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We present a two-component formulation of algebraic diagrammatic construction theory for simulating spin-orbit coupling and electron correlation in charged electronic states and photoelectron spectra. Our implementation supports Hartree-Fock and multiconfigurational reference wavefunctions, enabling efficient correlated calculations of relativistic effects using single-reference (SR-) and multireference (MR-) ADC. We combine the SR- and MR-ADC methods with three flavors of spin-orbit two-component Hamiltonians and benchmark their performance for a variety of atoms and small molecules. When multireference effects are not important, the SR-ADC approximations are competitive in accuracy to MR-ADC, often showing closer agreement with experimental results. However, for electronic states with multiconfigurational character and in non-equilibrium regions of potential energy surfaces, the MR-ADC methods are more reliable, predicting accurate excitation energies and zero-field splittings. Our results demonstrate that the two-component ADC methods are promising approaches for interpreting and predicting the results of modern spectroscopies.

Theoretical simulations of electron detachment processes are vital for understanding chemical redox reactions, semiconductor and electrochemical properties, and high-energy radiation damage. However, accurate calculations of ionized electronic states are very challenging due to their open-shell nature, importance of electron correlation effects, and strong interactions with chemical environment. In this work, we present an efficient approach based on algebraic diagrammatic construction theory with polarizable embedding that allows to accurately simulate ionized electronic states in condensed-phase or biochemical environments (PE-IP-ADC). We showcase the capabilities of PE-IP-ADC by computing the vertical ionization energy (VIE) of thymine molecule solvated in bulk water. Our results show that the second- and third-order PE-IP-ADC methods combined with the basis of set of triple-zeta quality yield a solvent-induced shift in VIE of -0.92 and -0.93 eV, respectively, in an excellent agreement with experimental estimate of -0.9 eV. This work demonstrates the power of PE-IP-ADC approach for simulating charged electronic states in realistic chemical environments and motivates its further development.

We present an efficient implementation of multireference algebraic diagrammatic construction theory (MR-ADC) for simulating core-ionized states and X-ray photoelectron spectra (XPS). Taking advantage of spin adaptation, automatic code generation, and density fitting, our implementation can perform calculations for molecules with more than 1500 molecular orbitals, incorporating static and dynamic correlation in the ground and excited electronic states. We demonstrate the capabilities of MR-ADC methods by simulating the XPS spectra of substituted ferrocene complexes and azobenzene isomers. For the ground electronic states of these molecules, the XPS spectra computed using the extended second-order MR-ADC method (MR-ADC(2)-X) are in a very good agreement with available experimental results. We further show that MR-ADC can be used as a tool for interpreting or predicting the results of time-resolved XPS measurements by simulating the core ionization spectra of azobenzene along its photoisomerization, including the XPS signatures of excited states and the minimum energy conical intersection. This work is the first in a series of publications reporting the efficient implementations of MR-ADC methods.

We present a formulation and implementation of second-order quasidegenerate N-electron valence perturbation theory (QDNEVPT2) that provides a balanced and accurate description of spin-orbit coupling and dynamic correlation effects in multiconfigurational electronic states. In our approach, the energies and wavefunctions of electronic states are computed by treating electron repulsion and spin-orbit coupling operators as equal perturbations to the non-relativistic complete active-space wavefunctions and their contributions are incorporated fully up to the second order. The spin-orbit effects are described using the Breit-Pauli (BP) or exact two-component Douglas-Kroll-Hess (DKH) Hamiltonians within spin-orbit mean-field approximation. The resulting second-order methods (BP2- and DKH2-QDNEVPT2) are capable of treating spin-orbit coupling effects in nearly degenerate electronic states by diagonalizing an effective Hamiltonian expanded in a compact non-relativistic basis. For a variety of atoms and small molecules across the entire periodic table, we demonstrate that DKH2-QDNEVPT2 is competitive in accuracy with variational two-component relativistic theories. BP2-QDNEVPT2 shows high accuracy for the second- and third-period elements, but its performance deteriorates for heavier atoms and molecules. We also consider the first-order spin-orbit QDNEVPT2 approximations (BP1- and DKH1-QDNEVPT2), among which DKH1-QDNEVPT2 is reliable but less accurate than DKH2-QDNEVPT2. Both DKH1- and DKH2-QDNEVPT2 hold promise as efficient and accurate electronic structure methods for treating electron correlation and spin-orbit coupling in a variety of applications.

