Relativistic Real-Time Time-Dependent Equation-of-Motion Coupled-Cluster

Dirac-Coulomb-Breit Molecular Mean-Field Exact-Two-Component Relativistic Equation-of-Motion Coupled-Cluster Theory

We present a relativistic equation-of-motion coupled-cluster with single and double excitation formalism within the exact two-component framework (X2C-EOM-CCSD), where both scalar relativistic effects and spin-orbit coupling are variationally included at the reference level. Three different molecular mean-field treatments of relativistic corrections, including the one-electron, Dirac-Coulomb, and Dirac-Coulomb-Breit Hamiltonian, are considered in this work. Benchmark calculations include atomic excitations and fine-structure splittings arising from spin-orbit coupling. Comparison with experimental values and relativistic time-dependent density functional theory is also carried out. The computation of the oscillator strength using the relativistic X2C-EOM-CCSD approach allows for studies of spin-orbit-driven processes, such as the spontaneous phosphorescence lifetime.

Cholesky Decomposition-Based Implementation of Relativistic Two-Component Coupled-Cluster Methods for Medium-Sized Molecules

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field spin-orbit integrals (the X2CAMF scheme) is reported. The present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. Our CD-based implementation extends the applicability of X2CAMF-CC and EOM-CC methods to medium-sized molecules with the possibility to correlate around 1000 spinors. Benchmark calculations for uranium-containing small molecules were performed to assess the dependence of the CC results on the Cholesky threshold. A Cholesky threshold of 10-4 is shown to be sufficient to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond-dissociation energy of the uranium hexafluoride molecule, UF6, with up to quadruple-ζ basis sets, and the lowest excitation energy in the solvated uranyl ion [UO22+(H2O)12].

Analytic gradients for relativistic exact-two-component equation-of-motion coupled-cluster singles and doubles method

A first implementation of analytic gradients for spinor-based relativistic equation-of-motion coupled-cluster singles and doubles method using an exact two-component Hamiltonian augmented with atomic mean-field spin-orbit integrals is reported. To demonstrate its applicability, we present calculations of equilibrium structures and harmonic vibrational frequencies for the electronic ground and excited states of the radium mono-amide molecule (RaNH2) and the radium mono-methoxide molecule (RaOCH3). Spin-orbit coupling is shown to quench Jahn-Teller effects in the first excited state of RaOCH3, resulting in a C3v equilibrium structure. The calculations also show that the radium atoms in these molecules serve as efficient optical cycling centers.

Coupled Cluster Simulation of Impulsive Stimulated X-ray Raman Scattering

Time-dependent equation-of-motion coupled cluster (TD-EOM-CC) is used to simulate impulsive stimulated X-ray Raman scattering (ISXRS) of ultrashort laser pulses by neon, carbon monoxide, pyrrole, and p-aminophenol. The TD-EOM-CC equations are expressed in the basis of field-free EOM-CC states, where the calculation of the core-excited states is simplified through the use of the core-valence separation (CVS) approximation. The transfer of electronic population from the ground state to the core- and valence-excited states is calculated for different numbers of included core- and valence-excited states, as well as for electric field pulses with different polarizations and carrier frequencies. The results indicate that Gaussian pulses can transfer significant electronic populations to the valence states through the Raman process. The sensitivity of this population transfer to the model parameters is analyzed. The time-dependent electronic density for p-aminophenol is also showcased, supporting the interpretation that ISXRS involves localized core excitations and can be used to rapidly generate valence wavepackets.

Time-dependent equation-of-motion coupled-cluster simulations with a defective Hamiltonian

Simulations of laser-induced electron dynamics in a molecular system are performed using time-dependent (TD) equation-of-motion (EOM) coupled-cluster (CC) theory. The target system has been chosen to highlight potential shortcomings of truncated TD-EOM-CC methods [represented in this work by TD-EOM-CC with single and double excitations (TD-EOM-CCSD)], where unphysical spectroscopic features can emerge. Specifically, we explore driven resonant electronic excitations in magnesium fluoride in the proximity of an avoided crossing. Near the avoided crossing, the CCSD similarity-transformed Hamiltonian is defective, meaning that it has complex eigenvalues, and oscillator strengths may take on negative values. When an external field is applied to drive transitions to states exhibiting these traits, unphysical dynamics are observed. For example, the stationary states that make up the time-dependent state acquire populations that can be negative, exceed one, or even complex-valued.

