Ideas of Quantum Chemistry

Ideas of Quantum Chemistry

by Lucjan Piela

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Overview

Ideas of Quantum Chemistry shows how quantum mechanics is applied to chemistry to give it a theoretical foundation. The structure of the book (a TREE-form) emphasizes the logical relationships between various topics, facts and methods. It shows the reader which parts of the text are needed for understanding specific aspects of the subject matter. Interspersed throughout the text are short biographies of key scientists and their contributions to the development of the field.

Ideas of quantum Chemistry has both textbook and reference work aspects. Like a textbook, the material is organized into digestable sections with each chapter following the same structure. It answers frequently asked questions and highlights the most impotant conclusions and the essential mathematical formulae in the text. In its reference aspects, it has a broader range than traditional quantum chemistry books and reviews virtually all of the pertinent literature. It is useful both for beginners as well as specialists in advanced topics of quantum chemistry. The book is supplemented by an appendix on the Internet at http://www.chem.uw.edu.pl/ideas/. Key features: Presents the widest range of quantum chemical problems covered in one book, Unique structure allows material to be tailored to the specific needs of the reader, Informal language facilitates the understanding of difficult topics.

Product Details

ISBN-13: 9780444594570
Publisher: Elsevier Science
Publication date: 11/21/2013
Sold by: Barnes & Noble
Format: NOOK Book
Pages: 1078
File size: 36 MB
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About the Author

Professor Piela received his bachelor degree in 1960 from the histroric Konarski College in his home town of Rzeszow, Poland. In 1965, he graduated with a Masters of Science from the University of Warsaw and, after obtaining his Ph.D. from the same university 5 years later, went on to became a professor in 1976. In addition to his work in Warsaw, he has carried out research in the Centre Européen de Calcul Atomique et Moléculaire (France), Facultés Universitaires de Namur (Belgium) and Cornell University (USA). In addition to being the author of about hundred papers published in international journals, Professor Piela is an elected member of the Academie Royale des Sciences, Lettres et Beaux-Arts de Belgique, and a member of the European Academy of Sciences.

