Assessing the Partial Hessian Approximation in QM/MM-Based Vibrational Analysis

The partial Hessian approximation is often used in vibrational analysis of quantum mechanics/molecular mechanics (QM/MM) systems because calculating the full Hessian matrix is computationally impractical. This approach aligns with the core concept of QM/MM, which focuses on the QM subsystem. Thus, using the partial Hessian approximation implies that the main interest is in the local vibrational modes of the QM subsystem. Here, we investigate the accuracy and applicability of the partial Hessian vibrational analysis (PHVA) approach as it is typically used within QM/MM, i.e., only the Hessian belonging to the QM subsystem is computed. We focus on solute-solvent systems with small, rigid solutes. To separate two of the major sources of errors, we perform two separate analyses. First, we study the effects of the partial Hessian approximation on local normal modes, harmonic frequencies, and harmonic IR and Raman intensities by comparing them to those obtained using full Hessians, where both partial and full Hessians are calculated at the QM level. Then, we quantify the errors introduced by QM/MM used with the PHVA by comparing normal modes, frequencies, and intensities obtained using partial Hessians calculated using a QM/MM-type embedding approach to those obtained using partial Hessians calculated at the QM level. Another aspect of the PHVA is the appearance of normal modes resembling the translation and rotation of the QM subsystem. These pseudotranslational and pseudorotational modes should be removed as they are collective vibrations of the atoms in the QM subsystem relative to a frozen MM subsystem and, thus, not well-described. We show that projecting out translation and rotation, usually done for systems in isolation, can adversely affect other normal modes. Instead, the pseudotranslational and pseudorotational modes can be identified and removed.

Tinned: A symbolic library for response theory and high-order derivatives

A symbolic C++ library-Tinned-has been developed for symbolic differentiation and manipulation in response theory. By recognizing different key building blocks in the density matrix-based (Thorvaldsen et al., J. Chem. Phys. 2008, 129, 214108) and coupled-cluster response theories, we have implemented their corresponding C++ symbolic classes, including but not limited to one- and two-electron operators, exchange-correlation energy and potential, and coupled-cluster operator. Formulas of response theory can be well expressed in terms of the symbolic classes in the library Tinned. Their high-order perturbation-strength derivatives can be straightforwardly computed and extracted afterwards for numerical evaluation. The library Tinned will greatly facilitate the development work of response theory and may lead to a unified framework for response theory at different levels of electronic structure theory.

After 50 Years of Vibrational Circular Dichroism Spectroscopy: Challenges and Opportunities of Increasingly Accurate and Complex Experiments and Computations

VCD research continues to thrive, driven by ongoing experimental and theoretical advances. Modern studies deal with increasingly complex samples featuring weak intermolecular interactions and shallow potential energy surfaces. Likewise, the combination of VCD measurements with, for instance, cryo-spectroscopic techniques has significantly increased their sensitivity. The extent to which such modern measurements enhance the informative value of VCD depends significantly on the quality of the theoretical models, which must adequately account for anharmonicity, solvation and molecular dynamics. We herein discuss how experimental advancements engage in a stimulating interplay with recent theoretical developments, pursuing either the static or the dynamic computational route. Both paths have their own strengths and limitations, each addressing fundamentally different problems. We give an outlook on future challenges of VCD research, including the possibility to combine static and dynamic approaches to obtain a full picture of the sample.

