.75 ? 10250 Cm3/V2

These values correspond to the gas phase conforma-tion of the molecules at zero Kelvin.

Methods of Crystal Design to ProduceNon-Centrosymmetry

Several preparative methods or synthesis procedurescan be used to produce the required crystals without acenter of symmetry. They are listed below:

1. Molecular chirality to avoid molecular and crystalsymmetry

2. Hydrogen bonding to produce chiral arrays3. Reduction of ground state dipole to prevent anti-

parallel arrangement of molecules4. Crystal growth in an electric field to align molecules

in the same direction5. Monolayer Langmuir Blodgett films in which all

molecules point in the same direction6. Liquid crystals to break symmetry.

Screening Crystals for NLO EffectImages showing the effect of frequency doubling (in

this case infrared = green) can be obtained using anSHG-microscope (Loos-Wildenauer et al., 1995). Thecore of the setup is an Olympus measuring microscope,which is adapted for imaging the SHG signal as well asfor dark field polarisation and fluorescence microscopy.The fundamental beam of a Q-switched Nd:YLF laser(l 5 1,047 nm) provides light pulses in the range of 100mJ (15 ns, at a repetition rate of 3,000 Hz). The incidentlaser beam (TEM00 beam diameter 0.9 mm, polarisation100:1) is focussed via a beam-steering mirror onto thecrystal in diascopic geometry. The resulting fundamen-tal intensity in the plane of the crystal is of the order of108 W/cm2. In order to protect the lens system and thesample against the high-power laser pulses, an infraredfilter (Schott BG 40, transmission at 1,047 nm 5 1024)was placed in front of the objective lens. An additionalband pass filter guarantees that only the second har-monic light reaches the detector. We have demon-strated the effect with colour micrographs in the specia-lised literature (Loos-Wildenauer et al., 1995).

Electron DiffractionIn order to relate molecular parameters to macro-

scopic optical properties, we have developed a complexprogram of interrelated computational and electronmicroscopic techniques. This is indicated schematicallyin Figure 3.

The experimental electron diffraction patterns fromthe organic thin crystals must be compared with thecalculated diffraction patterns obtained from a model ofthe molecules in the unit cell belonging to a specificspace group. In order to determine the space group, it isnecessary to obtain a large number of projectionsexperimentally. Generally we try to establish a basiczone by aligning along one of the principal reciprocalaxes and tilting both to the right and to the left until themaximum d-values, i.e., crystallographic axes, arefound, and then repeating this using two or morespecific crystallographic axes. Tilting in both directionsand using two axes is necessary to distinguish betweenorthorhombic and monoclinic cells. This can be verydemanding experimentally for unknown structures andbeam sensitive samples. In this way, it is possible toestablish the cell constants and, by applying the re-quired trigonometric relationships, the cell angles.

From the extinctions we determine the space group(Voigt-Martin et al., 1995a,b, 1997b; Yakimanski et al.,1997). It has been shown in many publications thatintensities are affected and extinctions may be maskedby a large number of factors such as dynamic scattering(Cowley, 1986), crystal bend or buckling (Dorset, 1995),and beam damage (Voigt-Martin et al., 1992). Bendingcan be recognised from extinction contours in theimage; therefore, flat regions should be chosen and asmall probe used. Beam damage is reduced by low dosemethods and by using low temperatures. The appropri-ate precautions, discussed at length in the literature,must be observed. They depend on the molecular struc-ture and therefore the specialised literature should beconsulted if specific details are required. Generally it ispossible to recognise symmetry forbidden reflections ifa tilting series is taken. The symmetry forbiddenreflections frequently disappear at appropriate tiltingangles and are thus revealed as such. Once the cellconstants and the extinctions in different zones havebeen carefully determined, the International Tables ofCrystallography frequently reveal that only a few spacegroups fulfill the necessary requirements. Modellingand molecular symmetry then further reduce possiblespace groups. Frequently physical information is used.For example, if the sample produces a second harmonic,then the space group must be non-centrosymmetric.

In all cases, problems concerning specimen prepara-tion are encountered (Dorset et al., 1998). For organicsamples, a number of particularly useful techniquesinvolving epitaxial deposition of crystals have beendeveloped (Wittmann and Lodz, 1990).

In addition to this, it is by no means easy to record theintensities correctly. The next chapter will summarisesome of the difficulties related to recording intensities.

For the ensuing structure analysis, it is essential notto lose additional intensity values unnecessarily be-cause a lot of data is already lost by the tilt anglelimitation of 60° imposed by older specimen holders.With modern specimen holders, it is possible to obtaintilting angles of about 70°.

181EC AND NLO

Further information about the structure is obtainedby high resolution imaging. The space group obtainedfrom electron diffraction can be checked in the imagesby using, for example, specialised software such as

CRISP (Hovmoller, 1992). Since the image is a 2-Dprojection, it is important to use the appropriate planegroup for this projection obtained from the Interna-tional Tables for X-Ray Crystallography. Properties

Fig. 2. Molecular conformation of DMABC, CNBA, and NPHU in gas phase. Calculated values of µand b are indicated on the diagram. (reproduced from Voigt-Martin and Kolb 1997) with permission of thepublisher.

