Reducing Non-Linearity in Spectral Evaluation via a Modified Lorentz

Abstract

Quantitative infrared spectroscopy of liquids often assumes linear Beer–Lambert behavior, although local-field interactions and dipole–dipole coupling introduce intrinsic nonlinearities. We investigate whether a modified Lorentz–Lorenz relation, motivated by electromagnetic theory, can reduce these deviations and restore approximate linearity in binary mixtures. Using benzene–toluene and benzene–cyclohexane as model systems, we evaluate linearity through RMSE metrics, hybrid 2D-correlation analysis, asynchronous residual sums of squares, and complex-valued classical least squares (CLS) regression. While correlation-based measures provide qualitative insights, they fail to reliably identify optimal linearization parameters. In contrast, CLS in the Lorentz–Lorenz–transformed domain, particularly with systematic-error correction, yields clear minima and substantially improves prediction accuracy, reducing mean absolute errors by more than a factor of three relative to the Beer–Lambert domain. These results demonstrate that the modified Lorentz–Lorenz transformation provides a physics-informed chemometric domain in which spectral mixing becomes more linear. The approach suggests extending inverse least square regression, principal component regression, and partial least squares regression into this domain to further enhance quantitative mixture analysis.

Graphical abstract

This is a visual representation of the abstract.

Keywords

Two-dimensional correlation spectroscopy 2D-COS chemometrics complex refractive index liquid mixtures

Introduction

The quantitative interpretation of light absorption is historically rooted in the Bouguer–Beer–Lambert (BBL) law, one of the earliest empirical attempts to relate optical attenuation to material properties. Its origin reaches back to the 18th century, when Pierre Bouguer observed that light intensity decreases exponentially with path length in a homogeneous absorbing medium.¹ Johann Heinrich Lambert formalized this relation in his Photometria (1760),² introducing the expression,

I = I_{0} e^{- μ d}

(1)

where

I_{0}

and

I

denote the light intensities at the entrance and exit of a slab of thickness

d

, and

μ

is the absorption coefficient. This formulation, now known as the Bouguer–Lambert law, assumes propagation of a plane wave in an infinite, homogeneous medium, free from interfaces or scattering boundaries. August Beer extended the concept in 1852 by studying colored solutions.³ He showed that when the product of solute volume fraction and cuvette thickness remains constant, the transmission remains unchanged. This enabled quantitative spectroscopic analysis of solutions by introducing an empirical dependence of attenuation on solute fraction. However, the familiar linear relation

A = ε cd

(2)

linking absorbance

A

to molar concentration

c

, path length

d

, and molar absorption coefficient

ε

, was formulated only later, around 1913, most likely by Luther and Nikolopulos.⁴

Although the BBL law became foundational in analytical chemistry, it is not exact from the standpoint of electromagnetic theory. Bouguer and Lambert’s exponential decay is strictly valid only for a plane wave traveling through an infinite medium without boundaries. In real condensed materials, interfaces between regions of different refractive index modify amplitude and phase through Fresnel reflection and refraction, leading to deviations from simple exponential behavior.^5–9

For ideal gases, however, the assumptions underlying the Bouguer–Lambert law are well satisfied. The refractive index is close to unity, scattering is negligible, and intermolecular interactions are weak. Each molecule experiences an almost homogeneous field, and both light-induced and permanent dipole–dipole interactions can typically be ignored.¹⁰ Hence, the Bouguer–Lambert relation and Beer’s extension hold with high accuracy in low-pressure gas-phase spectroscopy.

In condensed matter the situation differs fundamentally.^5,10 Shortly after Beer’s death, Maxwell’s electromagnetic field equations (1865),¹¹ established the theoretical basis for light–matter interaction. Building on this framework, Hendrik Antoon Lorentz developed a microscopic model linking the macroscopic dielectric function to the microscopic polarizability of atoms and molecules.¹² Independently, he derived the Clausius–Mossotti or Lorentz–Lorenz relation,

\frac{n^{2} - 1}{n^{2} + 2} = \frac{4 π}{3} N α

(3)

where

n

is the refractive index,

N

the number density of polarizable units, and

α

their polarizability. The form shown corresponds to Gaussian (cgs) units, in which the factor 4π appears explicitly; in SI units, the right-hand side would be written as

N α / (3 ε_{0})

. This relation shows that the local electromagnetic field experienced by a molecule differs from the external field, because each induced dipole interacts with those of its neighbors. In 1904, Planck explicitly demonstrated that Beer's law is only an approximation, valid for sufficiently dilute systems in which such local-field effects can be neglected.¹³

The Lorentz–Lorenz relation became a cornerstone for the modern electrodynamics of materials. It guided the development of effective-medium models such as the Maxwell–Garnett (1904) and Bruggeman (1935) formulations,^14,15 which describe how the effective permittivity of composite media arises from electromagnetic coupling between constituents. These models explicitly incorporate local-field effects and mutual polarization and remain essential for modeling mixtures, suspensions, and structured materials.¹⁰

The same physical principle, electromagnetic coupling among polarizable entities, also underlies intermolecular forces. London showed in 1930 that instantaneous correlations of fluctuating dipoles produce dispersion forces.¹⁶ More broadly, modern quantum field theory interprets these as arising from vacuum fluctuations of the electromagnetic field. This deep connection between optical dispersion, local fields, and intermolecular forces is emphasized in Paragraph 32 of the Feynman Lectures on Physics,¹⁷ where the Lorentz–Lorenz relation is used to illustrate how microscopic coupling governs macroscopic optical behavior.

Despite this theoretical lineage, most modern molecular spectroscopy diverges from it. In routine practice, condensed-phase systems are often interpreted within the weak-coupling approximation valid for gases. Absorbers are treated as independent, and absorbances are assumed to be additive. This simplification, while convenient, is physically inconsistent for liquids and liquid mixtures, where local-field effects, molecular anisotropy, and microscopic structure strongly influence the electromagnetic response.

Recognizing these limitations, the present work aims to reconnect spectroscopic analysis with its electrodynamic foundation. Because even thermodynamically ideal liquid mixtures cannot be described by a single universal expression, we investigate whether a modified Lorentz–Lorenz relation can better represent their optical behavior. We focus on binary mixtures such as benzene–toluene and benzene–cyclohexane, where refractive index contrasts and local interactions are moderate but measurable. Using hybrid 2D-correlation analysis and complex-valued classical least-squares (CLS) regression,¹⁸ we examine how deviations from Lorentz–Lorenz behavior reflect the underlying light–matter coupling. In doing so, this work seeks to build a bridge from Bouguer, Lambert, and Beer to Maxwell, Lorentz, and Planck—and ultimately to a physics-informed chemometrics framework that accounts for the collective electromagnetic interactions governing even simple liquid mixtures.