Harmonic and anharmonic vibrational computations for biomolecular building blocks: Benchmarking DFT and basis sets by theoretical and experimental IR spectrum of glycine conformers

Advanced vibrational spectroscopic experiments have reached a level of sophistication that can only be matched by numerical simulations in order to provide an unequivocal analysis, a crucial step to understand the structure-function relationship of biomolecules. While density functional theory (DFT) has become the standard method when targeting medium-size or larger systems, the problem of its reliability and accuracy are well-known and have been abundantly documented. To establish a reliable computational protocol, especially when accuracy is critical, a tailored benchmark is usually required. This is generally done over a short list of known candidates, with the basis set often fixed a priori. In this work, we present a systematic study of the performance of DFT-based hybrid and double-hybrid functionals in the prediction of vibrational energies and infrared intensities at the harmonic level and beyond, considering anharmonic effects through vibrational perturbation theory at the second order. The study is performed for the six-lowest energy glycine conformers, utilizing available "state-of-the-art" accurate theoretical and experimental data as reference. Focusing on the most intense fundamental vibrations in the mid-infrared range of glycine conformers, the role of the basis sets is also investigated considering the balance between computational cost and accuracy. Targeting larger systems, a broad range of hybrid schemes with different computational costs is also tested.

Influence of fourth-order vibrational corrections on semi-experimental (reSE) structures of linear molecules

Semi-experimental structures (reSE) are derived from experimental ground state rotational constants combined with theoretical vibrational corrections. They permit a meaningful comparison with equilibrium structures based on high-level ab initio calculations. Typically, the vibrational corrections are evaluated with second-order vibrational perturbation theory (VPT2). The amount of error introduced by this approximation is generally thought to be small; however, it has not been thoroughly quantified. Herein, we assess the accuracy of theoretical vibrational corrections by extending the treatment to fourth order (VPT4) for a series of small linear molecules. Typical corrections to bond distances are on the order of 10-5 Å. Larger corrections, nearly 0.0002 Å, are obtained to the bond lengths of NCCN and CNCN. A borderline case is CCCO, which will likely require variational computations for a satisfactory answer. Treatment of vibrational effects beyond VPT2 will thus be important when one wishes to know bond distances confidently to four decimal places (10-4 Å). Certain molecules with shallow bending potentials, e.g., HOC+, are not amenable to a VPT2 description and are not improved by VPT4.

responsefun: Fun with Response Functions in the Algebraic Diagrammatic Construction Framework

We present the open-source responsefun package, which implements a universally applicable procedure for computing molecular response properties within the algebraic diagrammatic construction (ADC) framework, exploiting the intermediate state representation (ISR) approach. With symbolic mathematics, the user can simply enter textbook sum-over-states (SOS) expressions from time-dependent perturbation theory, which are then automatically translated into the corresponding symbolic ADC/ISR formulations. Using the data structures provided by the hybrid Python/C++ module adcc for calculating excited states with ADC, the specified response property is directly evaluated, and the result is returned to the user. Employing the novel responsefun package, we present the first ADC/ISR calculations of second-order hyperpolarizability tensors and three-photon-absorption matrix elements.

Automatic derivation of many-body theories based on general Fermi vacua

This paper describes Wick&d, an implementation of the algebra of second-quantized operators normal ordered with respect to general correlated references and the corresponding Wick theorem [D. Mukherjee, Chem. Phys. Lett. 274, 561 (1997) and W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. Wick&d employs a compact representation of operators and a backtracking algorithm to efficiently evaluate Wick contractions. Since Wick&d can handle both fully and partially contracted terms, it can be applied to both projective and Fock-space many-body formalisms. To demonstrate the usefulness of  Wick&d, we use it to evaluate the single-reference coupled cluster equations up to octuple excitations and report an automated derivation and implementation of the second-order driven similarity renormalization group multireference perturbation theory.

Harmonic Infrared and Raman Spectra in Molecular Environments Using the Polarizable Embedding Model

We present a fully analytic approach to calculate infrared (IR) and Raman spectra of molecules embedded in complex molecular environments modeled using the fragment-based polarizable embedding (PE) model. We provide the theory for the calculation of analytic second-order geometric derivatives of molecular energies and first-order geometric derivatives of electric dipole moments and dipole-dipole polarizabilities within the PE model. The derivatives are implemented using a general open-ended response theory framework, thus allowing for an extension to higher-order derivatives. The embedding-potential parameters used to describe the environment in the PE model are derived through first-principles calculations, thus allowing a wide variety of systems to be modeled, including solvents, proteins, and other large and complex molecular environments. Here, we present proof-of-principle calculations of IR and Raman spectra of acetone in different solvents. This work is an important step toward calculating accurate vibrational spectra of molecules embedded in realistic environments.

