Natural orbitals and two-particle correlators as tools for the analysis of effective exchange couplings in solids.

Using generalizations of spin-averaged natural orbitals and two-particle charge correlators for solids, we investigate the electronic structure of antiferromagnetic transition-metal oxides with a fully self-consistent, imaginary-time GW method. Our findings disagree with the Goodenough-Kanamori (GK) rules that are commonly used for the qualitative interpretation of such solids. First, we found a strong dependence of the natural orbital occupancies on momenta, contradicting GK assumptions. Second, along the momentum path, the character of natural orbitals changes. In particular, the contributions of oxygen 2s orbitals are important, which has not been considered in the GK rules. To analyze the influence of the electronic correlation on the values of effective exchange coupling constants, we use both natural orbitals and two-particle correlators and show that electronic screening modulates the degree of superexchange by stabilizing the charge-transfer contributions, which greatly affects these coupling constants. Finally, we give a set of predictions and recommendations regarding the use of density functional, Green's function, and wave-function methods for evaluating effective magnetic couplings in molecules and solids.

State-Interaction Approach for Evaluating g-Tensors within EOM-CC and RAS-CI Frameworks: Theory and Benchmarks.

Among various techniques designed for studying open-shell species, electron paramagnetic resonance (EPR) spectroscopy plays an important role. The key quantity measured by EPR is the g-tensor, describing the coupling between an external magnetic field and molecular electronic spin. One theoretical framework for quantum chemistry calculations of g-tensors is based on response theory, which involves substantial developments that are specific to the underlying electronic structure models. A simplified and easier-to-implement approach is based on the state-interaction scheme, in which perturbation is included by considering a small number of states. We describe and benchmark the state-interaction approach using equation-of-motion coupled-cluster and restricted-active-space configuration interaction wave functions. The analysis confirms that this approach can deliver accurate results and highlights caveats of applying it, such as a choice of the reference state, convergence with respect to the number of states used in calculations, etc. The analysis also contributes toward a better understanding of challenges in calculations of higher-order properties using approximate wave functions.

Broken-symmetry self-consistent GW approach: Degree of spin contamination and evaluation of effective exchange couplings in solid antiferromagnets.

We adopt a broken-symmetry strategy for evaluating effective magnetic constants J within the fully self-consistent GW method. To understand the degree of spin contamination present in broken-symmetry periodic solutions, we propose several extensive quantities demonstrating that the unrestricted self-consistent GW preserves the broken-symmetry character of the unrestricted Hartree-Fock solutions. The extracted J are close to the ones obtained from multireference wave-function calculations. In this paper, we establish a robust computational procedure for finding magnetic coupling constants from self-consistent GW calculations and apply it to solid antiferromagnetic nickel and manganese oxides.

Iterative subspace algorithms for finite-temperature solution of Dyson equation.

One-particle Green's functions obtained from the self-consistent solution of the Dyson equation can be employed in the evaluation of spectroscopic and thermodynamic properties for both molecules and solids. However, typical acceleration techniques used in the traditional quantum chemistry self-consistent algorithms cannot be easily deployed for the Green's function methods because of a non-convex grand potential functional and a non-idempotent density matrix. Moreover, the optimization problem can become more challenging due to the inclusion of correlation effects, changing chemical potential, and fluctuations of the number of particles. In this paper, we study acceleration techniques to target the self-consistent solution of the Dyson equation directly. We use the direct inversion in the iterative subspace (DIIS), the least-squared commutator in the iterative subspace (LCIIS), and the Krylov space accelerated inexact Newton method (KAIN). We observe that the definition of the residual has a significant impact on the convergence of the iterative procedure. Based on the Dyson equation, we generalize the concept of the commutator residual used in DIIS and LCIIS and compare it with the difference residual used in DIIS and KAIN. The commutator residuals outperform the difference residuals for all considered molecular and solid systems within both GW and GF2. For a number of bond-breaking problems, we found that an easily obtained high-temperature solution with effectively suppressed correlations is a very effective starting point for reaching convergence of the problematic low-temperature solutions through a sequential reduction of temperature during calculations.

Spin-orbit couplings within spin-conserving and spin-flipping time-dependent density functional theory: Implementation and benchmark calculations.

We present a new implementation for computing spin-orbit couplings (SOCs) within a time-dependent density-functional theory (TD-DFT) framework in the standard spin-conserving formulation as well in the spin-flip variant (SF-TD-DFT). This approach employs the Breit-Pauli Hamiltonian and Wigner-Eckart's theorem applied to the reduced one-particle transition density matrices, together with the spin-orbit mean-field treatment of the two-electron contributions. We use a state-interaction procedure and compute the SOC matrix elements using zero-order non-relativistic states. Benchmark calculations using several closed-shell organic molecules, diradicals, and a single-molecule magnet illustrate the efficiency of the SOC protocol. The results for organic molecules (described by standard TD-DFT) show that SOCs are insensitive to the choice of the functional or basis sets, as long as the states of the same characters are compared. In contrast, the SF-TD-DFT results for small diradicals (CH2, NH2 +, SiH2, and PH2 +) show strong functional dependence. The spin-reversal energy barrier in a Fe(III) single-molecule magnet computed using non-collinear SF-TD-DFT (PBE0, ωPBEh/cc-pVDZ) agrees well with the experimental estimate.