Theoretical Aspects

Spectral Mixing in Liquids and Motivation for a Modified Lorentz–Lorenz Relation

Here, $φ_{1}$ and $φ_{2} = 1 - φ_{1}$ denote the volume fractions of components 1 and 2. The index $q$ labels the different spectral representations (experimental spectra, Beer–Lambert linear combinations, Lorentz–Lorenz–transformed spectra, etc.). Each spectrum is sampled at $n$ wavenumbers ${\tilde{ν}}_{i}$ and measured for $m$ mixture compositions. Throughout the manuscript, a tilde indicates wavenumber dependence, and a hat denotes a complex quantity, e.g., $\hat{n} (\tilde{ν}) = n (\tilde{ν}) + ik (\tilde{ν})$ , where $i$ is the imaginary unit and $k (\tilde{ν})$ is the absorption index.

In condensed phases, the optical response cannot be described by simply adding the contributions of isolated molecules. Even in thermodynamically ideal mixtures, the macroscopic refractive index reflects a collective response in which each polarizable unit interacts with the local electromagnetic field generated by its neighbors. For gases under low pressure, these interactions are negligible, but for liquids they introduce systematic deviations from Beer–Lambert behavior.

The Lorentz–Lorenz relation,^12,19,20

\frac{{\hat{n}}^{2} - 1}{{\hat{n}}^{2} + 2} = \sum_{i} \frac{{\hat{n}}_{i}^{2} - 1}{{\hat{n}}_{i}^{2} + 2} φ_{i}

(4)

connects the macroscopic refractive index to the microscopic polarizability by assuming a spherical Lorentz cavity and isotropic surroundings. However, real molecules in liquids exhibit anisotropy, packing constraints, and non-spherical solvation environments. These deviations modify the local field and therefore the optical response. To account for this, we introduce the modified Lorentz–Lorenz relation:

\frac{{\hat{n}}^{2} - 1}{{\hat{n}}^{2} + η} = \sum_{i} \frac{{\hat{n}}_{i}^{2} - 1}{{\hat{n}}_{i}^{2} + η} φ_{i}

(5)

In the present work, the term Lorentz–Lorenz transformation does not denote a new physical law. It simply refers to rewriting the experimentally determined complex refractive index function $\hat{n} (\tilde{ν})$ in terms of the Lorentz–Lorenz expression

F (\tilde{ν}) = \frac{\hat{n} {(\tilde{ν})}^{2} - 1}{\hat{n} {(\tilde{ν})}^{2} + η}

(6)

and using this quantity as the spectral variable for subsequent analysis. Thus, the Lorentz–Lorenz relation (Eq. 3) is the physical equation linking refractive index and polarizability, whereas the Lorentz–Lorenz transformation denotes the mathematical mapping

\hat{n} (\tilde{ν}) \to F (\tilde{ν})

(7)

applied to measured spectra for the purpose of linearizing mixture behavior.

In the classical derivation of the Lorentz–Lorenz relation, the denominator term $n^{2} + 2$ arises from the local-field correction associated with a spherical Lorentz cavity, which yields a depolarization factor of $1 / 3$ . For a general depolarization factor $L$ , the denominator term becomes $n^{2} + \frac{1 - L}{L}$ , which reduces to $n^{2} + 2$ only for the spherical case $L = 1 / 3$ . Modifying this denominator therefore corresponds physically to relaxing the assumption of a spherical, isotropic cavity and replacing the fixed depolarization factor by an effective one.¹⁰ The adjustable parameter η is thus introduced specifically in the denominator because this term encodes the local-field contribution; altering the numerator would instead correspond to changing the intrinsic molecular polarizability, which is not the purpose of the present modification.

Physical Meaning of η

The parameter η introduced in Eq. 5 can therefore be interpreted as an effective depolarization factor that accounts for deviations from the ideal spherical Lorentz cavity. In condensed phases, however, real molecules frequently exhibit:

(i) anisometric molecular shapes,

(ii) anisotropic polarizabilities,

(iii) non-spherical solvation shells,

(iv) packing-induced field inhomogeneities,

(v) short-range order

These factors modify the local electromagnetic field acting on each molecule. The parameter $η$ effectively tunes the relationship between the microscopic and macroscopic fields, thereby compensating for these deviations. It is unitless, like $n$ and k, and serves as a phenomenological measure of how strongly molecular anisotropy, packing, and intermolecular interactions perturb the ideal Lorentz field.

Although $η$ is not uniquely tied to a specific microscopic parameter, the value that minimizes a suitable linearity criterion correlates with the degree of local-field distortion in a given mixture and thus indicates the extent to which the Lorentz–Lorenz model must be modified to linearize the spectral response.

Although η originates from depolarization considerations analogous to effective-medium models, it should not be regarded as a universal constant of the constituents themselves. A binary mixture defines a specific local-field environment that depends on molecular anisotropy, packing constraints, and the solvation structure of both components. Therefore, η is system-specific rather than component-specific; it need not be transferable between two different mixtures that share one of the endmembers. Formally, η could vary with wavenumber because different normal modes experience different local-field distortions. In this work, however, η is treated as constant to obtain a single effective linearization parameter.

We consider a set of m spectra recorded for systematically varied mixture compositions. In the terminology of two-dimensional correlation spectroscopy, such a composition-dependent spectral series is referred to as dynamic spectra. Here, the perturbation variable is the volume fraction $φ_{1}$ of component 1 (not time). We denote these dynamic spectra by ${\tilde{y}}_{q} = {\tilde{y}}_{q} ({\tilde{ν}}_{i}, φ_{1})$ , sampled at wavenumbers ${\tilde{ν}}_{i}$ . The index $q$ distinguishes different spectral representations used throughout this work (experimental spectra and corresponding model spectra in either the Beer–Lambert or modified Lorentz–Lorenz domain). We define four forms of dynamic spectra:

(i) the experimentally determined complex refractive indices, possibly offset by the refractive index of the second endmember, ${\hat{n}}_{2}$ (q = 1),

(ii) linear combinations such as ${\hat{n}}_{1} φ_{1} + {\hat{n}}_{2} φ_{2}$ , possibly offset by ${\hat{n}}_{2}$ (q = 2),

(iii) terms of the form $\frac{{\hat{n}}^{2} - 1}{{\hat{n}}^{2} + η}$ , possibly offset by $\frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η}$ (q = 1),

(iv) linear combinations such as $\frac{{\hat{n}}_{1}^{2} - 1}{{\hat{n}}_{1}^{2} + η} φ_{1} + \frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η} φ_{2}$ , possibly offset by $\frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η}$ (q = 2).

