Statement proof and applications of the Hohenberg Kohn theorem
In 1964, a seminal study by Hohenberg and Kohn that was published in the Physical Review gave birth to density functional theory as we know it today.
Hohenberg-Kohn Theorem
Density functional theory is based on the Hohenberg-Kohn theorem, which claims that the energy of the ground electronic state is a special function of the electron density. This feature, together with a related variational theorem, sparked the creation of practical empirical representations of the density functional that even for intricate chemical systems produce accurate estimates of the electronic energy.
These functionals were initially thought to merely be dependent on the local density. The gradient of the densities, the kinetic energy density, the occupied orbitals, and finally, the unoccupied orbitals were all assumed to be independent of the empirical functionals throughout time.
History of The Scientist
Pierre C. Hohenberg (1934 – 2017)
Name: Pierre C. Hohenberg
Born: 3 October 1934
Died: 15 December 2017
Field: Physicist
Nationality: French-American
Walter Kohn (1923 – 2016)
Name: Walter Kohn
Born: March 9, 1923
Died: April 19, 2016
Field: Physicist, Chemist
Nationality: Austrian-American
What is the Density Functional Theory?
A low-cost, time-saving quantum mechanical theory called density functional theory is utilised to precisely compute several physical properties of materials. Research in this area covers a wide range of topics, including the creation of novel analytical techniques centred on the creation of precise exchange-correlation functionals and the application of this method to the prediction of the molecular and electronic configuration of atoms, molecules, complexes, and solids in both gas and solution phases.
Early advances in DFT focused on the most fundamental chemistry problems, namely the ability to create functionals that could explain both molecular geometries and dissociation energy. Hohenberg-proof Kohn's that each characteristic of a system in its ground state corresponds to a distinct ground state density functional laid the groundwork for DFT, which is presently utilised to investigate the uniqueness of materials.
Fundamentals of Density Functional Theory
The Kohn-Sham equation, a single particle independent Schrodinger equation, can be used to solve the Schrodinger equation for a multi-body system numerically using density functional theory. However, this concept is based on electron density rather than wave functions, for which physicist Walter Kohn received the 1998 Nobel Prize. This computational process results in the physical properties of solids.
Thomas and Fermi asserted that density is the crucial quantity in many body problems in 1927, despite the absence of exchange-correlation effects at the time. The framework for DFT was set by the Hohenberg, Kohn, and Sham theorems, which claimed that in the absence of a magnetic field, the function of a many-body problem's ground state electron charge density could completely define all features.
Hohenberg-Kohn Theorem I
The ground-state energy is a distinct function of the electron density, according to the first Hohenberg-Kohn theorem. The ground state wave function and ground state charge density are mapped one to one by this theorem. All of a system's ground state characteristics can be uniquely described by the ground state charge density. The core idea behind density functional theory is that, rather than utilising a wave function, charge density (three-dimensional) can accurately describe the ground state of an N-particle.
$E\left[\Psi\left[n_{0}\right]\right]=\left\langle\Psi\left[n_{0}\right]|\hat{T}+\hat{V}+\hat{U}| \Psi\left[n_{0}\right]\right\rangle$
Hohenberg-Kohn Theorem II
The second Hohenberg Kohn (HK) theorem states that the actual electron density is that which minimises the energy of the overall functionality. If it were possible to determine the genuine functional form of energy in terms of density, one might adjust the electron density until the energy from the functional was reduced to a minimum, providing the necessary ground state density. This works with approximation forms of the functional and is essentially a variational principle. The most straightforward option for a functioning can is a uniform electron density throughout the space.
$E\left[\Psi\left[n_{0}\right]\right]=\left\langle\Psi\left[n_{0}\right]|\hat{T}+\hat{V}+\hat{U}| \Psi\left[n_{0}\right]\right\rangle$
Solved Examples
1. What in DFT is SCF?
Ans: Both the Hartree-Fock (HF) theory and the Kohn-Sham (KS) density functional theory are self-consistent field (SCF) techniques (DFT). The most basic level of quantum chemical models is called self-consistent field theories, and it solely depends on the electronic density matrices.
2. What problem does DFT solve?
Ans: Using model interactions between particles, classical DFT enables the determination of the equilibrium particle density as well as the thermodynamic property and behaviour prediction of a many-body system. The local composition and structure of the material are determined by the spatially dependent density.
3. Why is the theory of density functionals significant?
Ans: DFT is a member of the first principles (ab initio) approach family, so termed for their ability to predict material properties for unidentified systems without the use of experiments. DFT has become well-known among these due to the minimal computational effort needed.