Algebraic diagrammatic construction (ADC) is a computationally efficient approach for simulating excited electronic states, absorption spectra, and electron correlation. Due to their origin in perturbation theory, the single-reference ADC methods may be susceptible to spin contamination when applied to molecules with unpaired electrons. In this work, we develop an approach to quantify spin contamination in the ADC calculations of electronic excitations and apply it to a variety of open-shell molecules starting with either the unrestricted (UHF) or restricted open-shell (ROHF) Hartree-Fock reference wavefunctions. Our results show that the accuracy of low-order ADC approximations (ADC(2), ADC(3)) significantly decreases when the UHF reference spin contamination exceeds 0.05 a.u. Such strongly spin-contaminated molecules exhibit severe excited-state spin symmetry breaking that contributes to decreasing the quality of computed excitation energies and oscillator strengths. In a case study of phenyl radical, we demonstrate that spin contamination can significantly affect the simulated UV/Vis spectra, altering the relative energies, intensities, and order of electronic transitions. The results presented here motivate the development of spin-adapted ADC methods for open-shell molecules.

Accurate simulations of transient X-ray photoelectron spectra (XPS) provide unique opportunities to bridge the gap between theory and experiment in understanding the photoactivated dynamics in molecules and materials. However, simulating X-ray photoelectron spectra along a photochemical reaction pathway is challenging as it requires accurate description of electronic structure incorporating core-hole screening, orbital relaxation, electron correlation, and spin-orbit coupling in excited states or at nonequilibrium ground-state geometries. In this work, we employ the recently developed multireference algebraic diagrammatic construction theory (MR-ADC) to investigate the core-ionized states and X-ray photoelectron spectra of Fe(CO)5 and its photodissociation products (Fe(CO)4, Fe(CO)3) following excitation with 266 nm light. The simulated transient Fe 3p and CO 3{\sigma} XPS spectra incorporating spin-orbit coupling and high-order electron correlation effects are shown to be in a good agreement with the experimental measurements by Leitner et al. [J. Chem. Phys. 149, 044307 (2018)]. Our calculations suggest that core-hole screening, spin-orbit coupling, and ligand-field splitting effects are similarly important in reproducing the experimentally observed chemical shifts in transient Fe 3p XPS spectra of iron carbonyl complexes. Our results also demonstrate that the MR-ADC methods can be very useful in interpreting the transient XPS spectra of transition metal compounds.

The concept of Dyall zeroth-order Hamiltonian [Dyall, K. G. J. Chem. Phys., 102, 4909-4918 (1995)] has been instrumental in the development of intruder- and parameter-free multireference perturbation theories for the efficient treatment of static and dynamic correlation in molecular systems. In this review, we discuss two theoretical approaches based on the Dyall Hamiltonian: (i) N-electron valence perturbation theory and (ii) multireference algebraic diagrammatic construction theory. We briefly cover the technical aspects behind these theories and discuss their relationship. We conclude with a short description of alternative approaches based on the Dyall Hamiltonian, a summary, and an outlook on future developments.

We report the development and benchmark of multireference algebraic diagrammatic construction theory (MR-ADC) for the simulations of core-excited states and X-ray absorption spectra (XAS). Our work features an implementation that incorporates core-valence separation into the strict and extended second-order MR-ADC approximations (MR-ADC(2) and MR-ADC(2)-X), providing an efficient access to high-energy excited states without including inner-shell orbitals in the active space. Benchmark results on a set of small molecules indicate that at equilibrium geometries the accuracy of MR-ADC is similar to that of single-reference ADC theory when static correlation effects are not important. In this case, MR-ADC(2)-X performs similarly to single- and multireference coupled cluster methods in reproducing the experimental XAS peak spacings. We demonstrate the potential of MR-ADC for chemical systems with multiconfigurational electronic structure by calculating the K-edge XAS spectrum of the ozone molecule with a multireference character in its ground electronic state and the dissociation curve of core-excited molecular nitrogen. For ozone, the MR-ADC results agree well with the data from experimental and previous multireference studies of ozone XAS, in contrast to the results of single-reference methods, which underestimate relative peak energies and intensities. The MR-ADC methods also predict the correct shape of core-excited nitrogen potential energy curve, in a good agreement with accurate calculations using driven similarity renormalization group approaches. These findings suggest that MR-ADC(2) and MR-ADC(2)-X are promising methods for the XAS simulations of multireference systems and pave the way for their efficient computer implementation and applications.