Core-Projected Hybrids Fix Systematic Errors in Time-Dependent Density Functional Theory Predicted Core-Electron Excitations

Linear response time-dependent density functional theory (TDDFT) is widely applied to valence, Rydberg, and charge-transfer excitations but, in its current form, makes large errors for core-electron excitations. This work demonstrates that the admixture of nonlocal exact exchange in atomic core regions significantly improves TDDFT-predicted core excitations. Exact exchange admixture is accomplished using projected hybrid density functional theory [ J. Chem. Theory Comput. 2023, 19, 837-847]. Scalar relativistic TDDFT calculations using core-projected B3LYP accurately model core excitations of second-period elements C-F and third-period elements Si-Cl, without sacrificing performance for the relative shifts of core excitation energies. Predicted K-edge X-ray near absorption edge structure (XANES) of a series of sulfur standards highlight the value of this approach. Core-projected hybrids appear to be a practical solution to TDDFT's limitations for core excitations, in the way that long-range-corrected hybrids are a practical solution to TDDFT's limitations for Rydberg and charge-transfer excitations.

Real-Time Equation-of-Motion Coupled-Cluster Cumulant Green's Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-Body Methods Infrastructure

We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function method [ J. Chem. Phys. 2020, 152, 174113] within the Tensor Algebra for Many-body Methods (TAMM) infrastructure. TAMM is a massively parallel heterogeneous tensor library designed for utilizing forthcoming exascale computing resources. The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin-explicit forms of the various operators when evaluating the tensor contractions. Unlike our previous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation supports fully complex algebra. The RT-EOM-CC singles (S) and doubles (D) time-dependent amplitudes are propagated using a first-order Adams-Moulton method. This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis functions, with parallel efficiencies above 90% up to 400 GPUs. The TAMM RT-EOM-CCSD was used to study core photoemission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules. Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals. The relative quasiparticle ionization energies and overall spectral functions agree well with available experimental results.

Adiabatic extraction of nonlinear optical properties from real-time time-dependent electronic-structure theory

Real-time simulations of laser-driven electron dynamics contain information about molecular optical properties through all orders in response theory. These properties can be extracted by assuming convergence of the power series expansion of induced electric and magnetic multipole moments. However, the accuracy relative to analytical results from response theory quickly deteriorates for higher-order responses due to the presence of high-frequency oscillations in the induced multipole moment in the time domain. This problem has been ascribed to missing higher-order corrections. We here demonstrate that the deviations are caused by nonadiabatic effects arising from the finite-time ramping from zero to full strength of the external laser field. Three different approaches, two using a ramped wave and one using a pulsed wave, for extracting electrical properties from real-time time-dependent electronic-structure simulations are investigated. The standard linear ramp is compared to a quadratic ramp, which is found to yield highly accurate results for polarizabilities, and first and second hyperpolarizabilities, at roughly half the computational cost. Results for the third hyperpolarizability are presented along with a simple, computable measure of reliability.

Accurate Relativistic Real-Time Time-Dependent Density Functional Theory for Valence and Core Attosecond Transient Absorption Spectroscopy

First principles theoretical modeling of out-of-equilibrium processes observed in attosecond pump-probe transient absorption spectroscopy (TAS) triggering pure electron dynamics remains a challenging task, especially for heavy elements and/or core excitations containing fingerprints of scalar and spin-orbit relativistic effects. To address this, we formulate a methodology for simulating TAS within the relativistic real-time, time-dependent density functional theory (RT-TDDFT) framework, for both the valence and core energy regimes. Especially for TAS, full four-component (4c) RT simulations are feasible but computationally demanding. Therefore, in addition to the 4c approach, we also introduce the atomic mean-field exact two-component (amfX2C) Hamiltonian accounting for one- and two-electron picture-change corrections within RT-TDDFT. amfX2C preserves the accuracy of the parent 4c method at a fraction of its computational cost. Finally, we apply the methodology to study valence and near-L_2,3-edge TAS processes of experimentally relevant systems and provide additional physical insights using relativistic nonequilibrium response theory.

