Kuhn, 2000

Försterling, 2000

Kuhn wrote the following (unofficial - he'll dislike this, to be sure) note to this website about his latest book : Translated: I took great care here, more than anywhere else, to properly describe what is important in the eletron gas model ... etc.

 
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           Problems ................................................................ 137
              Problem 5.1  -  Partial Charges and Dipole Moment ...................... 137
              Problem 5.2  -  Stretching Force Constant of H2+ Ion ................... 137
              Problem 5.3  -  Bond Angle of H20 ...................................... 139

         6 Bonding Described By Hybrid and Molecular Orbitals ...................... 140

           6.1 Degeneracy of Energy Levels ......................................... 140
              6.1.1 Hybrid Functions ............................................... 142
              6.1.2 Hybridization of H Atom Functions .............................. 144

           6.2 Localized Electrons: Hybrid Atomic Orbitals ......................... 147
              6.2.1 Li Atom (No Hybridization) ..................................... 147
              6.2.2 BeH2 (Linear(sp)Hybridization: Two s1/2p1/2 Hybrid Orbitals) ..... 148
              6.2.3 H2S and H20 (Two Orbitals Between p and s1/4p3/4 Hybrid) ......... 148
              6.2.4 CH4 (Tetrahedral (sp3) Hybridization:
                    Four s1/4p3/4 Hybrid Orbitals) ................................... 150

           6.3 Properties of Electron Pair Bonds .................................... 151
              6.3.1 Formal and Effective Charges .................................... 152
              6.3.2 Polymerization of BeH2 .......................................... 154

           6.4 Delocalized Electrons: Molecular Orbitals ............................ 154
              6.4.1 Box Wavefunctions ............................................... 154
              6.4.2 LCAO Wavefunctions .............................................. 155
              6.4.3 Improvement of Trial Functions .................................. 159

           Problems ................................................................. 159
              Problem 6.1 - Linear Combinations of Wave Functions ................... 159
              Problem 6.2 - Degeneracy and Hybridization of Box Functions ........... 159
              Problem 6.3 - Orthogonality and Normalization of Hybrid Functions ..... 160
              Problem 6.4 - Symmetry Properties of Linear Hybrid Functions .......... 160
              Problem 6.5 - Energy of Hybrid States ................................. 160
              Problem 6.6 - Structure of B(CH3)3 and Hg2Cl2 .......................... 161
              Problem 6.7 - Dative Bond ............................................. 161
              Problem 6.8 - Three-center Bond (CH5 +) ................................ 161
              Problem 6.9 - LCAO Model for 02 ........................................ 162
              Problem 6.10 - Molecular Orbitals of Some Diatomic Molecules .......... 162


         7 Molecules with pi Electron Systems ............................... 163

           7.1 Bonding Properties of π Electrons ..................................... 163
 
           7.2 Free-electron Model ................................................... 165
                7.2.1 Linear π Electron Systems ...................................... 165
                7.2.2 Cyclic π Electron Systems ...................................... 171
                7.2.3 Charge Density dQ/ds ........................................... 173
                7.2.4 Resonance ...................................................... 175
                7.2.5 Branched Molecules ............................................. 177
 
           7.3 HMO Model ............................................................. 181
                7.3.1 Wavefunctions and Energies ..................................... 181
                7.3.2 Charge Density dQ/ds ........................................... 184

           7.4 Bond Lengths, Dipole Moments  ......................................... 184
                7.4.1 Bond Length and Charge Density ................................. 184
                7.4.2 Bond Alternation in Polyenes and Fullerenes .................... 187
                7.4.3 Dipole Moment .................................................. 187
         
           Problems  ................................................................. 189
                Problem 7.1 - HMO Method: Ethene ..................................... 189
                Problem 7.2 - HMO Method: Butadiene .................................. 190
                Problem 7.3 - HMO Method: Fulvene .................................... 191
                Problem 7.4 - Resonance of Hückel (4n + 2) Rings ..................... 192
                Problem 7.5 - Bond Lengths from Bond Orders .......................... 192
                Problem 7.6 - Cyclobutadiene: Bond Lengths ........................... 192
                Problem 7.7 - Dipole Moment of Fulvene ............................... 193
 
            Foundation 7.1 - Free Electron Model ..................................... 195
            Foundation 7.2 - HMO Model  .............................................. 198
            Foundation 7.3 - Self-consistency in Bond Alternation .................... 200


        8 Absorption and Emission of Light  ............................... 204 

            8.1 Basic Experimental Facts ............................................. 204
                8.1.1 Transmittance and Absorbance ................................... 204
                8.1.2 Polyenes and Cyanines .......................................... 207

