A density matrix-based method for the linear-scaling calculation of dynamic second- and third-order properties at the Hartree-Fock and Kohn-Sham density functional theory levels

Gauge-Invariant Excited-State Linear and Quadratic Response Properties within the Meta-Generalized Gradient Approximation

Gauge invariance is a fundamental symmetry connected to charge conservation and is widely accepted as indispensable for any electronic structure method. Hence, the gauge variance of the time-dependent kinetic energy density τ used in many meta-generalized gradient approximations (MGGAs) to the exchange-correlation (XC) functional presents a major obstacle for applying MGGAs within time-dependent density functional theory (TDDFT). Replacing τ by the gauge-invariant generalized kinetic energy density τ̂ significantly improves the accuracy of various functionals for vertical excitation energies [R. Grotjahn, F. Furche, and M. Kaupp. J. Chem. Phys. 2022, 157, 111102]. However, the dependence of the resulting current-MGGAs (cMGGAs) on the paramagnetic current density gives rise to new exchange-correlation kernels and hyper-kernels ignored in previous implementations of quadratic and higher-order response properties. Here we report the first implementation of cMGGAs and hybrid cMGGAs for excited-state gradients and dipole moments, as well as an extension to quadratic response properties including dynamic hyperpolarizabilities and two-photon absorption cross sections. In the first comprehensive benchmark study of MGGAs and cMGGAs for two-photon absorption cross sections, the M06-2X functional is found to be superior to the GGA hybrid PBE0. Additionally, two case studies from the literature for the practical prediction of nonlinear optical properties are revisited and potential advantages of hybrid (c)MGGAs compared to hybrid GGAs are discussed. The effect of restoring gauge invariance varies depending on the employed MGGA functional, the type of excitation, and the property under investigation: While some individual excited-state equilibrium structures are significantly affected, on average, these changes result in marginal improvements when compared against high-level reference data. Although the gauge-variant MGGA quadratic response properties are generally close to their gauge-invariant counterparts, the resulting errors are not bounded and significantly exceed typical method errors in some of the cases studied. Despite the limited effects seen in benchmark studies, gauge-invariant implementations of cMGGAs for excited-state properties are desirable from a fundamental perspective, entail little additional computational cost, and are necessary for response properties consistent with cMGGA linear response calculations such as excitation energies.

An in-silico NMR laboratory for nuclear magnetic shieldings computed via finite fields: Exploring nucleus-specific renormalizations of MP2 and MP3

We developed and implemented a method-independent, fully numerical, finite difference approach to calculating nuclear magnetic resonance shieldings, using gauge-including atomic orbitals. The resulting capability can be used to explore non-standard methods, given only the energy as a function of finite-applied magnetic fields and nuclear spins. For example, standard second-order Møller-Plesset theory (MP2) has well-known efficacy for 1H and 13C shieldings and known limitations for other nuclei such as 15N and 17O. It is, therefore, interesting to seek methods that offer good accuracy for 15N and 17O shieldings without greatly increased compute costs, as well as exploring whether such methods can further improve 1H and 13C shieldings. Using a small molecule test set of 28 species, we assessed two alternatives: κ regularized MP2 (κ-MP2), which provides energy-dependent damping of large amplitudes, and MP2.X, which includes a variable fraction, X, of third-order correlation (MP3). The aug-cc-pVTZ basis was used, and coupled cluster with singles and doubles and perturbative triples [CCSD(T)] results were taken as reference values. Our κ-MP2 results reveal significant improvements over MP2 for 13C and 15N, with the optimal κ value being element-specific. κ-MP2 with κ = 2 offers a 30% rms error reduction over MP2. For 15N, κ-MP2 with κ = 1.1 provides a 90% error reduction vs MP2 and a 60% error reduction vs CCSD. On the other hand, MP2.X with a scaling factor of 0.6 outperformed CCSD for all heavy nuclei. These results can be understood as providing renormalization of doubles amplitudes to partially account for neglected triple and higher substitutions and offer promising opportunities for future applications.

Graph-based quantum response theory and shadow Born-Oppenheimer molecular dynamics

Graph-based linear scaling electronic structure theory for quantum-mechanical molecular dynamics simulations [A. M. N. Niklasson et al., J. Chem. Phys. 144, 234101 (2016)] is adapted to the most recent shadow potential formulations of extended Lagrangian Born-Oppenheimer molecular dynamics, including fractional molecular-orbital occupation numbers [A. M. N. Niklasson, J. Chem. Phys. 152, 104103 (2020) and A. M. N. Niklasson, Eur. Phys. J. B 94, 164 (2021)], which enables stable simulations of sensitive complex chemical systems with unsteady charge solutions. The proposed formulation includes a preconditioned Krylov subspace approximation for the integration of the extended electronic degrees of freedom, which requires quantum response calculations for electronic states with fractional occupation numbers. For the response calculations, we introduce a graph-based canonical quantum perturbation theory that can be performed with the same natural parallelism and linear scaling complexity as the graph-based electronic structure calculations for the unperturbed ground state. The proposed techniques are particularly well-suited for semi-empirical electronic structure theory, and the methods are demonstrated using self-consistent charge density-functional tight-binding theory both for the acceleration of self-consistent field calculations and for quantum-mechanical molecular dynamics simulations. Graph-based techniques combined with the semi-empirical theory enable stable simulations of large, complex chemical systems, including tens-of-thousands of atoms.

