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 needed to obtain quantum properties.

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

**II**

**Section 6**

Collisions of Heavy Ions

*by A.T. D'yachenko and I.A. Mitropolsky*

Non-equilibrium Equation and Density Functional **91**

**Chapter 6 93** Non-equilibrium Equation of State in the Approximation of the Local Density Functional and Its Application to the Emission of High-Energy Particles in

acted in a path less efficient for molecules with a stronger correlation effect; and, the molecular geometry and electronic properties were poorly calculated. The results were hugely improved from the Generalized Gradient Analytical (GGA) introduction in functional formalism, showing the importance of describing the electronic correlation from an electronic density distribution. A pure functional description was initiated; the foremost functional was the Perdew-Burke-Ernzerhof (PBE).

Meanwhile, functional complexes were created. An example of this new implementation was the PBE0 functional called a hybrid functional. After the hybrid functionals were introduced, the parameterized hybrid functional describes the functional by factors from experimental data to correct the GGA approach. In truth, the great challenge forced by the search of the final functional drove a huge scientific effort for a significant functional until recent days.

In this book, the chapters show research in a broad range of DFT applications. The first chapter is dedicated to one of the essential topics in Computational Chemistry: chemical reactivity. The analysis of the frontier molecular orbitals (FOM), more specifically, the high occupied molecular orbital (HOMO) and low unoccupied molecular orbital (LUMO), indicates molecular properties beneficial to support the chemical reactivity of compounds. The investigations on metallic complexes are very useful because of the molecular geometry, bonding, and antibonding molecular orbitals, charge transfer, and other molecular features. Ionic liquids have a great property of stability concerning significant electric potential variation. The second chapter describes Koopmans' theorem and Fukui index on three liquid ionic families discussing each family, an example of this new technological area's computational concepts. Chemoinformatic or in silico are words to indicate computational simulations in discrete or multiscale levels often applied to biological systems to investigate interactions, and molecular structures are the goal of these calculations. This book then presents one chapter to show an essential tool that can be worked in drug and bioactivity themes. In the same direction, one chapter is on molecular docking studies of the antioxidant effect of the Coumarin compound. The DFT is deeply connected to mathematical formalism; thus, it can be expanded to other areas. The chapter on the non-equilibrium equation is a sample of this expansion, applying the DFT to create a representative model associated with non-equilibrium states. In this case, the proposed model investigated heavy atoms and pions.

Welcome and have a good read!

**Sérgio Ricardo de Lázaro**

State University of Ponta Grossa, Department of Chemistry, Uvaranas, Ponta Grossa, Brazil

#### **Renan Augusto Pontes Ribeiro**

State University of Minas Gerais, Department of Chemistry, Divinópolis, Brazil

#### **Luis Henrique da Silveira Lacerda**

State University of Campinas, Institute of Chemistry, Campinas, Brazil

**1**

Section 1

Introduction

Section 1 Introduction

**3**

Chemistry of Materials area.

**2. Short points in pressure**

**Chapter 1**

*Sérgio Ricardo de Lázaro*

**1. Introduction**

Introductory Chapter: A Brief

Mention for High-Pressure in

Oxides from DFT Simulations

High pressure simulations are employed in materials because of the resistance and advanced properties. The material modifications from high pressures are essential to reach new crystalline structures. In geophysics, many high pressure processes happen inside Earth. Diamonds are a clear example on the matter transformations under high pressures. In this case, the diamond is the natural material more resistance already found; several studies were performed to understand how carbon in graphite form can be changed to diamond and the result was a synthesis procedure based on high pressure in a short time. The initial studies using pressure were performed in gas phase. The definition of the pressure is clearly connected to how a surface is modified from an applied force. All gaseous kinectic theory has the pressure as a central point to understand the atomic and molecular movement in macroscale. In solids investigations, the pressure is the property that deform or change the surface of the material. The high material resistance regarding to gas, the deformations were classified as total and partially elastic or inelastic showing a point of view on reversible and irreversible processes. Such concepts origin the idea on malleability of the solids since metals to plastic. Amorphous and crystalline solids have similar resistance; however, the cleavage of each one is dominated from the molecular ordering. The high pressure finds to investigate the phase transitions in crystalline solids without to reach the crystal's cleavage. The compression modulus theory are the scientific approaches more developed to understand solid deformations from applied external pressure. Since one-dimensional (1D) deformation denominated as Young Modulus, and, posteriorly, Shear (2D) and Bulk Modulus (3D); it is knowledge that Equation of States (EOS) are associated to the energy x volume curve to calculate Bulk Modulus. From coming Quantum Mechanic, more specifically, the Computational Chemistry in solid state; the making of the energy x volume curve becomes a possibility to calculate the Bulk Modulus from only theoretical proceedings. In this terms, the Density Functional Theory (DFT) has been broadly applied to predict and to investigate material's compressilibity from chemical substitutions or new crystalline structures. Such new approach reaches to a new level for research in

The pressure play an important role in chemical and physical systems. It is defined as force on area (1) showing as an external force acts on substances or materials in solids, liquid, and gas states. Automatically, the pressure is a physical

### **Chapter 1**
