Density Functional Theory: A Comprehensive Guide

Density Functional Theory: A Comprehensive Guide#

Introduction#

Density Functional Theory (DFT) is a fundamental tool in computational materials science and quantum chemistry. It provides a practical method for calculating the electronic structure of atoms, molecules, and solids with a reasonable balance between accuracy and computational cost. This guide explores the theoretical foundations, practical applications, and limitations of DFT.

The Many-Body Problem#

The primary challenge in quantum mechanics is solving the Schrödinger equation for systems with multiple electrons:

\[\hat{H}\Psi = E\Psi\]

For a system of N electrons and M nuclei, the Hamiltonian contains:

  • Kinetic energy of electrons

  • Kinetic energy of nuclei

  • Electron-electron repulsion

  • Nucleus-nucleus repulsion

  • Electron-nucleus attraction

The exact solution scales exponentially with system size, making it intractable for all but the smallest systems.

The Hohenberg-Kohn Theorems#

DFT is built on two fundamental theorems proved by Hohenberg and Kohn in 1964:

Theorem 1: Existence#

The ground-state electron density \(n(\mathbf{r})\) uniquely determines the external potential \(v(\mathbf{r})\) (up to a constant).

Implication: All ground-state properties are functionals of the electron density.

Theorem 2: Variational Principle#

The ground-state energy can be obtained by minimising the energy functional:

\[E[n] = F[n] + \int n(\mathbf{r})v(\mathbf{r})d\mathbf{r}\]

where \(F[n]\) is a universal functional containing kinetic and electron-electron interaction energies.

The Kohn-Sham Approach#

Kohn and Sham (1965) made DFT practical by mapping the interacting system onto a fictitious non-interacting system with the same ground-state density.

The Kohn-Sham Equations#

\[\left[-\frac{\hbar^2}{2m}\nabla^2 + v_{\text{eff}}(\mathbf{r})\right]\phi_i(\mathbf{r}) = \epsilon_i\phi_i(\mathbf{r})\]

where the effective potential is:

\[v_{\text{eff}}(\mathbf{r}) = v(\mathbf{r}) + \int \frac{n(\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|}d\mathbf{r'} + v_{\text{xc}}[n](\mathbf{r})\]

The density is constructed from the occupied orbitals:

\[n(\mathbf{r}) = \sum_{i}^{\text{occ}} |\phi_i(\mathbf{r})|^2\]

Self-Consistent Field Procedure#

  1. Initial Guess: Start with trial density \(n^{(0)}(\mathbf{r})\)

  2. Construct Potential: Calculate \(v_{\text{eff}}[n^{(k)}]\)

  3. Solve KS Equations: Find orbitals \(\{\phi_i\}\)

  4. Update Density: \(n^{(k+1)}(\mathbf{r}) = \sum_i |\phi_i(\mathbf{r})|^2\)

  5. Check Convergence: If not converged, return to step 2

Exchange-Correlation Functionals#

The exchange-correlation functional \(E_{\text{xc}}[n]\) contains all the many-body effects. Its exact form is unknown, necessitating approximations.

Local Density Approximation (LDA)#

This approximation assumes the exchange-correlation energy density depends only on the local density:

\[E_{\text{xc}}^{\text{LDA}}[n] = \int n(\mathbf{r})\epsilon_{\text{xc}}(n(\mathbf{r}))d\mathbf{r}\]

where \(\epsilon_{\text{xc}}(n)\) is the exchange-correlation energy per particle of a uniform electron gas.

Advantages:

  • Simple and computationally efficient

  • Works well for many systems

Limitations:

  • Overbinds molecules and solids

  • Underestimates band gaps

  • Poor for systems with rapidly varying density

Generalised Gradient Approximation (GGA)#

Includes dependence on the density gradient:

\[E_{\text{xc}}^{\text{GGA}}[n] = \int f(n(\mathbf{r}), |\nabla n(\mathbf{r})|)d\mathbf{r}\]

Popular GGA functionals:

  • PBE (Perdew-Burke-Ernzerhof): Widely used for solids

  • PW91 (Perdew-Wang 91): Earlier standard

  • BLYP: Popular in chemistry

Improvements over LDA:

  • Better molecular binding energies

  • Improved barrier heights

  • More accurate for inhomogeneous systems

Meta-GGA Functionals#

Include second derivatives of the density or kinetic energy density:

\[E_{\text{xc}}^{\text{meta-GGA}}[n] = \int f(n, |\nabla n|, \nabla^2 n, \tau)d\mathbf{r}\]

where \(\tau = \sum_i |\nabla\phi_i|^2\) is the kinetic energy density.