Interpretation of multiple solutions in fully iterative GF2 and GW schemes using local analysis of two-particle density matrices.

Due to the presence of non-linear equations, iterative Green's function methods can result in multiple different solutions even for simple molecular systems. In contrast to the wave-function methods, a detailed and careful analysis of such molecular solutions was not performed before. In this work, we use two-particle density matrices to investigate local spin and charge correlators that quantify the charge resonance and covalent characters of these solutions. When applied within the unrestricted orbital set, spin correlators elucidate the broken symmetry of the solutions, containing necessary information for building effective magnetic Hamiltonians. Based on GW and GF2 calculations of simple molecules and transition metal complexes, we construct Heisenberg Hamiltonians, four-spin-four-center corrections, and biquadratic spin-spin interactions. These Hamiltonian parameterizations are compared to previous wave-function calculations.

Evaluation of two-particle properties within finite-temperature self-consistent one-particle Green's function methods: Theory and application to GW and GF2.

One-particle Green's function methods can model molecular and solid spectra at zero or non-zero temperatures. One-particle Green's functions directly provide electronic energies and one-particle properties, such as dipole moment. However, the evaluation of two-particle properties, such as ⟨S2⟩ and ⟨N2⟩, can be challenging because they require a solution of the computationally expensive Bethe-Salpeter equation to find two-particle Green's functions. We demonstrate that the solution of the Bethe-Salpeter equation can be completely avoided. Applying the thermodynamic Hellmann-Feynman theorem to self-consistent one-particle Green's function methods, we derive expressions for two-particle density matrices in a general case and provide explicit expressions for GF2 and GW methods. Such density matrices can be decomposed into an antisymmetrized product of correlated one-electron density matrices and the two-particle electronic cumulant of the density matrix. Cumulant expressions reveal a deviation from ensemble representability for GW, explaining its known deficiencies. We analyze the temperature dependence of ⟨S2⟩ and ⟨N2⟩ for a set of small closed-shell systems. Interestingly, both GF2 and GW show a non-zero spin contamination and a non-zero fluctuation of the number of particles for closed-shell systems at the zero-temperature limit.

Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package.

This article summarizes technical advances contained in the fifth major release of the Q-Chem quantum chemistry program package, covering developments since 2015. A comprehensive library of exchange-correlation functionals, along with a suite of correlated many-body methods, continues to be a hallmark of the Q-Chem software. The many-body methods include novel variants of both coupled-cluster and configuration-interaction approaches along with methods based on the algebraic diagrammatic construction and variational reduced density-matrix methods. Methods highlighted in Q-Chem 5 include a suite of tools for modeling core-level spectroscopy, methods for describing metastable resonances, methods for computing vibronic spectra, the nuclear-electronic orbital method, and several different energy decomposition analysis techniques. High-performance capabilities including multithreaded parallelism and support for calculations on graphics processing units are described. Q-Chem boasts a community of well over 100 active academic developers, and the continuing evolution of the software is supported by an "open teamware" model and an increasingly modular design.

The elusive dynamics of aqueous permanganate photochemistry.

Despite decades of investigation, mechanistic details of aqueous permanganate photo-decomposition remain unclear. Here we follow photoinduced dynamics of aqueous permanganate with femtosecond spectroscopy. Photoexcitation of KMnO4(aq) in the visible unleashes a sub-picosecond cascade of non-radiative transitions, leading to a distinct species which relaxes to S0 with a lifetime of 16 ps. Tuning excitation to the UV shows increasing formation of a metastable intermediate, which outlives our ∼1 ns window of detection. Guided by electronic structure calculations and observations from three pulse excitation experiments, we assign the 16 ps species as the lowest Jahn-Teller component of the 3T1 triplet state and suggest a plausible sequence of radiationless transitions, which rapidly populate it. In conjunction with photodecomposition quantum yields obtained from the literature, these results demonstrate that aqueous permanganate photo-decomposition proceeds through a long-lived intermediate which is formed in parallel to the triplet in less than one ps upon UV absorption. The possibility that this is the postulated highly oxidative peroxo species, a fraction of which leads to the stable (MnO2- + O2) fragments, is discussed. Finally, periodic modulations detected in the pump-probe signal are assigned to ground-state vibrational coherences excited by impulsive Raman. Their wavelength-dependent absolute phases outline the borders between adjacent electronic transitions in the linear spectrum of permanganate.

