Discover theoretical, methodological, and applied perspectives on electron density studies and density functional theory Electron density or the single particle density is a 3D function even for a many-electron system. Electron density contains all information regarding the ground state and also about some excited states of an atom or a molecule. All the properties can be written as functionals of electron density, and the energy attains its minimum value for the true density. It has been used as the basis for a quantum chemical computational method called Density Functional Theory, or DFT, which can be used to determine various properties of molecules. DFT brings out a drastic reduction in computational cost due to its reduced dimensionality. Thus, DFT is considered to be the workhorse for modern computational chemistry, physics as well as materials science. Electron Density: Concepts, Computation and DFT Applications offers an introduction to the foundations and applications of electron density studies and analysis. Beginning with an overview of major methodological and conceptual issues in electron density, it analyzes DFT and its major successful applications. The result is a state-of-the-art reference for a vital tool in a range of experimental sciences. Readers will also find: A balance of fundamentals and applications to facilitate use by both theoretical and computational scientists Detailed discussion of topics including the Levy-Perdew-Sahni equation, the Kohn Sham Inversion problem, and more Analysis of DFT applications including the determination of structural, magnetic, and electronic properties Electron Density: Concepts, Computation and DFT Applications is ideal for academic researchers in quantum, theoretical, and computational chemistry and physics.

Table of Contents:
List of Contributors xvii Preface xxv 1 Levy–Perdew–Sahni Equation and the Kohn–Sham Inversion Problem 1 Ashish Kumar and Manoj K. Harbola 1.1 Introduction 1 1.2 One Equation ⟹ Several Methods; Universal Nature of Different Density-Based Kohn–Sham Inversion Algorithms 2 1.2.1 Generating Functional S[ρ] of Density-Based Kohn–Sham Inversion 2 1.2.2 Condition on Generating Functional S[ρ] 4 1.2.3 Examples of Different Generating Functionals 5 1.2.4 Application to Spherical Systems 7 1.2.5 Using Random Numbers to do Density-to-Potential Inversion 10 1.3 General Penalty Method for Density-to-Potential Inversion 12 1.4 Understanding Connection Between Density and Wavefunction-Based Inversion Methods Using LPS Equation 16 1.5 Concluding Remarks 19 Acknowledgments 19 References 20 2 Electron Density, Density Functional Theory, and Chemical Concepts 27 Swapan K. Ghosh 2.1 Introduction 27 2.2 Viewing Chemical Concepts Through a DFT Window 27 2.3 Electron Fluid, Quantum Fluid Dynamics, Electronic Entropy, and a Local Thermodynamic Picture 30 2.4 Miscellaneous Offshoots from Electron Density Experience 31 2.5 Concluding Remarks 31 Acknowledgments 32 References 32 3 Local and Nonlocal Descriptors of the Site and Bond Chemical Reactivity of Molecules 35 José L. Gázquez, Paulino Zerón, Maurizio A. Pantoja-Hernández and Marco Franco-Pérez 3.1 Introduction 35 3.2 Local and Nonlocal Reactivity Indexes 