For clarity, q = 1 denotes experimental spectra in a given domain and q = 2 the corresponding linear model spectra.

Each spectrum consists of n spectral points at the wavenumbers ${\tilde{ν}}_{i}$ . The different dynamic spectra are arranged into a matrix Y_q as follows:

Y_{q} = [\begin{matrix} {\tilde{y}}_{q} ({\tilde{ν}}_{1}, φ_{1}^{(1)}) & {\tilde{y}}_{q} ({\tilde{ν}}_{2}, φ_{1}^{(1)}) & \dots & {\tilde{y}}_{q} ({\tilde{ν}}_{n}, φ_{1}^{(1)}) \\ {\tilde{y}}_{q} ({\tilde{ν}}_{1}, φ_{1}^{(2)}) & {\tilde{y}}_{q} ({\tilde{ν}}_{2}, φ_{1}^{(2)}) & \dots & {\tilde{y}}_{q} ({\tilde{ν}}_{n}, φ_{1}^{(2)}) \\ \dots & \dots & \dots & \dots \\ {\tilde{y}}_{q} ({\tilde{ν}}_{1}, φ_{1}^{(m)}) & {\tilde{y}}_{q} ({\tilde{ν}}_{2}, φ_{1}^{(m)}) & \dots & {\tilde{y}}_{q} ({\tilde{ν}}_{n}, φ_{1}^{(m)}) \end{matrix}]

(8)

In this work, linearity refers to the condition that the spectral response is a linear function of the perturbation variable (volume fraction φ₁), meaning that mixture spectra can be represented as linear combinations of pure-component spectra in a given spectral representation (e.g., absorbance, refractive index, or Lorentz–Lorenz domain). For the present analysis, the real and imaginary parts of the complex refractive index, $n (\tilde{ν}, φ_{1})$ and $k (\tilde{ν}, φ_{1})$ , are treated as separate spectral datasets. Conventional 2D correlation analysis is applied independently to each of these quantities, without introducing any modification of the underlying formalism.

RMSE-Based Linearity Measure

Deviations from linearity can be quantified via the RMSE,

Δ Y_{R M S E} = \sqrt{\frac{1}{m \cdot n} \sum_{i} \sum_{j} {| Y_{1, i j} - Y_{2, i j} |}^{2}}

(9)

This measure is simple but insensitive to phase information in complex-valued spectra.

2D Correlation-Based Measures (SES1–SES3)

As an alternative to magnitude-based metrics, 2D correlation-based measures can also be employed. From the matrices Y_q, one can compute the corresponding variance–covariance matrices,

Φ_{Y_{1} Y_{2}} = \frac{1}{m - 1} Y_{1}^{T} Y_{2}

(10)

which are widely used in hybrid correlation analysis, where two series of spectra are compared. In the present context, for the conventional smart error sum approach, one series corresponds to the experimental spectra and the other to the corresponding simulated spectra.^21–23

As pointed out in Noda and Ozaki,²⁴ each element of the variance–covariance matrix expresses the similarity between an experimental and a simulated intensity variation at different wavenumbers. In order to calculate the asynchronous spectrum, the Hilbert–Noda transformation matrix N must first be determined.²⁴ Its elements are defined as:

N_{x y} = {\begin{matrix} 0 if x = y \\ \frac{1}{π (y - x)} otherwise \end{matrix}

(11)

The elements of the asynchronous spectra can then be calculated as,

Ψ_{Y_{1} Y_{2}} = \frac{1}{m - 1} Y_{1}^{T} N Y_{2}

(12)

For conventional 2D correlation spectra or maps, certain symmetry relations apply. Specifically, the asynchronous spectra are always antisymmetric with respect to the diagonal elements $Ψ ({\tilde{ν}}_{i}, {\tilde{ν}}_{i})$ :

\begin{matrix} Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k}) = - Ψ ({\tilde{ν}}_{k}, {\tilde{ν}}_{j}) \\ \to \sum_{k = 1}^{l} \sum_{j = k}^{l} [Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k}) + Ψ ({\tilde{ν}}_{k}, {\tilde{ν}}_{j})] = 0 \end{matrix}

(13)

Here,

Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k})

denotes the element in row j and column k of the asynchronous map. It should be noted that, in contrast to conventional 2D correlation spectroscopy, the present formulation corresponds to hybrid correlation between two spectral datasets Y₁ and Y₂ following Refs. 21–23.

For the hybrid 2D-correlation case, this condition becomes (SES1),

D_{A}^{p} = \sum_{k = 1}^{l} \sum_{j = k}^{l} [Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k}) + Ψ ({\tilde{ν}}_{k}, {\tilde{ν}}_{j})]^{p}

(14)

For conventional 2D correlation (Y₁ = Y₂), antisymmetry holds exactly and

D_{A}^{p} = 0

In the hybrid case (Y₁ ≠ Y₂), antisymmetry is fulfilled only if the experimental spectra are proportional to the model spectra, i.e., $Y_{1} = a Y_{2}$ ; otherwise, $D_{A}^{p} \neq 0$ .

In the present notation, the matrices $Y_{1}$ and $Y_{2}$ contain the spectral values at each wavenumber and mixture composition. Specifically, $Y_{1}$ contains the experimentally determined quantities (e.g., $\hat{n} (φ_{1})$ or their Lorentz–Lorenz–transformed counterparts), while $Y_{2}$ contains the corresponding model spectra constructed from the pure-component quantities ${\hat{n}}_{1}$ and ${\hat{n}}_{2}$ according to the assumed linear mixing rule. Thus, each matrix element corresponds directly to the spectral expressions involving ${\hat{n}}_{1}$ , ${\hat{n}}_{2}$ , and the volume fractions $φ_{1}$ and $φ_{2}$ shown below.