Effective Hamiltonians derived from equation-of-motion coupled-cluster wave functions: Theory and application to the Hubbard and Heisenberg Hamiltonians.

Effective Hamiltonians, which are commonly used for fitting experimental observables, provide a coarse-grained representation of exact many-electron states obtained in quantum chemistry calculations; however, the mapping between the two is not trivial. In this contribution, we apply Bloch's formalism to equation-of-motion coupled-cluster wave functions to rigorously derive effective Hamiltonians in Bloch's and des Cloizeaux's forms. We report the key equations and illustrate the theory by application to systems with two or three unpaired electrons, which give rise to electronic states of covalent and ionic characters. We show that Hubbard's and Heisenberg's Hamiltonians can be extracted directly from the so-obtained effective Hamiltonians. By establishing a quantitative connection between many-body states and simple models, the approach facilitates the analysis of the correlated wave functions. We propose a simple diagnostic for assessing the validity of the model space choice based on the overlaps between the target- and model-space states. Artifacts affecting the quality of electronic structure calculations such as spin contamination are also discussed.

Extension of frozen natural orbital approximation to open-shell references: Theory, implementation, and application to single-molecule magnets.

Natural orbitals are often used to achieve a more compact representation of correlated wave-functions. Using natural orbitals computed as eigenstates of the virtual-virtual block of the state density matrix instead of the canonical Hartree-Fock orbitals results in smaller errors when the same fraction of virtual space is frozen. This strategy, termed frozen natural orbital (FNO) approach, is effective in reducing the cost of regular coupled-cluster (CC) calculations and some multistate methods, such as EOM-IP-CC (equation-of-motion CC for ionization potentials). This contribution extends the FNO approach to the EOM-SF-CC ansatz (EOM-CC with spin-flip). In contrast to EOM-IP-CCSD, EOM-SF-CCSD relies on high-spin open-shell references. Using FNOs computed for an open-shell reference leads to an erratic behavior of the EOM-SF-CC energies and properties due to an inconsistent truncation of the α and ß orbital spaces. A general solution to problems arising in the EOM-CC calculations utilizing open-shell references, termed OSFNO (open-shell FNO), is proposed. By means of singular value decomposition (SVD) of the overlap matrix of the α and ß orbitals, the OSFNO algorithm identifies the corresponding orbitals and determines virtual orbitals corresponding to the singly occupied space. This is followed by SVD of the singlet part of the state density matrix in the remaining virtual orbital subspace. The so-computed FNOs preserve the spin purity of the open-shell orbital subspace to the extent allowed by the original reference, thus facilitating a safe truncation of the virtual space. The performance of OSFNO is benchmarked for selected diradicals and triradicals.

Calculation of spin-orbit couplings using RASCI spinless one-particle density matrices: Theory and applications.

This work presents the formalism and implementation for calculations of spin-orbit couplings (SOCs) using the Breit-Pauli Hamiltonian and non-relativistic wave functions described by the restricted active space configuration interaction (RASCI) method with general excitation operators of spin-conserving spin-flipping, ionizing, and electron-attaching types. The implementation is based on the application of the Wigner-Eckart theorem within the spin space, which enables the calculation of the entire SOC matrix based on the explicit calculation of just one transition between the two spin multiplets. Numeric results for a diverse set of atoms and molecules highlight the importance of a balanced treatment of correlation and adequate basis sets and illustrate the overall robust performance of RASCI SOCs. The new implementation is a useful addition to the methodological toolkit for studying spin-forbidden processes and molecular magnetism.

Spin-Forbidden Channels in Reactions of Unsaturated Hydrocarbons with O(3P).

The electronic structure of four prototypical Cvetanovic diradicals, species derived by addition of O(3P) to unsaturated compounds, is investigated by high-level electronic structure calculations and kinetics modeling. The main focus of this study is on the electronic factors controlling the rate of intersystem crossing (ISC): minimal energy crossing points (MECPs) and spin-orbit couplings (SOCs). The calculations illuminate significant differences in the electronic structure of ethene- and ethyne-derived compounds and explain the effect of methylation. The computed MECP heights and SOCs reveal different mechanisms of ISC in ethene- and ethyne-derived species, thus explaining variations in the observed branching ratios between singlet and triplet products and a puzzling effect of methyl substitution. In the ethene- and propene-derived species, the MECP is very low and the rate is controlled by variations of SOC, whereas in the ethyne- and propyne-derived species the MECP is high and the changes in the ISC rate due to methyl substitutions are driven by the variations of MECP heights.

General framework for calculating spin-orbit couplings using spinless one-particle density matrices: Theory and application to the equation-of-motion coupled-cluster wave functions.