            8.2 Absorption Maxima of Dyes ............................................ 209
                8.2.1 Band Broadening ................................................ 210
                8.2.2 Single-molecule Absorption ..................................... 210
                8.2.3 Cyanine Dyes ................................................... 211

            8.3 Strength and Polarization of Absorption Bands ........................ 213
                8.3.1 Oscillator Strength ............................................ 213
                8.3.2 Polarization of Absorption Bands ............................... 215

            8.4 Heteroatoms as Probes for Electron Distribution ...................... 216

            8.5 HOMO - LUMO Gap by Bond Alternation .................................... 219

            8.6 Dyes with Cyclic Electron Cloud: Phthalocyanine ...................... 223

            8.7 Coupling of pi Electrons ............................................. 226   
                   
            8.8 Light Absorption of Biomolecules ..................................... 228
                8.8.1 ß-Carotene ..................................................... 228
                8.8.2 Retinal ........................................................ 230
                8.8.3 Vitamin B12 .................................................... 231
                8.8.4 Chlorophyll, Bacteriochlorophyll ............................... 232

            8.9 Spontaneous Emission ................................................. 233
                8.9.1 Fluorescence and Phosphorescence ............................... 233
                8.9.2 Single Molecule Emission ....................................... 235
                8.9.3 Singlet and Triplet States ..................................... 236
                8.9.4 Shift of Fluorescence and Phosphorescence
                      Relative to Absorption ......................................... 238
                8.9.5 Absorption from Excited States ................................. 240
                8.9.6 Quenching of Fluorescence ...................................... 242

           8.10 Stimulated Emission .................................................. 242
                8.10.1 Inversion of Population ....................................... 242
                8.10.2 Dye Laser ..................................................... 243
                8.10.3 Excimer Laser ................................................. 246

           8.11 Optical Activity ..................................................... 247
                8.11.1 Rotatory Dispersion ........................................... 247
                8.11.2 Ellipticity ................................................... 248
                8.11.3 Circular Dichroism ............................................ 249
                8.11.4 Circular Dichroism of Spirobisanthracene ...................... 250
                8.11.5 Circular Dichroism of Chiral Cyanine Dye ...................... 253

           Problems .................................................................. 253
                Problem 8.1 - Lone Electron Pair at the Nitrogen ..................... 253
                Problem 8.2 - Light Absorption of Different Classes of Dyes .......... 254
                Problem 8.3 - Energy Shift in Azacyanines ............................ 254
                Problem 8.4 - Shift of Energy Levels by Bond Alternation ............. 256
                Problem 8.5 - Light Absorption of Phthalocyanine and Porphyrin ....... 256
                Problem 8.6 - Oscillatory Strength in Phthalocyanine and Porphyrin ... 257                
                Problem 8.7 - Cis-Peak in ß-Carotene ................................. 258
                Problem 8.8 - Splitting of Absorption Band ........................... 258
                Problem 8.9 - Circular Dichroism of Cyanine Dye ...................... 259

            Foundation 8.1 - Integrated Absorption: Classical Oscillator ............. 261
            Foundation 8.2 - Oscillator Strength: Quantum mechanical Treatment ....... 265
            Foundation 8.3 - Coupling Transitions with Parallel Transition Moment .... 267
            Foundation 8.4 - Normal modes of Coupled ................................. 270
            Foundation 8.5 - Pluorescence Life Time .................................. 274
            Foundation 8.6 - Proof of Relation for g (Anisotropy Factor) ............. 276


          9 Nuclei: Particle and Wave Properties .................................... 278

            9.1 Quantum Mechanical Rotator ........................................... 278
               9.1.1 Exact Solution .................................................. 279
               9.1.2 Simplified Model ................................................ 280

           9.2 Rotational Spectra .................................................... 282

           9.3 Quantum Mechanical Oscillator ......................................... 287
               9.3.1 Exact solution .................................................. 288
               9.3.2 Box Model for Oscillator ........................................ 291
               9.3.3 Comparison of a Quantum Mechanical Oscillator with a
                     Classical Oscillator ............................................ 293

           9.4 Vibrational - Rotational Spectra ........................................ 294
               9.4.1 Diatomic Molecules .............................................. 294
               9.4.2 Polyatomic Molecules ............................................ 297

           9.5 Raman Spectra ......................................................... 303
               9.5.1 Rayleigh Scattering ............................................. 303
               9.5.2 Rotational Raman Spectra ........................................ 305
               9.5.3 Vibrational-Rotational Raman Spectra of Diatomic Molecules ...... 307 
               9.5.4 Raman Spectra of Polyatomics .................................... 310

