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Density Functional Theory

Density Functional Theory (DFT) is a fundamental method in computational quantum chemistry, extensively used for calculating the electronic structure of atoms, molecules, and solids. Its main strength lies in achieving a balance between computational efficiency and accuracy, making it a vital tool for exploring a wide range of chemical systems and phenomena.

TeraChem is interfaced to the DFT functional library Libxc to support a flexible combination from a wide range of options. Details about the job control options can be found on this page.

TeraChem also has 20 hardcoded DFT functionals that can be specified via the method keyword, which will be ignored if the keyword libxc is specified. A full list is detailed here.

Theoretical background

DFT is based on quantum mechanics and the many-electron Schrödinger equation. Unlike methods focusing on wavefunctions, which become computationally complex due to the exponential increase in the number of electrons, DFT simplifies the problem by concentrating on the electron density, a function of three spatial coordinates. This approach is grounded in the Hohenberg-Kohn theorems, which state that the ground-state properties of a many-electron system are uniquely determined by its electron density. The Kohn-Sham formalism further simplifies this by introducing a system of non-interacting electrons that reproduce the same density as the interacting system, allowing for the solution of a set of self-consistent field equations.

  1. Hohenberg-Kohn Theorem:
\[ E[\rho] = F[\rho] + \int V_{\text{ext}}(\mathbf{r})\rho(\mathbf{r}) \, d\mathbf{r} \]

where \(E[\rho]\) is the total energy functional, \(F[\rho]\) is a universal functional of the density, and \(V_{\text{ext}}\) is the external potential.

  1. Kohn-Sham Equations:
\[ \left( -\frac{1}{2} \nabla^2 + V_{\text{eff}}(\mathbf{r}) \right) \psi_i(\mathbf{r}) = \epsilon_i \psi_i(\mathbf{r}) \]

where \(V_{\text{eff}}(\mathbf{r}) = V_{\text{ext}}(\mathbf{r}) + \int \frac{\rho(\mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|} d\mathbf{r'} + V_{\text{xc}}(\mathbf{r})\) includes the external potential, the Hartree potential, and the exchange-correlation potential \(V_{\text{xc}}(\mathbf{r})\).

Types of Functionals

The accuracy of DFT calculations hinges on the choice of the exchange-correlation functional, which approximates the complex many-body interactions. Functionals can be broadly categorized into:

  • Local Density Approximation (LDA): Assumes the exchange-correlation energy depends only on the electron density at each point. LDA is often accurate for systems with slowly varying densities.
  • Generalized Gradient Approximation (GGA): Incorporates the density gradient, improving accuracy for a broader range of systems compared to LDA. Popular GGA functionals include PBE (Perdew-Burke-Ernzerhof) and BLYP (Becke-Lee-Yang-Parr).
  • Hybrid Functionals: Combine exact exchange from Hartree-Fock theory with GGA or LDA exchange-correlation. Hybrid functionals like B3LYP are especially successful for molecular systems.
  • Meta-GGA and Beyond: Include higher-order derivatives of the density or additional variables, further enhancing accuracy for complex systems. Examples include the M06 series and range-separated hybrids like HSE.

Comparison and Relations to Other Methods

DFT differs from wavefunction-based methods like Hartree-Fock (HF) and post-Hartree-Fock methods (MP2, CCSD), which directly tackle the many-electron wavefunction. While these methods can provide higher accuracy for specific cases, they are often computationally expensive. DFT offers a practical compromise, delivering a good balance between computational cost and accuracy, especially for larger systems.

The computational scaling of DFT and other methods can be summarized as follows:

  • DFT (Kohn-Sham DFT): \(\mathcal{O}(N^3)\), where \(N\) is the number of electrons. GPU acceleration can significantly reduce the prefactor, enabling the study of larger systems.
  • Hartree-Fock (HF): \(\mathcal{O}(N^4)\)
  • MP2 (Second-order Møller-Plesset perturbation theory): \(\mathcal{O}(N^5)\)
  • CCSD (Coupled Cluster with Single and Double excitations): \(\mathcal{O}(N^6)\)

Time-Dependent Density Functional Theory (TDDFT)

Time-Dependent Density Functional Theory (TDDFT) extends DFT to study the excited states and dynamic properties of systems under the influence of time-dependent potentials. TDDFT is particularly useful for exploring electronic excitations, optical properties, and non-equilibrium processes. The central equation in TDDFT is the time-dependent Kohn-Sham equation:

\[ i \frac{\partial \psi_i(\mathbf{r}, t)}{\partial t} = \left( -\frac{1}{2} \nabla^2 + V_{\text{eff}}(\mathbf{r}, t) \right) \psi_i(\mathbf{r}, t) \]

where \(V_{\text{eff}}(\mathbf{r}, t)\) includes the time-dependent external potential, Hartree potential, and exchange-correlation potential. TDDFT provides a powerful framework for studying the real-time evolution of electronic systems and simulating spectroscopic properties with computational efficiency comparable to ground-state DFT.