The HSE exchange-correlation functional degenerates to the PBE0 hybrid functional for. Is the short range Hartree–Fock exact exchange functional, and are the short and long range components of the PBE exchange functional, and is the PBE correlation functional. 2.5 Density functional theory Density functional theory (DFT) is a powerful, formally exact theory (see Refs. 8,9,3 and references within). It is distinct from quantum chemical methods in that it is a non-interacting theory and does not yield a correlated -body wavefunction.
The density functional theory is now used by scientists around the world to determine the characteristics and properties of various molecules and atoms based on the electron density of the molecule. Some of these properties include the electronic structure of orbitals, atomic reactivity, and even the UV-Vis spectra. Unlike the wavefunction, electron density can be measured. Density functional theory has turned into the most popular way of computationally finding molecular properties for many reasons. The method has high structure accuracy, does an impressive job figuring out Gibbs free energy values for certain reactions, is not as difficult to run computationally as compared to other methods, and takes into account electron correlation, which provides an accurate representation of a system’s energy. The interaction that takes place between certain electrons in a give electronic structure is known as the electron correlation. Electron correlation can be used to find the discrepancies of the Hartree-Fock model.
The Hartree-Fock method only finds repulsion energy as an average over the entire molecular orbital, which is not accurate. The correlation energy is determined by finding the difference between the true energy and the Hartree-Fock energy.
Hohenberg, born in 1934 in France, is a world-renowned theoretical physicist. Earning his bachelor’s degree, master’s degree, and doctorate from Harvard University, Hohenberg began working at Bell Laboratories in Murray Hill, where from 1989 to 1995 he became the directer of the department of theoretical physics. Hohenberg has a unique fondness for his home country, as he was guess professor in Paris in 1963, 1964, and 1988. 2004 saw him advance to Senior Vice Provost of Research at NYU 1, but he soon stepped down to be a professor in the Physics Department. Hohenberg was a leading figure outside academia as well, as he was on the human rights committee of the New York Academy of Sciences and a proud member of the National Academy of Sciences and American Academy of Arts and Sciences.
His theorems, with Walter Kohn, in a paper written in the Physical Review would give birth to the density functional theory we have today. Walter Kohn, born in 1923 in Austria, is one of the most important people in the area of theoretical chemistry.
After receiving a PhD from Harvard, in the field of physics, Kohn would go on to become a physics professor at Carnegie Mellon University, University of California at San Diego, and University of California at Santa Barbara 2. Like Hohenberg, Kohn had a large impact outside of academia, as he was a member of the National Academy of Sciences and American Academy of Arts and Sciences. He is, however, best known for winning the 1998 Nobel Prize in Chemistry, who he shared with physicist John A. He was recognized with the prestigious award for his 'development of the density functional theory”. He passed away last year due after his battle with jaw cancer.
Before the discovery of the density functional theory, scientists worked to find the wavefunction of a system, as with it they could completely describe the properties of that molecular system. Every single particle of an atom is given a specific wavefunction, which gives all the calculable information about the particle. The Schrodinger equation, in layman’s terms, predicts the behavior of a dynamic system in the future.
It represents how a wavefunction of a system changes and evolves over time. In order to find the wavefunction in the Schrodinger equation, the computer needs initial numbers, which are found using basis set and basis functions. In the Schrodinger equation, which is shown below, the H stands for the Hamiltonian. Of course, there were issues surrounding solely using the wavefunction to provide information about a certain system. First of all, the wavefunction is not something that can be physically analyzed or quantified in a laboratory setting.
It is also a function that is confusing, due it having multi-dimensions. In addition, only the square of the wavefunction has any physical value, as it gives information about the probability density. The probability density is made up of numerous variables, adding to the complexity of this technique. There are other ways of making predictions in quantum mechanics, including path integral formulation and matrix mechanics. Erwin Schrodinger, born in 1887 in Austria, made significant contributions to quantum mechanics. Earning a doctorate from the University of Vienna in 1910, he later went on to serve in the military during World War I. At the University of Zurich in 1926, Schrodinger published his work that would set the stage for the rise of quantum wave mechanics.
