Calculation of optical rotation with time-periodic magnetic-field-dependent basis functions in approximate time-dependent density-functional theory

The Amsterdam Modeling Suite

In this paper, we present the Amsterdam Modeling Suite (AMS), a comprehensive software platform designed to support advanced molecular and materials simulations across a wide range of chemical and physical systems. AMS integrates cutting-edge quantum chemical methods, including Density Functional Theory (DFT) and time-dependent DFT, with molecular mechanics, fluid thermodynamics, machine learning techniques, and more, to enable multi-scale modeling of complex chemical systems. Its design philosophy allows for seamless coupling between components, facilitating simulations that range from small molecules to complex biomolecular and solid-state systems, making it a versatile tool for tackling interdisciplinary challenges, both in industry and in academia. The suite also emphasizes user accessibility, with an intuitive graphical interface, extensive scripting capabilities, and compatibility with high-performance computing environments.

Theoretical Study of the Isotope Effect in Optical Rotation

In this work, the isotope effect in optical rotation (OR) is examined by exploring structure-property relationships for H → D substitutions in chiral molecules. While electronic effects serve as the dominant source of optical activity, there is a non-negligible contribution from nuclear vibrations, which changes with isotopic substitution. We employ a test set of 50 small organic molecules: three-membered rings with varying heteroatoms (PCl, PH, S, NCl, NH, O, and NBr) and functional groups (Me, F), and simulations were run at the B3LYP/aug-cc-pVDZ level of theory. The objectives of this work are to determine locations of isotopic substitution that result in significant changes in the vibrational correction to the OR and to evaluate which vibrational modes and electronic response are the major contributors to the isotope effect. Molecules with more polarizable heteroatoms in the ring (e.g., S and P) have the largest change in the vibrational correction compared to the unsubstituted parent molecules. In many cases, isotopic substitution made to the hydrogens on the opposite side of the ring from the functional group provides the largest change in the OR. H/D wagging modes and C vibrations (for D-C centers) are the largest contributors to the isotope effect. This is explained with a molecular orbital decomposition analysis of the OR. The relevant vibrational modes affect the orbital transitions that are already significant at the equilibrium geometry. However, this effect is only large when polarizable heteroatoms are involved because the electron density surrounding them is diffuse enough to feel the subtle effect of change in mass due to isotopic substitution on the relevant vibrational modes.

First Principles Simulations of Optical Rotation of Chiral Molecular Crystals

In this work, we present simulations of the optical rotation (OR) for five molecular crystals at density functional theory level with periodic boundary conditions (DFT-PBC). Calculations are compared with experimental measurements and show semi-quantitative agreement with experimental data for three of the crystals: tartatic acid, benzil, and pentaerythritol. For the other two crystals, aspartic acid and glutamic acid, the calculated data are in qualitative agreement with, but two orders of magnitude smaller than, the experimental data. We provide some arguments that support the theoretical predictions and suggest that the experiments should be revisited. We also find that the position of H centers provided in experimental X-ray data is not sufficiently reliable for simulating OR, and better results are obtained when H atoms are allowed to relax while keeping heavier elements fixed at the experimental positions. Comparison with molecular cluster calculations with a better functional and a larger basis set indicate that the role of intermolecular interactions (reproduced with the PBC technique) is as or more important than the choice of model chemistry. Despite the current limitations in the level of theory that can be employed, these simulations provide a promising avenue to investigate the effect of intermolecular interactions on this sensitive electronic property of molecules and materials.

