Computational quantum chemistry:
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Boca Raton ; London ; New York
CRC Press, Taylor & Francis Group
[2021]
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| Ausgabe: | Second edition |
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| Beschreibung: | "First edition published by Foundation Books, an imprint of Cambridge University Press" |
| Beschreibung: | xxi, 692 Seiten Illustrationen, Diagramme |
| ISBN: | 9780367679699 |
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| 245 | 1 | 0 | |a Computational quantum chemistry |c Ram Yatan Prasad and Pranita |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton ; London ; New York |b CRC Press, Taylor & Francis Group |c [2021] | |
| 264 | 4 | |c © 2021 | |
| 300 | |a xxi, 692 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a "First edition published by Foundation Books, an imprint of Cambridge University Press" | ||
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| 650 | 0 | 7 | |a Quantenchemie |0 (DE-588)4047979-1 |2 gnd |9 rswk-swf |
| 653 | 0 | |a Quantum chemistry / Data processing | |
| 653 | 0 | |a Quantum chemistry / Mathematics / Data processing | |
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Datensatz im Suchindex
| _version_ | 1804182549551906816 |
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| adam_text | Contents Foreword................................................................................................................................................ xvii Preface.................................................................................................................................................... xix Authors..................................................................................................... xxi 1 Quantum Theory...............................................................................................................................1 1.1 Black-Body Radiation............................................................................................................... 1 1.2 Wien’s Radiation Law...............................................................................................................2 1.3 Rayleigh-Jeans Law................................................................................................................3 1.4 Planck’s Radiation Law...........................................................................................................4 1.5 Quantum Theory.......................................................................................................................7 1.6 Photoelectric Effect.................................................................................................................7 1.7 Compton Effect......................................................................................................................10 1.8 Atomic Hydrogen
Spectra......................................................................................................12 1.9 The Bohr Model.................................................................................. 14 1.9.1 Energy of an Electron Revolving around the Nucleus in a Permitted Orbit...........14 1.9.2 Velocity of an Electron............................................................................................16 1.9.3 Radius of the Orbit...................................................................................................16 1.9.4 Shortcoming of Bohr’s Model.................................................................................. 18 Bibliography...................................................................................................................................... 18 Solved Problems................................................................................................................................ 18 Questions on Concepts..................................................................................................................... 28 2 Wave-Particle Duality................................................................................................................... 31 2.1 Dual Nature of Electron/de Broglie Wave..............................................................................31 2.2 Davisson and Germer’s Experiment..................................................................................... 33 2.3 Quantisation of Angular
Momentum.................................................................................... 35 2.4 Heisenberg’s Uncertainty Principle................................................................. 36 2.5 Phase Velocity....................................................................................................................... 36 2.6 Group Velocity...................................................................................................................... 37 2.7 Uncertainty Relation between Energy and Time.................................................................. 37 2.8 Experimental Evidence of Heisenberg’s Uncertainty Principle............................................ 38 2.8.1 Diffraction of Electrons through a Slit................................................................... 38 2.8.2 Gamma Ray Microscope Thought Experiment...................................................... 39 2.8.3 Physical Significance of Uncertainty Principle......................................................40 Bibliography......................................................................................................................................41 Solved Problems................................................................................................................................41 Questions on Concepts..................................................................................................................... 46 3 Mathematical
Techniques............................................................................................................. 49 3.1 Differential Equations............................................................................................................49 3.1.1 Ordinary Differential Equation.............................................................................. 49 3.1.2 Partial Differential Equation.................................................................................. 49 3.1.3 Order and Degree of a Differential Equation......................................................... 50 3.1.3.1 Order....................................................................................................... 50 3.1.3.2 Degree..................................................................................................... 50 3.1.4 Linear and Non-Linear Differential Equation........................................................ 50 vii
viii Contents 3.1.5 3.1.6 3.1.7 General Solution, Particular Solution, and Arbitrary Constants............................ 51 3.1.5.1 General Solution........................................................................................51 3.1.5.2 Particular Solution.....................................................................................51 3.1.5.3 Arbitrary Constants...................................................................................51 Differential Equation of the First Order and the First Degree............................... 51 3.1.6.1 Worked Out Examples............................................................................... 51 Linear Differential Equation.................................................................................... 57 3.1.8 Equation of the Type — + Py = Qyn.........................................................................58 dx 3.1.9 Linear Differential Equation with Constant Coefficient/Second-Order Differential Equation with Constant Coefficient.....................................................60 3.1.10 Solving Differential Equations by Power Series.................................................... 64 3.2 Matrices.................................................................................................................................. 66 3.2.1 Types of Matrices......................................................................................................67 3.2.1.1 Rectangular Matrix................................................................................... 67 3.2.1.2 Square
Matrix............................................................................................ 67 3.2.1.3 Non-Singular and Singular Matrices.........................................................68 3.2.1.4 Unit Matrix................................................................................................ 68 3.2.1.5 Null Matrix or Zero Matrix..................................................................... 68 3.2.1.6 Row Matrix................................................................................................ 68 3.2.1.7 Column Matrix.......................................................................................... 68 3.2.1.8 Diagonal Matrix........................................................................................ 69 3.2.1.9 Scalar Matrix............................................................................................. 69 3.2.2 Operation of Matrices.............................................................................................. 69 3.2.2.1 Addition of Two Matrices.......................................................................... 69 3.2.2.2 Subtraction of Two Matrices................................................................... 70 3.2.2.3 Multiplication of Two Matrices............................................................... 70 3.2.3 Transpose of a Matrix.............................................................................................. 73 3.2.4 Symmetric Matrix.....................................................................................................73 3.2.5
Skew-Symmetric Matrix...........................................................................................74 3.2.6 Complex Matrix........................................................................................................75 3.2.7 Complex Conjugate of a Matrix............................................................................... 75 3.2.8 Hermitian Matrix..................................................................................................... 75 3.2.9 Skew-Hermitian Matrix........................................................................................... 76 3.2.10 Adjoint of a Matrix.................................................................................................... 76 3.2.11 Inverse of a Matrix................................ ;................................................................ 77 3.2.12 Orthogonal Matrices................................................................................................ 77 3.3 Determinants................................................................................... 78 3.3.1 Properties of Determinants...................................................................................... 78 3.3.2 Minors and Co-factors............................................................................................. 79 3.3.3 Uses of Determinants in Quantum Chemistry........................................................ 79 3.4 Characteristics Value Problem............................................................................................... 80 3.5 Similarity
Transformation......................................................................................................81 3.6 Block Diagonalisation of Matrices........................................................................................ 83 Bibliography................................................................................. 84 Solved Problems.................................................................................................................................84 Questions on Concepts.......................................................................................................................89 4 Quantum Mechanical Operators................................................................................................... 93 4.1 Linear Operator and Non-Linear Operator............................................................................94 4.2 Commutator............................................................................................................................95 4.2.1 Facts about Commutation.......................................................................................... 97 4.3 Hermitian Operator................................................................................................................ 97