If we set p = 2, the term $D_{A}^{2}$ represents a special form of the residual sum of squares, referred to as the asynchronous residual sum of squares (ARSS).^21–23 Depending on how the entries of Y_q, are defined, two situations are of interest. When the values of the second endmember have been subtracted, one obtains for example:

\underset{\hat{n}}{\underset{⏟}{{\hat{n}}_{1} φ_{1} + {\hat{n}}_{2} φ_{2}}} - {\hat{n}}_{2} = {\hat{n}}_{1} φ_{1} + {\hat{n}}_{2} (1 - φ_{1}) - {\hat{n}}_{2} = ({\hat{n}}_{1} - {\hat{n}}_{2}) φ_{1}

(15)

In the ideally linear case, the entries depend only on

φ_{1}

; all variations of

\hat{n}

with

φ_{1}

are perfectly in phase. Consequently, the asynchronous spectrum vanishes if the experimental values of

\hat{n}

are truly linear combinations of

{\hat{n}}_{1}

and

{\hat{n}}_{2}

. This holds independently for the real and imaginary parts, and therefore

D_{A}^{2}

becomes zero. In all other cases,

D_{A}^{2}

is supposed to be a measure of linearity. In this case, the entries for Y₁ and Y₂ must be identical and Eq. 14 must be replaced by (SES2):

D_{A}^{2} = \sum_{k = 1}^{l} \sum_{j = 1}^{l} [Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k})]^{2}

(16)

Such a situation is generally not expected for liquids or solids. Instead, we assume that the following entries,

\underset{\frac{{\hat{n}}^{2} - 1}{{\hat{n}}^{2} + η}}{\underset{⏟}{\frac{{\hat{n}}_{1}^{2} - 1}{{\hat{n}}_{1}^{2} + η} φ_{1} + \frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η} φ_{2}}} - \frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η}

(17)

enable minimization of

D_{A}^{2}

for a given value of η. From Eqs. 15 and 17, the entries of the matrices Y₁ and Y₂ used in the hybrid 2D-correlation analysis can be written explicitly in terms of the pure-component quantities and the volume fraction φ₁. Substituting these expressions into Eqs. 10–12 yields the corresponding hybrid synchronous and asynchronous spectra. In this formulation, deviations from linearity arise directly from the nonlinear dependence of the transformed spectral quantities on φ₁, which manifests as non-zero asynchronous contributions.

As an alternative, if the second endmember spectrum is not subtracted, $D_{A}^{2}$ from Eq. 14 itself serves as a measure of linearity. In this case, the entries of the asynchronous map $Ψ$ do not become zero. The antisymmetry condition of Eq. 13 is fulfilled only if the entries of Y₁ are proportional to those of Y₂. Specifically, Y₁ contains entries of the form $\frac{\hat{n} {(φ_{1})}^{2} - 1}{\hat{n} {(φ_{1})}^{2} + η}$ while Y₂ contains the corresponding linear combination model $\frac{{\hat{n}}_{1}^{2} - 1}{{\hat{n}}_{1}^{2} + η} φ_{1} + \frac{{\hat{n}}_{2}^{2} - 1}{{\hat{n}}_{2}^{2} + η} (1 - φ_{1})$ .

A further extension can be introduced in the form of the 2T2D smart error sum (hybrid two-trace 2D-COS). In this approach, one spectrum (“s”) corresponds to the experimental spectrum and the other (“m”) to the modelled spectrum. Both hybrid 2D-COS and 2T2D therefore compare the experimentally measured mixture spectra (contained in $Y_{1}$ ) with spectra generated from the pure-component optical constants ${\hat{n}}_{1}$ and ${\hat{n}}_{2}$ via the assumed mixing rule (contained in $Y_{2}$ ). The difference between the two approaches lies only in how this comparison is organized: Hybrid 2D-COS operates on the full dynamic matrices, whereas 2T2D evaluates paired experimental and model traces at each composition. The asynchronous spectrum or map is then calculated as

Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k}) = \frac{1}{2} [s ({\tilde{ν}}_{j}) \cdot m ({\tilde{ν}}_{k}) - s ({\tilde{ν}}_{k}) \cdot m ({\tilde{ν}}_{j})]

(18)

According to Eq. 18, the hybrid synchronous map is always symmetric, while the hybrid asynchronous map is always antisymmetric. In analogy to Eq. 16 we can define (SES3)

D_{A 2 T}^{2} = \sum_{k = 1}^{l} \sum_{j = k}^{l} [Ψ ({\tilde{ν}}_{j}, {\tilde{ν}}_{k})]^{2}

(19)

and sum the resulting

D_{A 2 T}^{2}

values over all volume fractions to obtain an additional quantitative measure of linearity.

Complex-Valued CLS Regression

Finally, linearity can also be assessed by applying classical least-squares regression and comparing the mean absolute error (MAE) or the RMSE as a function of η. To this end, we employ complex-valued CLS, as introduced in Mayerhöfer et al.,¹⁸ both with and without error correction. The mathematical framework of complex-valued CLS is identical to that of conventional CLS, except that the Hermitian transpose (i.e., the complex conjugate of the transpose) must be used. The error correction relies on the fact that the volume fraction is inherently a real quantity but becomes complex in the presence of systematic errors, i.e., deviations from linearity. The imaginary parts of the estimated volume fractions closely resemble the deviations between the calculated real parts and their corresponding calibration values, differing only by a common proportionality factor. Hence, by applying a leave-one-out (LOO) scheme and calibrating this common factor, linearity can be improved by subtracting the determined systematic error. The improvement in linearity is then evaluated as a function of η.

Experimental

Fourier transform infrared (FT-IR) spectra of benzene–toluene and benzene–cyclohexane mixtures were recorded using a Thermo Scientific Nicolet iS50 FT-IR spectrometer equipped with an integrated diamond attenuated total reflection (ATR) accessory. The toluene volume fraction varied from 0 to 1 ( $φ_{1} = 0, 0.1, 0.2, \dots, 1$ ). Spectra were acquired over the range 400–4000 cm⁻¹ with a resolution of 4 cm⁻¹ and subsequently interpolated to 0.5 cm⁻¹. Further experimental details are provided in Mayerhöfer et al.²⁵

The recorded ATR spectra correspond to reflectance at 45° under unpolarized illumination. To retrieve the complex refractive index function $\hat{n} (\tilde{ν}) = n (\tilde{ν}) + i k (\tilde{ν})$ , we applied the non-iterative ATR correction formalism described in Mayerhöfer et al.²⁶ The procedure can be summarized as follows. In brief, the effective polarization angle of the ATR accessory is determined, and the measured reflectance spectrum is converted into the s-polarized reflectance using the analytical relation valid for 45° incidence. The phase spectrum is then obtained via the modified Bertie–Lan Kramers–Kronig transformation including the appropriate correction terms. From the amplitude and phase, the complex refractive index function is calculated directly. Independently determined values of $n_{\infty}$ serve as boundary information to ensure physically consistent reconstruction of the optical constants. A detailed step-by-step derivation of the ATR correction and reconstruction of the complex refractive index is provided in the Supplemental Material.

