**2. Brief description of the G-HAUP method**

A schematic representation of the conventional HAUP system is shown in **Figure 1**, which also applies to the G-HAUP. The polarization optics is in both cases composed of two optical elements, a linear polarizer (*P*) and a linear analyzer (*A*).

The axes of *P* and *A* are set in the crossed Nicols position. The monochromatic light beam emerging from a monochromator or a laser travels through *P*, the analyte on a sample stage (*S*), and *A* successively. Then, the normalized intensity *I/I*<sup>0</sup> of the light transmitted through *P*, *S*, and *A* is detected with a photomultiplier. *θ* represents the azimuthal angle of *P* with respect to a principal axis of the sample, as shown in **Figure 1a**. The positions of *P* and *A* are varied systematically from the crossed Nicols position. In addition, *Υ* is defined as the deflection angle of *A* from perfect extinction. The transmittance can be expressed using the Jones calculus as a function of a matrix for the polarizer, a generalized Jones matrix containing the linear optical effects propagated through a distance, and a Jones matrix for the analyzer [16]. This expression can be reformulated as a quadratic function of the two variables, *θ* and *Υ* angles.

In the HAUP measurement, because the angle between polarization directions of *P* and *A* is set close to 90°, *I* is weak, such that changes are pronounced and can be well determined by an optical chopper and a lock-in amplifier with a photon-counting system.

The HAUP method requires the accurate evaluation and elimination of systematic errors, notably the parasitic ellipticities *p* and *q* originating from *P* and *A*, respectively, [3, 4] and a small error angle δ*Υ* [5]. Here, δ*Υ* is caused by the slight deviation from the perfect crossed Nicols condition by inserting the sample between *P* and *A*.

In summary, the relative intensity ratio Γ of the transmitted light *I* to the incident light *I*<sup>0</sup> is obtained by multiplying the Jones vectors of **P** and **A** and the Jones matrix **M**<sup>H</sup> and several approximations and coordinate transformations, as follows:

Γ *θ*<sup>0</sup> , *<sup>Υ</sup>*<sup>0</sup> <sup>ð</sup> , LB, LD, CB, CD, *<sup>p</sup>*, *<sup>q</sup>*, <sup>δ</sup>*Υ*Þ ¼ *<sup>I</sup>=I*<sup>0</sup> <sup>¼</sup> <sup>j</sup>**A**<sup>T</sup>**M**H**P**<sup>j</sup> <sup>2</sup> <sup>¼</sup> *<sup>A</sup>*<sup>00</sup> <sup>þ</sup> *<sup>B</sup>*00*Υ*<sup>0</sup> <sup>þ</sup> *<sup>C</sup>*00*Υ*0<sup>2</sup> (1)

where

$$A'' = H\_{11}'' + H\_{12}''\theta' + H\_{13}''\theta'^2.$$

$$B'' = H\_{21}'' + H\_{22}''\theta'.$$

$$C'' = H\_{31}''.$$

Each coefficient of the quadratic function of Γ contains the information of *θ'*, *Υ'*, LB, LD, CB, CD, and systematic errors *p*, *q*, and δ*Υ.* Therefore, LB, LD, CB, and CD can be recovered by measuring *I* at various *θ'* and *Υ'* and eliminating the systematic errors.

Here, we would like to point out the requirements of the samples for G-HAUP measurement. As for other optical measurements, transparent, homogeneous, surface-flat, and defect-free samples are preferred for G-HAUP measurements. In addition, although the details are described elsewhere [9], to avoid anomalous behavior near the unstable and low-sensitivity wavelength regions, samples with a small change in the total phase difference with respect to wavelength are also preferred. To fulfill such requirements, we prepared very thin samples for G-HAUP measurements.

The HAUP has been applied to investigate OA of various crystals such as amino acids [17, 18], proteins [19], chiral co-crystals [20], triglycine sulfate (TGS) [21, 22], as well as KH2PO4 (KDP) [23] and NH4H2PO4 [24] and their isomorphs.

Furthermore, to measure CD and LD in absorbing crystals of low symmetry, the HAUP method has been extended by various researchers [7, 25, 26]. We have

*Chiroptical Studies on Anisotropic Condensed Matter: Principle and Recent Applications… DOI: http://dx.doi.org/10.5772/intechopen.108721*

developed the extended HAUP method for measuring temperature dependencies of LB, LD, CB, and CD simultaneously in a tris(ethylenediamine) cobalt(III) triiodide monohydrate crystal [27]. The recent applications of G-HAUP are wavelength dependences of chioptical measurements for laminated collagen membranes with highly preferred orientation [28] and crystals of azobenzene-intercalated K4Nb6O17 [8], *γ*glycine [29], salicylidenephenylethylamines [13], *L*-alanine [14], benzil [15], and CeF3 [9]. In the following sections, analyses of the latter four systems by the G-HAUP are summarized.