Contents ІХ 4.3.1 4.4 4.5 4.6 4.7 Properties of Hermitian Operator................................................................................ 98 4.3.1.1 The Eigen Values of a Hermitian Operator Are Real............................ 98 4.3.1.2 Non-Degenerate Eigen Functions of a Hermitian Operator Form an Orthogonal Set....................................................................................... 99 4.3.1.3 If a Hermitian Operator A Commutes with an Arbitrary Operator B, and T i. and Ψ( Are Two Eigen Functions of A with NonDegenerate Eigen Values, then Bra-Ket Notation, Prove That Tkl ΒΙΨ։ = 0....................................... 100 4.3.1.4 If Two Hermitian Operators Ä and В Possess a Common Eigen Function, Then They Commute................................................................101 4.3.1.5 If Two Hermitian Operators  and В Commute, Then They Must Have a Common Eigen Function..............................................................101 Schmidt Orthogonalisation................................. 102 V and V2 Operators.................................................................................................................. 103 Linear Momentum Operator....................................................................................................103 4.6.1 Operators of Every Two Components of the Momentum Commute....................104 4.6.2 Momentum Components Commute with Unlike Co-Ordinates.............................104 4.6.3 Momentum Components Do Not Commute with Their Relative Co-
Ordinates............................................................................................... 105 Angular Momentum Operator or Angular Momentum Vector ^Lj..................................... 105 4.7.1 4.7.2 Operators of the Angular Momentum Components Do Not Commute.............. 106 Operators of the Angular Momentum Components Do Commute with the Operator of the Square of the Angular Momentum................................................107 4.7.3 Angular Momentum in Spherical Polar Co-Ordinates........................................... 109 4.7.4 Ladder Operators or Step-Up and Step-Down Operators for Angular Momentum................................................................................................... Ill 4.8 Hamiltonian Operator.............................................................................................................. 113 4.9 Commutation Relation of Angular Momentum Operators with Hamiltonian Operators and with Each Other...............................................................................................114 4.10 Projection Operators................................................................................................................ 115 4.11 Parity Operator (It operator).................................................................................................118 Bibliography..........................................................................................................................................119 Solved
Problems................................................................................................................................... 119 Questions on Concepts........................................................................................................................ 132 5 Postulates of Quantum Mechanics..................................................................................................135 5.1 Postulatei................................................................................................................................. 135 5.2 Postulate 2.................................................................................................................................136 5.2.1 Construction of Quantum Mechanical Operator.....................................................137 5.3 Postulate 3.................................................................................................................................138 5.4 Postulate 4................................................................................................................................. 139 5.5 Postulate5................................................................................................................................. 141 5.6 Postulate 6................................................................................................................................. 142 Bibliography...................................................................................................... 143 Solved Problems..........................................
143 Questions on Concepts.........................................................................................................................145 6 The Schrödinger Equation............................................................................................................... 147 6.1 Equation of Wave Motion.........................................................................................................147 6.1.1 Time-Independent Schrödinger Equation.................................................................149 6.1.2 Time-Dependent Schrödinger Equation...................................................................151 6.1.3 Interpretation of Wave Function, Ψ..........................................................................152
x Contents 6.1.4 Acceptable Wave Function......................................................................................153 Normalisation....................................................................................................................... 154 Orthogonality........................................................................................................................ 155 6.3.1 Orthonormality....................................................................................................... 156 6.3.2 Eigen Function and Eigen Value.............................................................................157 6.3.3 Degeneracy.............................................................................................................. 157 6.4 Transformation of the Laplacian intoSpherical Polar Co-Ordinates...................................157 6.5 Ehrenfest’s Theorem............................................................................................................. 160 6.6 Matrix Representation of Wave Function.............................................................................163 6.7 Matrix Representation of Operator.......................................................................................164 6.8 Properties of Matrix Elements..............................................................................................165 6.9 Matrix Form of the Schrôdinger Equation...........................................................................165 6.9.1 Time-Dependent Schrôdinger Equation in Matrix
Form....................................... 166 Bibliography......................................................................................................................................167 Solved Problems................................................................................................................................167 Questions on Concepts..................................................................................................................... 171 6.2 6.3 7 Playing with the Schrôdinger Equation....................................................................................... 173 7.1 Particle in a One-Dimensional Box....................................................................................... 173 7.1.1 Energy Level Diagram............................................................................................ 176 7.2 Particle in a Rectangular Three-Dimensional Box or Particle in a Three-Dimensional Box....................................................................................................... 177 7.2.1 Energy Levels for a Cubic Potential Box................................................................181 7.2.2 The Thnnel Effect or Thnnelling.............................................................................182 7.2.3 Importance of Thnnel Effect................................................................................... 187 7.2.4 Quantum Mechanical Explanation of Emission of a-Particles............................. 187 7.3 Particle on a
Ring.................................................................................................................. 188 7.3.1 Particle on a Ring (Considering the Spherical Polar Co-Ordinates)..................... 191 7.4 Particle on a Sphere.............................................................................................................. 196 7.4.1 The Legendre Polynomials...................................................................................... 198 7.4.1.7 Normalisation of the Legendre Polynomial.......................................... 200 7.4.1.2 Orthogonality of the Legendre Polynomials..........................................201 7.4.2 Associated Legendre Equation.............................................................................. 202 7.4.3 Associated Legendre Functions............................................................................. 203 7.4.4 Spherical Harmonics.............................................................................................. 204 7.4.5 Particle on a Sphere............................................................................................... 205 7.5 Rigid Rotors..........................................................................................................................208 7.5.1 F Equation............................................................................................................... 211 7.5.2 Г Equation...............................................................................................................212 7.5.3 Energy
Levels.......................................................................................................... 214 7.6 Hermite Polynomials............................................................................................................ 214 7.6.1 Orthogonal Properties of Hermite Polynomials.................................................... 216 7.7 Simple Harmonic Oscillator.................................................................................................218 7.7.1 Classical Treatment.................................................................................................218 7.7.2 Quantum Mechanical Treatment........................................................................... 220 7.7.2.7 Asymptotic Solution............................................................................... 220 7.7.2.2 Series Solution........................................................................................221 7.7.3 Wave Function of Linear Harmonic Oscillator.....................................................225 Bibliography.....................................................................................................................................228 Solved Problems...............................................................................................................................228 Questions on Concepts.....................................................................................................................236 Numerical
Problems.........................................................................................................................238