He stated that matter particles, in some circumstances, behave like waves. His equation, the Schrodinger equation, outlines how a wave equation can predict the behaviour of a certain system. Schrodinger argued that his equation could also determine measurable energies of a system. Not surprisingly, many physicists around the world did not believe Schrodinger’s work and were very reluctant to accept his theory. His most well known objection has become known as Schrodinger’s cat, which states a that poisonous flask, a cat, and a radioactive source are placed in a box.
In the event of a decaying atom, the cat is killed due to the release of the poison. When someone opens the box, the cat will be either alive or dead, not both dead and alive, supporting Schrodinger’s claim that questions the validity. Focused only on the electron density at a certain point in space, local density approximations are a simple type of exchange-correlation functional. Hohenberg and Kohn first came up with local density approximations in their density functional theory paper, making this functional type one of the oldest out there. To make use of local density approximations, the exchange-correlation energy of a electron gas at a certain density must be found 5. These approximations tend to work, as small errors in correlation and exchange energy densities cross each other out.
In Marcel Swart’s 2016 density functional theory poll, local density approximations was ranked the eighth most popular. Local density approximations, and their low to average accuracy, are not useful for many scientific applications.
Therefore, for many years, the field of computational chemistry was never impacted by density functional theory. Soon gradient-corrected functionals came along, also known as generalized gradient approximation functionals (GGA).
These functionals are sometimes called non-local functionals, which is not true since gradient and electron density provide only local information about a system. A gradient-corrected approximation solely relies on local density and the gradient of it 6. The gradient-corrected approximation is an improvement over local density approximations since information of the gradient of the charge density is known.
A gradient, in mathematics, calculates the rate of change of a property in question. Gradient-corrected functionals aim take into consideration the inconsistencies of electron density. These type of functionals. A type of gradient-corrected functional, PW91 has a Perdew-Wang exchange functional and a Perdew-Wang correlation functional.
With various functionals in his name, it is easy to see the massive role Dr. Yang has played in the development of the density functional theory (picture provided below). In Marcel Swart’s 2016 density functional theory poll, PW91 was ranked the eleventh most popular functional in the first division. To form the hybrid functional B3LYP, the PW91 correlation functional gets replaced by the Lee-Yang-Parr functional. For those in the computational chemistry field, hybrid functionals are the most popular and common type of DFT method. They are even sometimes known as ACM functionals, which stands for adiabatic connection method.
Hybrid functionals are very accurate at predicting the change of Gibb’s free energy in a system and at finding a geometry optimization but are more computationally costly than local density approximations and gradient-corrected functionals. These types of functionals bring together ab initio methods (majority of the time Hartree-Fock methods) and density functional theory mathematics. In other words, a hybrid functional is a functional that has electron correlation, exchange energy with a density functional theory approximation, and exchange energy with a Hartree-Fock approximation all combined together 7.
Ab initio methods are those in which Schrodinger equation is used to create the entire model mathematically. More specifically, Hartree-Fock methods take into consideration the average effect of correlation of one electron on another electron in a system. Hybrid functionals usually overpredict bond lengths of molecules. In Marcel Swart’s 2016 density functional theory poll, B3LYP was ranked the third most popular functional in the first division. For the majority of systems, the B3LYP/6-31G model chemistry will get the best approximations of a certain system, especially those involving organics. This functional approximates properties related to geometries really well 8. B3LYP should not be chosen as a functional if one is trying to approximate the excitation energies, reaction barriers, transition metals, and relativistic elements of a system.
The B3LYP exchange-correlation functional, a mixing scheme of 3 parameters, is ”made” from pieces of the Hartree-Fock exchange functional, local spin density approximation exchange functional, Becke88 exchange functional, Lee-Yang-Parr correlation functional, and Vosko-Wilk-Nusair correlation functional. As the search for the best functional to approximate the exchange-correlation functional progresses, it can be said that the easiest way to get a PhD in chemistry is to create a better functional than those that exist today.
From the information in the previous section, it can be seen that the two best functionals are the PBE gradient-corrected functional and PBE0 hybrid functional, due to their versatility to find numerous properties of a system. Whenever in doubt about what functional to use for a given system, the B3LYP hybrid functional, as well as the two functionals stated above, should be chosen. Keeping in mind that each functional class has certain applications and scenarios it works best in, in general, scientists should stick away from local density approximations, due to their primitiveness and by the availability of more useful functionals. As density functional theory continues to grow and become more and more popular, the field of computational chemistry will soon be pushed into the spotlight.