The time-dependent equation-of-motion coupled-cluster singles and doubles (TD-EOM-CCSD) and time-dependent coupled-cluster singles and doubles (TDCCSD) methods are compared by simulating Rabi oscillations for distant atoms in a classical electromagnetic field. While the TD-EOM-CCSD simulations are numerically stable, the oscillating time-dependent energy expectation value displays an incorrect scaling with the number of atoms resonant with the field. On the other hand, the TDCCSD simulations exhibit the correct scaling in the initial stages of the Rabi cycle but break down when the multiatom system approaches complete population inversion. The numerical stability of the TDCCSD simulations is shown to be very sensitive to any interactions between the atoms and to left amplitudes that do not participate in the expression for time-dependent expectation values for noninteracting atoms. Moreover, we introduce a general theoretical framework for describing the two methods, where the cluster amplitude time derivatives serve as auxiliary conditions related to a shift of the time-dependent Hamiltonian matrix. In this framework, time-dependent coupled-cluster methods exhibit a shifted Hamiltonian matrix with a block upper triangular structure in terms of the number of excited noninteracting subsystems, explaining the correct scaling properties of the TDCCSD method.
The time-dependent equation-of-motion coupled-cluster singles and doubles (TD-EOM-CCSD) and time-dependent coupled-cluster singles and doubles (TDCCSD) methods are compared by simulating Rabi oscillations for distant atoms in a classical electromagnetic field. While the TD-EOM-CCSD simulations are numerically stable, the oscillating time-dependent energy expectation value displays an incorrect scaling with the number of atoms resonant with the field. On the other hand, the TDCCSD simulations exhibit the correct scaling in the initial stages of the Rabi cycle but break down when the multiatom system approaches complete population inversion. The numerical stability of the TDCCSD simulations is shown to be very sensitive to any interactions between the atoms and to left amplitudes that do not participate in the expression for time-dependent expectation values for noninteracting atoms. Moreover, we introduce a general theoretical framework for describing the two methods, where the cluster amplitude time derivatives serve as auxiliary conditions related to a shift of the time-dependent Hamiltonian matrix. In this framework, time-dependent coupled-cluster methods exhibit a shifted Hamiltonian matrix with a block upper triangular structure in terms of the number of excited noninteracting subsystems, explaining the correct scaling properties of the TDCCSD method.
We advance the Coupled Cluster method’s in the time-dependent realm, by imple- menting a robust solver based on the orbital-adaptive time-dependent Coupled Cluster (OATDCC)[1] method. This involves implementing both a simplified static orbital time-dependent coupled cluster solver with single and double excitations (TDCCSD) and an orbital-adaptive scheme with double excitations (OATDCCD). To supplement the time-dependent methods we implement several ground state solvers based on the Lagrangian Coupled Cluster formulation, with single and double excitations, as well as a non-orthogonal orbital-optimised Coupled Cluster (NOCC) solver[2]. We construct several quantum dot basis sets with different potential functions in one- and two dimensions, including interactions with magnetic fields. What is more, we also implement an interface with popular quantum chemistry software modules PySCF[3] and Psi4[4] for extraction of additional basis sets for atoms and molecules. The quantum systems are allowed to vary with time by addition of a time-dependent term to the Hamiltonian, with which we simulate a laser field in the dipole approxima- tion. As a validation we reproduce results from the scientific literature, both for atoms, molecules and quantum dots. We show that our methods lead to convergence in the ever-increasing basis set size limit, for simple quantum dot systems. For the same quantum dot system, we show how sensitive a system is to changes in the frequency of a driving oscillating field. Frequencies closer to the resonant frequency lead to exctiations and increased energy. We are able to simulate systems that are fairly large - quantum dots in one- and two dimensions with up to twelve electrons. For systems that meander far from the reference state, we show that the orbital-adaptive method has far superior stability, compared with the method with static orbitals. For all quantum dot systems we find strong comformity with the harmonic potential theorem[5], yet we see a slight many-body effect for a two-dimensional double dot system. By subjecting the two-dimensional quantum dots to a homogoenous, static magnetic field in the form of an angular momentum operator, we see two frequencies in the dipole spectrum, instead of one frequency. This is also in accordance with the harmonic potential theorem. The difference between the two frequencies in the new spectrum is the same as the Larmor frequency of the magnetic field, within acceptable tolerance levels.
