Advances in Chemical Physics

By Stuart A. Rice

Advances in Chemical Physics is the only series of volumes available that explores the cutting edge of research in chemical physics.

• This is the only series of volumes available that presents the cutting edge of research in chemical physics.
• Includes contributions from experts in this field of research.
• Contains a representative cross-section of research that questions established thinking on chemical solutions.
• Structured with an editorial framework that makes the book an excellent supplement to an advanced graduate class in physical chemistry or chemical physics.

• Language: English
• Publisher: Wiley
• Release date: Nov 20, 2014
• ISBN: 9781118949726

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EDITORIAL BOARD

KURT BINDER Condensed Matter Theory Group, Institut Für Physik, Johannes Gutenberg-Universität, Mainz, Germany WILLIAM T. COFFEY Department of Electronic and Electrical Engineering, Printing House, Trinity College, Dublin, Ireland KARL F. FREED Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois USA DAAN FRENKEL Department of Chemistry, Trinity College, University of Cambridge, Cambridge, United Kingdom PIERRE GASPARD Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, Brussels, Belgium MARTIN GRUEBELE Departments of Physics and Chemistry, Center for Biophysics and Computational Biology, University of Illinois at Urbana-Champaign, Urbana, Illinois USA GERHARD HUMMER Theoretical Biophysics Section, NIDDK-National Institutes of Health, Bethesda, Maryland USA RONNIE KOSLOFF Department of Physical Chemistry, Institute of Chemistry and Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Israel KA YEE LEE Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois USA TODD J. MARTINEZ Department of Chemistry, Photon Science, Stanford University, Stanford, California USA SHAUL MUKAMEL Department of Chemistry, School of Physical Sciences, University of California, Irvine, California USA JOSE N. ONUCHIC Department of Physics, Center for Theoretical Biological Physics, Rice University, Houston, Texas USA STEPHEN QUAKE Department of Bioengineering, Stanford University, Palo Alto, California USA MARK RATNER Department of Chemistry, Northwestern University, Evanston, Illinois USA DAVID REICHMAN Department of Chemistry, Columbia University, New York City, New York USA GEORGE SCHATZ Department of Chemistry, Northwestern University, Evanston, Illinois USA STEVEN J. SIBENER Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois USA ANDREI TOKMAKOFF Department of Chemistry, James Franck Institute, University of Chicago, Chicago, Illinois USA DONALD G. TRUHLAR Department of Chemistry, University of Minnesota, Minneapolis, Minnesota USA JOHN C. TULLY Department of Chemistry, Yale University, New Haven, Connecticut, USA

ADVANCES IN CHEMICAL PHYSICS

VOLUME 156

Edited BY

STUART A. RICE

Department of Chemistry and The James Franck Institute,

The University of Chicago, Chicago, Illinois

AARON R. DINNER

Department of Chemistry and The James Franck Institute,

The University of Chicago, Chicago, Illinois

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Library of Congress Catalog Number: 58-9935

ISBN: 978-1-118-94969-6

CONTRIBUTORS TO VOLUME 156

ANASTASSIA N. ALEXANDROVA Department of Chemistry and Biochemistry, University of California, Los Angeles, CA, 90095-1569, USA LOUIS-S. BOUCHARD California NanoSystems Institute, Los Angeles, CA, 90095, USA DECLAN J. BYRNE School of Physics, University College Dublin, Belfield, Dublin 4, Ireland WILLIAM T. COFFEY Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland MARJOLEIN DIJKSTRA Soft Condensed Matter group, Debye Institute for Nanomaterials Science, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands WILLIAM J. DOWLING Department of Electronic and Electrical Engineering, Trinity College, Dublin 2, Ireland M. HAYASHI Condensed Matter Center, National Taiwan University, Taipei, Taiwan YURI P. KALMYKOV Laboratoire de Mathèmatiques et Physique, Universitè de Perpignan Via Domitia, 54, Avenue Paul Alduy, F-66860 Perpignan, France C.K. LIN Condensed Matter Center, National Taiwan University, Taipei, Taiwan S.H. LIN Department of Applied Chemistry, National Chiao-Tung University, Hsinchu, Taiwan G. ALI MANSOORI Department of Bioengineering, University of Illinois at Chicago, Chicago, IL 60607-7052, USA Y.L. NIU The State Key Laboratory of Molecular Reaction Dynamics, Institute of Chemistry, Chinese Academy of Sciences, Beijing, China RANKO RICHERT Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ, 85287-1604, USA STUART A. RICE Department of Chemistry and the James Franck Institute, The University of Chicago, Chicago, IL 60637, USA ASAF SHIMSHOVITZ Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel NORIO TAKEMOTO Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel DAVID J. TANNOR Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100 Israel SERGUEY V. TITOV Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Vvedenskii Square 1, Fryazino, Moscow Region, 141190, Russian Federation L. YANG Institute of Theoretical and Simulation Chemistry, Academy of Fundamental and Interdisciplinary Science, Harbin Institute of Technology, Harbin, China C.Y. ZHU Department of Applied Chemistry, National Chiao-Tung University, Hsinchu, Taiwan

