Irreducible Cartesian tensor decomposition: A computational approach

Cartesian tensors are widely used in physics and chemistry, especially for the formulation of linear and nonlinear spectroscopies as well as for molecular response properties. In this work, we review the problem of irreducible Cartesian tensor (ICT) decomposition of a generic Cartesian tensor of rank n into its irreducible parts, each characterized by a specific symmetry. The matrix formulation of the ICT decomposition is structurally similar to the problem of rotational averaging using isotropic Cartesian tensors. Analogously to the latter, the ICT decomposition can be considered as a problem of selecting a set of permutations of n indices that provides a linearly independent set of mappings between Cartesian tensor subspaces. This selection can be performed using a simple computational approach based on the reduced row echelon form (rref) algorithm. This protocol has been implemented in a computer code used to re-derive the already known ICT decomposition for 2 ≤ n ≤ 4. Finally, for the first time, we performed the explicit ICT decomposition of a Cartesian tensor of rank n = 5.

Strong fluorescence-detected two-photon circular dichroism of chiral gold nanoclusters

Progress in syntheses and understanding of the intriguing properties of chiral noble metal nanoclusters sparks interest to extend investigations of their chiroptical response to the nonlinear optics regime. We present a quantitative determination of two-photon circular dichroism of chiral gold nanoclusters with ATT and L- or D-Arg ligands (ATT = 6-aza-2-thiotymine and Arg = arginine). Introduction of arginine ligands enables the formation of two enantiomers of the nanoclusters, with strong chiroptical effects in both linear and nonlinear regime. We present two-photon absorption and luminescent properties measured in a wide range of wavelengths, with the two-photon absorption cross section reaching 1743 GM and two-photon brightness ∼1102 GM at 825 nm. We report strong, 245-fold enhancement of the two-photon circular dichroism of nanoclusters with respect to the one-photon absorption counterpart - the dissymmetry factor. The presence of multiple advantages of nanoclusters: high fluorescence quantum yield, strong nonlinear optical properties and well-controlled chirality is a powerful combination for applications of such clusters in multiphoton microscopy.

New opportunities for ultrafast and highly enantio-sensitive imaging of chiral nuclear dynamics enabled by synthetic chiral light

Synthetic chiral light [D. Ayuso et al., Nat. Photon., 2019, 13, 866-871] has opened up new opportunities for ultrafast and highly efficient imaging and control of chiral matter. Here we show that the giant enantio-sensitivity enabled by such light could be exploited to probe chiral nuclear rearrangements during chemical reactions in a highly enantio-sensitive manner. Using a state-of-the-art implementation of real-time time-dependent density functional theory, we explore how the nonlinear response of the prototypical chiral molecule H₂O₂ changes as a function of its dihedral angle, which defines its handedness. The macroscopic intensity emitted from randomly oriented molecules at even harmonic frequencies (of the fundamental) depends strongly on this nuclear coordinate. Because of the ultrafast nature of such nonlinear interactions, the direct mapping between the dissymmetry factor and the nuclear geometry provides a way to probe chiral nuclear dynamics at their natural time scales. Our work paves the way for ultrafast and highly efficient imaging of enantio-sensitive dynamics in more complex chiral systems, including biologically relevant molecules.

Chirality in fluorescence and energy transfer

Optical polarization features associated with the fundamental processes of molecular fluorescence and resonance energy transfer are in general studied with reference to plane polarizations. When any of the species involved is chiral, the associated emission processes may exhibit an element of circular polarization-a degree of optical helicity. Although usually a minor effect, some systems can exhibit a sizeable component of circularly polarized luminescence, whose helicity correlates with the enantiomeric form. In studies of multi-component systems, in which initial excitation of a donor species-followed by energy transfer-leads to emission from an acceptor molecule, the handedness of both donor and acceptor may influence output circularity. In systems with an achiral acceptor, a degree of fluorescence circularity may be influenced by the handedness of a chiral donor, but this should not be construed in terms of 'conveying' chirality. Chiral molecules may also play a passive role by inducing helicity in the fluorescence from achiral neighbours, and further tiers of complexity arise if the initial excitation is itself of circular polarization. In all such processes, symmetry principles play a major role in determining a sensitivity to molecular handedness, and their detailed consideration enables a range of new experimental procedures to be identified. Casting the fundamental theory in terms of formal photon-molecule couplings enables the quantum mechanisms involved in all such phenomena to be clearly resolved. The results provide fresh physical insights, and establish connections across a range of indirectly related chiroptical phenomena including induced circular dichroism.

