Double ExcitationEdit

Double excitation refers to a class of electronic transitions in which two electrons are simultaneously promoted from occupied to virtual orbitals in a many-electron system. This creates states that cannot be described by a single-electron promotion alone and is a signature of strong electron correlation. In chemistry and physics, double excitations play a critical role in the interpretation of certain ultraviolet–visible spectra of molecules, in photochemistry, and in the study of excited-state dynamics. They are a central concern in quantum chemistry because many standard, beginner-level approaches focus on single-electron promotions and miss the physics of these more complex transitions. In practical terms, a double excitation means that the excited state differs from the ground-state configuration by two electron promotions, i.e., two electrons are moved from occupied orbitals to unoccupied (virtual) orbitals.

Because of their inherently correlated character, doubly excited states challenge the simplest approximate methods. They are often weak in intensity for standard one-photon absorption, but can be bright in two-photon processes or in vibronic spectra. In many systems, especially larger organic molecules and polyatomic species, doubly excited characters mix with singly excited configurations, complicating the assignment of spectral features. Understanding and predicting these states requires going beyond basic CIS or simplest Hartree–Fock-based pictures and employing more sophisticated theories that can treat two-electron excitations with accuracy. Related concepts and methods include two-electron promotion and the broader framework of excited-state theory in quantum chemistry.

Definition and scope

A double excitation is formally defined as an excited configuration that differs from the ground-state reference determinant by two electron promotions. In practical language, it is a two-particle–two-hole (2p–2h) excitation relative to the reference. This distinguishes it from a single excitation (1p–1h) and from higher-order excitations that involve more than two electrons. In many molecules, doubly excited configurations can be valence in character (involving promotions among frontier orbitals) or Rydberg in character (involving promotions to high-lying diffuse orbitals). See for example discussions of valence vs Rydberg state excitations for context. The multiplicity (singlet, triplet, etc.) of a doubly excited state arises from the spin coupling of the two excited electrons and the remaining core; the overall symmetry is dictated by the molecular point group and the particular occupation pattern.

In practice, the presence of significant double-excitation character is identified by the failure of methods that assume predominantly single-particle promotions, and by diagnostic tools that quantify the fraction of the excitation that is double in character. In many systems, the doubly excited component is intertwined with singly excited configurations, requiring careful analysis to assign spectral features unambiguously. For discussions of how this relates to experimental observables, see the sections below on experimental observation and spectroscopy.

Theoretical frameworks

The description of double excitations sits at the intersection of several theoretical approaches in quantum chemistry and solid-state physics. Each framework has its own strengths and limitations when it comes to capturing two-electron promotions.

• Ground-state reference and configuration interaction

  • In simple configuration interaction (CI) hierarchies, people move from singles-only treatments (CIS) to include doubles (CISD) and triples (CISDT), etc. Doubles are essential for capturing many correlation effects, but CISD is not size-extensive and can be problematic for larger systems without careful treatment. See configuration interaction and related discussions of size-extensivity.
  • For a more complete treatment, one can use multi-reference approaches such as complete active space self-consistent field (CASSCF) and subsequent multireference configuration interaction (MRCI). These methods explicitly incorporate configurations with multiple excited electrons and are particularly valuable when the doubly excited character dominates near avoided crossings or conical intersections.

• Coupled-cluster and equation-of-motion methods

  • Coupled-cluster theory with singles and doubles (CCSD) and its extensions (CCSDT, CCSD(T)) provide a highly accurate framework for correlation, and when used to describe excited states via the equation-of-motion formalism (EOM-CCSD, EOM-CCSDT), they can treat many doubly excited states with good accuracy. See coupled-cluster theory and equation-of-motion coupled-cluster.
  • EOM-CC methods are particularly valuable because they systematically build excited states from a coupled-cluster ground state, but their computational cost grows rapidly with system size and the level of excitations included.

• Time-dependent density functional theory and its limitations

  • Time-dependent density functional theory (time-dependent density functional theory) is widely used for excited-state chemistry due to favorable scaling. However, in its conventional (adiabatic) form, TDDFT struggles to describe pure double excitations because the basic linear-response formalism lacks the necessary two-electron promotion channels. This has spurred the development of remedies such as spin-flip TDDFT, dressed TDDFT, and other approaches that aim to recover some double-excitation character within a density-functional framework. See spin-flip TDDFT and dressed TDDFT.
  • When doubles are important, standard TDDFT often misplaces or entirely misses the corresponding states, which motivates using multi-reference or wavefunction-based methods for reliable results.