PREFACE TO THE SERIES

Advances in science often involve initial development of individual specialized fields of study within traditional disciplines followed by broadening and overlap, or even merging, of those specialized fields, leading to a blurring of the lines between traditional disciplines. The pace of that blurring has accelerated in the last few decades, and much of the important and exciting research carried out today seeks to synthesize elements from different fields of knowledge. Examples of such research areas include biophysics and studies of nanostructured materials. As the study of the forces that govern the structure and dynamics of molecular systems, chemical physics encompasses these and many other emerging research directions. Unfortunately, the flood of scientific literature has been accompanied by losses in the shared vocabulary and approaches of the traditional disciplines, and there is much pressure from scientific journals to be ever more concise in the descriptions of studies, to the point that much valuable experience, if recorded at all, is hidden in supplements and dissipated with time. These trends in science and publishing make this series, Advances in Chemical Physics, a much needed resource.

The Advances in Chemical Physics is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field.

STUART A. RICE

AARON R. DINNER

CONTENTS

Chapter 1: Phase Space Approach to Solving The SchrÖdinger Equation: Thinking Inside the Box

I. Introduction

II. Theory

III. Application to Ultrafast Pulses

IV. Applications to Quantum Mechanics

V. Applications to Audio and Image Processing

VI. Conclusions and Future Prospects

Acknowledgments

References

Chapter 2: Entropy-Driven Phase Transitions In Colloids: From spheres to anisotropic particles

I. Introduction

II. Predicting Candidate Crystal Structures

III. Free-Energy Calculations

IV. Bulk Phase Diagram and Kinetic Pathways

V. Phase Diagrams of Binary Hard-Sphere Mixtures

VI. Phase Diagrams of Anisotropic Hard Particles

VII. Entropy Strikes Back Once More

Acknowledgments

References

Chapter 3: Sub-Nano Clusters: The Last Frontier of Inorganic Chemistry

I. Introduction

II. Chemical Bonding Phenomena in Clusters

III. Cluster-Based Technologies and Opportunities

IV. Conclusions

Acknowledgments

References

Chapter 4: Supercooled Liquids and Glasses by Dielectric Relaxation Spectroscopy

I. Introduction

II. Permittivity Fundamentals

III. Response Functions

IV. Linear Experimental Techniques

V. Nonlinear Experimental Techniques

VI. Applications

VII. Concluding Remarks and Outlook

Acknowledgments

References

Chapter 5: Confined Fluids: Structure, Properties and Phase Behavior

I. Introduction

II. Macroscopic Description of Nanoconfined Fluids

III. The Density Functional Theory Description of Confined Fluids

IV. Structure and Phase Behavior in Confined Colloid Suspensions

V. Nanoconfined Water

VI. Epilogue

References

Chapter 6: Theories and Quantum Chemical Calculations of Linear and Sum-Frequency Generation Spectroscopies, and Intramolecular Vibrational Redistribution and Density Matrix Treatment of Ultrafast Dynamics

I. Introduction

II. Recent Developments of Spectroscopies and Dynamics of Molecules

III. Theory and Applications of SFG

IV. Intramolecular Vibrational Redistribution

V. Ultrafast Dynamics and Density Matrix Method

References

Chapter 7: On The Kramers Very Low Damping Escape Rate for Point Particles and Classical Spins

