Notes: Density Functional Theory

Posted by Jingyi on March 11, 2018

Thanks for the notes of course MSAE6085 by Renata Wentzcovitch, and wiki page DFT.

DFT is a computational quantum mechanical modelling method,

• investigate the electronic structure (principally the ground state)
• obtain an approximate solution of the Schrödinger Equation for many-body systems.

Introduction

graph TD; A("Quantum many body problem")-->B("Wavefunction"); A-->C("Density"); B-->B1("Hartree-Fock"); B-->B2("QMC"); B-->B3("Coupled Cluster"); C-->C1("Density Functional Theory");

Derivations

Schrödinger Equation

The time-independent Schrödinger Equation:

$H \Psi(x_1,x_2,...,x_N,R_1,R_2,...,R_M) = E \Psi(x_1,x_2,...,x_N,R_1,R_2,...,R_M)$

Where $H$ is the Hamiltonian for a system with $M$ nuclei and $N$ electrons.

$H = -\frac{1}{2m_e}\sum^{N}_{i=1}\nabla^2_i-\frac{1}{2m_n}\sum^{M}_{j=1}\nabla^2_j \\ + \frac{1}{2}\sum^{N}_{i=1} \sum^{N}_{k=1,k\neq i} \frac{e^2}{\mid r_i-r_k \mid} - \sum^{N}_{i=1} \sum^{M}_{j=1,i\neq} \frac{Z_J e^2}{\mid r_i-r_k \mid} \\ + \frac{1}{2}\sum^{M}_{j=1} \sum^{M}_{w=1,w\neq j} \frac{Z_j Z_w e^2}{\mid r_j-r_w \mid}$

And it can be simplifed as

$H = T_e + T_n + V_{ee} + V_{en} + V_{nn}$

where

• $T_e$ is the kinetic energy of electrons,
• $T_n$ is the kinetic energy of nuclei,
• $V_{ee}$ is the electron-electron potential energy,
• $V_{en}$ is the electron-nuclei potential energy,
• $V_{nn}$ is the nuclei-nuclei potential energy.

Born-Oppenheimer (BO) Approximation: fixed nuclei

Born–Oppenheimer (BO) Approximation is the assumption that the motion of atomic nuclei and electrons in a molecule can be separated.

In many-body electronic structure calculations, the nuclei of are treated as fixed, by generating a static external potential $V$ in which the electrons are moving. A stationary electronic state is then described by a wavefunction satisfying the many-electron time-independent Schrödinger equation.

With $m_n \ll m_e$, $T_n = 0$, $V_{nn} = Constant$.

Thus, the new Hamiltonian for many body system is:

$H = T_e + V_{ee} + V_{en}$

Kohn-Sham Theory

Solve selfconsistent one-electron Schroedinger equations for the obitals or one-electron wavefunctions –> Ground-state density and energy of a many-electron system.

Approximation methods in early history

Lowest energy solution of Schroedinger equation ↓ Grounnd-state energy $E$ and electron density $n(\overrightarrow{r})$ of N-electron system

Thomas-Fermi Approximation

Replaces wavefunctions as the variational object by electron density $n(\overrightarrow{r})$.

Hartree-Fock Approximation

Replaces the wavefunctions by a single Slater determinant of the best possible orbitals or one-electron wavefunctions ↓ Solutions of a one-electron Schroedinger equation with an effective one-electron potential

Disvantage: Include exact exchange, but no correlation

Hohenberg-Kohn (H-K) Theorems

Hohenberg-Kohn (H-K) Theorem relate to any system consisting of electrons moving under the influence of an external potential $v_{ext}(r)$.

The first H–K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial coordinates. It set down the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to three spatial coordinates, through the use of functionals of the electron density. This theorem has since been extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states.

The second H–K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional.

Exchange-Correlation Functionals

The exchange-correlation energy is a negative energy that represents the lowering of the energy of the system due to the fact that the electrons avoid each other as they move through the density.

In DFT the key variable is the electron density $n(\overrightarrow{r})$, which for a normalized $\Psi$ is given by

$n(\overrightarrow{r}) = N \int d^3 r_2 \int d^3 r_3 ... \int d^3 r_N \mid \Psi(\overrightarrow{r},\overrightarrow{r_2},\overrightarrow{r_N}) \mid ^2$
graph TD; A("LDA")-->B("GGA"); B-->C("Meta-GGA"); C-->D("Hybrid"); D-->E("RPA-like");

Local Density Approximation (LDA): $n(\overrightarrow{r})$

$E^{LDA}_{XC}[n] = \int \varepsilon_{XC}(n) n(\overrightarrow{r})d^3r$

the exchange–correlation energy is typically separated into the exchange part and the correlation part:

$\varepsilon_{XC} = \varepsilon_{X} + \varepsilon_{C}$

Generalized Gradient Approximations (GGA): $\nabla n(\overrightarrow{r})$

$E^{GGA}_{XC}[n\uparrow , n\downarrow] = \int \varepsilon_{XC}(n\uparrow , n\downarrow, \nabla n\uparrow , \nabla n\downarrow) n(\overrightarrow{r})d^3r$

Software: Quantum ESPRESSO

• view my notes: here