Equation-of-motion Coupled-cluster Singles And Doubles Research Articles

Coupled-cluster treatment of complex open-shell systems: the case of single-molecule magnets.

We investigate the reliability of two cost-effective coupled-cluster methods for computing spin-state energetics and spin-related properties of a set of open-shell transition-metal complexes. Specifically, we employ the second-order approximate coupled-cluster singles and doubles (CC2) method and projection-based embedding that combines equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) with density functional theory (DFT). The performance of CC2 and EOM-CCSD-in-DFT is assessed against EOM-CCSD. The chosen test set includes two hexaaqua transition-metal complexes containing Fe(II) and Fe(III), and a large Co(II)-based single-molecule magnet with a non-aufbau ground state. We find that CC2 describes the excited states more accurately, reproducing EOM-CCSD excitation energies within 0.05 eV. However, EOM-CCSD-in-DFT excels in describing transition orbital angular momenta and spin-orbit couplings. Moreover, for the Co(II) molecular magnet, using EOM-CCSD-in-DFT eigenstates and spin-orbit couplings, we compute spin-reversal energy barriers, as well as temperature-dependent and field-dependent magnetizations and magnetic susceptibilities that closely match experimental values within spectroscopic accuracy. These results underscore the efficiency of CC2 in computing state energies of multi-configurational, open-shell systems and highlight the utility of the more cost-efficient EOM-CCSD-in-DFT for computing spin-orbit couplings and magnetic properties of complex and large molecular magnets.

Electronic Characterization of Glycolaldehyde: Experimental and Theoretical Insights from the Core- and Valence-Level Spectroscopy.

The core-level electron excitation and ionization spectra of glycolaldehyde have been investigated by photoabsorption and photoemission spectroscopy at both carbon and oxygen K-edges; the valence ionization spectra were also recorded by photoelectron spectroscopy in the UV-vis region. The spectra are interpreted by means of ab initio calculations based on the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) and coupled cluster singles, doubles, and perturbative are in good agreement with the experimental results, and many of the observed features are assigned. The photoabsorption spectra are not only dominated by transitions from core-level orbitals to unoccupied π and σ orbitals but also show structures due to Rydberg transitions.

The electronic structure of 2(5H)-thiophenone investigated by vacuum ultraviolet synchrotron radiation and theoretical calculations

The electronic state spectroscopy of 2(5H)-thiophenone, C4H4OS, has been investigated by high-resolution vacuum ultraviolet photoabsorption in the 3.76–10.69 eV energy range using synchrotron radiation, together with novel quantum chemical calculations performed at the equation of motion coupled cluster singles and doubles (EOM-CCSD) level of theory. The major electronic transitions have been assigned to valence and Rydberg character, with relevant C=O, C=C and C–C stretching vibrations across the entire absorption spectrum. Photolysis lifetimes in the Earth’s atmosphere (0–50 km altitude) have been estimated from the absolute photoabsorption cross-sections, indicating that solar photolysis can be expected to be a strong sink mechanism.Graphical abstract

Valence and Rydberg excitations of 2-fluorotoluene in the 4.4–10.8 eV photoabsorption energy region

The electronic state spectroscopy of 2-fluorotoluene in the gas phase has been investigated for the first time using high-resolution vacuum ultraviolet photoabsorption experiments in the 4.4–10.8 eV energy-range, with absolute cross-section measurements obtained. Additionally, we also present a novel set of ab initio calculations (vertical excitation energies and oscillator strengths) at two different levels of theory, equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) and time-dependent density functional theory (TD-DFT). These are used in the assignment of valence, mix valence-Rydberg and Rydberg transitions, with the associated vibronic series analysed. The measured absolute photoabsorption cross-sections have been used to calculate the photolysis lifetime of 2-fluorotoluene in the Earth's atmosphere.

Building on the strengths of a double-hybrid density functional for excitation energies and inverted singlet-triplet energy gaps.

