Contents


*Mohamed Shahin Thayyil and Muraleedharan Karuvanthodi*


Preface

The wave function theory based on the Schrödinger equation was first approximately solved by Hartree-Fock-Roothan (HFR) formalism, initiating a wonderful time of computational chemistry. The possibility of modeling and simulating the geometry, electronic density, and other properties in the quantum level for molecules is today, one of the essential areas of natural science. In this new area, the variational principle's resolution is the central question to reach the energies' electronic levels associated with a set of particles; such an approach was very successful for modeling for atoms to molecules. The HFR approach is a numeric approximation method developed to calculate a set of equations similar to the Schrödinger equation for many bodies, for example, the helium atom, the hydrogen molecule, and others. The evolution of the HFR approach was impressive. Its expansion was fast. Many scientists used the method to create the fundamental details of the molecular concepts and properties simulated today. Hartree proposes a modified wave function from the monoelectronic wave functions product or the self-consistent field (SCF). This approximation was essential, but not precise, to create a numerical path to calculate approximately the many body system's wave functions. The consequence was the development of the canonical molecular orbitals with many problems and high potential. Fock contributed significantly to building the quantum operator associated with Hartrees' wave function denominated Fock operator. Now, the Fock's operator would act on the Hartree's wave function extracting quantum information or properties; more importantly, from the monoelectronic wave function product and Focks' operator, the basis for the numerical implementation of the approach has been created.

Meanwhile, Hartree's' wave function is not determined analytically. It is a numerical approach; thus, it always needs to be built. Roothan used the Linear Combination of mathematical functions to create the Linear Combination of Atomic Orbitals (LCAO). This implementation was essential. It increases the precision of the SCF cycle through basis set functions. There is no unique mathematical form for the basis sets since the mathematical equation chooses to build the LCAO used in the SCF satisfies the Fock operator; it is a basis set valid. Then, the simulation of a more complex system was

The Density Functional Theory (DFT) is an improved quantum theory in recent times. However, such an idea was a development from 1960 by Hohenberg-Khon-Sham. Contrary to wave function or many bodies theory, the scientists create a new quantum approach to simulate many-body systems changing the wave function approach using the electronic density (r) concept. The evolution of the DFT showed that the operators have potential, and the monoelectronic formalism is equal to that found in the HFR approach; however, for the bi-electronic interactions in electron-electron repulsion, there was a significant change. In the HFR approach, the correlation energy is not available, resulting in lost information; in the DFT, the correlation energy is exact. It can be calculated from a derivative between energy and the electronic density creating a function. In a sense, since the functional creation, the DFT has three dimensions (x, y, z) for calculations; while the HFR approach has four dimensions (x, y, z, spin). Indeed, the find of the analytical form for the functional is the goal in the DFT. Then, from the last 60 years, scientists have researched numerous functional methods to solve such a mathematical problem. Initially, the functional was described to understand the functional's contribution to the molecular results. The Local Density Approximation (LDA) and Local Spin Density Approximation (LSDA) are examples. Very quickly, the researchers noted this task was not simple. A simple form to describe the functional

needed to obtain quantum properties.
