Calculation Of Correlation Energy Research Articles

Towards adiabatic-connection interpolation model with broader applicability

The adiabatic connection integrand interpolation (ACII) method represents a general path for calculating correlation energies in electronic systems within the density functional theory. ACII functionals include both exact-exchange and the second-order correlation energy, as well as an interpolating function toward the strictly correlated electron (SCE) regime. Several interpolating functions have been proposed in the last years targeting different properties, yet an accurate ACII approach with broad applicability is still missing. Recently, we have proposed an ACII functional that was made accurate for the three-dimensional (3D) uniform electron gas as well as for model metal clusters. In this paper we present an ACII functional (named genISI2), which is very accurate for both three-dimensional (3D) and two-dimensional (2D) uniform electron gases and for the quasi-2D infinite-barrier model, where most of the exchange-correlation functionals fail badly, as well as for strongly correlated two-electrons systems. Using the exact-exchange Kohn-Sham orbitals, we have also assessed the genISI2 for various molecular systems, showing a superior performance with respect to the other ACII methods for total energies, atomization energies, and ionization potentials. The genISI2 functional can thus find application in a broad range of systems and properties. Published by the American Physical Society 2024

Linear Scaling Incremental Scheme for Correlation Energies with Embedding Generated Virtuals.

A novel incremental scheme is presented including an incremental expansion of the virtual space for the calculation of electron correlation energies, which is compatible with any size-extensive correlation method and scales asymptotically linear for large molecules. The performance is studied for organic molecules, water clusters, and a La(III)-water complex, where the compatibility with pseudopotentials is also examined. The computational requirements are already reduced tremendously for medium-sized water clusters and hydrocarbons with respect to the canonical CCSD as well as the ordinary incremental scheme references. Correlation energies within chemical accuracy have been observed for all studied systems. The novelty of the method is that relatively small virtual spaces are used in combination with tuples of localized occupied spaces. The corresponding orthonormal occupied and virtual orbitals are obtained from QM/QM embedding calculations and can thus be used with standard quantum chemistry codes for correlation calculations. It is presented how relevant virtual spaces are selected and the correlation energies are linked in the new virtual space expansion.

Basis-Set-Error-Free Random-Phase Approximation Correlation Energies for Atoms Based on the Sternheimer Equation.

The finite basis set errors for all-electron random-phase approximation (RPA) correlation energy calculations are analyzed for isolated atomic systems. We show that, within the resolution-of-identity (RI) RPA framework, the major source of the basis set errors is the incompleteness of the single-particle atomic orbitals used to expand the Kohn-Sham eigenstates, instead of the auxiliary basis set (ABS) to represent the density response function χ0 and the bare Coulomb operator v. By solving the Sternheimer equation for the first-order wave function on a dense radial grid, we are able to eliminate the major error─the incompleteness error of the single-particle atomic basis set─for atomic RPA calculations. The error stemming from a finite ABS can be readily rendered vanishingly small by increasing the size of the ABS, or by iteratively determining the eigenmodes of the χ0v operator. The variational property of the RI-RPA correlation energy can be further exploited to optimize the ABS in order to achieve fast convergence of the RI-RPA correlation energy. These numerical techniques enable us to obtain basis-set-error-free RPA correlation energies for atoms, and in this work, such energies for atoms from H to Kr are presented. The implications of the numerical techniques developed in the present work for addressing the basis set issue for molecules and solids are discussed.

How the Exchange Energy Can Affect the Power Laws Used to Extrapolate the Coupled Cluster Correlation Energy to the Thermodynamic Limit.

