Mathematics Applied to Engineering in Action

Advanced Theories, Methods, and Models

Nazmul Islam, Satya Bir Singh, Prabhat Ranjan, A. K. Haghi, Nazmul Islam, Satya Bir Singh, Prabhat Ranjan, A. K. Haghi

• Publisher: Apple Academic Press
• Year: 2021
• eBook ISBN: 9781000089974
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Applied Mathematics
• Index: Technology & Engineering

Mathematics Applied to Engineering in Action: Advanced Theories, Methods, and Models focuses on material relevant to solving the kinds of mathematical problems regularly confronted by engineers. This new volume explains how an engineer should properly define the physical and mathematical problem statements, choose the computational approach, and solve the problem by a proven reliable approach. It presents the theoretical background necessary for solving problems, including definitions, rules, formulas, and theorems on the particular theme.

The book aims to apply advanced mathematics using real-world problems to illustrate mathematical ideas. This approach emphasizes the relevance of mathematics to engineering problems, helps to motivate the reader, and gives examples of mathematical concepts in a context familiar to the research students.

The volume is intended for professors and instructors, scientific researchers, students, and industry professionals. It will help readers to choose the most appropriate mathematical modeling method to solve engineering problems.

CHAPTER 1
Quantum Information Perspective on Chemical Reactivity^†

† The following notation is adopted throughout: A denotes a scalar, A is the row or column vector, A represents a square or rectangular matrix, and symbol A stands for the quantum-mechanical operator of the physical property A. The logarithm of the Shannon information measure is taken to an arbitrary but fixed base: log = log2 corresponds to the information content measured in bits (binary digits), while log = ln expresses the amount of information in nats (natural units): 1 nat = 1.44 bits.

ROMAN F. NALEWAJSKI^*

* Professor Emeritus.

Department of Theoretical Chemistry, Jagiellonian University, Gronostajowa 2, 30–387 Cracow, Poland, E-mail: [email protected]

ABSTRACT

A need for quantum information measures combining the probability and current contributions is emphasized and continuity relations are summarized. Resultant gradient information accounts for the classical (probability) and nonclassical (phase) contributions and reflects the kinetic energy content of molecular electronic state. The phase-information description enables one to distinguish the hypothetical states of the mutually bonded (entangled) and nonbonded (disentangled) reactants in donor-acceptor systems. This generalized information-theoretic perspective is applied to chemical reactivity phenomena within the grand-ensemble description of thermodynamic conditions in open molecular systems. The physical equivalence of variational principles for the system electronic energy and resultant gradient information is demonstrated. The virial theorem decomposition of energy profiles is used to index the Hammond rule of reactivity theory, information changes in chemical reactions are addressed, and charge transfer between reactants is explored. The frontier-electron approach to molecular interactions is used to rationalize the Hard (Soft) Acid and Bases (HSAB) principle of structural chemistry, and the rule implications for the intra- and inter-reactant communications are commented upon.

1.1 INTRODUCTION

The classical Information Theory (IT) of Fisher and Shannon [1–8] has been successfully applied to interpret in chemical terms the electron probability distributions of molecular systems, e.g., [9–12]. Information principles have been explored [13–18], density pieces attributed to Atoms-in-Molecules (AIM) have been tackled [9, 13, 1...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• About the Editors
• Contents
• Contributors
• Abbreviations
• Preface
• 1. Quantum Information Perspective on Chemical Reactivity
• 2. A Computational Modeling of the Structure, Frontier Molecular Orbital (FMO) Analysis, and Global and Local Reactive Descriptors of a Phytochemical ‘Coumestrol’
• 3. Theoretical Analysis of CuTiS2 and CuTiSe2 Invoking Density Functional Theory-Based Descriptors
• 4. Synergistic Effect of E. crassipes Biomass/Chitosan for As(III) Remediation from Water
• 5. Computational Investigations on Metal Oxide Clusters and Graphene-Based Nanomaterials for Heterogeneous Catalysis
• 6. Effect of Dye Concentration on Series Resistance of Thionin Dye-Based Organic Diode
• 7. On Gower’s Inverse Matrix
• 8. Applications of Noether’s Theorem
• 9. Mathematical Modeling of Elastic-Plastic Transitional Stresses in Human Femur and Tibia Bones Exhibiting Orthotropic Macro Structural Symmetry
• 10. Lorentz Transformations, Dirac Matrices, and 3-Rotations via Quaternions
• 11. Multiscale Approach Towards the Modeling and Simulation of Carbon Nanotubes Networks
• 12. When Perovskites Memorize
• Index