Quantum Mechanical Versus Polarizable Embedding Schemes: A Study of the Xray Absorption Spectra of Aqueous Ammonia and Ammonium.

The X-ray absorption spectra of aqueous ammonia and ammonium are computed using a combination of coupled cluster singles and doubles (CCSD) with different quantum mechanical and molecular mechanical embedding schemes. Specifically, we compare frozen Hartree-Fock (HF) density embedding, polarizable embedding (PE), and polarizable density embedding (PDE). Integrating CCSD with frozen HF density embedding is possible within the CC-in-HF framework, which circumvents the conventional system-size limitations of standard coupled cluster methods. We reveal similarities between PDE and frozen HF density descriptions, while PE spectra differ significantly. By including approximate triple excitations, we also investigate the effect of improving the electronic structure theory. The spectra computed using this approach show an improved intensity ratio compared to CCSD-in-HF. Charge transfer analysis of the excitations shows the local character of the pre-edge and main-edge, while the post-edge is formed by excitations delocalized over the first solvation shell and beyond.

Understanding X-ray absorption in liquid water using triple excitations in multilevel coupled cluster theory.

X-ray absorption (XA) spectroscopy is an essential experimental tool to investigate the local structure of liquid water. Interpretation of the experiment poses a significant challenge and requires a quantitative theoretical description. High-quality theoretical XA spectra require reliable molecular dynamics simulations and accurate electronic structure calculations. Here, we present the first successful application of coupled cluster theory to model the XA spectrum of liquid water. We overcome the computational limitations on system size by employing a multilevel coupled cluster framework for large molecular systems. Excellent agreement with the experimental spectrum is achieved by including triple excitations in the wave function and using molecular structures from state-of-the-art path-integral molecular dynamics. We demonstrate that an accurate description of the electronic structure within the first solvation shell is sufficient to successfully model the XA spectrum of liquid water within the multilevel framework. Furthermore, we present a rigorous charge transfer analysis of the XA spectrum, which is reliable due to the accuracy and robustness of the electronic structure methodology. This analysis aligns with previous studies regarding the character of the prominent features of the XA spectrum of liquid water.

Particle-Breaking Unrestricted Hartree-Fock Theory for Open Molecular Systems.

We recently introduced the particle-breaking restricted Hartree-Fock (PBRHF) model, a mean-field approach to address the fractional charging of molecules when they interact with an electronic environment. In this paper, we present an extension of the model referred to as particle-breaking unrestricted Hartree-Fock (PBUHF). The unrestricted formulation contains odd-electron states necessary for a realistic description of fractional charging. Within the PBUHF parametrization, we use two-body operators as they yield convenient operator transformations. However, two-body operators can change only the particle number by two. Therefore, we include noninteracting zero-energy bath orbitals to generate a linear combination of even and odd electron states. Depending on whether the occupied or virtual orbitals of a molecule interact with the environment, the average number of electrons is either decreased or increased. Without interaction, PBUHF reduces to the unrestricted Hartree-Fock wave function.

Core-ionization spectrum of liquid water.

We present state-of-the-art calculations of the core-ionization spectrum of water. Despite significant progress in procedures developed to mitigate various experimental complications and uncertainties, the experimental determination of ionization energies of solvated species involves several non-trivial steps such as assessing the effect of the surface potential, electrolytes, and finite escape depths of photoelectrons. This provides a motivation to obtain robust theoretical values of the intrinsic bulk ionization energy and the corresponding solvent-induced shift. Here we develop theoretical protocols based on coupled-cluster theory and electrostatic embedding. Our value of the intrinsic solvent-induced shift of the 1sO ionization energy of water is -1.79 eV. The computed absolute position and the width of the 1sO peak in photoelectron spectrum of water are 538.47 eV and 1.44 eV, respectively, agreeing well with the best experimental values.

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.

Triplet Excited States with Multilevel Coupled Cluster Theory.

