Density functional theory (DFT) is a powerful computational method for studying electronic structures. It simplifies complex many-electron systems by using electron density as the key variable, making it faster and more efficient than traditional wavefunction-based approaches.

DFT's foundation lies in the Hohenberg-Kohn theorems and Kohn-Sham equations. Various exchange-correlation functionals, from simple LDA to more advanced hybrid methods, offer a balance between accuracy and computational cost for different systems and properties.

Theoretical Foundations

Hohenberg-Kohn Theorems

• State the electron density uniquely determines the ground state properties of a many-electron system
  • The external potential (up to a constant) is a unique functional of the electron density
  • The ground state energy can be obtained variationally with the electron density as the basic variable
• Provide a formal justification for using the electron density as the fundamental variable in electronic structure calculations
• Lay the groundwork for the development of density functional theory (DFT) as a practical computational method

Kohn-Sham Equations

• Introduce a fictitious system of non-interacting electrons that generate the same density as the real, interacting system
• Map the interacting many-electron problem onto a non-interacting single-electron problem
• Consist of a set of single-particle Schrödinger-like equations for the non-interacting system
  • Kohn-Sham orbitals are the single-particle wavefunctions that are solutions to these equations
  • Kohn-Sham eigenvalues are the corresponding orbital energies
• Include the exchange-correlation potential, which accounts for the many-body effects of exchange and correlation
  • Exact form of the exchange-correlation potential is unknown and must be approximated ($V_{xc}[n(r)]$)

Self-Interaction Error

• Arises from the incomplete cancellation of the electron self-interaction in the Hartree term by the approximate exchange-correlation functional
• Leads to incorrect behavior of the exchange-correlation potential, particularly for systems with localized electrons (molecules, transition metal compounds)
• Causes issues such as overdelocalization of electron density and underestimation of band gaps in solids
• Can be mitigated by using self-interaction corrected functionals or by applying a posteriori corrections to the Kohn-Sham eigenvalues (DFT+U method)

Exchange-Correlation Functionals

Local Density Approximation (LDA)

• Approximates the exchange-correlation energy density at each point in space as that of a homogeneous electron gas with the same density
• Depends solely on the electron density at each point ($E_{xc}^{LDA}[n] = \int n(r) \epsilon_{xc}^{hom}(n(r)) dr$)
• Works well for systems with slowly varying electron densities (simple metals, semiconductors)
• Tends to overbind molecules and solids, underestimate bond lengths and lattice constants

Generalized Gradient Approximation (GGA)

• Incorporates the gradient of the electron density in addition to the local density ($E_{xc}^{GGA}[n] = \int f(n(r), \nabla n(r)) dr$)
• Improves upon LDA by accounting for the spatial variation of the electron density
• Popular GGA functionals include PBE (Perdew-Burke-Ernzerhof) and BLYP (Becke-Lee-Yang-Parr)
• Generally provides better agreement with experiment for molecular geometries, binding energies, and reaction barriers compared to LDA

Hybrid Functionals

• Incorporate a portion of exact exchange from Hartree-Fock theory with the exchange and correlation from DFT
• Improve upon GGA functionals by partially correcting for self-interaction error and providing more accurate band gaps for solids
• Popular hybrid functionals include B3LYP, PBE0, and HSE (Heyd-Scuseria-Ernzerhof)
• B3LYP (Becke, 3-parameter, Lee-Yang-Parr) is widely used in chemistry for its good performance on molecular properties
  • Mixes 20% exact exchange with 80% GGA exchange and 100% GGA correlation
  • Has been extensively benchmarked and shown to provide reliable results for a wide range of chemical systems