I. Introduction

II. The Contribution of Kramers to Escape Rate Theory

III. Energy-Controlled Diffusion Equation for Particles with Separable and Additive Hamiltonians

IV. Energy-Controlled Diffusion of Classical Spins

V. Conclusion

Appendix A: Longest Relaxation Time for a Double-Well Potential, Eq. (13), in the VLD Limit

Appendix B: Undamped Limit for Biaxial Anisotropy

References

Author Index

Subject Index

End User License Agreement

List of Tables

Chapter 2

Table I

Chapter 4

Table I

Table II

Table III

Chapter 5

Table I

Chapter 6

Table I

Table II

Table III

Table IV

Table V

Table VI

Table VII

Table VIII

Table IX

Table X

Table XI

List of Illustrations

Chapter 1

Figure 1 (a) A section of Beethoven’s fifth symphony, showing that if a musical score is viewed as a plot of the time–frequency plane there is strong correlation between frequency and time. Note that most of the time–frequency phase space cells are empty. (b) A schematic representation of the von Neumann lattice in which one Gaussian is placed in every phase space cell of area h. For a color version of this figure, see the color plate section.

Figure 2 Classical phase space contours for (a) harmonic oscillator Hamiltonian, (b) Coulomb Hamiltonian.

Figure 3 (a) A schematic diagram of the development of the von Neumann/Gabor method in the quantum mechanics and signal processing communities. The development proceeded largely independently. (b) Quotes from the quantum mechanics and signal processing literatures indicating that the von Neumann/Gabor basis on a truncated lattice does not converge.

Figure 4 Illustration of the Dirichlet or periodic sinc functions. These functions are the underlying basis of the fast Fourier transform (discrete Fourier transform with periodic boundary conditions). They go to 1 at one of the Fourier grid points and to 0 at all the other Fourier grid points. The various members of the basis are orthonormal. For a color version of this figure, see the color plate section.

Figure 5 (a) N = 9 coordinate grid points and N = 9 Gabor unit cells cover the same area in phase space, S = 2πN. Superimposed is a typical Gabor function. Note that its boundary conditions are not appropriate for the rectangular area. (b) The periodic Gabor (pg) basis is a complete set for the truncated space. The pg basis functions are, loosely speaking, periodic and band-limited Gaussians whose centers are located at the center of each unit cell.

Figure 6 Magnitude of S (a) and S− 1 (b) matrices on a logarithmic scale. For a color version of this figure, see the color plate section.

Figure 7 A typical biorthogonal basis function.

Figure 8 Example of a non-intuitive Fourier transform pair. The solid line shows the amplitude and the dashed line the phase. The V phase profile in frequency leads to a double-pulse structure in time.

Figure 9 The Wigner representation (a), the Husimi representation (b), and the von Neumann representation (c) of the pulse in Fig. 8. All three representations allow the visualization of the pulse in time and frequency simultaneously. For a color version of this figure, see the color plate section.

Figure 10 Transformation of a Gaussian pulse from frequency to the von Neumann representation and back without periodic boundary conditions. The error in the back-transformed signal is quite significant (blue vs. red curve in (c)). Panel (a) shows the amplitude of the von Neumann representation and panel (b) the phase. Adapted from Ref. [15]. For a color version of this figure, see the color plate section.

Figure 11 Transformation of a Gaussian pulse from frequency to the von Neumann representation and back with periodic boundary conditions. The back-transformed signal agrees with the original signal to the accuracy of the computer (blue vs. red curve in (c)). Panel (a) shows the amplitude of the von Neumann representation and panel (b) the phase. Adapted from Ref. [15]. For a color version of this figure, see the color plate section.

Figure 12 (a) Error in the seventh eigenvalue of the harmonic oscillator for a rectangular phase space grid as a function of the basis set size N. The pvN, pvb, and Fourier grid methods all give identical results (solid), 14 orders of magnitude more accurate than the usual vN basis (dashed). (b) Kinetic energy spectrum using the vN basis(dashed) and using the FGH, pvN, and bvN basis (solid). (c). Error in the 24th eigenvalue of the Morse potential as one discards basis functions from a rectangular phase space lattice. The pvb (solid), pvN (dashed), and Fourier grid (dotted) behave completely differently. Removing even one basis function from the pvN introduces significant error, while more than two-thirds of the pvb basis functions can be removed without introducing any significant error. Adapted from Ref. [16].

