Damped Response Theory in Combination with Polarizable Environments: The Polarizable Embedding Complex Polarization Propagator Method

Complex Linear Response Functions for a Multiconfigurational Self-Consistent Field Wave Function in a High Performance Computing Environment

We present novel developments for the highly efficient evaluation of complex linear response functions of a multiconfigurational self-consistent field (MCSCF) wave function as implemented in MultiPsi. Specifically, expressions for the direct evaluation of linear response properties at given frequencies using the complex polarization propagator (CPP) approach have been implemented, within both the Tamm-Dancoff approximation (TDA) and the random phase approximation (RPA). Purely real algebra with symmetric and antisymmetric trial vectors in a shared subspace is used wherein the linear response equations are solved. Two bottlenecks of large scale MC-CPP calculations, namely, the memory footprint and computational time, are addressed. The former is addressed by limiting the size of the subspace of trial vectors by using singular value decomposition (SVD) on either orbital or CI subspaces. The latter is addressed using an efficient parallel implementation as well as the strategy of dynamically adding linear response equations at near-convergence to neighboring roots. Furthermore, a novel methodology for decomposing MC-CPP spectra in terms of intuitive orbital excitations in an approximate fashion is presented. The performance of the code is illustrated with several numerical examples, including the X-ray spectrum of a molecule with nearly one hundred atoms. Additionally, for X-ray spectroscopy, the effect of including or excluding the core orbital in the active space on small covalent metal complexes is discussed.

Polarizable Embedding without Artificial Boundary Polarization

We present a fully self-consistent polarizable embedding (PE) model that does not suffer from unphysical boundary polarization. This is achieved through the use of the minimum-image convention (MIC) in the induced electrostatics. It is a simple yet effective approach that includes a more physically accurate description of the polarization throughout the molecular system. Using PE with MIC (PE-MIC), we shed new light on the limitations of commonly employed cutoff models, such as the droplet model, when used in PE calculations. Specifically, we investigate the effects of the unphysical polarization at the outer boundary by comparing induced dipoles and the associated electrostatic potentials, as well as some optical properties of solute-solvent and biomolecular systems. We show that the magnitude of the inaccuracies caused by the unphysical polarization depends on multiple parameters: the nature of the quantum subsystem and of the environment, the cutoff model and distance, and the calculated property.

A method to capture the large relativistic and solvent effects on the UV-vis spectra of photo-activated metal complexes

We have recently developed a method based on relativistic time-dependent density functional theory (TD-DFT) that allows the calculation of electronic spectra in solution (Creutzberg, Hedegård, J. Chem. Theory Comput.18, 2022, 3671). This method treats the solvent explicitly with a classical, polarizable embedding (PE) description. Furthermore, it employs the complex polarization propagator (CPP) formalism which allows calculations on complexes with a dense population of electronic states (such complexes are known to be problematic for conventional TD-DFT). Here, we employ this method to investigate both the dynamic and electronic effects of the solvent for the excited electronic states of trans-trans-trans-[Pt(N₃)₂(OH)₂(NH₃)₂] in aqueous solution. This complex decomposes into species harmful to cancer cells under light irradiation. Thus, understanding its photo-physical properties may lead to a more efficient method to battle cancer. We quantify the effect of the underlying structure and dynamics by classical molecular mechanics simulations, refined with a subsequent DFT or semi-empirical optimization on a cluster. Moreover, we quantify the effect of employing different methods to set up the solvated system, e.g., how sensitive the results are to the method used for the refinement, and how large a solvent shell that is required. The electronic solvent effect is always included through a PE potential.

Comparison of Length, Velocity, and Symmetric Gauges for the Calculation of Absorption and Electric Circular Dichroism Spectra with Real-Time Time-Dependent Density Functional Theory

A velocity and symmetric gauge implementation for real-time time-dependent density functional theory (RT-TDDFT) in the CP2K package using a Gaussian and plane wave approach is presented, including the explicit gauge-transformed contributions due to the nonlocal part of pseudopotentials. Absorption spectra of gas-phase α-pinene are calculated in length and velocity gauges in the long-wavelength approximation for the application of a δ pulse in linear and full order. The velocity gauge implementation is also applied to a solvated uracil molecule to showcase its use within periodic boundary conditions (PBC). For the calculation of the expectation value of the electric dipole moment in PBC, both the velocity representation and the modern theory of polarization give equivalent absorption spectra if a distributed reference point is used for the nonlocal term of the velocity operator. The discussion of linear response theory takes place in a unified framework in terms of linear response functions in propagator notation, distinguishing the parts of the linear response functions associated with perturbation and response. To further investigate gauge dependence, electric circular dichroism (ECD) spectra of α-pinene were calculated either as magnetic response to an electric field perturbation, in length or velocity gauge, or as electric response to a magnetic field perturbation in the symmetric gauge. Both approaches, electric and magnetic perturbations, have been found to yield equivalent ECD spectra.