It is demonstrated that a double hybrid density functional approximation, ωB88PTPSS, that incorporates equipartition of density functional theory and the non-local correlation, however with a meta-generalized gradient approximation correlation functional, as well as with the range-separated exchange of ωB2PLYP, provides accurate excitation energies for conventional systems, as well as correct prescription of negative singlet-triplet gaps for non-conventional systems with inverted gaps, without any necessity for parametric scaling of the same-spin and opposite-spin non-local correlation energies. Examined over "safe" excitations of the QUESTDB set, ωB88PTPSS performs quite well for open-shell systems, correctly and fairly accurately [relative to equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) reference] predicts negative gaps for 50 systems with inverted singlet-triplet gaps, and is one of the leading performers for intramolecular charge-transfer excitations and achieves near-second-order approximate coupled cluster (CC2) and second-order algebraic diagrammatic construction quality for the Q1 and Q2 subsets. Subsequently, we tested ωB88PTPSS on two sets of real-life examples from recent computational chemistry literature-the low energy bands of chlorophyll a (Chl a) and a set of thermally activated delayed fluorescence (TADF) systems. For Chl a, ωB88PTPSS qualitatively and quantitatively achieves DLPNO-STEOM-CCSD-level performance and provides excellent agreement with experiment. For TADF systems, ωB88PTPSS agrees quite well with spin-component-scaled CC2 (SCS-CC2) excitation energies, as well as experimental values, for the gaps between the S1 and T1 excited states.

Electronic, vibrational, and rotational analysis of 1,2-benzanthracene by high-resolution spectroscopy referenced to an optical frequency comb.

The electronic and vibrational structures of 1,2-benzanthracene-h12 (aBA-h12) and 1,2-benzanthracene-d12 (aBA-d12) were elucidated by analyzing fluorescence excitation spectra and dispersed fluorescence spectra in a supersonic jet on the basis of DFT calculation. We also observed the high-resolution and high-precision fluorescence excitation spectrum of the S1←S000 0 band, and determined the accurate rotational constants in the zero-vibrational levels of the S0 and S1 states. In this high-resolution measurement, we used a single-mode UV laser whose frequencies were controlled with reference to an optical frequency comb. The inertial defect is negligibly small, the molecule is considered to be planar, and the obtained rotational constants were well reproduced by the equation-of-motion coupled cluster singles and doubles (EOM-CCSD) calculation. Both a-type and b-type transitions are found to be included in the rotationally resolved spectrum, and the a-type contribution is dominant, that is, the transition moment is nearly parallel to the long axis of the aBA molecule. We concluded that the S1 state is mainly composed of the Φ(B) configuration. The observed fluorescence lifetime (106ns) is considerably longer than that of the Φ(A) system, such as anthracene (18ns). The transition moment for the lower state of mixed states becomes small, reflecting a near-cancelation of the contributions from the parts of the wavefunction corresponding to the two electronic configurations. The bandwidth of the S2 ← S0 transition is large, and the structure is complicated. It is attributed to vibronic coupling with the high vibrational levels of the S1 state.

Cherry-Picking Resolvents: Recovering the Valence Contribution in X-ray Two-Photon Absorption within the Core-Valence-Separated Equation-of-Motion Coupled-Cluster Response Theory.

Calculations of first-order response wave functions in the X-ray regime often diverge within correlated frameworks such as equation-of-motion coupled-cluster singles and doubles (EOM-CCSD), a consequence of the coupling with the valence ionization continuum. Here, we extend our strategy of introducing a hierarchy of approximations to the EOM-EE-CCSD resolvent (or, inversely, the model Hamiltonian) involved in the response equations for the calculation of X-ray two-photon absorption (X2PA) cross sections. We exploit the frozen-core core-valence separation (fc-CVS) scheme to first decouple the core and valence Fock spaces, followed by a separate approximate treatment of the valence resolvent. We demonstrate the robust convergence of X-ray response calculations within this framework and compare X2PA spectra of small benchmark molecules with the previously reported density functional theory results.

Embedded equation-of-motion coupled-cluster theory for electronic excitation, ionisation, electron attachment, and electronic resonances

The projection-based quantum embedding method is applied to a comprehensive set of electronic states. We embed different variants of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) theory in density functional theory and investigate electronically excited states of valence, Rydberg, and charge-transfer character, valence- and core-ionised states, as well as bound and temporary radical anions. The latter states, which are unstable towards electron loss, are treated by means of a complex-absorbing potential. Besides transition energies, we present Dyson orbitals and natural transition orbitals for embedded EOM-CCSD. We benchmark the performance of the embedded EOM-CCSD methods against full EOM-CCSD using small organic molecules microsolvated by a varying number of water molecules as test cases. Our results illustrate that embedded EOM-CCSD describes ionisation and valence excitation very well and that these transitions are quite insensitive towards technical details of the embedding procedure. On the contrary, more care is required when dealing with Rydberg excitations or electron attachment. For the latter type of transition in particular, the use of long-range corrected density functionals is mandatory and truncation of the virtual orbital space proves to be difficult.