Finite size error is commonly removed from coupled cluster theory calculations by N-1 extrapolations over correlation energy calculations of different system sizes (N), where the N-1 scaling comes from the total energy rather than the correlation energy. However, previous studies in the quantum Monte Carlo community suggest an exchange-energy-like power law of N-2/3 should also be present in the correlation energy when using the conventional Coulomb interaction. The rationale for this is that the total energy goes as N-1 and the exchange energy goes as N-2/3; thus, the correlation energy should be a combination of these two power laws. Further, in coupled cluster theory, these power laws are related to the low G scaling of the transition structure factor, S(G), which is a property of the coupled cluster wave function calculated from the amplitudes. We show here that data from coupled cluster doubles calculations on the uniform electron gas fit a function with a low G behavior of S(G) ∼ G. The prefactor for this linear term is derived from the exchange energy to be consistent with an N-2/3 power law at large N. Incorporating the exchange structure factor into the transition structure factor results in a combined structure factor of S(G) ∼ G2, consistent with an N-1 scaling of the exchange-correlation energy. We then look for the presence of an N-2/3 power law in the energy. To do so, we first develop a plane-wave cutoff scheme with less noise than the traditional basis set used for the uniform electron gas. Then, we collect data from a wide range of electron numbers and densities to systematically test five methods using N-1 scaling, N-2/3 scaling, or combinations of both scaling behaviors. We find that power laws that incorporate both N-1 and N-2/3 scaling perform better than either alone, especially when the prefactor for N-2/3 scaling can be found from exchange energy calculations.

First-principles calculations to investigate structural, elastic, electronic, optical, and magnetic properties of Hg2WO4 for photocatalytic applications

In the present work, structural, elastic, electronic, optical, and magnetic properties of Hg2WO4 are investigated using CASTEP simulation tool. Generalized Gradient Approximation (GGA) along with Perdew-Burke-Ernzerhof (PBE) are employed for the calculation of electronic exchange and correlation energy. In structural analysis, Hg2WO4 is found to be stable with C-centered monoclinic structure having space group C2/c. Material’s ductile nature is confirmed with Pugh’s approximation where calculated B/G ratio is greater than 0.75 and Frantsevich’s rule where v is greater than 0.26. The calculations of electronic properties show that Hg2WO4 has indirect band gap of 2.14 eV. While the optical study reveals that Hg2WO4 has a good photosensitivity in visible light region. Furthermore, the spin-polarized treatment shows that Hg2WO4 is antiferromagnetic in nature. The current findings can be of great importance for the development of efficient photocatalytic materials for water splitting and associated applications.

A Minimum Quantum Chemistry CCSD(T)/CBS Data Set of Dimeric Interaction Energies for Small Organic Functional Groups: Heterodimers.

We extend our previous quantum chemistry calculations of interaction energies for 31 homodimers of small organic functional groups (the SOFG-31 data set) by including 239 heterodimers with monomers selected within the SOFG-31 data set, thus resulting in the SOFG-31+239 data set. The minimum-level theoretical scheme contains (1) the basis set superposition error corrected supermolecule (BSSE-SM) approach for intermolecular interactions; (2) the second-order Møller–Plesset perturbation theory (MP2) with the Dunning’s aug-cc-pVXZ (X = D, T, Q) basis sets for the geometry optimization and correlation energy calculations; and (3) the single-point energy calculations with the coupled cluster with single, double, and perturbative triple excitations method at the complete basis set limit [CCSD(T)/CBS] using the well-tested extrapolation methods for the MP2 energy calibrations. In addition, we have performed a parallel series of energy decomposition calculations based on the symmetry adapted perturbation theory (SAPT) in order to gain chemical insights. That the above procedure cannot be further reduced has been proven to be very crucial for constructing reliable data sets of interaction energies. The calculated CCSD(T)/CBS interaction energy data can serve as a benchmark for testing or training less accurate but more efficient calculation methods, such as the electronic density functional theory. As an application, we employ a segmental SAPT model previously developed for the SOFG-31 data set to predict binding energies of large heterodimer complexes. These model energy “quanta” can be used in coarse-grained molecular dynamics simulations by avoiding large-scale calculations.

Staggered Mesh Method for Correlation Energy Calculations of Solids: Random Phase Approximation in Direct Ring Coupled Cluster Doubles and Adiabatic Connection Formalisms