Charged excitations are electronic transitions that involve a change in the total charge of a molecule or material. Understanding the properties and reactivity of charged species requires insights from theoretical calculations that can accurately describe orbital relaxation and electron correlation effects in open-shell electronic states. In this perspective, we review the current state of algebraic diagrammatic construction (ADC) theory for simulating charged excitations and its recent developments. We start with a short overview of ADC formalism for the one-particle Green's function, including its single- and multireference formulations and extension to periodic systems. Next, we focus on the capabilities of ADC methods and discuss recent findings about their accuracy for calculating a wide range of excited-state properties. We conclude our perspective by outlining possible directions for future developments of this theoretical approach.

We present the first implementation of spin-orbit coupling effects in fully internally contracted second-order quasidegenerate N-electron valence perturbation theory (SO-QDNEVPT2). The SO-QDNEVPT2 approach enables the computations of ground- and excited-state energies and oscillator strengths combining the description of static electron correlation with an efficient treatment of dynamic correlation and spin-orbit coupling. In addition to SO-QDNEVPT2 with the full description of one- and two-body spin-orbit interactions at the level of two-component Breit-Pauli Hamiltonian, our implementation also features a simplified approach that takes advantage of spin-orbit mean-field approximation (SOMF-QDNEVPT2). The accuracy of these methods is tested for the group 14 and 16 hydrides, 3d and 4d transition metal ions, and two actinide dioxides (neptunyl and plutonyl dications). The zero-field splittings of group 14 and 16 molecules computed using SO-QDNEVPT2 and SOMF-QDNEVPT2 are in a good agreement with the available experimental data. For the 3d transition metal ions, the SO-QDNEVPT2 method is significantly more accurate than SOMF-QDNEVPT2, while no substantial difference in the performance of two methods is observed for the 4d ions. Finally, we demonstrate that for the actinide dioxides the results of SO-QDNEVPT2 and SOMF-QDNEVPT2 are in a good agreement with the data from previous theoretical studies of these systems. Overall, our results demonstrate that SO-QDNEVPT2 and SOMF-QDNEVPT2 are promising multireference methods for treating spin-orbit coupling with a relatively low computational cost.

We present the first implementation and applications of non-Dyson algebraic diagrammatic construction theory for charged excitations in three-dimensional periodic solids (EA/IP-ADC). The EA/IP-ADC approach has a computational cost similar to the ground-state M{\o}ller-Plesset perturbation theory, enabling efficient calculations of a variety of crystalline excited-state properties (e.g., band structure, band gap, density of states) sampled in the Brillouin zone. We use EA/IP-ADC to compute the quasiparticle band structures and band gaps of several materials (from large-gap atomic and ionic solids to small-gap semiconductors) and analyze the errors of EA/IP-ADC approximations up to the third order in perturbation theory. Our work also reports the first-ever calculations of ground-state properties (equation-of-state and lattice constants) of three-dimensional crystalline systems using a periodic implementation of third-order M{\o}ller-Plesset perturbation theory (MP3).

Algebraic diagrammatic construction (ADC) theory is a computationally efficient and accurate approach for simulating electronic excitations in chemical systems. However, for the simulations of excited states in molecules with unpaired electrons the performance of ADC methods can be affected by the spin contamination in unrestricted Hartree-Fock (UHF) reference wavefunctions. In this work, we benchmark the accuracy of ADC methods for electron attachment and ionization of open-shell molecules with the UHF reference orbitals (EA/IP-ADC/UHF) and develop an approach to quantify the spin contamination in the charged excited states. Following this assessment, we demonstrate that the spin contamination can be reduced by combining EA/IP-ADC with the reference orbitals from restricted open-shell Hartree-Fock (ROHF) or orbital-optimized M{\o}ller-Plesset perturbation (OMP) theories. Our numerical results demonstrate that for open-shell systems with strong spin contamination in the UHF reference the third-order EA/IP-ADC methods with the ROHF or OMP reference orbitals are similar in accuracy to equation-of-motion coupled cluster theory with single and double excitations.