38 3.3 Site and Bond Reactivities 42 3.4 Concluding Remarks 46 Acknowledgment 47 References 47 4 Relativistic Treatment of Many-Electron Systems Through DFT in CCG 53 Shamik Chanda and Amlan K. Roy 4.1 Introduction 53 4.2 Theoretical Framework 56 4.2.1 Dirac Equation 56 4.2.2 Relativistic Density Functional Theory: Dirac–Kohn–Sham Method 58 4.2.3 Decoupling of Dirac Hamiltonian: DKH Methodology 60 4.2.4 DFT in Cartesian Grid 62 4.2.4.1 Basic Methodology 62 4.2.4.2 Hartree Potential in CCG 63 4.2.4.3 Hartree Fock Exchange Through FCT in CCG 65 4.2.4.4 Orbital-Dependent Hybrid Functionals via RS-FCT 65 4.3 Computational Details 66 4.4 Results and Discussion 67 4.4.1 One-Electron Atoms 67 4.4.2 Many-Electron Systems 68 4.4.2.1 Grid Optimization 68 4.4.2.2 Ground-State Energy of Atoms and Molecules 70 4.4.3 Application to Highly Charged Ions: He- and Li-Isoelectronic Series 71 4.5 Future and Outlook 74 Acknowledgement 76 References 76 5 Relativistic Reduced Density Matrices: Properties and Applications 83 Somesh Chamoli, Malaya K. Nayak and Achintya Kumar Dutta 5.1 Introduction 83 5.2 Relativistic One-Body Reduced Density Matrix 84 5.3 Properties of Relativistic 1-RDM 85 5.3.1 Natural Spinors: An Efficient Framework for Low-cost Calculations 87 5.3.1.1 Correlation Energy 88 5.3.1.2 Bond Length and Harmonic Vibrational Frequency 90 5.3.2 Natural Spinors as an Interpretive Tool 93 5.4 Concluding Remarks 93 Acknowledgments 93 References 94 6 Many-Body Multi-Configurational Calculation Using Coulomb Green’s Function 97 Bharti Kapil, Shivalika Sharma, Priyanka Aggarwal, Harsimran Kaur, Sunny Singh and Ram Kuntal Hazra 6.1 Introduction 97 6.2 Theoretical Development 98 6.2.1 Presence of Magnetic Field 99 6.2.1.1 3D Electron Gas Model 99 6.2.1.2 2D Electron Gas Model 103 6.2.1.3 3D Exciton Model 107 6.2.1.4 2D Exciton Model 109 6.2.2 Absence of Magnetic Field 114 6.2.2.1 3D He-Isoelectronic Ions 114 6.2.2.2 2D He-Isoelectronic Ions 119 6.2.2.3 Energy Calculation Through Perturbation 122 6.2.2.4 Current Density of 2-e System 123 6.3 Results and Discussion 123 6.3.1 3D Interacting Electron Gas 123 6.3.2 2D Interacting Electron Gas 125 6.3.3 3D Exciton Complexes 126 6.3.4 2D Exciton Complexes 127 6.3.5 3D He-Isoelectronic Species 128 6.3.5.1 Analysis of E(2)0 of He-Isoelectronic Ions 129 6.3.5.2 Analysis of E(3)0 of He-Isoelectronic Ions 129 6.3.6 2D He-Isoelectronic Species 130 6.4 Concluding Remarks 131 Acknowledgments 131 6.A Standard Equations and Integrals 132 References 133 7 Excited State Electronic Structure – Effect of Environment 137 Supriyo Santra and Debashree Ghosh 7.1 Introduction 137 7.2 Methodology 138 7.2.1 Quantum Mechanical Methods 138 7.2.1.1 Time-Dependent Density Functional Theory 138 7.2.1.2 Active Space-Based Methods 138 7.2.1.3 Configuration Interaction-Based Approaches 139 7.2.1.4 Equation of Motion Coupled Cluster 140 7.2.2 Molecular Mechanical Methods 140 7.2.2.1 Oniom 141 7.2.2.2 Mechanical Embedding 141 7.2.2.3 Electronic Embedding 142 7.2.2.4 Polarizable Embedding 142 7.3 Representative Examples 143 7.3.1 Photo-Isomerization of Rhodopsin 143 7.3.2 DNA-Base Excited States in Solution 143 7.3.3 Green Fluorescent Proteins 145 7.4 Conclusion 146 