182 I.G. VOIGT-MARTIN

Fig. 3. Schematic diagram showing experimental procedure.

183E

CA

ND

NL

O

Fig. 4. Tilt series and simulated diffraction patterns obtained for CNBA. Zones are indicated ondiffraction patterns. Column a 5 experimental, b 5 simulated, electron diffraction patterns, c 5corresponding projection model.

184 I.G. VOIGT-MARTIN

such as second harmonic generation also depend oncrystal defects, and high resolution images may givecrucial information in this respect (Voigt-Martin et al.,1997c).

When all the different projections obtained from thesamples had been evaluated as we have described inthe specialised literature, the following cell constantsand space groups were determined for the examplesdiscussed here:

DMABC Pna21 a 5 10.28 Å a 5 b 5 g 5 90°

(Voigt-Martin et al., b 5 22.64 Å

1997a) c 5 5.27 Å

CNBA P21/c a 5 14.7 Å a 5 g 5 90°

(Voigt-Martin et al., b 5 9.47 Å b 5 112°

1995a) c 5 15.42 Å

NPHU P21 a 5 5.607 Å a 5 g 5 90°

(Yakimanski et al., b 5 5.756 Å b 5 96.94°

1997) c 5 21.49 Å

Figure 4a shows, as an example, part of the experi-mental tilt series obtained for CNBA. From such experi-ments the third axis and the missing angles can bedetermined.

The experimental and simulated diffraction patternsfor DMABC, CNBA, and NPHU are published in thespecialised literature (Voigt-Martin et al., 1995a,b,1997b; Yakimanski et al., 1997).

At this stage of the investigation, the followinginformation is available (method of obtaining the infor-mation indicated at right):

Fig. 4. (Continued.)

185EC AND NLO

The information that is still required is the following:

1. Position and orientation of molecule with respect tocrystal axes

2. Conformation of molecule in crystal3. Value and direction of dipole vector components and

hyperpolarisability tensor components

Routine Check of d-Values Using X-RayPowder Data

Standard X-ray powder diffraction patterns werealways obtained to refine the d-values obtained fromelectron diffraction. A Siemens D 500 diffractometer(Cu Ka, l 5 0.1542 nm) in u/2u X-Ray reflectivity modewas used (Voigt-Martin et al., 1995a,b, 1997b; Yakiman-ski et al., 1997). We have tried to use Rietveld methods,but were successful only in refining d-values but notatomic positions. This is because frequently we haveindependent molecules each consisting of several tensof atoms in general positions in space groups of lowsymmetry. It seems possible that Rietveld methodsmight be used successfully to refine known organicstructures of this complexity, but that would not beaddressing the problem discussed in this article. This isone of the reasons why electron crystallography needsto be improved.

Quantifying Electron Diffraction DataQuantification of the experimental intensity data is

one of the most crucial steps in structure determina-tion. Several methods are available in order to obtainquantitative values of the intensities:

1. Microdensitometer2. Image plates3. On-line slow scan CCD camera

4. Off-line CCD camera5. Scanner

The microdensitometer has been used successfully byDorset (1995) for many years and many of the theoreti-cal concepts involved have been discussed by him.However, future trends point towards CCD technology.On-line slow scan CCD cameras are available with a12-bit or 16-bit dynamic range and a resolution of 2,0483 2,048 pixels. The use of both slow scan cameras andimage plates involves enormous expense and thesemethods are therefore not available to everyone. Conse-quently, it is important to discuss the available, rela-tively cheap, off-line technology, namely CCD camerasand scanners. In an extensive series of experiments(Kolb and Kothe, 1997; Kothe, 1998), it was possible toshow that scanners and CCD cameras with comparablespecifications deliver results that are also comparable.Calibration of non-linearities in the film and the CCDsensors needs to be performed in both cases, but largeamounts of data are easier to handle with a scanner.This quickly becomes an important criterion becauseeach zone requires an exposure series and several scansin order to evaluate high and low intensities correctlyand to eliminate fluctuations during scanning. Usuallyat least 8 carefully chosen zones are required (tilting tothe right and left about at least two specific crystallo-graphic axes in at least two zones). At the present timeproblems arise when programs are required to handlethe enormous amounts of data that are produced withhigh resolution, high dynamic range cameras or scan-ners. Since these data are essential for quantitativeestimation of intensities, it is expected that theseproblems will be solved in the near future.

In the example shown in Figure 5, the electrondiffraction intensities were quantified using the ELDsystem (Zou et al., 1993). The electron diffractionpatterns were transferred to a PC from a CCD cameraor scanner via a frame grabber and the intensities wereevaluated by the ELD software. At present we are usingEXCEL 7.0 (Microsoft Office 95) to handle the enor-mous quantity of data, specifically to calculate averagevalues of several scans and the exposure series. Theseparate zones were merged into a single set by normal-izing the common axis. A typical series of calibrationcurves is shown in Figure 5.

