Casida EquationEdit

The Casida equation is a foundational result in the field of computational electronic structure, providing a practical route to obtain electronic excitation energies and oscillator strengths from ground-state information. It sits at the core of time-dependent approaches to density functional theory, most commonly used to predict ultraviolet–visible spectra and to analyze excited-state properties of molecules and materials. The formalism expresses the linear response of a system to an external perturbation as a matrix eigenvalue problem that couples single-particle transitions through an exchange–correlation kernel. In practice, solving the Casida equation yields both the energies of excited states and how strongly those states interact with light.

Because it emerges from time-dependent, perturbative formulations, the Casida equation ties together ground-state information (such as Kohn–Sham or Hartree–Fock orbital energies and orbitals) with a dynamical kernel that encodes electron–electron interactions beyond the mean field. This bridge between ground-state calculations and excited-state predictions makes it a workhorse for interpreting and predicting photophysical and photochemical behavior across a broad range of systems, from small organic molecules to extended materials. The method is typically implemented within the framework of Time-Dependent Density Functional Theory, though closely related formalisms exist in other theories of electronic structure.

History and origin

The Casida equation is named for its developer and the formalism he introduced to cast TD-DFT response theory into an eigenvalue problem suitable for routine computation. The approach builds on the linear-response framework that underpins many-body theories of electronic excitations and leverages information from the ground-state Kohn–Sham set of orbitals. Over time, the Casida formulation has become the standard route for turning TD-DFT into a practical spectroscopy tool, complementing other methods such as the Random Phase Approximation and providing a clear path to include exchange–correlation effects via the xc kernel.

Mathematical formulation

At the heart of the Casida equation is a coupled set of matrices, commonly denoted A and B, built from single-particle orbital energy differences and two-electron integrals that involve the exchange–correlation kernel. The basic idea is to work in a basis of single excitations, that is, transitions from occupied orbitals i to virtual (unoccupied) orbitals a. For a closed-shell system, the matrices have dimensions equal to the number of such single excitations, typically labeled N_occ × N_virtual.

A representative form of the Casida eigenvalue problem is a two-block matrix equation for the excitation amplitudes (X, Y):

A X + B Y = ω X
B X + A Y = ω Y

Here: - ε_a and ε_i are the Kohn–Sham (or Hartree–Fock) orbital energies of virtual and occupied orbitals, and (ε_a − ε_i) provides the source term for each single excitation. - A encodes the orbital-energy differences plus the direct coupling through the Hartree–exchange–correlation kernel. - B encodes the coupling between excitations and de-excitations, a feature that distinguishes the full Casida equation from simpler approximations. - ω is the excitation energy, and the eigenvectors (X, Y) give the corresponding excitation amplitudes. The solutions come in ±ω pairs, with the physically relevant positive frequencies describing actual excitations.

In practice, the exchange–correlation kernel f_xc, the second functional derivative of the xc energy with respect to density, appears in the matrix elements that couple different single-particle transitions. In many common implementations, the kernel is approximated at the adiabatic level (frequency-independent), yielding the so-called adiabatic Casida equation. When the kernel is approximated as frequency-independent, the equation reduces to a well-behaved, real-symmetric eigenproblem under appropriate conditions, and a widely used simplification called the Tamm–Dancoff approximation (TDA) sets B to zero, removing the de-excitation coupling and simplifying the computation.

Because the matrix size grows with the number of occupied and virtual orbitals, practical calculations rely on efficient linear-algebra strategies and often employ localized or iterative solvers to handle systems with thousands of single excitations. The approach is closely tied to the choice of basis set and the underlying ground-state method (e.g., Kohn–Sham density functional theory or Hartree–Fock theory), and it inherits their strengths and limitations.

Assumptions, variants, and practical use

Spin and multiplicity: The formalism naturally accommodates singlet and triplet excitations, with spin selection rules reflected in the matrix elements and the structure of A and B.
Adiabatic approximation: Most widely used implementations assume a frequency-independent xc kernel, which simplifies the problem and often yields accurate valence excitations but can miss certain dynamical correlation effects, such as double excitations.
Tamm-Dancoff approximation: By dropping B (the de-excitation coupling), one obtains a simpler, real-valued eigenproblem for A, called the TDA. This often improves numerical stability and reduces cost, while still producing reasonable excitation energies for many systems.
Basis and reference: The accuracy of the Casida equation depends on the quality of the ground-state orbitals and energies, so the choice of functional (in TD-DFT) or level of theory (in HF-based variants) matters. This is why functionals with good long-range behavior are preferred for certain excitations.
Extensions: Non-adiabatic kernels (frequency-dependent f_xc) and more sophisticated many-body kernels can be used to improve accuracy for challenging excitations, including charge-transfer and core-level transitions. The Casida framework remains a flexible backbone for these developments.
Comparisons: The Casida approach is often compared to real-time TD-DFT, which propagates the time-dependent Kohn–Sham equations directly in the time domain. Both methods have their own advantages; Casida is typically convenient for state-by-state analysis, while real-time TD-DFT can capture ultrafast dynamics.

Applications and impact

Spectroscopy: The primary use of the Casida equation is to predict excitation energies and oscillator strengths that translate into UV–visible absorption spectra for molecules and materials. This helps interpret experimental spectra and guide synthesis.
Photochemistry and photophysics: By revealing the energies and characters of excited states, the method informs understanding of photochemical reaction pathways, fluorescence quantum yields, and intersystem crossing tendencies.
Materials science: For organic electronics, dyes, and pigments, Casida-based TD-DFT calculations assist in screening materials with desirable light-absorption properties.
Solvent and environment effects: In many cases, simulations incorporate solvent models or embedding schemes to improve agreement with experiment, recognizing that the local environment can shift excitation energies and alter oscillator strengths.
Cross-method context: The Casida framework is complementary to other advanced electronic-structure methods such as GW-based approaches and the Bethe–Salpeter equation for solids, providing a tractable route to picosecond-to-femtosecond spectroscopic properties in a variety of systems.

Controversies and debates

Accuracy for challenging excitations: While TD-DFT with common functionals performs well for many valence excitations, it can struggle with charge-transfer and Rydberg states, particularly when using standard (semi)local functionals. The community often debates which long-range-corrected functionals or tuned-range schemes best remedy these issues.
Role of the exchange–correlation kernel: The adiabatic, local or semi-local kernels miss certain dynamical correlation effects, including double excitations. This has spurred ongoing work to develop frequency-dependent or more sophisticated kernels, as well as hybrid and range-separated approaches.
TDA vs full Casida: The choice between the full Casida equation and the Tamm–Dancoff approximation reflects a trade-off between accuracy and computational cost. In some cases, TDA yields surprisingly good results and better numerical stability, while in others the full formulation is necessary to capture specific features of the spectrum.
Interpretability and reliability: As with any approximate method, interpretation of the results requires caution. Differences between functionals, basis sets, and kernel choices can lead to qualitatively different predictions for the same system, which motivates systematic benchmarking and cross-method comparisons.
Alternatives for excited states: For systems where standard TD-DFT struggles, researchers compare results against higher-level methods such as Equation-of-motion coupled-cluster theories or the Bethe–Salpeter equation, to better capture correlation effects, especially in larger or more strongly correlated systems.