Predicting the Second-Order Nonlinear Optical Responses of Organic Materials: The Role of Dynamics

The last 30 years have witnessed an ever-growing application of computational chemistry for rationalizing the nonlinear optical (NLO) responses of organic chromophores. More specifically, quantum chemical calculations proved highly helpful in gaining fundamental insights into the factors governing the magnitude and character of molecular first hyperpolarizabilities (β), be they either intrinsic to the chromophore molecular structure and arising from symmetry, chemical substitution, or π-electron delocalization, or induced by external contributions such as the laser probe or solvation and polarization effects. Most theoretical reports assumed a rigid picture of the investigated systems, the NLO responses being computed solely at the most stable geometry of the chromophores. Yet, recent developments combining classical molecular dynamics (MD) simulations and DFT calculations have evidenced the significant role of structural fluctuations, which may induce broad distributions of NLO responses, and even generate them in some instances.This Account presents recent case studies in which theoretical simulations have highlighted these effects. The discussion specifically focuses on the simulation of the second-order NLO properties that can be measured experimentally either from Hyper-Rayleigh Scattering (HRS) or Electric-Field Induced Second Harmonic Generation (EFISHG). More general but technical topics concerning several aspects of the calculations of hyperpolarizabilities are instead discussed in the Supporting Information.Selected examples include organic chromophores, photochromic systems, and ionic complexes in the liquid phase, for which the effects of explicit solvation, concentration, and chromophore aggregation are emphasized, as well as large flexible systems such as peptide chains and pyrimidine-based helical polymers, in which the relative variations of the responses were shown to be several times larger than their average values. The impact of geometrical fluctuations is also illustrated for supramolecular architectures with the examples of nanoparticles formed by organic dipolar dyes in water solution, whose soft nature allows for large shape variations translating into huge fluctuations in time of their NLO response, and of self-assembled monolayers (SAMs) based on indolino-oxazolidine or azobenzene switches, in which the geometrical distortions of the photochromic molecules, as well as their orientational and positional disorder within the SAMs, highly impact their NLO response and contrast upon switching. Finally, the effects of the rigidity and fluidity of the surrounding are evidenced for NLO dyes inserted in phospholipid bilayers.

Notes on density matrix perturbation theory

Density matrix perturbation theory (DMPT) is known as a promising alternative to the Rayleigh-Schrödinger perturbation theory, in which the sum-over-states (SOS) is replaced by algorithms with perturbed density matrices as the input variables. In this article, we formulate and discuss three types of DMPT, with two of them based only on density matrices: the approach of Kussmann and Ochsenfeld [J. Chem. Phys. 127, 054103 (2007)] is reformulated via the Sylvester equation and the recursive DMPT of Niklasson and Challacombe [Phys. Rev. Lett. 92, 193001 (2004)] is extended to the hole-particle canonical purification (HPCP) from Truflandier et al. [J. Chem. Phys. 144, 091102 (2016)]. A comparison of the computational performances shows that the aforementioned methods outperform the standard SOS. The HPCP-DMPT demonstrates stable convergence profiles but at a higher computational cost when compared to the original recursive polynomial method.

Matter-wave interference of a native polypeptide

The de Broglie wave nature of matter is a paradigmatic example of quantum physics and it has been exploited in precision measurements of forces and fundamental constants. However, matter-wave interferometry has remained an outstanding challenge for natural polypeptides, building blocks of life, which are fragile and difficult to handle. Here, we demonstrate the wave nature of gramicidin, a natural antibiotic composed of 15 amino acids. Its center of mass is delocalized over more than 20 times the molecular size in our time-domain Talbot-Lau interferometer. We compare the observed interference fringes with a model that includes both a rigorous treatment of the peptide’s quantum wave nature as well as a quantum chemical assessment of its optical properties to distinguish our result from classical predictions. The realization of quantum optics with this prototypical biomolecule paves the way for quantum-assisted measurements on a large class of biologically relevant molecules.

Matter-wave interferometry of complex molecules is challenging due to difficulties in preparing and detecting molecular beams. Here the authors demonstrate quantum behavior of a polypeptide using matter-wave interference in an all-optical time-domain Talbot-Lau interferometer.