Figure 13 (a) The triangle potential. (b) Comparison of the error in the highest eigenvalue of the Fourier (dashed) and pvb (solid) methods as a function of basis set size. Adapted from Ref. [16]. For a color version of this figure, see the color plate section.

Figure 14 (a) Phase space area spanned in the bvN method (magenta) and in the pvN (or FGH) method (full rectangle) for a 1D Morse oscillator Morse. (b) Efficiency ratio (defined as number of basis functions per converged eigenstates) of the bvN (solid) and FGH (dashed) methods for the 1D Morse oscillator as a function of h. (c) Efficiency ratio of the bvN (triangles) and FGH (circles) methods for the 2D triangle potential of Fig. 13 as a function of h. The solid triangle and circle are the efficiencies in the classical limit. Adapted from Ref. [16]. For a color version of this figure, see the color plate section.

Figure 15 (a) The periodic von Neumann method does not require identical, evenly spaced Gaussians. One may tile the phase space any way one likes as long as the rectangular tiles have area h. Then a basis of Gaussians whose centers and aspect ratios are matched to the rectangles will be a complete and stable basis. This flexibility in the positions and widths of the Gaussians can improve the efficiency of the pvN method significantly, particularly for problems that have multiple length scales. (b) The error in E3 for the Coulomb potential as a function of basis set size. FGH (dashed), pvb (circles), and wpvb (solid). Adapted from Ref. [17].

Figure 16 Schematic diagram of electron motion on the attosecond time scale. Left: Strong field ionization and recollision, leading to high harmonic generation. Right: Strong field manipulation of electronic motion in a diatomic molecule. For a color version of this figure, see the color plate section.

Figure 17 Vector potentials, and , of the NIR and XUV laser pulses applied to the model 1D atom. For a color version of this figure, see the color plate section.

Figure 18 Snapshots of the wavepacket coefficients shown by ellipses located at the Gaussian centers . The colors of the ellipses indicate the magnitude of |cj|² according to the scale above the figure. The sequence of dark blue dots represent the simple-man trajectories (i.e., classical trajectories evolving in the presence of the field without the Coulomb potential) for direct ionization; the light blue dots represent the rescattered simple-man trajectories. The dark blue + marks represent the simple-man trajectories absorbing one XUV photon in the presence of the NIR field. The snapshots were taken at (a) t = −2.06, (b) t = 0.69, and (c) t = 2.06 in units of NIR cycles. These times are indicated by the green × marks in Fig. 17. Adapted from Ref. [18]. For a color version of this figure, see the color plate section.

Figure 19 Comparison of the photoelectron momentum distributions obtained with the reduced pvb basis (blue solid line) and full pvb basis (red dashed line). The momentum distribution from a simulation without the XUV pulse (using the full pvb basis) is also shown (gray solid line). The vertical dashed lines indicate the cut-offs of the direct (N1 and N1′) and rescattered (N2 and N2′) photoelectrons, as well as the NIR-streaked single-XUV-photon ionization peaks (X1 and X1′), estimated by the simple-man model. Adapted from Ref. [18]. For a color version of this figure, see the color plate section.

Figure 20 The error ε as a function of (black × marks). The horizontal error bars indicate the range of in . The data marked by the red filled circle is from the simulation shown in Figs. 18 and 19. Adapted from Ref. [18]. For a color version of this figure, see the color plate section.

Figure 21 The splat signal in time (a), frequency (b), and pgb (c) representations. Adapted from Ref. [19]. For a color version of this figure, see the color plate section.

Figure 22 The norm of the error of the reconstructed signal as a function of the number of basis functions using the DFT (red), the DGE with an additional correction due to Porat (green), the pgb (blue), and the pgb with a correction developed by Porat (black). Adapted from Ref. [19]. For a color version of this figure, see the color plate section.

Figure 23 Reconstruction of the Barbara image using about 8% (20,552) of the coefficients. (a) Original picture. (b) DGE method. (c) DCT transformation. (d) DCT on 8 × 8 blocks. (e) Daubechies wavelet. (f) pgb method. Adapted from Ref. [19].