Table of Contents


Introduction     XXI
The Magic of Quantum Mechanics     1
History of a revolution     4
Postulates     15
The Heisenberg uncertainty principle     34
The Copenhagen interpretation     37
How to disprove the Heisenberg principle? The Einstein-Podolsky-Rosen recipe     38
Is the world real?     40
Bilocation     40
The Bell inequality will decide     43
Intriguing results of experiments with photons     46
Teleportation     47
Quantum computing     49
The Schrodinger Equation     55
Symmetry of the Hamiltonian and its consequences     57
The non-relativistic Hamiltonian and conservation laws     57
Invariance with respect to translation     61
Invariance with respect to rotation     63
Invariance with respect to permutation of identical particles (fermions and bosons)     64
Invariance of the total charge     64
Fundamental and less fundamental invariances     65
Invariance with respect to inversion - parity     65
Invariance with respect to charge conjugation     68
Invariance with respect to the symmetry of the nuclear framework     68
Conservation of total spin     69
Indices of spectroscopic states     69
Schrodinger equation for stationary states     70
Wave functions of class Q     73
Boundary conditions     73
An analogy     75
Mathematical and physical solutions     76
The time-dependent Schrodinger equation     76
Evolution in time     7
Normalization is preserved     78
The mean value of the Hamiltonian is preserved     78
Linearity     79
Evolution after switching a perturbation     79
The two-state model     81
First-order perturbation theory     82
Time-independent perturbation and the Fermi golden rule     83
The most important case: periodic perturbation     84
Beyond the Schrodinger Equation     90
A glimpse of classical relativity theory     93
The vanishing of apparent forces     93
The Galilean transformation     96
The Michelson-Morley experiment     96
The Galilean transformation crashes     98
The Lorentz transformation     100
New law of adding velocities     102
The Minkowski space-time continuum      104
How do we get E = mc[superscript 2]?     106
Reconciling relativity and quantum mechanics     109
The Dirac equation     111
The Dirac electronic sea     111
The Dirac equations for electron and positron     115
Spinors and bispinors     115
What next?     117
Large and small components of the bispinor     117
How to avoid drowning in the Dirac sea     118
From Dirac to Schrodinger - how to derive the non-relativistic Hamiltonian?     119
How does the spin appear?     120
Simple questions     122
The hydrogen-like atom in Dirac theory     123
Step by step: calculation of the ground state of the hydrogen-like atom within Dirac theory     123
Relativistic contraction of orbitals     128
Larger systems     129
Beyond the Dirac equation     130
The Breit equation     130
A few words about quantum electrodynamics (QED)     132
Exact Solutions - Our Beacons     142
Free particle     144
Particle in a box     145
Box with ends     145
Cyclic box     149
Comparison of two boxes: hexatriene and benzene      152
Tunnelling effect     153
A single barrier     153
The magic of two barriers     158
The harmonic oscillator     164
Morse oscillator     169
Morse potential     169
Solution     170
Comparison with the harmonic oscillator     172
The isotope effect     172
Bond weakening effect     174
Examples     174
Rigid rotator     176
Hydrogen-like atom     178
Harmonic helium atom (harmonium)     185
What do all these solutions have in common?     188
Beacons and pearls of physics     189
Two Fundamental Approximate Methods     195
Variational method     196
Variational principle     196
Variational parameters     200
Ritz Method     202
Perturbational method     203
Rayleigh-Schrodinger approach     203
Hylleraas variational principle     208
Hylleraas equation     209
Convergence of the perturbational series     210
Separation of Electronic and Nuclear Motions     217
Separation of the centre-of-mass motion      221
Space-fixed coordinate system (SFCS)     221
New coordinates     221
Hamiltonian in the new coordinates     222
After separation of the centre-of-mass motion     224
Exact (non-adiabatic) theory     224
Adiabatic approximation     227
Born-Oppenheimer approximation     229
Oscillations of a rotating molecule     229
One more analogy     232
The fundamental character of the adiabatic approximation - PES     233
Basic principles of electronic, vibrational and rotational spectroscopy     235
Vibrational structure     235
Rotational structure     236
Approximate separation of rotations and vibrations     238
Polyatomic molecule     241
Kinetic energy expression     241
Simplifying using Eckart conditions     243
Approximation: decoupling of rotation and vibrations     244
The kinetic energy operators of translation, rotation and vibrations     245
Separation of translational, rotational and vibrational motions     246
Non-bound states     247
Adiabatic, diabatic and non-adiabatic approaches     252
Crossing of potential energy curves for diatomics     255
The non-crossing rule     255
Simulating the harpooning effect in the NaCl molecule     257
Polyatomic molecules and conical intersection     260
Conical intersection     262
Berry phase     264
Beyond the adiabatic approximation     268
Muon catalyzed nuclear fusion     268
"Russian dolls" - or a molecule within molecule     270
Motion of Nuclei     275
Rovibrational spectra - an example of accurate calculations: atom - diatomic molecule     278
