Consequences of Kramers-Kronig Relations on Kleinman Antisymmetric Spectral Observables: Chiral-Specific Sum-Frequency Generation Spectroscopy

Abstract

Investigating water structures at aqueous interfaces has been a central focus in the field of nonlinear surface spectroscopy over decades. This large body of work leads to a conclusion that asymmetric OH stretches should be largely silent in sum frequency generation (SFG) spectra. Nonetheless, our recent studies show chiral-specific SFG response of water originating from the first hydration shell of proteins and DNA arising from a sum of equal-magnitude and opposite-phase symmetric and asymmetric OH stretches, resulting in an oppositely-signed couplet. Motivated initially by this apparently inconsistent behavior for chiral-specific versus conventional (achiral) SFG, we demonstrate herein a fundamental argument for the appearance of the oppositely-signed couplets in chiral-specific vibrational SFG. The interplay between two foundational optical relations, Kramers-Kronig relations and index-interchange symmetry arising in the adiabatic zero-frequency limit (often referred to as Kleinman symmetry), is shown to require the imaginary resonant contributions to the tensors describing chiral-specific SFG responses to collectively sum to zero. In brief, all indices within the surface susceptibility must become interchangeable in the degenerate, zero-frequency adiabatic limit. From Kramers-Kronig relations connecting the real and imaginary susceptibility, this asymptotic limit in the real susceptibility can only be met if the imaginary-valued resonant contributions over the relevant spectral range sum to zero. These symmetry constraints were found to agree with density functional theory calculations on small water clusters and with experimental and computational phase-resolved chiral-specific SFG spectra. These symmetry requirements provide constraints for analyzing phase-resolved chiral-specific SFG spectra for extracting structural information about chiral molecules at interfaces.

Introduction

Nonlinear optical spectroscopy enables observations inaccessible by conventional linear spectroscopic means, with correspondingly unique symmetry relationships. An excellent example is chiral-specific sum-frequency generation (SFG) spectroscopy, which supports surface-selective spectroscopy with chiral signals rivaling their achiral counterparts in magnitude. Chiral-specific SFG is highly sensitive to molecular and macromolecular symmetry, and previous work in the field has interrogated biomolecular structure^1–3 and hydration.^4–6 The dual characteristics of interfacial and chiral selectivity forbid contributions from isotropic bulk in the absence of electronic resonance with incident visible light. This allows for vibrational detection of comparably dilute analytes in crowded spectral regions (e.g., the peptide NH stretches, whose signal would typically be obscured by the intense absorption of bulk water between 3000 and 4000 ${\text{cm}}^{-1}$) without need for spectroscopic or isotopic labels that are often employed in linear vibrational spectroscopies.⁷ Phase-resolved experiments can also provide insight into the symmetry and orientation of chromophores and aid in disentangling complex spectra with resonances that are similar in frequency. Recent studies have established chiral-specific SFG as capable of detecting water vibrational structures within the first hydration shell of biopolymers.^5,8,9 Due to its dual selectivity to interfaces and chirality, chiral-specific SFG can uniquely probe the interplay of water, chirality, and interfaces at the molecular level in situ and in real time under ambient conditions.⁸ This interplay is vital to the function of biomolecules, and the insight that chiral-specific SFG offers can help to elucidate the mechanisms underlying these functions. Thus, chiral-specific SFG shows promise in addressing a wide range of fundamental and engineering challenges,⁸ such as molecular mechanisms of biocatalysts and membrane protein folding as well as molecular design of drug carriers and biosensors.

Interestingly, both simulations and phase-sensitive heterodyne experiments have regularly reported the appearance of spectral couplets with equal amplitude and opposite sign for the chiral-specific SFG of the OH stretching modes of water in contact with interfacial biomolecules (Figure 1).^5,9,10 Here, we define a couplet as a collection of two or more spectral features whose amplitudes sum to zero, and a doublet is a couplet composed of two peaks (with equal and opposite intensity). Why this couplet structure appears is nonobvious from the vibrational resonances of water under consideration. Understanding the origins of this couplet formation at the molecular level has the potential to inform on analogous relations in broad classes of linear and nonlinear spectroscopic measurements.

Figure 1:

Simulated chiral-specific SFG spectrum of water in the first hydration shell of ${\text{LK}}_7\beta$, a model polypeptide for interfacial spectroscopy. The pink spectrum forms an oppositely signed couplet lineshape, which can be represented as the sum of two oppositely signed Lorentzian peaks (red and blue). This spectrum was previously reported in Ref.⁴

In recent work, Konstantinovsky et al. posited a coupled oscillator model to provide a molecular-level explanation for formation of oppositely signed doublets in the chiral-specific vibrational SFG of OH stretching modes of water molecules hydrating polypeptides at aqueous interfaces.^5,11,12 In brief, coupling between OH vibrational transitions within a single water molecule into symmetric and asymmetric modes was shown to produce doublets in the chiral-specific SFG activity. Appropriate orientation of such nominally ${\text{C}}_{2\text{v}}$ symmetry structures about a chiral biomolecular architecture was found through simulation and experiment to provide an explanation for the appearance of positive/negative doublets in the chiral-specific SFG within the first hydration shell.^5,9,11 Although Hu et al.³ did not offer experimental data to solidly conclude an observation of chiral-selective phase resolved OH signals of water hydrating a β-sheet protein, they proposed that, if such signals could indeed be observed, these signals could be interpreted as arising from a microscopic origin wherein the interactions of the OH groups and lone pairs of a given water molecule with a chiral environment creates a chiral center at the oxygen atom. This interpretation is different from our interpretation, which is based on a chiral supramolecular assembly of water molecules surrounding the folded macroscopically chiral biopolymers.

The coupled oscillator model of Konstantinovsky et al. is supported by molecular dynamics (MD) simulations combined with electrostatic mapping calculations of chiral-specific SFG responses from symmetric and asymmetric stretches.¹¹ This work raises the question about whether a similar relation exists in general beyond ${\text{C}}_{2\text{v}}$-symmetric molecules. If generally applicable, the requirement of separable classes of spectral features (e.g., water OH stretching modes or protein amide I modes) to sum to zero in chiral-specific SFG spectra could serve as an invaluable constraint in spectral fitting and subsequent interpretation of vibrational SFG of chiral interfaces. For transitions of similar energy, the degree of coupling from perturbation theory scales inversely with the energy difference,¹³ enabling groupings of spectroscopic features into separable clusters of related transitions (e.g., H stretching modes). However, decades of research in conventional (achiral) SFG studies of water at interfaces suggest that the complex network of OH stretching motions at aqueous interfaces does not produce symmetric and asymmetric modes of comparable amplitude.^14–20 Since both the chiral and achiral SFG responses of vibrational resonances are fully allowed within the electric dipole approximation, it may not be obvious why such disparate behavior should be expected for the chiral versus achiral interfacial SFG activities.

