Variational Quantum Eigensolver Boosted by Adiabatic Connection.

In this work, we integrate the variational quantum eigensolver (VQE) with the adiabatic connection (AC) method for efficient simulations of chemical problems on near-term quantum computers. Orbital-optimized VQE methods are employed to capture the strong correlation within an active space, and classical AC corrections recover the dynamical correlation effects comprising electrons outside of the active space. On two challenging strongly correlated problems, namely, the dissociation of N2 and the electronic structure of the tetramethyleneethane biradical, we show that the combined VQE-AC approach enhances the performance of VQE dramatically. Moreover, since the AC corrections do not bring any additional requirements on quantum resources or measurements, they can actually boost the VQE algorithms. Our work paves the way toward quantum simulations of real-life problems on near-term quantum computers.

On-Surface Synthesis and Determination of the Open-Shell Singlet Ground State of Tridecacene.

The character of the electronic structure of acenes has been the subject of longstanding discussion. However, convincing experimental evidence of their open-shell character has so far been missing. Here, we present the on-surface synthesis of tridecacene molecules by thermal annealing of octahydrotridecacene on a Au(111) surface. We characterized the electronic structure of the tridecacene by scanning probe microscopy, which reveals the presence of an inelastic signal at 126 meV. We attribute the inelastic signal to spin excitation from the singlet diradical ground state to the triplet excited state. To rationalize the experimental findings, we carried out many-body ab initio calculations as well as model Hamiltonians to take into account the effect of the metallic substrate. Moreover, we provide a detailed analysis of how the dynamic electron correlation and virtual charge fluctuation between the molecule and metallic surface reduces the singlet-triplet band gap. Thus, this work provides the first experimental confirmation of the magnetic character of tridecacene.

Toward more accurate adiabatic connection approach for multireference wavefunctions.

A multiconfigurational adiabatic connection (AC) formalism is an attractive approach to compute the dynamic correlation within the complete active space self-consistent field and density matrix renormalization group (DMRG) models. Practical realizations of AC have been based on two approximations: (i) fixing one- and two-electron reduced density matrices (1- and 2-RDMs) at the zero-coupling constant limit and (ii) extended random phase approximation (ERPA). This work investigates the effect of removing the "fixed-RDM" approximation in AC. The analysis is carried out for two electronic Hamiltonian partitionings: the group product function- and the Dyall Hamiltonians. Exact reference AC integrands are generated from the DMRG full configuration interaction solver. Two AC models are investigated, employing either exact 1- and 2-RDMs or their second-order expansions in the coupling constant in the ERPA equations. Calculations for model molecules indicate that lifting the fixed-RDM approximation is a viable way toward improving the accuracy of existing AC approximations.

Experimental-computational approach to investigate elastic properties of struvite.

The research presented in the paper concerns the elastic properties of struvite. The article combines theoretical and experimental research. Experimental studies were carried out on struvite single crystals grown in sodium metasilicate gel by single diffusion. This unique method leads to obtaining crystals of sufficiently large size to conduct, for the first time, experimental measurement of elastic properties of monocrystalline struvite. Using the nanoindentation method, the Ez = 29.1 ± 0.7 GPa value of the component of Young's modulus was determined for a struvite single crystal. In addition, the elastic constants C11, C22, and C33 were determined using micro-Brillouin spectroscopy. Theoretical calculations of the abovementioned properties have been carried out by employing density functional theory methods. Scaling of the theoretical elastic constants leads to obtaining good agreement with the experimental values. Values of the Ex and Ey components of the Young's modulus, not available from the experimental nanoindentation technique, have been determined theoretically as 23 GPa and 27 GPa, respectively. Differences in the values of elastic components and Young's modulus components are related to the layered crystal structure of struvite and directional character of the hydrogen-bonding pattern.

TREXIO: A file format and library for quantum chemistry.

