Electronic Structure: Density Functional Theory ¶

Density Functional Theory (DFT) is a quantum mechanical method for calculating electronic structure of materials from first principles. Unlike classical force fields that treat atoms as charged spheres, DFT explicitly considers electrons and their interactions, enabling accurate predictions of material properties. In this document we will cover the main features of DFT, including how to:

Calculate properties from first principles without empirical parameters for the material
Access electronic properties that classical force fields cannot predict (band gaps, charge transfer, magnetism)
Understand the trade-off: High accuracy and electronic detail vs. limited system size and timescales

1. Introduction: What is Electronic Structure?¶

From Force Fields to Electrons¶

In the previous page on Force Fields & Molecular Dynamics, we treated atoms as classical particles:

Solid spheres with fixed masses
Fixed charges (or no charges at all)
Interactions described by simple mathematical functions

But atoms are not solid spheres. Each atom consists of:

A tiny, massive nucleus (protons + neutrons)
Electrons moving around the nucleus
The nucleus is ~10,000× smaller than the atom - this means the electrons occupy most of the “volume” of the atom

Why Electrons Matter¶

The electrons determine many important material properties:

Chemical bonding: Electrons shared or transferred between atoms
Electrical conductivity: Mobile electrons carry current
Optical properties: Electrons interact with light via absorption and emission processes
Magnetism: Electron spins create magnetic moments
Mechanical properties: Bond strengths depend on electron distribution
Reactivity: Chemical reactions involve electron rearrangement

The nucleus provides:

Positive charge that attracts electrons
Most of the atom’s mass
Identity of the element (atomic number Z = # protons)

Electronic Structure: The Electron Distribution¶

Electronic structure refers to:

How electrons are distributed around and between atoms
The energy levels (quantum states) electrons occupy

Key concept: The electron density \(\rho(r)\) tells us the probability of finding an electron at position \(r\):

High \(\rho(r)\) near nuclei (electrons attracted to positive charge)
High \(\rho(r)\) between bonded atoms (chemical bonds)
Low \(\rho(r)\) far from atoms
Changes when atoms form molecules or solids as the electrons rearrange

Electron configuration determines:

Bond lengths and strengths
Whether the material is a metal, semiconductor, or insulator
Magnetic properties
Chemical reactivity

From Classical to Quantum¶

Classical force fields assume:

Electrons are not explicitly present
Atomic charges are fixed and pre-assigned
No electron rearrangement during simulation

Electronic structure methods (DFT) calculate:

Where electrons actually are
How electrons respond to environment
Charge transfer between atoms
Chemical bond formation and breaking

Example: Water molecule (H₂O)

Classical view:

3 atoms with fixed charges (O: -0.8e, H: +0.4e each)
Connected by “springs” (OH bonds)
Charges chosen to reproduce experimental properties

Electronic structure view:

10 electrons distributed around 3 nuclei
Electron density concentrated in OH bonds (covalent)
Oxygen has more electron density (more electronegative)
Lone pairs on oxygen (non-bonding electrons)
Charge distribution emerges from quantum mechanics, not assigned

Why Calculate Electronic Structure?¶

Use DFT when you need to calculate electronic structure:

New materials where no force field parameters exist
Electronic properties (band gaps, conductivity, optical)
Chemical reactions (bonds break and form)
Charge transfer (electrons move between atoms)
Magnetic properties (electron spin arrangements)
Validation of force field parameters

Classical force fields might be a better choice when:

Working with well-studied materials with good parameters
No electronic property predictions needed
Large systems and long timescales required
Temperature-dependent dynamics are needed
Structure doesn’t change much qualitatively (no reactions)

2. The Quantum Mechanical Many-Body Problem ¶

The Schrödinger Equation¶

The fundamental equation of quantum mechanics describes how particles behave:

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

Where:

\(\hat{H}\) = Hamiltonian operator (total energy operator)
\(\Psi\) = wavefunction (describes the quantum state of the full system)
\(E\) = total energy of the system