#### **3. Chiral photomechanical crystals**

The chiral single crystals of *S*- and *R*-enantiomers of *N*-3,5-di-*tert*-butylsalicylidene-1-phenylethylamine in the enol form [enol-(*S*)-**1** and enol-(*R*)-**1**] (**Figure 2**) show photomechanical motion under UV light irradiation [30], that is, light-driven macroscale mechanical motion. Photochromic crystals are often good candidates for photomechanical motion. Upon photoirradiation, the making or breaking of chemical bonds can lead to color changes as well as stresses manifest as macroscale mechanical motion. Photomechanical motion involves the direct conversion of light energy to mechanical energy, and thus, photomechanical crystals may be beneficial for energy conversion. Previous reports show that the dissymmetry of chiral crystals can be manifested in the photomechanical behavior of chiral crystals as opposed to racemic crystals [31, 32].

Enantiomeric salicylidenephenylethylamines enol-(*S*)-**1** and enol-(*R*)-**1** (**Figure 2**) before and under UV light irradiation were analyzed by the G-HAUP method in order

**Figure 2.**

*Photoinduced hydrogen transfer reaction of salicylidenephenylethylamines enol-(*S*)-1 (a) and enol-(*R*)-1 (b). Reproduced from ref. [13] with permission from the American Chemical Society.*

to obtain LB, LD, CB, and CD spectra and subsequently to correlate the changes of the optical properties to the changes of the crystal structures during the photoreaction [13].

Compounds enol-(*S*)-**1** and enol-(*R*)-**1** were synthesized according to a published protocol, and the single crystals of enol-(*S*)-**1** and enol-(*R*)-**1** were grown by sublimation at 10–20°C below the melting points (92–93°C).

At the outset, the crystal structure of enol-(*S*)-**1** had already been determined, but not that of *trans*-keto-(*S*)-**1** [30]. Hence, *in situ* crystallographic analyses were carefully performed under continuous UV irradiation to analyze the crystal structure of *trans* photoproduct. Because no disorder was found in the crystals, we calculated the crystal structure of *trans*-keto-(*S*)-**1** by dispersion-corrected density functional theory (DFT) calculations (**Figure 3b**). As shown in **Table 1**, the *a* and *b* axes contracted, and the *c* axis extended.

**Figure 4** shows the LB, LD, CB, and CD spectra of the enol-(*S*)-**1** (thickness: 6.5 μm) and enol-(*R*)-**1** (thickness: 7.6 μm) crystals through the (001) face before UV irradiation. The LB and LD spectra between the *S* and *R* enantiomeric crystals are coincident (**Figure 4a** and **b**), because linear anisotropies are not affected by enantiomorphism. The negative LD peak at 330 nm corresponds to the π-π\* transition of the intramolecularly hydrogen-bonded salicylidenimino moiety. The LB spectra exhibited anomalous dispersion of negative peaks at 360 nm with a change in sign at the strong LD peak. These results show that the LB and LD spectra satisfy the

*Calculated crystal structures: Ball-and-stick drawings of (a) enol-(*S*)-1 (yellow) and (b)* trans*-keto-(*S*)-1 (orange). Tautomer structures overlaid on the (c) (100) and (d) (00*1*) faces. The red arrows in (d) show the direction of contraction of the* trans*-keto-(*S*)-1 crystal along* a *and* b *axes. Reproduced from ref. [13] with permission from the American Chemical Society.*

*Chiroptical Studies on Anisotropic Condensed Matter: Principle and Recent Applications… DOI: http://dx.doi.org/10.5772/intechopen.108721*


#### **Table 1.**

*Unit cell dimensions of enol-(*S*)-1 by X-ray diffraction and* trans*-keto-(*S*)-1 crystals obtained from DFT calculations. Reproduced from ref. [13] with permission from the American Chemical Society.*

#### **Figure 4.**

*Optical anisotropic and chiroptical spectra of enol-(*S*)-1 and enol-(*R*)-1 crystals on the (001) face: (a) LB, (b) LD, (c) CB, and (d) CD. These properties were measured with the G-HAUP in the dark and under continuous UV light irradiation at 365 nm. The curved lines are fitted gaussian functions intended to guide the eye (a, c). Reproduced from ref. [13] with permission from the American Chemical Society.*

Kramers–Kronig relathionship [33]. The CD spectra of enol-(*S*)-**1** and enol-(*R*)-**1** crystals revealed, respectively, a strong negative and positive Cotton effect at 330 nm, which mirror each other (**Figure 4d**). The CB spectra of enol-(*S*)-**1** and enol-(*R*)-**1** crystals also exhibited anomalous dispersion of negative and positive peaks at 360 nm with changes in sign at the CD peaks (**Figure 4c**), respectively, indicating that the Kramers–Kronig relationship also holds between CB and CD.