TREXIO is an open-source file format and library developed for the storage and manipulation of data produced by quantum chemistry calculations. It is designed with the goal of providing a reliable and efficient method of storing and exchanging wave function parameters and matrix elements, making it an important tool for researchers in the field of quantum chemistry. In this work, we present an overview of the TREXIO file format and library. The library consists of a front-end implemented in the C programming language and two different back-ends: a text back-end and a binary back-end utilizing the hierarchical data format version 5 library, which enables fast read and write operations. It is compatible with a variety of platforms and has interfaces for Fortran, Python, and OCaml programming languages. In addition, a suite of tools have been developed to facilitate the use of the TREXIO format and library, including converters for popular quantum chemistry codes and utilities for validating and manipulating data stored in TREXIO files. The simplicity, versatility, and ease of use of TREXIO make it a valuable resource for researchers working with quantum chemistry data.

Interplay between π-Conjugation and Exchange Magnetism in One-Dimensional Porphyrinoid Polymers.

The synthesis of novel polymeric materials with porphyrinoid compounds as key components of the repeating units attracts widespread interest from several scientific fields in view of their extraordinary variety of functional properties with potential applications in a wide range of highly significant technologies. The vast majority of such polymers present a closed-shell ground state, and, only recently, as the result of improved synthetic strategies, the engineering of open-shell porphyrinoid polymers with spin delocalization along the conjugation length has been achieved. Here, we present a combined strategy toward the fabrication of one-dimensional porphyrinoid-based polymers homocoupled via surface-catalyzed [3 + 3] cycloaromatization of isopropyl substituents on Au(111). Scanning tunneling microscopy and noncontact atomic force microscopy describe the thermal-activated intra- and intermolecular oxidative ring closure reactions as well as the controlled tip-induced hydrogen dissociation from the porphyrinoid units. In addition, scanning tunneling spectroscopy measurements, complemented by computational investigations, reveal the open-shell character, that is, the antiferromagnetic singlet ground state (S = 0) of the formed polymers, characterized by singlet-triplet inelastic excitations observed between spins of adjacent porphyrinoid units. Our approach sheds light on the crucial relevance of the π-conjugation in the correlations between spins, while expanding the on-surface synthesis toolbox and opening avenues toward the synthesis of innovative functional nanomaterials with prospects in carbon-based spintronics.

DFT exchange: sharing perspectives on the workhorse of quantum chemistry and materials science.

In this paper, the history, present status, and future of density-functional theory (DFT) is informally reviewed and discussed by 70 workers in the field, including molecular scientists, materials scientists, method developers and practitioners. The format of the paper is that of a roundtable discussion, in which the participants express and exchange views on DFT in the form of 302 individual contributions, formulated as responses to a preset list of 26 questions. Supported by a bibliography of 777 entries, the paper represents a broad snapshot of DFT, anno 2022.

Dispersion Interactions between Molecules in and out of Equilibrium Geometry: Visualization and Analysis.

The London dispersion interactions between systems undergoing bond breaking, twisting, or compression are not well studied due to the scarcity and the high computational cost of methods being able to describe both the dynamic correlation and the multireference character of the system. Recently developed methods based on the Generalized Valence Bond wave function, such as EERPA-GVB and SAPT(GVB) (SAPT = symmetry-adapted perturbation theory), allow one to accurately compute and analyze noncovalent interactions between multireference systems. Here, we augment this analysis by introducing a local indicator for dispersion interactions inspired by Mata and Wuttke's Dispersion Interaction Density [ J. Comput. Chem. 2017, 38, 15-23] applied on top of an EERPA-GVB computation. Using a few model systems, we show what insights into the nature and evolution of the dispersion interaction during bond breaking and twisting such an approach is able to offer. The new indicator can be used at a minimal cost additional to an EERPA-GVB computation and can be complemented by an energy decomposition employing the SAPT(GVB) method. We explain the physics behind the initial increase, followed by a decrease in the interaction of linear molecules upon bond stretching. Namely, the elongation of covalent bonds leads to the enhancement of attractive dispersion interactions. For even larger bond lengths, this effect is canceled by the increase of the repulsive exchange forces resulting in a suppression of the interaction and finally leading to repulsion between monomers.