Dalton Project: A Python platform for molecular- and electronic-structure simulations of complex systems

The Dalton Project provides a uniform platform access to the underlying full-fledged quantum chemistry codes Dalton and LSDalton as well as the PyFraME package for automatized fragmentation and parameterization of complex molecular environments. The platform is written in Python and defines a means for library communication and interaction. Intermediate data such as integrals are exposed to the platform and made accessible to the user in the form of NumPy arrays, and the resulting data are extracted, analyzed, and visualized. Complex computational protocols that may, for instance, arise due to a need for environment fragmentation and configuration-space sampling of biochemical systems are readily assisted by the platform. The platform is designed to host additional software libraries and will serve as a hub for future modular software development efforts in the distributed Dalton community.

Efficient calculation of NMR isotopic shifts: Difference-dedicated vibrational perturbation theory

We present difference-dedicated second-order vibrational perturbation theory (VPT2) as an efficient method for the computation of nuclear magnetic resonance (NMR) isotopic shifts, which reflect the geometry dependence of the NMR property in combination with different vibration patterns of two isotopologues. Conventional calculations of isotopic shifts, e.g., by standard VPT2, require scanning the geometry dependence over the whole molecule, which becomes expensive rapidly as the molecule size increases. In DD-VPT2, this scan can be restricted to a small region around the substitution site. At the heart of DD-VPT2 is a set of localized vibration modes common to the two isotopologues and designed such that the difference between the vibration patterns is caught by a small subset of them (usually fewer than 10). We tested the DD-VPT2 method for a series of molecules with increasing size and found that this method provides results with the same quality as VPT2 and in good agreement with the experiment, with computational savings up to 95% and less numerical instabilities. The method is easy to automatize and straightforward to generalize to other molecular properties.

Time-Dependent Formulation of Resonance Raman Optical Activity Spectroscopy

In this work, we extend the theoretical framework recently developed for the simulation of resonance Raman (RR) spectra of medium-to-large sized systems to its chiral counterpart, namely, resonance Raman optical activity (RROA). The theory is based on a time-dependent (TD) formulation, with the transition tensors obtained as half-Fourier transforms of the appropriate cross-correlation functions. The implementation has been kept as general as possible, supporting adiabatic and vertical models for the PES representation, both in Cartesian and internal coordinates, with the possible inclusion of Herzberg-Teller (HT) effects. Thanks to the integration of this TD-RROA procedure within a general-purpose quantum-chemistry program, both solvation and leading anharmonicity effects can be included in an effective way. The implementation is validated on one of the smallest chiral molecule (methyloxirane). Practical applications are illustrated with three medium-size organic molecules (naproxen-OCD₃, quinidine and 2-Br-hexahelicene), whose simulated spectra are compared to the corresponding experimental data.

Open-ended formulation of self-consistent field response theory with the polarizable continuum model for solvation

The study of high-order absorption properties of molecules is a field of growing importance. Quantum-chemical studies can help design chromophores with desirable characteristics. Given that most experiments are performed in solution, it is important to devise a cost-effective strategy to include solvation effects in quantum-chemical studies of these properties. We here present an open-ended formulation of self-consistent field (SCF) response theory for a molecular solute coupled to a polarizable continuum model (PCM) description of the solvent. Our formulation relies on the open-ended, density matrix-based quasienergy formulation of SCF response theory of Thorvaldsen, et al., [J. Chem. Phys., 2008, 129, 214108] and the variational formulation of the PCM, as presented by Lipparini et al., [J. Chem. Phys., 2010, 133, 014106]. Within the PCM approach to solvation, the mutual solute-solvent polarization is represented by means of an apparent surface charge (ASC) spread over the molecular cavity defining the solute-solvent boundary. In the variational formulation, the ASC is an independent, variational degree of freedom. This allows us to formulate response theory for molecular solutes in the fixed-cavity approximation up to arbitrary order and with arbitrary perturbation operators. For electric dipole perturbations, pole and residue analyses of the response functions naturally lead to the identification of excitation energies and transition moments. We document the implementation of this approach in the Dalton program package using a recently developed open-ended response code and the PCMSolver libraries and present results for one-, two-, three-, four- and five-photon absorption processes of three small molecules in solution.