We then attempted to measure the LB, LD, CB, and CD spectra of both enantiomeric enol-**1** crystals under UV irradiation with a 365 nm LED directed normal to the G-HAUP light path at low power (5 mW cm<sup>2</sup> ) to minimize incident UV light reaching the detector. UV irradiation-induced bending was inhibited by fixing the crystals to a

plate with silicone grease. Note that salicylidenephenylethylamines, which were used in this study, do not show thermochromism. **Figure 4** also shows the LB, LD, CB, and CD spectra under continuous UV irradiation at 365 nm, which represents the spectra at the photostationary state of the reactant and product. New LD peaks corresponding to *trans*-keto-(*S*)-**1** and *trans*-keto-(*R*)-**1** appeared at around 460 nm, and the magnitudes of the LD peaks at 330 nm decreased slightly. The LD spectra of the *S* and *R trans*-keto isomers were coincident (**Figure 4b**). New, small negative and positive CD peaks appeared at 460 nm due to the formation of *trans*-keto-(*S*)-**1** and *trans*-keto-(*R*)-**1** crystals, respectively, and the magnitudes of CD peaks at 330 nm decreased slightly (**Figure 4d**). The CB spectra also exhibited anomalous dispersions of negative [enol-(*S*)-**1**] and positive [enol-(*R*)-**1**] peaks at 500 and 360 nm with a change in sign at the new CD peak (**Figure 4c**), as with the LB and LD spectra. The Kramers–Kronig transformation related the CB to the CD and the LB to the LD.

The LB of both enantiomeric enol-**1** crystals along the *c-*axis showed a relatively lower value (0.02) above 600 nm than the previously reported organic crystals [8, 17–20, 28, 34, 35]. The total intermolecular interaction in the enol-(*S*)-**1** crystal along all directions is the van der Waals force alone. This very weak molecular interaction may have induced such a small LB. In fact, we have reported a much larger value (0.4) for the LB of the chiral cocrystal composed of tryptamine and 4 chlorobenzoic acid [20]. This cocrystal exhibits strong intermolecular interactions, such as ionic bridging and the hydrogen bonding. The optical rotatory power (ORP) value of enol-(*S*)-**1** crystals along the *c-*axis at 632.8 nm before UV irradiation was calculated to be 5.2 deg./mm. The signs of ORP dispersion along the *c-*axis are opposite to those in the hexane solution, the orientationally averaged value [13]. This suggests that the contribution from the ORP dispersion along the *c-*axis might be small, or the ORP dispersions along the *a-* and/or *b-*axes might be largely positive in sign. The dissymmetry parameter, *g*, in the crystalline state is defined as the ratio in absolute magnitude of CD to absorbance. The *g* value of enol-(*S*)-**1** crystal along the *c-*axis was calculated to be 0.013. On the other hand, the *g* value of enol-(*S*)-**1** in hexane solution was calculated to be 0.0010, revealing that the *g* value of the enol-(*S*)-**1** crystal obtained by the G-HAUP measurement without UV irradiation is around 10 times larger than *g* values in the solution and by the calculation in Ref [13].

## **4. Chiral alanine crystal**

*L*-alanine is the smallest chiral natural amino acid. As an additive in ferroelectric triglycine sulfate crystals, it can control the crystal polarity, which is of practical use in infrared detectors. *L*-alanine alone grows as large, hard, transparent crystals [36] from evaporating solutions in the space group *P*212121 [37, 38]. Thus, alanine is an optically biaxial chiral crystal with four zwitterionic molecules (<sup>+</sup> H3NCH(CH3)COO) in the unit cell. Misoguti *et al.* [39] measured the dispersion of refractive indices, among the many other physicochemical properties that have been investigated [40–43], but the anisotropy of the optical activity has not been established.

We measured the wavelength dependence of the CB of alanine crystals along each crystallographic axis, by G-HAUP, and assigned the absolute structure of the crystals examined by the method of anomalous dispersion to determine the absolute chirality of alanine crystals, by correlating the absolute structure obtained with the X-ray diffraction method with the CB measured using the G-HAUP.