An efficient implementation of time-dependent linear-response theory for strongly orthogonal geminal wave function models.

We present an implementation of time-dependent linear-response equations for strongly orthogonal geminal wave function models: the time-dependent generalized valence bond (TD-GVB) perfect-pairing theory and the antisymmetrized product of strongly orthogonal geminals. The geminal wave functions are optimized using a restricted-step second-order algorithm suitable for handling many geminals, and the linear-response equations are solved in an efficient way using a direct iterative approach. The wave function optimization algorithm features an original scheme to create initial orbitals for the geminal functions in a black-box fashion. The implementation is employed to examine the accuracy of the geminal linear response for singlet excitation energies of small and medium-sized molecules. In systems dominated by dynamic correlation, geminal models constitute only a minor improvement with respect to time-dependent Hartree-Fock. Compared to the linear-response complete active space self-consistent field, TD-GVB either misses or gives large errors for states dominated by double excitations.

Extension of an Atom-Atom Dispersion Function to Halogen Bonds and Its Use for Rational Design of Drugs and Biocatalysts.

A dispersion function Das in the form of a damped atom-atom asymptotic expansion fitted to ab initio dispersion energies from symmetry-adapted perturbation theory was improved and extended to systems containing heavier halogen atoms. To illustrate its performance, the revised Das function was implemented in the multipole first-order electrostatic and second-order dispersion (MED) scoring model. The extension has allowed applications to a much larger set of biocomplexes than it was possible with the original Das. A reasonable correlation between MED and experimentally determined inhibitory activities was achieved in a number of test cases, including structures featuring nonphysically shortened intermonomer distances, which constitute a particular challenge for binding strength predictions. Since the MED model is also computationally efficient, it can be used for reliable and rapid assessment of the ligand affinity or multidimensional scanning of amino acid side-chain conformations in the process of rational design of novel drugs or biocatalysts.

Excited states in the adiabatic connection fluctuation-dissipation theory: Recovering missing correlation energy from the negative part of the density response spectrum.

The adiabatic connection (AC) theory offers an alternative to the perturbation theory methods for computing correlation energy in the multireference wavefunction framework. We show that the AC correlation energy formula can be expressed in terms of the density linear response function as a sum of components related to positive and negative parts of the transition energy spectrum. Consequently, generalization of the adiabatic connection fluctuation-dissipation theory to electronically excited states is obtained. The component of the linear response function related to the negative-transition energy enters the correlation energy expression with an opposite sign to that of the positive-transition part and is non-negligible in the description of excited states. To illustrate this, we analyze the approximate AC model in which the linear response function is obtained in the extended random phase approximation (ERPA). We demonstrate that AC can be successfully combined with the ERPA for excited states, provided that the negative-excitation component of the response function is rigorously accounted for. The resulting AC0D model, an extension of the AC0 scheme introduced in our earlier works, is applied to a benchmark set of singlet excitation energies of organic molecules. AC0D constitutes a significant improvement over AC0 by bringing the excitation energies of the lowest excited states to a satisfactory agreement with theoretical best estimates, which parallels or even exceeds the accuracy of the n-electron valence state perturbation theory method. For higher excitations, AC0D is less reliable due to the gradual deterioration of the underlying ERPA linear response.

Embracing local suppression and enhancement of dynamic correlation effects in a CASΠDFT method for efficient description of excited states.