Figure 24 A detailed part of the original image of Fig. 23 (a) and the reconstructions using DCT on 8 × 8 blocks (b) and the pgb method (c). The artificial effect of blocking is much less severe in the pgb method. Adapted from Ref. [19].

Figure 25 A detailed part of the original image of Fig. 23 (a) and the reconstructions using Daubechies wavelets (b) and the pgb method (c). The pgb method is seen to be much closer to the original than wavelet compression which is known to cause blurring for areas with rich sharp edges [73]. Adapted from Ref. [19].

Chapter 2

Figure 1 Close-packed unit cell for the Great Stellated Dodecahedron. Adapted from the data presented in the supplementary information of Ref. [69] for the densest packings of a huge variety of anisotropic particle shapes as obtained from the floppy-box Monte Carlo method. For each shape we give a figure depicting the particle, the unit cell, and a small piece of the crystal, the maximum packing fraction ϕUB that we obtained, the number of particles in the unit cell, the lattice vectors, and positions and orientations of the particles in the unit cell, etc.

Figure 2 Schematic illustration of the common tangent construction to determine phase coexistence at varying temperatures as denoted by the horizontal dashed lines in the phase diagram in the temperature T–density ρσ³ representation in (d). The free-energy density f = F/V versus the density ρσ³, showing (a) the existence of a symmetry-breaking liquid–solid transition at high temperature, (b) a symmetry-conserving gas–liquid transition at low densities and a symmetry-breaking liquid–solid transition at higher densities at intermediate temperature, (c) a metastable gas–liquid transition with respect to a stable liquid–solid transition at low temperature.

Figure 3 Phase diagram for a binary hard-sphere mixture with size ratio q = 0.82 in the reduced pressure p–composition x representation with p = βPσ³L, x = NS/(NS + NL), N(S)L the number of (small) large spheres, and σ(S)L the diameter of (small) large spheres. Adapted from Ref. [38]. FCC(S) denotes a face-centered cubic crystal of small spheres, FCC(L) denotes a face-centered cubic crystal of large spheres. The phase coexistence regions are labeled FCC(L) + Laves, FCC(S) + Laves, Laves + Fluid, etc.

Figure 4 Phase diagram for a binary hard-sphere mixture with size ratio q = 0.3 in the reduced pressure p–composition xs representation with p = βPσ³L, xS = NS/(NS + NL), N(S)L the number of (small) large spheres, and σ(S)L the diameter of (small) large spheres [131]. The interstitial solid solution is denoted by ISS, FCC(S) denotes a face-centered cubic crystal of small spheres, FCC(L) denotes a face-centered cubic crystal of large spheres, and LS6 denotes a binary superlattice structure. A typical configuration of the ISS phase is shown in the inset of the phase diagram. (a) shows a configuration of the pure FCC of large spheres, (b) of the NaCl phase, and (c) the LS6 phase. The top inset in the phase diagram shows that the filling fraction of the octahedral holes in the coexisting ISS phase increases with pressure from 0 (pure FCC of large spheres) to 1 (NaCl phase). The trajectory of a single small sphere in the FCC lattice of big spheres at a volume fraction ηL = 0.6. Note that the small particle in an octahedral hole (d) hops first to a tetrahedral hole (e), and then to the next octahedral hole (f). Adapted from Ref. [131].

Figure 5 The phase diagram of hard dumbbells in the reduced density ρ* (and packing fraction ϕ) versus L* = L/σ representation, where L is the distance between the centers of the spheres and σ is the diameter of the spheres as denoted in the schematic picture of a dumbbell [56, 57, 167, 168]. Hence the model reduces to hard spheres for L* = 0 and to tangent spheres for L* = 1. The dimensionless density is defined as ρ* = d³N/V with N the number of particles, V the volume, and d³/σ³ = 1 + 3L*/2 − 1/2(L*)³ is the volume of a dumbbell divided by that of a sphere with diameter σ, so that d is the diameter of a sphere with the same volume as the dumbbell. F denotes the fluid phase and CP1 the periodic crystal. The aperiodic phase aper is stable only in a narrow region of the phase diagram. The stable face-centered cubic type plastic crystal is denoted by filled squares, the hexagonal-close-packed plastic crystal phase is denoted by empty squares. Adapted from Ref. [57].