The values of n_∞ for pure benzene and toluene were taken from Myers et al.,²⁷ while for the mixtures, n_∞ was calculated using the Lorentz–Lorenz relation.^18,28 For cyclohexane, n_∞ was extrapolated from experimental visible range data using the Sellmeier equation.²⁹ The same procedure was applied to its mixtures, analogous to the benzene–toluene system. All computations were performed using custom-developed Mathematica programs.

More generally, it is important to recognize that extracting the complex refractive index from infrared spectra involves solving an inverse electromagnetic problem. In infrared spectroscopy, quantitative interpretation ultimately requires access to the complex refractive index $\hat{n} (\tilde{ν}) = n (\tilde{ν}) + i k (\tilde{ν})$ , which represents the fundamental optical response of the material. Different experimental techniques (e.g., transmission, ATR, specular reflectance, ellipsometry) provide indirect measurements from which $\hat{n}$ must be reconstructed by solving an inverse electromagnetic problem.

In the present work, ATR spectra are used in combination with a Fresnel-based, Kramers–Kronig-consistent reconstruction scheme (Mayerhöfer et al.²⁶) to obtain $\hat{n} (\tilde{ν})$ . The determination of the complex refractive index thus follows a standard inverse electromagnetic formalism linking reflectance, phase, and optical constants. For clarity, the procedure is summarized here as: (i) determination of the effective polarization state, (ii) conversion to s-polarized reflectance, (iii) Kramers–Kronig phase retrieval, and (iv) calculation of the complex refractive index from amplitude and phase. The underlying formalism is described in detail in Mayerhöfer et al.,²⁶ while a concise step-by-step summary of the procedure, including the key equations for each step, is provided in the Supplemental Material (Section S1). However, the subsequent analysis is not specific to ATR; any method that yields physically consistent optical constants can serve as input for the modified Lorentz–Lorenz transformation. The Lorentz–Lorenz relation itself is not restricted to binary systems and can be extended to multicomponent mixtures. However, the present implementation employs a single effective parameter η, whose transferability and adequacy for systems with more than two components remain to be systematically investigated.

A comprehensive treatment of the underlying inverse electromagnetic problem and its connection to wave optics has been discussed in detail elsewhere (see Ref. 10 and references therein) and is not reproduced here.

Results and Discussion

Figure 1 shows hybrid 2D-correlation spectra derived from the real or imaginary parts of the experimental complex refractive index for the benzene–cyclohexane system. In the upper row, the experimental spectra are compared with their corresponding Beer–Lambert linear combinations. In the lower row, the Lorentz–Lorenz–transformed experimental spectra are compared with the respective linear combinations of the Lorentz–Lorenz terms of the pure-component spectra. For the Lorentz–Lorenz transformation, we chose η = 0.45, which—as shown later—minimizes the nonlinearity.

Figure 1.

Hybrid 2D-correlation spectra obtained by correlating experimental spectra (Y₁) with corresponding model spectra (Y₂) for the benzene–cyclohexane system, based on the real or imaginary parts of the complex refractive index. In the upper row (Beer–Lambert domain), Y₁ contains the experimental spectra, while Y₂ contains spectra constructed from the experimental pure-component spectra using the Beer–Lambert linear mixing rule. In the lower row (modified Lorentz–Lorenz domain), Y₁ contains the Lorentz–Lorenz-transformed experimental spectra, while Y₂ contains the corresponding linear combinations of the transformed pure-component spectra. The parameter η = 0.45 is used for the Lorentz–Lorenz transformation.

Inspection of the spectra shows that neither approach satisfies the criterion for linearity, namely that points above and below the diagonal (from lower to higher wavenumbers) must be strictly antisymmetric with the diagonal acting as the mirror plane. When considering only features associated with the absorption bands in the imaginary part, the antisymmetry condition appears better fulfilled after applying the Lorentz–Lorenz transformation. This does not hold for the real-part spectra, which exhibit a larger number of features with greater structural complexity. Overall, it seems practically impossible to extract reliable quantitative information from these spectra alone. In Figure 1, antisymmetry of the hybrid asynchronous map is the strict criterion for linear Beer–Lambert behavior because the spectra are compared directly without subtraction. In this representation, deviations from perfect antisymmetry directly indicate nonlinear mixing.

Figure 2 suggests that at least semi-quantitative differences are visible. As in Figure 1, these spectra are obtained from hybrid correlation between experimental and model datasets (Y₁ ≠ Y₂). However, due to the representation used here, antisymmetry is no longer the appropriate criterion. Here, intensity, rather than antisymmetry, is the relevant criterion. This is expected, because spectra based solely on real or imaginary parts obey perfect antisymmetry, whereas hybrid real/imaginary spectra do not. Displaying the hybrid spectrum in the way used in Figure 1 would therefore not be meaningful, since hybrid asynchronous maps are inherently non-antisymmetric. Instead, decreasing nonlinearity manifests as decreasing intensity, which becomes apparent when the modified Lorentz–Lorenz transformation is applied. Nevertheless, whether an optimal η exists cannot be determined reliably from visual inspection alone. It must also be noted that, by the structure of the modified Lorentz–Lorenz relation, the magnitude of all terms decreases for decreasing η, because the denominator increases. Thus, any intensity decrease is only meaningful if it occurs for η < 2.

Figure 2.

Hybrid 2D-correlation spectra obtained by correlating experimental spectra (Y₁) with corresponding model spectra (Y₂) for the benzene–cyclohexane system, based on the real or imaginary parts of the complex refractive index. In the upper row (Beer–Lambert domain), Y₁ contains the experimental spectra (each offset by the spectrum of cyclohexane), while Y₂ contains spectra constructed from the experimental pure-component spectra using the Beer–Lambert linear mixing rule. In the lower row, Y₁ contains the modified Lorentz–Lorenz-transformed experimental spectra (each offset accordingly), while Y₂ contains the corresponding linear combinations of the transformed pure-component spectra.

