The curse of dimensionality (COD) limits the current state-of-the-art ab initio propagation methods for non-relativistic quantum mechanics to relatively few particles. For stationary structure calculations, the coupled-cluster (CC) method overcomes the COD in the sense that the method scales polynomially with the number of particles while still being size-consistent and extensive. We generalize the CC method to the time domain while allowing the single-particle functions to vary in an adaptive fashion as well, thereby creating a highly flexible, polynomially scaling approximation to the time-dependent Schrödinger equation. The method inherits size-consistency and extensivity from the CC method. The method is dubbed orbital-adaptive time-dependent coupled-cluster, and is a hierarchy of approximations to the now standard multi-configurational time-dependent Hartree method for fermions. A numerical experiment is also given.
The curse of dimensionality (COD) limits the current state-of-the-art ab initio propagation methods for non-relativistic quantum mechanics to relatively few particles. For stationary structure calculations, the coupled-cluster (CC) method overcomes the COD in the sense that the method scales polynomially with the number of particles while still being size-consistent and extensive. We generalize the CC method to the time domain while allowing the single-particle functions to vary in an adaptive fashion as well, thereby creating a highly flexible, polynomially scaling approximation to the time-dependent Schrödinger equation. The method inherits size-consistency and extensivity from the CC method. The method is dubbed orbital-adaptive time-dependent coupled-cluster, and is a hierarchy of approximations to the now standard multi-configurational time-dependent Hartree method for fermions. A numerical experiment is also given.
A dynamical formulation of coupled cluster theory is derived using a variational principle. By allowing time-dependent single-particle functions, a high degree of adaptivity is introduced, allowing complex systems to be simulated with high accuracy. Equations of motion are derived which are shown to be suitable for computer implementation. The method, called adaptive time-dependent coupled cluster, is a strict generalization of the formulation used in standard coupled cluster response theory, and also represents a systematic hierarchy of size-consistent approximations, including standard time-dependent Hartree--Fock as a trivial case.
Accurate descriptions of many-particle quantum systems subject to laser interactions can be found using real-time ab initio methods. Of these the arguably most popular and exact is the multi-configuration time-dependent Hartree-Fock (MCTDHF) method. However, MCTDHF suffers from computational limitations in that it quickly becomes too time consuming. The orbital-adaptive time-dependent coupled-cluster (OATDCC) method represents a hierarchy of approximations to MCTDHF that are less computationally expensive while retaining as much accuracy as possible. Building on an existing codebase we have in this thesis generalized the OATDCC method to include Q-space orbital equations. A novel ground state solver is implemented, employing adiabatic switching, since imaginary time propagation is not feasible "out of the box". Furthermore, we implement a sinc-discrete variable representation basis for one-dimensional model systems. We demonstrate that ionization and high-harmonic processes can be described using the OATDCCD method. Comparison with the more accurate, yet more expensive, multiconfigurational time-dependent Hartree-Fock method indicates that the OATDCCD method is an excellent approximation.
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 workhorse method of computational materials science is undeniably density functional theory in the Kohn-Sham framework of approximate exchange and correlation energy functionals. However, the need for highly accurate predictions of ground and excited state properties in materials science motivates the further development and exploration of alternative as well as complementary techniques. Among these alternative approaches, quantum chemical wavefunction based theories and in particular coupled cluster theory hold the promise to fill a gap in the toolbox of computational materials scientists. Coupled cluster (CC) theory provides a compelling framework of approximate infinite-order perturbation theory in the form of an exponential of cluster operators describing the true quantum many-body effects of the electronic wave function at a computational cost that, despite being significantly more expensive than DFT, scales polynomially with system size. The hierarchy of size-extensive approximate methods established in the framework of CC theory achieves systematic improvability for many materials properties. This is in contrast to currently available density functionals that often suffer from uncontrolled approximations that limit the accuracy in the prediction of materials properties. In this tutorial-style review we will introduce basic concepts of coupled cluster theory and recent developments that increase its computational efficiency for calculations of molecules, solids and materials in general. We will touch upon the connection between coupled cluster theory and the random-phase approximation to bridge the gap between traditional quantum chemistry and many-body Green’s function theories that are widely-used in the field of solid state physics. We will discuss various approaches to improve the computational performance without compromising accuracy. These approaches include large-scale parallel design as well as techniques that reduce the prefactor of the computational complexity. A central part of this article discusses the convergence of calculated properties to the thermodynamic limit which is of significant importance for reliable predictions of materials properties and constitutes an additional challenge compared to calculations of large molecules. Furthermore we mention technical aspects of computer code implementations of periodic coupled cluster theories in different numerical frameworks of the one-electron orbital basis; the projector-augmented-wave formalism using a plane wave basis set and the numeric atom-centered-orbital with resolution-of-identity.
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.
Five time-dependent orbital optimized coupled-cluster methods, of which four can converge to the time-dependent complete active space self-consistent-field method, are presented for fermion-mixtures with arbitrary fermion kinds and numbers. Truncation schemes maintaining the intragroup orbital rotation invariance, as well as equations of motion of coupled-cluster (CC) amplitudes and orbitals, are derived. Present methods are compact CC-parameterization alternatives to the time-dependent multiconfiguration self-consistent-field method for systems consisting of arbitrarily different kinds and numbers of interacting fermions. Theoretical analysis of applications of present methods to various chemical systems is reported.