We present a new theoretical approach for the simulations of X-ray photoelectron spectra of strongly correlated molecular systems that combines multireference algebraic diagrammatic construction theory (MR-ADC) [J. Chem. Phys., 2018, 149, 204113] with core-valence separation (CVS) technique. The resulting CVS-MR-ADC approach has a low computational cost while overcoming many challenges of the conventional multireference theories associated with the calculations of excitations from inner-shell and core molecular orbitals. Our results demonstrate that the CVS-MR-ADC methods are as accurate as single-reference ADC approximations for predicting core ionization energies of weakly-correlated molecules, but are more accurate and reliable for systems with multireference character, such as stretched nitrogen molecule, ozone, and isomers of benzyne diradical. We also highlight the importance of multireference effects for the description of core-hole screening that determines the relative spacing and order of peaks in the XPS spectra of strongly correlated systems.

We present an implementation and benchmark of new approximations in multireference algebraic diagrammatic construction theory for simulations of neutral electronic excitations and UV/Vis spectra of strongly correlated molecular systems (MR-ADC). Following our work on the first-order MR-ADC approximation [J. Chem. Phys. 2018, 149, 204113], we report the strict and extended second-order MR-ADC methods (MR-ADC(2) and MR-ADC(2)-X) that combine the description of static and dynamic electron correlation in the ground and excited electronic states without relying on state-averaged reference wavefunctions. We present an extensive benchmark of the new MR-ADC methods for excited states in several small molecules, including the carbon dimer, ethylene, and butadiene. Our results demonstrate that for weakly-correlated electronic states the MR-ADC(2) and MR-ADC(2)-X methods outperform the third-order single-reference ADC approximation and are competitive with the results from equation-of-motion coupled cluster theory. For states with multireference character, the performance of the MR-ADC methods is similar to that of an N-electron valence perturbation theory. In contrast to conventional multireference perturbation theories, the MR-ADC methods have a number of attractive features, such as a straightforward and efficient calculation of excited-state properties and a direct access to excitations outside of the frontier (active) orbitals.

We present an efficient implementation of the second- and third-order single-reference algebraic diagrammatic construction theory for electron attachment (EA) and ionization (IP) energies and spectra (EA/IP-ADC(n), n = 2, 3). Our new EA/IP-ADC program features spin adaptation for closed-shell systems, density fitting for efficient handling of the two-electron integral tensors, as well as vectorized and parallel implementation of tensor contractions. We demonstrate capabilities of our efficient implementation by applying the EA/IP-ADC(n) (n = 2, 3) methods to compute the photoelectron spectrum of the TEMPO radical, as well as the vertical and adiabatic electron affinities of TEMPO and two DNA base pairs (guanine-cytosine and adenine-thymine). The spectra and electron affinities computed using large diffuse basis sets with up to 1028 molecular orbitals are found to be in a good agreement with the best available results from the experiment and theoretical simulations.

The previously proposed ansatz for density cumulant theory that combines orbital-optimization and a parameterization of the 2-electron reduced density matrix cumulant in terms of unitary coupled cluster amplitudes (OUDCT) is carefully examined. Formally, we elucidate the relationship between OUDCT and orbital-optimized unitary coupled cluster theory and show the existence of near-zero denominators in the stationarity conditions for both the exact and some approximate OUDCT methods. We implement methods of the OUDCT ansatz restricted to double excitations for numerical study, up to the fifth commutator in the Baker-Campbell-Hausdorff expansion. We find that methods derived from the ansatz beyond the previously known ODC-12 method tend to be less accurate for equilibrium properties and less reliable when attempting to describe $H2$ dissociation. New developments are needed to formulate more accurate DCT variants., Comment: 17 pages, 4 figures, appendices added

