Density Functional Theory

Elementary quantum mechanics

The Schrodinger equation

The Schrodinger equation $$\hat{\mathcal{H}}\Psi_{i} = E_{i}\Psi_{i}$$ The Hamiltonian for a system consisting of $M$ nuclei and $N$ electrons $$\hat{\mathcal{H}} = -\frac{1}{2} \sum_{i=1}^{N} {\nabla^{2}_{i}} -\frac{1}{2} \sum_{A=1}^{M} {\frac{1}{M_{A}}\nabla^{2}_{A}} -\sum_{i=1}^{N} \sum_{A=1}^{M} {\frac{Z_{A}}{r_{iA}}} + \sum_{i=1}^{N} \sum_{j>i}^{N} {\frac{1}{r_{ij}}} + \sum_{A=1}^{M} \sum_{B>A}^{M} {\frac{Z_{A}Z_{B}}{R_{AB}}}$$ Born-Oppenheimer approximation $$\hat{\mathcal{H}}_{elec}\Psi_{elec} = E_{elec}\Psi_{elec}$$ $$\hat{\mathcal{H}}_{elec} = -\frac{1}{2} \sum_{i=1}^{N} {\nabla^{2}_{i}} -\sum_{i=1}^{N} \sum_{A=1}^{M} {\frac{Z_{A}}{r_{iA}}} + \sum_{i=1}^{N} \sum_{j>i}^{N} {\frac{1}{r_{ij}}}$$ $$E_{tot}=E_{elec}+E_{nuc}\qquad \text{where } E_{nuc}=\sum_{A=1}^{M}\sum_{B>A}^{M}\frac{Z_{A}Z_{B}}{R_{AB}}$$

The variational principle for the ground state

$$E[\Psi]=\frac{\left \langle\Psi|\hat{\mathcal{H}}|\Psi\right \rangle}{\left \langle\Psi|\Psi\right \rangle} \qquad \text{where} \left \langle \Psi|\hat{\mathcal{H}}|\Psi \right \rangle = \int {\Psi^{*}\hat{\mathcal{H}}\Psi}\, d\overrightarrow{x} $$ $$E_{0}=\min_{\Psi \rightarrow N}{E[\Psi]} = \min_{\Psi \rightarrow N}{\left \langle | T + V_{Ne} + V_{ee} | \right \rangle}$$

The Hartree-Fock approximation

The Slater determinant $$\Psi_{0} \approx \Psi_{HF} = \frac{1}{\sqrt{N!}} \begin{vmatrix} \psi_{1}(\overrightarrow{x_{1}}) & \psi_{2}(\overrightarrow{x_{1}}) & \cdots & \psi_{N}(\overrightarrow{x_{1}})\\ \psi_{1}(\overrightarrow{x_{2}}) & \psi_{2}(\overrightarrow{x_{2}}) & \cdots & \psi_{N}(\overrightarrow{x_{2}}) \\ \vdots & \vdots & & \vdots \\ \psi_{1}(\overrightarrow{x_{N}}) & \psi_{2}(\overrightarrow{x_{N}}) & \cdots & \psi_{N}(\overrightarrow{x_{N}}) \\ \end{vmatrix} $$ The Hartree-Fock approximation $$ E_{HF}=min_{\Psi_{HF}\rightarrow N}E[\Psi_{HF}] $$ $$ E_{HF} = \left \langle \Psi_{HF} | \hat{\mathcal{H}} | \Psi_{HF} \right \rangle = \sum_{i=1}^{N} {\hat{\mathcal{H}}_{i}} + \frac{1}{2} \sum_{i,j=1}^{N} {(J_{ij}-K_{ij})} $$ $$ \hat{\mathcal{H}}_{i} \equiv \int {\psi^{*}_{i}(\overrightarrow{x})[-\frac{1}{2}\nabla^{2}- V_{ext}(\overrightarrow{x})]\psi_{i}(\overrightarrow{x})} \, d\overrightarrow{x} $$ Non-Classical Contribution(Self-Interaction correction) : Coulomb integrals and Exchange integrals $$\begin{align*} J_{ij} &= \iint \psi_{i}(\overrightarrow{x_{1}}) \psi^{*}_{i}(\overrightarrow{x_{1}}) \frac{1}{r_{12}} \psi_{j}^{*}(\overrightarrow{x_{2}}) \psi_{j}(\overrightarrow{x_{2}}) \, d\overrightarrow{x_{1}} \, d\overrightarrow{x_{2}} \\ K_{ij} &= \iint \psi_{i}^{*}(\overrightarrow{x_{1}}) \psi_{j}(\overrightarrow{x_{1}}) \frac{1}{r_{12}} \psi_{i}(\overrightarrow{x_{2}}) \psi_{j}^{*}(\overrightarrow{x_{1}}) \, d\overrightarrow{x_{1}} \, d\overrightarrow{x_{2}}\\ \end{align*}$$ $$ J_{ij} \ge K_{ij} \ge 0 $$ The minimization of the energy functional with the normalization conditions $$ \int {\psi_{i}^{*}(\overrightarrow{x})\psi_{j}(\overrightarrow{x})} \, d\overrightarrow{x} = \delta_{ij} $$ The Hartree-Fock differential equations $$ f\psi_{i} = \epsilon_{i}\psi_{i} $$ $$ f = -\frac{1}{2}\nabla^{2}_{i} - \sum_{A}^{M} {\frac{Z_{A}}{r_{iA}}} + V_{HF}(i) $$ Hartree-Fock potential $$\begin{align*} V_{HF}(\overrightarrow{x_{1}}) &= \sum_{j}^{N} {(J_{j}(\overrightarrow{x_{1}})-K_{j}(\overrightarrow{x_{1}}))} \\ J_{j}(\overrightarrow{x_{1}}) &= \int \left \vert \psi_{j}(\overrightarrow{x_{2}}) \right \vert^{2} \frac{1}{r_{12}} \, d\overrightarrow{x_{2}} \\ K_{j}(\overrightarrow{x_{1}}) \psi_{i}(\overrightarrow{x_{1}}) &= \int \psi_{j}^{*}(\overrightarrow{x_{2}}) \frac{1}{r_{12}} \psi_{i}(\overrightarrow{x_{2}}) \, d\overrightarrow{x_{2}} \psi_{j}(\overrightarrow{x_{1}}) \\ \end{align*}$$