Figure 6 The phase diagram of hard snowman particles in the size ratio q–packing fraction η representation, with q = σs/σl ranging from 0 (the hard sphere) to q = 1 (the tangential dumbbell), σs is the diameter of the smaller sphere and σl is the diameter of the larger sphere. Adapted from Ref. [170]. The packing fraction is defined as η = Nv0/V, where v0 is the particle volume for a given q value. Circles indicate coexisting phases, while the lines are intended to guide the eye. At the top of the plot we indicate the density of closest packing, with triangles indicating the crossover from one close-packed structure to another. Isotropic denotes the isotropic fluid phase, NaCl, CrB, γCuTi, αIrV, and FCC* denotes aperiodic crystal structures, which are stabilized by the degeneracy of the crystal structure (i.e., the number of bond configurations). Rotator denotes a plastic crystal phase.

Figure 7 The phase diagram of hard asymmetric dumbbell particles with size ratio q = σs/σl = 0.5 in the reduced center-of-mass distance L*–packing fraction η representation. Adapted from Ref. [171]. The diameter of the (smaller) larger sphere is denoted by (σs) σl and the reduced center-of-mass distance is defined as L* = (2L + σs − σl)/2σl ranging from 0 (hard spheres) to 0.5 (tangential snowman-shaped particles). APC denotes the aperiodic CrB phase, CrB denotes the periodic CrB crystal, NaCl denotes the periodic NaCl crystal phase. Circles indicate coexisting phases, while the lines are guides to the eye. The density of the maximum packing is denoted by the line at the top of the figure, and the triangles indicate crossover points from one close-packed structure to another.

Figure 8 (a) Side view of an oblate spherocylinder for L/σ = 0.2, where L denotes the thickness of the plate and σ the diameter. An oblate spherocylinder is obtained by padding a disk of diameter D, as indicated by the black line, with a layer of uniform thickness L/2. (b) The phase diagram of hard oblate spherocylinders in the packing fraction ϕ–reduced thickness L/σ representation. The state points in the dark grey area are inaccessible since they lie above the maximum close packing line. "Xaligned and Xtilted denote the aligned and tilted crystal structures as shown in (c) and (d), iso denotes the isotropic fluid, nem the nematic phase, and col" the columnar phase. The solid lines are a guide to the eye, connecting coexistence points. The data for L/σ = 0 are taken from Ref. [182]. (c) The unit cell of the tilted crystal phase for L/σ = 0.3 and (d) the aligned crystal phase for L/σ = 0.5. Adapted from Ref. [53].

Figure 9 (a) The bulk phase diagram of hard cubes as a function of packing fraction η. A stable fluid phase is found for η < 0.45, and a stable simple cubic crystal phase with vacancies is observed for η > 0.50. Coexistence between the crystal and fluid is found for 0.45 < η < 0.50. (b) A typical configuration of a simple cubic crystal phase of hard cubes at η = 0.52 and vacancy concentration of 1.6%. The particles surrounding the delocalized defects are yellow. The defect at the top has six cubes sharing seven lattice positions, the defect at the right bottom has three cubes sharing four lattice positions, and the defect at the left bottom shows seven cubes spread over eight lattice positions. Adapted from Ref. [54].

Figure 10 (a) The shape of superballs interpolates between octahedra (q = 0.5) and cubes (q = ∞) via spheres (q = 1). (b) The bulk phase diagram of hard superballs as a function of packing fraction ϕ versus 1/q (bottom axis) and q (top axis) representation where q is the shape parameter [82]. The "C1 and C0" crystal phases are defined in Refs. [50, 51], where the particles of the same color are in the same layer of stacking. The solid diamonds indicate close packing, and the locations of triple points are determined by extrapolation as shown by the dashed lines. The phase boundaries for hard cubes are taken from Ref. [54]. Adapted from Ref. [82].

Figure 11 Phase diagram of hard bowl-shaped particles in the packing fraction (ϕ) versus thickness (D/σ) representation. Adapted from Refs. [90, 91]. The light grey areas denote the coexistence regions, while the dark grey area indicates the forbidden region as it exceeds the maximum packing fraction of the bowls. The lines are a guide to the eye. The inset in the phase diagram shows the theoretical model of the bowl-shaped particle, which is the solid of revolution of a crescent around the axis as indicated by the dashed line. The thickness of the bowl is denoted by D and the diameter of the bowl by σ. The stable crystal phases, IX, IX′, IB, and fcc², the fluid, and hexagonal columnar phase col are drawn schematically below the phase diagram.

