"Version: 20210301"--Title page verso.

Includes bibliographical references.

1. Introduction to molecular-scale electronics -- 1.1. Background -- 1.2. The theoretical challenges addressed by this book -- 1.3. De Broglie wave patterns -- 1.4. The Landauer formula -- 1.5. Resonant transport -- 1.6. Thermoelectricity -- 1.7. Off-resonance transport -- 1.8. Intuitive picture of Green's functions -- 1.9. Magic ratio theory -- 1.10. Controlling quantum interference -- 1.11. Summary

2. Connectivity theory and noisy neighbour equations for quantum transport -- 2.1. Introduction -- 2.2. Green's functions for beginners -- 2.3. The principle of superposition -- 2.4. Destructive versus constructive interference -- 2.5. Green's functions produced by noisy neighbours -- 2.6. NNEs for transport through more complex coupled quantum structures -- 2.7. Identifying quantum interference within the five-component NNE -- 2.8. Magic ratio theory -- 2.9. Green's function of a molecular core -- 2.10. Mid-gap energies versus E = 0 -- 2.11. Summary

3. A beginner's guide to solving the Schr�odinger equation -- 3.1. Introduction -- 3.2. Mathematical aim : an introduction to linear algebra -- 3.3. A simple harmonic oscillator -- 3.4. The equations of motion of two harmonic oscillators -- 3.5. Matrix form of the equations of motion of two harmonic oscillators -- 3.6. Combining linear independence with the best choice of basis to solve equation (3.7) -- 3.7. Eigenvalues and eigenvectors -- 3.8. Solution to the general N x N dynamical problem -- 3.9. Solution to the general N x N Schr�odinger equation -- 3.10. How many linearly-independent eigenvectors does a dynamical matrix H possess? -- 3.11. A two-level quantum system -- 3.12. The Hamiltonian of a two-level quantum system -- 3.13. Eigenvalues and eigenvectors of N x N Hermitian matrices -- 3.14. Proof that a N x N Hermitian matrix possesses N linearly independent eigenvectors -- 3.15. Completeness and initial conditions revisited -- 3.16. The eigenstates of a two-level quantum system -- 3.17. Summary -- Appendix A. Basic properties of matrices and complex numbers

4. Quantum properties of linear chains and simple molecules -- 4.1. Introduction -- 4.2. The eigenvalues and eigenvectors of a two-level molecule -- 4.3. The eigenvalues and eigenvectors of a doubly infinite chain of identical atoms -- 4.4. Effective mass and group velocity -- 4.5. Degeneracies and the allowed values of k -- 4.6. The most general solution to the Schr�odinger equation for a linear chain -- 4.7. A finite chain of N sites with periodic boundary conditions -- 4.8. Group velocity revisited -- 4.9. The eigenvalues and eigenvectors of a linear chain of N identical atoms -- 4.10. The two-level system revisited -- 4.11. A three-level system -- 4.12. Normal modes of a vibrating atomic chain -- 4.13. Radicals and Huckel's rule -- 4.14. Summary of results for the eigenstates and eigenvalues of one-dimensional chains -- 4.15. A quantum spider's web -- 4.16. Eigenstates of a semi-infinite one-dimensional chain -- 4.17. Summary -- Appendix A. Relationship to Fourier analysis -- Appendix B. Continuity equations

5. Quantum properties of electrodes in higher dimensions -- 5.1. Introduction -- 5.2. Peierl's distortion and oligoynes -- 5.3. Conduction bands versus valence bands -- 5.4. Flexibility in labelling states -- 5.5. A finite-width electrodes formed from a linear chain of cells -- 5.6. A linear chain of cells, each containing two sites -- 5.7. A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix -- 5.8. A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix and an intra-cell Hamiltonian H0 describing a linear chain with free ends -- 5.9. A linear chain of cells, each containing N sites, with an intra-cell Hamiltonian H1 proportional to the unit matrix and an intra-cell Hamiltonian H0 describing a linear chain with periodic boundary conditions -- 5.10. A two-dimensional crystal on a square lattice -- 5.11. The most general solution to the Schr�odinger equation for the periodic structure of figure 5.7, for which H1 is proportional to the unit matrix -- 5.12. The equivalence between a finite-width lead and many one-dimensional leads -- 5.13. Summary

6. Scattering theory of electrical conductance and thermopower -- 6.1. Introduction -- 6.2. A single impurity in one dimension -- 6.3. The scattering matrix -- 6.4. Bond currents and conservation of probability -- 6.5. Current carried by counter-propagating plane waves -- 6.6. Unitarity of the scattering matrix -- 6.7. Physical meaning of scattering matrix elements -- 6.8. Consequences of unitarity of the scattering matrix -- 6.9. Bound states -- 6.10. The S matrix in higher dimensions -- 6.11. The Landauer formula -- 6.12. Derivation of the Landauer formula and formulae for thermoelectric properties of a quantum object connected to external electrodes -- 6.13. Summary -- Appendix A. A general expression for bond currents -- Appendix B. Unitarity of the multi-channel S matrix