Acknowledgement 146 References 146 8 Electron Density in the Multiscale Treatment of Biomolecules 149 Soumyajit Karmakar, Sunita Muduli, Atanuka Paul, and Sabyashachi Mishra 8.1 Introduction 149 8.2 Theoretical Background 150 8.2.1 Hybrid Quantum Mechanics–Molecular Mechanics Approach 152 8.3 Polarizable Density Embedding 155 8.4 Multi-Scale QM/MM with Extremely Localized Molecular Orbitals 157 8.5 Multiple Active Zones in QM/MM Modelling 159 8.6 Reactivity Descriptors with QM/MM Modeling 161 8.7 Treatment of Hydrogen Bonding with QM/MM 163 8.8 Quantum Refinement of Crystal Structure with QM/MM 164 8.9 Concluding Remarks 166 Acknowledgments 167 References 167 9 Subsystem Communications and Electron Correlation 173 Roman F. Nalewajski 9.1 Introduction 173 9.2 Discrete and Local Probability Networks in Molecular Bond Systems 174 9.3 Bond Descriptors of Molecular Communication Channels 177 9.4 Hartree–Fock Communications and Fermi Correlation 179 9.5 Communication Partitioning of Two-Electron Probabilities 181 9.6 Communications in Interacting Subsystems 184 9.7 Illustrative Application to Reaction HSAB Principle 188 9.8 Conclusion 191 References 192 10 Impacts of External Electric Fields on Aromaticity and Acidity for Benzoic Acid and Derivatives: Directionality, Additivity, and More 199 Meng Li, Xinjie Wan, Xin He, Chunying Rong, Dongbo Zhao, and Shubin Liu 10.1 Introduction 199 10.2 Methodology 199 10.3 Computational Details 202 10.4 Results and Discussion 203 10.5 Conclusions 213 Acknowledgments 213 References 213 11 A Divergence and Rotational Component in Chemical Potential During Reactions 217 Jean-Louis Vigneresse 11.1 Introduction 217 11.2 Chemical Descriptors 218 11.3 Charge and Energy Exchange 219 11.4 Fitness Landscape Diagrams 219 11.5 Chemical Reactions 220 11.6 Examining the Charge Exchange 221 11.6.1 Path pχη(ζ) and Charge Exchange 221 11.6.2 Systematic Changes Depending on the Starting Points on pχη(ζ) 223 11.6.3 Specific Solutions Using a pηω Path 224 11.7 Significance and Applications 225 11.8 Conclusions 227 Acknowledgments 227 References 228 12 Deep Learning of Electron Density for Predicting Energies: The Case of Boron Clusters 231 Pinaki Saha and Minh Tho Nguyen 12.1 Introduction 231 12.2 Deep Learning of Electron Density 233 12.3 Neural Networks for Neutral Boron Clusters 235 12.4 Concluding Remarks 242 Acknowledgements 243 References 243 13 Density-Based Description of Molecular Polarizability for Complex Systems 247 Dongbo Zhao, Xin He, Paul W. Ayers and Shubin Liu 13.1 Introduction 247 13.2 Methodology and Computations 248 13.2.1 Information-Theoretic Approach (ITA) Quantities 248 13.2.2 The GEBF Method 249 13.3 Results and Discussion 250 13.4 Conclusions and Perspectives 260 Acknowledgment 261 References 261 14 Conceptual Density Functional Theory-Based Study of Pure and TMs-Doped cdx (X = S, Se, Te; TMs = Cu, Ag, and Au) Nano Cluster for Water Splitting and Spintronic Applications 265 Prabhat Ranjan, Preeti Nanda, Ramon Carbó-Dorca, and Tanmoy Chakraborty 14.1 Introduction 265 14.2 Methodology 266 14.3 Results and Discussion 267 14.3.1 Electronic Properties and CDFT-Based Descriptors 267 14.4 Conclusion 275 Acknowledgments 275 Funding 276 