In situ Second-Harmonic Generation Circular Dichroism with Submonolayer Sensitivity

In this work, we present an experimental setup for the in situ and ex situ study of the optical activity of samples, which can be prepared under ultra-high vacuum (UHV) conditions by second-harmonic generation circular dichroism (SHG-CD) over a broad spectral range. The use of a racemic mixture as a qualified reference for the anisotropy factor is described and, as an example, the chiroptical properties of 1.5 μm thick (multilayers) as well as sub-monolayer thin films of the R- and S-enantiomer of 1,1'-Bi-2-naphthol (BINOL) evaporated onto BK7 substrates were investigated.

Two-photon chiro-optical properties of gold Au25 nanoclusters

Atomically-precise chiral gold nanoclusters hold promise for an accurate manipulation of chiro-optical properties, both in linear and nonlinear optics regimes. Here, we present the determination of two-photon linear-circular dichroism and two-photon circular dichroism (TPCD) of Au25[(Capt)18]- (where Capt is captopril). TPCD is found to be two orders of magnitude higher than one-photon circular dichroism.

Origin-independent two-photon circular dichroism calculations in coupled cluster theory

We present the first origin-independent approach for the treatment of two-photon circular dichroism (TPCD) using coupled cluster methods. The approach is assessed concerning its behavior on the choice of the basis set and different coupled cluster methods. We also provide a comparison of results from CC2 with those from density functional theory using the CAM-B3LYP functional. Concerning the basis set we note that in most cases an augmented triple zeta basis or a doubly augmented double zeta basis is needed for reasonably converged results. In the comparison of different coupled cluster methods results from CCSD, CC3 and CC2 have been found to be quite similar in most cases, while CCS results differ remarkably from the results at the higher levels. However, this proof-of-principle study also shows that further benchmarking of DFT and CC2 against accurate coupled cluster reference values (e.g. CCSD or CC3) is needed.

Three-photon circular dichroism: towards a generalization of chiroptical non-linear light absorption

We present the theory of three-photon circular dichroism (3PCD), a novel non-linear chiroptical property not yet described in the literature. We derive the observable absorption cross section including the orientational average of the necessary seventh-rank tensors and provide origin-independent expressions for 3PCD using either a velocity-gauge treatment of the electric dipole operator or a length-gauge formulation using London atomic orbitals. We present the first numerical results for hydrogen peroxide, 3-methylcyclopentanone (MCP) and 4-helicene, including also a study of the origin dependence and basis set convergence of 3PCD. We find that for the 3PCD-brightest low-lying Rydberg state of hydrogen peroxide, the dichroism is extremely basis set dependent, with basis set convergence not being reached before a sextuple-zeta basis is used, whereas for the MCP and 4-helicene molecules, the basis set dependence is more moderate and at the triple-zeta level the 3PCD contributions are more or less converged irrespective of whether the considered states are Rydberg states or not. The character of the 3PCD-brightest states in MCP is characterized by a fairly large charge-transfer character from the carbonyl group to the ring system. In general, the quadrupole contributions to 3PCD are found to be very small.

Ab initio study of the enantio-selective magnetic-field-induced second harmonic generation in chiral molecules

We present a systematic ab initio study of enantio-selective magnetic-field-induced second harmonic generation (MFISHG) on a set of chiral systems ((l)-alanine, (l)-arginine and (l)-cysteine; 3,4-dehydro-(l)-proline; (S)-α-phellandrene; (R,S)- and (S,S)-cystine disulphide; N-(4-nitrophenyl)-(S)-prolinol, N-(4-(2-nitrovinyl)-phenyl)-(S)-prolinol, N-(4-tricyanovinyl-phenyl)-(S)-prolinol, (R)-BINOL, (S)-BINAM and 6-(M)-helicene). The needed electronic frequency dependent cubic response calculations are performed within a density functional theory (DFT) approach. A study of the dependence of the property on the choice of electron correlation, on one-electron basis set extension and on the choice of magnetic gauge origin is carried out on a prototype system (twisted oxygen peroxide). The magnetic gauge dependence analysis is extended also to the molecules of the set. An attempt to analyze the structure-property relationships is also made, based on the results obtained for biphenyl (in a frozen twisted conformation), for prolinol and for some of their derivatives. The strength of the effect is discussed, in order to establish its measurability with a proposed experimental setup.