Contents xi 8 Hydrogen Atom...............................................................................................................................239 8.1 The Hydrogen Atom (Simple Solution of the Schrödinger Equation)............................... 239 8.2 Generalised Solution of the Schrödinger Equation for Hydrogen Atom/Hydrogen-Like Species............................................................................................. 242 8.3 Solution of the F Equation................................................................................................... 245 8.4 Solution of the T Equation or the Polar Wave Equation.....................................................247 8.5 The Laguerre Differential Equation....................................................................................249 8.5.1 Laguerre Polynomials............................................................................................ 251 8.5.2 The Rodrigues Formula for the Laguerre Polynomials.........................................251 8.5.3 The Laguerre Associated Equation and Its Solution.............................................253 8.5.4 Associated Laguerre Polynomials.........................................................................253 8.5.5 The Rodrigues Formula for the Associated Laguerre Polynomials..................... 255 8.6 Solution of the Radial Equation...........................................................................................256 8.6.1 Normalisation of the Radial Wave Function.........................................................261
8.6.2 Complete Wave Function for the H Atom..............................................................265 8.6.3 Hydrogenic Atomic Orbital...................................................................................267 8.6.4 Radial Wave Function............................................................................................ 274 8.7 Most Probable Distance of Electron from the Nucleus of H Atom................................... 276 8.7.1 Average Distance of Electron from the Nucleus of H Atom................................ 276 Bibliography.................................................................................................................................... 277 Solved Problems.............................................................................................................................. 277 Questions on Concepts.................................................................................................................... 287 9 Approximate Methods.................................................................................................................. 289 9.1 Perturbation Theory/Method for Nondegenerate States......................................................289 9.1.1 First-Order Perturbation........................................................................................290 9.1.1.1 Correction to Energy.............................................................................290 9.1.1.2 Correction to Wave Function.................................................................292 9.1.2 Second-Order
Perturbation.....................................................................................292 9.1.2.1 Correction to Energy.............................................................................292 9.1.2.2 Second-Order Correction to Wave Functions........................................................ 294 9.2 Bra-ket Notation or Dirac’s Notation..................................................................................295 9.2.1 Expression for First-Order Correction to Energy for Nondegenerate State Using Dirac’s Notation...........................................................................................297 9.2.2 First-Order Correction to Wave Function for Nondegenerate State Using Dirac’s Notation.....................................................................................................298 9.2.3 Second-Order Correction to the Energy Using Dirac’s Notation......................... 299 9.2.4 Alternatively: Second-Order Correction to the Energy Using Dirac’s Notation...........................................................................................300 9.2.5 Second-Order Correction to Wave Function Using Dirac’s Notation.................. 301 9.3 Perturbation Theory: A Degenerate Case............................................................................302 9.3.1 First-Order Correction to Energy...........................................................................302 9.3.2 First-Order Correction to Wave Function..............................................................304 9.3.3 Alternative Way to Handle Degenerate
Perturbation Theory: Twofold Degeneracy...............................................................................................304 9.4 Application of Perturbation Theory.....................................................................................307 9.4.1 Anharmonic Oscillator.......................................................................................... 307 9.4.2 Electronic Polarisability of Hydrogen Atom........................................................ 309 9.4.3 Helium Atom.......................................................................................................... 313 9.4.4 Alternatively: The Helium Atom........................................................................... 316 9.5 Variation Theorem/Method................................................................................................. 318 9.5.1 Variation Method.................................................................................................... 318 9.5.2 Variation Theorem................................................................................................. 319
Contents xii 9.5.3 Computation of Energy Eigen Value and Wave Function by Variation Method................................................................................................... 323 9.5.4 Computation of Wave Function............................................................................. 325 9.6 Application of Variation Principle/Method....................................................................... 326 9.6.1 Estimation of Energy of the Ground State of the Simple Harmonic Oscillator Using the Trial Function Ae “*.............................................................................. 326 9.6.2 Ground State of Helium Atom............................................................................... 330 9.6.3 Ground State of Hydrogen Atom..........................................................................331 Bibliography................................................................................................................................... 333 Solved Problems............................................................................................................................. 333 Based on Variation Theory............................................................................................................. 349 Questions on Concepts.................................................................................................................... 363 10 Diatomic Molecules........................................................................................................................367 10.1 Bom-Oppenhelmer
Approximation.................................................................................... 367 10.2 Hydrogen Molecule Ion......................................................................................................370 10.2.1 Evaluation of Overlap Integral..............................................................................379 10.2.2 Evaluation of the Coulomb Integral......................................................................380 10.2.3 Evaluation of Resonance Integral or Exchange Integral.....................................381 10.3 Evaluation of Ψ and Ψ2 (Probability)..................................................................................382 10.4 Hydrogen Molecule (Spin Independent)..............................................................................384 10.5 Linear Combination of Atomic Orbitals.............................................................................. 391 10.6 Molecular Orbital Theory.................................................................................................... 392 10.7 Valence Bond Treatment of H2 Molecule............................................................................ 395 10.8 Configuration Interaction.....................................................................................................400 10.9 Comparison of the Molecular Orbital and Valence Bond Theories................................... 402 10.10 Symmetric and Antisymmetric Wave Functions.................................................................403 10.11 Pauli’s Exclusion
Principle.................................................................................................. 405 10.12 Antisymmetric Wave Function and Slater Determinant....................................................406 10.13 Bonding and Antibonding Orbitals....................................................................................409 10.14 Electron Density in Molecular Hydrogen........................................................................... 410 10.15 Excited State of H2 Molecule...............................................................................................412 10.16 Electronic Transition in Hydrogen Molecule.......................................................................417 10.17 Homopolar Diatomic or Homonuclear Diatomic Molecules...............................................419 10.17.1 Molecules with s and p Valence Atomic Orbitals..................................................420 10.17.2 Electronic Configuration of Homonuclear Diatomic Molecules.......................... 423 10.18 Heteropolar Diatomic or Heteronuclear Diatomic Molecules.............................................423 Bibliography.................................................................................................................................... 424 Solved Problems.............................................................................................................................. 424 Questions on Concepts..... ............................................................................................................... 432
Numerical Problems........................................................................................................................ 434 11 Multielectronic Systems.............................................................................................................. 437 11.1 Energy of the Many-Electron System.................................................................................. 437 11.2 Fock Equation and Hartree Equation..................................................................................441 11.2.1 Application in Two-Electron Systems - For Getting Hartree Equation and Energy of Two-Electron System............................................................................442 11.3 Hartree and Hartree-Fock Self-Consistent Field Methods................................................ 444 11.4 Excited State of Helium.......................................................................................................447 11.5 Lithium in the Ground State................................................................................................ 452 11.6 Atomic Magnets and Magnetic Quantum Numbers............................................................456 11.6.1 Atomic Magnets..................................................................................................... 457
