Goals of this lecture

DFT introduction #1


Team Rutgers

(Valeria, Ezekiel and Michele)
An interactive session

Retrieve this presentation at:


https://github.com/asesma-org/miniASESMA2026

mini ASESMA 2026 \(\bullet\) Accra, Ghana \(\bullet\) June 16, 2026

Goals of this lecture#

  • Reasons for considering DFT

  • Basics of the theory behind DFT and Kohn-Sham DFT

  • KS equations and XC functional

Split into N groups#

  • Assign number 1-N to each student

  • Groups sit together

  • Possibly have 1 instructor per group

The Real World#

Photocatalyst

Catalytic nanoparticles
Chem. Comm., 43, 6551 (2009) PCCP, 21, 15080 (2019)

Available electronic structure methods#


Challenge 1#

Let’s check your QM knowledge.

  • What is the molecular Hamiltonian?

Solution to Challenge #1 and a bit more…#

Molecular Hamiltonian for interacting electrons and nuclei of charge \(Z_\alpha\)

\[ \hat{H} = \underbrace{-\frac{1}{2}\sum_i^{N_e} \nabla^2_i}_{\hat T} + \underbrace{\sum_i^{N_e}\sum_\alpha^{N_n} \frac{-Z_\alpha}{|r_i-R_\alpha|}}_{\hat V_{eN}} + \underbrace{\frac{1}{2}\sum_{i\neq j}^{N_e} \frac{1}{|r_i-r_j|}}_{\hat V_{ee}} + E_{NN} \]

The Schrödinger equation for the ground state of the interacting system

\[ \hat H \Psi_0 = E_0 \Psi_0 \]

Expectation values of each term of the Hamiltonian

\[ T[\Psi_0] = \langle \Psi_0 | \hat T | \Psi_0 \rangle, ~~ E_{ee}[\Psi_0] = \langle \Psi_0 | \hat V_{ee} | \Psi_0 \rangle \]
\[ E_{eN}[\Psi_0] = \color{red}{E_{eN}[n]} = \langle \Psi_0 | \hat V_{eN} | \Psi_0 \rangle = \int n(r) v_{eN}(r) dr \]

The Hohenberg and Kohn theorems#

\[ \Psi_0 \longleftrightarrow n(r) \longleftrightarrow v_{eN}(r) \longleftrightarrow \Psi_0 \]

Therefore \(n(r)\), \(v_{eN}(r)\) or \(\Psi_0\) hold the same information.

HK paper 1964

In particular:

\[ E \equiv E[\Psi_0] \equiv E[v_{eN}] \equiv E[n] \]

DFT exploits the latter as follows:

$$ E[n] = T[n] + E_{ee}[n]+E_{eN}[n]+E_{NN} $$

Challenge 2#

What do the energy functionals mean?

  • Write an expression for each of the functionals (\(T\), \(E_{ee}\), \(E_{eN}\)) in terms of \(n(r)\) and/or of \(\Psi_0\).

Solution to Challenge #2 and a bit more…#


Recalling:

\[ \color{red}{E_{eN}[n]} = \langle \Psi_0 | \hat V_{eN} | \Psi_0 \rangle = \int n(r)\,v_{eN}(r)\,dr \]
Because \(v_{eN}\) is fixed by the system's geometry, \(E_{eN}\) is already a density functional.

For the other terms in the Hamiltonian:


$\( T[\Psi_0] = \langle \Psi_0 | \hat T | \Psi_0 \rangle, ~~ E_{ee}[\Psi_0] = \langle \Psi_0 | \hat V_{ee} | \Psi_0 \rangle \)$

Apparently, \(T\) and \(E_{ee}\) cannot be written as denisty functionals...

What can we do with the HK theorems?



\(n(r)\)



...the density determines everything... in principle.... but is this useful? ... Are the functionals involved known?

Introducing: the Kohn-Sham (KS) system of noninteracting electrons#

KS paper 1965

The density of the KS system is the same as the interacting system.
\[ v_s(r) \longleftrightarrow n(r) \longleftrightarrow v_{eN}(r) \]

Energy functional for the KS system#

\[ E[n] = \color{red}{T[n] + E_{ee}[n]}+E_{eN}[n]+E_{NN} \]
\[E[n] = \color{red}{T_s[n] + E_{H}[n] + E_{xc}[n]}+ E_{eN}[n]+ E_{NN}\]
where the density $n(r) = \sum_i^N |\phi_i(r)|^2$.