*Chiroptical Studies on Anisotropic Condensed Matter: Principle and Recent Applications… DOI: http://dx.doi.org/10.5772/intechopen.108721*

*L*- and *D*-alanine crystals were grown by solvent evaporation using the enantiomeric *L*- and *D*-alanine powder as solute and DIW as solvent. Samples for crystal structure determination and the measurement of CB (and optical activity, OA = CB/2) were prepared from these crystals.

The results from the X-ray crystal structure analyses are shown in **Table 2**. The unit cell parameters are almost the same as those from the previous study. Furthermore, we succeeded in determining the absolute structure of the *L*- and *D*-alanine crystals.

For the measurements of chiroptical properties, crystals were cut perpendicular to the <010> direction and (010) faces of *L*- and *D*-alanine crystals were polished to 21 μm and 13 μm, respectively, by using lapping films with SiC (5.0 μm), Al2O3 (1.0 μm), and FeO (0.3 μm) abrasive, successively. The polished crystals were fixed on a pinhole and chiroptical properties were measured. The values of LB, LD, and OA in the <010> direction were obtained (**Figure 5**). LB clearly decreases and approaches zero with decreasing wavelength (**Figure 5a**), while LD is almost zero from 280 to 680 nm (**Figure 5b**). LB in the <010> direction is significantly smaller than that in the <100> direction, which allows G-HAUP to detect OA more sensitively. The magnitude of OA is almost the same in *L*- and *D*-alanine crystals, but their signs are opposite as shown in **Figure 5c**. Consequently, *L*- and *D*-alanine crystals exhibit positive and negative OA in the <010> direction, respectively, which means *L*- and *D*-alanine crystals are *levorotatory* and *dextrorotatory*, respectively, in this direction, over the spectrum evaluated.

The interpretation of the results of HAUP studies remains a challenge. In a study of the CB of crystals of *L*-glutamic acid, we introduced the *chirality index, r* =1-|*ρ*s|/|*ρ*c|, where |*ρ*s| is the absolute optical rotation per molecule in solution and |*ρ*c| is the optical rotation per molecule in the crystal averaged over the eigenvalues of the gyration tensor. Values close to 1 are dominated by the effects of crystallization. For *L*-alanine, *r* = 0.999, indicating that the OA is principally a crystal-optical effect.

No general, quantum chemical methods [44] have yet been implemented in widely distributed electronic structure computing programs for interpreting the chiroptical effects of molecular crystals. However, progress is on the horizon. Linear response theories with periodic boundary conditions are required because the OA of molecules is strongly affected by the environment, as confirmed by experimental and computational


#### **Table 2.**

*Crystal data for* L- *and* D*-alanine crystals.*

#### **Figure 5.**

*Wavelength dependence of LB (a), LD (b), and OA (c) in the <010> direction of* L*- and* D*-alanine crystals. Closed and opened circles are results for* L*- and* D*-alanine, respectively. Reproduced from ref. [14] with permission from Elsevier.*

studies on the solvent dependence of OR [45], in addition to many studies of computational investigations of crystallographic supercells that we performed over the years. To minimize the effects of interfacial molecules, larger and larger aggregates of molecules must be computed, a process that becomes intractable, as illustrated in the following section for benzil. Balduf and Caricato made this convergence problem explicit in silico for F2 and HF molecules arranged as model helices [46]. Unit cells were inadequate representations of large helices. More recently, Rérat and Kirtman have introduced computed results of the chiroptical properties of periodic systems using the selfconsistent coupled-perturbed method in the program suite CRYSTALS [47].

The relationship between the absolute structure and OA along the *b*-axis was considered parallel to the twofold screw axes considered for *L*-alanine. The hydrogen bond chains for *L*- and *D*-alanine were, respectively, right and left handed (**Figure 6**). However, there is no simple prescription for correlating configuration with the sense of optical rotation. This is true for crystals as for molecules.

From a classical perspective, a right-handed helix of atoms/molecules might be dextrorotatory or levorotatory depending on the polarizability of the groups decorating the helix. From a quantum mechanical perspective, identifying the relevant chromophores is requisite. Furthermore, individual bands may contribute to the CB positively or negatively.