7. Thermoelectricity in nanostructures and molecules -- 7.1. Introduction -- 7.2. A qualitative description of thermoelectricity -- 7.3. Linear response formulae for thermoelectric coefficients -- 7.4. An expressions for thermoelectric efficiency -- 7.5. Relationship between thermoelectric efficiency and ZT -- 7.6. Expressions for thermoelectric coefficients -- 7.7. Proof that ZTe is positive -- 7.8. Strategies for maximising thermoelectric performance -- 7.9. Summary

8. A very useful formula (VUF) for the transmission coefficient of an arbitrary scatterer connected to one-dimensional leads and a magic ratio theory for intra-molecular currents -- 8.1. Introduction -- 8.2. The Schr�odinger equation for a scatterer connected to two one-dimensional leads -- 8.3. Note about the choice of wave vectors -- 8.4. Solution to the Schr�odinger equation -- 8.5. Expression for the transmission amplitude in terms of the full Green's function G -- 8.6. Expression for the transmission amplitude in terms of the Green's function of the isolated scatterer -- 8.7. Expression for the reflection amplitude -- 8.8. Expression for the wave function inside the scatterer -- 8.9. Expressions for bond currents -- 8.10. Magic ratio theory for intra-molecular currents -- 8.11. Properties of the very useful formula for the transmission coefficient -- 8.12. The relationship between magic ratio theory and the VUF -- 8.13. The relationship between the Breit-Wigner formula, on-resonance transport and the VUF -- 8.14. Summary

9. A quantum system connected to many scattering channels -- 9.1. Introduction -- 9.2. Solving the Schr�odinger equation -- 9.3. Unitarity of the many-channel scattering matrix -- 9.4. Expressions for the transmission coefficients -- 9.5. Expressions for the reflection coefficients -- 9.6. The self-energy matrix -- 9.7. Dyson's equation -- 9.8. Example : a scatterer connected to two 1-dimensional leads -- 9.9. A scatterer attached to M leads, in which each lead connects to only one site of the scatterer -- 9.10. Scattering theory in the presence of an open scatterer -- 9.11. The effect of closed channels -- 9.12. A scatterer connected to finite-dimensional leads -- 9.13. Summary

10. Relationship between Green's functions, wave functions and scattering amplitudes -- 10.1. Introduction -- 10.2. Green's functions of closed systems -- 10.3. Approximations to Green's functions of closed systems -- 10.4. Green's functions of a doubly infinite linear chain -- 10.5. Relationship between a Green's function of a linear chain and the wave function of a T-shaped junction -- 10.6. Green's functions of a semi-infinite linear chain -- 10.7. The relationship between Green's functions and wave functions in the presence of a scatterer -- 10.8. Green's function of a linear chain with periodic boundary conditions -- 10.9. Green's functions of a linear chain with free-end boundary conditions -- 10.10. Green's functions of a finite-width nanoribbon -- 10.11. Relationships between Green's functions and transmission amplitudes -- 10.12. Summary

11. Connectivity theory revisited : heteroatom substitution, decimation and the Breit-Wigner formula -- 11.1. Introduction -- 11.2. Recursively describing the effect of changes to a molecular junction -- 11.3. Decimation and the effect of pendant groups -- 11.4. Illustration of destructive quantum interference due to pendant atoms -- 11.5. The effect of a local perturbation -- 11.6. The effect of introducing a single heteroatom -- 11.7. The effect of introducing two heteroatoms -- 11.8. The Green's function of a semi-infinite lead revisited -- 11.9. Fusion of substructures via Dyson's equation -- 11.10. Constructing a lead-scatterer-lead system using Dyson's equation -- 11.11. From Green's functions to transmission functions -- 11.12. Simplification of the Green's function of a scatterer in the presence of electrodes -- 11.13. Summary of Green's function equations for the transmission coefficient -- 11.14. The Breit-Wigner formula for an arbitrary scatterer connected to finite-width leads -- 11.15. Derivation of the five-component NNE -- 11.16. Summary -- Appendix A. Similarity transformations -- Appendix B. An alternative derivation of the core Green's function

12. Linear molecules -- 12.1. Introduction -- 12.2. Single-site decimation and the variation of conductance with length -- 12.3. A negative eigenvalue theorem -- 12.4. Recurrence relations via the noisy neighbour equations -- 12.5. Decay of conductance with length of an oligoyne and a diatomic chain -- 12.6. Quantum circuit rules for the conductance and Seebeck coefficient of linear molecules -- 12.7. Summary