On the choice of coordinate origin in length gauge optical rotation calculations

In this work, we explore the issue of origin dependence in optical rotation (OR) calculations in the length dipole gauge (LG) using standard approximate methods belonging to density functional theory (DFT) and coupled cluster (CC) theory. We use the origin-invariant LG approach, LG(OI), that we recently proposed as reference for the calculations, and we study whether a proper choice of coordinate origin and molecular orientation can be made such that diagonal elements of the LG-OR tensor match those of the LG(OI) tensor. Using a numerical search algorithm, we show that multiple spatial orientations can be found where the LG and LG(OI) results match. However, a simple analytical procedure provides a spatial orientation where the origin of the coordinate system is close to the center of mass of the molecule. At the same time, we also show that putting the origin at the center of mass is not an ideal choice for every molecule (relative errors in the OR up to 70% can be obtained in out test set). Finally, we show that the choice of coordinate origin based on the analytical procedure is transferable across different methods and it is superior to putting the origin in the center of mass or center of nuclear charge. This is important because the LG(OI) approach is trivial to implement for DFT, but not necessarily for nonvariational methods in the CC family. Therefore, one can determine an optimal coordinate origin at DFT level and use it for standard LG-CC response calculations.

X-ray and Optical Circular Dichroism as Local and Global Ultrafast Chiral Probes of [12]Helicene Racemization

Chirality is a fundamental molecular property that plays a crucial role in biophysics and drug design. Optical circular dichroism (OCD) is a well-established chiral spectroscopic probe in the UV-visible regime. Chirality is most commonly associated with a localized chiral center. However, some compounds such as helicenes (Figure 1) are chiral due to their screwlike global structure. In these highly conjugated systems, some electric and magnetic allowed transitions are distributed across the entire molecule, and OCD thus probes the global molecular chirality. Recent advances in X-ray sources, in particular the control of their polarization and spatial profiles, have enabled X-ray circular dichroism (XCD), which, in contrast to OCD, can exploit the localized and element-specific nature of X-ray electronic transitions. XCD therefore is more sensitive to local structures, and the chirality probed with it can be referred to as local. During the racemization of helicene, between opposite helical structures, the screw handedness can flip locally, making the molecule globally achiral while retaining a local handedness. Here, we use the racemization mechanism of [12]helicene as a model to demonstrate the capabilities of OCD and XCD as time-dependent probes for global and local chiralities, respectively. Our simulations demonstrate that XCD provides an excellent spectroscopic probe for the time-dependent local chirality of molecules.

Comparative Study of Vibrational Raman Optical Activity with Different Time-Dependent Density Functional Approximations: The VROA36 Database

A new database, VROA36, is introduced to investigate the performance of computational approaches for vibrational Raman optical activity (VROA) calculations. The database is composed of 36 molecules with known experimental VROA spectra. It includes 93 conformers. Normal modes calculated with B3LYP-D3(BJ)/def2-TZVP are used to compute the VROA spectra with four functionals, B3LYP-D3(BJ), ωB97X-D, M11, and optimally tuned LC-PBE, as well as several basis sets. SimROA indices and frequency scaling factors are used to compare calculated spectra with each other and with experimental data. The four functionals perform equally well independently of the basis set and usually achieve good agreement with the experimental data. For molecules in near- or at-resonance conditions, the inclusion of a complex (damped) linear response approach is important to obtain physically meaningful VROA intensities. The use of any of the tested functional approximations with the def2-SVPD Gaussian-type basis set, or a basis of similar flexibility, can be recommended for efficient and reliable theoretical VROA studies.

Basis Set Dependence of Optical Rotation Calculations with Different Choices of Gauge

In this work, the basis set dependence of optical rotation (OR) calculations is examined for various choices of gauge/level of theory. The OR is calculated for a set of 50 molecules using B3LYP and CAM-B3LYP and 17 molecules using coupled cluster with single and double excitations (CCSD). The calculations employ the correlation-consistent basis sets, aug-cc-pVζZ with ζ = D, T, Q. An inverse-power extrapolation formula is then utilized to obtain OR values at the complete basis set (CBS) limit. We investigate the basis set convergence for these methods and three choices of gauge: length gauge (with gauge-including atomic orbitals, LG(GIAOs), for DFT), the origin-invariant length gauge [LG(OI)], and the modified velocity gauge (MVG). The results show that all methods converge smoothly to the CBS limit and that the LG(OI) approach has a slightly faster convergence rate than the other choices of gauge. While the DFT methods reach gauge invariance at the CBS limit, CCSD does not. The significant difference between the MVG and LG(OI) results at the CBS limit, 26%, indicates that CCSD is not quite at convergence in the description of electron correlation for this property. On the other hand, gauge invariance at the CBS limit for DFT does not lead to the same OR values for the two density functionals, which is also due to electron correlation incompleteness. A limited comparison to gas-phase experimental OR values for the DFT methods shows that CAM-B3LYP seems more accurate than B3LYP. Overall, this study shows that the LG(OI) approach with the aug-cc-pVTZ basis set for DFT, and with the CBS(DT) extrapolation for CCSD, provides a good cost/accuracy balance.