We propose a staggered mesh method for correlation energy calculations of periodic systems under the random phase approximation (RPA), which generalizes the recently developed staggered mesh method for periodic second order Møller-Plesset perturbation theory (MP2) calculations [Xing; Li; Lin J. Chem. Theory Comput. 2021]. Compared to standard RPA calculations, the staggered mesh method introduces negligible additional computational cost. It avoids a significant portion of the finite-size error and can be asymptotically advantageous for quasi-1D systems and certain quasi-2D and 3D systems with high symmetries. We demonstrate the applicability of the method using two different formalisms: the direct ring coupled cluster doubles (drCCD) theory, and the adiabatic-connection (AC) fluctuation-dissipation theory. In the drCCD formalism, the second order screened exchange (SOSEX) correction can also be readily obtained using the staggered mesh method. In the AC formalism, the staggered mesh method naturally avoids the need of performing "head/wing" corrections to the dielectric operator. The effectiveness of the staggered mesh method for insulating systems is theoretically justified by investigating the finite-size error of each individual perturbative term in the RPA correlation energy, expanded as an infinite series of terms associated with ring diagrams. As a side contribution, our analysis provides proof that the finite-size error of each perturbative term of standard RPA and SOSEX calculations scales as , where Nk is the number of grid points in a Monkhorst-Pack mesh.

Reparametrization of the Colle-Salvetti formula.

We investigate the Colle–Salvetti (CS) formula, the basis of the Lee, Yang and Parr (LYP) correlation functional used in approximate density functional theory. The CS formula is reparametrized using high-accuracy Hartree–Fock (HF) wavefunctions to determine the accuracy of the formula to calculate anions. Fitting to the hydride ion or the two-electron system just prior to electron detachment at the HF level of theory does not, in general, improve the calculated correlation energies using the parameters derived from the CS/LYP method. An analysis of the CS parameters used in the popular LYP functional demonstrates the ingenuity and perhaps fortuitousness of the original formulation by CS.

Staggered Mesh Method for Correlation Energy Calculations of Solids: Second-Order Møller-Plesset Perturbation Theory.

The calculation of the MP2 correlation energy for extended systems can be viewed as a multidimensional integral in the thermodynamic limit, and the standard method for evaluating the MP2 energy can be viewed as a trapezoidal quadrature scheme. We demonstrate that the existing analysis neglects certain contributions due to the nonsmoothness of the integrand and may significantly underestimate finite-size errors. We propose a new staggered mesh method, which uses two staggered Monkhorst-Pack meshes for occupied and virtual orbitals, respectively, to compute the MP2 energy. The staggered mesh method circumvents a significant error source in the standard method in which certain quadrature nodes are always placed on points where the integrand is discontinuous. One significant advantage of the proposed method is that there are no tunable parameters, and the additional numerical effort needed can be negligible compared to the standard MP2 calculation. Numerical results indicate that the staggered mesh method can be particularly advantageous for quasi-1D systems as well as quasi-2D and 3D systems with certain symmetries.

A Machine Learning Approach for MP2 Correlation Energies and Its Application to Organic Compounds.

A proper treatment of electron correlation effects is indispensable for accurate simulation of compounds. Various post-Hartree-Fock methods have been adopted to calculate correlation energies of chemical systems, but time complexity usually prevents their usage in a large scale. Here, we propose a density functional approximation, based on machine learning using neural networks, which can be readily employed to produce results comparable to second-order Møller-Plesset perturbation (MP2) ones for organic compounds with reduced computational cost. Various systems have been tested and the transferability across basis sets, structures, and nuclear configurations has been evaluated. Only a small number of molecules at the equilibrium structure has been needed for the training, and generally less than 5% relative error has been achieved for structures outside the training domain and systems containing about 140 atoms. In addition, this approach has been applied to make predictions on correlation energies of nuclear configurations extracted from density functional theory-based molecular dynamics trajectories with only one or two structures as training data.

Crystal Structures and Electrical Resistivity of Three Exotic TMTSF Salts with I 3 − : Determination of Valence by DFT and MP2 Calculations

