Glossary

7.1 Hohenberg-Kohn theorems and Kohn-Sham approach

3 min read•august 9, 2024

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

The 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

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  • 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

Kohn-Sham Approach

Kohn-Sham Equations and Their Significance

  • Set of one-electron Schrödinger-like equations for non-interacting particles
  • take form: [−12∇2+Veff(r)]ϕi(r)=ϵiϕi(r)[-\frac{1}{2}\nabla^2 + V_{eff}(\mathbf{r})]\phi_i(\mathbf{r}) = \epsilon_i\phi_i(\mathbf{r})
  • Veff(r)V_{eff}(\mathbf{r}) represents effective potential experienced by Kohn-Sham particles
  • ϕi(r)\phi_i(\mathbf{r}) denotes Kohn-Sham orbitals
  • ϵi\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: Veff(r)=Vext(r)+VH(r)+Vxc(r)V_{eff}(\mathbf{r}) = V_{ext}(\mathbf{r}) + V_H(\mathbf{r}) + V_{xc}(\mathbf{r})
  • Vext(r)V_{ext}(\mathbf{r}) denotes external potential
  • VH(r)V_H(\mathbf{r}) represents Hartree potential
  • Vxc(r)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
  • (LDA) assumes uniform electron gas
  • (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

Key Terms to Review (18)

Ab initio methods: Ab initio methods are computational techniques used in quantum chemistry that rely on fundamental principles of quantum mechanics to predict molecular properties and behaviors from first principles, without empirical parameters. These methods provide a theoretical framework that allows chemists to explore the electronic structure of molecules and their interactions, which has evolved alongside the development of computational chemistry as a field.
Basis Set Expansion: Basis set expansion is a mathematical technique used in quantum chemistry to approximate the wave function of a system by expressing it as a linear combination of basis functions. This approach allows for the calculation of molecular properties and energies by using a finite set of functions that can describe the electronic structure accurately enough for practical purposes. Basis set expansion is essential in methods like Density Functional Theory (DFT) and Hartree-Fock, connecting directly to the principles outlined in the Hohenberg-Kohn theorems and Kohn-Sham approach.
Density Functional Theory: Density Functional Theory (DFT) is a quantum mechanical method used to investigate the electronic structure of many-body systems, primarily atoms, molecules, and the condensed phases. It simplifies the complex many-electron problem by using electron density rather than wave functions as the central variable, which makes it computationally efficient and widely applicable in various fields.
Electron Affinity: Electron affinity is the amount of energy released when an electron is added to a neutral atom in the gaseous state, forming a negatively charged ion. This concept is crucial in understanding how atoms interact with electrons, influencing their chemical reactivity and the formation of ions. In the context of different atomic structures, such as hydrogen and multi-electron atoms, electron affinity helps explain the stability and behavior of various elements when they acquire additional electrons.
Electron density: Electron density refers to the probability distribution of electrons within a given space around an atom or molecule. It is a crucial concept in quantum chemistry, as it helps to describe how electrons are arranged and how they interact with one another, influencing molecular properties and reactivity. Understanding electron density allows for insights into molecular orbitals, chemical bonding, and the behavior of electrons in various environments.
Energy functional: An energy functional is a mathematical expression that represents the total energy of a quantum mechanical system in terms of its electron density. This concept is crucial in density functional theory (DFT), where the energy functional is minimized to obtain the ground-state properties of a system. It connects the microscopic behaviors of electrons to macroscopic observables through functionals that depend on the spatial distribution of electrons.
Exchange-correlation functional: The exchange-correlation functional is a key component of density functional theory (DFT) that accounts for the complex interactions between electrons in a many-body quantum system. It combines both exchange and correlation effects, which arise from the indistinguishable nature of electrons and their interactions. This functional is essential for accurately describing the electronic structure of systems and plays a critical role in the Hohenberg-Kohn theorems and the Kohn-Sham approach.
Generalized gradient approximation: The generalized gradient approximation (GGA) is a refinement of the local density approximation (LDA) used in density functional theory (DFT) to improve the accuracy of calculations on molecular and solid-state systems. By incorporating the electron density's gradient, GGA provides better descriptions of exchange-correlation energy, leading to improved results for geometries, energies, and properties of many-body systems.
Hohenberg-Kohn Theorems: The Hohenberg-Kohn theorems are fundamental principles in density functional theory (DFT) that establish a relationship between the electron density of a many-body quantum system and its ground state properties. The first theorem states that the ground state electron density uniquely determines the external potential acting on the system, while the second theorem indicates that the ground state energy can be expressed as a functional of the electron density. These theorems form the backbone of DFT, leading to powerful computational methods in quantum chemistry and materials science.
Kohn-Sham Approach: The Kohn-Sham approach is a method within density functional theory (DFT) that simplifies the many-body problem of interacting electrons in a system by introducing an auxiliary system of non-interacting electrons. This approach enables the calculation of the electron density and the properties of a quantum system using a single-particle framework while still accounting for the effects of electron-electron interactions through an exchange-correlation functional.
Kohn-Sham Equations: The Kohn-Sham equations are a set of fundamental equations in density functional theory (DFT) that describe the behavior of many-electron systems in terms of a non-interacting system of particles. They provide a practical framework for calculating the electronic structure of atoms, molecules, and solids, linking the complex many-body problem to a simpler single-particle problem through the concept of an effective potential.
Local Density Approximation: The local density approximation (LDA) is a method used in density functional theory (DFT) to simplify the exchange-correlation energy of an electron system by assuming that it can be expressed as a function of the electron density at each point in space. This approach allows for efficient calculations of many-body systems while providing reasonably accurate results for various properties, particularly in systems where electron densities are slowly varying.
Molecular modeling: Molecular modeling is a computational technique used to represent, analyze, and predict the behavior and properties of molecular systems through visualization and simulations. By employing mathematical models and algorithms, this technique allows chemists to understand complex molecular interactions, explore conformational space, and evaluate the effects of molecular structure on reactivity and properties.
Pierre Hohenberg: Pierre Hohenberg is a prominent physicist known for his contributions to the field of density functional theory, particularly through the formulation of the Hohenberg-Kohn theorems. His work laid the groundwork for understanding how electron density can be used to determine the properties of many-body systems, which is essential in computational chemistry for accurately predicting molecular behavior and interactions.
Quantum mechanical calculations: Quantum mechanical calculations are computational methods used to solve the equations of quantum mechanics, allowing scientists to predict the behavior and properties of molecular systems. These calculations are crucial for understanding electronic structure, molecular interactions, and reaction mechanisms, which play a significant role in areas such as material science and environmental chemistry.
Self-consistent field: Self-consistent field (SCF) is a computational approach used in quantum chemistry to solve the many-body Schrödinger equation iteratively, where the interaction of particles is treated consistently within a given approximation. This method allows for the calculation of electron distributions and energy states by updating potential fields until they converge, making it crucial for understanding molecular electronic structures and the behavior of electrons in various systems.
Total energy: Total energy refers to the sum of all forms of energy in a system, encompassing both kinetic and potential energy contributions. In the context of quantum mechanics and computational methods, total energy is crucial as it determines the stability and interactions of molecular systems. Understanding total energy helps in predicting molecular behavior and the results of calculations performed in density functional theory (DFT).
Walter Kohn: Walter Kohn was an influential physicist and chemist best known for his contributions to density functional theory (DFT), which revolutionized the way chemists understand the electronic structure of atoms and molecules. His work laid the foundation for the Hohenberg-Kohn theorems and the Kohn-Sham approach, which provide powerful methods for calculating properties of many-body systems in a computationally efficient manner.
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