To compare the different measures of nonlinearity introduced above, we evaluate the RMSE-based and 2D-correlation-based criteria. The SES3 measure corresponds to the 2T2D formulation introduced in Eq. 19 and therefore reflects the two-trace comparison between experimental and model spectra. Figure 3 shows that these expectations are fulfilled, at least for the benzene–toluene system. All methods predict a minimum around $η \approx 0.75$ . Although the exact position differs slightly among the methods, the results for the real and imaginary parts are essentially identical. All correlation-based methods agree on a maximum near $η \approx 2$ , whereas the RMSE-based method predicts a maximum around $η \approx 4$ . This overall agreement is encouraging. For benzene–cyclohexane, the 2D-correlation-based indicators decrease monotonically with η and do not show a distinct minimum. One might incorrectly interpret this as evidence for near-linearity of this system. However, the CLS results in Figure 4 clearly exhibit a sharp minimum of the regression error as a function of η. This reveals that the system is not close to linear; instead, it indicates that spectral measures of nonlinearity (based on shape deviations) are not equivalent to measures based on regression accuracy in the perturbation variable. CLS quantifies the deviation of the estimated volume fraction from its true value, whereas 2D-based metrics quantify deviations in spectral structure. The fact that only CLS reveals a minimum suggests that minimizing systematic chemometric error exposes nonlinear effects that purely spectral measures do not currently reveal.

Figure 3.

Comparison of the different methods proposed in this work for quantitatively assessing the degree of linearity. RMSE refers to the conventional approach defined in Eq. 9, whereas the 2D-correlation-based methods are denoted SES1-3 (SES1: Eq. 14, SES2: Eq. 16, and SES3: Eq. 19, corresponding to the 2T2D formulation). The comparison is performed for the benzene–toluene and benzene–cyclohexane systems, and separately for the real and imaginary parts of the complex refractive index as well as for the modified Lorentz–Lorenz terms. All curves are normalized to their maximum value for easier comparison.

Figure 4.

Results of complex-valued CLS regression with (black curves) and without error correction (red and green curves) in the systems benzene–toluene and benzene–cyclohexane in dependence of η. For comparison, results are given for CLS and ILS based on the complex-valued CLS and ILS of the complex refractive index function and its real and imaginary part alone with and without error correction.

Although the real and imaginary parts of the complex refractive index are connected by the Kramers–Kronig (KK) relations, complex-valued regression does not simply duplicate information. The KK transformation used here relies not only on the infrared absorption index but also on independent information about n_∞ from the visible range. Its variation with volume fraction is not contained in the absorption spectrum and introduces additional physical information related to the high-frequency response of the mixture. Furthermore, in regression, real and imaginary components behave differently under systematic error, noise, and collinearity. Even when no fundamentally new information is added, the redundancy created by using both components allows systematic deviations to become visible in the imaginary part of the coefficients, enabling their empirical correction. Thus, the complex representation assists chemometrics not primarily by increasing information content, but by creating a domain in which deviations from linearity manifest in a correctable and numerically advantageous manner. This observation shifts the emphasis from purely spectral considerations to regression performance with respect to the perturbation variable itself. At this point, we shifted our focus to a more pragmatic question: Why do we want the nonlinear terms to be as small as possible? The answer is the applicability of the Beer–Lambert approximation and, ultimately, the avoidance of systematic errors when using classical chemometric regression methods. In this spirit, we define the optimal η as the value that minimizes the regression error with respect to the perturbation variable, i.e., the volume fraction. Since the modified Lorentz–Lorenz relation is expected to be linear in this parameter, classical least-squares regression can be used. The corresponding results are shown in Figure 4.

This pragmatic approach substantially improves CLS performance not only for benzene–toluene but also for benzene–cyclohexane, despite the latter not showing favorable behavior in Figure 3. For benzene–toluene, the mean absolute error (MAE) of complex-valued CLS based on the complex refractive index is reduced to less than one third. With error correction, the minimum MAE becomes comparable to that achieved by ILS and conventional PLS, while complex-valued PLS yields a slightly lower MAE. Similar improvements are observed for benzene–cyclohexane. In this system, PLS applied to the real part (refractive index) alone can even outperform conventional PLS because the refractive-index contrast at high wavenumbers is significantly larger than in the benzene–toluene system. For this system, as well as for benzene–CCl₄ (not shown), linearization using the modified Lorentz–Lorenz relation combined with error correction produces MAE values comparable to those obtained from PLS based solely on the absorption index. These results demonstrate that the modified Lorentz–Lorenz relation, together with complex-valued CLS and error correction, offers a practical and effective route toward minimizing systematic errors in spectroscopic mixture analysis.

Practical Workflow

Based on the above considerations, the following workflow summarizes the proposed approach:

Acquire ATR spectra of calibration mixtures spanning the relevant composition range, as well as the corresponding pure-component spectra.

Apply a rigorous ATR correction to the measured spectra in order to retrieve the complex refractive index function $\hat{n} (\tilde{ν}) = n (\tilde{ν}) + i k (\tilde{ν})$ . In this procedure, independently determined values of $n_{\infty}$ are used as boundary information, ensuring physically consistent reconstruction of the optical constants.

Apply the modified Lorentz–Lorenz transformation (Eq. 5) to all calibration and unknown spectra.

Perform regression (e.g., CLS, ILS, or PLS) in the transformed domain. The optimal value of η can be selected by minimizing cross-validated prediction error, as demonstrated in Figure 4. In this framework, the concentration (volume fraction φ₁) is obtained directly from the regression model applied in the Lorentz–Lorenz–transformed domain, analogous to conventional Beer–Lambert-based calibration but with improved linearity of the spectral response.

Apply the resulting calibrated model to unknown samples measured under identical ATR conditions.

Conclusion

In this work, we revisited the problem of spectral linearity in condensed-phase mixtures by explicitly incorporating light–matter coupling through a modified Lorentz–Lorenz relation. Using benzene–toluene and benzene–cyclohexane as representative systems, we systematically evaluated several strategies for quantifying nonlinearity, including RMSE-based metrics, hybrid 2D-correlation analysis, asynchronous residual sums of squares, and complex-valued CLS regression with and without error correction.