References 276 15 “Phylogenetic” Screening of External Potential Related Response Functions 279 Paweł Szarek 15.1 Introduction 279 15.2 Alchemical Approach 281 15.3 The “Family Tree” 281 15.4 First-order Sensitivities 282 15.5 Second-Order Sensitivities 283 15.5.1 Electric Dipole Polarizability 283 15.5.2 “Polarizability Potential” – Local Polarization 284 15.6 Alchemical Hardness 285 15.6.1 Local Alchemical Hardness 287 15.7 Alchemical Characteristic Radius 289 15.8 Linear Response Function 291 15.9 Conclusions 292 References 293 16 On the Nature of Catastrophe Unfoldings Along the Diels–Alder Cycloaddition Pathway 299 Leandro Ayarde-Henríquez, Cristian Guerra, Mario Duque-Noreña, Patricia Pérez, Elizabeth Rincón and Eduardo Chamorro 16.1 Introduction 299 16.2 Molecular Symmetry and Elementary Catastrophe Unfoldings 301 16.2.1 The Case of Normal- and Inverse-Electron-Demand Diels–Alder Reactions 301 16.2.2 The C—C Bond Breaking in a High Symmetry Environment 304 16.2.3 The Photochemical Ring Opening of 1,3-Cyclohexadiene 305 16.3 Concluding Remarks 306 Acknowledgments 307 References 307 17 Designing Principles for Ultrashort H···H Nonbonded Contacts and Ultralong C—C Bonds 313 Nilangshu Mandal and Ayan Datta 17.1 Introduction 313 17.1.1 The Art of the Chemical Bond 314 17.1.2 Designing and Decoding Chemical Bond 314 17.2 Governing Factors for Ultrashort H···H Nonbonded Contacts 315 17.2.1 London Dispersion Interaction 316 17.2.2 Polarity and Charge Separation 317 17.2.3 Conformations and Orientations 317 17.2.4 Iron Maiden Effect 318 17.3 Elongation Strategies for C—C Bonds 319 17.3.1 Steric Crowding Effect 320 17.3.2 Core–Shell Strategy and Scissor Effect 321 17.3.3 Negative Hyperconjugation Effect 321 17.4 Concluding Remarks 323 Acknowledgments 324 References 324 18 Accurate Determination of Materials Properties: Role of Electron Density 329 Anup Pramanik, Sourav Ghoshal, Santu Biswas, Biplab Rajbanshi and Pranab Sarkar 18.1 Introduction 329 18.2 Materials Properties: Structure and Electronic Properties 330 18.2.1 Classification of Materials 330 18.2.2 Electronic Properties of Materials 332 18.3 Molecules to Materials, Essential Role of Electron Density 333 18.3.1 The Density Functional Theory (DFT) 334 18.3.2 The Hohenberg–Kohn Theorems 334 18.3.3 The Hohenberg–Kohn Variational Theorems 335 18.3.4 The Kohn–Sham (KS) Method 335 18.3.5 Local Density Approximation 337 18.3.6 Generalized Gradient Approximation 337 18.3.7 Meta-GGA and Hybrid Functionals 338 18.4 Further Approximations in DFT 339 18.4.1 The Density Functional Tight-Binding Theory 339 18.4.2 Self-Consistent-Charge Density-Functional Tight-Binding (SCC-DFTB) Method 340 18.5 Solar Cell Materials, Interfacial Charge Transfer Phenomena 340 18.5.1 The Time-Dependent Density Functional Theory 342 18.5.2 TDDFT and Linear Response 343 18.5.3 Excitation Energy and Excited State Properties 344 18.5.3.1 Exciton Binding Energy 346 18.5.3.2 Reorganization Energy 346 18.5.3.3 The Rates of Charge Transfer and Recombination Processes 347 18.6 Concluding Remarks 348 Acknowledgements 349 References 349 19 A Conceptual DFT Analysis of Mechanochemical Processes 355 Ruchi Jha, Shanti Gopal Patra, Debdutta Chakraborty, and