Coordinate system and Hamiltonian     279
Anisotropy of the potential V     280
Adding the angular momenta in quantum mechanics     281
Application of the Ritz method     282
Calculation of rovibrational spectra     283
Force fields (FF)     284
Local Molecular Mechanics (MM)     290
Bonds that cannot break     290
Bonds that can break     291
Global molecular mechanics     292
Multiple minima catastrophe     292
Is it the global minimum which counts?     293
Small amplitude harmonic motion - normal modes     294
Theory of normal modes     295
Zero-vibration energy      303
Molecular Dynamics (MD)     304
The MD idea     304
What does MD offer us?     306
What to worry about?     307
MD of non-equilibrium processes     308
Quantum-classical MD     308
Simulated annealing     309
Langevin Dynamics     310
Monte Carlo Dynamics     311
Car-Parrinello dynamics     314
Cellular automata     317
Electronic Motion in the Mean Field: Atoms and Molecules     324
Hartree-Fock method - a bird's eye view     329
Spinorbitals     329
Variables     330
Slater determinants     332
What is the Hartree-Fock method all about?     333
The Fock equation for optimal spinorbitals     334
Dirac and Coulomb notations     334
Energy functional     334
The search for the conditional extremum     335
A Slater determinant and a unitary transformation     338
Invariance of the J and K operators     339
Diagonalization of the Lagrange multipliers matrix     340
The Fock equation for optimal spinorbitals (General Hartree-Fock method - GHF)     341
The closed-shell systems and the Restricted Hartree-Fock (RHF) method     342
Iterative procedure for computing molecular orbitals: the Self-Consistent Field method     350
Total energy in the Hartree-Fock method     351
Computational technique: atomic orbitals as building blocks of the molecular wave function     354
Centring of the atomic orbital     354
Slater-type orbitals (STO)     355
Gaussian-type orbitals (GTO)     357
Linear Combination of Atomic Orbitals (LCAO) Method     360
Basis sets of Atomic Orbitals     363
The Hartree-Fock-Roothaan method (SCF LCAO MO)     364
Practical problems in the SCF LCAO MO method     366
Results of the Hartree-Fock Method     369
Back to foundations     369
When does the RHF method fail?     369
Fukutome classes     372
Mendeleev Periodic Table of Chemical Elements     379
Similar to the hydrogen atom - the orbital model of atom     379
Yet there are differences     380
The nature of the chemical bond     383
H[Characters not reproducible] in the MO picture     384
Can we see a chemical bond?     388
Excitation energy, ionization potential, and electron affinity (RHF approach)     389
Approximate energies of electronic states     389
Singlet or triplet excitation?     391
Hund's rule     392
Ionization potential and electron affinity (Koopmans rule)     393
Localization of molecular orbitals within the RHF method     396
The external localization methods     397
The internal localization methods     398
Examples of localization     400
Computational technique     401
The [sigma], [pi], [delta] bonds     403
Electron pair dimensions and the foundations of chemistry     404
Hybridization     407
A minimal model of a molecule     417
Valence Shell Electron Pair Repulsion (VSEPR)     419
Electronic Motion in the Mean Field: Periodic Systems     428
Primitive lattice     431
Wave vector     433
Inverse lattice     436
First Brillouin Zone (FBZ)     438
Properties of the FBZ     438
A few words on Bloch functions     439
Waves in 1D     439
Waves in 2D     442
The infinite crystal as a limit of a cyclic system     445
A triple role of the wave vector     448
Band structure     449
Born-von Karman boundary condition in 3D     449
Crystal orbitals from Bloch functions (LCAO CO method)     450
SCF LCAO CO equations     452
Band structure and band width     453
Fermi level and energy gap: insulators, semiconductors and metals     454
Solid state quantum chemistry     460
Why do some bands go up?     460
Why do some bands go down?     462
Why do some bands stay constant?     462
How can more complex behaviour be explained?     462
The Hartree-Fock method for crystals     468
Secular equation     468
Integration in the FBZ     471
Fock matrix elements     472
Iterative procedure     474
Total energy     474
Long-range interaction problem     475
Fock matrix corrections     476
Total energy corrections     477
Multipole expansion applied to the Fock matrix     479
Multipole expansion applied to the total energy     483
Back to the exchange term     485
Choice of unit cell     488
Field compensation method     490
The symmetry of subsystem choice     492
Correlation of the Electronic Motions      498
Variational Methods Using Explicitly Correlated Wave Function     502
Correlation cusp condition     503
The Hylleraas function     506
The Hylleraas CI method     506
The harmonic helium atom     507
James-Coolidge and Kolos-Wolniewicz functions     508
Neutrino mass     511
Method of exponentially correlated Gaussian functions     513
Coulomb hole ("correlation hole")     513
Exchange hole ("Fermi hole")     516
Variational Methods with Slater Determinants     520
Valence bond (VB) method     520
Resonance theory - hydrogen molecule     520
Resonance theory - polyatomic case     523
Configuration interaction (CI) method     525
Brillouin theorem     527
Convergence of the CI expansion     527
Example of H[subscript 2]O     528