Early density functional theories

The electron density

The electron density $$\rho(\overrightarrow{r}) = N \int \cdots \int {\left \vert \Psi(\overrightarrow{x_{1}},\overrightarrow{x_{2}},\cdots,\overrightarrow{x_{N}}) \right \vert^{2} } ,\ ds_{1}d\overrightarrow{x_{1}}d\overrightarrow{x_{2}} \cdots d\overrightarrow{x_{N}}$$ $$ \rho( \overrightarrow{r} \rightarrow \infty ) = 0 \qquad \int {\rho(\overrightarrow{r})} \, d\overrightarrow{r}= N $$ $$ \lim_{r_{i}A \to 0} {[\nabla_{r}+2Z_{A}]\overline{\rho}(\overrightarrow{r})} = 0 $$ $$ \rho(\overrightarrow{r}) \sim e^{-2\sqrt{2I}\left \vert \overrightarrow{r} \right \vert} $$

The Thomas-Fermi model

$$ T_{TF}[\rho(\overrightarrow{r})] = \frac{3}{10}(3\pi^{2})^{2/3} \int \rho^{5/3}(\overrightarrow{r}) \, d\overrightarrow{r} $$ $$ E_{TF}[\rho(\overrightarrow{r})] = \frac{3}{10}(3\pi^{2})^{2/3} \int \rho^{5/3}(\overrightarrow{r}) \, d\overrightarrow{r} -Z \int {\frac{\rho(\overrightarrow{r})}{r}}\, d\overrightarrow{r} + \frac{1}{2} \iint \frac{\rho(\overrightarrow{r_{1}})\rho(\overrightarrow{r_{2}})}{r_{12}} \,d\overrightarrow{r_{1}} \,d\overrightarrow{r_{2}} $$

The Hohenberg-Kohn theorems

The first Hohenberg-Kohn theorem

The exact kinetic energy of a non-interacting reference system with the same density as the real $$\begin{align*} T_{S} &= -\frac{1}{2} \sum_{i}^{N} \left \langle \psi_{i} | \nabla^2 | \psi_{i} \right \rangle\\ \rho_{S}(\vec{r}) &= \sum_{i}^{N} \sum_{s} \left \vert \psi_{i}(\vec{r},s) \right \vert^{2} = \rho(\vec{r}) \\ \end{align*}$$ Separation of the functional $F[\rho]$ $$E[\rho] = E_{Ne}[\rho] + T[\rho] + E_{ee}[\rho] = \int {\rho(\vec{r})V_{Ne}(\vec{r})} \, d\vec{r} + F_{HK}[\rho]$$ $$\color{red}{F_{HF}[\rho]} = \color{red}{T[\rho]} + E_{ee}[\rho]$$ $$E_{ee}[\rho] = \frac{1}{2} \iint {\frac{\rho(\vec{r_{1}})\rho(\vec{r_{2}})}{r_{12}}} \,d\vec{r_{1}} \,d\vec{r_{2}} + E_{ncl} = J[\rho] + \color{red}{E_{ncl}[\rho]}$$

