Today, coupled-cluster theory offers the most accurate results among the practical ab initio electronic-structure theories applicable to moderate-sized molecules. Though it was originally proposed for problems in physics, it has seen its greatest development in chemistry, enabling an extensive range of applications to molecular structure, excited states, properties, and all kinds of spectroscopy. In this review, the essential aspects of the theory are explained and illustrated with informative numerical results.

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Coupled-cluster theory in quantum chemistry

Dalton is a powerful general-purpose program system for the study of molecular electronic structure at the Hartree-Fock, Kohn-Sham, multiconfigurational self-consistent-field, MOller-Plesset, confi ...

The Dalton quantum chemistry program system

A benchmark set of 28 medium-sized organic molecules is assembled that covers the most important classes of chromophores including polyenes and other unsaturated aliphatic compounds, aromatic hydrocarbons, heterocycles, carbonyl compounds, and nucleobases. Vertical excitation energies and one-electron properties are computed for the valence excited states of these molecules using both multiconfigurational second-order perturbation theory, CASPT2, and a hierarchy of coupled cluster methods, CC2, CCSD, and CC3. The calculations are done at identical geometries (MP26-31G*) and with the same basis set (TZVP). In most cases, the CC3 results are very close to the CASPT2 results, whereas there are larger deviations with CC2 and CCSD, especially in singlet excited states that are not dominated by single excitations. Statistical evaluations of the calculated vertical excitation energies for 223 states are presented and discussed in order to assess the relative merits of the applied methods. CC2 reproduces the CC3 reference data for the singlets better than CCSD. On the basis of the current computational results and an extensive survey of the literature, we propose best estimates for the energies of 104 singlet and 63 triplet excited states.

https://www.chemie.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Chemie/SFB_663/Veroeffentlichungen/C4_043.pdf

Benchmarks for electronically excited states: CASPT2, CC2, CCSD, and CC3

The equation-of-motion coupled-cluster (EOM-CC) approach is a versatile electronic-structure tool that allows one to describe a variety of multiconfigurational wave functions within single-reference formalism. This review provides a guide to established EOM methods illustrated by examples that demonstrate the types of target states currently accessible by EOM. It focuses on applications of EOM-CC to electronically excited and open-shell species. The examples emphasize EOM's advantages for selected situations often perceived as multireference cases [e.g., interacting states of different nature, Jahn-Teller (JT) and pseudo-JT states, dense manifolds of ionized states, diradicals, and triradicals]. I also discuss limitations and caveats and offer practical solutions to some problematic situations. The review also touches on some formal aspects of the theory and important current developments.

Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker's Guide to Fock Space

Near-infrared-emissive polymer-carbon nanodots (PCNDs) are fabricated by a newly developed facile, high-output strategy. The PCNDs emit at a wavelength of 710 nm with a quantum yield of 26.28%, which is promising for deep biological imaging and luminescent devices. Moreover, the PCNDs possess two-photon fluorescence; in vivo bioimaging and red-light-emitting diodes based on these PCNDs are demonstrated.

Near-Infrared Photoluminescent Polymer-Carbon Nanodots with Two-Photon Fluorescence

Using a Lagrangian formulation an integral-density direct implementation of the analytic CCSD(T) molecular gradient is presented, which circumvents the bottleneck of storing either O(N4) two-electron integrals or O(N4) density matrix elements on disk. Canonical orbitals are used to simplify the implementation of the frozen-core approximation and the CCSD gradient is obtained as a special case. Also a new, simplified approach to (geometrical) derivative integrals is presented. As a first application we report a full geometry optimization for the most stable isomer of SiC3 using the cc-pV5Z basis set with 368 contracted basis functions and the frozen-core approximation.

A Lagrangian, integral-density direct formulation and implementation of the analytic CCSD and CCSD(T) gradients

An implementation is reported for first-order properties of excited triplet states within the approximate coupled cluster model CC2 using an explicitly spin coupled basis for the triplet excitation manifold and the resolution of the identity (RI) approximation for the electron repulsion integrals. Results are presented for the change of the second moment of charge upon excitation in the ππ* valence and n=3 Rydberg states of benzene. Employing large basis sets with up to 828 functions, we obtain results close to the CC2 basis set limit and are able to resolve an uncertainty in the assignment of the lowest 1E1u states. It is found that the often used %T1 measure for the single excitation contribution to excited states is not reliable for a comparison across different excitation operator manifolds. An alternative diagnostic is proposed which provides a unique measure for the single excitation contribution that is independent of the chosen representation of the excitation operator manifold.

First-order properties for triplet excited states in the approximated coupled cluster model CC2 using an explicitly spin coupled basis

CC3 is a member of the coupled-cluster model hierarchy CCS, CC2, CCSD, CC3, and CCSDT which is especially designed to describe frequency-dependent properties. CCS is the coupled-cluster singles model, in CCSD doubles are added and in CC2 the doubles of the CCSD model are approximated using the same strategy as for triples when CCSDT is approximated to give CC3. Excitation energies have been calculated successfully using this hierarchy. The error in the excitation energies is reduced by about a factor 3 at each level for the models CCS, CC2, CCSD, and CC3, and the CC3 excitation energies closely approximate the ones of the CCSDT model. 14 Calculation of frequency-dependent polarizabilities and hyperpolarizabilities have shown similar systematic improvements to the excitation energies. 15‐17

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Calculation of frequency-dependent polarizabilities using the approximate coupled-cluster triples model CC3

Triplet excitation energies are calculated from the response eigenvalue equation for the coupled cluster singles and doubles (CCSD) model using an integral direct approach and an explicit spin coupled triplet excitation space. The cost of one linear transformation for the triplet excitation energy is about two times the cost of one linear transformation for the singlet excitation energy. The triplet excitation spectrum of benzene is calculated using from 147 to 432 basis functions. The calculated triplet excitation energies are compared with experimental and other theoretical values.

Triplet excitation energies in the coupled cluster singles and doubles model using an explicit triplet spin coupled excitation space

A general scheme is presented for the calculation of excitation energies using the standard coupled cluster hierarchy and a simple implementation is described for the higher standard models. An error analysis is performed to find to what order excitation energies in different coupled cluster models are correct. The analysis includes both the standard coupled cluster hierarchy as well as the approximate models and considers excitations to states that are dominated by one, two, and three electron replacements compared to the reference state. Calculations are presented up to the quadruple excitation level for the open shell B2 molecule using an excited closed shell state as reference state to emphasize the usefulness of the order analysis. The coupled cluster excitation energies are compared to full configuration interaction results.

Kasper Hald

Papers

A Lagrangian, integral-density direct formulation and implementation of the analytic CCSD and CCSD(T) gradients

First-order properties for triplet excited states in the approximated coupled cluster model CC2 using an explicitly spin coupled basis

Calculation of frequency-dependent polarizabilities using the approximate coupled-cluster triples model CC3

Triplet excitation energies in the coupled cluster singles and doubles model using an explicit triplet spin coupled excitation space

An analysis and implementation of a general coupled cluster approach to excitation energies with application to the B2 molecule