           9.6 Vibrational Structure of Electronic Spectra ........................... 312
               9.6.1 Diatomic Molecule ............................................... 312
               9.6.2 Photoelectron Spectroscopy ...................................... 312
               9.6.3 Polyatomic Molecules ............................................ 317

           9.7 Nuclear Spin (Orthohydrogen and Parahydrogen) ......................... 318
               9.7.1 Spin of Protons in H2 ........................................... 319
               9.7.2 Nuclear Wavefunctions ........................................... 319
               9.7.3 Antisymmetry Postulate in H2 .................................... 320
   
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Priciples of Physical Chemistry

Preface and Guided Tour

Preface


This textbook for an undergraduate course in physical chemistry is addressed equally to chemists, polymer chemists, biologists, chemical engineers, material scientists, and physicists.

Chemical structures and processes can be interpreted as an interplay of atoms and molecules. Matter can be regarded as a collective interaction of these atoms and molecules, with macroscopic properties based on a few fundamental principles governing chemistry and related sciences. To emphasize this situation, we have adopted in this book a presentation where physical chemistry is first introduced by the treatment of atoms and chemical bonds, followed by a discussion of how these lead to molecules and, subsequently, to more and more complex representations of matter. By this structure in the sequence of chapters, the student is encouraged to focus on a systematic understanding of the subject. Simple theoretical models will illustrate many of these concepts.

Although the presentation is logical, the course can proceed in almost any order because the chapters are relatively self-contained (see "Guided Tour" below). The concepts determining the processes at the atomic level are based on the postulates of quantum mechanics. Many of these concepts seem to be counterintuitive and strange at a first glance. The bookstarts with a simple introduction to quantum mechanics in a way that should be easy to grasp.

We feel that an early confrontation with quantum mechanics is advantageous to the student. Thinking in terms of quantum mechanics is needed in all parts of physical chemistry and is crucial in understanding the nature of the chemical bond. Students in chemistry and biochemistry are trained to think in terms of chemical bonds and processes at the atomic level; it is important for them to see immediately the interlinkage of physical chemistry with inorganic and organic chemistry. Obviously, this aim can be much more easily reached by emphasizing the atomic approach instead of treating thermodynamics without reference to molecular interplay.

Inventing idealized theoretical descriptions of real situations is crucial in understanding increasingly complex organized systems, and ingenious experiments are of particular importance in modern fields of physical chemistry. Studying typical examples is an excellent opportunity for learning how to design simple experiments and theoretical models. With this goal in mind, we do not attempt to cover as many topics as possible. Experimental methods must stay in the background. Selected cases are emphasized in order to focus on fundamental aspects, for example, pi electron systems are discussed in more detail than other subjects because they illustrate particularly well the use of simple theoretical models.

The treatment of organized systems of molecules, including supramolecular machines, is a fast-developing subject in physical chemistry. We attempt to introduce the student to this fascinating and dramatically growing field by discussing selected examples illustrating the new approaches. Of course, an overview of the broad variety of interdisciplinary topics cannot be given. The final chapter should lead to an understanding of the basic processes in the origin of life; it demonstrates the usefulness of typical physicochemical modeling in the development of exciting new areas.


A Guided Tour Through Physical Chemistry


The sequence of chapters in this book - starting with quantum mechanics and its application to simple atoms and molecules and then proceeding stepwise to increasingly complex systems - should emphasize the fact that chemical phenomena can be considered as processes among molecules and that the properties of molecules can be deduced from the concepts of quantum mechanics. This straightforward organization of the book retains the logical structure of the subject of physical chemistry.

Various approaches to using this textbook can lead toward the goal that the postulates of quantum mechanics are to be recognized as the conceptual basis for rationalizing the great variety of chemical phenomena. Following the given sequence of chapters is one possible pathway. It requires that we start with quantum theory, which is unfamiliar to the beginner, but is ultimately needed for understanding chemical bonds and the world of structures based on chemical bonding.

The interplay of atoms and molecules, however, can be approximated by using the familiar laws of classical mechanics. Consequently, as an alternative, the student may begin with Chapter 11, which describes the thermal motion of molecules in the familiar classical way of thinking. This leads to a lively picture of the molecular interplay, and the student learns the concepts of temperature and heat.

After reading Chapter 11 it is advisable to continue with Chapters 1 and 2 in order to grasp the fundamental importance of quantum theory and chemical bonding. The student may then proceed with Chapters 3 - 10, which focus on molecular structure and intermolecular forces. He or she would subsequently read Chapters 12 - 24, which are concerned with criteria for chemical reactions, equilibria, rate of chemical reactions, and the properties of increasingly complex systems.

