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REVIEWS OF MODERN PHYSICS, VOLUME 79, JANUARY–MARCH 2007 Coupled-cluster theory in quantum chemistry Rodney J. Bartlett and Monika Musiał* Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, Florida 32611-8435, USA ͑Published 22 February 2007͒ 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. DOI: 10.1103/RevModPhys.79.291 PACS number͑s͒: 01.30.Rr, 31.15.Dv, 71.15.Qe, 31.25.Jf CONTENTS chemistry circles, we begin with a quote from Ken Wil- son ͑1989͒: I. Perspective on the Molecular Electronic Problem 291 “Ab initio quantum chemistry is an emerging com- II. Plan for Review 293 putational area that is fifty years ahead of lattice … III. Some Essential Preliminaries 294 gauge theory and a rich source of new ideas and A. Configuration interaction 295 new approaches to the computation of many fer- B. Perturbation theory 296 mion systems.” C. Coupled-cluster theory 297 Driving these developments are the types of problems IV. Normal-Ordered Hamiltonian 299 addressed by quantum chemists, as shown in Fig. 1. Pri- ͑ ͒ V. Coupled-Cluster Equations 302 mary among these are potential-energy surfaces PES A. Double excitations in CC theory 304 which describe the behavior of the electronic energy B. Choice of single determinant reference function 305 with respect to the locations of the nuclei, subject to the C. Single excitations in CC theory 307 underlying Born-Oppenheimer or clamped nuclei ap- D. Triple and quadruple excitations in CC theory 308 proximation. As shown, these PES can be for ground or E. Noniterative approximations 313 electronic excited states. At the equilibrium geometry, VI. Survey of Ground-State Numerical Results 315 ٌE͑R͒=0 defines the molecular structure. The second A. Equilibrium properties 315 derivatives ٌٌE͑R͒ determine whether the critical point B. Basis-set issue 319 is a minimum or a saddle point. C. Bond breaking 321 In addition, from the ground- and excited-state wave VII. The Coupled-Cluster Functional functions one obtains all properties that arise from a and the Treatment of Properties 327 solution to the vibrational Schrödinger equation that VIII. Equation-of-Motion Coupled-Cluster Method for gives the frequencies, and, with the derivatives of the Excited, Ionized, and Electron Attached States 330 dipole moment, the infrared intensities. The derivatives A. Numerical results 333 of the dipole polarizability define the Raman intensities. IX. Multireference Coupled-Cluster Method 338 Electronic excited states are also accessible along with A. Hilbert-space formulation of the MRCC approach 340 electronic and photoelectron spectra. B. Fock-space formulation of the MRCC approach 342 For the more kinetic aspects of chemistry, the basic C. Fock-space MRCC based on an intermediate concept is a reaction path that is defined as a multidi- Hamiltonian 346 mensional path along which all vibrational degrees of Acknowledgments 347 freedom are optimum except one, which defines a path References 347 toward products. The latter might have a saddle point as shown, which defines a transition state and its activation barriers. From that information, it is possible to obtain I. PERSPECTIVE ON THE MOLECULAR ELECTRONIC rate constants and state-to-state cross sections. PROBLEM In addition, properties that arise from the one-particle density matrix, such as dipole moments, hyperfine cou- As the recent developments in coupled-cluster ͑CC͒ pling constants, and electric-field gradients, are readily theory have been mostly accomplished in quantum available. Also, one obtains second- and higher-order properties such as the dipole polarizability and NMR chemical shifts and coupling constants. From even *Permanent address: Institute of Chemistry, University of higher-order electric-field derivatives, one obtains hy- Silesia, Szkolna 9, 40-006 Katowice, Poland. perpolarizabilities, which determine nonlinear optical 0034-6861/2007/79͑1͒/291͑62͒ 291 ©2007 The American Physical Society 292 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry 1955͒, the summation of ring diagrams ͑Gell-Mann and Brueckner, 1957͒, and an infinite-order generalization of many-body perturbation theory ͑MBPT͒͑Kelly, 1969; Bartlett and Silver, 1974a, 1976͒. Hence, it is a very pow- erful method for correlation in many-electron systems. Its principal rationale compared to other quantum chemical methods is its correct scaling with size, termed size extensivity ͑Bartlett and Purvis, 1978, 1980͒. This means it is a purely linked diagram theory that guaran- tees correct scaling with the number of particles or units in a system, and facilitates accurate relative energies along a potential-energy surface or between different electronic states. Only with this property are applica- tions to polymers, solids, or the electron gas possible, and, even for small molecules, its effects are numerically quite significant. Configuration interaction methods, long the focus of the correlation problem in quantum chemistry ͑Shavitt, 1998͒, do not, in general, have this property which is responsible for the emphasis on CC theory and its MBPT approximations ͑Kelly, 1969; Bartlett and Silver, 1974a, 1974b; Pople et al., 1976͒ in chemistry. The CC theory was introduced in 1960 ͑Coester and ͒ FIG. 1. The nature of quantum chemical problems. Kümmel, 1960 for calculating nuclear binding energies in nuclei that could be treated in the first approximation by a single configuration of neutrons or protons. The behavior. From derivatives relative to atomic displace- detailed equations for