Contents xiii 11.6.2 Magnetic Quantum Number.................................................................................. 458 11.6.2.1 The Fourth Quantum Number..............................................................458 11.6.2.2 Electron Spin......................................................................................... 458 11.6.3 Atoms Having Two or More than Two Electrons..................................................459 11.7 The Gyromagnetic Ratio and the Landė Splitting Factor.................................................. 459 11.7.1 Landė ‘g’ Factor or Splitting Factor...................................................................... 460 11.7.2 Landė Interval Rule............................................................................................... 461 11.7.3 Zeeman Effect....................................................................................................... 462 11.7.3.1 Origin of the Zeeman Effect................................................................462 11.7.3.2 The Normal Zeeman Effect........ .......................................................... 463 11.7.3.3 The Anomalous/Complex Zeeman Effect............................................ 464 11.8 Stark Effect................... 465 11.9 Coupling of Orbital Angular Momentum............................................................................467 11.10 Coupling of Spin Momenta..................................................................................................469 11.11 Coupling of Orbital and Spin Angular
Momenta................................................................470 11.11.1 L-S or the Russell-Saunders Coupling Scheme................................................... 470 11.11.2 ji/ -Coupling Scheme...............................................................................................472 11.12 Multiplicity and Atomic States............................................................................................473 11.13 Hund’s Rule.......................................................................................................................... 474 11.14 Atomic Terms and Symbols................................................................................................. 474 11.14.1 Terms of Nonequivalent Electrons.........................................................................476 11.14.2 Terms of Equivalent Electrons...............................................................................477 11.14.3 Use of jj Coupling.................................................................................................. 478 11.15 Slater Rules.......................................................................................................................... 478 11.16 Slater-Type Orbitals............................................................................................................. 479 11.17 Gaussian-Ţype Orbitals........................................................................................................480 11.17.1 Gaussian Basis
Set.................................................................................................482 11.18 Condon-Slater Rules: Evaluation of Matrix Elements....................................................... 483 11.19 Koopman’s Theorem............................................................................................................486 11.20 Brillouin’s Theorem.............................................................................................................489 11.21 Roothaan’s Equations: The Matrix Solution of the Hartree-Fock Equation..................... 490 Bibliography.................................................................................................................................... 492 Solved Problems.............................................................................................................................. 493 Questions on Concepts....................................................................................................................504 12 Polyatomic Molecules....................................................................................................................507 12.1 Matrix Form of Roothaan’s Equations............................................................... 507 12.2 Fock Matrix Elements..........................................................................................................508 12.3 Roothaan’s Method in One Dimension................................................................................ 511 12.4 Electronic
Energy................................................................................................................ 518 12.5 Solution of Roothaan’s Equation for He Atom....................................................................519 12.6 Hybridisation........................................................................................................................ 528 12.6.1 ip3 Hybridisation....................................................................................................528 12.6.2 sp2 Hybridisation....................................................................................................536 12.6.3 sp Hybridisation.....................................................................................................538 12.6.4 Hybridisation in H20............................................................................................ 540 12.7 Semi-Empirical Methods.....................................................................................................543 12.7.1 Valence Electrons...................................................................................................543 12.7.2 Zero Differential Overlap..................................................................................... 544 12.7.3 π,-Electron Evaluation............................................................................................545 12.7.4 Invariance under Transformation..........................................................................547 12.7.5 Complete Neglect of Differential
Overlap............................................................ 548 12.7.6 Parametrisation...................................................................................................... 550
xiv Contents 12.7.7 Intermediate Neglect of Differential Overlap.........................................................552 12.7.8 Neglect of Diatomic Differential Overlap..............................................................553 12.7.9 The Pariser-Parr-Pople Method............................................................................554 12.7.9.1 Evaluation of Integrals of Pariser-Parr-Pople Method....................... 555 Bibliography......................................................................................................................................558 Solved Problems................................................................................................................................558 Questions on Concepts......................................................................................................................561 13 Hiickel Molecular Orbital Theory/Method.................................................................................563 13.1 Application of the Hiickel Molecular Orbital Method to π Systems..................................565 13.1.1 Ethylene................................................................................................................... 566 13.1.2 Determination of the Hiickel Molecular Orbital Coefficients and Molecular Orbitals of Ethylene................................................................................................568 13.1.2.1 Graphical Representation: Plots of ψχ and ψ2 vs Distance................... 569 13.1.2.2 Three-Dimensional
Representation....................................................... 570 13.1.3 Allyl System............................................................................................................ 570 13.1.4 Delocalisation Energy of Allyl System...................................................................572 13.1.5 Determination of the Hiickel Molecular Orbital Coefficients and Molecular Orbitals of Aliyi System......................................................................................... 573 13.1.5.1 Graphical Representation.......................................................................576 13.1.5.2 Three-Dimensional Representation: Plots of ψγ, ψ2, and ψ3 vs Directions............................................................................................... 576 13.1.6 Butadiene................................................................................................................ 577 13.1.7 Delocalisation Energy of Butadiene.......................................................................581 13.1.8 Hiickel Molecular Orbital Coefficients and Molecular Orbitals...........................581 13.1.8.1 Graphical Representation.......................................................................583 13.1.8.2 Three-Dimensional Representation....................................................... 583 13.2 Application of the Hiickel Method to Some Cyclic Polyenes.............................................583 13.2.1 Cyclopropenyl System.............................................................................................584
13.2.2 Delocalisation of Cyclopropenyl System................................................................586 13.2.3 Hiickel Molecular Orbital Coefficients and Molecular Orbitals...........................587 13.2.4 Cyclobutadiene........................................................................................................ 588 13.2.5 Delocalisation Energy of Cyclobutadiene.............................................................. 590 13.2.6 Hiickel Molecular Orbital Coefficient and Molecular Orbitals.............................590 13.2.7 Cyclopentadienyl System........................................................................................593 13.2.8 Delocalisation Energy of Cyclopentadienyl Systems............................................596 13.2.9 Hiickel Molecular Orbital Coefficient and Molecular Orbitals.............................597 13.2.10 Benzene................................................................................................................... 600 13.2.11 Delocalisation Energy of Benzene......................................................................... 603 13.2.12 Hiickel Molecular Orbital Coefficients and Molecular Orbitals.......................... 603 13.2.13 Graphical Representation of Molecular Orbitals in Benzene............................... 609 13.3 Electron Density................................................................................................................... 609 13.3.1
Ethylene....................................................................................................................610 13.3.2 Butadiene................................................................................................................ 610 13.3.3 Benzene....................................................................................................................611 13.4 Bond Order............................................................................................................................ 612 13.4.1 Ethylene................................................................................................................... 612 13.4.2 Butadiene................................................................................................................ 612 13.4.3 Benzene................................................................................................................... 613 13.5 Free Valence..........................................................................................................................614 13.5.1 Ethylene....................................................................................................................615 13.5.2 Butadiene................................................................................................................ 615 13.5.3 Benzene................................................................................................................... 615