Which excitations are most important?     529
Natural orbitals (NO)     531
Size consistency     532
Direct CI method     533
Multireference CI method     533
Multiconfigurational Self-Consistent Field method (MC SCF)     535
Classical MC SCF approach      535
Unitary MC SCF method     536
Complete active space method (CAS SCF)     538
Non-Variational Methods with Slater Determinants     539
Coupled cluster (CC) method     539
Wave and cluster operators     540
Relationship between CI and CC methods     542
Solution of the CC equations     543
Example: CC with double excitations     545
Size consistency of the CC method     547
Equation-of-motion method (EOM-CC)     548
Similarity transformation     548
Derivation of the EOM-CC equations     549
Many body perturbation theory (MBPT)     551
Unperturbed Hamiltonian     551
Perturbation theory - slightly different approach     552
Reduced resolvent or the "almost" inverse of (E[Characters not reproducible] - H[superscript 0])     553
MBPT machinery     555
Brillouin-Wigner perturbation theory     556
Rayleigh-Schrodinger perturbation theory     557
Moller-Plesset version of Rayleigh-Schrodinger perturbation theory     558
Expression for MP2 energy     558
Convergence of the Moller-Plesset perturbation series     559
Special status of double excitations      560
Electronic Motion: Density Functional Theory (DFT)     567
Electronic density - the superstar     569
Bader analysis     571
Overall shape of p     571
Critical points     571
Laplacian of the electronic density as a "magnifying glass"     575
Two important Hohenberg-Kohn theorems     579
Equivalence of the electronic wave function and electron density     579
Existence of an energy functional minimized by p[subscript 0]     580
The Kohn-Sham equations     584
The Kohn-Sham system of non-interacting electrons     584
Total energy expression     585
Derivation of the Kohn-Sham equations     586
What to take as the DFT exchange-correlation energy E[subscript xc]?     590
Local density approximation (LDA)     590
Non-local approximations (NLDA)     591
The approximate character of the DFT vs apparent rigour of ab initio computations     592
On the physical justification for the exchange correlation energy     592
The electron pair distribution function     592
The quasi-static connection of two important systems     594
Exchange-correlation energy vs II[subscript laver]     596
Electron holes      597
Physical boundary conditions for holes     598
Exchange and correlation holes     599
Physical grounds for the DFT approximations     601
Reflections on the DFT success     602
The Molecule in an Electric or Magnetic Field     615
Hellmann-Feynman theorem     618
Electric Phenomena     620
The molecule immobilized in an electric field     620
The electric field as a perturbation     621
The homogeneous electric field     627
The inhomogeneous electric field: multipole polarizabilities and hyperpolarizabilities     632
How to calculate the dipole moment     633
Hartree-Fock approximation     633
Atomic and bond dipoles     634
Within the ZDO approximation     635
How to calculate the dipole polarizability     635
Sum Over States Method     635
Finite field method     639
What is going on at higher electric fields     644
A molecule in an oscillating electric field     645
Magnetic Phenomena     647
Magnetic dipole moments of elementary particles     648
Electron     648
Nucleus     649
Dipole moment in the field      650
Transitions between the nuclear spin quantum states - NMR technique     652
Hamiltonian of the system in the electromagnetic field     653
Choice of the vector and scalar potentials     654
Refinement of the Hamiltonian     654
Effective NMR Hamiltonian     658
Signal averaging     658
Empirical Hamiltonian     659
Nuclear spin energy levels     664
The Ramsey theory of the NMR chemical shift     666
Shielding constants     667
Diamagnetic and paramagnetic contributions     668
The Ramsey theory of NMR spin-spin coupling constants     668
Diamagnetic contributions     669
Paramagnetic contributions     670
Coupling constants     671
The Fermi contact coupling mechanism     672
Gauge invariant atomic orbitals (GIAO)     673
London orbitals     673
Integrals are invariant     674
Intermolecular Interactions     681
Theory of Intermolecular Interactions     684
Interaction energy concept     684
Natural division and its gradation     684
What is most natural?     685
Binding energy      687
Dissociation energy     687
Dissociation barrier     687
Supermolecular approach     689
Accuracy should be the same     689
Basis set superposition error (BSSE)     690
Good and bad news about the supermolecular method     691
Perturbational approach     692
Intermolecular distance - what does it mean?     692
Polarization approximation (two molecules)     692
Intermolecular interactions: physical interpretation     696
Electrostatic energy in the multipole representation and the penetration energy     700
Induction energy in the multipole representation     703
Dispersion energy in the multipole representation     704
Symmetry adapted perturbation theories (SAPT)     710
Polarization approximation is illegal     710
Constructing a symmetry adapted function     711
The perturbation is always large in polarization approximation     712
Iterative scheme of the symmetry adapted perturbation theory     713
Symmetry forcing     716
A link to the variational method - the Heitler-London interaction energy     720