Chapter 3

Figure 1 Representatives of unusual clusters: (a) B9− is a wheel [1]. (b) TaB10− [2]. (c) Au20 [3]. (d) Clusters containing tetracoordinated planar C and Si atoms [4]. (e) Stannaspherene [5].

Figure 2 (a) Left: LiAl4− structure; right: atomic charges, and populated valence MOs of Al4²− [43]. (b) Left: Li3Al4− structure and atomic charges; right: populated valence MOs of Al4⁴− [41]. MO types are labeled as σ-radial, σ-peripheral, π-, or LP (lone pair), as described in the text.

Figure 3 Boron wheels, and their delocalized valence MOs responsible for the doubly aromatic character of chemical bonding [1]. For a color version of this figure, see the color plate section.

Figure 4 Ta3O3− the first cluster possesses π- and δ-aromaticity [45]. Reproduced with permission from JWS, Angewandte Chemie International Edition, April 30 2007.

Figure 5 Structures of the gold clusters derived from the tetrahedral Au20 species and patterns of chemical bonding according to AdNDP [3]. Reproduced with permission from American Chemical Society, The Journal of Physical Chemistry A, Feb 2009. For a color version of this figure, see the color plate section.

Figure 6 B6²− and its valence MOs. The decomposition of the MOs onto those of π- and σ-type, and localizable as 2c–2e B–B bonds is shown [51].

Figure 7 B6²− and Al6²− have different structures rooted in the differences in chemical bonding [50, 56, 57]. The B cluster has covalent bonding that defines its planar shape, whereas in the Al cluster all bonding is delocalized. Reproduced with permission from American Chemical Society, The Journal of Physical Chemistry Letters, August 2011.

Figure 8 LiNa4− versus LiK4−: their different global minimum structures and valence MOs. The s–p hybridization of AOs on Li in one case but not in the other is illustrated [6].

Figure 9 (a) CAl4−, (b) SiAl4− as a representative of all tetraatomic C2v clusters of the group III doped with a single group IV atom. The AO hybridization is demonstrated in a, as opposed to b. Square clusters are CAl4− and SiIn4−. Distorted covalent clusters are SiAl4−, GeAl4−, and SiGa4− [4].

Figure 10 LiB6− and LiB8−, the prototypic clusters containing doubly antiaromatic and doubly aromatic all-boron ligands [22, 24].

Figure 11 Energy levels in atoms and clusters. Also shown are the electronic levels in a Cl atom and that in an Al13 cluster [61]. Reproduced with permission from American Chemical Society, The Journal Physical Chemistry C, Feb 2009.

Figure 12 (a) The Ti7Rh4Ir2B6 solid containing the flat, highly charged B6 core [28]. (b) Salts of Re and Ru containing flat, highly charged boranes in the core [27, 29].

Figure 13 (a) B13+ can undergo a low-barrier intra-cluster rotation of the inner cycle with respect to the outer cycle (blue line), however, the energy landscape changes in the presence of an electric field pointing to the right: the barrier to the rotation to the left disappears, and the barrier to the rotation to the right grows threefold (red line). (b) Thus, in the presence of a rotating electric field in the THz range, the cluster can be driven as a Wankel motor, but circularly polarized light [55]. Reproduced with permission from John Wiley & Sons, Angewandte Chemie International Edition, July 2012. For a color version of this figure, see the color plate section.

Figure 14 Catalytic activity of size-selected surface-deposited clusters as a function of cluster size. (a) Pd clusters on titania [12]. Reproduced with permission from The American Association for the Advancement of Science, Science, November 2009. (b) Au clusters on titania [13]. Reproduced with permission from American Chemical Society, Journal of the American Society, March 2014.

Figure 15 Chemical bonding in flat Pd4 deposited on titania is explained: (a) structure changes from 3D in the gas phase to 2D when the cluster is deposited on titania surface; (b) together with this, double s-aromaticity emerges in the cluster, which should correlate with specific stability and reactivity; (c) the flattening is facilitated by the matching with the surface oxygen atoms to which Pd binds [106]. For a color version of this figure, see the color plate section.