Time-Dependent Coupled Perturbed Hartree-Fock and Density-Functional-Theory Approach for Calculating Frequency-Dependent (Hyper)Polarizabilities with Nonorthogonal Localized Molecular Orbitals

The time-dependent coupled perturbed Hartree-Fock/density-functional-theory (TDHF/TDDFT) approach has been reformulated based on nonorthogonal localized molecular orbitals (NOLMOs). Based on the NOLMO Fock equation, we have derived the corresponding NOLMO-TDHF/TDDFT equations up to the third order, and the formula for the frequency-dependent (hyper)polarizabilities has been given. Our approach has been applied to calculate both static and dynamic (hyper)polarizabilities of molecules varying from small molecules to large molecules. The NOLMO-TDHF/TDDFT approach can reproduce the reference canonical molecular orbital (CMO) results for all of our testing calculations. With the help of ongoing development of optimized local virtual molecular orbitals, the NOLMO-TDHF/TDDFT approach would be a very efficient method for large system calculations and tp achieve linear scaling.

Efficient pole-search algorithm for dynamic polarizability: Toward alternative excited-state calculation for large systems

This study presents an efficient algorithm to search for the poles of dynamic polarizability to obtain excited states of large systems with nonlocal excitation nature. The present algorithm adopts a homogeneous search with a constant frequency interval and a bisection search to achieve high accuracy. Furthermore, the subtraction process of the information about the detected poles from the total dynamic polarizability is used to extract the undetected pole contributions. Numerical assessments confirmed the accuracy and efficiency of the present algorithm in obtaining the excitation energies and oscillator strengths of all dipole-allowed excited states. A combination of the present pole-search algorithm and divide-and-conquer-based dynamic polarizability calculations was found to be promising to treat nonlocal excitations of large systems. © 2016 Wiley Periodicals, Inc.

Canonical density matrix perturbation theory

Density matrix perturbation theory [Niklasson and Challacombe, Phys. Rev. Lett. 92, 193001 (2004)] is generalized to canonical (NVT) free-energy ensembles in tight-binding, Hartree-Fock, or Kohn-Sham density-functional theory. The canonical density matrix perturbation theory can be used to calculate temperature-dependent response properties from the coupled perturbed self-consistent field equations as in density-functional perturbation theory. The method is well suited to take advantage of sparse matrix algebra to achieve linear scaling complexity in the computational cost as a function of system size for sufficiently large nonmetallic materials and metals at high temperatures.

Robust and Efficient Auxiliary Density Perturbation Theory Calculations

A new iterative solver for the recently developed time-dependent auxiliary density perturbation theory is presented. It is based on the Eirola-Nevanlinna algorithm for large nonsymmetric linear equation systems. The new methodology is validated by static and dynamic polarizability calculations of small molecules. Comparison between the analytic and iterative solutions of the response equation system shows excellent agreement for the calculated static and dynamic polarizabilities. The new iterative solver reduces the formal scaling from [Symbol: see text](N(4)) to [Symbol: see text](N(3)). Furthermore, the observed computational scaling for linear alkane chains is N(1.6). This subquadratic behavior is possible in systems with a few hundred atoms because of the very small prefactors of the [Symbol: see text](N(3)) and [Symbol: see text](N(2)) steps remaining in the iterative solver. To demonstrate the potential of this new methodology, static polarizabilities of giant fullerenes up to C960, with more than 14,000 basis functions, are calculated.

Coupled-Perturbed SCF Approach for Calculating Static Polarizabilities and Hyperpolarizabilities with Nonorthogonal Localized Molecular Orbitals

Coupled-perturbed self-consistent-field (CPSCF) approach has been broadly used for polarizabilities and hyperpolarizabilities computation. To extend this application to large systems, we have reformulated the CPSCF equations with nonorthogonal localized molecular orbitals (NOLMOs). NOLMOs are the most localized representation of electronic degrees of freedom. Methods based on NOLMOs are potentially ideal for investigating large systems. In atomic orbital representation, with a static external electric field added, the wave function and SCF operator of unperturbed NOLMO-SCF wave function/orbitals are expanded to different orders of perturbations. We have derived the corresponding equations up to the third order, which are significantly different from those of a conventional CPSCF method because of the release of the orthogonal restrictions on MOs. The solution to these equations has been implemented. Several chemical systems are used to verify our method. This work represents the first step toward efficient calculations of molecular response and excitation properties with NOLMOs.

Theoretical approaches to the calculation of Raman optical activity spectra

In this article, we will give a brief account of the different approaches that have been presented in the literature for calculating Raman optical activity (ROA) spectra by ab initio methods. We will also outline the general structure of a self-consistent-field-based approach for analytic calculations of ROA spectra, including also contributions from London orbitals. The use of London orbitals ensures that the relevant ROA parameters are gauge origin independent. We will also give an outlook on the future of ab initio calculations of Raman optical activity spectra.