Automatic purpose-driven basis set truncation for time-dependent Hartree-Fock and density-functional theory

Real-time time-dependent density-functional theory (RT-TDDFT) and linear response time-dependent density-functional theory (LR-TDDFT) are two important approaches to simulate electronic spectra. However, the basis sets used in such calculations are usually the ones designed mainly for electronic ground state calculations. In this work, we propose a systematic and robust scheme to truncate the atomic orbital (AO) basis set employed in TDDFT and TD Hartree-Fock (TDHF) calculations. The truncated bases are tested for both LR- and RT-TDDFT as well as RT-TDHF approaches, and provide an acceleration up to an order of magnitude while the shifts of excitation energies of interest are generally within 0.2 eV. The procedure only requires one extra RT calculation with 1% of the total propagation time and a simple modification on basis set file, which allows an instant application in any quantum chemistry package supporting RT-/LR-TDDFT calculations. Aside from the reduced computational effort, this approach also offers valuable insight into the effect of different basis functions on computed electronic excitations and further ideas on the design of basis sets for special purposes.

Understanding Charge Dynamics in Dense Electronic Manifolds in Complex Environments

Photoinduced charge transfer (CT) excited states and their relaxation mechanisms can be highly interdependent on the environment effects and the consequent changes in the electronic density. Providing a molecular interpretation of the ultrafast (subpicosecond) interplay between initial photoexcited states in such dense electronic manifolds in condensed phase is crucial for improving and understanding such phenomena. Real-time time-dependent density functional theory is here the method of choice to observe the charge density, explicitly propagated in an ultrafast time domain, along with all time-dependent properties that can be easily extracted from it. A designed protocol of analysis for real-time electronic dynamics to be applied to time evolving electronic density related properties to characterize both in time and in space CT dynamics in complex systems is here introduced and validated, proposing easy to be read cross-correlation maps. As case studies to test such tools, we present the photoinduced charge-transfer electronic dynamics of 5-benzyluracil, a mimic of nucleic acid/protein interactions, and the metal-to-ligand charge-transfer electronic dynamics in water solution of [Ru(dcbpy)2(NCS)2]4-, dcbpy = (4,4'-dicarboxy-2,2'-bipyridine), or "N34-", a dye sensitizer for solar cells. Electrostatic and explicit ab initio treatment of solvent molecules have been compared in the latter case, revealing the importance of the accurate modeling of mutual solute-solvent polarization on CT kinetics. We observed that explicit quantum mechanical treatment of solvent slowed down the charge carriers mobilities with respect to the gas-phase. When all water molecules were modeled instead as simpler embedded point charges, the electronic dynamics appeared enhanced, with a reduced hole-electron distance and higher mean velocities due to the close fixed charges and an artificially increased polarization effect. Such analysis tools and the presented case studies can help to unveil the influence of the electronic manifold, as well as of the finite temperature-induced structural distortions and the environment on the ultrafast charge motions.

Route to Chemical Accuracy for Computational Uranium Thermochemistry

Benchmark spinor-based relativistic coupled-cluster calculations for the ionization energies of the uranium atom, the uranium monoxide molecule (UO), and the uranium dioxide molecule (UO₂) and for the bond dissociation energies of UO and UO₂ are reported. The accuracy of these calculations in the treatments of relativistic, electron-correlation, and basis-set effects is analyzed. The intrinsic convergence of the computed results and the favorable comparison with the experimental values demonstrate the unique applicability of the spinor representation of quantum-chemical methods to open-shell uranium-containing atomic and molecular species with uranium oxidation states ranging from U(0) to U(V).