The Coupled-Cluster (CC) method is one of the most popular and efficient correlation methods in quantum chemistry. \nEspecially the CCSD(T) approximation, which includes single and double excitations by means of the application of the cluster operator \nto a reference determinant and triple excitations via a perturbative treatment, has become a standard tool \nin quantum chemical applications. However, the method is restricted to relatively small system sizes due to its unfavourable scaling (N^7, where N is the number of basis functions applied). \nFor the full treatment of triple excitations the scaling advances to N^8 and every further excitation level increases the exponent by two. \n \nThe goal of this work is to reduce the calculation time for closed shell systems at least by a factor growing with the excitation level for arbitrary truncation levels. \nThis is done by restricting the spatial parts of the spin orbitals and thus treat pairs of spin orbitals on the same footing. The restrictions can be easily constructed \nfor the CCSD model by spin integration. \nThe derivation of the restrictions arising in higher excited case will be done employing \nthe spatial orbital excitation operators E. \n \nIn the first part of this work an algorithm is derived that is capable of the derivation of the energy and amplitude equations for arbitrary excitation levels. In the second part \nan implementation of this algorithm is presented.
One-electron reduced density matrices (1RDMs) from equation-of-motion (EOM) coupled-cluster with single and double excitations (CCSD) calculations are analyzed to assess their N-representability (i.e., whether they are derivable from a physical N-electron state). We identify EOM-CCSD stationary states whose 1RDMs violate either ensemble-state N-representability conditions or pure-state conditions known as generalized Pauli constraints. As such, these 1RDMs do not correspond to any physical N-electron state. Unphysical states are also encountered in the course of time-dependent EOM-CC simulations; when an external field drives transitions between a pair of stationary states with pure-state N-representable 1RDMs, the 1RDM of the time-dependent state can violate ensemble-state conditions. These observations point to potential challenges in interpreting the results of time-dependent EOM-CCSD simulations.
Equations are derived for the time evolution of time-dependent vibrational coupled cluster (TDVCC) wave functions covering both the TDVCC ket state and the associated so-called Λ bra state. The equations are implemented in the special case of both the Hamiltonian and the cluster operator containing at most two-mode coupling terms. The nontrivial behavior of the evolution of norm, energy, and expectation values due to the nonunitary time-evolution of the nonvariational TDVCC theory is analyzed theoretically and confirmed in numerical experiments that also include time-dependent Hamiltonians. In the spirit of time-independent size-consistency analysis, the separability of both the coupled cluster and Λ states for noninteracting systems is studied. While the coupled cluster state clearly has the correct behavior, the behavior of the Λ state is more intricate, and the consequence for different properties is shown theoretically and numerically. Overall, the numerical experiments show that TDVCC in incomplete expansions gives higher accuracy than a standard linear variational wave function parameterization with the same number of independent parameters, while equivalent results are obtained for complete expansions. The efficiency of the methodology is illustrated in computations on polycyclic aromatic hydrocarbons with up to 156 modes.
Recently, [JCP 161, 114114 (2024)] successfully derived numerically stable and converging (to the time-dependent complete active space self-consistent-field) time-dependent orthogonal coupled cluster (CC) methods, namely, time-dependent orbital optimized CC (TD-ooCC) methods for fermion-mixtures with arbitrary fermion kinds and numbers, via the time-dependent bivariational principle and setting constraints in the ansatz. In this work, we extend the TD-ooCC methods to eighteen non-bivariational time-dependent orbital-quasi-optimized coupled-cluster (TD-oqoCC-NB) methods. In TD-oqoCC-NB methods, the constraints in the ansatz are also used to avoid overparameterization in the complete active space limit and eliminate the numerical instability in the finite truncation case. The equations of motion are not determined by the time-dependent bivariational principle. Instead, all equations ensuring the convergence to the complete active space limit can be selected flexibly. The applications to electron dynamics and vibrational dynamics are also discussed. In particular, we find that all NB methods violate the Ehrenfest theorem even when they are gauge-invariant, which distinguishes them from TD-ooCC methods. The stationary condition of EOMs of each NB methods also provides novel CC methods for time-independent electronic structure calculations.
In this paper, we analyze the properties of the recently proposed real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function approach [Rehr et al., J. Chem. Phys. 152, 174113 (2020)]. We specifically focus on identifying the limitations of the original time-dependent coupled cluster (TDCC) ansatz and propose an enhanced double TDCC ansatz, ensuring the exactness in the expansion limit. In addition, we introduce a practical cluster-analysis-based approach for characterizing the peaks in the computed spectral function from the RT-EOM-CC cumulant Green's function approach, which is particularly useful for the assignments of satellite peaks when many-body effects dominate the spectra. Our preliminary numerical tests focus on reproducing, approximating, and characterizing the exact impurity Green's function of the three-site and four-site single impurity Anderson models using the RT-EOM-CC cumulant Green's function approach. The numerical tests allow us to have a direct comparison between the RT-EOM-CC cumulant Green's function approach and other Green's function approaches in the numerical exact limit.