In this work, causality-based arguments suggest precisely such unique differentiation between the chiral and achiral SFG responses, demonstrating that the collective resonance-enhanced chiral contributions to vibrational SFG spectroscopy are expected to sum to zero while the achiral contributions are not. Quantum chemical calculations of model water clusters were performed to evaluate the predictions of the model through decomposition of the molecular hyperpolarizabilities of the water clusters into Kleinman symmetric and antisymmetric contributions, yielding excellent agreement. This symmetry conjecture was also tested by reexamining previously published heterodyne chiral-specific SFG spectra. Chiral-specific SFG spectra simulated from MD agree very well with our proposed model. While the experimental internal heterodyne chiral-specific SFG spectra are in generally good agreement with the model, significant deviations from predictions are observed in some instances. We discuss the possible reasons for the discrepancy between our model and the experimental spectra. This work provides fundamental insights into validating and interpreting chiral-specific SFG response from interfaces, empowering further method development in detecting microscopic chirality and hydration of important molecular systems, such as folded proteins and nucleic acids at interfaces.

Theoretical Background

Kramers-Kronig Relations and Index-Interchange (Kleinman) Symmetry

The Kramers-Kronig relations allow determination of the real part of a continuous complex function from knowledge of the imaginary part and vice versa.^21,22 For the general function $\chi \left(\omega \right)={\chi }_{\text{Re}}\left(\omega \right)+i{\chi }_{\text{Im}}\left(\omega \right)$, the following relations hold.

$${\chi }_{\text{Re}}\left(\omega \right)=\frac{1}{\pi }P{\int }_{-\infty }^{\infty }\frac{{\chi }_{\text{Im}}\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$ (1)

$${\chi }_{\text{Im}}\left(\omega \right)=-\frac{1}{\pi }P{\int }_{-\infty }^{\infty }\frac{{\chi }_{\text{Re}}\left({\omega }^{\prime }\right)}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }$$ (2)

The symbol P in Eqs. 1 and 2 indicates the Cauchy principal value (i.e., evaluating the path integral excluding the discontinuity at ${\omega }^{\prime }=\omega$). For the specific case of vibrational SFG spectroscopy considered in this work, $\omega$ represents the frequency of infrared light. The Kramers-Kronig dispersion relations ultimately derive their origins in arguments of causality, namely that the temporal response of a system must follow a stimulus in time. Translating this restriction into the frequency domain ultimately leads to the complex relations in Eqs. 1 and 2, which collectively ensure frequency-domain descriptions consistent with principles of causality within the time domain. The expressions given in Eqs. 1 and 2 hold rigorously for integration over all frequencies, which poses practical challenges for implementation in real physical systems with responses measured over finite frequency windows. In practice, baseline-resolved resonances with imaginary contributions outside a given measurement window can contribute to slowly varying real-valued baseline offsets. Fortunately, the core concepts resulting in couplet formation detailed below can be illustrated theoretically through simulations requiring only a small number of nonzero resonances, all of which fall within the spectral range of interest. If one considers a simple molecular assembly possessing only a single resonance with a Lorentzian lineshape, the nonzero imaginary term on resonance necessarily imposes a nonzero real term far from resonance (Figure 2A). This real part manifests through the broad “wings” of the distribution. However, the expressions in Eqs. 1 and 2 are general for any lineshape function. For SFG, these Kramers-Kronig relations demand a matched real (refractive) susceptibility far from resonance scaling proportionally to the integrated amplitude of the imaginary (absorptive) resonant component. An explicit example of this expectation is evaluated for Lorentzian functions in the Supplementary Material, suggesting an asymptotic limiting value of the real component on the order of ~1% of the corresponding peak resonant imaginary tensor component, consistent with the calculations in Figure 2A.

Figure 2:

Illustration of the impact of dispersion on the nonresonant limit of an analytic complex function (e.g., a chiral-specific SFG tensor element): A) a single resonance at 900 ${\text{cm}}^{-1}$, whose real component does not approach zero in the zero-frequency limit and therefore cannot conform with Kleinman symmetry (vide infra), and B) an oppositely signed doublet at 900 ${\text{cm}}^{-1}$ and 920 ${\text{cm}}^{-1}$, whose real component can converge to zero. Calculations were performed using Lorentzian lineshape functions with a damping factor of $\Gamma =10{\text{cm}}^{-1}$.

The Kramers-Kronig dispersion requirement imposes constraints on chiral-specific SFG. As the frequencies (here, the incident infrared and visible beams and the emitted sum frequency) collectively approach zero, they approach degeneracy in the adiabatic limit, at which point the frequencies become indistinguishable and their corresponding indices interchangeable; this interchangeability of indices is commonly referred to as Kleinman symmetry.²³ Here, we use “adiabatic” to refer to the classical/low-frequency limit in which effects due to dispersion and absorption become negligible.

Before advancing further, it is prudent to provide additional clarifications on how “Kleinman symmetry” is defined and used in the present work. Within the theoretical framework in this study, we only assume validity for the symmetry of index interchangeability between frequency-dependent hyperpolarizability tensor elements as strictly arising in the adiabatic limit. These are consistent with the conditions necessary for recovery of index interchange symmetry detailed by Franken and Ward.²⁴ Practical realization of index-interchange symmetry can be challenging to observe experimentally, particularly in second harmonic generation for which it was originally considered. The virtual states are rarely sufficiently far from electronic resonance for the energy gap between the incident and doubled frequency to be negligible, consistent with prior critiques of Kleinman symmetry by Franken and Ward²⁴ and by one of the authors.²⁵ However, it is nevertheless useful to consider decomposition of the hyperpolarizability tensor into interchange-symmetric (i.e., Kleinman symmetric) and interchange-antisymmetric (Kleinman antisymmetric) components, just as any arbitrary linear function can be decomposed as the sum of even and odd components. This decomposition holds irrespective of the presumed validity of index interchangeability; indeed, it clearly requires departure from Kleinman symmetry as defined herein for there to be a nonzero antisymmetric component at all.