A periodic equation-of-motion coupled-cluster implementation applied to F-centers in alkaline earth oxides.

We present an implementation of the equation of motion coupled-cluster singles and doubles (EOM-CCSD) theory using periodic boundary conditions and a plane wave basis set. Our implementation of EOM-CCSD theory is applied to study F-centers in alkaline earth oxides employing a periodic supercell approach. The convergence of the calculated electronic excitation energies for neutral color centers in MgO, CaO, and SrO crystals with respect to the orbital basis set and system size is explored. We discuss extrapolation techniques that approximate excitation energies in the complete basis set limit and reduce finite size errors. Our findings demonstrate that EOM-CCSD theory can predict optical absorption energies of F-centers in good agreement with experiment. Furthermore, we discuss calculated emission energies corresponding to the decay from triplet to singlet states responsible for the photoluminescence properties. Our findings are compared to experimental and theoretical results available in the literature.

Cherry-picking resolvents: A general strategy for convergent coupled-cluster damped response calculations of core-level spectra

Damped linear response calculations within the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) framework usually diverge in the x-ray regime. This divergent behavior stems from the valence ionization continuum in which the x-ray response states are embedded. Here, we introduce a general strategy for removing the continuum from the response manifold while preserving important spectral properties of the model Hamiltonian. The strategy is based on decoupling the core and valence Fock spaces using the core-valence separation (CVS) scheme combined with separate (approximate) treatment of the core and valence resolvents. We illustrate this approach with the calculations of resonant inelastic x-ray scattering (RIXS) spectra of benzene and para-nitroaniline using EOM-CCSD wave functions and several choices of resolvents, which differ in their treatment of the valence manifold. The method shows robust convergence and extends the previously introduced CVS-EOM-CCSD RIXS scheme to systems for which valence contributions to the total cross section are important, such as the push-pull chromophores with charge-transfer states.

Accurate prediction of core-level spectra of radicals at density functional theory cost via square gradient minimization and recoupling of mixed configurations.

State-specific orbital optimized approaches are more accurate at predicting core-level spectra than traditional linear-response protocols, but their utility had been restricted due to the risk of "variational collapse" down to the ground state. We employ the recently developed square gradient minimization [D. Hait and M. Head-Gordon, J. Chem. Theory Comput. 16, 1699 (2020)] algorithm to reliably avoid variational collapse and study the effectiveness of orbital optimized density functional theory (DFT) at predicting second period element 1s core-level spectra of open-shell systems. Several density functionals (including SCAN, B3LYP, and ωB97X-D3) are found to predict excitation energies from the core to singly occupied levels with high accuracy (≤0.3 eV RMS error) against available experimental data. Higher excited states are, however, more challenging by virtue of being intrinsically multiconfigurational. We thus present a configuration interaction inspired route to self-consistently recouple single determinant mixed configurations obtained from DFT, in order to obtain approximate doublet states. This recoupling scheme is used to predict the C K-edge spectra of the allyl radical, the O K-edge spectra of CO+, and the N K-edge of NO2 with high accuracy relative to experiment, indicating substantial promise in using this approach for the computation of core-level spectra for doublet species [vs more traditional time dependent DFT, equation of motion coupled cluster singles and doubles (EOM-CCSD), or using unrecoupled mixed configurations]. We also present general guidelines for computing core-excited states from orbital optimized DFT.

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.

Unusual Complexes of P(CH)3 with FH, ClH, and ClF.