These quantitative values are first compared withthe kinematical values obtained from the model, whichis derived as described in Simulation of Electron Diffrac-tion Patterns and Packing Energy Calculations toobtain the Rkin factor using the relationship Rkin 5Shkl 00Fo 0 2 0Fc 00 /Shkl 0Fo 0. We use SHELX93 (G. Shedrick:‘‘A program for crystal structure refinement,’’ Univer-sity of Gottingen) to calculate the structure factors,incorporating electron scattering factors instead ofX-Ray scattering factors. If the organic sample is morethan 10 nm thick, dynamical scattering must be takeninto account (Voigt-Martin et al., 1990). The HRTEMmodule in CERIUS gives the Pendellosung plots for allreflections related to the model for different thicknessesand the appropriate intensities for different thick-nesses can be extracted from them. Experimental val-ues for the samples described here are tabulated in thespecialised literature (Voigt-Martin et al., 1995a,b,1997b; Yakimanski et al., 1997).

MOLECULENumber and type of

atomsNMR, IR, UV-VIS,

elemental analysis,FD-MS, thin layer chro-matography

Length and shape ofmolecule

MOPAC

Conformation of mol-ecule in gas

MOPAC

Symmetry of molecule Chemistry, MOPACDirection and strength

of dipoleMOPAC, permittivity,

EFISHHyperpolarisability

tensor componentsTURBOMOL, MOPAC,

EFISH, HRS

CRYSTALLattice type Electron diffractionSpace group Electron diffractionSize of unit cell Electron diffractionDensity Flotation methodNumber of molecules/

unit cellCalc. from known density

Number and type ofatoms in unit cell

Chemical analysis

Diffraction intensitiesfor consecutive zones

Electron diffraction

Angle between consecu-tive zones

Electron diffraction

186 I.G. VOIGT-MARTIN

Fig. 5. Typical calibration curves obtained from exposure series.

Simulation of Electron Diffraction Patterns andPacking Energy Calculations

The prediction of crystal structures from a knowledgeof molecular architecture only, without any experimen-tal diffraction patterns, was initiated by Kitaigorodsky(1961). In recent years, considerable effort has beenexpended in attempts to predict crystal structures froma knowledge of the molecular architecture (Gavezottiand Fillipini, 1996; Scaringe and Perez, 1987; Williams,1996). Attempts to calculate the potential energy hyper-surface of organic crystals have been quite successful.Gavezzotti uses the 6-exp empirical potential (Gavezottiand Fillipini, 1994):

E 5 Aexp(2Br)2Cr26

for organic crystals containing C, H, N, O, Cl, S.Computational simplification is achieved using theatom-atom approximation (Pertsin and Kitaigorodsky,1987). A vast amount of information on packing modesis available through thousands of crystal structurescollected in the Cambridge Database.

Generally several local minima are located withsimilar negative packing energies that differ by only afew kcal/mol, and it is difficult to predict which of thesestructures will occur in practice. This is a consequenceof the conformational polymorphism prevalent in or-ganic crystals, because complex equilibria betweenpolymorphic forms are already established in solutionand interconversions between them are frequent(Gavezotti and Fillipini, 1995). Consequently, crystalli-zation is extremely sensitive to experimental condi-tions.

The major contributions to the packing energy arethe non-bonded and bonded terms indicated below:

E 5 Evdw 1 Ecoul 1 Ehb 1 Etor

1. The non-bonded Van der Waals term (Evdw) is gener-ally treated using the Lennard-Jones 6–12 potentialform, Evdw 5 Ar212-Br26 but the exp(26) form canalso be chosen (Gavezotti and Fillipini, 1994).

Evdw 5 Aexp(2Br)2Cr26,

where A, B and C are empirical parameters and r isthe interatomic distance.

2. The Ewald summation technique is used to calculatethe Coulomb energy:

Ecoul 5 322.0637 ? oi.j

(QiQj/erij),

where the constant effects the conversion to kcal/mol, e is the dielectric constant and Qi,j are atomiccharges. We use the Qi values calculated by the PM3method implemented into the MOPAC 6.0 programpackage. For strongly ionic crystals, a reliable calcu-lation of Ecoul is a difficult problem because of thedivergence of the lattice summation series(Karasawa and Goddard, 1989). The Ewald summa-tion method employed in the CERIUS programpackage can produce large errors in these cases.

Therefore, calculated Ecoul values may only be consid-ered as rather approximate.

3. The energy of the hydrogen bonds (Ehb) is calculatedusing a CHARM-like potential:

Ehb 5 Dhb[5(rhb /rDA)12 2 6(rhb /rDA)10]cos4(qDHA)

qDHA is the bond angle hydrogen donor (D)-hydrogen-hydrogen acceptor (A), while rDA is the distancebetween the donor and acceptor. There is an enor-mous amount of statistical material about hydrogenbond patterns in the Cambridge Database.

4. A Dreiding force-field is used for the calculation ofsubrotation torsional interactions Etor (Mayo et al.,1989). However, it is important to note that weoptimized the molecular valence geometry (bondlengths and bond angles) previously by semi-empirical quantum mechanical calculations. Thechanges in molecular geometry during crystalliza-tion generally involve only the subrotations. Conse-quently, the Dreiding force-field is used only tooptimise a quantum mechanically calculated gas-phase conformation in the crystal.

The parameters involved in these expressions arewell known for the molecules and bonds in question andhave been continuously updated during the past de-cade.