We report a new implementation of multireference algebraic diagrammatic construction theory (MR-ADC) for simulations of electron attachment and ionization in strongly correlated molecular systems (EA/IP-MR-ADC). Following our recent work on IP-MR-ADC [J. Chem. Theory Comput. 2019, 15, 5908], we present the first implementation of the second-order MR-ADC method for electron attachment (EA-MR-ADC(2)), as well as two extended second-order approximations (EA- and IP-MR-ADC(2)-X) that incorporate a partial treatment of third-order electron correlation effects. Introducing a small approximation for the second-order amplitudes of the effective Hamiltonian, our implementation of EA- and IP-MR-ADC(2)-X has a low O(M^5) computational scaling with the basis set size M. Additionally, we describe an efficient algorithm for solving the first-order amplitude equations in MR-ADC and partially-contracted second-order N-electron valence perturbation theory (NEVPT2) that completely avoids computation of the four-particle reduced density matrices without introducing any approximations or imaginary-time propagation. For a benchmark set of eight small molecules, carbon dimer, and a twisted ethylene, we demonstrate that EA- and IP-MR-ADC(2)-X achieve accuracy similar to that of strongly-contracted NEVPT2, while having a lower computational scaling with the active space size and providing efficient access to transition properties.

We present implementation of second- and third-order algebraic diagrammatic construction theory for efficient and accurate computations of molecular electron affinities (EA), ionization potentials (IP), and densities of states (EA-/IP-ADC(n), n = 2, 3). Our work utilizes the non-Dyson formulation of ADC for the single-particle propagator and reports working equations and benchmark results for the EA-ADC(2) and EA-ADC(3) approximations. We describe two algorithms for solving EA-/IP-ADC equations: (i) conventional algorithm that uses iterative diagonalization techniques to compute low-energy EA, IP, and density of states, and (ii) Green's function algorithm (GF-ADC) that solves a system of linear equations to compute density of states directly for a specified spectral region. To assess accuracy of EA-ADC(2) and EA-ADC(3), we benchmark their performance for a set of atoms, small molecules, and five DNA/RNA nucleobases. As our next step, we demonstrate efficiency of our GF-ADC implementation by computing core-level K-, L-, and M-shell ionization energies of a zinc atom without introducing core-valence separation approximation. Finally, we use EA- and IP-ADC methods to compute band gaps of equally-spaced hydrogen chains Hn with n up to 150, providing their estimates near thermodynamic limit. Our results demonstrate that EA-/IP-ADC(n) (n = 2, 3) methods are efficient and accurate alternatives to widely used electronic structure methods for simulations of electron attachment and ionization properties.

We present a second-order formulation of multi-reference algebraic diagrammatic construction theory [Sokolov, A. Yu. J. Chem. Phys. 2018, 149, 204113] for simulating photoelectron spectra of strongly correlated systems (MR-ADC(2)). The MR-ADC(2) method uses second-order multi-reference perturbation theory (MRPT2) to efficiently obtain ionization energies and intensities for many photoelectron transitions in a single computation. In contrast to conventional MRPT2 methods, MR-ADC(2) provides information about ionization of electrons in all orbitals (i.e., core and active) and allows to compute transition intensities in straightforward and efficient way. Although equations of MR-ADC(2) depend on four-particle reduced density matrices, we demonstrate that computation of these large matrices can be completely avoided without introducing any approximations. The resulting MR-ADC(2) implementation has a lower computational scaling compared to conventional MRPT2 methods. We present results of MR-ADC(2) for photoelectron spectra of small molecules, carbon dimer, and equally-spaced hydrogen chains (H10 and H30) and outline directions for future developments.