Alternatively, reading the chapters in the sequence 11, 1, 2, 12 - 24, 3 - 10 is also acceptable. In both cases, it is important that the student is exposed to thinking in terms of quantum mechanics by reading Chapters 1 and 2 at an early stage. The content of each chapter is summarized below.


Chapters 1 and 2. Here we describe experiments (e.g., diffraction of electrons) that focus on the essence of quantum mechanics. At first glance, these considerations seem to be remote from physical chemistry. However, they show essential aspects of atoms and molecules. At the same time, these experiments demonstrate how developments in one field in scientific research can revolutionize other scientific areas.

Experiments demonstrating dramatic consequences of the quantum nature of matter are discussed and the simplest atom and the simplest molecule, the hydrogen atom and the hydrogen molecule ion, are considered. The variation principle, which is crucial in understanding the nature of the chemical bond, is introduced as a postulate. Helium-like systems and the hydrogen molecule are discussed.


Chapters 3 and 4. In Chapter 3 a deeper understanding of quantum mechanics is given by introducing the Schrödinger equation as a postulate and deducing the variation principle as a theorem. On this basis the stationary states of the H atom are discussed.

In Chapter 4 the treatment is applied to H₂⁺ and extended to the hydrogen molecule and to systems containing more than one electron. The Pauli principle is introduced as an additional postulate.

Beginners might restrict themselves to looking at the energy levels and wavefunctions of the excited states of the H atom and to the elementary form of the Pauli principle, proceed to Chapter 5, and then come back to Chapters 3 and 4 at a later stage.


Chapters 5 - 10. In Chapters 5 and 6 different types of chemical bonds, their occurrence, and their specific properties are discussed. Chapters 7 and 8 are devoted to pi electrons. These chapters exemplify the use of simple theoretical models which are crucial to understanding complex molecular assemblies, the heart of today’s physical chemistry. Phenomena such as light absorption and laser action establish the importance of quantum mechanics. Chapter 9 is concerned with the quantum mechanical view of nuclear motion. Chapter 10 deals with intermolecular bonding and the occurrence of molecular assemblies.


Chapters 11 - 16. Chapter 11 introduces the concept of temperature as a measure of the motion of molecules. The classical aspects of thermal motion are emphasized and the phase transitions gas-liquid-solid are discussed at a molecular level. In Chapter 12 the intramolecular distribution of thermal energy is considered; here the quantum mechanical aspects are crucial. Chapters 13 - 15 are concerned with quantities and theorems important for an understanding of macroscopic behavior of matter: internal energy, heat, work, and entropy. Chapter 16 introduces the quantities useful in discussing the criteria for chemical reactions besides temperature and entropy: enthalpy and Gibbs energy.


Chapters 17 - 20. These chapters deal with applications of the basic concepts developed in Chapters 11 - 16: chemical equilibrium, bioenergetics and electrochemistry.


Chapters 21 - 24. Chapter 21 is concerned with the kinetics of chemical reactions. Chapters 22 - 24 consider increasingly complex systems: interfaces, membranes, polymers, supramolecular machines, and the mechanisms involved in the origin of life.


A Note on Introducing Thermodynamics (Chapters 13 - 16). Historically, the macroscopic properties of matter were investigated without taking the atomic structure into account. Classical thermodynamics emerged in connection with the development of the steam engine, which drove the industrial revolution in the nineteenth century. The classical picture of molecules developed in the second half of the nineteenth century led to the molecular view of temperature and heat. The quantum theoretical approach driving toward an understanding of the particular properties of chemical species originated in the first half of the twentieth century. This traditional sequence is still kept in most introductory courses in physical chemistry, which start with the laws of thermodynamics. When proceeding to classical and then to quantum mechanical statistical thermodynamics, it is confusing for the student and difficult to accept that the laws of thermodynamics are first set as postulates, then understood as properties of collections of molecules, and finally deduced from the postulates of quantum mechanics.

This difficulty is avoided in our approach by first viewing the quantum mechanical nature of matter and then deducing the macroscopic properties as a consequence.

The significance of entropy is easily seen when introducing it as a measure of the number of possible configurations of a macroscopic system in a given state: this number does not decrease in an isolated system, so the entropy of an isolated system cannot decrease (Chapter 14). We then demonstrate that the entropy defined by the number of configurations is closely connected with the heat transfer in reversible processes in classical thermodynamics (Chapter 15).