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