Contents xv 13.6 Generalised Treatment of the Hiickel Molecular Orbital Theory to Open-Chain Conjugated System...........................................................................................615 13.6.1 Ethylene................................................................................................................... 618 13.6.2 Butadiene................................................................................................................618 13.7 Generalised Treatment of the Hiickel Molecular Orbital Theory to Cyclic Polyenes.........619 13.7.1 Cyclopropenyl Radical............................................................................................621 13.7.2 Cyclobutadiene....................................................................................................... 622 13.7.3 Cyclopentadienyl Radical...................................................................................... 622 13.7.4 Benzene.................................................................................................................. 623 13.8 Extended Hiickel Theory..................................................................................................... 624 13.8.1 Hetero Atom Substitutions..................................................................................... 625 13.8.2 General Improvement............................................................................................ 625 13.8.3 Extended Hiickel Theory Applied to Pyrrole........................................................626 13.8.4 Delocalisation Energy of
Pyrrole.......................................................................... 627 13.8.5 Hiickel Molecular Orbital Coefficients and Molecular Orbitals.......................... 628 13.8.6 Pyridine.................................................................................................................. 628 13.8.7 Hiickel Molecular Orbital Coefficients andMolecular Orbitals............................630 13.8.8 Electron Density......................................................................................................631 13.8.9 Bond Order................ . ........................................................................................... 631 13.8.10 HMO Treatment to Naphthalene............................................................................ 632 13.8.11 Hiickel Molecular Orbital Coefficients and Molecular Orbitals.......................... 634 References........................................................................................................................................635 Bibliography.....................................................................................................................................635 Solved Problems...............................................................................................................................635 Questions on Concepts.................................................................................................................... 643 14 Density Functional
Theory........................................................................................................... 647 14.1 Function................................................................................................................................647 14.2 Functional............................................................................................................................ 648 14.3 Hohenberg-Kohn Theorem................................................................................................. 649 14.3.1 Theorem 1.............................................................................................................. 649 14.3.2 Theorem 2.............................................................................................................. 651 14.3.3 Alternative Proof of Hohenberg-Kohn Theorems................................................652 14.3.3.1 Theorem 1.............................................................................................. 652 14.3.3.2 Theorem 2.............................................................................................. 653 14.4 Kohn-Sham Energy............................................................................................................. 654 14.5 Kohn-Sham Equations........................................................................................................ 656 14.5.1 Comments..............................................................................................................658 14.6 Local Density
Approximation............................................................................................. 658 14.6.1 Comments on LDA................................................................................................659 14.6.2 Application of the LDA.......................................................................................... 659 14.6.3 Electron Gas........................................................................................................... 659 14.6.4 The Local Spin Density Approximation.............................................. 660 14.6.5 Generalised Gradient Approximation or Gradient Correlated Functional........... 660 14.6.6 Meta-Generalised Gradient Approximation......................................................... 660 14.6.7 Hybrid Functionals................................................................................................. 661 14.6.8 Time-Dependent DFT............................................................................................662 14.6.9 Application of Density Functional Theory............................................................662 Bibliography.....................................................................................................................................662 Questions on Concepts.................................................................................................................... 663 Appendix 1..............................................................................................................................................665 Appendix
II.............................................................................................................................................669
XVI Contents Appendix ПІ................ 671 Model Question Papers 673 Glossary....................... 687 Index............................ 689
Computational Quantum Chemistry, Second Edition, is an extremely useful tool for teaching and research alike. It stipulates information in an accessible manner for scientific investigators, researchers and entrepreneurs. The book supplies an overview of the field and explains the fondamental underlying principles. It also gives the knowledge of numerous comparisons of different methods. The book consists of a wider range of applications in each chapter. It also provides a number of references which will be useful for academic and industrial researchers. It includes a large number of worked-out examples and unsolved problems for enhancing the computational skill of the users. FEATURES • Includes comprehensive coverage of most essential basic concepts • Achieves greater clarity with improved planning of topics and is reader-friendly • Deals with the mathematical techniques which will help readers to more efficient problem solving • Explains a structured approach for mathematical derivations • A reference book for academicians and scientific investigators Ram Yatan Prasad, PhD, DSc (India), DSc (he) Colombo, is a Professor of Chemistry and former Vice Chancellor of S.K.M University, Jharkhand, India. Pranita, PhD, DSc (he) Sri Lanka, F1CS, is an Assistant Professor of Chemistry at Vinoba Bhave University, India. CRC Press Taylor և Francis Group an informa business www.routledge.com CRC Press titles are available as eBook editions in a range of digital formats CHEMISTRY ISBN: 978-0-367-67969-9 9780367679699
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Contents Foreword. xvii Preface. xix Authors. xxi 1 Quantum Theory.1 1.1 Black-Body Radiation. 1 1.2 Wien’s Radiation Law.2 1.3 Rayleigh-Jeans Law.3 1.4 Planck’s Radiation Law.4 1.5 Quantum Theory.7 1.6 Photoelectric Effect.7 1.7 Compton Effect.10 1.8 Atomic Hydrogen
Spectra.12 1.9 The Bohr Model. 14 1.9.1 Energy of an Electron Revolving around the Nucleus in a Permitted Orbit.14 1.9.2 Velocity of an Electron.16 1.9.3 Radius of the Orbit.16 1.9.4 Shortcoming of Bohr’s Model. 18 Bibliography. 18 Solved Problems. 18 Questions on Concepts. 28 2 Wave-Particle Duality. 31 2.1 Dual Nature of Electron/de Broglie Wave.31 2.2 Davisson and Germer’s Experiment. 33 2.3 Quantisation of Angular
Momentum. 35 2.4 Heisenberg’s Uncertainty Principle. 36 2.5 Phase Velocity. 36 2.6 Group Velocity. 37 2.7 Uncertainty Relation between Energy and Time. 37 2.8 Experimental Evidence of Heisenberg’s Uncertainty Principle. 38 2.8.1 Diffraction of Electrons through a Slit. 38 2.8.2 Gamma Ray Microscope Thought Experiment. 39 2.8.3 Physical Significance of Uncertainty Principle.40 Bibliography.41 Solved Problems.41 Questions on Concepts. 46 3 Mathematical
Techniques. 49 3.1 Differential Equations.49 3.1.1 Ordinary Differential Equation. 49 3.1.2 Partial Differential Equation. 49 3.1.3 Order and Degree of a Differential Equation. 50 3.1.3.1 Order. 50 3.1.3.2 Degree. 50 3.1.4 Linear and Non-Linear Differential Equation. 50 vii
viii Contents 3.1.5 3.1.6 3.1.7 General Solution, Particular Solution, and Arbitrary Constants. 51 3.1.5.1 General Solution.51 3.1.5.2 Particular Solution.51 3.1.5.3 Arbitrary Constants.51 Differential Equation of the First Order and the First Degree. 51 3.1.6.1 Worked Out Examples. 51 Linear Differential Equation. 57 3.1.8 Equation of the Type — + Py = Qyn.58 dx 3.1.9 Linear Differential Equation with Constant Coefficient/Second-Order Differential Equation with Constant Coefficient.60 3.1.10 Solving Differential Equations by Power Series. 64 3.2 Matrices. 66 3.2.1 Types of Matrices.67 3.2.1.1 Rectangular Matrix. 67 3.2.1.2 Square
Matrix. 67 3.2.1.3 Non-Singular and Singular Matrices.68 3.2.1.4 Unit Matrix. 68 3.2.1.5 Null Matrix or Zero Matrix. 68 3.2.1.6 Row Matrix. 68 3.2.1.7 Column Matrix. 68 3.2.1.8 Diagonal Matrix. 69 3.2.1.9 Scalar Matrix. 69 3.2.2 Operation of Matrices. 69 3.2.2.1 Addition of Two Matrices. 69 3.2.2.2 Subtraction of Two Matrices. 70 3.2.2.3 Multiplication of Two Matrices. 70 3.2.3 Transpose of a Matrix. 73 3.2.4 Symmetric Matrix.73 3.2.5
Skew-Symmetric Matrix.74 3.2.6 Complex Matrix.75 3.2.7 Complex Conjugate of a Matrix. 75 3.2.8 Hermitian Matrix. 75 3.2.9 Skew-Hermitian Matrix. 76 3.2.10 Adjoint of a Matrix. 76 3.2.11 Inverse of a Matrix. ;. 77 3.2.12 Orthogonal Matrices. 77 3.3 Determinants. 78 3.3.1 Properties of Determinants. 78 3.3.2 Minors and Co-factors. 79 3.3.3 Uses of Determinants in Quantum Chemistry. 79 3.4 Characteristics Value Problem. 80 3.5 Similarity
Transformation.81 3.6 Block Diagonalisation of Matrices. 83 Bibliography. 84 Solved Problems.84 Questions on Concepts.89 4 Quantum Mechanical Operators. 93 4.1 Linear Operator and Non-Linear Operator.94 4.2 Commutator.95 4.2.1 Facts about Commutation. 97 4.3 Hermitian Operator. 97
Contents ІХ 4.3.1 4.4 4.5 4.6 4.7 Properties of Hermitian Operator. 98 4.3.1.1 The Eigen Values of a Hermitian Operator Are Real. 98 4.3.1.2 Non-Degenerate Eigen Functions of a Hermitian Operator Form an Orthogonal Set. 99 4.3.1.3 If a Hermitian Operator A Commutes with an Arbitrary Operator B, and T'i. and Ψ( Are Two Eigen Functions of A with NonDegenerate Eigen Values, then Bra-Ket Notation, Prove That Tkl ΒΙΨ։ = 0. 100 4.3.1.4 If Two Hermitian Operators Ä and В Possess a Common Eigen Function, Then They Commute.101 4.3.1.5 If Two Hermitian Operators  and В Commute, Then They Must Have a Common Eigen Function.101 Schmidt Orthogonalisation. 102 V and V2 Operators. 103 Linear Momentum Operator.103 4.6.1 Operators of Every Two Components of the Momentum Commute.104 4.6.2 Momentum Components Commute with Unlike Co-Ordinates.104 4.6.3 Momentum Components Do Not Commute with Their Relative Co-
Ordinates. 105 Angular Momentum Operator or Angular Momentum Vector ^Lj. 105 4.7.1 4.7.2 Operators of the Angular Momentum Components Do Not Commute. 106 Operators of the Angular Momentum Components Do Commute with the Operator of the Square of the Angular Momentum.107 4.7.3 Angular Momentum in Spherical Polar Co-Ordinates. 109 4.7.4 Ladder Operators or Step-Up and Step-Down Operators for Angular Momentum. Ill 4.8 Hamiltonian Operator. 113 4.9 Commutation Relation of Angular Momentum Operators with Hamiltonian Operators and with Each Other.114 4.10 Projection Operators. 115 4.11 Parity Operator (It operator).118 Bibliography.119 Solved
Problems. 119 Questions on Concepts. 132 5 Postulates of Quantum Mechanics.135 5.1 Postulatei. 135 5.2 Postulate 2.136 5.2.1 Construction of Quantum Mechanical Operator.137 5.3 Postulate 3.138 5.4 Postulate 4. 139 5.5 Postulate5. 141 5.6 Postulate 6. 142 Bibliography. 143 Solved Problems.