We extend the multilevel coupled cluster framework with triplet excitation energies at the singles and perturbative doubles (MLCC2) and singles and doubles (MLCCSD) levels of theory. In multilevel coupled cluster theory, we partition the orbitals and restrict the higher-order excitations in the cluster operator to a set of active orbitals. With an appropriate choice of these orbitals, the multilevel approach can give significant computational savings while maintaining the high accuracy of standard coupled cluster theory. In this work, we generated active orbitals from approximate correlated natural transition orbitals (CNTOs). The CNTOs form a compact orbital space specifically tailored to describe the triplet excited states of interest. We compare the performance of MLCCSD and MLCC2, in terms of cost and accuracy, to those of their standard coupled cluster counterparts (CC2 and CCSD) and finally show proof-of-concept calculations of the singlet-triplet gaps of molecules that are of interest for their potential use in organic light-emitting diodes.

Entanglement coupled cluster theory: Exact spin-adaptation.

We present a novel framework for spin-adapted coupled cluster theory. The approach exploits the entanglement of an open-shell molecule with electrons in a non-interacting bath. Together, the molecule and the bath form a closed-shell system, and electron correlation can be included using the standard spin-adapted closed-shell coupled cluster formalism. A projection operator, which enforces conditions on the electrons in the bath, is used to obtain the desired state of the molecule. This entanglement coupled cluster theory is outlined, and proof-of-concept calculations for doublet states are reported. The approach is further extendable to open-shell systems with other values of the total spin.

New Implementation of an Equation-of-Motion Coupled-Cluster Damped-Response Framework with Illustrative Applications to Resonant Inelastic X-ray Scattering.

We present an implementation of a damped response framework for calculating resonant inelastic X-ray scattering (RIXS) at the equation-of-motion coupled-cluster singles and doubles (CCSD) and second-order approximate coupled-cluster singles and doubles (CC2) levels of theory in the open-source program eT. This framework lays the foundation for future extension to higher excitation methods (notably, the coupled-cluster singles and doubles with perturbative triples, CC3) and to multilevel approaches. Our implementation adopts a fully relaxed ground state and different variants of the core-valence separation projection technique to address convergence issues. Illustrative results are compared with those obtained within the frozen-core core-valence separated approach, available in Q-Chem, as well as with experiment. The performance of the CC2 method is evaluated in comparison with that of CCSD. It is found that, while the CC2 method is noticeably inferior to CCSD for X-ray absorption spectra, the quality of the CC2 RIXS spectra is often comparable to that of the CCSD level of theory, when the same valence excited states are probed. Finally, we present preliminary RIXS results for a solvated molecule in aqueous solution.

Particle-Breaking Hartree-Fock Theory for Open Molecular Systems.

In this work we present the particle-breaking Hartree-Fock (PBHF) model which is a mean-field approach to open molecular systems. The interaction of a system with the environment is parametrized through a particle-breaking term in the molecular Hamiltonian. The PBHF wave function is constructed through an exponential unitary transformation of a Slater determinant with a given number of electrons. We consider only the closed-shell formalism. The parametrization results in a linear combination of Slater determinants with different numbers of electrons, i.e., the PBHF wave function is not an eigenfunction of the number operator. As a result, the density matrix may have fractional occupations which are, because of the unitary parametrization, always between 0.0 and 2.0. The occupations are optimized simultaneously with the orbitals, using the trust-region optimization procedure. In the limit of a particle-conserving Hamiltonian, the PBHF optimization will converge to a standard Hartree-Fock wave function. We show that the average number of electrons may be decreased or increased depending on whether the particle-breaking term affects occupied or virtual orbitals.

Oscillator Strengths in the Framework of Equation of Motion Multilevel CC3.

We present an efficient implementation of the equation of motion oscillator strengths for the closed-shell multilevel coupled cluster singles and doubles with perturbative triples method (MLCC3) in the electronic structure program eT. The orbital space is split into an active part treated with CC3 and an inactive part computed at the coupled cluster singles and doubles (CCSD) level of theory. Asymptotically, the CC3 contribution scales as O(nVnv3no3) floating-point operations, where nV is the total number of virtual orbitals while nv and no are the number of active virtual and occupied orbitals, respectively. The CC3 contribution, thus, only scales linearly with the full system size and can become negligible compared to the cost of CCSD. We demonstrate the capabilities of our implementation by calculating the ultraviolet-visible spectrum of azobenzene and a core excited state of betaine 30 with more than 1000 molecular orbitals.