For a system with \(N\) electrons and \(M\) nuclei:

\[\hat{H} = \underbrace{-\sum_i \frac{\hbar^2}{2m_e}\nabla_i^2}_{\text{electron kinetic}} + \underbrace{-\sum_I \frac{\hbar^2}{2M_I}\nabla_I^2}_{\text{nuclear kinetic}} + \underbrace{\sum_{i<j}\frac{e^2}{|r_i-r_j|}}_{\text{e-e repulsion}} + \underbrace{\sum_{I<J}\frac{Z_IZ_Je^2}{|R_I-R_J|}}_{\text{nuclear repulsion}} + \underbrace{-\sum_{i,I}\frac{Z_Ie^2}{|r_i-R_I|}}_{\text{e-nuclear attraction}}\]

The Problem: This equation is impossible to solve exactly for more than a few particles, say anything beyond the hydrogen atom (1 electron + 1 proton).

Key Approximations¶

To make the problem tractable, we use approximations:

1. Born-Oppenheimer Approximation¶

Idea: Nuclei are ~2000× heavier than electrons
Consequence: Electrons respond instantly to nuclear positions
Result: Separate electronic and nuclear motion

This reduces the problem to solving for electron behavior with nuclei fixed.

2. Density Functional Theory (DFT)¶

Idea: Replace the complex many-electron wavefunction with electron density
Key insight: Ground state properties determined by electron density \(\rho(r)\)
Result: 3 coordinates instead of 3N coordinates

This is the foundation of DFT!

DFT: The Practical Solution¶

Hohenberg-Kohn Theorems (1964): Proved that the ground state energy is uniquely determined by the electron density \(\rho(r)\):

\[E_0 = E[\rho(r)]\]

Kohn-Sham Approach (1965): Map the real interacting electrons onto fictitious non-interacting electrons that give the same density. We solve:

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

Where \(\psi_i(r)\) are single-electron orbitals (much easier to solve than the full many-body problem).

The DFT Total Energy:

This paved the way to express the total energy as a functional of the electron density:

\[E[\rho] = T_s[\rho] + \int V_{\text{ext}}(r)\rho(r)\,dr + E_H[\rho] + E_{xc}[\rho]\]

Where:

\(T_s[\rho]\) = Kinetic energy of non-interacting electrons
\(V_{\text{ext}}(r)\) = External potential from nuclei (attracts electrons)
\(E_H[\rho]\) = Hartree energy = classical electron-electron repulsion

\[E_H[\rho] = \frac{1}{2}\int\int \frac{\rho(r)\rho(r')}{|r-r'|} dr\,dr'\]
\(E_{xc}[\rho]\) = Exchange-correlation energy = complicated quantum many-body effects
- Exchange: Pauli exclusion (same-spin electrons avoid each other)
- Correlation: Electron-electron interactions beyond classical repulsion

The Challenge: While the other terms are known, we don’t know the exact form of \(E_{xc}[\rho]\)!

This means that we have now isolated the approximations in this term, the so-called exchange-correlation functional. Defining accurate, transferable and efficient exchange-correlation functionals is a major area of research and has been since DFT was originally proposed. Different exchange-correlation functionals provide different approximations, each with different pros and cons:

LDA (Local Density Approximation): Simplest, depends only on \(\rho(r)\). Fast, good for crystal structures, poor for molecules and band gaps.
GGA (Generalized Gradient Approximation): Includes gradients \(\nabla\rho(r)\), most common (e.g., PBE). More generally applicable than LDA and more transferable.
Hybrid functionals: Mix in exact exchange, much better for band gaps (e.g., HSE06) and overall accuracy, but computationally more expensive.
+vdW corrections: Add the non-local van der Waals interactions missing in standard DFT. Can be combined with LDA/GGA/hybrids.

Key Takeaway: DFT reduces the impossible many-body problem to a solvable set of single-particle equations. The only approximation is the exchange-correlation functional \(E_{xc}[\rho]\), and choosing the right functional is critical for a successful project.