143 Questions on Concepts.145 6 The Schrödinger Equation. 147 6.1 Equation of Wave Motion.147 6.1.1 Time-Independent Schrödinger Equation.149 6.1.2 Time-Dependent Schrödinger Equation.151 6.1.3 Interpretation of Wave Function, Ψ.152
x Contents 6.1.4 Acceptable Wave Function.153 Normalisation. 154 Orthogonality. 155 6.3.1 Orthonormality. 156 6.3.2 Eigen Function and Eigen Value.157 6.3.3 Degeneracy. 157 6.4 Transformation of the Laplacian intoSpherical Polar Co-Ordinates.157 6.5 Ehrenfest’s Theorem. 160 6.6 Matrix Representation of Wave Function.163 6.7 Matrix Representation of Operator.164 6.8 Properties of Matrix Elements.165 6.9 Matrix Form of the Schrôdinger Equation.165 6.9.1 Time-Dependent Schrôdinger Equation in Matrix
Form. 166 Bibliography.167 Solved Problems.167 Questions on Concepts. 171 6.2 6.3 7 Playing with the Schrôdinger Equation. 173 7.1 Particle in a One-Dimensional Box. 173 7.1.1 Energy Level Diagram. 176 7.2 Particle in a Rectangular Three-Dimensional Box or Particle in a Three-Dimensional Box. 177 7.2.1 Energy Levels for a Cubic Potential Box.181 7.2.2 The Thnnel Effect or Thnnelling.182 7.2.3 Importance of Thnnel Effect. 187 7.2.4 Quantum Mechanical Explanation of Emission of a-Particles. 187 7.3 Particle on a
Ring. 188 7.3.1 Particle on a Ring (Considering the Spherical Polar Co-Ordinates). 191 7.4 Particle on a Sphere. 196 7.4.1 The Legendre Polynomials. 198 7.4.1.7 Normalisation of the Legendre Polynomial. 200 7.4.1.2 Orthogonality of the Legendre Polynomials.201 7.4.2 Associated Legendre Equation. 202 7.4.3 Associated Legendre Functions. 203 7.4.4 Spherical Harmonics. 204 7.4.5 Particle on a Sphere. 205 7.5 Rigid Rotors.208 7.5.1 F Equation. 211 7.5.2 Г Equation.212 7.5.3 Energy
Levels. 214 7.6 Hermite Polynomials. 214 7.6.1 Orthogonal Properties of Hermite Polynomials. 216 7.7 Simple Harmonic Oscillator.218 7.7.1 Classical Treatment.218 7.7.2 Quantum Mechanical Treatment. 220 7.7.2.7 Asymptotic Solution. 220 7.7.2.2 Series Solution.221 7.7.3 Wave Function of Linear Harmonic Oscillator.225 Bibliography.228 Solved Problems.228 Questions on Concepts.236 Numerical
Problems.238
Contents xi 8 Hydrogen Atom.239 8.1 The Hydrogen Atom (Simple Solution of the Schrödinger Equation). 239 8.2 Generalised Solution of the Schrödinger Equation for Hydrogen Atom/Hydrogen-Like Species. 242 8.3 Solution of the F Equation. 245 8.4 Solution of the T Equation or the Polar Wave Equation.247 8.5 The Laguerre Differential Equation.249 8.5.1 Laguerre Polynomials. 251 8.5.2 The Rodrigues Formula for the Laguerre Polynomials.251 8.5.3 The Laguerre Associated Equation and Its Solution.253 8.5.4 Associated Laguerre Polynomials.253 8.5.5 The Rodrigues Formula for the Associated Laguerre Polynomials. 255 8.6 Solution of the Radial Equation.256 8.6.1 Normalisation of the Radial Wave Function.261
8.6.2 Complete Wave Function for the H Atom.265 8.6.3 Hydrogenic Atomic Orbital.267 8.6.4 Radial Wave Function. 274 8.7 Most Probable Distance of Electron from the Nucleus of H Atom. 276 8.7.1 Average Distance of Electron from the Nucleus of H Atom. 276 Bibliography. 277 Solved Problems. 277 Questions on Concepts. 287 9 Approximate Methods. 289 9.1 Perturbation Theory/Method for Nondegenerate States.289 9.1.1 First-Order Perturbation.290 9.1.1.1 Correction to Energy.290 9.1.1.2 Correction to Wave Function.292 9.1.2 Second-Order
Perturbation.292 9.1.2.1 Correction to Energy.292 9.1.2.2 Second-Order Correction to Wave Functions. 294 9.2 Bra-ket Notation or Dirac’s Notation.295 9.2.1 Expression for First-Order Correction to Energy for Nondegenerate State Using Dirac’s Notation.297 9.2.2 First-Order Correction to Wave Function for Nondegenerate State Using Dirac’s Notation.298 9.2.3 Second-Order Correction to the Energy Using Dirac’s Notation. 299 9.2.4 Alternatively: Second-Order Correction to the Energy Using Dirac’s Notation.300 9.2.5 Second-Order Correction to Wave Function Using Dirac’s Notation. 301 9.3 Perturbation Theory: A Degenerate Case.302 9.3.1 First-Order Correction to Energy.302 9.3.2 First-Order Correction to Wave Function.304 9.3.3 Alternative Way to Handle Degenerate
Perturbation Theory: Twofold Degeneracy.304 9.4 Application of Perturbation Theory.307 9.4.1 Anharmonic Oscillator. 307 9.4.2 Electronic Polarisability of Hydrogen Atom. 309 9.4.3 Helium Atom. 313 9.4.4 Alternatively: The Helium Atom. 316 9.5 Variation Theorem/Method. 318 9.5.1 Variation Method. 318 9.5.2 Variation Theorem. 319
Contents xii 9.5.3 Computation of Energy Eigen Value and Wave Function by Variation Method. 323 9.5.4 Computation of Wave Function. 325 9.6 Application of Variation Principle/Method. 326 9.6.1 Estimation of Energy of the Ground State of the Simple Harmonic Oscillator Using the Trial Function Ae"“*. 326 9.6.2 Ground State of Helium Atom. 330 9.6.3 Ground State of Hydrogen Atom.331 Bibliography. 333 Solved Problems. 333 Based on Variation Theory. 349 Questions on Concepts. 363 10 Diatomic Molecules.367 10.1 Bom-Oppenhelmer
Approximation. 367 10.2 Hydrogen Molecule Ion.370 10.2.1 Evaluation of Overlap Integral.379 10.2.2 Evaluation of the Coulomb Integral.380 10.2.3 Evaluation of Resonance Integral or Exchange Integral.381 10.3 Evaluation of Ψ and Ψ2 (Probability).382 10.4 Hydrogen Molecule (Spin Independent).384 10.5 Linear Combination of Atomic Orbitals. 391 10.6 Molecular Orbital Theory. 392 10.7 Valence Bond Treatment of H2 Molecule. 395 10.8 Configuration Interaction.400 10.9 Comparison of the Molecular Orbital and Valence Bond Theories. 402 10.10 Symmetric and Antisymmetric Wave Functions.403 10.11 Pauli’s Exclusion