Complete analytic anharmonic hyper-Raman scattering spectra

We present the first computational treatment of the complete second-order vibrational perturbation theory applied to hyper-Raman scattering spectroscopy. The required molecular properties are calculated in a fully analytic manner using a recently developed program [Ringholm, Jonsson and Ruud, J. Comp. Chem., 2014, 35, 622] that utilizes recursive routines. For some of the properties, these calculations are the first analytic calculations of their kind at their respective levels of theory. We apply this approach to the calculation of the hyper-Raman spectra of methane, ethane and ethylene and compare these to available experimental data. We show that the anharmonic corrections have a larger effect on the vibrational frequencies than on the spectral intensities, but that the inclusion of combination and overtone bands in the anharmonic treatment can improve the agreement with the experimental data, although the quality of available experimental data limits a detailed comparison.

Effective Inclusion of Mechanical and Electrical Anharmonicity in Excited Electronic States: VPT2-TDDFT Route

We present a reliable and cost-effective procedure for the inclusion of anharmonic effects in excited-state energies and spectroscopic intensities by means of second-order vibrational perturbation theory. This development is made possible thanks to a recent efficient implementation of excited-state analytic Hessians and properties within the time-dependent density functional theory framework. As illustrated in this work, by taking advantage of such algorithmic developments, it is possible to perform calculations of excited-state infrared spectra of medium-large isolated molecular systems, with anharmonicity effects included in both the energy and property surfaces. We also explore the use of this procedure for the inclusion of anharmonic effects in the simulation of vibronic bandshapes of electronic spectra and compare the results with previous, more approximate models.

Open-Ended Recursive Approach for the Calculation of Multiphoton Absorption Matrix Elements

We present an implementation of single residues for response functions to arbitrary order using a recursive approach. Explicit expressions in terms of density-matrix-based response theory for the single residues of the linear, quadratic, cubic, and quartic response functions are also presented. These residues correspond to one-, two-, three- and four-photon transition matrix elements. The newly developed code is used to calculate the one-, two-, three- and four-photon absorption cross sections of para-nitroaniline and para-nitroaminostilbene, making this the first treatment of four-photon absorption in the framework of response theory. We find that the calculated multiphoton absorption cross sections are not very sensitive to the size of the basis set as long as a reasonably large basis set with diffuse functions is used. The choice of exchange–correlation functional, however, significantly affects the calculated cross sections of both charge-transfer transitions and other transitions, in particular, for the larger para-nitroaminostilbene molecule. We therefore recommend the use of a range-separated exchange–correlation functional in combination with the augmented correlation-consistent double-ζ basis set aug-cc-pVDZ for the calculation of multiphoton absorption properties.

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

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

Analytic calculations of anharmonic infrared and Raman vibrational spectra

Using a recently developed recursive scheme for the calculation of high-order geometric derivatives of frequency-dependent molecular properties [Ringholm et al., J. Comp. Chem., 2014, 35, 622], we present the first analytic calculations of anharmonic infrared (IR) and Raman spectra including anharmonicity both in the vibrational frequencies and in the IR and Raman intensities. In the case of anharmonic corrections to the Raman intensities, this involves the calculation of fifth-order energy derivatives-that is, the third-order geometric derivatives of the frequency-dependent polarizability. The approach is applicable to both Hartree-Fock and Kohn-Sham density functional theory. Using generalized vibrational perturbation theory to second order, we have calculated the anharmonic infrared and Raman spectra of the non- and partially deuterated isotopomers of nitromethane, where the inclusion of anharmonic effects introduces combination and overtone bands that are observed in the experimental spectra. For the major features of the spectra, the inclusion of anharmonicities in the calculation of the vibrational frequencies is more important than anharmonic effects in the calculated infrared and Raman intensities. Using methanimine as a trial system, we demonstrate that the analytic approach avoids errors in the calculated spectra that may arise if numerical differentiation schemes are used.