The KS kinetic energy is

\[ T_{s}[n] \equiv T_s[\{\phi_i\}]= -\frac{1}{2}\sum_i \langle \phi_i | \nabla^2 | \phi_i\rangle = -\frac{1}{2}\sum_i \int \phi_i^*(r) \nabla^2 \phi_i(r) dr \]

Mind: \(T_s \neq T\).

The classical e-e repulsion (Hartree) and e-N attraction:

\[ E_H[n]=\frac{1}{2}\int \frac{n(r)n(r')}{|r-r'|}drdr' \qquad\qquad E_{eN}[n] = \int n(r) v_{eN}(r) dr \]

Challenge 3: What is \(E_{xc}\)?#

Derive an expression for \(E_{xc}\)

  • Derive an expression for \(E_{xc}\) in terms of \(T\), \(E_{ee}\), \(T_s\) and \(E_H\).

Solution to Challenge #3#

The energy functional, whether KS or interacting, should yield the same energy value, therefore

\[ E_{xc}[n] = E_{ee}[n]-E_H[n] + T[n]-T_s[n] \]

Can \(E_{xc}[n]\) be approximated?


Usually it is approached by separating exchange, \(E_x\), and correlation, \(E_c\): \[ E_{xc}[n] = E_x[n] + E_c[n] \]
Local Density Approximation from the UEG. Example from Dirac's exchange: \[ E_{x}^{\rm UEG} = - c_x \bar n^{4/3} V \longrightarrow E_x[n] \simeq -c_x \int n^{4/3}(r) dr \]

  • QEpy uses same structure as QE to specify the xc functional

qe_options = {}
qe_options["&system"] = {}
qe_options["&system"]["input_dft"] = 'LDA'

  • Typically xc is already specified in the pseudopotential files.

Solving for the electronic structure (find the KS orbitals, \(\{\phi_i\}\))#

To minimize the energy, we define an appropriate Lagrangian:

\[ \mathcal{L}_{KS}[\{\phi_i\}] = E[\{\phi_i\}] - \sum_{ij} \varepsilon_{ij}\left(\langle \phi_j|\phi_i \rangle - \delta_{ij}\right) \]

At the minimum, we impose \(\frac{\delta \mathcal{L}_{KS}[\{\phi_i\}]}{\delta \langle \phi_j|}=0\) or just \(\frac{\delta \mathcal{L}_{KS}[\{\phi_i\}]}{\delta \phi_j^*(r)}=0\), and choose the so-called canonical orbitals (i.e., \(\varepsilon_{ij}=\varepsilon_{i}\delta_{ij}\)).

This yields the Kohn-Sham equations:

\[ -\frac{1}{2}\nabla^2 \phi_i(r) + v_s[n](r)\phi_i(r) = \varepsilon_i\phi_i(r) \]

where \(v_s(r)\) includes effects of e-N attraction and e-e repulsion. But what is \(v_s(r)\)?

Challenge 4#

Considering the chain rule of functional differentiation (\(\phi^*\) and \(\phi\) are considered independent variables):

\[ \frac{\delta F[n]}{\delta \phi_j^*(r)} = \int \frac{\delta F[n]}{\delta n(r')} \frac{\delta n(r')}{\delta \phi_j^*(r)} \,dr' = \frac{\delta F[n]}{\delta n(r)}\phi_j(r). \]
Show that the KS potential is given by:
\[\begin{split} \color{green}{ \begin{aligned} v_s[n](r) &= \frac{\delta E_H[n]}{\delta n(r)} + \frac{\delta E_{xc}[n]}{\delta n(r)} + v_{eN}(r) \\[0.5em] &= \int \frac{n(r')}{|r-r'|} \,dr' + \frac{\delta E_{xc}[n]}{\delta n(r)} + v_{eN}(r). \end{aligned} } \end{split}\]

Can you derive the KS equations?

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