We present a new approach for simulating X-ray absorption spectra based on linear-response density cumulant theory (LR-DCT) [A. V. Copan and A. Yu. Sokolov, J. Chem. Theory Comput., 2018, 14, 4097 - 4108]. Our new method combines the LR-ODC-12 formulation of LR-DCT with core-valence separation approximation (CVS) that allows to efficiently access high-energy core-excited states. We describe our computer implementation of the CVS-approximated LR-ODC-12 method (CVS-ODC-12) and benchmark its performance by comparing simulated X-ray absorption spectra to those obtained from experiment for several small molecules. Our results demonstrate that the CVS-ODC-12 method shows a good agreement with experiment for relative spacings between transitions and their intensities, but the excitation energies are systematically overestimated. When comparing to results from excited-state coupled cluster methods with single and double excitations, the CVS-ODC-12 method shows a similar performance for intensities and peak separations, while coupled cluster spectra are less shifted, relative to experiment. An important advantage of CVS-ODC-12 is that its excitation energies are computed by diagonalizing a Hermitian matrix, which enables efficient computation of transition intensities.

We present a multi-reference generalization of the algebraic diagrammatic construction theory (ADC) [J. Schirmer, Phys. Rev. A 26, 2395 (1982)] for excited electronic states. The resulting multi-reference ADC approach (MR-ADC) can be efficiently and reliably applied to systems, which exhibit strong electron correlation in the ground or excited electronic states. In contrast to conventional multi-reference perturbation theories, MR-ADC describes electronic transitions involving all orbitals (core, active, and external) and enables efficient computation of spectroscopic properties, such as transition amplitudes and spectral densities. Our derivation of MR-ADC is based on the effective Liouvillean formalism of Mukherjee and Kutzelnigg [D. Mukherjee, W. Kutzelnigg, in Many-Body Methods in Quantum Chemistry (1989), pp. 257--274], which we generalize to multi-determinant reference states. We discuss a general formulation of MR-ADC, perform its perturbative analysis, and present an implementation of the first-order MR-ADC approximation, termed MR-ADC(1), as a first step in defining the MR-ADC hierarchy of methods. We show results of MR-ADC(1) for the excitation energies of the Be atom, an avoided crossing in LiF, doubly excited states in C2, and outline directions for our future developments.

We present a linear-response formulation of density cumulant theory (DCT) that provides a balanced and accurate description of many electronic states simultaneously. In the original DCT formulation, only information about a single electronic state (usually, the ground state) is obtained. We discuss the derivation of linear-response DCT, present its implementation for the ODC-12 method (LR-ODC-12), and benchmark its performance for excitation energies in small molecules (N$_2$, CO, HCN, HNC, C$_2$H$_2$, and H$_2$CO), as well as challenging excited states in ethylene, butadiene, and hexatriene. For small molecules, LR-ODC-12 shows smaller mean absolute errors in excitation energies than equation-of-motion coupled cluster theory with single and double excitations (EOM-CCSD), relative to the reference data from EOM-CCSDT. In a study of butadiene and hexatriene, LR-ODC-12 correctly describes the relative energies of the singly-excited $1^1\mathrm{B_{u}}$ and the doubly-excited $2^1\mathrm{A_{g}}$ states, in excellent agreement with highly accurate semistochastic heat-bath configuration interaction results, while EOM-CCSD overestimates the energy of the $2^1\mathrm{A_{g}}$ state by almost 1 eV. Our results demonstrate that linear-response DCT is a promising theoretical approach for excited states of molecules.

We discuss the time-dependent formulation of perturbation theory in the context of the interacting zeroth-order Hamiltonians that appear in multi-reference situations. As an example, we present a time-dependent formulation and implementation of second-order n-electron valence perturbation theory. The resulting t-NEVPT2 method yields the fully uncontracted n-electron valence perturbation wavefunction and energy, but has a lower computational scaling than the usual contracted variants, and also avoids the construction of high-order density matrices and the diagonalization of metrics. We present results of t-NEVPT2 for the water, nitrogen, carbon, and chromium molecules, and outline directions for the future.

We describe how multirefence dynamic correlation theories can be naturally obtained as single-reference correlation theories in a canonically transformed frame. Such canonically transformed correlation theories are very simple and involve identical expressions to their single-reference counterparts. The corresponding excitations involve quasiparticles rather than the bare particles of the system. High-order density matrices (or their approximations) and the numerical metric instabilities common to multireference correlation theories do not appear. As an example, we formulate the Bogoliubov canonically transformed version of second-order M{\o}ller-Plesset perturbation theory and demonstrate its performance in hydrogen, water, nitrogen, and BeH$_2$ bond dissociation.