Density Functional Theory revolutionized quantum chemistry by using electron density instead of wavefunctions. The Hohenberg-Kohn theorems prove that electron density determines all ground-state properties, simplifying calculations from 3N to 3 spatial variables.

The Kohn-Sham approach makes DFT practical by introducing non-interacting particles with the same density as the real system. This clever trick allows us to solve simpler equations iteratively, making DFT calculations feasible for many-electron systems.

Hohenberg-Kohn Theorems and Electron Density

Fundamental Principles of Hohenberg-Kohn Theorems

• First Hohenberg-Kohn theorem establishes electron density as unique determinant of ground-state properties
• Proves one-to-one correspondence between external potential and ground-state electron density
• Demonstrates all ground-state properties can be expressed as functionals of electron density
• Reduces complexity from 3N variables (N electrons) to 3 spatial variables
• Provides theoretical foundation for density functional theory (DFT)

Electron Density and Its Significance

• Electron density represents probability of finding an electron in a specific volume element
• Integrates to total number of electrons in the system
• Contains all information needed to describe ground state of a system
• Can be measured experimentally through X-ray diffraction or electron microscopy
• Serves as central quantity in DFT calculations
• Simplifies computational approach compared to wavefunction-based methods

Non-interacting Reference System and Kohn-Sham Formalism

• Non-interacting reference system consists of fictitious particles with same density as real system
• Allows separation of kinetic energy into non-interacting and interacting components
• Introduces concept of Kohn-Sham orbitals to represent non-interacting particles
• Enables calculation of exact kinetic energy for non-interacting system
• Forms basis for Kohn-Sham approach in practical DFT calculations
• Bridges gap between interacting and non-interacting systems through exchange-correlation functional

Kohn-Sham Approach

Kohn-Sham Equations and Their Significance

• Set of one-electron Schrödinger-like equations for non-interacting particles
• Kohn-Sham equations take form: $[-\frac{1}{2}\nabla^2 + V_{eff}(\mathbf{r})]\phi_i(\mathbf{r}) = \epsilon_i\phi_i(\mathbf{r})$
• $V_{eff}(\mathbf{r})$ represents effective potential experienced by Kohn-Sham particles
• $\phi_i(\mathbf{r})$ denotes Kohn-Sham orbitals
• $\epsilon_i$ represents orbital energies
• Solve Kohn-Sham equations iteratively to obtain ground-state electron density

Self-Consistent Field Procedure

• Iterative process to solve Kohn-Sham equations
• Start with initial guess for electron density
• Calculate effective potential from guessed density
• Solve Kohn-Sham equations to obtain new set of orbitals
• Construct new electron density from obtained orbitals
• Repeat process until convergence criteria met (energy or density change below threshold)
• Convergence indicates self-consistency between density and potential
• Typically requires 10-100 iterations for most systems

Effective Potential and Its Components

• Effective potential in Kohn-Sham equations consists of several terms
• Includes external potential from nuclei (Coulomb attraction)
• Hartree potential represents classical electron-electron repulsion
• Exchange-correlation potential accounts for quantum mechanical effects
• Total effective potential expressed as: $V_{eff}(\mathbf{r}) = V_{ext}(\mathbf{r}) + V_H(\mathbf{r}) + V_{xc}(\mathbf{r})$
• $V_{ext}(\mathbf{r})$ denotes external potential
• $V_H(\mathbf{r})$ represents Hartree potential
• $V_{xc}(\mathbf{r})$ signifies exchange-correlation potential

Exchange-Correlation Functional and Approximations

• Exchange-correlation functional accounts for many-body effects in DFT
• Exact form of exchange-correlation functional unknown
• Various approximations developed to estimate exchange-correlation energy
• Local Density Approximation (LDA) assumes uniform electron gas
• Generalized Gradient Approximation (GGA) includes density gradients
• Hybrid functionals incorporate exact exchange from Hartree-Fock theory
• Meta-GGA functionals include kinetic energy density
• Choice of functional depends on system and desired accuracy
• Continuous development of new functionals improves DFT accuracy