Excited-State Methods for Multireference Systems

Q-Chem contains an ever-growing suite of many-body methods for describing open-shell molecules and excited states.¹⁷² The EOM-CC^224–226 (link) and ADC^227,241 (link) formalisms are two powerful approaches for describing multiconfigurational wave functions within a black-box single-reference formalism. Target states |Ψ_ex⟩ are described as excitations from a reference state |Ψ₀⟩,

| Ψ_{ex} 〉 = \hat{R} | Ψ_{0} 〉,

where

\hat{R}

is an excitation operator parameterized via amplitudes that are determined by solving an eigenvalue problem. In EOM-CC, these amplitudes are eigenvectors of the effective Hamiltonian

\bar{H} = e^{- \hat{T}} \hat{H} e^{\hat{T}},

in which

\hat{T}

is either the CC or the MP2 operator for the reference state. Currently, EOM-CCSD and EOM-MP2 models are available. In ADC, an effective shifted Hamiltonian is constructed using perturbation theory and the intermediate state representation (ISR) formalism,^227,241 (link) similar to Eq. (10), to afford

M = ⟨ Ψ_{ex} | \hat{H} - E_{0} | Ψ_{ex} ⟩,

where E₀ is the energy of the MPn reference state. Diagonalization of the Hermitian matrix M yields excitation energies, and the ADC eigenvectors give access to the excited-state wave function. Second-order standard ADC(2), extended ADC(2)-x, and ADC(3) are available.²⁴¹ For the second-order ADC schemes, spin-opposite-scaled (SOS) variants are also implemented.²⁴² (link)
Various EOM-CC and ADC models are defined by the choice of reference state |Ψ₀⟩ and excitation operator

\hat{R}

, as illustrated in Fig. 10. The following models are available:^224,227,241 (link) EE (excitation energies), IP (ionization potentials), EA (electron affinities), SF (spin–flip, for triplet and quartet references), 2SF (double SF, for quintet references); DIP (double IP), and DEA (double EA). At present, the 2SF, DIP, and DEA variants are only available in combination with an EOM treatment.²⁴³ (link)
Analytic gradients^244,245 (link) and properties^246–248 (link) are available for most of these models, including transition properties between different target states (e.g., transition dipoles, angular momentum, and electronic circular dichroism rotatory strengths),²⁴⁹ (link) nonadiabatic couplings,²⁵⁰ (link) spin–orbit couplings,^220,251,252 (link) and nonlinear optical properties, including two-photon transition moments and (hyper)polarizabilities for both ground and excited states.^253–256 (link) Extensions of these theories to metastable states²⁵⁷ (link) (resonances) and to core-level excitations^258–260 (link) are also available and are highlighted in Sec. V.
The IP and EA variants of these models afford spin-pure descriptions of ground and excited doublet states and are useful for modeling charge-transfer processes. EOM-SF and SF-ADC methods are suitable for treating diradicals, triradicals, and conical intersections. The DEA and DIP ansätze further expand the scope of applicability.²⁴³ (link) Spin–flip methods can be used to treat strongly correlated systems within an effective Hamiltonian formalism,^221,261,262 (link) with applications to single-molecule magnets and even infinite spin chains.²²² (link)
For visualization purposes, both Dyson orbitals²⁶⁴ (link) and natural transition orbitals²⁶⁵ (link) (NTOs) are available,^{15,88,220,266–269} (link) including NTOs of the response density matrices for analyzing two-photon absorption²⁷⁰ (link) and resonant inelastic x-ray scattering.²⁷¹ (link)
Figure 11 highlights the application of these tools to model magnetic properties and spin-forbidden chemistry. Exciton analyses,^{267,268,272–274} (link) bridging the gap between the quasiparticle and MO pictures of excited states, enable the calculation and visualization of electron–hole correlation.^{89,267,268,272,273} (link)

