We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.
We present a time-dependent formulation of coupled cluster theory. This theory allows for direct computation of the free energy of quantum systems at finite temperature by imaginary time integration and is closely related to the thermal cluster cumulant theory of Mukherjee and co-workers [ Chem. Phys. Lett. 1992 , 192 , 55 -61 ; Phys. Rev. E 1993 , 48 , 3373 -3389 ; Chem. Phys. Lett. 2001 , 335 , 281 -288 ; Chem. Phys. Lett. 2002 , 352 , 63 -69 ; Int. J. Mod. Phys. B 2003 , 17 , 5367 -5377 ]. Our derivation of the finite-temperature theory highlights connections to perturbation theory and to zero-temperature coupled cluster theory. We show explicitly how the finite-temperature coupled cluster singles and doubles amplitude equations can be derived in analogy with the zero-temperature theory and how response properties can be efficiently computed using a variational Lagrangian. We discuss the implementation for realistic systems and showcase the potential utility of the method with calculations of the exchange correlation energy of the uniform electron gas under warm dense matter conditions.
At zero temperature, coupled cluster theories are widely used to predict total energies, ground state expectation values, and even excited states for molecules and extended systems. However, for systems with a small band gap, such as metals, the zero-temperature approximation does not necessarily hold. Thermal effects may even give rise to interesting chemistry on metal surfaces. Most approaches to temperature dependent electronic properties employ finite temperature perturbation theory in the Matsubara frequency formulation. Computations require a large number of Matsubara frequencies to yield sufficiently accurate results, especially at low temperatures. This work, and independently the work of White and Chan J. Chem. Theory Comput. 2018 , DOI: 10.1021/acs.jctc.8b00773 , proposes a coupled cluster implementation directly in the imaginary time domain on the compact interval [0, β], closely related to the thermal cluster cumulant approach of Sanyal et al. [ Chem. Phys. Lett. 1992 , 192 , 55 - 61 ] , Sanyal et al. [ Phys. Rev. E 1993 , 48 , 3373 - 3389 ], and Mandal et al. [ Int. J. Mod. Phys. B 2003 , 17 , 5367 - 5377 ]. Here, the arising imaginary time dependent coupled cluster amplitude integral equations are solved in the linearized direct ring doubles approximation, also referred to as Tamm-Dancoff approximation with second order (linearized) screened exchange. In this framework, the transition from finite to zero temperature is uniform and comes at no extra costs, allowing to go to temperatures as low as room temperature. In this approximation, correlation grand potentials are calculated over a wide range of temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.
At zero temperature coupled cluster theory is widely used to predict total energies, ground state expectation values and even excited states for molecules and extended systems. Generalizations to finite temperature exist, however, they are in practice several orders of magnitude more demanding compared to the zero temperature case since the amplitudes must be computed and stored for many Matsubara frequencies to yield sufficiently accurate results. Instead of using Matsubara frequencies one can also work directly in the imaginary time domain on the compact interval [0, beta]. The arising imaginary time dependent coupled cluster amplitude integral equations are solved numerically on a non-uniform grid. About a dozen grid points provide sufficient accuracy if the Hamiltonian is repartitioned to include particle/hole interactions in the non-perturbative part. In this framework the transition from finite to zero temperature is uniform and comes at no extra costs, allowing, for instance, to benchmark fractional occupancy formulations of coupled cluster theories. Finite temperature direct ring coupled cluster doubles free energies are calculated for various temperatures for solid lithium, a metallic system, and for solid silicon, a semiconductor.
We leverage the Keldysh formalism to extend our implementation of finite temperature coupled cluster theory [ J. Chem. Theory Comput. 2018, 14, 5690–5700] to thermal systems that have been driven out of equilibrium. The resulting Keldysh coupled cluster theory is discussed in detail. We describe the implementation of the equations necessary to perform Keldysh coupled cluster singles and doubles calculations of finite-temperature dynamics, and we apply the method to some simple systems including a Hubbard model with a Peierls phase and an ab initio model of warm-dense silicon subject to an ultrafast XUV pulse.
We present in this paper a systematic nonperturbative cluster-cumulant method for deriving thermal averages of operators in quantum many-body systems. The method combines the advantages of the cumulant expansion scheme of thermodynamic perturbation theory, the approach of thermofield dynamics as a finite-temperature field theory, and the time-dependent coupled-cluster theory extended to ``imaginary time.'' We have generalized the concepts of cumulants in a nonperturbative manner and have posited on the statistical operator an exponential-like ansatz containing connected, size-extensive operators in the exponent. These latter cumulantlike operators have been termed ``cluster cumulants'' by us. For a compact treatment, we have derived an alternative thermal field theory in which a time-ordered product is expanded in terms of ``thermal normal products'' of operators and thermal contractions---leading to a ``thermal Wick's theorem.'' The thermal normal products are the finite-temperature analogs of the ordinary normal products and have zero thermal averages. Operators in these products commute (anticommute) under permutations for bosons (fermions). This thermal representation is shown to be unitarily related to the traditional thermofield dynamics formulation, but has the advantage of using only the physical variables. The imaginary-time evolution of the statistical operator is treated by our recently formulated time-dependent cluster-cumulant theory. The partition function is evaluated as an exponential of a connected quantity. As an illustrative example, we have computed the partition function of an anharmonic oscillator with equally weighted cubic and quartic perturbation for a wide range of coupling, extending to the strongly nonperturbative regime. We study the behavior of free energy in the low-temperature limit and verify numerically the validity of the Kohn-Luttinger theorem [Phys. Rev. 118, 41 (1960)] for this system. We also show that our formalism is a natural nonperturbative analog of the thermodynamic perturbative theory by showing that a perturbative solution of the thermal-cluster-cumulant equations generates a variation of the Bloch--Balian--de Dominicis theory.
We discuss the theory and implementation of the finite temperature coupled cluster singles and doubles (FT-CCSD) method including the equations necessary for an efficient implementation of response properties. Numerical aspects of the method including the truncation of the orbital space and integration of the amplitude equations are tested on some simple systems, and we provide some guidelines for applying the method in practice. The method is then applied to the 1D Hubbard model, the uniform electron gas (UEG) at warm, dense conditions, and some simple materials. The performance of model systems at high temperatures is encouraging: for the one-dimensional Hubbard model, FT-CCSD provides a qualitatively accurate description of finite-temperature correlation effects even at U = 8, and it allows for the computation of systematically improvable exchange-correlation energies of the warm, dense UEG over a wide range of conditions. We highlight the obstacles that remain in using the method for realistic ab initio calculations on materials.
We extend the finite-temperature Keldysh non-equilibrium coupled cluster theory (Keldysh-CC) [A. F. White and G. K.-L. Chan, J. Chem. Theory Comput. 15, 6137–6253 (2019)] to include a time-dependent orbital basis. When chosen to minimize the action, such a basis restores local and global conservation laws (Ehrenfest’s theorem) for all one-particle properties while remaining energy conserving for time-independent Hamiltonians. We present the time-dependent Keldysh orbital-optimized coupled cluster doubles method in analogy with the formalism for zero-temperature dynamics, extended to finite temperatures through the time-dependent action on the Keldysh contour. To demonstrate the conservation property and understand the numerical performance of the method, we apply it to several problems of non-equilibrium finite-temperature dynamics: a 1D Hubbard model with a time-dependent Peierls phase, laser driving of molecular H2, driven dynamics in warm-dense silicon, and transport in the single impurity Anderson model.