Local-Field Effects in Linear Response Properties within a Polarizable Frozen Density Embedding Method

In this work, we present a polarizable frozen density embedding (FDE) method for calculating polarizabilities of coupled subsystems. The method (FDE-pol) combines a FDE method with an explicit polarization model such that the expensive freeze/thaw cycles can be bypassed, and approximate nonadditive kinetic potentials are avoided by enforcing external orthogonality between the subsystems. To describe the polarization of the frozen environment, we introduce a Hirshfeld partition-based density-dependent method for calculating the atomic polarizabilities of atoms in molecules, which alleviates the need to fit the atomic parameters to a specific system of interest or to a larger general set of molecules. We show that the Hirshfeld partition-based method predicts molecular polarizabilities close to the basis set limit, and thus, a single basis set-dependent scaling parameter can be introduced to improve the agreement against the reference polarizability data. To test the model, we characterized the uncoupled and coupled response of small interacting molecular complexes. Here, the coupled response properties include the perturbation of the frozen system due to the external perturbation which is ignored in the uncoupled response. We show that FDE-pol can accurately reproduce both the exact uncoupled polarizability and the coupled polarizabilities of the supermolecular systems. Using damped response theory, we also demonstrate that the coupled frequency-dependent polarizability can be described by including local field effects. The results emphasize the necessity of including local-field effects for describing the response properties of coupled subsystems, as well as the importance of accurate atomic polarizability models.

Origin invariant full optical rotation tensor in the length dipole gauge without London atomic orbitals

We present an origin-invariant approach to compute the full optical rotation tensor (Buckingham/Dunn tensor) in the length dipole gauge without recourse to London atomic orbitals, called LG(OI). The LG(OI) approach is simpler and less computationally demanding than the more common length gauge (LG)-London and modified velocity gauge (MVG) approaches, and it can be used with any approximate wave function or density functional method. We report an implementation at the coupled cluster with single and double excitations level (CCSD), for which we present the first simulations of the origin-invariant Buckingham/Dunn tensor in the LG. We compare LG(OI) and MVG results on a series of 22 organic molecules, showing good linear correlation between the approaches, although for small tensor elements, they provide values of opposite sign. We also attempt to decouple the effects of electron correlation and basis set incompleteness on the choice of gauge for specific rotation calculations on simple test systems. The simulations show a smooth convergence of the LG(OI) and MVG results with the basis set size toward the complete basis set limit. However, these preliminary results indicate that CCSD may not be close to a complete description of the electron correlation effects on this property even for small molecules and that basis set incompleteness may be a less important cause of discrepancy between choices of gauge than electron correlation incompleteness.