In our method of proceeding, the atomic co-ordinatesof the molecules are obtained from semi-empiricalquantum mechanical MOPAC 6.0 calculations as de-scribed in Generation of a Molecular Model and Deter-mination of Gas Phase Dipole Moment µ and Hyperpo-larisability b. The molecule is initially placed in theunit cell that was estimated previously from the experi-mental diffraction evidence on the basis of the observedd-spacings and symmetries as described previously andcarefully choosing the origin according to the rules setout in the International Tables of Crystallography. Themolecules are then rotated and shifted in the unit cellusing the packing energy calculations incorporated inCERIUS (Cerius 2, San Diego, Molecular Simulations,1996). At the early stage of the calculation the kine-matic approximation is used and this model structureinitiates the calculations.

It is impossible to estimate a space group from onlyone projection of the crystal lattice. A series of diffrac-tion patterns is obtained experimentally at different tiltangles and compared with the corresponding calculatedcrystallographic zones obtained from the CERIUS calcu-lation. When the unit cell in question contains atomsforming rather flexible molecules with considerablefreedom of bond rotation, it is by no means a trivialmatter to obtain good agreement between calculatedand experimental diffraction patterns (Voigt-Martin etal., 1995a,b, 1997a,b; Yakimanski et al., 1997).

The initiating procedure is as follows: From theexperimental diffraction patterns in different projec-

Fig. 6. Orientation of molecules with respect to crystal axes. Thenew values of µ and b with respect to the crystal axes are indicated.Reproduced from Voigt-Martin and Kolb (1997) with permission of thepublisher.

188 I.G. VOIGT-MARTIN

Fig. 6.

tions a unit cell is proposed. If the intensity maximafrom an X-ray powder pattern can be consistentlyindexed, the cell parameters and angles are improvedand possible space groups established keeping in mindthe required density. In the next step, the molecularconformation in the gas phase is calculated by semi-empirical quantum mechanical methods using MOPAC6.0. Frequently the conformations of parts of the mol-ecule are well established in the literature from X-raycrystallography, and the remaining molecule is care-fully built up, atom by atom. There are always severallocal energy minima, so that all these molecular confor-mations must be placed into the unit cell using CERIUSsuch that the density and symmetry conditions of thespace group are satisfied. In our experience, this proce-dure rapidly reduces the number of possible conforma-tions. However, even the most favourable still produces

non-allowed close contacts. These are eliminated bycareful control of the packing energy, while makingsure that the simulated diffraction patterns in all zonesremain sufficiently close to the experimental diffractionpatterns. Finally, by an iterative procedure, the bestconformation and orientation of the molecule in theunit cell are established. Subsequently the packingenergy is calculated and the structure is consideredsatisfactory on condition that this is a negative energyminimum.

The symmetry and packing restrictions imposed bythe experimental electron diffraction patterns and theneed to simulate these in many different zones reducethe number of variable parameters considerably andhave enabled us to solve several unknown structures(Voigt-Martin et al., 1995a,b, 1997a,b; Yakimanski etal., 1997).

Fig. 7. Relationship between molecular orientation, molecular parameters, and macroscopic crystalmorphology of NPHU. Reproduced from Voigt-Martin and Kolb (1997) with permission of the publisher.

190 I.G. VOIGT-MARTIN

Finally the quantitative values of the intensities arecompared with the theoretical values and the R-factorscalculated.

For our samples, it was possible to obtain modelstructures giving good agreement between calculatedand experimental diffraction patterns with R-factors inthe range between 0.28 and 0.30. The precise values indifferent zones vary and also depend on how manyreflections are included. Intensities are merged into onedata set by multiplication with a constant kHKL deter-mined by Sfc

2/Sfo2, or (less satisfactory) by using common

reflections. While these R-values would be unaccept-able in X-ray diffraction, they are quite normal inelectron diffraction. This problem has been discussed atlength by Dorset (1995). By improving the statistics ofquantitative intensity evaluation or by using on-lineslow scan CCD facilities, we expect to reduce theR-factors considerably.

The resulting molecular orientation and packing ofthe molecules discussed here with reference to their

unit cells is shown in Figure 6. The new values of µ andb are also indicated. How these can be calculated isdiscussed in Relating Molecular a, µ, and b to CrystalCo-Ordinates.

The relationship between the molecular orientation,the unit cell, and the macroscopic needle-shaped crystalof NPHU is shown in Figure 7. It is immediatelyobvious that the major components of µ and b lie alongthe needle axis. This is crucial information for thecharacterisation of SHG properties.

High Resolution Imaging and Image RestorationFurther information about the structure is obtained

by high resolution imaging. The symmetry group ob-tained from electron diffraction can be checked using,for example, specialised software such as CRISP (Hov-moller, 1992). Properties such as SHG also depend oncrystal defects, and high resolution images may givecrucial information in this respect (Voigt-Martin et al.,1995a,b, 1997a,c; Yakimanski et al., 1997).

Fig. 8. Crystallographic image processing of CNBA showing original image (lower LHS), itsdiffractogram (RHS), and processed image (upper LHS). Reproduced from Voigt-Martin et al. (1995a) withpermission of the publisher.

191EC AND NLO

Fig. 9.