Polarizable Embedding Complex Polarization Propagator in Four- and Two-Component Frameworks

Explicit embedding methods combined with the complex polarization propagator (CPP) enable the modeling of spectroscopy for increasingly complex systems with a high density of states. We present the first derivation and implementation of the CPP in four- and exact-two-component (X2C) polarizable embedding (PE) frameworks. We denote the developed methods PE-4c-CPP and PE-X2C-CPP, respectively. We illustrate the methods by estimating the solvent effect on ultraviolet-visible (UV-vis) and X-ray atomic absorption (XAS) spectra of [Rh(H₂O)₆]³⁺ and [Ir(H₂O)₆]³⁺ immersed in aqueous solution. We moreover estimate solvent effects on UV-vis spectra of a platinum complex that can be photochemically activated (in water) to kill cancer cells. Our results clearly show that the inclusion of the environment is required: UV-vis and (to a lesser degree) XAS spectra can become qualitatively different from vacuum calculations. Comparison of PE-4c-CPP and PE-X2C-CPP methods shows that X2C essentially reproduces the solvent effect obtained with the 4c methods.

Environment Effects on X-Ray Absorption Spectra With Quantum Embedded Real-Time Time-Dependent Density Functional Theory Approaches

In this work we implement the real-time time-dependent block-orthogonalized Manby-Miller embedding (rt-BOMME) approach alongside our previously developed real-time frozen density embedding time-dependent density functional theory (rt-TDDFT-in-DFT FDE) code, and investigate these methods' performance in reproducing X-ray absorption spectra (XAS) obtained with standard rt-TDDFT simulations, for model systems comprised of solvated fluoride and chloride ions ([X@( H 2 O ) 8 - , X = F, Cl). We observe that for ground-state quantities such as core orbital energies, the BOMME approach shows significantly better agreement with supermolecular results than FDE for the strongly interacting fluoride system, while for chloride the two embedding approaches show more similar results. For the excited states, we see that while FDE (constrained not to have the environment densities relaxed in the ground state) is in good agreement with the reference calculations for the region around the K and L1 edges, and is capable of reproducing the splitting of the 1s1 (n + 1)p1 final states (n + 1 being the lowest virtual p orbital of the halides), it by and large fails to properly reproduce the 1s1 (n + 2)p1 states and misses the electronic states arising from excitation to orbitals with important contributions from the solvent. The BOMME results, on the other hand, provide a faithful qualitative representation of the spectra in all energy regions considered, though its intrinsic approximation of employing a lower-accuracy exchange-correlation functional for the environment induces non-negligible shifts in peak positions for the excitations from the halide to the environment. Our results thus confirm that QM/QM embedding approaches are viable alternatives to standard real-time simulations of X-ray absorption spectra of species in complex or confined environments.

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.

A discrete interaction model/quantum mechanical method for simulating surface-enhanced Raman spectroscopy in solution

Since surface-enhanced Raman scattering (SERS) is of considerable interest for sensing applications in aqueous solution, the role that solvent plays in the spectroscopy must be understood. However, these efforts are hindered due to a lack of simulation approaches for modeling solvent effects in SERS. In this work, we present an atomistic electrodynamics-quantum mechanical method to simulate SERS in aqueous solution based on the discrete interaction model/quantum mechanical method. This method combines an atomistic electrodynamics model of the nanoparticle with a time-dependent density functional theory description of the molecule and a polarizable embedding method for the solvent. The explicit treatment of solvent molecules and nanoparticles results in a large number of polarizable dipoles that need to be considered. To reduce the computational cost, a simple cut-off based approach has been implemented to limit the number of dipoles that need to be treated without sacrificing accuracy. As a test of this method, we have studied how solvent affects the SERS of pyridine in the junction between two nanoparticles in aqueous solution. We find that the solvent leads to an enhanced SERS due to an increased local field at the position of the pyridine. We further demonstrate the importance of both image field and local field effects in determining the enhancements and the spectral signatures. Our results show the importance of describing the local environment due to the solvent molecules when modeling SERS.

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.

A zeroth-order active-space frozen-orbital embedding scheme for multireference calculations

Multireference computations of large-scale chemical systems are typically limited by the computational cost of quantum chemistry methods. In this work, we develop a zeroth-order active space embedding theory [ASET(0)], a simple and automatic approach for embedding any multireference dynamical correlation method based on a frozen-orbital treatment of the environment. ASET(0) is combined with the second-order multireference driven similarity renormalization group and tested on several benchmark problems, including the excitation energy of 1-octene and bond-breaking in ethane and pentyldiazene. Finally, we apply ASET(0) to study the singlet-triplet gap of p-benzyne and 9,10-anthracyne diradicals adsorbed on a NaCl surface. Our results show that despite its simplicity, ASET(0) is a powerful and sufficiently accurate embedding scheme applicable when the coupling between the fragment and the environment is in the weak to medium regime.

CPPE: An Open-Source C++ and Python Library for Polarizable Embedding

We present a modular open-source library for polarizable embedding (PE) named CPPE. The library is implemented in C++, and it additionally provides a Python interface for rapid prototyping and experimentation in a high-level scripting language. Our library integrates seamlessly with existing quantum chemical program packages through an intuitive and minimal interface. Until now, CPPE has been interfaced to three packages, Q-Chem, Psi4, and PySCF. Furthermore, we show CPPE in action using all three program packages for a computational spectroscopy application. With CPPE, host program interfaces only require minor programming effort, paving the way for new combined methodologies and broader availability of the PE model.