First-Principles Calculation of the Optical Rotatory Power of Periodic Systems: Application on α-Quartz, Tartaric Acid Crystal, and Chiral (n,m)-Carbon Nanotubes

The self-consistent coupled-perturbed (SC-CP) method in the CRYSTAL program has been adapted to obtain electromagnetic optical rotation properties of chiral periodic systems based on the calculation of the magnetic moment induced by the electric field. Toward that end, an expression for the magnetic transition moment is developed, which involves an appropriate electronic angular momentum operator. This operator is forced to be hermitian so that the chiroptical properties are real. In our formulation, the trace of the optical rotatory power matrix is gauge-origin-invariant as long as the electric dipole transition matrix elements are obtained using the velocity (rather than position) operator. On the other hand, the component along the optic axis is invariant in general for uniaxial and biaxial crystals. Under the same conditions, these properties also do not depend on the so-called missing integers that occur in the treatment of the electric dipole moment of quasi-one-dimensional periodic systems or the analogue of missing integers for the case of higher dimensionality. Tests on a model H₂O₂ polymer confirm the formalism and, as desired, show that the calculated properties are independent of the size and definition of the unit cell. In addition, an empirical relation to a finite oligomer gauge-including atomic orbital (GIAO) calculation is found. Applications, with comparison to experiment, are carried for α-quartz, tartaric acid crystal, and carbon nanotubes. Future developments of this initial approach to chiroptical properties in the solid state are noted.

Gauge Dependence of the S̃ Molecular Orbital Space Decomposition of Optical Rotation

The measurement of optical rotation (OR) is a foundational technique for the detection and characterization of chiral molecules, but it is poorly understood how the observed property relates to the structure of the molecule. Over the years, several schemes have been developed to decompose the OR into more chemically intuitive contributions. In this paper, we introduce two alternative formulations of our previously developed S̃ molecular orbital space decomposition. These new expressions use the modified velocity gauge-magnetic (MVG-M) and -electric (MVG-E) definitions of OR, rather than the length gauge-magnetic (LG-M) definition used in the original paper. Comparing these formulations across a small set of previously studied chiral molecules, we find that these different definitions produce consistent physical interpretations of the OR. These results demonstrate that the S̃ methodology for investigations of structure-property relationships in chiral molecules is insensitive to the choice of gauge.

Full optical rotation tensor at coupled cluster with single and double excitations level in the modified velocity gauge

This work presents the first simulations of the full optical rotation (OR) tensor at coupled cluster with single and double excitations (CCSD) level in the modified velocity gauge (MVG) formalism. The CCSD-MVG OR tensor is origin independent, and each tensor element can in principle be related directly to experimental measurements on oriented systems. We compare the CCSD results with those from two density functionals, B3LYP and CAM-B3LYP, on a test set of 22 chiral molecules. The results show that the functionals consistently overestimate the CCSD results for the individual tensor components and for the trace (which is related to the isotropic OR), by 10%-20% with CAM-B3LYP and 20%-30% with B3LYP. The data show that the contribution of the electric dipole-magnetic dipole polarizability tensor to the OR tensor is on average twice as large as that of the electric dipole-electric quadrupole polarizability tensor. The difficult case of (1S,4S)-(-)-norbornenone also reveals that the evaluation of the former polarizability tensor is more sensitive than the latter. We attribute the better agreement of CAM-B3LYP with CCSD to the ability of this functional to better reproduce electron delocalization compared with B3LYP, consistent with previous reports on isotropic OR. The CCSD-MVG approach allows the computation of reference data of the full OR tensor, which may be used to test more computationally efficient approximate methods that can be employed to study realistic models of optically active materials.

Simplified Ab Initio Molecular Dynamics-Based Raman Spectral Simulations

We describe a simplified approach to simulating Raman spectra from ab initio molecular dynamics (AIMD) calculations. The protocol relies on on-the-fly calculations of approximate molecular polarizabilities using the well-known sum over orbitals (as opposed to states) method. This approach bypasses the more accurate but computationally expensive approach to calculating molecular polarizabilities along AIMD trajectories, i.e., solving the coupled perturbed Hartree-Fock/Kohn-Sham equations. We demonstrate the advantages and limitations of our method through a few case studies targeting molecular systems of interest to surface- and/or tip-enhanced Raman spectroscopy practitioners.