Implementation of Occupied and Virtual Edmiston-Ruedenberg Orbitals Using Cholesky Decomposed Integrals.

We present a trust-region optimization of the Edmiston-Ruedenberg orbital localization function. The approach is used to localize both the occupied and the virtual orbitals and is the first demonstration of general virtual orbital localization using the Edmiston-Ruedenberg localization function. In the Edmiston-Ruedenberg approach, the sum of the orbital self-repulsion energies is maximized to obtain the localized orbitals. The Cholesky decomposition reduces the cost of transforming the electron repulsion integrals, and the overall scaling of our implementation is O(N4). The optimization is performed with all quantities in the molecular orbital basis, and the localization of the occupied orbitals is often less expensive than the corresponding self-consistent field (SCF) optimization. Furthermore, the occupied orbital localization scales linearly with the basis set. For the virtual space, the cost is significantly higher than the SCF optimization. The orbital spreads of the resulting virtual Edmiston-Ruedenberg orbitals are larger than for other, less expensive, orbital localization functions. This indicates that other localization procedures are more suitable for applications such as local post-Hartree-Fock calculations.

Multilevel CC2 and CCSD in Reduced Orbital Spaces: Electronic Excitations in Large Molecular Systems.

We present efficient implementations of the multilevel CC2 (MLCC2) and multilevel CCSD (MLCCSD) models. As the system size increases, MLCC2 and MLCCSD exhibit the scaling of the lower-level coupled cluster model. To treat large systems, we combine MLCC2 and MLCCSD with a reduced-space approach in which the multilevel coupled cluster calculation is performed in a significantly truncated molecular orbital basis. The truncation scheme is based on the selection of an active region of the molecular system and the subsequent construction of localized Hartree-Fock orbitals. These orbitals are used in the multilevel coupled cluster calculation. The electron repulsion integrals are Cholesky decomposed using a screening protocol that guarantees accuracy in the truncated molecular orbital basis and reduces computational cost. The Cholesky factors are constructed directly in the truncated basis, ensuring low storage requirements. Systems for which Hartree-Fock is too expensive can be treated by using a multilevel Hartree-Fock reference. With the reduced-space approach, we can handle systems with more than a thousand atoms. This is demonstrated for paranitroaniline in aqueous solution.

Equation-of-Motion MLCCSD and CCSD-in-HF Oscillator Strengths and Their Application to Core Excitations.

We present an implementation of equation-of-motion oscillator strengths for the multilevel CCSD (MLCCSD) model where CCS is used as the lower level method (CCS/CCSD). In this model, the double excitations of the cluster operator are restricted to an active orbital space, whereas the single excitations are unrestricted. Calculated nitrogen K-edge spectra of adenosine, adenosine triphosphate (ATP), and an ATP-water system are used to demonstrate the performance of the model. Projected atomic orbitals (PAOs) are used to partition the virtual space into active and inactive orbital sets. Cholesky decomposition of the Hartree-Fock density is used to partition the occupied orbitals. This Cholesky-PAO partitioning is cheap, scaling as O(N3), and is suitable for the calculation of core excitations, which are localized in character. By restricting the single excitations of the cluster operator to the active space, as well as the double excitations, the CCSD-in-HF model is obtained. A comparison of the two models-MLCCSD and CCSD-in-HF-is presented for the core excitation spectra of the adenosine and ATP systems.

Multilevel CC2 and CCSD Methods with Correlated Natural Transition Orbitals.

In the multilevel coupled cluster approach, an active orbital space is treated at a higher level of coupled cluster theory than the remaining inactive orbitals. We introduce the multilevel CC2 method where CC2 is used for the active orbital space. Furthermore, we present a simplified formulation of the multilevel CCSD method where CCSD is used for the active space. The simplification lies in the evaluation of the CC2 amplitudes in the inactive space; these CC2 amplitudes have previously been determined iteratively. We use correlated natural transition orbitals to determine the active orbital spaces. The convergence of the multilevel CC2 and multilevel CCSD valence excitation energies is established with proof-of-concept calculations. The methods are also applied to two larger systems: p-nitroaniline in water and amoxicillin. The calculations on the p-nitroaniline-water system illustrate the usefulness of multilevel coupled cluster methods for molecules in solution and for charge transfer excitations.