When we do not have at our disposal the ideal [Psi subscript A,0] and [Psi subscript B,0]     720
Convergence problems     721
Non-additivity of intermolecular interactions     726
Many-body expansion of interaction energy     727
Additivity of the electrostatic interaction     730
Exchange non-additivity     731
Induction energy non-additivity     735
Additivity of the second-order dispersion energy     740
Non-additivity of the third-order dispersion interaction     741
Engineering of Intermolecular Interactions     741
Noble gas interaction     741
Van der Waals surface and radii     742
Pauli hardness of the van der Waals surface     743
Quantum chemistry of confined space - the nanovessels     743
Synthons and supramolecular chemistry     744
Bound or not bound     745
Distinguished role of the electrostatic interaction and the valence repulsion     746
Hydrogen bond     746
Coordination interaction     747
Hydrophobic effect     748
Molecular recognition - synthons     750
"Key-lock", template and "hand-glove" synthon interactions     751
Intermolecular Motion of Electrons and Nuclei: Chemical Reactions     762
Hypersurface of the potential energy for nuclear motion      766
Potential energy minima and saddle points     767
Distinguished reaction coordinate (DRC)     768
Steepest descent path (SDP)     769
Our goal     769
Chemical reaction dynamics (a pioneers' approach)     770
Accurate solutions for the reaction hypersurface (three atoms)     775
Coordinate system and Hamiltonian     775
Solution to the Schrodinger equation     778
Berry phase     780
Intrinsic reaction coordinate (IRC) or statics     781
Reaction path Hamiltonian method     783
Energy close to IRC     783
Vibrationally adiabatic approximation     785
Vibrationally non-adiabatic model     790
Application of the reaction path Hamiltonian method to the reaction H[subscript 2] + OH [rightarrow] H[subscript 2]O + H     792
Acceptor-donor (AD) theory of chemical reactions     798
Maps of the molecular electrostatic potential     798
Where does the barrier come from?     803
MO, AD and VB formalisms     803
Reaction stages     806
Contributions of the structures as reaction proceeds     811
Nucleophilic attack H[superscript -] + ETHYLENE [rightarrow] ETHYLENE + H[superscript -]      816
Electrophilic attack H[superscript +] + H[subscript 2] [rightarrow] H[subscript 2] + H[superscript +]     818
Nucleophilic attack on the polarized chemical bond in the VB picture     818
What is going on in the chemist's flask?     821
Role of symmetry     822
Barrier means a cost of opening the closed-shells     826
Barrier for the electron-transfer reaction     828
Diabatic and adiabatic potential     828
Marcus theory     830
Information Processing - the Mission of Chemistry     848
Complex systems     852
Self-organizing complex systems     853
Cooperative interactions     854
Sensitivity analysis     855
Combinatorial chemistry - molecular libraries     855
Non-linearity     857
Attractors     858
Limit cycles     859
Bifurcations and chaos     860
Catastrophes     862
Collective phenomena     863
Scale symmetry (renormalization)     863
Fractals     865
Chemical feedback - non-linear chemical dynamics     866
Brusselator - dissipative structures     868
Hypercycles     873
Functions and their space-time organization     875
The measure of information     875
The mission of chemistry     877
Molecular computers based on synthon interactions     878
Appendices     887
A Remainder: Matrices and Determinants     889
Matrices     889
Determinants     892
A Few Words on Spaces, Vectors and Functions     895
Vector space     895
Euclidean space     896
Unitary space     897
Hilbert space     898
Eigenvalue equation     900
Group Theory in Spectroscopy     903
Group     903
Representations     913
Group theory and quantum mechanics     924
Integrals important in spectroscopy     929
A Two-State Model     948
Dirac Delta Function     951
Approximations to [delta](x)     951
Properties of [delta](x)     953
An application of the Dirac delta function     953
Translation vs Momentum and Rotation vs Angular Momentum     955
The form of the U operator     955
The Hamiltonian commutes with the total momentum operator     957
The Hamiltonian, J[superscript 2] and J[subscript z] do commute     958
Rotation and translation operators do not commute     960
Conclusion     960
Vector and Scalar Potentials     962
Optimal Wave Function for a Hydrogen-Like Atom     969
Space- and Body-Fixed Coordinate Systems     971
Orthogonalization     977
Schmidt orthogonalization     977
Lowdin symmetric orthogonalization     978
Diagonalization of a Matrix     982
Secular Equation (H - [epsilon]S)c = 0     984
Slater-Condon Rules     986
Lagrange Multipliers Method     997
Penalty Function Method     1001
Molecular Integrals with Gaussian Type Orbitals 1s     1004
Singlet and Triplet States for Two Electrons     1006
The Hydrogen Molecular Ion in the Simplest Atomic Basis Set     1009
Population Analysis     1015
The Dipole Moment of a Lone Electron Pair     1020
Second Quantization     1023
The Hydrogen Atom in the Electric Field - Variational Approach     1029
NMR Shielding and Coupling Constants - Derivation     1032
Shielding constants     1032
Coupling constants     1035
Multipole Expansion      1038
Pauli Deformation     1050
Acceptor-Donor Structure Contributions in the MO Configuration     1058
Name Index     1065
Subject Index     1077

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