For the Raman tensor, the condition of index interchangeability is met when the energy of the virtual state in the polarizability evaluation is sufficiently low relative to the electronic excited states defining the nonresonant polarizability. The low energies and narrow resonances of vibrational transitions relative to electronic transitions generally result in comparable Raman cross-sections for the Stokes and anti-Stokes transitions, consistent with a symmetric Raman matrix. In this adiabatic limit, the higher frequency excited states can respond instantaneously to the slower electronic motion of the virtual state, which in quantum chemical calculations is synonymous with evaluation of the Raman polarizability from the change in energy with respect to DC fields. In the case of vibrational SFG, in this limit of a symmetric Raman matrix (i.e., with the visible beam far from electronic resonance), the first two indices of the resonance-enhanced nonlinear response are inherently interchangeable in both the hyperpolarizability and correspondingly the nonlinear susceptibility. If the incident infrared light is sufficiently far from resonance as well, the Kleinman symmetry extends to the third hyperpolarizability tensor index. In this limit, all six chiral-specific susceptibility tensor elements become degenerate: ${\chi }_{XYZ}\cong {\chi }_{YXZ}\cong {\chi }_{YZX}\cong {\chi }_{XZY}\cong {\chi }_{ZXY}\cong {\chi }_{ZYX}$. These chiral-specific tensor elements become zero when changing from ${\text{C}}_{\infty }$ symmetry for a chiral interface to ${\text{C}}_{\infty \text{v}}$ symmetry for an achiral interface, as the additional mirror plane removes tensor elements with either a single X or single Y index. In this work, lowercase letters ($xyz$) index the hyperpolarizability $\mathbf{\beta }$ as calculated within the molecular frame. Uppercase letters ($XYZ$) index the second-order nonlinear susceptibility $\mathbf{\chi }$ in the laboratory/ensemble frame (i.e., following application of orientational factors). We define our laboratory frame as is conventional for ${\text{C}}_{\infty }$ symmetry systems: the Z axis is coincident with the interfacial surface normal vector, and the X and Y axes are mutually orthogonal vectors in the plane of the interface.

In addition, uniaxial ${\text{C}}_{\infty }$ symmetry typical in chiral-specific vibrational SFG experiments imposes a rigorous symmetry constraint of ${\chi }_{XYZ}=-{\chi }_{YXZ};{\chi }_{YZX}=-{\chi }_{XZY};{\chi }_{ZXY}=-{\chi }_{ZYX}$. The intrinsic interchangeability within the first two indices of the molecular hyperpolarizability from the symmetry in the Raman tensor demands that ${\chi }_{XYZ}={\chi }_{YXZ}$, while uniaxial symmetry requires ${\chi }_{XYZ}=-{\chi }_{YXZ}$. These constraints can only be satisfied if ${\chi }_{XYZ}={\chi }_{YXZ}=0$. No such constraints are imposed on the other two pairs of chiral-specific interfacial tensor elements, and this constraint applies even in the presence of vibrational resonance. For infrared frequencies close to vibrational resonance but far enough away for the response function to be predominantly real-valued, Kleinman symmetry is generally not expected to hold.²⁵ However, sufficiently far from both electronic and vibrational resonance, Kleinman symmetry requires all indices to be interchangeable, and in turn all chiral tensor elements to be equal and therefore approach zero. This fact can be combined with the Kramers-Kronig relation given in Eq. 1 to give a general relation for chiral-specific tensor elements, exemplified in Eq. 3.

$$\underset{\omega \to 0}{\text{lim}}{\chi }_{ZYX,\text{Re}}\left(\omega \right)\cong 0\cong \frac{1}{\pi }{\int }_0^{\infty }\frac{{\chi }_{ZYX,\text{Im}}\left({\omega }^{\prime }\right)}{{\omega }^{\prime }}d{\omega }^{\prime }$$ (3)

Since ${\omega }^{\prime }$ is positive in Eq. 3, the collective set of resonance-enhanced chiral-specific imaginary contributions corresponding to the molecular resonances must therefore sum to zero to satisfy the causality implicit in the Kramers-Kronig relations. This restriction on the chiral-specific tensor contributions was first reported by Giordmaine²⁶ and subsequently used to explore the range over which Kleinman symmetry holds locally near electronic resonance.^25,27–30 Importantly, no analogous constraint arises from Kleinman symmetry for the achiral components from assemblies of ${\text{C}}_{\infty \text{v}}$ symmetry. Although these tensor elements also converge to Kleinman symmetric values in the limit of zero frequency (e.g., ${\chi }_{ZXX}\cong {\chi }_{XZX}\cong {\chi }_{XXZ}$), these achiral terms differ from their chiral counterparts by having no zeros in the limiting equalities (as illustrated in Figure 2A).

This disparate trend in the asymptotic behavior between tensor element combinations adhering to or defying Kleinman symmetry carries corresponding consequences in the collective resonance-enhanced nonlinear optical responses. Specifically, this trend requires Kleinman antisymmetric contributions to approach zero much faster than their Kleinman symmetric counterparts away from resonance. This requirement arises by consideration of the extreme limits of zero frequency, in which the molecular hyperpolarizability remains well defined and calculable in quantum chemistry, but the chiral-specific contributions in a uniaxial assembly are necessarily removed. Of course, setting the molecular hyperpolarizability to zero trivially satisfies these requirements, but not in a particularly interesting way. From inspection of the functional form of Eq. 1 and illustrated in Figure 2A, a single isolated resonance exhibits an identical frequency-dependent scaling for all tensor elements and cannot nontrivially satisfy this requirement.

One nontrivial solution to this apparent paradox is to consider the collective contributions of multiple resonances. Arguably, the simplest possible scenario is one in which the Kleinman antisymmetric components include a pair of resonances of equal magnitude and opposite sign but shifted resonance frequencies, such as was considered previously by Konstantinovsky et al.¹¹ In this instance, the nonresonant Kleinman antisymmetric contributions of the doublet quickly sum to zero away from the two resonance frequencies, recovering Kleinman symmetry in the asymptotic low frequency limit (Figure 2B). Furthermore, the Kleinman symmetric resonant contributions do not face a similar imposition, potentially remaining nonzero well outside of the resonance-enhanced spectral range as depicted in Figure 2A. This simple example of an equal and opposite doublet is a specific example of a more general phenomenon, with the combined requirements of Kramers-Kronig relations and index-interchange symmetry arising in the adiabatic zero-frequency limit demanding all Kleinman antisymmetric resonance-enhanced contributions to collectively integrate to zero.

The degree to which this high-level global symmetry constraint may impact specific localized resonances within the molecular tensor, such as those recovered in simulations and measurements by Konstantinovksy et al.,^5,11,12 may not be obvious. To assess the generalizability of this strategy, a series of quantum chemical calculations was performed for a set of model water clusters. Analogous cluster calculations have served previously as comparatively simple models for assisting in interpreting chiral-specific vibrational SFG spectroscopic measurements of chiral hydration of biopolymers.¹² Decomposing the resulting quantum chemical calculations for vibrational SFG into Kleinman symmetric and antisymmetric resonance-enhanced tensors provides a route for exploring the degree to which summations to zero are general phenomena in chiral-specific vibrational SFG spectroscopy.