Standard implementations of nonrelativistic excited-state calculations compute only one component of spin multiplets (i.e., Ms = 0 triplets); however, matrix elements for all components are necessary for deriving spin-dependent experimental observables. Wigner-Eckart's theorem allows one to circumvent explicit calculations of all multiplet components. We generate all other spin-orbit matrix elements by applying Wigner-Eckart's theorem to a reduced one-particle transition density matrix computed for a single multiplet component. In addition to computational efficiency, this approach also resolves the phase issue arising within Born-Oppenheimer's separation of nuclear and electronic degrees of freedom. A general formalism and its application to the calculation of spin-orbit couplings using equation-of-motion coupled-cluster wave functions are presented. The two-electron contributions are included via the mean-field spin-orbit treatment. Intrinsic issues of constructing spin-orbit mean-field operators for open-shell references are discussed, and a resolution is proposed. The method is benchmarked by using several radicals and diradicals. The merits of the approach are illustrated by a calculation of the barrier for spin inversion in a high-spin tris(pyrrolylmethyl)amine Fe(II) complex.

Evaluation of Neel Temperatures from Fully Self-Consistent Broken-Symmetry GW and High-Temperature Expansion: Application to Cubic Transition-Metal Oxides.

Using fully self-consistent thermal broken-symmetry GW, we construct effective magnetic Heisenberg Hamiltonians for a series of transition metal oxides (NiO, CoO, FeO, and MnO), capturing a rigorous but condensed description of the magnetic states. Then applying high-temperature expansion, we find the decomposition coefficients for spin susceptibility and specific heat. The radius of convergence of the found series determines the Neel temperature. The NiO, CoO, and FeO contain a small ferromagnetic interaction between the nearest neighbors (NNs) and the dominant antiferromagnetic interaction between the next-nearest neighbors (NNNs). For them, the derived Neel temperatures are in good agreement with experiment. The case of MnO is different because both NN and NNN couplings are antiferromagnetic and comparable in magnitude, for which the error in the estimated Neel temperature is larger, which is a signature of additional effects not captured by electronic structure calculations.

Equation-of-Motion Coupled-Cluster Theory to Model L-Edge X-ray Absorption and Photoelectron Spectra.

We present an extension of the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory for computing X-ray L-edge spectra, both in the absorption (XAS) and in the photoelectron (XPS) regimes. The approach is based on the perturbative evaluation of spin-orbit couplings using the Breit-Pauli Hamiltonian and nonrelativistic wave functions described by the fc-CVS-EOM-CCSD ansatz (EOM-CCSD within the frozen-core core-valence separated (fc-CVS) scheme). The formalism is based on spinless one-particle density matrices. The approach is illustrated by modeling XAS and XPS of several model systems ranging from Ar to small molecules containing sulfur and silicon.

Quantitative El-Sayed Rules for Many-Body Wave Functions from Spinless Transition Density Matrices.

One-particle transition density matrices and natural transition orbitals enable quantitative description of electronic transitions and interstate properties involving correlated many-body wave functions within the molecular orbital framework. Here we extend the formalism to the analysis of tensor properties, such as spin-orbit couplings (SOCs), which involve states of different spin projection. By using spinless density matrices and Wigner-Eckart's theorem, the approach allows one to treat the transitions between states with arbitrary spin projections in a uniform way. In addition to a pictorial representation of the transition, the analysis also yields quantitative contributions of hole-particle pairs into the overall many-body matrix elements. In particular, it helps to rationalize the magnitude of computed SOCs in terms of El-Sayed's rules. The capabilities of the new tool are illustrated by the analysis of the equation-of-motion coupled-cluster calculations of two transition metal complexes.

Double Precision Is Not Needed for Many-Body Calculations: Emergent Conventional Wisdom.

Using single-precision floating-point representation reduces the size of data and computation time by a factor of 2 relative to double precision conventionally used in electronic structure programs. For large-scale calculations, such as those encountered in many-body theories, reduced memory footprint alleviates memory and input/output bottlenecks. Reduced size of data can lead to additional gains due to improved parallel performance on CPUs and various accelerators. However, using single precision can potentially degrade the accuracy of the computed quantities. Here we report an implementation of coupled-cluster and equation-of-motion coupled-cluster methods with single and double excitations in single precision. We consider both standard implementation and one using Cholesky decomposition or resolution-of-the-identity representation of electron-repulsion integrals. Numerical tests illustrate that when single precision is used in correlated calculations, the loss of accuracy is insignificant, and pure single-precision implementation can be used for computing energies, analytic gradients, excited states, and molecular properties. In addition to pure single-precision calculations, our implementation allows one to follow a single-precision calculation by cleanup iterations, fully recovering double-precision results while retaining significant savings.