Ab initio MP2/aug’-cc-pVTZ calculations have been performed to determine the structures and binding energies of complexes formed by phosphatetrahedrane, P(CH)3, and HF, HCl, and ClF. Four types of complexes exist on the potential energy surfaces. Isomers A form at the P atom near the end of a P-C bond, B at a C-C bond, C at the centroid of the C-C-C ring along the C3 symmetry axis, and D at the P atom along the C3 symmetry axis. Complexes A and B are stabilized by hydrogen bonds when FH and ClH are the acids, and by halogen bonds when ClF is the acid. In isomers C, the dipole moments of the two monomers are favorably aligned but in D the alignment is unfavorable. For each of the monomers, the binding energies of the complexes decrease in the order A > B > C > D. The most stabilizing Symmetry Adapted Perturbation Theory (SAPT) binding energy component for the A and B isomers is the electrostatic interaction, while the dispersion interaction is the most stabilizing term for C and D. The barriers to converting one isomer to another are significantly higher for the A isomers compared to B. Equation of motion coupled cluster singles and doubles (EOM-CCSD) intermolecular coupling constants J(X-C) are small for both B and C isomers. J(X-P) values are larger and positive in the A isomers, negative in the B isomers, and have their largest positive values in the D isomers. Intramolecular coupling constants 1J(P-C) experience little change upon complex formation, except in the halogen-bonded complex FCl:P(CH3) A.

Calculated coupling constants 1 J(X-Y) and 1 K(X-Y), and fundamental relationships among the reduced coupling constants for molecules Hm X-YHn , with X, Y ═ 1 H, 7 Li, 9 Be, 11 B, 13 C, 15 N, 17 O, 19 F, 31 P, 33 S, and 35 Cl.

Equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) calculations have been performed to determine coupling constants 1 J(X-Y) for 65 molecules Hm X-YHn , with X,Y ═ 1 H, 7 Li, 9 Be, 11 B, 13 C, 15 N, 17 O, 19 F, 31 P, 33 S, and 35 Cl. The computed 1 J(X-Y) values are in good agreement with available experimental data. The reduced coupling constants 1 K(X-Y) have been derived from 1 J(X-Y) by removing the dependence on the magnetogyric ratios of X and Y. Patterns are found for the reduced coupling constants on a 1 K(X-Y) surface that are related to the positions of X and Y in the periodic table.

Applicability of density functional and wave function theories combined with the three-dimensional reference interaction site model self-consistent field method to the d–d transitions of a transition metal aqua complex

The applicability of density functional theory (DFT) and wave function theory combined with the three-dimensional reference interaction site model self-consistent field method to the d–d transitions of transition metal aqua complexes was examined using [Cr(H2O)6]3+ in aqueous solution as an example. DFTs with hybrid functionals, multiconfigurational self-consistent field followed by perturbation theory, and coupled-cluster singles and doubles (CCSD) followed by the equation of motion CCSD, gave reasonable d–d transition energies.

Rank reduced coupled cluster theory. II. Equation-of-motion coupled-cluster singles and doubles.

Equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) is a reliable and popular approach to the determination of electronic excitation energies. Recently, we have developed a rank-reduced CCSD (RR-CCSD) method that allows the ground-state coupled-cluster energy to be determined with low-rank cluster amplitudes. Here, we extend this approach to excited-state energies through a RR-EOM-CCSD method. We start from the EOM-CCSD energy functional and insert low-rank approximations to the doubles amplitudes. The result is an approximate EOM-CCSD method with only a quadratic number (in the molecular size) of free parameters in the wavefunction. Importantly, our formulation of RR-EOM-CCSD preserves the size intensivity of the excitation energy and size extensivity of the total energy. Numerical tests of the method suggest that accuracy on the order of 0.05-0.01 eV in the excitation energy is possible with 1% or less of the original number of wavefunction coefficients; accuracy of better than 0.01 eV can be achieved with about 4% or less of the free parameters. The amount of compression at a given accuracy level is expected to increase with the size of the molecule. The RR-EOM-CCSD method is a new path toward the efficient determination of accurate electronic excitation energies.

N…C and S…S Interactions in Complexes, Molecules, and Transition Structures HN(CH)SX:SCO, for X = F, Cl, NC, CCH, H, and CN.