The Molecular Tensor

The molecular hyperpolarizability describing resonance-enhanced vibrational SFG spectroscopy is related to the direct product of the Raman polarizability matrix $\mathbf{\alpha }$ with the transition moment for infrared absorption $\mathbf{\mu }$:

$${\beta }_{ijk}(-{\omega }_{\text{sum}};{\omega }_{\text{vis}},{\omega }_{\text{IR}})\cong \sum _n{S}_n\left({\omega }_{\text{IR}}\right)\left[\right({\alpha }_{0n}^{ij}{)}_{\text{AR}}{\mu }_{n0}^k]=\sum _n\frac{-({\alpha }_{0n}^{ij}{)}_{\text{AR}}{\mu }_{n0}^k}{2h({\omega }_n-{\omega }_{\text{IR}}-i{\Gamma }_n)}$$ (4)

where $S_n$ is the complex-valued line-shape function for the $n^{\text{th}}$ resonance. For homogeneously broadened peaks, the lineshape function of a damped driven oscillator is described by a Lorentzian, defined by a center frequency ${\omega }_n$ and a damping term ${\Gamma }_n$ related to the linewidth.

In the limit of negligible intermolecular interactions (or if intermolecular interactions can be integrated into local-frame calculations), the second-order nonlinear susceptibility $\mathbf{\chi }$ of an ensemble can be initially approximated by consideration of orientational averages connecting the molecular and ensemble frames. In the case of a uniaxial assembly of $N_s$ molecules with identical hyperpolarizability $\mathbf{\beta }$, these orientational averages adopt well-established forms.³¹ For example,

$${\chi }_{ZYX}=\frac{N_s}{2}\left\{\begin{array}{l}⟨{\text{cos}}^2\theta ⟩({\beta }_{zyx}-{\beta }_{zxy})\\ +⟨{\text{sin}}^2\theta {\text{sin}}^2\psi ⟩({\beta }_{yxz}-{\beta }_{yzx})+⟨{\text{sin}}^2\theta {\text{cos}}^2\psi ⟩({\beta }_{xzy}-{\beta }_{xyz})\\ +⟨{\text{sin}}^2\theta \text{sin}\psi \text{cos}\psi ⟩({\beta }_{xzx}-{\beta }_{xxz}+{\beta }_{yyz}-{\beta }_{yzy})\\ +⟨\text{sin}\theta \text{cos}\theta \text{sin}\psi ⟩({\beta }_{yyx}-{\beta }_{yxy}+{\beta }_{zxz}-{\beta }_{zzx})\\ +⟨\text{sin}\theta \text{cos}\theta \text{cos}\psi ⟩({\beta }_{xxy}-{\beta }_{xyx}+{\beta }_{zyz}-{\beta }_{zzy})\end{array}\right\}$$ (5)

This expression is for the most general case of a uniaxial system; in the event that the molecular system has higher symmetry (e.g., an isolated water molecule with no intrinsic chirality), some of these elements of $\mathbf{\beta }$ may be negligible or zero.^31,32 The relations in Eq. 5 contain pairs of terms within the molecular hyperpolarizability (e.g., ${\beta }_{xzy}$ and $-{\beta }_{yzx}$) that will collectively sum to zero in the limit of full index interchangeability, consistent with the Kleinman symmetric component. ${\chi }_{ZYX}$ was selected as an example, as it is the tensor element probed using the psp beam polarization for obtaining the experimental data used to test the model here.² Analogous expressions can be written for ${\chi }_{ZXY}$ and the other chiral elements of $\mathbf{\chi }$. Of most relevance, it should be clear that complete interchangeability of the indices (i.e., ${\beta }_{xxz}={\beta }_{zxx},{\beta }_{yyz}={\beta }_{zyy}$, etc.) results in cancellation of all contributions and a zero-valued ${\chi }_{ZYX}$ tensor element.

The resonance-enhanced tensor measured experimentally or computationally can be decomposed into symmetric and antisymmetric (S and A, respectively) components by first evaluating the Kleinman symmetric tensor from the average of each set of tensor elements connected through index-interchange (e.g., ${\beta }_{xxz},{\beta }_{xzx}$, and ${\beta }_{zxx}$), then subtracting the symmetric tensor from the original tensor to isolate the antisymmetric terms as follows:

$$\mathbf{\beta }=\mathbf{S}+\mathbf{A}$$ (6)

Additional details on the symmetric/antisymmetric decomposition are provided in the Appendix. The rotational transformation mapping $\mathbf{\beta }$ to ${\chi }_{ZYX}$ (or to any other chiral element of $\mathbf{\chi }$) is linear. Accordingly, the contributions of the two symmetrized tensors to ${\chi }_{XZY}$ can be decomposed additively:

$$\begin{gathered} {\chi }_{ZYX}={\chi }_{ZYX}\left(\beta \right)={\chi }_{ZYX}\left(S+A\right) \\ ={\chi }_{ZYX}\left(S\right)+{\chi }_{ZYX}\left(A\right) \\ =0+{\chi }_{ZYX}\left(A\right)={\chi }_{ZYX}\left(A\right) \end{gathered}$$ (7)

Due to the interchangeability of the tensor elements in S, its contribution to the chiral nonlinear susceptibility is 0. As such, the Kleinman antisymmetric contributions alone are symmetry-allowed to contribute to chiral-specific SFG. Their isolation in both quantum chemical calculations and experimental measurements can support theoretical evaluation of the generalizability of Eq. 3 and reveal trends within the molecular and ensemble responses that are obscured if only the total hyperpolarizability is examined.

The terms in Eq. (5) help illustrate the connections bridging index-interchangeability within the local hyperpolarizability and the ensemble response in the limit of an assembly of non-interacting interfacial molecules. However, analogous connections must also arise within more complex assemblies with strong intermolecular interactions, such as the extended H-bonded network of water. Only the interchange-antisymmetric hyperpolarizability in the local frame can contribute to the macroscopic interchange-antisymmetry susceptibility describing the ensemble response of the interfacial assembly.

Methods

Quantum chemical calculations were performed in Gaussian 16.³³ A water dimer, trimer, and pentamer were built with GaussView 6.³⁴ All structures were optimized to an energy minimum, followed by normal mode calculations, with density functional theory (DFT) using the $\omega$B97X-D functional³⁵ and the 6–311+G(d,p) basis set.³⁶ All calculations were performed in the gas phase. Harmonic frequencies for the water clusters were scaled by 0.95, which is standard for $\omega$B97X-D.³⁷ Normal mode visualizations were prepared with UCSF Chimera.³⁸ All calculations were limited to electric dipole-allowed effects only, with symmetric Raman polarizabilities evaluated in the absence of electronic resonance enhancement (i.e., from the derivative of the zero-frequency polarizability with respect to normal mode motion).