The recently proposed CASΠDFT method combines the reliable description of nondynamic electron correlation with the complete active space (CAS) wavefunction and the efficient treatment of dynamic correlation by density functional theory (DFT). This marriage is accomplished by adopting the DFT correlation energy functional modified with the local correction function of the on-top pair density (Π). The role of the correction function is to sensitize the correlation functional to local effects of suppression and enhancement of dynamic correlation and to account for an adequate amount of dynamic correlation energy. In this work we show that the presence of covalent and ionic configurations in a wavefunction gives rise to spatial regions where the effects of suppression and enhancement of correlation energy, respectively, dominate. The results obtained for the potential energy curves of the excited states of the hydrogen molecule prove that CASΠDFT is reliable for states that change their character along the dissociation curve. The method is also applied to the lowest excited states of six-membered heterocyclic nitrogen compounds such as pyridine, pyrazine, pyrimidine, and pyridazine. The obtained excitation energies for the n → π* and π → π* excitations confirm the good performance of CASΠDFT for excited states. The absolute average error of the method is 0.1 eV lower than that of the CCSD method and higher by the same amount than that of the more expansive CC3 variant. Compared with the coupled cluster methods, this encouraging performance of CASΠDFT is achieved at the negligible computational cost of obtaining the correlation energy.

Structural and Optical Properties of Struvite. Elucidating Structure of Infrared Spectrum in High Frequency Range.

Study of structure and optical properties of magnesium ammonium phosphate hexahydrate crystal known as struvite is presented. Experimentally determined infrared (IR) and ultraviolet-visible (UV-vis) spectra are compared with the theoretical predictions of density functional methods. Examination of the interatomic bond lengths, Mulliken atomic charges, and binding energies of water in the magnesium hexahydrate cation, together with the analysis of the hydrogen bond pattern have allowed us to explain a special feature of the IR spectrum of struvite, a blueshift of the band corresponding to the O-H stretching mode. This mode has been assigned to a "dangling" hydroxyl group in one of the water molecules in magnesium hexahydrate. Using experimentally obtained UV-vis spectrum and performing Tauc plots analysis, optical bandgap of struvite has been narrowed to a range from 5.92 to 6.06 eV.

Molecular multibond dissociation with small complete active space augmented by correlation density functionals.

Molecular multibond dissociation displays a variety of electron correlation effects posing a challenge for theoretical description. We propose a CASΠ(M)DFT approach, which includes these effects in an efficient way by combining the complete active space self-consistent field method with density functional theory (DFT). Within CASΠ(M)DFT, a small complete active space (CAS) accounts for the long-range intrabond and middle-range interbond nondynamic correlation in the stretched bonds. The common short-range dynamic correlation is calculated with the Lee-Yang-Parr (LYP) correlation DFT functional corrected for the suppression of dynamic correlation with nondynamic correlation. The remaining middle-range interbond dynamic correlation is evaluated with the modified LYP functional of the bond densities. As a result, CASΠ(M)DFT potential energy curves (PECs) calculated in the relatively small triple-zeta basis closely reproduce the benchmark complete basis set PECs for the following prototype multibonded molecules: N2, CO, H2O, and C2.

Long-range-corrected multiconfiguration density functional with the on-top pair density.

We propose a multiconfiguration density functional combining a short-range density functional approximation with a novel long-range correction for dynamic correlation effects. The correction is derived from the adiabatic connection formalism so that the resulting functional requires access only to one- and two-electron reduced density matrices of the system. In practice, the functional is formulated for wavefunctions of the complete active space (CAS) type and the short-range density functional part is made dependent on the on-top pair density via auxiliary spin densities. The latter allows for reducing the self-interaction and the static correlation errors without breaking the spin symmetry. We study the properties and the performance of the non-self-consistent variant of the method, termed lrAC0-postCAS. Numerical demonstration on a set of dissociation energy curves and excitation energies shows that lrAC0-postCAS provides accuracy comparable with more computationally expensive ab initio rivals.

Complete active space and corrected density functional theories helping each other to describe vertical electronic π → π* excitations in prototype multiple-bonded molecules.

The CASΠDFT method, which combines the complete active space (CAS) wave function approach and density functional theory (DFT), offers an efficient description of important excitations to the lowest excited states. CASΠDFT employs a correlation DFT functional corrected with a function P[x] of the ratio xr of the conditional and conventional electron densities obtained with the CAS on-top pair density Π(r). The sectors of P[x] for x(r) ≤ 1 and x(r) > 1 represent the opposite effects of the suppression of dynamic correlation with nondynamic correlation and its enhancement due to the ionic-type excitation. The present combination of the self-consistent-field CAS and the corrected Lee-Yang-Parr correlation functional closely reproduces in the relatively small double-zeta basis the benchmark experimental lowest singlet vertical π → π* excitations in the prototype multiple-bonded molecules N2, CO, C2H2, and C2H4.