Further reading for a general understanding of DFT: Walter Kohn’s nobel prize lecture

3. What Can DFT Calculate?¶

DFT is a ground-state theory, designed to calculate the lowest energy configuration of electrons for a given arrangement of nuclei. From this ground state, many material properties can be derived:

Structural Properties:

Lattice parameters and crystal structures
Bond lengths and angles
Elastic constants

Energetic Properties:

Formation energies
Cohesive energies
Surface energies
Defect formation energies
Reaction energies and barriers

Electronic Properties:

Band structure and band gaps (for the right xc-functionals)
Density of states (DOS)
Charge density distributions, including Partial charges on atoms
Work functions

Magnetic Properties:

Magnetic moments
Magnetic ground states (ferromagnetic, antiferromagnetic)

Vibrational Properties:

Phonon frequencies
Infrared and Raman spectra
Zero-point energies
Thermodynamic properties via phonons

4. DFT Workflow and Best Practices ¶

DFT is solved iteratively using self-consistent field (SCF) methods. The electronic density and potential are alternately updated until the two are consistent and convergence is reached, and one must specify a tolerance for this. Additionally, one has to choose parameters governing the description of the atomic cores, the basis functions describing the electronic states, the real-space grid, the sampling of the Brillouin Zone. Here is a typical workflow for performing a DFT calculation:

Typical DFT Calculation Workflow¶

Build Structure

Import from database or build manually
Set up unit cell and atomic positions

Choose Computational Parameters

XC-Functional (PBE (GGA-type) is standard starting point)
Pseudopotentials and basis sets
Density mesh cutoff (resolution of real-space grid)
K-point sampling
SCF convergence criteria

Geometry Optimization (if needed)

Relax atomic positions and cell parameters until forces are small (<~0.05 eV/Å)
Make any special modifications (defects, surfaces, adsorbates)
Relax atomic positions again

Calculate desired properties

Use fine k-point sampling for electronic properties
Extract and analyze results

Validation

Compare with experiment (if available)
Check physical reasonableness
Test sensitivity to parameters

5. DFT vs. Force Fields: Making the Choice ¶

Comparison Table¶

Aspect	Classical Force Fields	DFT
Physics	Classical mechanics, empirical	Quantum mechanics, first-principles
Electrons	Implicit (fixed charges)	Explicit (calculated)
System size	Up to ~10⁷ atoms	Up to ~10⁴ atoms
Time scale	ps - 100s of ns	fs - ps
Speed	Fast	Slower
Accuracy	Depends on parameterization	Generally high
Electronic properties	No	Yes
Chemical reactions	Difficult	Natural
New materials	Need parameters	No parameters needed
Temperature	Easy (MD)	Challenging (ab initio MD, EPC)

6. Brief Introduction to SemiEmpirical Methods ¶

SemiEmpirical methods are quantum mechanical approaches that simplify electronic structure calculations by incorporating empirical parameters derived from experimental data or higher-level theory. Unlike Density Functional Theory (DFT), which aims for first-principles accuracy, SemiEmpirical methods use approximations to reduce computational cost, making them suitable for larger systems and longer timescales, but at the cost of some accuracy and especially transferability.

Key Features:

Use a simplified Hamiltonian with parameters fitted to reproduce experimental results or ab initio calculations.
Retain quantum mechanical treatment of electrons, but neglect or approximate certain integrals.

Advantages:

Much faster than ab initio methods (DFT, Hartree-Fock)
Can handle larger systems
Reasonable accuracy for many applications

Limitations:

Accuracy depends on quality and transferability of parameters
Less reliable for transition metals, excited states, or systems outside the parameterization set
Not suitable for predicting new physics or properties far from fitted data

Summary:

SemiEmpirical methods provide a practical compromise between classical force fields and fully quantum mechanical approaches like DFT. They are valuable for initial screening and electronic properties of large systems, but should be used with caution as transferability is generally low.

Electronic Structure: Density Functional Theory¶