Ab initio Møller–Plesset perturbation theory (MP2)/aug’-cc-pVTZ calculations have been carried out in search of complexes, molecules, and transition structures on HN(CH)SX:SCO potential energy surfaces for X = F, Cl, NC, CCH, H, and CN. Equilibrium complexes on these surfaces have C1 symmetry, but these have binding energies that are no more than 0.5 kJ·mol–1 greater than the corresponding Cs complexes which are vibrationally averaged equilibrium complexes. The binding energies of these span a narrow range and are independent of the N–C distance across the tetrel bond, but they exhibit a second-order dependence on the S–S distance across the chalcogen bond. Charge-transfer interactions stabilize all of these complexes. Only the potential energy surfaces HN(CH)SF:SCO and HN(CH)SCl:SCO have bound molecules that have short covalent N–C bonds and significantly shorter S…S chalcogen bonds compared to the complexes. Equation-of-motion coupled cluster singles and doubles (EOM-CCSD) spin-spin coupling constants 1tJ(N–C) for the HN(CH)SX:SCO complexes are small and exhibit no dependence on the N–C distance, while 1cJ(S–S) exhibit a second-order dependence on the S–S distance, increasing as the S–S distance decreases. Coupling constants 1tJ(N–C) and 1cJ(S–S) as a function of the N–C and S–S distances, respectively, in HN(CH)SF:SCO and HN(CH)SCl:SCO increase in the transition structures and then decrease in the molecules. These changes reflect the changing nature of the N…C and S…S bonds in these two systems.

Exact two-component equation-of-motion coupled-cluster singles and doubles method using atomic mean-field spin-orbit integrals.

A new semi-atomic-orbital- based algorithm for a two-component spin-orbit (SO) equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method using mean-field SO integrals is reported. The new algorithm removes the major computational bottlenecks of a SO-EOM-CCSD calculation associated with the evaluation, storage, and processing of the H¯ab,cd elements in the similarity-transformed Hamiltonian involving four virtual orbital labels. The partial recovery of spin symmetry in the present algorithm reduces the storage requirement by an order of magnitude and the floating point operation count for the evaluation of the ladder-like term by a factor of three to four. EOM-CCSD calculations of excited states in the triiodide ion (I3 -) using the exact two-component Hamiltonian in combination with atomic mean-field SO integrals (X2CAMF) are reported as a validation of the implementation and also as a demonstration of the capability of the new algorithm to correlate extended virtual spaces. X2CAMF-EOM-CCSD calculations of the ground and excited states in As2, Sb2, and Bi2 are also presented and compared with the available experimental studies. An analysis based on the computed spectroscopic constants as well as the compositions of the excited-state wavefunctions strongly supports a new assignment for the lowest 2u and 0u - levels in the photoelectron spectrum of Bi2.

Partition of electronic excitation energies: the IQA/EOM-CCSD method.

Different developments in chemistry and emerging technologies have generated a renewed interest in the properties of molecular excited states. We present herein the partition of black-box, size-consistent equation-of-motion coupled cluster singles and doubles (EOM-CCSD) excitation energies within the framework of the interacting quantum atoms (IQA) formalism. We denote this method as IQA/EOM-CCSD. We illustrate this approach by considering small molecules used often in the study of excited states. This investigation shows how the combination of IQA and EOM-CCSD may provide valuable insights into the molecular changes induced by electron excitation via the real space distribution of the energy of an absorbed photon in a molecular system. Our results reveal (i) the most energetically deformed atomic basins and (ii) the most affected covalent and non-covalent interactions within a molecule due to a given photoexcitation. In other words, this kind of analysis provides insights into the spatial energetic redistribution accompanying an electronic excitation, with interesting foreseeable applications in the rational design of photoexcitations with tailored chemical effects. Altogether, we expect that the IQA/EOM-CCSD excitation energy partition will prove useful in the understanding of systems and processes of interest in photophysics and photochemistry.

State-Averaged Pair Natural Orbitals for Excited States: A Route toward Efficient Equation of Motion Coupled-Cluster.

A reduced-complexity variant of equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) method is formulated in terms of state-averaged excited-state pair natural orbitals (PNO) designed to describe manifolds of excited states. State-averaged excited-state PNOs for the target manifold are determined by averaging CIS(D) pair densities over the model manifold. The performance of the PNO-EOM-CCSD approach has been tested with the help of a distributed-memory parallel canonical EOM-CCSD implementation within the Massively Parallel Quantum Chemistry program that allows treatment of systems with 50+ atoms using realistic basis sets with 1000+ functions. The use of state-averaged PNOs offers several potential advantages relative to the recently proposed state-specific PNOs: our approach is robust with respect to root flipping and state degeneracies, it is more economical when computing large manifolds of states, and it simplifies evaluation of transition-specific observables such as dipole moments. With the PNO truncation threshold of 10-7, the errors in excitation energies are on average below 0.02 eV for the first six singlet states of 28 organic molecules included in the standard test set of Thiel and co-workers ( J. Chem. Phys. 2008, 128, 134110) with 50-70 state-averaged PNOs per pair.