Our results show that 2D-correlation-based measures capture qualitative signatures of nonlinearity but do not reliably reveal an optimal depolarization parameter η, particularly for systems such as benzene–cyclohexane where the spectra display monotonic behavior. RMSE-based metrics exhibit similar limitations. A more pragmatic and physically meaningful criterion arises when the modified Lorentz–Lorenz transformation is embedded directly into complex-valued CLS regression. In this case, η can be optimized by minimizing the regression error with respect to the perturbation variable (volume fraction). This approach proved highly robust: CLS errors were substantially reduced for both systems, and with systematic-error correction the results approached or matched the performance of ILS and PLS. Importantly, improvements were obtained even in systems where correlation-based analyses did not indicate any benefit of the Lorentz–Lorenz transformation. Having established that η can be optimized via regression error minimization, we now briefly outline how this framework can be implemented in practical ATR-based mixture analysis.

In conventional quantitative infrared spectroscopy, Beer–Lambert behavior is assumed by directly modeling absorbance spectra as linear combinations of pure-component absorbances, i.e.,

A (\tilde{ν}, ϕ_{1}) \approx A_{1} (\tilde{ν}) φ_{1} + A_{2} (\tilde{ν}) (1 - φ_{1})

(20)

The present work replaces this assumption not by abandoning linearity, but by shifting it to a physically more appropriate domain. Instead of assuming linearity of absorbance, we first reconstruct the complex refractive index

\hat{n} (\tilde{ν})

from ATR measurements and then apply the modified Lorentz–Lorenz transformation

F (\tilde{ν}) = \frac{{\hat{n}}^{2} (\tilde{ν}) - 1}{{\hat{n}}^{2} (\tilde{ν}) + η}

(21)

Linearity is then imposed in this transformed domain, where electromagnetic local-field effects are approximately condensed into the parameter η. In this sense, the Lorentz–Lorenz framework replaces Beer–Lambert not at the level of regression methodology, but at the level of the spectral representation in which linear mixing is assumed.

In practical applications, quantitative analysis of liquid mixtures is frequently performed using ATR FT-IR spectroscopy. The present results suggest that such calibration procedures can be improved by performing them in the modified Lorentz–Lorenz domain rather than directly in the Beer–Lambert absorbance domain.

The practical implementation of the approach is summarized in the Discussion section. In this way, the modified Lorentz–Lorenz transformation acts as another physics-informed preprocessing step that reduces systematic nonlinearities before multivariate regression is performed.

These findings illustrate that the modified Lorentz–Lorenz domain provides a physics-informed chemometric space in which spectral mixing is much closer to linear and systematic deviations are greatly reduced. By incorporating physically grounded principles, i.e., local fields, mutual polarization, and wave-optics-based mixing rules, this approach creates a chemometric framework that is not only empirically robust but also consistent with the electrodynamics of condensed phases. The modified Lorentz–Lorenz framework should therefore be regarded as a physics-informed approximation rather than an exact microscopic description. Its purpose is not to eliminate local-field complexity but to condense it into a single effective parameter that restores approximate linearity for chemometric purposes.

Because the transformation restores approximate linearity in a physically interpretable way, it is natural to extend this concept beyond CLS. The results strongly suggest that ILS, PCA, and PLS should be revisited in this physics-informed Lorentz–Lorenz domain, where their calibration spaces may become more linear, more stable, and ultimately more predictive. Such an extension may significantly improve quantitative spectroscopy of liquid mixtures and provide a unified methodology for connecting classical chemometrics with the underlying physics of light–matter interaction. Future work will therefore focus on implementing these physics-informed chemometric methods—CLS, ILS, PCA, and PLS—directly in the modified Lorentz–Lorenz space and systematically evaluating their performance across a broader range of molecular systems.

Beyond its conceptual implications, the modified Lorentz–Lorenz framework introduced here has several practical consequences for quantitative vibrational spectroscopy. Any application that relies on Beer–Lambert-type linearity in condensed phases, such as quantitative infrared analysis of liquid mixtures, process analytical technology (PAT), reaction monitoring, or multicomponent calibration in ATR FT-IR, may benefit from operating in a Lorentz–Lorenz–transformed chemometric domain. In such settings, systematic deviations from linearity often dominate the prediction error and limit the transferability of calibration models. In particular, nonlinear mixing is a major obstacle for calibration transfer across instruments, sampling conditions, or concentration ranges. By condensing local-field effects, molecular anisotropy, and packing-related interactions into a single effective parameter, the present approach offers a pragmatic route to reduce these systematic errors without resorting to fully microscopic modeling.

More generally, the results suggest that physics-informed transformations of spectroscopic data can significantly improve the stability and interpretability of chemometric models. This is particularly relevant for complex-valued regression approaches, where the availability of both real and imaginary components allows systematic deviations from ideal mixing behavior to be detected and corrected. Potential applications therefore extend beyond infrared spectroscopy to other optical techniques that probe complex refractive indices, including Raman spectroscopy, ellipsometry-based analysis of liquids, and broadband optical sensing of multicomponent systems.

Supplemental Material

sj-docx-1-asp-10.1177_00037028261454699 - Supplemental material for Reducing Non-Linearity in Spectral Evaluation via a Modified Lorentz–Lorenz Relation

Supplemental material, sj-docx-1-asp-10.1177_00037028261454699 for Reducing Non-Linearity in Spectral Evaluation via a Modified Lorentz–Lorenz Relation by Thomas G. Mayerhöfer, Isao Noda and Jürgen Popp in Applied Spectroscopy

Footnotes

Acknowledgments and Funding

Financial support from the EU, the “Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft”, the “Thüringer Aufbaubank”, the Federal Ministry of Education and Research, Germany (BMBF), the German Science Foundation, the “Fonds der Chemischen Industrie” and the Carl-Zeiss Foundation is gratefully acknowledged.

Declaration of Conflicting Interests

Thomas G. Mayerhöfer is an Associate Editor for Applied Spectroscopy. The authors declared no potential conflicts of interest with respect to the research, authorship, review, and/or publication of this article.

Supplemental Material

All supplemental material mentioned in the text accompanies this paper online. All supplemental material mentioned in the text is available in the online version of the journal.

ORCID iDs

Thomas G. Mayerhöfer

Isao Noda

Jürgen Popp

References

Bouguer

. Essai d'Optique sur la Gradation de la Lumiere, par M. Bouguer. Paris: Chez Claude Jombert, 1729.

Lambert

J.H.

Anding

. Lambert's Photometrie: (Photometria sive de mensura et gradibus luminis, colorum et umbrae). (1760). Erstes Heft 3, Tl. 6 und 7. Leipzig: Verlag von Wilhelm Engelmann, 1892.