The polarization dependence of the resonance-enhanced hyperpolarizability tensor of the $n^{\text{th}}$ vibrational resonance was evaluated using Eq. 4 without explicit assumption of the functional form of the lineshape function $S_n$ (i.e., as a simple tensor product of the transition polarizability and transition dipole). Vibrational tensor elements were visualized using NLOPredict,³⁹ which is an open-source plug-in for UCSF Chimera. The amplitudes H of the hyperellipsoid tensor representations correspond to the efficiency of generating an SFG field in a given direction when driven by coincident fields of the same polarization, with sign information conveyed by color (blue for positive, red for negative). The form of these amplitudes is given by

$$H\left(\theta ,\varphi \right)=E_I\left(\theta ,\varphi \right){\beta }_{\mathit{\text{III}}}{E}_I{(\theta ,\varphi )}^2$$ (8)

where $H(\theta ,\varphi )$ is the signed amplitude of the hyperellipsoid and $E_I(\theta ,\varphi )$ is the I-polarized component of a unit vector in the direction dictated by the usual spherical polar coordinates $\theta$ and $\varphi$, mapping the amplitude of the molecular hyperpolarizability ${\beta }_{III}$ for generation of coparallel polarization in that same direction.

To evaluate the extent of cancellation predicted by Eq. 3 observed in previously published simulated and experimental heterodyne chiral-specific SFG spectra, a normalized integration was computed. The normalized integrated magnitude is calculated as follows:

$$F\left(\omega \right)=\frac{100\%}{{\int }_{{\omega }_i}^{{\omega }_f}\frac{\left|\chi \right({\omega }^{\prime }\left)\right|}{{\omega }^{\prime }}d{\omega }^{\prime }}{\int }_{{\omega }_i}^{\omega }\frac{\chi \left({\omega }^{\prime }\right)}{{\omega }^{\prime }}d{\omega }^{\prime }$$ (9)

where $F\left(\omega \right)$ is the normalized integrated magnitude, ${\omega }_i$ and ${\omega }_f$ are the beginning and end, respectively, of the spectral range considered, and $\chi \left({\omega }^{\prime }\right)$ is the computed/measured imaginary spectrum. The first term is a normalization constant such that $F\left({\omega }_f\right)=\pm 100\%$ indicates no cancellation at all, and a value of $F\left({\omega }_f\right)=0\%$ indicates complete cancellation. Strictly speaking, this integration should be performed starting from high frequency backward to evaluate the limiting behavior near zero in accordance with the Kramers-Kronig relations. However, for clarity of presentation, we have chosen to display the integration starting from the lower end of each spectral range.

Note that the Kramers-Kronig relations also requires division of the imaginary component of $\chi$ by ${\omega }^{\prime }$ during the integration. In the case of sharp resonances that are close in frequency (such as those in Figure 2B), this denominator can be easily neglected, and cancellation is more obvious. However, when peaks are broad or vibrational resonances are significantly delocalized, the summation to zero may be obscured, and the denominator must be considered.

Results

Water Trimer Calculations

The Kramers-Kronig relations apply equally well to both individual local hyperpolarizability tensor elements as well as the ensemble susceptibility. As such, the merits of the predictions of couplet-formation are arguably most straightforward to assess in simulations of small clusters. The results of the DFT calculations for a water trimer are represented in Figure 3 for a set of three hydrogen-bonded OH stretches. Results are shown only for the normal modes dominated by the OH stretches closely coupled through the H-bonding network. For results from the entire set of OH stretching modes, see the Supplementary Material. The OH stretching normal modes (two per water molecule) within the water molecules are close in energy and reasonably spectrally isolated from other vibrational normal modes of the discrete clusters. For spectrally overlapping peaks, the degree of coupling scales inversely with the energy difference between the local modes, such that strongly coupled vibrational modes tend to be similar in energy and spectrally grouped (e.g., H stretching modes). Contributions from Fermi resonances with the OH bend overtone were not considered in this work, which centers primarily on fundamental understanding of spectroscopic symmetry relations rather than precise reproduction of experimental observables. No implicit assumptions of molecular symmetry were imposed on the simulations; all clusters were allowed to adopt C₁ symmetry, with normal mode vibrational resonances typically involving extended motions across the clusters. Indeed, the O-atom of each water molecule depicted in Figure 3 serves as a chiral center, with four unique environments (free OH, H-bonded OH, free lone pair, and H-bonded lone pair). However, the details of the particular molecular arrangements are arguably not as important as the collective frequency-dependent response of the coupled assembly. Notably, these calculated structures are not intended to represent those anticipated at the air/water interface nor those expected from interactions from chiral structures such as DNA or polypeptides. Indeed, the chiral-specific response of the individual clusters depicted in Figure 3 is not particularly relevant for the evaluation of the theoretical predictions. More importantly, any resonance-enhanced hyperpolarizability contributions can be directly decomposed into Kleinman symmetric and antisymmetric components for assessing the reliability of the predictions from Eq. 3.

Figure 3:

A subset of three H-bonded OH stretching normal modes in a water trimer. A) Displacement vectors for each mode scaled by 1.5 for visualization. Vectors with magnitude $\le 0.1$ a.u. omitted for clarity. B) Hyperellipsoid representations of the Kleinman symmetric resonant tensors for each of the modes (rescaled as noted to aid in visualization). See the Methods section for more detail on the hyperellipsoid calculation. C) Total hyperpolarizability ($\mathbf{\beta }$) for the three modes. D) Kleinman antisymmetric hyperpolarizability (A) for the three modes. The hyperpolarizability contributions of the three normal modes considered are stacked on top of one another and distinguished by color as indicated in the legend. Hyperpolarizabilities for the normal modes are calculated within the molecular frame (x, y, z), with Cartesian axes shown in A, left. Hydrogen bond lengths are also shown in A, left.

As shown in Figure 3C, the total hyperpolarizability $\mathbf{\beta }$ in the local reference frame shown associated with mode 17 (red) is dominated by the ${\beta }_{xyy}={\beta }_{yxy}$ tensor elements, scaling with the projection of the infrared transition moment along the y-axis in the reference frame of the figure. Modes 16 and 18 (blue and yellow, respectively) have their transition dipoles aligned closely to the x-axis, with ${\beta }_{xxx}$ and ${\beta }_{yyx}$ as the largest components of their corresponding tensors. As illustrated by the scaling for the hyperellipsoids depicted in Figure 3B, which depict the Kleinman symmetric amplitude, and the corresponding plots shown in Figure 3C, the hyperpolarizability is significantly larger for modes 17 and 18 relative to 16. While the dominant tensor elements can be rationalized though examination of the normal modes considered, no particular trends between the nonzero elements are obvious.

In stark contrast, isolating the Kleinman antisymmetric tensor A for each mode, shown in Figure 3D, reveals that each dominant tensor element yields a sum over the collective contributions in the relevant frequency window approximately equal in magnitude and opposite in sign. As ${\beta }_{xxx}$ cannot contribute to the Kleinman antisymmetric tensor (and thus the resulting chiral-specific SFG spectrum), the contributions from ${\beta }_{xyy},{\beta }_{yxy}$, and ${\beta }_{yyx}$ dominate A. Despite the substantial reduction in local symmetry relative to an isolated water molecule, the coupling of OH modes produces the expected cancellation from Eq. 3. Analogous trends arise for the set of three free OH stretching modes (shown in the Supplementary Material).