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Epifanovsky E., Gilbert A.T., Feng X., Lee J., Mao Y., Mardirossian N., Pokhilko P., White A.F., Coons M.P., Dempwolff A.L., Gan Z., Hait D., Horn P.R., Jacobson L.D., Kaliman I., Kussmann J., Lange A.W., Lao K.U., Levine D.S., Liu J., McKenzie S.C., Morrison A.F., Nanda K.D., Plasser F., Rehn D.R., Vidal M.L., You Z.Q., Zhu Y., Alam B., Albrecht B.J., Aldossary A., Alguire E., Andersen J.H., Athavale V., Barton D., Begam K., Behn A., Bellonzi N., Bernard Y.A., Berquist E.J., Burton H.G., Carreras A., Carter-Fenk K., Chakraborty R., Chien A.D., Closser K.D., Cofer-Shabica V., Dasgupta S., de Wergifosse M., Deng J., Diedenhofen M., Do H., Ehlert S., Fang P.T., Fatehi S., Feng Q., Friedhoff T., Gayvert J., Ge Q., Gidofalvi G., Goldey M., Gomes J., González-Espinoza C.E., Gulania S., Gunina A.O., Hanson-Heine M.W., Harbach P.H., Hauser A., Herbst M.F., Hernández Vera M., Hodecker M., Holden Z.C., Houck S., Huang X., Hui K., Huynh B.C., Ivanov M., Jász Á., Ji H., Jiang H., Kaduk B., Kähler S., Khistyaev K., Kim J., Kis G., Klunzinger P., Koczor-Benda Z., Koh J.H., Kosenkov D., Koulias L., Kowalczyk T., Krauter C.M., Kue K., Kunitsa A., Kus T., Ladjánszki I., Landau A., Lawler K.V., Lefrancois D., Lehtola S., Li R.R., Li Y.P., Liang J., Liebenthal M., Lin H.H., Lin Y.S., Liu F., Liu K.Y., Loipersberger M., Luenser A., Manjanath A., Manohar P., Mansoor E., Manzer S.F., Mao S.P., Marenich A.V., Markovich T., Mason S., Maurer S.A., McLaughlin P.F., Menger M.F., Mewes J.M., Mewes S.A., Morgante P., Mullinax J.W., Oosterbaan K.J., Paran G., Paul A.C., Paul S.K., Pavošević F., Pei Z., Prager S., Proynov E.I., Rák Á., Ramos-Cordoba E., Rana B., Rask A.E., Rettig A., Richard R.M., Rob F., Rossomme E., Scheele T., Scheurer M., Schneider M., Sergueev N., Sharada S.M., Skomorowski W., Small D.W., Stein C.J., Su Y.C., Sundstrom E.J., Tao Z., Thirman J., Tornai G.J., Tsuchimochi T., Tubman N.M., Veccham S.P., Vydrov O., Wenzel J., Witte J., Yamada A., Yao K., Yeganeh S., Yost S.R., Zech A., Zhang I.Y., Zhang X., Zhang Y., Zuev D., Aspuru-Guzik A., Bell A.T., Besley N.A., Bravaya K.B., Brooks B.R., Casanova D., Chai J.D., Coriani S., Cramer C.J., Cserey G., DePrince AE I.I.I., DiStasio RA J.r., Dreuw A., Dunietz B.D., Furlani T.R., Goddard WA I.I.I., Hammes-Schiffer S., Head-Gordon T., Hehre W.J., Hsu C.P., Jagau T.C., Jung Y., Klamt A., Kong J., Lambrecht D.S., Liang W., Mayhall N.J., McCurdy C.W., Neaton J.B., Ochsenfeld C., Parkhill J.A., Peverati R., Rassolov V.A., Shao Y., Slipchenko L.V., Stauch T., Steele R.P., Subotnik J.E., Thom A.J., Tkatchenko A., Truhlar D.G., Van Voorhis T., Wesolowski T.A., Whaley K.B., Woodcock HL I.I.I., Zimmerman P.M., Faraji S., Gill P.M., Head-Gordon M., Herbert J.M, & Krylov A.I. (2021). Software for the frontiers of quantum chemistry: An overview of developments in the Q-Chem 5 package. The Journal of Chemical Physics, 155(8), 084801.

Publication 2021

Body Circular dichroism Conical Electron Intersections Matrix m Optical Orbit Resonances Triplet X ray

Corresponding Organization : University of Southern California

Other organizations : Q Chem (United States), Australian National University, University of Sydney, University of California, Berkeley, California Institute of Technology, The Ohio State University, Ludwig-Maximilians-Universität München, Hefei National Center for Physical Sciences at Nanoscale, University of Science and Technology of China, Loughborough University, Technical University of Denmark, Institute of Chemistry, Academia Sinica, University of Pittsburgh, University of Pennsylvania, University of Luxembourg, Kent State University, University of Cambridge, Donostia International Physics Center, Lawrence Berkeley National Laboratory, University of Michigan–Ann Arbor, University of Nottingham, National Taiwan University, University of Utah, The University of Texas Rio Grande Valley, University of Notre Dame, Boston University, Gonzaga University, University of Geneva, Graz University of Technology, Virginia Tech, Xiamen University, Korea Advanced Institute of Science and Technology, Massachusetts Institute of Technology, Digital Wave (United States), Purdue University West Lafayette, Florida State University, University of Helsinki, University of Minnesota, Harvard University, University of Groningen, Florida Institute of Technology, Yale University, University of Oklahoma, University of Bremen, Fudan University, National Center for Theoretical Sciences, Cornell University, University at Buffalo, State University of New York, University of California, Davis, University of South Carolina, University of South Florida

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Variable analysis

independent variables

Choice of reference state |Ψ₀⟩ and excitation operator R̂
EOM-CC and ADC model variants (EE, IP, EA, SF, 2SF, DIP, DEA)

dependent variables

Excitation energies
Ionization potentials
Electron affinities
Magnetic properties
Spin-forbidden chemistry
Electron-hole correlation
Transition properties (dipoles, angular momentum, electronic circular dichroism rotatory strengths)
Nonadiabatic couplings
Spin-orbit couplings
Nonlinear optical properties (two-photon transition moments, (hyper)polarizabilities)

control variables

MP₂ reference state |Ψ₀⟩
Coupled cluster (CC) and many-body perturbation theory (ADC) formalisms

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