Introduction to 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
1. What are a few Hohenberg-Kohn theorem-related problems?
Even though they seem straightforward, the astonishing notion that there is a one-to-one correspondence between densities and external potentials was brought up by the Hohenberg and Kohn theorems. It turns out that, in some cases, the Hohenberg and Kohn theorems hold. Fortunately, we rarely think about these issues because, in reality, atomic system densities adhere to these restrictions. One can no longer discuss the uniqueness of the ground state expectation value of operators if there are multiple wavefunctions that each correspond to a system's ground state energy. The first theorem, in particular, is impossible to demonstrate. The aforementioned problem will be reformulated differently in the following subsection about another issue, which will also, as a byproduct, answer the degeneracy problem.
2. What are a few difficulties with the density functional theory?
DFT is exact in theory, but to be useful, exchange-correlation functionals must advance. This can be done by optimising against larger data sets and employing superior functional arrangements that are more flexible and contain more elements. Recent improvements have also given smoothness a higher priority, which helps to solve issues with grid-size convergence and self-consistent field iterations. We shall discuss a few DFT difficulties in this area, ranging from those that seem to be "solved" to those that are currently being investigated. There are many more that are less well-known but as important to the development and application of DFT in the future.
3. Mention some of the formal density-functional theory's core tenets
The renowned Hohenberg-Kohn theorem, which, for non-degenerate ground states, can be summed up as follows, is the fundamental proposition of formal density-functional theory.
The external potential in which the electrons flow, as well as the Hamiltonian and all other physical features of the system, are all uniquely determined by the ground state electron density of a system of interacting electrons.
In other words, the ground-state energy can be written as a function of the density, which gives the ground-state energy if and only if the true ground-state density is inserted, and the ground-state energy and ground-state density of a system characterised by the potential can be obtained from a variational principle which involves only the density. The disparity persists at all additional densities.