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10^-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnF_n, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Accelerating Real-Time Coupled Cluster Methods with Single-Precision Arithmetic and Adaptive Numerical Integration

We explore the framework of a real-time coupled cluster method with a focus on improving its computational efficiency. Propagation of the wave function via the time-dependent Schrödinger equation places high demands on computing resources, particularly for high level theories such as coupled cluster with polynomial scaling. Similar to earlier investigations of coupled cluster properties, we demonstrate that the use of single-precision arithmetic reduces both the storage and multiplicative costs of the real-time simulation by approximately a factor of 2 with no significant impact on the resulting UV/vis absorption spectrum computed via the Fourier transform of the time-dependent dipole moment. Additional speedups─of up to a factor of 14 in test simulations of water clusters─are obtained via a straightforward GPU-based implementation as compared to conventional CPU calculations. We also find that further performance optimization is accessible through sagacious selection of numerical integration algorithms, and the adaptive methods, such as the Cash-Karp integrator, provide an effective balance between computing costs and numerical stability. Finally, we demonstrate that a simple mixed-step integrator based on the conventional fourth-order Runge-Kutta approach is capable of stable propagations even for strong external fields, provided the time step is appropriately adapted to the duration of the laser pulse with only minimal computational overhead.

Linear and Nonlinear Optical Properties from TDOMP2 Theory

We present a derivation of real-time (RT) time-dependent orbital-optimized Møller–Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.

Quadratic Unitary Coupled-Cluster Singles and Doubles Scheme: Efficient Implementation, Benchmark Study, and Formulation of an Extended Version

An efficient implementation of the quadratic unitary coupled-cluster singles and doubles (qUCCSD) scheme for calculations of electronic ground and excited states using an unrestricted molecular spin-orbital formulation and an efficient tensor contraction library is reported. The accuracy of the qUCCSD scheme and the efficiency of the present implementation are demonstrated using extensive benchmark calculations of excitation energies and an application to S₀ → S₁ vertical excitation energies for cis- and trans-4a,4b-dihydrotriphenylene. The qUCCSD scheme has been shown to provide improved excitation energies compared with the UCC3 scheme formulated based on perturbation theory. A UCC truncation scheme that can provide excitation energies correct through the fourth order is also presented to further improve the accuracy of the qUCCSD scheme.

Real-Time Equation-of-Motion CCSD Cumulant Green's Function

Many-body excitations in X-ray photoemission spectra have been difficult to simulate from first principles. We have recently developed a cumulant-based one-electron Green's function method using the real-time coupled-cluster-singles equation-of-motion approach (RT-EOM-CCS) that provides a general framework for treating these problems. Here we extend this approach to include double excitations in the ground-state energy and in the coupled cluster amplitudes, which have been implemented using subroutines generated by the Tensor Contraction Engine (TCE). As in the case of the singles approximation, RT-EOM-CCSD yields a nonperturbative cumulant form of the Green's function in terms of the time-dependent cluster amplitudes, adding nonlinear corrections to the traditional cumulant forms. The extended approach is applied to the core-hole spectral function for small molecular systems. We find that, when core-optimized basis sets are used, the doubles contributions reduce the mean absolute errors in the core binding energies of the 10e systems from 0.8 to 0.3 eV. They also significantly improve the quasiparticle-satellite gap by reducing its overestimation from about 3-5 to about 0-1 eV in CH₄, NH₃, and H₂O, and also improving the overall shape of the satellite features. Finally, we demonstrate the application of the new implementation to the larger, classical XPS ESCA series of molecules and show that the singles approximation can be paired with a modest basis set to study carbon speciation.