13. Quantum interference in molecules with parallel paths and pendant groups -- 13.1. Introduction -- 13.2. The effect of pendant groups -- 13.3. The Green's function of an isolated molecule containing a bridging unit -- 13.4. A molecular backbone coupled to two different atoms of a bridging unit -- 13.5. A molecule with three moieties in series -- 13.6. A molecule with two bridging groups in parallel -- 13.7. The electrical conductance of a molecule with weakly coupled bridges -- 13.8. The electrical conductance of a molecule with many parallel bridges -- 13.9. Summary

14. Connectivity theory and equations of motion -- 14.1. Introduction -- 14.2. Stationary states and Green's functions -- 14.3. Green's function of a 2-component system revisited -- 14.4. Connectivity equations from combinations of active and passive viewpoints -- 14.5. Stationary states of vibrating structures -- 14.6. Connectivity theory, equality of currents and unitarity of the scattering matrix -- 14.7. The limit [eta] [right arrowp] 0 -- 14.8. The case [eta] [not equal] 0 -- 14.9. Comparison between advanced and retarded Green's functions -- 14.10. Summary -- Appendix A. Explicit integration of equation (14.4) -- Appendix B. Green's functions of open systems -- Appendix C. Green's functions of a double infinite tight-binding chain revisited -- Appendix D. Iterative analysis of the 3-component structure of figure 14.3 -- Appendix E. The Green's function of a 5-component system

15. Relationship between Green's functions, molecular orbitals and densities of states -- 15.1. Introduction -- 15.2. The Coulson-Rushbrooke theorem -- 15.3. An orbital symmetry rule -- 15.4. Densities of states -- 15.5. van Hove singularities -- 15.6. A more precise form of the density of states -- 15.7. Properties of delta functions -- 15.8. Examples of delta functions -- 15.9. Relationship between Green's functions and densities of states -- 15.10. Summary

16. Solving the time-dependent Schr�odinger equation and the theory of representations -- 16.1. Introduction -- 16.2. The theory of representations -- 16.3. Representations involving orthonormal basis functions -- 16.4. Representations involving non-orthogonal basis functions -- 16.5. Transformations between representations -- 16.6. A more compact notation for the transformation equation -- 16.7. Atomic orbitals versus pseudo-atomic orbitals -- 16.8. Contribution of overlap integrals to nearest-neighbour couplings -- 16.9. Dyson's equation in a non-orthogonal basis -- 16.10. Summary

17. Scattering theory in the presence of material-specific leads -- 17.1. Introduction -- 17.2. The most general solution to the Schr�odinger equation for an arbitrary crystalline lead -- 17.3. Matrix elements between channel eigenvectors -- 17.4. Cancellation of off-diagonal contributions to currents -- 17.5. Note about degeneracies -- 17.6. Analysis of a general scatterer connected to a periodic structure -- 17.7. Analysis of currents -- 17.8. Expressions for transmission amplitudes -- 17.9. Determining the allowed wave vectors for a given energy E -- 17.10. The Green's function of an arbitrary material-specific, finite-width, doubly infinite lead -- 17.11. The Green's function of an arbitrary finite-width, semi-infinite lead -- 17.12. The surface Green's function of an arbitrary finite-width, semi-infinite lead -- 17.13. A simple derivation of the surface Green's function of an arbitrary finite-width, semi-infinite lead -- 17.14. Dyson's equation for the surface Green's function of an arbitrary finite-width, semi-infinite lead -- 17.15. Expressions for transmission and reflection amplitudes of a scatterer connected to arbitrary, material-specific leads -- 17.16. Summary -- Appendix A. Solving the H1 problem -- Appendix B. Translation operators, velocity operators, Green's functions, time reversal symmetry and the transmission coefficient -- Appendix C. A note about degeneracies.

This reference text presents a conceptual framework for understanding room-temperature electron and phonon transport through molecules and other quantum objects. The flow of electricity through molecules is explained at the boundary of physics and chemistry, providing an authoritative introduction to molecular electronics for physicists, and quantum transport for chemists. Professor Lambert provides a pedagogical account of the fundamental concepts needed to understand quantum transport and thermoelectricity in molecular-scale and nanoscale structures. The material provides researchers and advanced students with an understanding of how quantum transport relates to other areas of materials modelling, condensed matter and computational chemistry. After reading the book, the reader will be familiar with the basic concepts of molecular-orbital theory and scattering theory, which underpin current theories of quantum transport.

Researchers interested in molecular-scale and nanoscale transport, particularly postgraduate students. The interdisciplinary audience includes chemists, physicists and materials scientists.

Also available in print.

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.

Professor Colin J Lambert is a research professor in the Department of Physics at Lancaster University, and a world leader in the field of single-molecule electronics. He has been a professor at Lancaster since 1990 and was awarded a research professorship in 2010. He is also a visiting professor in the Materials Department at the University of Oxford, and an elected member of Academia Europaea.

Title from PDF title page (viewed on April 12, 2021).