The results for the trimer are in good agreement with a relatively simple coupled oscillator model by Konstantinovsky et al.¹¹ Since local mode motions with nominal ${\text{C}}_{\infty }$-symmetry are symmetry-forbidden to contribute to chiral-specific SFG,³¹ local-mode coupling to produce normal modes should produce chiral-specific tensor elements that sum to the same integrated result of the three zero-valued local modes, resulting in a positive/negative couplet.¹¹ The combined trends illustrated in this figure are mirrored in dimer and pentamer structures, the results of which are detailed in the Supplementary Material. In all cases, the Kleinman antisymmetric tensors for the normal modes associated with H-bond stretching motions collectively produced tensors with nominally equal and opposite amplitudes, which are therefore expected to produce spectral couplets. Notably, these simulations include the full set of intermolecular interactions arising withing the extended H-bonded water network, rather than local-mode descriptions based on symmetric and asymmetric vibrational stretching modes for water.

When interpreting the results of the simulations in Figure 3, the tensor decomposition into index-interchange symmetric/antisymmetric components should not be confused with symmetric / asymmetric assignments of local-mode descriptions of molecular motions of isolated water molecules. The following model is cast in terms of the molecular hyperpolarizability for the strongly H-bonded networks within the simulations. The trimer cluster depicted in Figure 3 exhibits C₁ symmetry, naturally eluding simplistic symmetric/asymmetric local mode classification for the vibrational resonances. In contrast, any hyperpolarizability tensor (including those describing symmetric and asymmetric local modes) can be decomposed into index-interchange symmetric and antisymmetric contributions.

The symmetry constraints suggested by the dual requirements of Kramers-Kronig relations connecting resonant and nonresonant behavior and Kleinman symmetry far from resonance have several implications for nonlinear optical analyses. Most directly, the set of vibrational transitions associated with a suite of strongly coupled spectrally overlapping modes, such as the three OH-stretching normal modes in Figures 3, are generally expected to produce couplets for the Kleinman antisymmetric contributions that collectively sum to zero. Chiral-specific vibrational SFG is an excellent model for testing this hypothesis computationally and experimentally, as the observables depend exclusively on the Kleinman antisymmetric components given in Eq. 7. This restriction is expected to hold for both the local molecular hyperpolarizability and the ensemble-averaged interfacial susceptibility. In the case of the latter, a direct consequence is the prediction of oppositely signed couplets in the chiral-specific SFG susceptibilities.^9,11,12

Simulated and Experimental Spectra

Both computational and experimental heterodyne phase-resolved chiral-specific SFG studies^9,10,40,41 provide direct access to observables for assessing the proposed Kramers-Kronig/Kleinman antisymmetry relations. In computational studies, chiral-specific SFG spectra of proteins or DNA were simulated previously by analyzing MD trajectories.^{5,9–12,40,41} The OH stretch spectra of water solvating proteins and DNA were simulated using the OH electrostatic map developed by the Skinner group,^14,42,43 and the protein NH spectra were calculated using the NH electrostatic map developed by Konstantinovsky et al.⁴¹ In experimental studies, the amplitude of the chiral-specific response is isolated by interference not with the achiral components but with a nonresonant local oscillator that is nominally uniform in amplitude and phase.^44,45

In these spectra, the presence of couplets predicted by the symmetry relations can potentially be isolated. To test the symmetry model, the chiral-specific SFG signals from computational and experimental spectra can be integrated, as the model in Eq. 3 predicts that the spectral signals divided by the vibrational frequency should integrate to zero over a spectral range encompassing a suite of coupled vibrational modes. The normalized integrated magnitude for each spectrum can be calculated according to Eq. 9.

Figure 4 presents previously published simulated chiral-specific SFG spectra, including the spectra of the antiparallel $\beta$-sheet proteins (L-)${\text{LK}}_7\beta$ and (L-)${\text{LE}}_7\beta$ at the vacuum-water interface for the water OH stretches (Figure 4A), the backbone NH (amide A) stretches (Figure 4B), and the NH/OH coupled spectra (Figure 4C).⁴¹ Figure 4D shows the water OH stretching spectra of the first hydration shell of a ${\text{dA}}_{12}\cdot {\text{dT}}_{12}$ duplex.⁹ Remarkably, the black traces representing the cumulative integrals F(w) of the simulated spectra converge close to zero at the high frequency end of each spectral range [i.e., $F\left({\omega }_f\right)=0\%$] in all cases except Figure 4A, which shows a small deviation from zero around 3800 cm⁻¹. This is most likely due to the limited range of integration. Nonetheless, these simulated spectra show excellent agreement with the model. These trends are noteworthy, as the only symmetry constraint imposed on the simulations was enforcement of uniaxial (${\text{C}}_{\infty }$) symmetry, and the simulations were performed using distinctly different methods than the quantum chemical calculations summarized in Figure 3 and the Supplementary Material. It is notable that the high frequency range of the water doublets in Figure 4A extends beyond what is typically expected for aqueous interfaces and approaches the frequencies anticipated for free OH stretches or water at electrode interfaces.⁴⁶ Previous work⁵ has suggested that this is due to water that has one OH forming a hydrogen bond with the backbone and one free OH group as well as water molecules near the lysine sidechains, where a positive charge disrupts local hydrogen bond networks.

Figure 4:

Simulated phase-resolved chiral-specific vibrational SFG spectra. Simulated spectra of (L-)${\text{LK}}_7\beta$ (red) and (L-)${\text{LE}}_7\beta$ (blue) for A) the water OH stretches of the first hydration shell, B) the NH stretches of the protein backbone, and C) the coupled NH and OH stretches. D) Simulated spectra of double-stranded ${\text{dA}}_{12}\cdot {\text{dT}}_{12}$ DNA (green) for the water OH stretches. The black curves represent the normalized integrated magnitude, F(ω), calculated using Eq. 9 with values given on the right axes. The spectra shown in A-C were previously published by Konstantinovsky et al⁴¹⁷ and the spectrum shown in D was previously published by Perets et al.⁹