Short Iterative Lanczos Integration in Time-Dependent Equation-of-Motion Coupled-Cluster Theory

A time-dependent (TD) formulation of equation-of-motion coupled-cluster (EOM-CC) theory can provide excited-state information over an arbitrarily wide energy window with a reduced memory footprint relative to conventional, frequency-domain EOM-CC theory. However, the floating-point costs of the time-integration required by TD-EOM-CC are generally far larger than those of the frequency-domain form of the approach. This work considers the potential of the short iterative Lanczos (SIL) integration scheme [J. Chem. Phys. 1986, 85, 5870-5876] to reduce the floating-point costs of TD-EOM-CC simulations. Low-energy and K-edge absorption features for small molecules are evaluated using TD-EOM-CC with single and double excitations, with the time-integrations carried out via SIL and fourth-order Runge-Kutta (RK4) schemes. Spectra derived from SIL- and RK4-driven simulations are nearly indistinguishable, and with an appropriately chosen subspace dimension, the SIL requires far fewer floating-point operations than are required by RK4. For K-edge spectra, SIL is the more efficient scheme by an average factor of 7.2.

Electron Dynamics with the Time-Dependent Density Matrix Renormalization Group

In this work, we simulate the electron dynamics in molecular systems with the time-dependent density matrix renormalization group (TD-DMRG) algorithm. We leverage the generality of the so-called tangent-space TD-DMRG formulation and design a computational framework in which the dynamics is driven by the exact nonrelativistic electronic Hamiltonian. We show that by parametrizing the wave function as a matrix product state, we can accurately simulate the dynamics of systems including up to 20 electrons and 32 orbitals. We apply the TD-DMRG algorithm to three problems that are hardly targeted by time-independent methods: the calculation of molecular (hyper)polarizabilities, the simulation of electronic absorption spectra, and the study of ultrafast ionization dynamics.

Interpretation of Coupled-Cluster Many-Electron Dynamics in Terms of Stationary States

We demonstrate theoretically and numerically that laser-driven many-electron dynamics, as described by bivariational time-dependent coupled-cluster (CC) theory, may be analyzed in terms of stationary-state populations. Projectors heuristically defined from linear response theory and equation-of-motion CC theory are proposed for the calculation of stationary-state populations during interaction with laser pulses or other external forces, and conservation laws of the populations are discussed. Numerical tests of the proposed projectors, involving both linear and nonlinear optical processes for He and Be atoms and for LiH, CH+, and LiF molecules show that the laser-driven evolution of the stationary-state populations at the coupled-cluster singles-and-doubles (CCSD) level is very close to that obtained by full configuration interaction (FCI) theory, provided that all stationary states actively participating in the dynamics are sufficiently well approximated. When double-excited states are important for the dynamics, the quality of the CCSD results deteriorates. Observing that populations computed from the linear response projector may show spurious small-amplitude, high-frequency oscillations, the equation-of-motion projector emerges as the most promising approach to stationary-state populations.

Real-Time Coupled-Cluster Approach for the Cumulant Green's Function

Green's function methods within many-body perturbation theory provide a general framework for treating electronic correlations in excited states and spectra. Here, we develop the cumulant form of the one-electron Green's function using a real-time coupled-cluster equation-of-motion approach, in an extension of our previous study (Rehr J.; et al. J. Chem. Phys. 2020, 152, 174113). The approach yields a nonperturbative expression for the cumulant in terms of the solution to a set of coupled first-order, nonlinear differential equations. The method thereby adds nonlinear corrections to traditional cumulant methods, which are linear in the self-energy. The approach is applied to the core-hole Green's function and is illustrated for a number of small molecular systems. For these systems, we find that the nonlinear contributions yield significant improvements, both for quasiparticle properties such as core-level binding energies and for inelastic losses that correspond to satellites observed in photoemission spectra.

Equation of motion coupled-cluster cumulant approach for intrinsic losses in x-ray spectra

We present a combined equation of motion coupled-cluster cumulant Green's function approach for calculating and understanding intrinsic inelastic losses in core level x-ray absorption spectra (XAS) and x-ray photoemission spectra. The method is based on a factorization of the transition amplitude in the time domain, which leads to a convolution of an effective one-body absorption spectrum and the core-hole spectral function. The spectral function characterizes intrinsic losses in terms of shake-up excitations and satellites using a cumulant representation of the core-hole Green's function that simplifies the interpretation. The one-body spectrum also includes orthogonality corrections that enhance the XAS at the edge.