Decomposition of molecular properties

We review recent work on property decomposition techniques using quantum chemical methods and discuss some topical applications in terms of quantum mechanics-molecular mechanics calculations and the constructing of properties of large molecules and clusters. Starting out from the so-called LoProp decomposition scheme [Gagliardi et al., J. Chem. Phys., 2004, 121, 4994] for extracting atomic and inter-atomic contributions to molecular properties we show how this method can be generalized to localized frequency-dependent polarizabilities, to localized hyperpolarizabilities and to localized dispersion coefficients. Some applications of the generalized decomposition technique are reviewed - calculations of frequency-dependent polarizabilities, Rayleigh scattering of large clusters, and calculations of hyperpolarizabilities of proteins.

Theoretical insight into the photophysical properties of long-lifetime Ir(iii) and Rh(iii) complexes for two-photon photodynamic therapy

Two-photon photodynamic therapy (TP-PDT) plays crucial roles in curing tumors because it involves deep penetration of drugs into the tissue and has minimal damage to the surrounding cells.

Absorption Spectra of FAD Embedded in Cryptochromes

The magnetic compass sense utilized by migratory birds for long-distance navigation functions only once light of a certain wavelength is present. This piece of evidence fits partially with the popular hypothesis of chemical magnetoreception in cryptochrome proteins, located in the bird retina. According to this hypothesis a magnetosensitive radical pair is produced after photoexcitation of an FAD cofactor inside cryptochrome, and as such the absorption properties of FAD are of crucial importance for cryptochrome activation. However, we reveal that absorption spectra of FAD show very little variation between six different cryptochromes, suggesting that the electronic transitions are barely affected by the chemical differences in the proteins. This conclusion hints on the presence of a secondary photoreceptor or cofactor that could be necessary to explain green-light-activated magnetoreception in birds.

A quantum-mechanical perspective on linear response theory within polarizable embedding

We present a derivation of linear response theory within polarizable embedding starting from a rigorous quantum-mechanical treatment of a composite system. To this aim, two different subsystem decompositions (symmetric and nonsymmetric) of the linear response function are introduced and the pole structures as well as residues of the individual terms are discussed. In addition to providing a thorough justification for the descriptions used in polarizable embedding models, this theoretical analysis clarifies which form of the response function to use and highlights complications in separating out subsystem contributions to molecular properties. The basic features of the presented expressions and various approximate forms are illustrated by their application to a composite model system.

Optical absorption and magnetic circular dichroism spectra of thiouracils: a quantum mechanical study in solution

The excited electronic states of thiouracils, the analogues of uracil where the carbonyl oxygens are substituted by sulphur atoms, have been investigated by computing the magnetic circular dichroism (MCD) and one-photon absorption (OPA) spectra at the TD-DFT level of theory.

Stabilizing and Modulating Color by Copigmentation: Insights from Theory and Experiment

Natural anthocyanin pigments/dyes and phenolic copigments/co-dyes form noncovalent complexes, which stabilize and modulate (in particular blue, violet, and red) colors in flowers, berries, and food products derived from them (including wines, jams, purees, and syrups). This noncovalent association and their electronic and optical implications constitute the copigmentation phenomenon. Over the past decade, experimental and theoretical studies have enabled a molecular understanding of copigmentation. This review revisits this phenomenon to provide a comprehensive description of the nature of binding (the dispersion and electrostatic components of π-π stacking, the hydrophobic effect, and possible hydrogen-bonding between pigment and copigment) and of spectral modifications occurring in copigmentation complexes, in which charge transfer plays an important role. Particular attention is paid to applications of copigmentation in food chemistry.

Excited states in large molecular systems through polarizable embedding

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

Requirements of first-principles calculations of X-ray absorption spectra of liquid water

A computational benchmark study on X-ray absorption spectra of water has been performed by means of transition-potential density functional theory (TP-DFT), damped time-dependent density functional theory (TDDFT), and damped coupled cluster (CC) linear response theory. For liquid water, using TDDFT with a tailored CAM-B3LYP functional and a polarizable embedding, we find that an embedding with over 2000 water molecules is required to fully converge spectral features for individual molecules, but a substantially smaller embedding can be used within averaging schemes. TP-DFT and TDDFT calculations on 100 MD structures demonstrate that TDDFT produces a spectrum with spectral features in good agreement with experiment, while it is more difficult to fully resolve the spectral features in the TP-DFT spectrum. Similar trends were also observed for calculations of bulk ice. In order to further establish the performance of these methods, small water clusters have been considered also at the CC2 and CCSD levels of theory. Issues regarding the basis set requirements for spectrum simulations of liquid water and the determination of gas-phase ionization potentials are also discussed.

Frequency-dependent force fields for QMMM calculations

The frequency-dependent localized polarizabilities are calculated for the first time using analytical response theory and benchmarked for different water clusters and the tryptophan residue embedded in a protein.