Metamaterials enable chiral-selective enhancement of two-photon luminescence from quantum emitters

The amplification of chirally modified, non-linear signals from quantum emitters is demonstrated by manipulating the geometric chirality of resonant plasmonic nanostructures. The chiral center of the metamaterial is opened and emitters occupy this light-confining and chirally sensitive region. Non-linear emission signals are enhanced by 40× that of the emitters not embedded in the metamaterial and display a 3× contrast for the opposite circular polarization.

Two-photon absorption and two-photon circular dichroism of hexahelicene derivatives: a study of the effect of the nature of intramolecular charge transfer

Effect of the strength and nature of the extension of the π-electronic delocalization on TPA and TPCD of helicenes derivatives.

Theoretical study of two-photon circular dichroism on molecular structures simulating aromatic amino acid residues in proteins with secondary structures

Calculation and comparative analysis of the theoretical two-photon circular dichroism (TPCD) spectra of l-His, l-Phe, and l-Tyr simulating residues in proteins with secondary structures (α-helix, β-strand and random coil), down to the far-UV region (FUV).

Ab initio study of the circular intensity difference in electric-field-induced second harmonic generation of chiral natural amino acids

We present a systematic computational study of circular intensity difference (CID) in electric-field-induced second harmonic generation (EFISHG) of some representative chiral natural amino acids (Alanine, Arginine, Aspartic Acid, Cysteine and Tryptophan), taking into account the electric-dipole, electric-quadrupole and magnetic-dipole interactions. The calculations are performed by applying cubic response theory at both Hartree-Fock and Density Functional theory levels, the latter with the popular Becke-three parameters, Lee, Yang and Parr (B3LYP) functional. Special focus is given to the basis set, electron correlation and origin dependence of the properties. The full set of molecular parameters defined by Lam and Thirunamachandran in their reference theoretical paper published in 1982 [J. Chem. Phys., 1982, 77, 3810] is obtained and discussed. This permits the prediction of the CID observable for different possible experimental setups.

Revealing the Electronic and Molecular Structure of Randomly Oriented Molecules by Polarized Two-Photon Spectroscopy

In this Letter, we explored the use of polarized two-photon absorption (2PA) spectroscopy, which brings additional information when compared to methods that do not use polarization control, to investigate the electronic and molecular structure of two chromophores (FD43 and FD48) based on phenylacetylene moieties. The results were analyzed using quantum chemical calculations of the two-photon transition strengths for circularly and linearly polarized light, provided by the response function formalism. On the basis of these data, it was possible to distinguish and identify the excited electronic states responsible for the lowest-energy 2PA-allowed band in both chromophores. By modeling the 2PA circular-linear dichroism, within the sum-over-essential states approach, we obtained the relative orientation between the dipole moments that are associated with the molecular structure of the chromophores in solution. This result allowed to correlate the V-shape structure of the FD48 chromophore and the quantum-interference-modulated 2PA strength.

Polarization effect on the two-photon absorption of a chiral compound

In this report, we investigate the polarization effect (linear, elliptical and circular) on the two-photon absorption (2PA) properties of a chiral compound based in azoaromatic moieties using the femtosecond Z-scan technique with low repetition rate and low pulse energy. We observed a strong 2PA modulation between 800 nm and 960 nm as a function the polarization changes from linear through elliptical to circular. Such results were interpreted employing the sum-over-essential states approach, which allowed us to model the 2PA circular-linear dichroism effect and to identifier the overlapping of the excited electronic states responsible by the 2PA allowed band.

The Effect of the π-Electron Delocalization Curvature on the Two-Photon Circular Dichroism of Molecules with Axial Chirality

Herein we report on the theoretical-experimental study of the effect of curvature of the π-electron delocalization on the two-photon circular dichroism (TPCD) of a family of optically active biaryl derivatives (S-BINOL, S-VANOL, and S-VAPOL). The comparative analysis of the influence of the different transition moments to their corresponding TPCD rotatory strength reveals an enhanced contribution of the magnetic transition dipole moment on VAPOL. This effect is hereby attributed to the additional twist in the π-electron delocalization on this compound. TPCD measurements were done using the double L-scan technique in the picosecond regime. Theoretical calculations were completed using modern analytical response theory, within a time-dependent density functional theory (TD-DFT) approach, at both, B3LYP and CAM-B3LYP levels, with the aug-cc-pVDZ basis set for S-BINOL and S-VANOL, and 6-31G* for S-VAPOL. Solvent effects were included by means of the polarizable continuum model (PCM) in CH2Cl2.