Three novel organic conductors (TMTSF)8(I3)5, (TMTSF)5(I3)2, and (TMTSF)4(I3)4·THF (THF = tetrahydrofuran) were synthesized and their crystal structures were characterized by X-ray diffraction analyses, where TMTSF denotes tetramethyltetraselenafulvalene. The crystals of both the (TMTSF)8(I3)5 and (TMTSF)5(I3)2 are composed of one-dimensional stacks of TMTSF trimers separated by TMTSF monomers. The crystal of the (TMTSF)4(I3)4·THF is composed of the TMTSF tetramers and I3− tetramers; and regarded as the elongated rock-salt structure. The electrical conductivity of the (TMTSF)8(I3)5 and (TMTSF)5(I3)2 is about 60 and 50 S·cm−1 at room temperature, respectively. The electrical resistivity of (TMTSF)8(I3)5 is weakly metallic below room temperature and rapidly increases below 88 and 53 K on cooling suggesting two possible phase transitions. The electrical resistivity of (TMTSF)5(I3)2 is semiconducting below room temperature but shows an anomaly around 190 K, below which the activation energy becomes small. The application of hydrostatic pressure up to 1.7 GPa do not change these behaviors of (TMTSF)8(I3)5 and (TMTSF)5(I3)2 very much. A method to evaluate the non-integer valence of crystallographically independent TMTSF molecules is developed by using the DFT (density-functional-theory) and MP2 (Hartree–Fock calculations followed by Møller–Plesset correlation energy calculations truncated at second order) calculations. It is shown that the method gives the valence of the TMTSF molecules of the I3 salts consistent with their electrical properties.

Efficient Treatment of Correlation Energies at the Basis-Set Limit by Monte Carlo Summation of Continuum States.

The calculation of electron correlation is vital for the description of atomistic phenomena in physics, chemistry, and biology. However, accurate wavefunction-based methods exhibit steep scaling and often sluggish convergence with respect to the basis set at hand. Because of their delocalization and ease of extrapolation to the basis-set limit, plane waves would be ideally suited for the calculation of basis-set limit correlation energies. However, the routine use of correlated wavefunction approaches in a plane-wave basis set is hampered by prohibitive scaling due to a large number of virtual continuum states and has not been feasible for all but the smallest systems, even if substantial computational resources are available and methods with comparably beneficial scaling, such as the Møller–Plesset perturbation theory to second order (MP2), are used. Here, we introduce a stochastic sampling of the MP2 integrand based on Monte Carlo summation over continuum orbitals, which allows for speedups of up to a factor of 1000. Given a fixed number of sampling points, the resulting algorithm is dominated by a flat scaling of . Absolute correlation energies are accurate to <0.1 kcal/mol with respect to conventional calculations for several hundreds of electrons. This allows for the calculation of unbiased basis-set limit correlation energies for systems containing hundreds of electrons with unprecedented efficiency gains based on a straightforward treatment of continuum contributions.

Time-dependent optimized coupled-cluster method for multielectron dynamics. II. A coupled electron-pair approximation.

We report the implementation of a cost-effective approximation method within the framework of the time-dependent optimized coupled-cluster (TD-OCC) method [T. Sato et al., J. Chem. Phys. 148, 051101 (2018)] for real-time simulations of intense laser-driven multielectron dynamics. The method, designated as TD-OCEPA0, is a time-dependent extension of the simplest version of the coupled-electron pair approximation with optimized orbitals [U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 139, 054104 (2013)]. It is size extensive, gauge invariant, and computationally much more efficient than the TD-OCC method with double excitations. We employed this method to simulate the electron dynamics in Ne and Ar atoms exposed to intense near infrared laser pulses with various intensities. The computed results, including high-harmonic generation spectra and ionization yields, are compared with those of various other methods ranging from uncorrelated time-dependent Hartree-Fock to fully correlated (within the active orbital space) time-dependent complete-active-space self-consistent field (TD-CASSCF). The TD-OCEPA0 results show good agreement with TD-CASSCF ones for moderate laser intensities. For higher intensities, however, TD-OCEPA0 tends to overestimate the correlation effect, as occasionally observed for CEPA0 in the ground-state correlation energy calculations.

Separable resolution-of-the-identity with all-electron Gaussian bases: Application to cubic-scaling RPA.

We explore a separable resolution-of-the-identity (RI) formalism built on quadratures over limited sets of real-space points designed for all-electron calculations. Our implementation preserves, in particular, the use of common atomic orbitals and their related auxiliary basis sets. The setup of the present density fitting scheme, i.e., the calculation of the system specific quadrature weights, scales cubically with respect to the system size. Extensive accuracy tests are presented for the Fock exchange and MP2 correlation energies. We finally demonstrate random phase approximation (RPA) correlation energy calculations with a scaling that is cubic in terms of operations, quadratic in memory, with a small crossover with respect to our standard RI-RPA implementation.