• Practical diagnostics and composition analysis

  • In practice, excited-state wavefunctions are analyzed to determine how much of the state is composed of doubly excited configurations versus singly excited ones. This diagnostic helps in choosing an appropriate computational strategy and in interpreting spectra. See discussions of excitation character in multireference configuration interaction and CASSCF analyses.

Computational methods and limitations

• CIS versus CISD and beyond

  • CIS captures only singly excited configurations and thus cannot describe true double excitations. Extending to CISD or higher (e.g., CISDT) introduces doubles (and higher) explicitly, improving accuracy for doubly excited states but at greater computational cost and, in some cases, without guaranteed size-extensivity unless paired with additional corrections.

• Coupled-cluster and EOM formulations

  • CCSD and CCSDT are among the most accurate general-purpose tools for ground and excited states, respectively. EOM-CCSD is particularly popular for excited states, including those with double-character admixtures, though truly pure double excitations can still pose challenges and may require higher-level methods (e.g., EOM-CCSDT) or tailored active-space treatments.

• TDDFT and its remedies

  • Standard TDDFT often misses pure double excitations due to the adiabatic approximation. Spin-flip TDDFT extends the method to access certain doubly excited states by flipping spin, while dressed TDDFT incorporates additional orbital information to borrow strength from higher-order excitations. Each remedy has its domain of applicability and limitations, and cross-method validation is common in practice.

• Multi-reference approaches

  • When doubly excited states are strongly correlated and near-degenerate with other configurations, single-reference methods (like standard CC or TDDFT) may be insufficient. In such cases, CASSCF/MRCI or other multi-reference schemes provide a more reliable description by explicitly including several important electronic configurations in the reference space.

• Practical considerations

  • The computational cost of methods capable of handling double excitations grows with system size and the level of explicit correlation included. Researchers balance accuracy against feasibility, often using lower-level methods for screening and reserving high-level multi-reference or EOM-CC approaches for the most challenging states.

Experimental observation

Doubly excited states can be accessed and characterized by a variety of spectroscopic techniques. Pure one-photon absorption features may be weak or obscured by mixing with singly excited states, but two-photon absorption and nonlinear optical processes can preferentially populate and reveal doubly excited states. Vibronic coupling can also borrow intensity into transitions that would otherwise be forbidden, enabling experimental observation of doubly excited character in some cases. Spectroscopic interpretation often relies on comparisons with high-level theoretical predictions (e.g., from CASSCF, EOM-CC, or dressed TDDFT) to assign features to DVB-like states and to distinguish them from purely singly excited transitions. See also discussions of two-photon absorption in molecular systems.

Controversies and debates

• Characterization and naming

  • A continuing debate in the literature concerns how best to classify states that have substantial contributions from both singly and doubly excited configurations. Some researchers emphasize a strict 2p–2h definition, while others describe such states as "doubly excited in character" or as mixtures requiring multi-reference treatment for accurate description.

• Methodological reliability

  • The reliability of popular electronic-structure methods for doubly excited states is a point of contention. While high-level CC methods often perform well, there are cases where even EOM-CCSD or CCSDT can struggle if the state is strongly multi-reference in nature. This drives ongoing development of multi-reference methods and hybrid approaches that aim to combine accuracy with tractable cost.

• TDDFT limitations and cures

  • The inability of adiabatic TDDFT to describe pure double excitations has been a central controversy in computational chemistry. While remedies like spin-flip TDDFT and dressed TDDFT offer practical workarounds, they introduce new approximations and are not universally applicable. The debate centers on when these methods provide trustworthy predictions and how best to benchmark them against higher-level theories.

Applications

• Photochemistry and excited-state dynamics

  • Doubly excited states influence nonradiative decay pathways, internal conversion, and photosensitive reactions. In molecules where these states couple strongly to the ground or singly excited states, accurate treatment can be essential for predicting quantum yields and reaction mechanisms.

• Organic electronics and photovoltaics

  • In conjugated systems and organic semiconductors, doubly excited character can appear in charge-transfer or locally excited states that affect energy gaps, absorption spectra, and exciton dynamics. Accurate modeling informs material design and interpretation of optoelectronic properties.

• Spectroscopic design and interpretation

  • For researchers designing experiments or interpreting spectra, understanding when and how doubly excited states contribute helps in assigning features and in choosing the most appropriate theoretical framework to support experimental conclusions.

See also

• electronic excitation
• two-photon absorption
• CASSCF
• multireference configuration interaction
• configuration interaction
• CISD
• CCSD
• EOM-CC
• spin-flip TDDFT
• dressed TDDFT
• Rydberg state
• valence state
• conical intersection