#### **5. Chiral benzil crystal**

Benzil (C6H5C(O)-C(O)C6H5) crystals have been considered the organic analogue of quartz; both substances have *D*<sup>3</sup> point symmetry and can be obtained *Chiroptical Studies on Anisotropic Condensed Matter: Principle and Recent Applications… DOI: http://dx.doi.org/10.5772/intechopen.108721*

**Figure 6.**

*The handedness of the twofold screw axis of alanine crystals along the* b*-axis. (a)* L*-alanine and (b)* D*-alanine. Reproduced from ref. [14] with permission from Elsevier.*

as large single crystals. The OA of benzil has been studied more than that of any other organic crystal; however, unlike quartz, its OA anisotropy has resisted characterization. Without measurements of the optical activity along the diad axes, as opposed to the easily measured optic axis where LB = 0, interpretations are incomplete. Here, we compare OA measurements along the low-symmetry direction of crystalline benzil by the G-HAUP accompanied by electronic structure calculations of the benzil molecule and aggregates of benzil molecules based on the crystal structure.

Single crystals of benzil were grown by slow evaporation from acetone at 25°C. Plates (5 mm � 5 mm � 1 mm) were cut with a razor blade, exposing large (001) or (100) faces. Samples were then polished sequentially with SiC (grain diameter 9 and 5 μm), Al2O3 (3 and 1 μm), and Fe2O3 (0.3 μm) lapping films. Single-crystal X-ray diffraction analysis confirmed the enantiomorphous space groups *P*31(2)21 (**Figure 7**). While the structure of benzil has been established previously [48], the absolute structure was determined first in the aforementioned citation.

A *c*-cut slab of *P*3121 benzil measuring 26.2 μm was analyzed. The OA along the optic axis was easily measured by rotating the analyzer to the extinction position in the G-HAUP. For the *P*3121 enantiomorph, benzil is *dextrorotatory* at optical frequencies (**Figure 8c**). The OA along the *c*-axis is 24.3°/mm at 590 nm, in good agreement with sodium D-line (25°/mm) measurements [49, 50].

An 88 μm thick *a*-slab of *P*3121 benzil was likewise polished. The LB, LD, OA, and CD were successfully extracted, and the dispersion in **Figure 8** was fit to a simple Drude oscillator:

$$
\rho = \frac{A}{\lambda^2 - \lambda\_i^2},
\tag{2}
$$

*Unit cell of* P*3121 crystalline benzil viewed along [001] with the 31 axes in green and orange. Reproduced from ref. [15] with permission from the American Chemical Society.*

#### **Figure 8.**

*Wavelength dependences of LB (a), LD (b), fitted OA (c), and CD (d) of the* P*3121 benzil crystal. (a) Published values at fixed wavelengths are red [51] and green [52] points. (c) The OA was fitted to an oscillator model (see text). For the* P*3121 enantiomorph, the OA along the* c- *and* a*-axes at 590 nm is 24.3 deg./mm and 23.8 deg./mm, respectively. Reproduced from ref. [15] with permission from the American Chemical Society.*

where *<sup>A</sup>* and *<sup>λ</sup><sup>I</sup>* are constants with fitted values of 7.9462 <sup>10</sup><sup>6</sup> and 117.18 nm, respectively. The LB perpendicular to the optic axis and OA along the optic axis agreed well with literature values [49–52].

The long wavelength (589 nm) OA tensors of one benzil molecule, three benzil molecules in the unit cell, and three benzil molecules related by a threefold rotation *Chiroptical Studies on Anisotropic Condensed Matter: Principle and Recent Applications… DOI: http://dx.doi.org/10.5772/intechopen.108721*

**Figure 9.**

*Representation surfaces of the computed long wavelength OA of benzil at 589 nm (plotted with the software WinTensor, W. Kaminsky). (a) One molecule in the gas phase, (b) one unit cell treated of three molecules, (c) one unit cell based on results for one molecule in (a) and symmetrized, and (d) experimental result based on G-HAUP data. Reproduced from ref. [15] with permission from the American Chemical Society.*

(as opposed to a threefold screw) were calculated using well-known methods [53, 54]. The results are summarized in **Figure 9**. The calculations of a small number of molecules are a poor mimic of the crystallographic response.

Benzil and 4-methylbenzophenone [55] are the only such examples of molecular crystals dominated by weak intermolecular interactions for which the longwavelength OA anisotropy has been determined. Unfortunately, because benzil is in dynamic equilibrium in solution, it is not amenable to a calculation of the chirality index, *r* (see above).

Interpreting this tensor in terms of a small number of excited states is difficult because unlike simple hydrocarbons investigated previously [56, 57], a great number of states contribute to the long-wavelength value of benzil. Requisite for the computation of chiroptical properties in crystals is the development of linear response theory with periodic boundary conditions to provide a framework for interpreting the results of single-crystal polarimetry, as discussed above.