Organic samples have virtually no contrast and arebeam sensitive. Low-dose imaging techniques must,therefore, be used to avoid beam damage. Contrast isobtained by using phase-contrast methods. In moderninstruments, autofocussing facilities are available. Inolder instruments, two different methods can be used toobtain the correct defocus value. (1) The Fourier trans-form of the image is first viewed by using a CCD cameraattached to the electron microscope and transferring toa suitable computer system. Generally we used theTIETZ VIPS computer system but in one case (TATB)the Gatan slow scan CCD camera was employed. Fromthe Fourier transform of the image, the microscopeparameters can be adjusted until the correct spatialfrequencies are transferred in the electron microscope.The system is also used to optimise for defocus andcorrect for astigmatism. (2) The required contrast trans-fer function is calculated beforehand and adjusted,using carbon particles placed directly on the sample forfocusing. This is sometimes more convenient because ofbeam damage. Subsequently, an adjacent, previouslyunexposed, region is photographed. As an example, thehigh resolution image of CNBA is shown in Figure 8.

The main problems arising in high resolution elec-tron microscopy of beam-sensitive organic samples arepoor contrast and poor signal/noise ratio. The methodselected to deal with this difficulty depends on theproblem that has to be solved. In the case of a regularlyrepeated structure, averaging methods, both frequencyfiltering in reciprocal space (Predere and Thomas,1990; Voigt-Martin et al., 1990) and image averaging inreal space (Henderson et al., 1990), have been success-fully used for many years. Appropriate software such asMRC in Cambridge (Amos et al., 1982; Henderson andUnwin, 1975) and the EMS system in Martinsried(Hawkes, 1980) were developed.

In this work, we used CRISP (Hovmoller, 1992),which is modelled on the MRC programs. The image isdigitised by a CCD camera or scanner and transferredvia a frame grabber to a PC, where it is analysed. Morerecently, we have found it advantageous to use a highresolution, high dynamic range scanner. A fast Fouriertransform routine calculates a transform of the selectedimage area. The intensities are displayed on the screenand amplitudes as well as phases are tabulated. Someof these phases can be used to initiate the ME crystallo-graphic programs. Comparison of the diffractogramwith the original electron diffraction patterns makes itpossible to determine the correct projection symmetry.The diffractogram exhibits, of course, a lower resolutionthan the electron diffraction pattern.

From the molecular modelling procedure (Simulationof Electron Diffraction Patterns and Packing EnergyCalculations), the conformation and orientation of themolecule in the unit cell for this projection, as well as itssymmetry, have already been calculated. Therefore, itis now possible to make an intelligent decision as towhich projection of the crystal offers the most suitableconditions for image processing, and, indeed, which isthe most suitable projection for the subsequent MEphasing procedure.

Finally, origin refinement based on the principlesimplemented in the MRC programs can be performed(Amos et al., 1982; Hovmoller, 1992; Zou et al., 1993).First, a suitable symmetry center is found and thephase residuals determined. Then CRISP automati-cally finds the position in the unit cell with the lowestphase residual. Subsequently, the constraints on ampli-tudes and phases corresponding to the determinedsymmetry are imposed and the image restored takingaccount of the required symmetry conditions.

Since all the necessary data are now available (i.e.,amplitudes and, more importantly, some of the phasesof small angle reflections from the Fourier transform ofthe restored image), the potential maps were calculatedby ME methods (see Ab Initio Structure DeterminationUsing ME Approach) using experimental intensities(see Quantifying Electron Diffraction Data). This iscompared with the projected potential calculated bymolecular modelling (see Simulation of Electron Diffrac-tion Patterns and Packing Energy Calculations) com-bined with multislice calculations using CERIUS. Theimage of the model is calculated for different thick-nesses and defocus values taking account of dynamicalscattering and then compared with the experimentalhigh-resolution images.

Simulation of ImagesSince all the atomic positions have now been deter-

mined by the simulation procedure, the images can becalculated using the HRTEM mode of CERIUS. Theimages are calculated in several steps using the mul-tislice method (Cowley, 1986) and the computer pro-grams developed by Saxton et al. (1983). If the modelcan be substantiated by taking account of secondaryscattering effects (Voigt-Martin et al., 1997b), a furtherrefinement of the observed intensities can be under-taken by accounting for dynamical scattering. Thiseffect can be calculated using the HRTEM mode inCERIUS, which calculates the image as a function ofthickness. We have shown the results of such calcula-tions in the specialised literature (Voigt-Martin et al.,1995a, 1997a,b).

Dynamical ScatteringA serious problem that must be handled and can be

calculated by CERIUS relates to dynamical scattering.We have shown previously (Voigt-Martin et al., 1990)that dynamical scattering already begins to affect thediffracted amplitudes of organic materials at 10 nm.CERIUS uses the well-known multi-slice methods inwhich the propagation and transmission functions arecalculated in reciprocal space (Cowley, 1986; Saxtonand Koch, 1982; Saxton et al., 1983; Self et al., 1983).

We have discussed the effect of dynamical diffractionon the diffraction patterns and images of DMACB,CNBA, and NPHU in the specialised literature (Voigt-Martin et al., 1995a,b, 1997a,b; Yakimanski et al.,1997). As an example we show the projected potential inthe b, c projection of NPHU in Figure 9a.