Chiral vibrational structures of proteins at interfaces probed by sum frequency generation spectroscopy

We review the recent development of chiral sum frequency generation (SFG) spectroscopy and its applications to study chiral vibrational structures at interfaces. This review summarizes observations of chiral SFG signals from various molecular systems and describes the molecular origins of chiral SFG response. It focuses on the chiral vibrational structures of proteins and presents the chiral SFG spectra of proteins at interfaces in the C-H stretch, amide I, and N-H stretch regions. In particular, a combination of chiral amide I and N-H stretches of the peptide backbone provides highly characteristic vibrational signatures, unique to various secondary structures, which demonstrate the capacity of chiral SFG spectroscopy to distinguish protein secondary structures at interfaces. On the basis of these recent developments, we further discuss the advantages of chiral SFG spectroscopy and its potential application in various fields of science and technology. We conclude that chiral SFG spectroscopy can be a new approach to probe chiral vibrational structures of protein at interfaces, providing structural and dynamic information to study in situ and in real time protein structures and dynamics at interfaces.

Polarization-induced control of two-photon excited fluorescence in a chiral polybinaphthyl

The fluorescence behavior of a chiral polybinaphthyl excited with 100 fs 800 nm laser pulses was investigated in tetrahydrofuran solution. The peak fluorescence intensity versus the input irradiance was measured to meet a square dependence, giving evidence for two-photon excited fluorescence (TPF). The variations of the TPF intensity were found to be strongly modulated by the different polarized incident lights and tightly depend on the linearly polarized component of the incident light. Furthermore, combining with the characteristics of chiral molecules, the two-photon polarization ratio was studied to reveal the symmetry of the involved excited states.

Two-photon polarization dependent spectroscopy in chirality: a novel experimental-theoretical approach to study optically active systems

Many phenomena, including life itself and its biochemical foundations are fundamentally rooted in chirality. Combinatorial methodologies for catalyst discovery and optimization remain an invaluable tool for gaining access to enantiomerically pure compounds in the development of pharmaceuticals, agrochemicals, and flavors. Some exotic metamaterials exhibiting negative refractive index at optical frequencies are based on chiral structures. Chiroptical activity is commonly quantified in terms of circular dichroism (CD) and optical rotatory dispersion (ORD). However, the linear nature of these effects limits their application in the far and near-UV region in highly absorbing and scattering biological systems. In order to surmount this barrier, in recent years we made important advancements on a novel non linear, low-scatter, long-wavelength CD approach called two-photon absorption circular dichroism (TPACD). Herein we present a descriptive analysis of the optics principles behind the experimental measurement of TPACD, i.e., the double L-scan technique, and its significance using pulsed lasers. We also make an instructive examination and discuss the reliability of our theoretical-computational approach, which uses modern analytical response theory, within a Time-Dependent Density Functional Theory (TD-DFT) approach. In order to illustrate the potential of this novel spectroscopic tool, we first present the experimental and theoretical results obtained in C(2)-symmetric, axially chiral R-(+)-1,1'-bi(2-naphthol), R-BINOL, a molecule studied at the beginning of our investigation in this field. Next, we reveal some preliminary results obtained for (R)-3,3'-diphenyl-2,2'-bi-1-naphthol, R-VANOL, and (R)-2,2'-diphenyl-3,3'-(4-biphenanthrol), R-VAPOL. This family of optically active compounds has been proven to be a suitable model for the structure-property relationship study of TPACD, because its members are highly conjugated yet photo-stable, and easily derivatized at the 5- and 6-positions. With the publication of these outcomes we hope to motivate more members of the scientist community to engage in state-of-the-art TPACD spectroscopy.

Polarization induced control of single and two-photon fluorescence

Modulation of two-photon absorption, two-photon fluorescence (TPF), as well as single-photon fluorescence (SPF), is shown through incident laser polarization for different fluorescent dyes. TPF intensity increases as the polarization changes from circular to linear irrespective of the dye, though the intensity and wavelength dependent studies of two-photon polarization ratio for any particular dye (e.g., Rhodamine 6G) reveal the nature of their excited state. SPF intensity of IR125 and IR144 dyes increases as the polarization changes from linear to circular. Thus, polarization studies indicate that in case of TPF, there is a preference toward the linear component while in case of SPF, the preference is toward the circular component of the incident laser beam.