Beer. “Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten”. Ann. Physik (Berlin). 1852. 162(5): 78–88. 10.1002/andp.18521620505

Luther

Nikolopulos

. “Über die Beziehungen zwischen den Absorptionsspektren und der Konstitution der komplexen Kobaltamminsalze”. Z. Phys. Chem. 1913. 82U(1): 361. 10.1515/zpch-1913-8229

Mayerhöfer

T.G.

Pahlow

Popp

. “The Bouguer–Beer–Lambert Law: Shining Light on the Obscure”. ChemPhysChem. 2020. 21(18): 2029–2046. 10.1002/cphc.202000464

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “Removing Interference-Based Effects from the Infrared Transflectance Spectra of Thin Films on Metallic Substrates: A Fast and Wave Optics Conform Solution”. Analyst. 2018. 143(13): 3164–3175. 10.1039/C8AN00526E

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “CaF₂: An Ideal Substrate Material for Infrared Spectroscopy?”. Anal. Chem. 2020. 92(13): 9024–9031. 10.1021/acs.analchem.0c01158

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “Removing Interference-Based Effects from Infrared Spectra: Interference Fringes Re-Revisited”. Analyst. 2020. 145(9): 3385–3394. 10.1039/D0AN00062K

Mayerhöfer

T.G.

Pahlow

Hübner

Popp

. “Deviations from the Beer–Lambert Approximation Investigated Using Two-Dimensional Correlation Infrared Spectroscopy”. Appl. Spectrosc. 2025. 80(6). 10.1177/00037028251368394

10.

Mayerhöfer

T.G.

. Wave Optics in Infrared Spectroscopy: Theory, Simulation, and Modeling. Philadelphia: Elsevier, 2024.

11.

Maxwell

J.C.

. “VIII. A Dynamical Theory of the Electromagnetic Field”. Philos. Trans. R. Soc. London. 1865. 155: 459–512. 10.1098/rstl.1865.0008

12.

Lorentz

H.A.

. “Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte”. Ann. Physik (Berlin). 1880. 245(4): 641–665. 10.1002/andp.18802450406

13.

Planck

. “Über die Extinction des Lichtes in einem optisch homogenen Medium von normaler Dispersion”. Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften. 1904. I: 740–750.

14.

Garnett

J.C.M.

Larmor

. “XII. Colours in Metal Glasses and in Metallic Films”. Philos. Trans. R. Soc., A. 1904. 203(359–371): 385–420. 10.1098/rsta.1904.0024

15.

Bruggeman

D.A.G.

. “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen”. Ann. Physik (Berlin). 1935. 416(7): 636–664. 10.1002/andp.19354160705

16.

London

. “Zur Theorie und Systematik der Molekularkräfte”. Z. Physik. 1930. 63(3): 245–279. 10.1007/BF01421741

17.

Feynman

R.P.

Leighton

R.B.

Sands

. The Feynman Lectures on Physics, Vol. II: The New Millennium Edition: Mainly Electromagnetism and Matter. Boulder, Colorado: Basic Books, 2015.

18.

Mayerhöfer

T.G.

Ilchenko

Kutsyk

Popp

. “Complex-Valued Chemometrics in Spectroscopy: Classical Least Squares Regression”. Appl. Spectrosc. 2025. 79(2): 1768–1775. 10.1177/00037028251343908

19.

Lorentz

H.A.

. Collected Papers: Volume II. Dordrecht, Netherlands: Springer-Science + Business Media, B.V., 1936.

20.

Mayerhöfer

T.G.

Popp

. “Beyond Beer's Law: Revisiting the Lorentz–Lorenz Equation”. ChemPhysChem. 2020. 21(12): 1218–1223. 10.1002/cphc.202000301.

21.

Mayerhöfer

T.G.

Noda

Pahlow

Heintzmann

Popp

. “Correcting Systematic Errors by Hybrid 2D Correlation Loss Functions in Nonlinear Inverse Modelling”. PLoS One. 2023. 18(4): e0284723. 10.1371/journal.pone.0284723

22.

Mayerhöfer

T.G.

Richard-Lacroix

Pahlow

Hübner

Popp

. “Smart Error Sum Based on Hybrid Two-Trace Two-Dimensional (2T2D) Correlation Analysis”. Appl. Spectrosc. 2023. 77(6): 583–592. 10.1177/00037028221077310

23.

Mayerhöfer

T.G.

Richard-Lacroix

Pahlow

Hübner

, et al.. “Hybrid 2D Correlation-Based Loss Function for the Correction of Systematic Errors”. Anal. Chem. 2022. 94(2): 695–703. 10.1021/acs.analchem.1c02830

24.

Noda

Ozaki

. Two-Dimensional Correlation Spectroscopy: Applications in Vibrational and Optical Spectroscopy. Chichester, UK: John Wiley and Sons, 2005.

25.

Mayerhöfer

T.G.

Ilchenko

Kutsyk

Popp

. “Beyond Beer’s Law: Quasi-Ideal Binary Liquid Mixtures”. Appl. Spectrosc. 2022. 76(1): 92–104. 10.1177/00037028211056293

26.

Mayerhöfer

T.G.

Costa

W.D.P.

Popp

. “Sophisticated Attenuated Total Reflection Correction Within Seconds for Unpolarized Incident Light at 45°”. Appl. Spectrosc. 2024. 78(3): 321–328. 10.1177/00037028231219528

27.

Myers

T.L.

Tonkyn

R.G.

Danby

T.O.

Taubman

M.S.

, et al. “Accurate Measurement of the Optical Constants n and k for a Series of 57 Inorganic and Organic Liquids for Optical Modeling and Detection”. Appl. Spectrosc. 2018. 72(4): 535–550. 10.1177/0003702817742848

28.

Mayerhöfer

T.G.

Ilchenko

Kutsyk

Popp

. “Complex-Valued Chemometrics in Spectroscopy: Inverse Least Square Regression”. Appl. Spectrosc. 2025. 80(1): 100–108. 10.1177/00037028251358392

29.

Sellmeier

. “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Koerpertheilchen und deren Rueckwirkung auf die ersteren, besonders zur Erklaerung der Dispersion und ihrer Anomalien”. Ann. Physik (Berlin). 1872. 223(12): 525–554. 10.1002/andp.18722231203

Supplementary Material

Please find the following supplemental material available below.

For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.

For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.

0.35 MB