In Figure 9 b and c, the images and diffractionpatterns of the same projection are shown for a samplethickness of 10.09 and 30.28 nm, respectively, thesevalues being an integer number of unit cells. The screwaxis along a is clearly reflected in both images, butdetails of the molecular conformation are completely

Fig. 9. Projected potential of NPHU (a) together with simulatedimage and diffraction pattern at t 5 10.1 nm (b) and at t 5 30.2 nm (c).

193EC AND NLO

Fig. 10. Centroid map of CNBA compared with model obtained from simulation.

194I.G

.V

OIG

T-MA

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IN

lost in the images. It is important to note that theintensity values in the diffraction patterns are dramati-cally affected as the sample thickness increases, show-ing, apart from the loss of higher order reflections andchanges in absolute values, a complete reversal of therelationship between the intensities of the (014) and(015) reflections.

It is clear that dynamical scattering effects can bevery misleading and must be considered during struc-ture determination. However, the packing energy calcu-lations place strong restrictions on the possible struc-tures. Once these have been found, it is relatively easyto calculate the dynamical effect by using the HRTEMmode in CERIUS and to check whether this could be acause of discrepancies with the experimental results.

Ab Initio Structure Determination Using MEApproach

In general, electron diffraction patterns from smallorganic molecules are incomplete insofar as they do notfully sample reciprocal space and have a resolution ofabout 0.15 nm, indicating some beam damage. Thesuccessful implementation of traditional direct meth-ods usually requires a complete data set to 0.11 nmresolution. This requirement can be relaxed somewhatwhen heavy scatterers are present or when the unit cellcontents are small. Dorset (1995) has overcome thesedifficulties by applying careful control over the phasingprocess. Nonetheless, the incomplete nature of theelectron diffraction data coupled with possible system-atic errors in the diffraction intensities arising from

Fig. 10. (Continued.)

195EC AND NLO

dynamical effects make it difficult to solve such struc-tures by traditional methods.

The ME method suffers less from these limitations,being stable regardless of data resolution and complete-ness; it is also robust with respect to errors in theintensity data. The quantitative theory is based on thesame assumptions as traditional direct methods in sofar as the ensemble of crystal structures from which thesolution is sought may be generated by placing atomsrandomly and independently of each other in the asym-metric unit of the crystal. From limit theorems ofprobability theory, we can estimate the joint probabilitydistribution of suitably chosen sets of structure factors,and substitute their observed amplitudes to obtainconditional joint distributions for the phase. Theseindicate that certain combinations of phases are moreprobable than others, and hence contain phase informa-tion (Voigt-Martin et al., 1995a).

In conventional direct methods, the approximatejoint distributions are usually obtained using the Gram-Chalier (Cramer, 1946) or the Edgeworth (Klug, 1958)series, but this is unsuitable for large structure factoramplitudes. In addition, these methods require that theatom distribution is uniform. These limitations can beovercome simultaneously by the use of the saddle pointmethod (Daniels, 1954).

The underlying theory on which this work is basedhas been described in detail elsewhere (Bricogne, 1984,1988, 1991; Gilmore et al., 1990, 1993) and will betreated in a separate article of this volume.

For this reason, we show only the results of our MEcalculations on CNBA (Voigt-Martin et al., 1995a) andDMBC (Voigt-Martin et al., 1997a) in the centroid mapsof Figures 10 and 11. They are compared with thepotential maps obtained by the simulation procedurepreviously described and are clearly in good agreementwith the results of these earlier calculations.

A prerequisite for all of these methods is a set of goodvalues for the intensities obtained from electron diffrac-tion. Note that the resolution obtained from the cen-troid maps is relatively low, because we rarely havehigh-angle data from beam sensitive samples. For thisreason, the results obtained from the simulations givesubstantially superior results and the ME method onlyserves to substantiate (or disprove) our model.

Relating Molecular a, m, and b to CrystalCo-Ordinates

Once the model has been confirmed, the molecularparameters in relation to the crystal axes can becalculated. Table 1 (Nye, 1967) shows which of thehyperpolarisability components are non-zero for thedifferent space groups.

The values of the various components of a, µ, and bare calculated by semi-empirical quantum mechanicalcalculations for the modified molecular conformation inthe crystalline state obtained from our model. Initially,the values are obtained in molecular co-ordinates i, j,and k and must then be related to crystallographicco-ordinates I, J, and K by applying a suitable rotationmatrix and using crystal symmetry conditions.

Once the crystal symmetry and the co-ordinates of allthe atoms in the unit cell have been calculated and, inaddition, when the structure has been substantiated by

high resolution images and ME calculations, we are in aposition to calculate the non-linear optical parametersof the crystal. In view of the complex nature of this

Fig. 11. Centroid map of DMBC showing model from simulationsprojected onto it.

196 I.G. VOIGT-MARTIN

problem, we recommend two books, one written by Nye(1967) and the other edited by Chemla and Zyss (1987).In a number of fundamental publications, Zyss hasderived the relationship between the molecular orienta-tion, crystal symmetry, and macroscopically measuredoptical susceptibility values dIJK (Zyss and Oudar,1982). In Table 2 we show an example of the expres-sions that he derived for the measured optical suscepti-bilities bIJK in different directions depending on thecalculated hyperpolarisability coefficients bijk in thedifferent symmetry groups and the angle a between themolecular axis and the crystal axis.