Open-ended response theory with polarizable embedding: multiphoton absorption in biomolecular systems

We present the theory and implementation of an open-ended framework for electric response properties that includes effects from the molecular environment modeled by the polarizable embedding model.

Excited states in large molecular systems through polarizable embedding

Using the polarizable embedding model enables rational design of light-sensitive functional biological materials.

Pyrrolo[3,2-b]pyrroles-From Unprecedented Solvatofluorochromism to Two-Photon Absorption

A combined experimental and theoretical study of the two-photon absorption (2PA) properties of a series of quadrupolar molecules possessing a highly electron-rich heterocyclic core, pyrrolo[3,2-b]pyrrole, is presented. In agreement with quantum-chemical calculations, large 2PA cross-section values, σ2PA ≈10(2) -10(3)  GM (1 GM=10(50)  cm(4)  s photon(-1) ), are observed at wavelengths of 650-700 nm, which correspond to the two-photon allowed but one-photon forbidden transitions. The calculations also predict that increased planarity of this molecule through removal of two N-substituents leads to further increase in the σ2PA values. Surprisingly, the most quadrupolar pyrrolo[3,2-b]pyrrole derivative, containing two 4-nitrophenyl substituents at positions 2 and 5, demonstrates a very strong solvatofluorochromic effect, with a fluorescence quantum yield as high as 0.96 in cyclohexane, whereas the fluorescence vanishes in DMSO.

Five-Photon Absorption and Selective Enhancement of Multiphoton Absorption Processes

We study one-, two-, three-, four-, and five-photon absorption of three centrosymmetric molecules using density functional theory. These calculations are the first ab initio calculations of five-photon absorption. Even- and odd-order absorption processes show different trends in the absorption cross sections. The behavior of all even- and odd-photon absorption properties shows a semiquantitative similarity, which can be explained using few-state models. This analysis shows that odd-photon absorption processes are largely determined by the one-photon absorption strength, whereas all even-photon absorption strengths are largely dominated by the two-photon absorption strength, in both cases modulated by powers of the polarizability of the final excited state. We demonstrate how to selectively enhance a specific multiphoton absorption process.

Analytic density functional theory calculations of pure vibrational hyperpolarizabilities: the first dipole hyperpolarizability of retinal and related molecules

We present a general approach for the analytic calculation of pure vibrational contributions to the molecular (hyper)polarizabilities at the density functional level of theory. The analytic approach allows us to study large molecules, and we apply the new code to the study of the first dipole hyperpolarizabilities of retinal and related molecules. We investigate the importance of electron correlation as described by the B3LYP exchange-correlation functional on the pure vibrational and electronic hyperpolarizabilities and compare the computed hyperpolarizabilities with available experimental data. The effects of electron correlation on the pure vibrational corrections vary signficantly even between these structurally very similar molecules, making it difficult to estimate these effects without explicit calculations at the density functional theory level. As expected, the frequency-dependent first hyperpolarizability, which determines the experimentally observed second-harmonic generation, is dominated by the electronic term, whereas for the static hyperpolarizability, the vibrational contribution is equally important. As a consequence, frequency extrapolation of the measured optical hyperpolarizabilities can only provide an estimate for the electronic contribution to the static hyperpolarizability, not its total value. The relative values of the hyperpolarizabilities for different molecules, obtained from the calculations, are in reasonable agreement with experimental data.