Figure 5 presents experimental internal heterodyne chiral-specific SFG spectra of two antiparallel $\beta$-sheets, ${\text{LK}}_7\beta$ and ${\text{LE}}_7\beta$.^10,41 Figure 5A exhibits the amide I spectra of the two enantiomers of ${\text{LK}}_7\beta$,¹⁰ and Figure 5B presents the spectra containing NH stretches of the protein backbone and OH stretches of water in the hydration shells of the antiparallel ${\text{LK}}_7\beta$ and ${\text{LE}}_7\beta$ in the native (L-) form.⁴¹ Notably, the experimental spectra of ${\text{LK}}_7\beta$ and ${\text{LE}}_7\beta$ above 3400 cm⁻¹ appear to be substantially different. This is unsurprising, as ${\text{LK}}_7\beta$’s lysine residues are positively charged and ${\text{LE}}_7\beta$’s glutamate residues are negatively charged, which can influence the orientation of water molecules surrounding both the backbone and sidechains of the peptides, resulting in different water responses. In each spectrum, the values of the normalized integrated magnitude computed according to Eq. 9 over the spectral range are displayed on the right axes. The amide I spectra (Figure 5A) clearly show a doublet line shape. The integration converges close to zero at the high-frequency limit, which is in good agreement with the model. The deviation of the NH/OH spectra (Figure 5B) is similarly small, converging to normalized integrals of $~10\%$ for (L-)${\text{LK}}_7\beta$. However, the experimental results for (L-)${\text{LE}}_7\beta$ depart significantly from expectations, converging to $~-60\%$. This nonzero integration observation may not be too surprising, as the results are necessarily measured over finite spectral ranges. In the case of the NH/OH spectra, the spectral compositions can be more complex than just pure coupled NH and OH stretches because it is possible that the spectra also contain overtone contributions and Fermi resonance contributions of lower-frequency vibrations such as amide I,^47–49 NH bending,⁵⁰ and OH bending.^51–54

Figure 5:

Experimental internal heterodyne chiral-specific SFG spectra of antiparallel beta-sheets. A) The spectra for (L-)${\text{LK}}_7\beta$ (red) and (D-)${\text{LK}}_7\beta$ (purple) in the amide I region. B) The spectra for (L-)${\text{LK}}_7\beta$ (red) and (L-)${\text{LE}}_7\beta$ (blue) in the NH/OH stretching region. The black curves represent the normalized integrated magnitude, F(ω), calculated using Eq. 9 with values given on the right axes. The amide I spectra were previously published by Perets et al.,¹⁰ and the NH/OH spectra were previously published by Konstantinovsky et al.⁴¹

Figure 6 shows the experimental internal heterodyne chiral-specific SFG spectrum of a ${\text{dA}}_{12}\cdot {\text{dT}}_{12}$ DNA duplex covering the spectral regions for the CH stretches, NH stretches, and OH stretches ranging from 2800 ${\text{cm}}^{-1}$ to 3800 ${\text{cm}}^{-1}$. The integration of this spectrum converges to a relatively small but nonzero value. This residual could be due to the overtone contributions, as discussed above. Additionally, Fermi resonances are well known to contribute extra spectral signals in the CH stretching region.⁵⁵ To avoid the complexity due to this Fermi resonance, one can limit the spectral region to smaller regions and attempt to isolate more limited classes of spectral contributions. This analysis is presented in the Supplementary Material and highlights the experimental challenges in testing the proposed model from SFG spectra acquired within a finite spectral window for complex, spectrally congested samples.

Figure 6:

Experimental internal heterodyne chiral-specific SFG spectra of the double-stranded ${\text{dA}}_{12}\cdot {\text{dT}}_{12}$ DNA (green traces) in the spectral region of 2800–3800 ${\text{cm}}^{-1}$, previously published by Perets et al.⁹ The black curve represents the normalized integrated magnitude, F(ω), calculated using Eq. 9 with values given on the right axis.

The analyses above suggest that the computational spectra agree better with the proposed model than do the experimental spectra. The difference in the extent of agreement between simulation and experiment can potentially be explained by the assumptions underlying the proposed Kleinman antisymmetry model. The model predicts a value of zero for the integral over all frequencies in the chiral-specific SFG spectra, but experimental measurements are necessarily constrained to finite spectral regions, such as amide I, water OH stretches, or coupled NH/OH stretches. Contributions from overtones and Fermi resonances are often not recovered in simulations but could potentially account for residual chiral-specific signals not cancelled by integration over the experimentally accessible stretching regions. In addition, subtle experimental offsets in the baseline can also produce relatively large deviations in the integrals through compounded errors absent in the simulations. These extra spectral contributions likely account for the deviation from the model prediction. Within this context, although the analyses of experimental spectra of the proteins (Figure 5) and DNA (Figure 6) do not refute the proposed symmetry models, more chiral-specific SFG data and improved measurements are needed to affirm the proposed model. However, the analyses of simulated spectra (Figure 4) offer strong support to the proposed symmetry conjecture.

Discussion

These same index-interchange (Kleinman) antisymmetry constraints also offer possible insights into chiral-specific SFG trends observed in studies centering on other vibrational transitions. Stokes et al. have reported on the observation of a positive/negative doublet on chiral-specific vibrational SFG of surface-immobilized DNA.¹ Switching complementarity in the DNA resulted in changes in the signs of some contributions in the achiral signal and a corresponding change in the spectral interference with the chiral response. Coherent mixing of the chiral-specific and achiral polarization combinations enables isolation of a signal directly dependent on the chiral-specific SFG activity of the interface. Although a positive/negative doublet of equal amplitude was not directly observed in either of the two base-pairing configurations, the chiral/achiral interference method has the potential to affect the relative amplitudes and signs of the chiral-specific features from corresponding phase changes in the achiral features. Simple modeling of complex lineshape functions consistent with Eq. 3 detailed in the Supplementary Material recovers peak shapes that are qualitatively similar to those observed experimentally for the methyl modes of DNA studies in that prior work. As such, those results are consistent with expectations predicted from the arguments presented herein. It is worth noting that many possible alternative explanations can also potentially recover similar peak shapes, such as contributions from Fermi resonances not explicitly considered in the modeling calculations presented in this work.⁴

Similar arguments agree with prior chiral-specific SFG studies of water occupying the major/minor grooves of DNA by McDermott et al.⁶ As in the work by Stokes et al.,¹ McDermott et al. isolated the chiral-specific response either by direct detection of the squared magnitude (i.e., “homodyne” detection) or through interference with the achiral response. In that work, the authors observed spectral features in the chiral-specific SFG scaling smoothly with the achiral response rather than producing positive/negative doublets.⁶ However, sign changes across the achiral OH stretching band have been reported previously in SFG measurements^20,56,57 and modeling,⁴ depending in part on the interfacial charge density and local field. For cases in which the chiral-specific doublet inverts in sign at a crossover frequency comparable to the larger achiral sign inversion, like-signed chiral/achiral interference spectra such as those reported by McDermott et al.⁶ are anticipated to arise, despite sign inversion of the chiral-specific signal. Modeling calculations summarized in the Supplementary Material illustrate conditions capable of producing like-signed couplets from chiral/achiral interference in homodyne-detected SFG, despite an equal and opposite couplet within the imaginary chiral-specific resonant response. As such, these prior observations are not inconsistent with the Kleinman antisymmetry couplet constraints predicted in the present study.