Examples:

  • TPSS: Tao-Perdew-Staroverov-Scuseria

  • SCAN: Strongly Constrained and Appropriately Normed

Hybrid Functionals#

These functionals mix exact Hartree-Fock exchange with DFT exchange:

\[E_{\text{xc}}^{\text{hybrid}} = \alpha E_{\text{x}}^{\text{HF}} + (1-\alpha)E_{\text{x}}^{\text{DFT}} + E_{\text{c}}^{\text{DFT}}\]

Popular hybrid functionals:

  • B3LYP: 20% HF exchange, popular in chemistry

  • PBE0: 25% HF exchange

  • HSE06: Range-separated hybrid for solids

Advantages:

  • Better band gaps

  • Improved reaction barriers

  • More accurate for charge transfer

Disadvantages:

  • Computationally expensive

  • Parameter fitting required

Practical Considerations#

Basis Sets#

  1. Plane Waves: Natural for periodic systems

    • Systematic improvement with cutoff energy

    • Efficient with FFT

    • Requires pseudopotentials

  2. Gaussian Functions: Common in molecular calculations

    • Localised basis

    • Efficient for molecules

    • Basis set superposition error

  3. Augmented Plane Waves: Hybrid approach

    • All-electron method

    • Accurate but complex

Pseudopotentials#

These replace core electrons with an effective potential:

  • Norm-Conserving: Accurate but harder

  • Ultrasoft: Softer, more efficient

  • PAW (Projector Augmented Wave): All-electron accuracy

k-Point Sampling#

For periodic systems, Brillouin zone integration is approximated by sampling:

  • Monkhorst-Pack: Regular grid

  • Gamma-Centred: For certain symmetries

  • Convergence testing essential

Applications in Materials Science#

1. Structure Prediction#

  • Geometry optimisation

  • Crystal structure determination

  • Surface reconstructions

2. Electronic Properties#

  • Band structures

  • Density of states

  • Work functions

  • Charge densities

3. Mechanical Properties#

  • Elastic constants

  • Bulk moduli

  • Phonon spectra

4. Chemical Properties#

  • Reaction pathways

  • Adsorption energies

  • Defect formation energies

5. Magnetic Properties#

  • Spin polarisation

  • Magnetic moments

  • Exchange coupling

Limitations of DFT#

1. Band Gap Problem#

Standard functionals typically underestimate band gaps by 30-50% due to:

  • Self-interaction error

  • Derivative discontinuity

  • Lack of many-body effects

2. Van der Waals Interactions#

Standard functionals do not capture long-range dispersion interactions:

  • Requires corrections (DFT-D, vdW-DF)

  • Important for molecular crystals, layered materials

3. Strongly Correlated Systems#

Fails for systems with localised d/f electrons:

  • Transition metal oxides

  • Rare earth compounds

  • Requires DFT+U or beyond

4. Excited States#

Ground-state theory, limited for:

  • Optical properties

  • Photochemistry

  • Requires TD-DFT or many-body methods

Advanced Topics#

DFT+U#

Adds on-site Coulomb repulsion for localised electrons:

\[E = E_{\text{DFT}} + \frac{U}{2}\sum_{i,\sigma} n_{i\sigma}(1-n_{i\sigma})\]

Time-Dependent DFT (TD-DFT)#

Extension to excited states and dynamics:

\[i\hbar\frac{\partial}{\partial t}\phi_i(\mathbf{r},t) = \hat{H}_{\text{KS}}[n(t)]\phi_i(\mathbf{r},t)\]

Constrained DFT#

Fix certain properties (magnetisation, charge) to study specific states.

Comparison with Other Methods#

Method

Scaling

Accuracy

System Size

Hartree-Fock

O(N⁴)

Poor correlation

~100 atoms

DFT

O(N³)

Good

~1000 atoms

MP2

O(N⁵)

Better

~50 atoms

CCSD(T)

O(N⁷)

Excellent

~20 atoms

Quantum Monte Carlo

O(N³-N⁴)

Excellent

~100 atoms

Software Packages#

Plane-Wave Codes#

  • VASP: Commercial, widely used

  • Quantum ESPRESSO: Open source

  • CASTEP: Commercial, UK-based

  • Abinit: Open source

Localised Basis Codes#

  • Gaussian: Molecular focus

  • CP2K: Mixed Gaussian/plane-wave

  • SIESTA: Numerical orbitals

  • FHI-aims: All-electron, numeric orbitals

Best Practices#

  1. Convergence Testing:

    • Cutoff energy/basis set size

    • k-point density

    • SCF convergence criteria

  2. Functional Choice:

    • PBE for general solids

    • Hybrids for band gaps

    • Meta-GGA for accuracy

    • Benchmark when possible

  3. Validation:

    • Compare with experiment

    • Test different functionals

    • Check literature precedent

  4. Computational Efficiency:

    • Exploit symmetry

    • Appropriate parallelisation

    • Restart from previous calculations

Conclusion#

DFT has transformed computational materials science by providing:

  • Practical quantum mechanical calculations

  • Predictive power for materials properties

  • Foundation for materials design

Whilst not without limitations, ongoing developments in functionals, algorithms, and computing power continue to expand DFT’s capabilities. Understanding both its strengths and weaknesses is crucial for effective application in materials research.

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