electrons were first presented in ments in molecules, one obtains anharmonic effects on 1966 ͑Cˇ ížek, 1966͒ and its initial applications to elec- vibrational-rotational spectra. The two-particle density tronic structure were reported ͑Cˇ ížek, 1966; Paldus et al., matrix, besides having a role in the energy, is also essen- 1972͒. Starting in 1978 general purpose programs and ͑ tial for spin operators such as Sˆ 2 and spin-orbit effects in applications of CC theory were developed Bartlett and particular. Purvis, 1978, 1980; Pople et al., 1978; Purvis and Bartlett, ͒ Consequently, the objective is an accurate solution of 1982 . Early in the history of the CC, it was shown to be the Schrödinger equation for molecules composed of remarkably accurate for describing the correlation en- comparatively light elements. When relativistic effects ergy of the electron gas compared to the random-phase ͑ ͒͑ are essential, the solution of the Dirac equation might approximation RPA Freeman, 1977; Bishop and Lühr- ͒ be preferred. Of course, Darwin, mass-velocity, and mann, 1978 . spin-orbit effects can be added to the nonrelativistic so- For more details, the history of CC theory is best told lution to provide a wealth of approximations lying be- from the viewpoint of some of its principal developers. tween the Schrödinger equation and the four- In the proceedings from the workshop “Coupled Cluster component Dirac equation. Theory of Electron Correlation” papers of Kümmel ˇ For modest-sized molecules of up to ϳ15 light atoms ͑1991͒ and Cížek ͑1991͒ address this issue. More re- ͑ ͒ or ϳ100 electrons and small-molecule relativistic calcu- cently, papers on this topic Bartlett, 2005; Paldus, 2005 lations, coupled-cluster theory ͑Coester, 1958; Coester are pertinent. For the physics viewpoint, in the previous and Kümmel, 1960; Cˇ ížek, 1966, 1969; Paldus et al., 1972; proceedings see Arponen ͑1991͒ and Bishop ͑1991͒. Bartlett and Purvis, 1978, 1980; Bishop and Lührmann, For larger molecules and solids, far more approximate 1978͒ has become the preeminent tool to introduce the but more easily applied methods such as density- instantaneous effects of electron correlation that are not functional theory ͑DFT͒ or from the wave-function included in a mean-field approximation ͑Urban et al., world the simplest correlated model MBPT͑2͒͑also 1987; Bartlett, 1989, 1995; Paldus, 1991; Lee and Scuse- sometimes known as MP2 when using a Hartree-Fock ria, 1995; Gauss, 1998͒. There are many textbook and reference function͒ are preferred. There are CC solu- similar accounts, each with a different focus ͑Lindgren tions for some simple polymers ͑Hirata, Grabowski, et and Morrison, 1986; Harris et al., 1992; Bartlett and al., 2001; Hirata, Podeszwa, et al., 2004͒. For nuclei, after Stanton, 1994; Bishop, 1998; Crawford and Schaefer, an appropriate regularization of the strong interaction, 2000; Helgaker et al., 2000; Bishop et al., 2002; Shavitt CC theory can be applied almost as it is for molecules and Bartlett, 2006͒. ͑Coester and Kümmel, 1960; Kümmel et al., 1978; In short, from the viewpoint of a physicist, coupled- Guardiola et al., 1996; Bishop et al., 1998; Heisenberg cluster theory offers a synthesis of cluster expansions, and Mihaila, 1999; Mihaila and Heisenberg, 2000; Kow- Brueckner’s summation of ladder diagrams ͑Brueckner, alski et al., 2004; Włoch et al., 2005͒. Rev. Mod. Phys., Vol. 79, No. 1, January–March 2007 Rodney J. Bartlett and Monika Musiał: Coupled-cluster theory in quantum chemistry 293 Heavy-atom relativistic
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  • Coupled Cluster Theory for Molecular Polaritons: Changing Ground and Excited States

    Coupled Cluster Theory for Molecular Polaritons: Changing Ground and Excited States

    PHYSICAL REVIEW X 10, 041043 (2020) Featured in Physics Coupled Cluster Theory for Molecular Polaritons: Changing Ground and Excited States Tor S. Haugland ,1 Enrico Ronca ,2,3 Eirik F. Kjønstad ,1 Angel Rubio ,3,4,5 and Henrik Koch 6,1,* 1Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2Istituto per i Processi Chimico Fisici del CNR (IPCF-CNR), Via G. Moruzzi, 1, 56124, Pisa, Italy 3Max Planck Institute for the Structure and Dynamics of Matter and Center Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany 4Center for Computational Quantum Physics (CCQ), The Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA 5Nano-Bio Spectroscopy Group, Departamento de Física de Materiales, Universidad del País Vasco, 20018 San Sebastian, Spain 6Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56124 Pisa, Italy (Received 9 May 2020; revised 6 September 2020; accepted 14 October 2020; published 2 December 2020) We present an ab initio correlated approach to study molecules that interact strongly with quantum fields in an optical cavity. Quantum electrodynamics coupled cluster theory provides a nonperturbative description of cavity-induced effects in ground and excited states. Using this theory, we show how quantum fields can be used to manipulate charge transfer and photochemical properties of molecules. We propose a strategy to lift electronic degeneracies and induce modifications in the ground-state potential energy surface close to a conical intersection. DOI: 10.1103/PhysRevX.10.041043 Subject Areas: Atomic and Molecular Physics, Chemical Physics, Photonics I. INTRODUCTION quantized electromagnetic field interacts with the molecu- lar system, producing new hybrid light-matter states called Manipulation by strong electron-photon coupling [1] polaritons [19].