Theory and algorithms for chiroptical properties and spectroscopies of aqueous systems

Chiroptical properties and spectroscopies are valuable tools to study chiral molecules and assign absolute configurations. The spectra that result from chiroptical measurements may be very rich and complex, and hide much of their information content. For this reason, the interplay between experiments and calculations is especially useful, provided that all relevant physico-chemical interactions that are present in the experimental sample are accurately modelled. The inherent difficulty associated to the calculation of chiral signals of systems in aqueous solutions requires the development of specific tools, able to account for the peculiarities of water-solute interactions, and especially its ability to form hydrogen bonds. In this perspective we discuss a multiscale approach, which we have developed and challenged to model the most used chiroptical techniques.

Origin invariant optical rotation in the length dipole gauge without London atomic orbitals

We present an approach to perform origin invariant optical rotation calculations in the length dipole gauge without recourse to London atomic orbitals, called origin invariant length gauge [LG(OI)]. The LG(OI) approach works with any approximate wave function or density functional method, but here we focus on the implementation with the coupled cluster (CC) with single and double excitations method because of the lack of production-level alternatives. Preliminary numerical tests show the efficacy of the LG(OI) procedure and indicate that putting the origin in the center of mass of a molecule may not be an optimal choice for conventional CC-LG calculations.

Direct Surface Enhanced Raman Spectroscopic Detection of Cortisol at Physiological Concentrations

Cortisol is an important steroid hormone in human physiology. Variations or abnormalities in the physiological cortisol levels control acute and chronic stress response, as well as contribute to diseases and syndromes including Addison's disease and Cushing syndrome. The ability to monitor cortisol levels in the physiological range is key in diagnosis and monitoring of these conditions, where current methodology for determination of cortisol levels relies on instrumentation that requires extensive sample preparation, long run times, and is destructive to the sample. Raman spectroscopy provides rapid sample analysis with relatively simple instrumentation; however, Raman spectroscopy is an inherently weak technique. To provide an enhanced Raman signal, we use surface enhanced Raman spectroscopy (SERS) which utilizes oscillating electric fields of metal nanoparticles, enhancing the overall electric field and therefore resulting in an enhanced signal. We demonstrate SERS-based detection of cortisol in the physiologically relevant range using colloidal silver nanoparticles in ethanolic solutions and bovine serum albumin. The SERS spectra obtained in an ethanol matrix demonstrate a sigmoidal concentration response over the physiologically relevant concentration range, with a limit of detection established at 177 nM. Analysis of cortisol solutions in a complex matrix (bovine serum albumin in phosphate buffered saline) is also demonstrated through the use of principal components analysis, a multivariate technique, which shows the separation of cortisol in a linear fashion with respect to cortisol concentration.

Redox-Active Chiroptical Switching in Mono- and Bis-Iron Ethynylcarbo[6]helicenes Studied by Electronic and Vibrational Circular Dichroism and Resonance Raman Optical Activity

Introducing one or two alkynyl-iron moieties onto a carbo[6]helicene results in organometallic helicenes (2 a,b) that display strong chiroptical activity combined with efficient redox-triggered switching. The neutral and oxidized forms have been studied in detail by electronic and vibrational circular dichroism, as well as by Raman optical activity (ROA) spectroscopy. The experimental results were analyzed and spectra were assigned with the help of first-principles calculations. In particular, a recently developed method for ROA calculations under resonance conditions has been used to study the intricate resonance effects on the ROA spectrum of mono-iron ethynylhelicene 2 a.

Electronic orbital response of regular extended and infinite periodic systems to magnetic fields. I. Theoretical foundations for static case

A theoretical treatment for the orbital response of an infinite, periodic system to a static, homogeneous, magnetic field is presented. It is assumed that the system of interest has an energy gap separating occupied and unoccupied orbitals and a zero Chern number. In contrast to earlier studies, we do not utilize a perturbation expansion, although we do assume the field is sufficiently weak that the occurrence of Landau levels can be ignored. The theory is developed by analyzing results for large, finite systems and also by comparing with the analogous treatment of an electrostatic field. The resulting many-electron Hamilton operator is forced to be hermitian, but hermiticity is not preserved, in general, for the subsequently derived single-particle operators that determine the electronic orbitals. However, we demonstrate that when focusing on the canonical solutions to the single-particle equations, hermiticity is preserved. The issue of gauge-origin dependence of approximate solutions is addressed. Our approach is compared with several previously proposed treatments, whereby limitations in some of the latter are identified.