As an example of such a calculation, we show ourderivation of the macroscopically measured value dIJK

for NPHU in Table 3 (Yakimanski et al., 1997).It should be noted that the quantum mechanically

calculated values of aij in Table 3 are the polarisabilitytensor components and not the angle between themolecular axis and crystal axis as in the previousexample. The calculation indicates that we can expectvery high values of SHG in these crystals. The qualita-tive measurements undertaken with the SHG micro-scope confirm this expectation (Yakimanski et al., 1997).

TABLE 1. Non-zero hyperpolarisability components in different space groups1

1From Nye (1957).

197EC AND NLO

A VERY STRANGE CASE: WHY NON-DIPOLARMOLECULES CAN BE NLO ACTIVE

The molecular engineering concept described aboverelies on the combination of an electron-donating groupand an electron-accepting group, linked together by asuitable conjugated p-system. In such systems, whichcan be described by a two state model, the second orderpolarisability b within the dipolar approximation isproportional to Dµ, the difference between the dipolemoments in the ground and the first allowed singletcharge transfer excited state (Oudar, 1977). However,extremely strong SHG was observed in 1,3,5-triamino-

2,4,6-trinitrobenzene in solution, in spite of the factthat symmetry of the individual molecules leads tocomplete mutual cancellation of all dipole moments, inboth the excited and ground states (see schematicdiagram below).

In a number of fundamental publications, Zyss hasshown that the origin of the NLO effect in such mol-ecules in solution is then an octopolar contribution to band x, while the vectorial components cancel (Zyss andLedoux, 1994). However, the crystals also demon-strated very strong SHG, although they were shown tobe centrosymmetric by X-ray analysis (Cady and Lar-son, 1993), so that also the octopolar componentsshould cancel. The structure analysis, in which thespace group was proposed to be P1, Z 5 2, and the cellconstants a 5 9.01Å, b 5 9.03 Å, c 5 6.81 Å, a 5 108.6°,b 5 91.82°, g 5 119.97° was never repeated because it isextremely difficult to grow single crystals of TATB.Several explanations for the SHG were attempted, suchas the presence of small non-centrosymmetric P1 do-mains or torsional librations of the nitro groups aroundthe C-N bond by about 12° (Fillipini and Gavezotti,1994) or other structural defects.

We believe that we have found the explanation forthis phenomenon by electron crystallography (Voigt-Martin et al., 1996, 1997d): High resolution imagesrevealed that there are no structural defects in thecrystals. Electron diffraction analysis using the simula-tion methods described here indicated two crystal struc-tures, one of which was trigonal, space group P31 withthe following cell constants: a 5 b 5 9.0 Å, c 5 40.9 Å,

TABLE 2. Expression of the Macroscopic Nonlinear Susceptibilities bIJK in Terms of the Four Two-Dimensional Molecular HyperpolarizabilityCoefficients bijk for Lower Crystal Symmetry1

1 2 m mm2 222

bXXX bxxx 0 bxxx 0 0bYYY byyy byyy cos3 a 0 0 0bZZZ 0 0 2byyy sin3 a byyy cos3 a 0bXYY bxyy 0 byyx cos2 a 0 0bYXX byxx 2 0 0 0bYZZ 0 byyy cos a sin2 a 0 0 0

bZYY 0 0 2byyy cos2 a sin a

2bxyy(sin 2F sin 2a)/21 byxx sin 2 F cos a

1 byyy cos2 F cos a sin2 a 0bXZZ 0 0 bxyy sin2 a 0 0

bZXX 0 0 2byxx sin a

2bxyy(sin 2F sin 2a)/21 byxx cos 2 F cos a

1 byyy sin2 F cos a sin2 a 0

bXYZ 0 2bxyy cos a sin a 0 0sin 2F cos a(byxx 2 byyy sin2 a)

2 bxyy cos 2F sin 2a

1From Zyss and Oudar (1982) with permission of the publisher.

TABLE 3. Calculation of Non-Linear Crystal Tensor Componentsfor NPHU

Space group P21; V 5 688.57 Å3, N 5 2, q 5 44.7°1. Polarisability tensor a calculated by MOPAC

axx 5 15.7

axy 5 7.9 ayy 5 21.5

axz 5 1.0 ayz 5 28.8 azz 5 18.2

Summation over 2 molecules related by P21 leads to cancellationof axy and ayz.