A heuristic estimate of molecular correlation energies using pair correlation energies of localized molecular orbitals

The biggest problem for the application of wavefunction-based quantum chemical ab initio methods is the calculation of the electronic correlation energy. Advanced methods such as coupled-cluster theory using iterative single and double excitations as well as perturbative triple excitations [CCSD(T)] are able to achieve an accuracy of ± 5 kJ/mol (‘chemical accuracy’) for small molecular systems, but become too time-consuming for routine applications to larger systems containing twenty or more heavy atoms. We propose a heuristic approach for a rapid estimate of correlation energies for larger molecules, based on parametrized pair correlation energies (PCEs) for localized molecular orbitals. Such PCEs were calculated for a training set of 112 small- and medium-sized neutral molecules from the G2/97 data set, using an approximate coupled-cluster method with single and double excitations (CCSD) and a basis set of quadruple zeta quality (def2-QZVP). The PCEs were then fitted to appropriate functional forms, taking into account the properties of the localized orbitals: type (lone pair, single bond, multiple bond), bond length, hybridization of the atoms involved in the bond, spatial extent, and distance between two orbitals. The ab initio results for the total correlation energies of the molecules in the training set could be reproduced within 1%. For most of the molecules in an extended test set containing also molecular ions a slightly lower accuracy of about 1% to 3% could be obtained.

Colle-Salvetti type correlation functionals for two-dimensional quantum dot systems

Colle-Salvetti type energy functionals for dynamical correlation are constructed for two-dimensional quantum systems. These correlation functionals depend on the 2D-Becke88 exchange energy functional and contain a progressive part which can be updated by taking a perfect approximation of the pair density and redefining the adjustable parameters. Assessment of these proposed functionals is endorsed through parabolic quantum dot systems. The exact correlation energies along with self-consistent results corresponding to two existing correlation functionals are calculated for comparison. The calculated correlation energies for the considered quantum dot systems agree well with the reference results having the minimum error among all considered functionals.

Assessment of MC-PDFT Excitation Energies for a Set of QM/MM Models of Rhodopsins.

A methodology for the automatic production of quantum mechanical/molecular mechanical (QM/MM) models of retinal-binding rhodopsin proteins and subsequent prediction of their spectroscopic properties has been proposed recently by some of the authors. The technology employed for the evaluation of the excitation energies is called Automatic Rhodopsin Modeling (ARM), and it involves the use of the complete active space self-consistent field (CASSCF) method followed by a multiconfiguration second-order perturbation theory (in particular, CASPT2) calculation of external correlation energies. Although it was shown that ARM is capable of successfully reproducing and predicting spectroscopic property trends in chromophore-embedding protein sets, practical applications of such technology are limited by the high computational costs of the multiconfiguration perturbation theory calculations. In the present work we benchmark the more affordable multiconfiguration pair-density functional theory (MC-PDFT) method whose accuracy has been recently validated for retinal chromophores in the gas phase, indicating that MC-PDFT could potentially be used to analyze large (e.g., few hundreds) sets of rhodopsin proteins. Here, we test this theory for a set of rhodopsin QM/MM models whose experimental absorption maxima (λ a max) have been measured. The results indicate that MC-PDFT may be employed to calculate λ a max values for this important class of photoresponsive proteins.