Pratim Kumar Chattaraj 19.1 Introduction 355 19.2 Theoretical Background 356 19.2.1 The Constrained Geometries Simulate External Force (COGEF) 356 19.2.2 External Force is Explicitly Included (EFEI) 358 19.3 Results and Discussions 358 19.3.1 General Consideration 358 19.3.2 Constrained Geometries Simulate External Force (COGEF) 360 19.3.2.1 Mechanochemical CDFT Reactivity Descriptors and Their Application to Diatomic Molecules 362 19.3.3 Understanding Ball Milling Mechanochemical Processes with DFT Calculations and Microkinetic Modeling 365 19.3.4 Explicit Force 369 19.3.5 Dynamical Aspect of Mechanochemistry 369 19.4 Concluding Remarks 373 Acknowledgments 373 References 373 20 Molecular Electron Density and Electrostatic Potential and Their Applications 379 Shyam V.K. Panneer, Masiyappan Karuppusamy, Kanagasabai Balamurugan, Sathish K. Mudedla, Mahesh K. Ravva and Venkatesan Subramanian 20.1 Introduction 379 20.2 Topography Analysis of Scalar Fields 380 20.2.1 Molecular Electron Density 380 20.2.2 Topology of Molecular Electrostatic Potential 381 20.3 Usefulness of MESP and MED Analysis for Understanding Weak Interactions 382 20.3.1 MESP and MED Topography Analysis of Oligomers of Conjugated Polymers and their Interaction with PCBM Acceptors 382 20.3.2 Interaction of Small Molecules with Models of Single-Walled Carbon Nanotube and Graphene 386 20.3.2.1 Interaction of Nucleobases with Carbon Nanomaterials 386 20.3.2.2 Interaction of Chlorobenzene with Carbon Nanomaterials 392 20.3.2.3 Interaction of Carbohydrates with Carbon Nanomaterials 394 20.4 Conclusion 397 Acknowledgment 398 Conflict of Interest 398 References 398 21 Origin and Nature of Pancake Bonding Interactions: A Density Functional Theory and Information-Theoretic Approach Study 401 Dongbo Zhao, Xin He and Shubin Liu 21.1 Introduction 401 21.2 Methodology 402 21.2.1 Interaction Energy and Its Components in DFT 402 21.2.2 Information-Theoretic Approach Quantities 403 21.3 Computational Details 404 21.4 Results and Discussion 404 21.5 Concluding Remarks 410 Acknowledgment 411 References 411 22 Electron Spin Density and Magnetism in Organic Diradicals 415 Suranjan Shil, Debojit Bhattacharya and Anirban Misra 22.1 Introduction 415 22.2 Quantitative Relation Between Magnetic Exchange Coupling Constant and Spin Density 416 22.3 Spin Density Alternation 416 22.3.1 Phenyl Nitroxide 416 22.3.2 Methoxy Phenyl Nitroxide 417 22.3.3 Phenyl Nitroxide Coupled Through Methylene 417 22.3.4 Spin Density of Radical Systems 418 22.3.5 Distance Dependence of Spin Density 418 22.3.6 Geometry Dependence of Spin Density 423 22.3.7 Dependence on Connecting Atoms 423 22.4 Concluding Remarks 427 Acknowledgements 427 References 428 23 Stabilization of Boron and Carbon Clusters with Transition Metal Coordination – An Electron Density and DFT Study 431 Amol B. Rahane, Rudra Agarwal, Pinaki Saha, Nagamani Sukumar and Vijay Kumar 23.1 Introduction 431 23.2 Computational Details 434 23.3 Results and Discussion 435 23.3.1 Structures and Stability of Metal Atom Encapsulated Boron Clusters 435 23.3.2 Bonding Characteristics in M@B18, M@B20, M@B22, and M@B24 Clusters 440 23.3.3 Structures and Stability of Carbon Rings 447 23.3.4 Bonding