The second Hohenberg-Kohn theorem

$$ E_{0} \le E[\tilde{\rho}] = T[\tilde{\rho}] + E_{Ne}[\tilde{\rho}] + E_{ee}[\tilde{\rho}] $$

The Kohn-Sham approach

The Kohn-Sham equations : One-electron Schrödinger-like equations

$$F[\rho] = T_{S} + J[\rho] + E_{XC}[\rho]$$ $$\begin{align*} T_{S} &= \frac{1}{2} \sum_{i}^{N} \left \langle \psi_{i} | \nabla^{2} |\psi_{i} \right \rangle \\ E_{XC}[\rho] &\equiv (T[\rho] - T_{S}[\rho]) + (E_{ee}[\rho] - J[\rho]) \\ \end{align*}$$ The energy of the interacting system for uniquely determining the orbitals in our non-interacting reference system $$\begin{align*} E[\rho] &= T_{S} + J[\rho] +E_{XC}[\rho] + E_{Ne}[\rho] \\ &= T_{S} + \frac{1}{2} \iint \frac{\rho(\vec{r_{1}})\rho(\vec{r_{2}})}{r_{12}} \,d\vec{r_{1}} \,d\vec{r_{2}} + E_{XC}[\rho] + \int V_{Ne}\rho(\vec{r}) \, d\vec{r} \\ &= \frac{1}{2} \sum_{i}^{N} \left \langle \psi_{i} | \nabla^{2} |\psi_{i} \right \rangle + \frac{1}{2} \sum_{i}^{N} \sum_{j}^{N} \iint \left \vert \psi_{i}(\vec{r_{1}}) \right \vert^2 \frac{1}{r_{12}} \left \vert \psi_{j}(\vec{r_{2}}) \right \vert^2 \, d\vec{r_{1}} \,d\vec{r_{2}} \\ &\qquad + E_{XC}[\rho] - \sum_{i}^{N} \int \sum_{A}^{M} \frac{Z_{A}}{r_{1A}} \left \vert \psi_{i}(\vec{r_{1}}) \right \vert^2 \,d\vec{r_{1}} \end{align*}$$ Kohn-Sham equations $$ \left ( -\frac{1}{2}\nabla^{2} + \left [ \int {\frac{\rho(\vec{r_{2}})}{r_{12}}} \,d\vec{r_{2}} + V_{XC}(\vec{r_{1}}) - \sum_{A}^{M} {\frac{Z_{A}}{r_{1A}}} \right ] \right ) \psi_{i} = \left ( -\frac{1}{2}\nabla^{2} + V_{S}(\vec{r_{1}}) \right ) \psi_{i} = \epsilon_{i}\psi_{i} $$ $$ V_{S}(\vec{r_{1}}) = \int \frac{\rho(\vec{r_{2}})}{r_{12}} \, d\vec{r_{2}} + V_{XC}(\vec{r_{1}}) - \sum_{A}^{M}\frac{Z_{A}}{r_{1A}} $$ $$\text{constraint}\quad \left \langle \psi_{i} | \psi_{j}\right \rangle = \delta_{ij}$$

The exchange-correlation functionals

The local density approximation(LDA)

$$E_{XC}^{LDA}[\rho] = \int \rho(\vec{r})\epsilon_{XC}(\rho(\vec{r})) \, d\vec{r}$$ $$\begin{align*} \epsilon_{XC}(\rho(\vec{r})) &= \epsilon_{X}(\rho(\vec{r})) + \epsilon_{C}(\rho(\vec{r}))\\ \epsilon_{X} &= -\frac{3}{4} \left ( \frac{3\rho(\vec{r})}{\pi} \right )^{1/3} \\ \end{align*}$$

The generalized gradient approximation

$$E_{XC}^{GGA}[\rho_{\alpha}, \rho_{\beta}] = \int f(\rho_{\alpha}, \rho_{\beta}, \nabla_{\rho_{\alpha}}, \nabla_{\rho_{\beta}}) \, d\vec{r}$$

Hybrid functional

$$ E_{XC}^{hyb} = \alpha E_{X}^{KS} + (1-\alpha)E_{XC}^{GGA} $$

The basic machinary of DFT

The LCAO Ansatz in the The Kohn-Sham equations

The Kohn-Sham one-electron operator from Kohn-Sham equations $$ \hat{f}^{KS}\psi_{i} = \epsilon_{i}\psi_{i} $$ $$ \left ( \int {\frac{\rho(\vec{r_{2}})}{r_{12}}} \,d\vec{r_{2}} + V_{XC}(\vec{r_{1}}) - \sum_{A}^{M} {\frac{Z_{A}}{r_{1A}}} \right ) \psi_{i} = \epsilon_{i}\psi_{i} $$ K-S orbital $$ \psi_{i} = \sum_{\mu=1}^{L} c_{\mu i}\eta_{\mu} $$