2. Refractive index and local field factor

(nI2 2 1)/(nI

2 1 2) 54

3p 1NV2 aII

f1 5nI

2 1 2

35 1 ⁄ 31 2 1432 p 1NV2 aII4

fx 5 1.2 fy 5 1.4 fz 5 1.3

3. Hyperpolarisability (non-linear optical) components bcalculated by MOPAC

In P21 there are only 2 non-linear crystalline tensorcomponents b:

bYYY 5 byyy cos3 q 5 3.5 3 10230 esu

bYZZ 5 byyy cos q sin2 q 5 2.7 3 10230 esu

4. Relationship to measured macroscopic values d:

dYZZ 5 1NV2 fy(fz)2bYZZ 5 18.5 31029 esu 5 7.7 pm/V

dYYY not relevant for 3. wave mixing

Schematic diagram of TATB

198 I.G. VOIGT-MARTIN

g 5 120°. The long c-axis is remarkable because thedistance between the planar TATB molecules is only 3.4Å, clearly indicating a superstructure. Detailed quan-tum mechanical calculations combined with the simula-tion methods described in this contribution revealed

that the crystals belong to a P31 space group in which12 TATB molecules are arranged in 3 groups consistingof 4 molecules each and rotated with respect to oneanother as indicated in Figure 12.

Within each independent group, the molecules areshifted with respect to one another, so that the 3-foldsymmetry of the molecule is destroyed in the indepen-dent group. Detailed quantum mechanical calculationsusing the atomic co-ordinates obtained from the simula-tions of both electron diffraction patterns and highresolution images then revealed the following results:According to the PM3 data, the charge transfer in theindependent unit of four TATB molecules is mostlyrealized in its zy-plane, byyy and byzz being the majorcomponents of the b-tensor of the independent unit.Therefore, for this independent unit the two-dimen-sional model (Zyss and Oudar, 1982) relating the crys-talline nonlinear tensor coefficients, bIJK, to the molecu-lar b-tensor coefficients, bijk, is valid. According to theaxes convention used by Zyss and Oudar (1982), theangle, a, between this zy-plane and the 3-fold screwZ-axis of the crystal is 62.78°.

The results of the PM3 calculations of the molecularb-tensor components necessary for the calculationswithin the two-dimensional model for space groups ofclass 3 are:

bzzz 5 22.0 ? 10230 esu

bzyy 5 4.4 ? 10230 esu

byyy 5 214.6 ? 10230 esu

byzz 5 12.2 ? 10230 esu

This leads to the following results:

bZZZ 5 byyycos3a 5 21.4 ? 10230esu

bYYY 5 (1/4) (byyysin2a 2 3byzz)

sina 5 210.7 ? 10230 esu

bXXX 5 (1/4) (bzzz 2 3bzyysin2a) 5 23.1 ? 10230 esu

bZXX 5 (1/2) (byzz 1 byyysin2a)cosa 5 0.2 ? 10230 esu

(In these equations, the bzzz, bzyy, byzz coefficients wereused instead of bxxx, bxyy, byxx, as defined by Zyss andOudar, 1982, because the molecular plane of chargetransfer in the independent unit is the zy-plane ratherthan the xy-plane).

Thus, the major coefficient of the crystalline nonlin-ear tensor is bYYY, which is observable in SHG-experiments, according to phase-matching conditionswith respect to the propagation direction. The calcu-lated dYYY coefficient is:

dYYY 5 (N/V)(fY)3bYYY 5 51.40 ? 1029 esu 5 21.4 pm/V

This value is considerably higher than the value ofdXYZ 5 2.3 pm/V obtained experimentally for ureasingle crystals (Betzler et al., 1978).

Fig. 12. Model showing structural arrangement of TATB moleculesin the unit cell. Reproduced from Voigt-Martin et al. (1998b) withpermission of the publisher.

199EC AND NLO

In addition, we have shown that there is an octopolarcontribution, which can be estimated from the ratio, r,of the magnitude of the octopolar to the dipolar contribu-tion (Zyss and Ledoux, 1994).

CONCLUSIONSWe have shown that detailed information about the

relationship between molecular parameters and macro-scopic physical properties can be obtained by electroncrystallography using a combination of quantum me-chanics and packing energy calculations to simulateexperimental electron diffraction patterns and highresolution images. For the packing energy calculations,it is essential to use information about molecularsymmetry as well as chemical knowledge about hydro-gen bond patterns and Coulomb interactions. Usuallyspectroscopic data also give important informationabout the molecules. The structural data were substan-tiated by ab initio ME methods and the physicalpredictions were substantiated by experimental mea-surements in solution and in the crystal. Such calcula-tions require that good quantitative data are availablefrom the diffraction patterns. Here considerable im-provement can be expected in the near future. Atpresent, we are performing experiments that show thatconsiderably better structural R-values can be obtainedby using electron microscopes with higher voltages andslow scan CCD facilities. A major consideration hasbeen to show that we can extract far more informationfrom a structural investigation than only atomic posi-tions, namely the molecular and crystallographic polar-isability, dipole moment, hyperpolarisability, and opti-cal susceptibility.

ACKNOWLEDGMENTSThis paper reviews work to which many of my past

research students and present co-workers have contrib-uted. I would like to name and thank them specifically:Drs. A. Yakimanski, U. Kolb, D.H. Yan, Z.X. Zhang, GaoLi, P. Simon, and Dipl. Geol. H. Kothe. At all timesthere was close collaboration with the Organic Chemis-try Department (Prof. R. Ringsdorf) and the Spectros-copy Group (Dr. R. Wortmann) in Mainz. The samplesfor the TATB problem were obtained from Dr. J.J. Wolff,Organic Chemistry Department in Heidelberg. Gener-ous support by the Deutsche Forschungsgemeinschaftis gratefully acknowledged.

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