Important Points to Remember
A potent and frequently used quantum mechanical tool for examining different features of the matter is density functional theory.
The research in this area covers a wide range of topics, including the creation of original analytical methods centred on the creation of precise exchange-correlation functionals.
The application of this method to the prediction of the molecular and electronic configuration of atoms, molecules, and solids in both gas and solution phases.
Conclusion
Since there are still problems to be solved, designing and evolving more effective density functionals is a continual process. Getting all of the qualities just right at a reasonable processing cost is a quantum fantasy. Future work will concentrate on creating density functionals that are even more consistently exact for particular applications, enabling researchers to take advantage of DFT's relatively high accuracy at low processing costs and the potential for even further advancements.
FAQs on Hohenberg Kohn Theorem in Density Functional Theory
1. What is the Hohenberg–Kohn theorem?
The Hohenberg–Kohn theorem states that the ground-state electron density uniquely determines all properties of a many-electron system, including its total energy. It is the foundation of Density Functional Theory (DFT) in quantum chemistry.
- The theorem applies to interacting electrons in an external potential.
- It proves that the external potential (and thus the Hamiltonian) is a unique functional of the electron density.
- Therefore, all ground-state observables are functionals of the electron density ρ(r).
2. What are the two Hohenberg–Kohn theorems?
The two Hohenberg–Kohn theorems establish the uniqueness of electron density and the variational principle in Density Functional Theory.
- First theorem: The ground-state electron density uniquely determines the external potential (except for a constant).
- Second theorem: The correct ground-state density minimizes the total energy functional.
3. Why is the Hohenberg–Kohn theorem important in chemistry?
The Hohenberg–Kohn theorem is important because it provides the theoretical foundation for modern computational methods used to calculate molecular structure and energy.
- It enables the development of Density Functional Theory (DFT).
- DFT is widely used to predict molecular geometries, reaction energies, and electronic properties.
- It reduces computational complexity compared to wavefunction-based methods like Hartree–Fock.
4. How does the Hohenberg–Kohn theorem relate to Density Functional Theory (DFT)?
The Hohenberg–Kohn theorem provides the theoretical basis of Density Functional Theory (DFT) by proving that the ground-state energy is a functional of electron density.
- DFT uses the electron density ρ(r) instead of the many-electron wavefunction Ψ.
- The theorem ensures that all ground-state properties can be derived from ρ(r).
- Practical DFT methods approximate the unknown exchange–correlation functional.
5. What does it mean that electron density uniquely determines the external potential?
It means that a given ground-state electron density corresponds to only one external potential (except for an additive constant).
- The external potential is usually due to atomic nuclei in molecules.
- If two different potentials produced the same ground-state density, it would violate the first theorem.
- Therefore, the electron density contains all information about the system.
6. What is the variational principle in the second Hohenberg–Kohn theorem?
The second Hohenberg–Kohn theorem states that the correct ground-state electron density minimizes the total energy functional.
- Any trial density ρ′(r) gives an energy greater than or equal to the true ground-state energy.
- The true density gives the lowest possible energy.
- This is analogous to the variational principle in wavefunction-based quantum mechanics.
7. Does the Hohenberg–Kohn theorem apply to excited states?
The original Hohenberg–Kohn theorem applies strictly to ground states, not excited states.
- It guarantees uniqueness only for the ground-state density.
- Extensions such as time-dependent DFT (TD-DFT) are used for excited states.
- Excited-state densities may not uniquely determine the external potential in the same way.
8. What is the difference between wavefunction methods and the Hohenberg–Kohn approach?
The key difference is that wavefunction methods use the many-electron wavefunction, while the Hohenberg–Kohn approach uses electron density as the fundamental variable.
- Wavefunction Ψ depends on 3N spatial coordinates for N electrons.
- Electron density ρ(r) depends only on three spatial coordinates.
- This reduction greatly simplifies computational cost.
9. What assumptions are made in the Hohenberg–Kohn theorem?
The Hohenberg–Kohn theorem assumes a non-degenerate ground state and an interacting electron system under an external potential.
- The proof assumes electrons obey quantum mechanics and Coulomb interactions.
- The external potential is typically due to fixed nuclei (Born–Oppenheimer approximation).
- Degenerate ground states require modified arguments.
10. What is the universal functional in the Hohenberg–Kohn theorem?
The universal functional is a density functional that includes kinetic and electron–electron interaction energies and is independent of the external potential.
- It is commonly denoted as F[ρ].
- It is the same for all many-electron systems.
- The exact form of F[ρ] is unknown, which leads to exchange–correlation approximations in DFT.