Principle. 405 10.12 Antisymmetric Wave Function and Slater Determinant.406 10.13 Bonding and Antibonding Orbitals.409 10.14 Electron Density in Molecular Hydrogen. 410 10.15 Excited State of H2 Molecule.412 10.16 Electronic Transition in Hydrogen Molecule.417 10.17 Homopolar Diatomic or Homonuclear Diatomic Molecules.419 10.17.1 Molecules with s and p Valence Atomic Orbitals.420 10.17.2 Electronic Configuration of Homonuclear Diatomic Molecules. 423 10.18 Heteropolar Diatomic or Heteronuclear Diatomic Molecules.423 Bibliography. 424 Solved Problems. 424 Questions on Concepts. . 432
Numerical Problems. 434 11 Multielectronic Systems. 437 11.1 Energy of the Many-Electron System. 437 11.2 Fock Equation and Hartree Equation.441 11.2.1 Application in Two-Electron Systems - For Getting Hartree Equation and Energy of Two-Electron System.442 11.3 Hartree and Hartree-Fock Self-Consistent Field Methods. 444 11.4 Excited State of Helium.447 11.5 Lithium in the Ground State. 452 11.6 Atomic Magnets and Magnetic Quantum Numbers.456 11.6.1 Atomic Magnets. 457
Contents xiii 11.6.2 Magnetic Quantum Number. 458 11.6.2.1 The Fourth Quantum Number.458 11.6.2.2 Electron Spin. 458 11.6.3 Atoms Having Two or More than Two Electrons.459 11.7 The Gyromagnetic Ratio and the Landė Splitting Factor. 459 11.7.1 Landė ‘g’ Factor or Splitting Factor. 460 11.7.2 Landė Interval Rule. 461 11.7.3 Zeeman Effect. 462 11.7.3.1 Origin of the Zeeman Effect.462 11.7.3.2 The Normal Zeeman Effect. . 463 11.7.3.3 The Anomalous/Complex Zeeman Effect. 464 11.8 Stark Effect. 465 11.9 Coupling of Orbital Angular Momentum.467 11.10 Coupling of Spin Momenta.469 11.11 Coupling of Orbital and Spin Angular
Momenta.470 11.11.1 L-S or the Russell-Saunders Coupling Scheme. 470 11.11.2 ji/'-Coupling Scheme.472 11.12 Multiplicity and Atomic States.473 11.13 Hund’s Rule. 474 11.14 Atomic Terms and Symbols. 474 11.14.1 Terms of Nonequivalent Electrons.476 11.14.2 Terms of Equivalent Electrons.477 11.14.3 Use of jj Coupling. 478 11.15 Slater Rules. 478 11.16 Slater-Type Orbitals. 479 11.17 Gaussian-Ţype Orbitals.480 11.17.1 Gaussian Basis
Set.482 11.18 Condon-Slater Rules: Evaluation of Matrix Elements. 483 11.19 Koopman’s Theorem.486 11.20 Brillouin’s Theorem.489 11.21 Roothaan’s Equations: The Matrix Solution of the Hartree-Fock Equation. 490 Bibliography. 492 Solved Problems. 493 Questions on Concepts.504 12 Polyatomic Molecules.507 12.1 Matrix Form of Roothaan’s Equations. 507 12.2 Fock Matrix Elements.508 12.3 Roothaan’s Method in One Dimension. 511 12.4 Electronic
Energy. 518 12.5 Solution of Roothaan’s Equation for He Atom.519 12.6 Hybridisation. 528 12.6.1 ip3 Hybridisation.528 12.6.2 sp2 Hybridisation.536 12.6.3 sp Hybridisation.538 12.6.4 Hybridisation in H20. 540 12.7 Semi-Empirical Methods.543 12.7.1 Valence Electrons.543 12.7.2 Zero Differential Overlap. 544 12.7.3 π,-Electron Evaluation.545 12.7.4 Invariance under Transformation.547 12.7.5 Complete Neglect of Differential
Overlap. 548 12.7.6 Parametrisation. 550
xiv Contents 12.7.7 Intermediate Neglect of Differential Overlap.552 12.7.8 Neglect of Diatomic Differential Overlap.553 12.7.9 The Pariser-Parr-Pople Method.554 12.7.9.1 Evaluation of Integrals of Pariser-Parr-Pople Method. 555 Bibliography.558 Solved Problems.558 Questions on Concepts.561 13 Hiickel Molecular Orbital Theory/Method.563 13.1 Application of the Hiickel Molecular Orbital Method to π Systems.565 13.1.1 Ethylene. 566 13.1.2 Determination of the Hiickel Molecular Orbital Coefficients and Molecular Orbitals of Ethylene.568 13.1.2.1 Graphical Representation: Plots of ψχ and ψ2 vs Distance. 569 13.1.2.2 Three-Dimensional
Representation. 570 13.1.3 Allyl System. 570 13.1.4 Delocalisation Energy of Allyl System.572 13.1.5 Determination of the Hiickel Molecular Orbital Coefficients and Molecular Orbitals of Aliyi System. 573 13.1.5.1 Graphical Representation.576 13.1.5.2 Three-Dimensional Representation: Plots of ψγ, ψ2, and ψ3 vs Directions. 576 13.1.6 Butadiene. 577 13.1.7 Delocalisation Energy of Butadiene.581 13.1.8 Hiickel Molecular Orbital Coefficients and Molecular Orbitals.581 13.1.8.1 Graphical Representation.583 13.1.8.2 Three-Dimensional Representation. 583 13.2 Application of the Hiickel Method to Some Cyclic Polyenes.583 13.2.1 Cyclopropenyl System.584
13.2.2 Delocalisation of Cyclopropenyl System.586 13.2.3 Hiickel Molecular Orbital Coefficients and Molecular Orbitals.587 13.2.4 Cyclobutadiene. 588 13.2.5 Delocalisation Energy of Cyclobutadiene. 590 13.2.6 Hiickel Molecular Orbital Coefficient and Molecular Orbitals.590 13.2.7 Cyclopentadienyl System.593 13.2.8 Delocalisation Energy of Cyclopentadienyl Systems.596 13.2.9 Hiickel Molecular Orbital Coefficient and Molecular Orbitals.597 13.2.10 Benzene. 600 13.2.11 Delocalisation Energy of Benzene. 603 13.2.12 Hiickel Molecular Orbital Coefficients and Molecular Orbitals. 603 13.2.13 Graphical Representation of Molecular Orbitals in Benzene. 609 13.3 Electron Density. 609 13.3.1
Ethylene.610 13.3.2 Butadiene. 610 13.3.3 Benzene.611 13.4 Bond Order. 612 13.4.1 Ethylene. 612 13.4.2 Butadiene. 612 13.4.3 Benzene. 613 13.5 Free Valence.614 13.5.1 Ethylene.615 13.5.2 Butadiene. 615 13.5.3 Benzene. 615
Contents xv 13.6 Generalised Treatment of the Hiickel Molecular Orbital Theory to Open-Chain Conjugated System.615 13.6.1 Ethylene. 618 13.6.2 Butadiene.618 13.7 Generalised Treatment of the Hiickel Molecular Orbital Theory to Cyclic Polyenes.619 13.7.1 Cyclopropenyl Radical.621 13.7.2 Cyclobutadiene. 622 13.7.3 Cyclopentadienyl Radical. 622 13.7.4 Benzene. 623 13.8 Extended Hiickel Theory. 624 13.8.1 Hetero Atom Substitutions. 625 13.8.2 General Improvement. 625 13.8.3 Extended Hiickel Theory Applied to Pyrrole.626 13.8.4 Delocalisation Energy of