Comparison of measured and predicted specific optical rotation in gas and solution phases: A test for the polarizable continuum model of solvation

A comparative theoretical and experimental study of dispersive optical activity is presented for a set of small, rigid organic molecules in gas and solution phases. Target species were chosen to facilitate wavelength-resolved measurements of specific rotation in rarefied vapors and in organic solvents having different polarities, while avoiding complications due to conformational flexibility. Calculations were performed with two density functionals (B3LYP and CAM-B3LYP) and with the coupled-cluster singles and doubles (CCSD) ansatz, and solvent effects were included through use of the polarizable continuum model (PCM). Across the various theoretical methods surveyed, CCSD with the modified velocity gauge provided the best overall performance for both isolated and solvated conditions. Zero-point vibrational corrections to equilibrium calculations of chiroptical response tended to improve agreement with gas-phase experiments, but the quality of performance realized for solutions varied markedly. Direct comparison of measured and predicted specific-rotation suggests that PCM, in general, is not able to reproduce attendant solvent shifts (neither between gas and solution phases nor among solvents) and fares better in estimating actual medium-dependent values of this property (although the error is rather system dependent). Thus, more elaborate solvation models seem necessary for a proper theoretical description of solvation in dispersive optical activity.

Gauge-Invariant Formulation of Circular Dichroism

Standard formulations of magnetic response properties, such as circular dichroism spectra, are plagued by gauge dependencies, which can lead to unphysical results. In this work, we present a general gauge-invariant and numerically efficient approach for the calculation of circular dichroism spectra from the current density. First we show that in this formulation the optical rotation tensor, the response function from which circular dichroism spectra can be obtained, is independent of the origin of the coordinate system. We then demonstrate that its trace is independent of the gauge origin of the vector potential. We also show how gauge invariance can be retained in practical calculations with finite basis sets. As an example, we explain how our method can be applied to time-dependent current-density-functional theory. Finally, we report gauge-invariant circular dichroism spectra obtained using the adiabatic local-density approximation. The circular dichroism spectra we thus obtain are in good agreement with experiment.

Infrared and Raman Spectroscopy from Ab Initio Molecular Dynamics and Static Normal Mode Analysis: The C-H Region of DMSO as a Case Study

Carbon-hydrogen (C-H) vibration modes serve as key probes in the chemical identification of hydrocarbons and in vibrational sum-frequency generation spectroscopy of hydrocarbons at the liquid/gas interface. Their assignments pose a challenge from a theoretical viewpoint. In this work, we present a detailed study of the C-H stretching region of dimethyl sulfoxide using a new ab initio molecular dynamics (AIMD) module that we have implemented in NWChem. Through a combination of AIMD simulations and static normal mode analysis, we interpret experimental infrared and Raman spectra and explore the role of anharmonic effects in this system. Comprehensive anharmonic normal mode analysis of the C-H stretching region casts doubt upon previous experimental assignments of the shoulder on the symmetric C-H stretching peak. In addition, our AIMD simulations also show significant broadening of the in-phase symmetric C-H stretching resonance, which suggests that the experimentally observed shoulder is due to thermal broadening of the symmetric stretching resonance.

Time-Dependent, Optically Controlled Dielectric Function

We suggest optical modulation of the dielectric function of a molecular monolayer adsorbed on a metal surface as a potential means of controlling plasmon resonance phenomena. The dielectric function is altered using a laser pulse of moderate intensity and linear polarization to align the constituent molecules. After the pulse, the monolayer returns to its initial state. Time-dependent, optically controlled dielectric function is illustrated by molecular dynamics calculations.