Roothaan equations $$\hat{F}^{KS}_{\mu\nu}\hat{C} = \hat{S}_{\mu \nu}\hat{C}\hat{\epsilon}$$

Kohn-Sham matrix and overlap matrix $$\begin{align*} F_{\mu \nu}^{KS} &= \int {\eta_{\mu} (\vec{r_{1}}) \hat{f}^{KS}(\vec{r_{1}}) \eta_{\nu}(\vec{r_{1}}) } \, d\vec{r_{1}} \\ S_{\mu \nu} &= \int {\eta_{\mu}(\vec{r_{1}})\eta_{\nu}(\vec{r_{1}}) } \, d\vec{r_{1}} \\ \end{align*}$$ Extended Kohn-Sham Operator $$\begin{align*} \hat{F}_{\mu \nu}^{KS} &= h_{\mu\nu} + \int { \eta_{\mu} (\vec{r_{1}}) \left ( \int { \frac{\rho(\vec{r_{2}})}{r_{12}} } \, d\vec{r_{2}} + V_{XC}(\vec{r_{1}}) \right ) \eta_{\nu} (\vec{r_{1}}) } \, d\vec{r_{1}} \\ &= \int { \eta_{\mu} (\vec{r_{1}}) \left ( -\frac{1}{2} \nabla^{2} - \sum_{A}^{M} \frac{Z_{A}}{r_{1A}} + \int { \frac{\rho(\vec{r_{2}})}{r_{12}} } \, d\vec{r_{2}} + V_{XC}(\vec{r_{1}}) \right ) \eta_{\nu} (\vec{r_{1}}) } \, d\vec{r_{1}} \\ h_{\mu\nu} &= \int { \eta_{\mu} (\vec{r_{1}}) \left ( -\frac{1}{2} \nabla^{2} - \sum_{A}^{M} \frac{Z_{A}}{r_{1A}} \right ) \eta_{\nu} (\vec{r_{1}}) } \, d\vec{r_{1}} \\ \end{align*}$$ Charge density in the LCAO scheme $$ \rho(\vec{r}) = \sum_{i}^{L}\left \vert \psi_{i}(\vec{r}) \right \vert^{2} = \sum_{\mu}^{L}\sum_{\nu}^{L}P_{\mu\nu}\eta_{\mu}(\vec{r})\eta_{\nu}(\vec{r}) $$ $$ \text{density matrix,}\quad P_{\mu\nu} = \sum_{i}^{N} c_{\mu i}c_{\nu i} $$ Coulomb contribution and exchange-correlation and Hartree-Fock exchange integral $$\begin{align*} J_{\mu\nu} &= \sum_{\lambda}^{L}\sum_{\sigma}^{L} P_{\lambda\sigma} \iint \eta_{\mu}(\vec{r_{1}})\eta_{\nu}(\vec{r_{1}}) \frac{1}{r_{12}} \eta_{\lambda}(\vec{r_{2}})\eta_{\sigma}(\vec{r_{2}}) \, d\vec{r_{1}}\,d\vec{r_{2}} \\ V_{\mu\nu}^{XC} &= \int \eta_{\mu}(\vec{r_{1}})V_{XC}(\vec{r_{1}})\eta_{\nu}(\vec{r_{1}})\,d\vec{r_{1}}\\ K_{\mu\nu} &= \sum_{\lambda}^{L}\sum_{\sigma}^{L} P_{\lambda\sigma} \iint \eta_{\mu}(\vec{x_{1}})\eta_{\lambda}(\vec{x_{1}}) \frac{1}{r_{12}} \eta_{\nu}(\vec{x_{2}})\eta_{\sigma}(\vec{x_{2}}) \, d\vec{x_{1}}\,d\vec{x_{2}} \\ \end{align*}$$

Basis sets

Slater-Type Orbitals (STO) $$\eta^{STO} = Nr^{n-1}e^{-\beta r}Y_{lm}(\theta, \phi)$$ Gaussian-Type Orbitals (GTO) $$\eta^{GTO} = Nx^{l}y^{m}z^{b}e^{-\alpha r}$$ Contracted Gaussian Functions (CGF) $$ \eta_{\tau}^{CGF} = \sum_{a}^{A} d_{a\tau}\eta_{a}^{GTO} $$

Self-Consistent Field

[REF|10]

DFT applications

Applications in quantum chemistry

Applications in solid state physics

DFT in Molecular Electronics

Reference

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