Arguments analogous to those invoked by Franken and Ward²⁴ for index-interchangeability in adiabatic nonlinear optical interactions also define the symmetric (adiabatic) and antisymmetric contributions to the electric dipole-allowed Raman polarizability, which in turn contributes to coherent SFG in isotropic chiral liquids in noncollinear beam configurations.^26,58–61 Antisymmetry in the Raman tensor can be interpreted as arising from differences in the cross-sections for the Stokes vs. anti-Stokes transitions.⁶² The virtual states for the two different transitions differ by one vibrational quantum in energy. Under electronic resonance-enhanced conditions, this difference can be sufficient to induce relatively large antisymmetric contributions. Since the Raman matrix must converge to a symmetric matrix far from electronic resonance (i.e., in the zero-frequency limit of electronic spectroscopy), the arguments outlined for vibrational SFG apply to Raman as well, suggesting the anticipation of positive and negative couplets in the electronic resonance enhancement for the antisymmetric Raman tensors to conform with Kramers-Kronig relations. The consequences of this limiting behavior directly impact the interpretation of vibrational SFG spectroscopy of isotropic chiral media, which requires asymmetry in the Raman matrix within the electric dipole approximation.^26,58–61 Specifically, positive and negative couplets are anticipated in the resonance-enhanced isotropic SFG spectral response, in excellent agreement with the imaginary (resonant) component in isotropic-allowed electronic SFG studies of binaphthol solutions.^61,63,64

Our conjecture on the appearance of couplets for the interchange-antisymmetric tensor components is sufficiently general to apply for both chiral-specific and achiral SFG processes scaling with the Kleinman antisymmetric tensor. As such, one may ask why such observations have not been reported previously in achiral vibrational SFG studies of interfacial layers, for which a significantly larger body of work exists that also depends on resonances generally containing significant interchange antisymmetric contributions. One likely possibility is the experimental challenge of isolating observables scaling exclusively with the interchange-antisymmetric components in achiral systems. In principle, polarization-dependent analysis can be used to recover the resonance-enhanced tensor elements driving the achiral SFG, which in turn could be decomposed into interchange-symmetric and antisymmetric components. However, such isolation can be challenging in achiral assemblies. First, interchange-symmetric contributions to the resonance-enhanced hyperpolarizability are regularly significantly greater in magnitude than their antisymmetric components, complicating their experimental recovery in achiral assemblies. In addition, experimental uncertainties in local field correction factors (e.g., due to ambiguities in the complex-valued optical constants within the interfacial layer, which is generally birefringent and absorptive)⁶⁵ could further complicate definitive experimental isolation of the antisymmetric response in achiral systems. Finally, it is also possible that the spectral behavior of the index-interchange-antisymmetric response in achiral SFG measurements has not been reported previously simply because it has not yet been sought. In contrast, chiral-specific SFG provides direct and unambiguous observables scaling exclusively with the interchange-antisymmetric response (e.g., through χ_ZYX).

Conclusion

This work expands on previous investigations of the interpretation of chiral-specific sum frequency generation spectroscopy by examining the interplay between index-interchange (Kleinman) symmetry and the Kramers-Kronig relations. First, we show that the requirement of complete index interchangeability in the zero-frequency (adiabatic) limit mandates that the elements of the nonlinear susceptibility tensor that correspond to chiral-specific SFG must approach zero in the same limit. From the Kramers-Kronig relations, it follows that the resonant absorptive components of a chiral-specific SFG spectrum must integrate to zero in order to satisfy this zero-frequency limiting behavior. The previous observation of equal and opposite doublets in past chiral-specific SFG of water hydrating biomolecules is a specific example of this more general phenomenon. This claim is evaluated using simulated and experimental phase-resolved spectra from the literature. Second, we show that the hyperpolarizability of a given normal mode can be decomposed into Kleinman-symmetric and Kleinman-antisymmetric tensors. Only the Kleinman-antisymmetric tensor contributes to the observed chiral-specific nonlinear susceptibility. DFT calculations on water clusters show that while there is no apparent pattern to the total hyperpolarizabilities of coupled OH stretching modes, the antisymmetric tensors align to produce equal and opposite contributions, which would create couplets in a measured spectrum. This theoretical understanding will guide interpretation of simulated and experimental spectra from chiral-specific SFG by providing constraints on the relative amplitudes and phases of peaks used for fitting.

Although the primary focus of the present study is on vibrational SFG, generalization to lower and higher order interactions suggests analogous spectral constraints in electric dipole-allowed chiral spectroscopy broadly and in Kleinman antisymmetric spectroscopic contributions more generally. As one example, electric dipole-allowed chiral-specific coherent anti-Stokes Raman spectroscopy has been predicted in uniaxial assemblies, scaling with the asymmetry within the Raman matrix.⁶⁶ As with isotropic SFG, the resulting vibrational resonances accessible through this method are anticipated to produce oppositely signed spectral couplets. Further, as the scope of nonlinear optical methods continues to expand, the constraints described herein may aid in the interpretation of additional future experiments not yet envisioned.

Supplementary Material

The Supplementary Material is available free of charge and contains total and Kleinman antisymmetric tensor contributions for a water dimer and pentamer, as well as an analysis of all OH stretching modes of the trimer discussed in the main text. Division of the experimental ${\text{dA}}_{12}\cdot {\text{dT}}_{12}$ DNA spectrum into smaller spectral regions is explored. Also present is further discussion of the role of coupling in chiral-specific SFG and lineshapes associated with mixed-polarization spectra.

Appendix: Analytical Expressions for Kleinman Symmetric and Antisymmetric Decomposition

The analytical expressions used to decompose the resonance-enhanced hyperpolarizabilities into symmetric and antisymmetric tensors (S and A, respectively) are given below. In brief, each element in a set of tensor elements connected through index interchange was replaced by the average of the set to ensure Kleinman symmetry. For the tensor elements containing two Cartesian coordinates, (e.g., ${\beta }_{xxz},{\beta }_{xzx},{\beta }_{zxx}$), the average was performed over three terms.

$$S_{iij}=S_{iji}=S_{jii}=\frac{1}{3}\left({\beta }_{iij}+{\beta }_{iji}+{\beta }_{jii}\right)$$ (A1)

For the tensor elements containing all three cartesian coordinates, (e.g., ${\beta }_{xyz},{\beta }_{yzx},{\beta }_{zxy}$, etc.), the average was performed over six terms.

$$S_{ijk}=S_{ikj}=S_{jik}=S_{jki}=S_{kij}=S_{kji}=\frac{1}{6}\left({\beta }_{ijk}+{\beta }_{ikj}+{\beta }_{jik}+{\beta }_{jki}+{\beta }_{kij}+{\beta }_{kji}\right)$$ (A2)

In Eqs. A1 and A2, $i,j,k=\{x,y,z\}$, with $i\ne j\ne k$. Once the Kleinman symmetric tensor has been determined, the subsequent isolation of the Kleinman antisymmetric tensor A was achieved by subtraction of the symmetric tensor S from the original full $\mathbf{\beta }$ tensor.

Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.