Pyrrole. 627 13.8.5 Hiickel Molecular Orbital Coefficients and Molecular Orbitals. 628 13.8.6 Pyridine. 628 13.8.7 Hiickel Molecular Orbital Coefficients andMolecular Orbitals.630 13.8.8 Electron Density.631 13.8.9 Bond Order. .'. 631 13.8.10 HMO Treatment to Naphthalene. 632 13.8.11 Hiickel Molecular Orbital Coefficients and Molecular Orbitals. 634 References.635 Bibliography.635 Solved Problems.635 Questions on Concepts. 643 14 Density Functional
Theory. 647 14.1 Function.647 14.2 Functional. 648 14.3 Hohenberg-Kohn Theorem. 649 14.3.1 Theorem 1. 649 14.3.2 Theorem 2. 651 14.3.3 Alternative Proof of Hohenberg-Kohn Theorems.652 14.3.3.1 Theorem 1. 652 14.3.3.2 Theorem 2. 653 14.4 Kohn-Sham Energy. 654 14.5 Kohn-Sham Equations. 656 14.5.1 Comments.658 14.6 Local Density
Approximation. 658 14.6.1 Comments on LDA.659 14.6.2 Application of the LDA. 659 14.6.3 Electron Gas. 659 14.6.4 The Local Spin Density Approximation. 660 14.6.5 Generalised Gradient Approximation or Gradient Correlated Functional. 660 14.6.6 Meta-Generalised Gradient Approximation. 660 14.6.7 Hybrid Functionals. 661 14.6.8 Time-Dependent DFT.662 14.6.9 Application of Density Functional Theory.662 Bibliography.662 Questions on Concepts. 663 Appendix 1.665 Appendix
II.669
XVI Contents Appendix ПІ. 671 Model Question Papers 673 Glossary. 687 Index. 689
Computational Quantum Chemistry, Second Edition, is an extremely useful tool for teaching and research alike. It stipulates information in an accessible manner for scientific investigators, researchers and entrepreneurs. The book supplies an overview of the field and explains the fondamental underlying principles. It also gives the knowledge of numerous comparisons of different methods. The book consists of a wider range of applications in each chapter. It also provides a number of references which will be useful for academic and industrial researchers. It includes a large number of worked-out examples and unsolved problems for enhancing the computational skill of the users. FEATURES • Includes comprehensive coverage of most essential basic concepts • Achieves greater clarity with improved planning of topics and is reader-friendly • Deals with the mathematical techniques which will help readers to more efficient problem solving • Explains a structured approach for mathematical derivations • A reference book for academicians and scientific investigators Ram Yatan Prasad, PhD, DSc (India), DSc (he) Colombo, is a Professor of Chemistry and former Vice Chancellor of S.K.M University, Jharkhand, India. Pranita, PhD, DSc (he) Sri Lanka, F1CS, is an Assistant Professor of Chemistry at Vinoba Bhave University, India. CRC Press Taylor և Francis Group an informa business www.routledge.com CRC Press titles are available as eBook editions in a range of digital formats CHEMISTRY ISBN: 978-0-367-67969-9 9780367679699 |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV047336265 |
| illustrated | Illustrated |
| index_date | 2024-07-03T17:32:46Z |
| indexdate | 2024-07-10T09:09:19Z |
| institution | BVB |
| isbn | 9780367679699 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032738774 |
| oclc_num | 1236096297 |
| open_access_boolean | |
| owner | DE-11 DE-703 DE-19 DE-BY-UBM |
| owner_facet | DE-11 DE-703 DE-19 DE-BY-UBM |
| physical | xxi, 692 Seiten Illustrationen, Diagramme |
| publishDate | 2021 |
| publishDateSearch | 2021 |
| publishDateSort | 2021 |
| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| spelling | Prasad, Ram Yatan ca. 20./21. Jh. Verfasser (DE-588)123727396X aut Computational quantum chemistry Ram Yatan Prasad and Pranita Second edition Boca Raton ; London ; New York CRC Press, Taylor & Francis Group [2021] © 2021 xxi, 692 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier "First edition published by Foundation Books, an imprint of Cambridge University Press" Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Quantenchemie (DE-588)4047979-1 gnd rswk-swf Quantum chemistry / Data processing Quantum chemistry / Mathematics / Data processing (DE-588)4123623-3 Lehrbuch gnd-content Quantenmechanik (DE-588)4047989-4 s DE-604 Quantenchemie (DE-588)4047979-1 s Pranita ca. 20./21. Jh. Verfasser (DE-588)1237273668 aut Erscheint auch als Online-Ausgabe 978-1-003-13360-5 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032738774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032738774&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Prasad, Ram Yatan ca. 20./21. Jh Pranita ca. 20./21. Jh Computational quantum chemistry Quantenmechanik (DE-588)4047989-4 gnd Quantenchemie (DE-588)4047979-1 gnd |
| subject_GND | (DE-588)4047989-4 (DE-588)4047979-1 (DE-588)4123623-3 |
| title | Computational quantum chemistry |
| title_auth | Computational quantum chemistry |
| title_exact_search | Computational quantum chemistry |
| title_exact_search_txtP | Computational quantum chemistry |
| title_full | Computational quantum chemistry Ram Yatan Prasad and Pranita |
| title_fullStr | Computational quantum chemistry Ram Yatan Prasad and Pranita |
| title_full_unstemmed | Computational quantum chemistry Ram Yatan Prasad and Pranita |
| title_short | Computational quantum chemistry |
| title_sort | computational quantum chemistry |
| topic | Quantenmechanik (DE-588)4047989-4 gnd Quantenchemie (DE-588)4047979-1 gnd |
| topic_facet | Quantenmechanik Quantenchemie Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032738774&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032738774&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT prasadramyatan computationalquantumchemistry AT pranita computationalquantumchemistry |