Theoretical study of laser cooling of potassium chloride anion

The potential energy curves and transition dipole moments (TDMs) for three Λ-S states (X&lt;sup&gt;2&lt;/sup&gt;Σ&lt;sup&gt;+&lt;/sup&gt;, A&lt;sup&gt;2&lt;/sup&gt;Π, and B&lt;sup&gt;2&lt;/sup&gt;Σ&lt;sup&gt;+&lt;/sup&gt;) of potassium chloride anion (KCl&lt;sup&gt;–&lt;/sup&gt;) are investigated by using multi-reference configuration interaction (MRCI) method. The def2-AQZVPP-JKFI of K atom and AV5Z-DK all-electron basis set of Cl atom are used in all calculations. The Davidson correction, core-valence (CV) correction, and spin-orbit coupling effect (SOC) are also considered. In the complete active self-consistent field (CASSCF) calculations, eight molecular orbitals are selected as active orbitals, which includ K 4s4p and Cl 3s3p shells; K 3p shell is closed orbital, and the remaining shells (K 1s2s3s and Cl 1s2s2p) are frozen orbitals. In the MRCI+&lt;i&gt;Q&lt;/i&gt; calculations, K 3p shell is used for the CV correction. There are 15 electrons in the correlation energy calculations. Then, their spectroscopic parameters, Einstein coefficients, Franck-Condon factors, and radiative lifetimes are obtained by solving the radial Schrödinger equation. The spectroscopic properties and transition properties for the Ω states are predicted. Highly diagonally distributed Franck-Condon factor &lt;i&gt;f&lt;/i&gt;&lt;sub&gt;00&lt;/sub&gt; values for the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 transition are 0.8816 and 0.8808, respectively. And the short radiative lifetimes for the (2)1/2 and (1)3/2 excited states are also obtained, i.e. &lt;i&gt;τ&lt;/i&gt;[(2)1/2] = 45.7 ns and &lt;i&gt;τ&lt;/i&gt;[(1)3/2] = 45.5 ns, which can ensure laser cooling of KCl&lt;sup&gt;–&lt;/sup&gt; anion rapidly. The results indicate that the (2)1/2↔(1)1/2 and (1)3/2↔(1)1/2 quasicycling transitions are suitable to the building of laser cooling projects. For driving the (2)1/2↔(1)1/2 transition, a main pump laser (λ&lt;sub&gt;00&lt;/sub&gt;) and two repumping lasers (λ&lt;sub&gt;10&lt;/sub&gt; and λ&lt;sub&gt;21&lt;/sub&gt;) are required. Their wavelengths are λ&lt;sub&gt;00&lt;/sub&gt; = 1065.77 nm, λ&lt;sub&gt;10&lt;/sub&gt; = 1090.13 nm and λ&lt;sub&gt;21&lt;/sub&gt; = 1087.76 nm. For driving the (1)3/2↔(1)1/2 transition, the wavelengths are λ&lt;sub&gt;00&lt;/sub&gt; = 1064.24 nm, λ&lt;sub&gt;10&lt;/sub&gt; = 1088.54 nm, and λ&lt;sub&gt;21&lt;/sub&gt; = 1086.17 nm. The cooling wavelengths of KCl&lt;sup&gt;-&lt;/sup&gt; anion for two transitions are both deep in the infrared range. Finally, the Doppler temperature and recoil temperature for two transitions are also calculated, respectively. The Doppler temperatures for (2)1/2↔(1)1/2 and (1)3/2(1)1/2 transitions are 83.57 μK and 83.93 μK, and the recoil temperatures for two transitions are 226 nK and 227 nK, respectively. for two transitions are 226 nK and 227 nK, respectively.

Efficient evaluation of electron correlation along the bond-dissociation coordinate in the ground and excited ionic states with dynamic correlation suppression and enhancement functions of the on-top pair density

Calculation of the electron correlation energy ${E}_{c}$ of ground and excited states through the partitioning and evaluation of the dynamic ${E}_{c}^{d}$ and nondynamic ${E}_{c}^{nd}$ components requires one to account for the interplay of these correlation modes as well as for the excitation effect. In this paper it is demonstrated that both local suppression of dynamic correlation (SDC) by nondynamic correlation and enhancement of dynamic correlation (EDC) due to excitation to a state of the ionic nature can be quantified with the ratio $x(\mathbit{r})$ between the correlated and uncorrelated on-top pair densities $\mathrm{\ensuremath{\Pi}}(\mathbit{r})$. A $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$ scheme is proposed, in which ${E}_{c}^{nd}$ is calculated with the complete active space approach in a small basis, while ${E}_{c}^{d}$ is calculated in the same basis with a functional of density functional theory corrected for SDC and EDC with an original correction function of $x(\mathbit{r})$. Correlation energies calculated with $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$ along the bond-dissociation coordinate for the paradigmatic ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ molecules as well as for the ${\mathrm{C}}_{2}$ molecule with strong nondynamic correlation at the equilibrium agree well with the reference data, thus providing a proof of concept for $\mathrm{CAS}\mathrm{\ensuremath{\Pi}}\mathrm{DFT}$.

A Time-Dependent Formulation of Coupled-Cluster Theory for Many-Fermion Systems at Finite Temperature.

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.