Characteristics in Carbon Rings 450 23.4 Conclusions 457 Acknowledgments 458 References 458 24 DFT-Based Computational Approach for Structure and Design of Materials: The Unfinished Story 465 Ravi Kumar, Mayank Khera, Shivangi Garg, and Neetu Goel 24.1 Introduction 465 24.2 Different Frameworks of DFT 466 24.2.1 Kohn Sham Density Functional Theory (KS-DFT) 466 24.2.2 Time-Dependent Density Functional Theory (TD-DFT) 467 24.2.3 Linear Response Time-Dependent Density-Functional Theory (LR-TDDFT) 469 24.2.4 Discontinuous Galerkin Density Functional Theory (DGDFT) 469 24.3 DFT Implemented Computational Packages 470 24.4 DFT as Backbone of Electronic Structure Calculations 472 24.4.1 Design of 2D Nano-Materials 472 24.4.2 Non-covalent Interactions and Crystal Packing 476 24.4.3 Designing of Organic Solar Cell 477 24.5 Concluding Remarks 480 Acknowledgment 481 References 481 25 Structure, Stability and Bonding in Ligand Stabilized C 3 Species 491 Sudip Pan and Zhong-hua Cui 25.1 Introduction 491 25.2 Computational Details 492 25.3 Structures and Energetics 493 25.4 Bonding 495 25.5 Conclusions 500 Acknowledgements 501 References 501 26 The Role of Electronic Activity Toward the Analysis of Chemical Reactions 505 Swapan Sinha and Santanab Giri 26.1 Introduction 505 26.2 Theoretical Backgrounds and Computational Details 506 26.3 Results and Discussions 509 26.3.1 Bimolecular Nucleophilic Substitution (SN2) Reaction 509 26.3.2 Alkylation of Zintl Cluster 512 26.3.3 Proton Transfer Reaction 515 26.3.4 Water Activation by Frustrated Lewis Pairs (FLPs) 519 26.4 Concluding Remarks 522 Acknowledgments 522 References 522 27 Prediction of Radiative Efficiencies and Global Warming Potential of Hydrofluoroethers and Fluorinated Esters Using Various DFT Functionals 527 Kanika Guleria, Suresh Tiwari, Dali Barman, Snehasis Daschakraborty, and Ranga Subramanian 27.1 Introduction 527 27.2 Computational Methodology 528 27.3 RE and GWP Calculation Methodology 528 27.4 Results and Discussions 529 27.4.1 (Difluoromethoxy)trifluoromethane (CF3OCHF2) 529 27.4.2 Difluoro(methoxy)methane (CH3OCHF2) 529 27.4.3 Trifluoro(methoxy)methane (CF3OCH3) 531 27.4.4 Bis(2,2,2-trifluoroethyl)ether (CF3CH2OCH2CF3) 531 27.4.5 1,1,1,2,2-Pentafluoro-2-Methoxyethane (CF3CF2OCH3) 534 27.4.6 Fluoro(fluoromethoxy)methane (CH2FOCH2F) 537 27.4.7 Methyl 2,2,2-Difluoroacetate (CHF2C(O)OCH3) 537 27.4.8 Ethyl 2,2,2-Trifluoroacetate (CF3C(O)OCH2CH3) 537 27.4.9 2,2,2-Trifluoroethyl 2,2,2-trifluoroacetate (CF3C(O)OCH2CF3) 540 27.4.10 1,1-Difluoroethyl Carbonofluoridate (FC(O)OCF2CH3) 543 27.4.11 Methyl 2,2,2-trifluoroacetate (CF3C(O)OCH3) 543 27.5 Concluding Remarks 547 Acknowledgment 547 References 548 28 Density Functional Theory-Based Study on Some Natural Products 551 Abhishek Kumar, Ambrish K. Srivastava, Ratnesh Kumar, and Neeraj Misra 28.1 Introduction 551 28.2 Computational Details 552 28.3 Results and Discussion 552 28.3.1 Geometrical Properties 552 28.3.2 Vibrational Properties 553 28.3.2.1 O–H Vibration 555 28.3.2.2 C–H Vibration 555 28.3.2.3 C–C Vibration 555 28.3.2.4 C=O Vibration 555 28.3.3 HOMO–LUMO and MESP Plots 555 28.3.4 Chemical Reactivity 557 28.4 Conclusion 558 Acknowledgments 558 References 558 Index 561