Reproducing benchmark potential energy curves of molecular bond dissociation with small complete active space aided with density and density-matrix functional corrections.

Various effects of electron correlation accompany molecular bond dissociation, which makes the efficient calculation of potential energy curves a notoriously difficult problem. In an attempt to reliably reproduce both absolute energies and shapes of the benchmark dissociation curves, calculations with the combined CASΠDFT method are carried out for the prototype molecules H2, BH, F2, and N2. The complete active space (CAS) part of CASΠDFT accounts for long-range nondynamic correlation, while short-range dynamic correlation is accounted for with the corrected Lee-Yang-Parr correlation functional of density functional theory (DFT). The correction represents the suppression of dynamic correlation with nondynamic correlation, and it is a function of the ratio x(r) between the conditional and conventional densities obtained with the CAS on-top pair density Π(r). For the single-bonded molecules H2, BH, and F2, CASΠDFT succeeds in reproducing the shapes and absolute energies (for H2 and BH) of the benchmark curves, while for the triple-bonded N2 molecule, the addition to CASΠDFT of a multibond correction is required. It accounts for the middle-range dynamic correlation of the same-spin electrons in the (symmetrized) high-spin atomic electron configurations of the dissociating N2.

Electron Correlation from the Adiabatic Connection for Multireference Wave Functions.

An adiabatic connection (AC) formula for the electron correlation energy is derived for a broad class of multireference wave functions. The AC expression recovers dynamic correlation energy and assures a balanced treatment of the correlation energy. Coupling the AC formalism with the extended random phase approximation allows one to find the correlation energy only from reference one- and two-electron reduced density matrices. If the generalized valence bond perfect pairing model is employed a simple closed-form expression for the approximate AC formula is obtained. This results in the overall M^{5} scaling of the computation cost making the method one of the most efficient multireference approaches accounting for dynamic electron correlation also for the strongly correlated systems.

Exact and approximate adiabatic connection formulae for the correlation energy in multireference ground and excited states.

Recently it has been shown how to employ the adiabatic connection (AC) formalism to obtain correlation energy for multireference wavefunctions [K. Pernal, Phys. Rev. Lett. 120, 013001 (2018)]. Approximations to the exact AC formulation have been based on assuming that a one-electron reduced density matrix is constant along the AC path and by employing the extended random phase approximation. In this paper, the importance of these approximations is examined by comparing approximate AC integrands with their exact counterparts obtained for the hydrogen molecule in its ground and excited states. Encouraging results obtained for H2 indicate that AC is a viable and promising approach to a correlation energy problem not only for ground but also for excited states of electronic systems.

Reduced Density Matrix Functional Theory (RDMFT) and Linear Response Time-Dependent RDMFT (TD-RDMFT).

Recent advances in reduced density matrix functional theory (RDMFT) and linear response time-dependent reduced density matrix functional theory (TD-RDMFT) are reviewed. In particular, we present various approaches to develop approximate density matrix functionals which have been employed in RDMFT. We discuss the properties and performance of most available density matrix functionals. Progress in the development of functionals has been paralleled by formulation of novel RDMFT-based methods for predicting properties of molecular systems and solids. We give an overview of these methods. The time-dependent extension, TD-RDMFT, is a relatively new theory still awaiting practical and generally useful functionals which would work within the adiabatic approximation. In this chapter we concentrate on the formulation of TD-RDMFT response equations and various adiabatic approximations. None of the adiabatic approximations is fully satisfactory, so we also discuss a phase-dependent extension to TD-RDMFT employing the concept of phase-including-natural-spinorbitals (PINOs). We focus on applications of the linear response formulations to two-electron systems, for which the (almost) exact functional is known.