Chapter 4

Figure 1 Result for the average orientation, ⟨cos θ⟩, on the basis of a Boltzmann population using Eq. (5), indicating the effect of saturation at ⟨cos θ⟩ = 1 and showing the regime of linearity with respect to the field for μE ≪ kBT.

Figure 2 Displacement retardation D(t) in time domain in response to a field step E(t). In this constant field situation, polarization P(t) is linearly related to both D(t) and ϵ(t).

Figure 3 Electric field relaxation E(t) in time domain in response to a displacement (or charge) step D(t). In this constant charge situation, polarization P(t) is linearly related to both E(t) and M(t).

Figure 4 Schematic illustration of the Boltzmann superposition, with the field E(t) being approximated by three step functions, see upper panel. The lower panel shows the individual responses to the steps, Pn(t−tn), which differ only in their amplitude and initial time. The total polarization Psum(t) is then the sum of the Pn(t−tn) curves.

Figure 5 (a) Gaussian fluctuation of a variable x calculated via Langevin random forces. (b) The solid line represents the autocorrelation function of the above noise, and the dashed line is the exponential correlation decay that would result in the limit of better statistics.

Figure 6 Dielectric permittivity and modulus of a Debye type system with ϵ∞ = 3, ϵs = 30, and a retardation time of τϵ. The system also has a dc conductivity at a level of σdc = 0.03 × ϵ0. Note that the relaxation time τM is one-tenth of the retardation time τϵ, and that the signature of conductivity changes from a low frequency wing with ϵ′′ ∝ 1/ω to a Debye peak at ωτσ = 1 that takes the static limit of M′ to zero.

Figure 7 Equivalent circuit of a Debye type permittivity with ϵ∞ ∝ C∞, Δϵ ∝ C, and τ = RC, based upon loss-free capacitors and a purely ohmic resistor. The relations among current and voltages are given for the RC series path.

Figure 8 Polarization, P(t), and power, p(t), within a single RC series circuit subject to the external voltage Vx(t) = sin(ωt), which is zero before and after the two periods shown [87]. Note the non-stationary behavior of the P(t) and p(t) curves and their decay after ωt = 4π, that is, after two periods of an external voltage with ω = 1 s−1.

Figure 9 Equivalent circuit for a dielectric with dispersive permittivity, modeled here as a superposition of RiCi type Debye responses. The capacitances reflect the probability density of retardation times, τ = RC.

Figure 10 Set of normalized KWW type decay curves for stretching exponents β = 1.0–0.3 in steps of 0.1, in the order from most to least steep curve at t = τ0. The inset shows the derivative dϕ/d ln t multiplied by (−e) for the β = 0.6 case, indicating that the curve peaks at t = τ0 with an amplitude equal to β.

Figure 11 Set of loss profiles for the Cole–Cole type permittivity, normalized to Δϵ/2 and on a logarithmic frequency scale relative to the characteristic time constant τ0. Different curves are for different values of the exponent α from 1.0 to 0.3 in steps of 0.1 in the order from narrowest to widest curve.

Figure 12 Set of loss profiles for the Cole–Davidson type permittivity, normalized to Δϵ/2 and on a logarithmic frequency scale relative to the characteristic time constant τ0. Different curves are for different values of the exponent γ from 1.0 to 0.3 in steps of 0.1 in the order from narrowest to widest curve.

Figure 13 Set of loss profiles for the Havriliak–Negami type permittivity, normalized to Δϵ/2 and on a logarithmic frequency scale relative to the characteristic time constant τ0. Different curves are for different values of the exponents α and γ, selected to match the KWW stretching exponent β from 1.0 to 0.3 in steps of 0.1 in the order from narrowest to widest curve.

Figure 14 Pairs of Havriliak–Negami exponents α and γ matched to the Kohlrausch–Williams–Watts stretching exponent β, based upon fitting synthetic data (solid symbols [104]) and upon aligning moments of the probability densities (open symbols [105]). Triangles pointing up are for α, those pointing down are for γ.

Figure 15 Complex permittivity of a Debye type retardation (solid symbols, solid lines) and of a dispersive counterpart (open symbols, dashed lines). The three projections are ϵ′(ω) versus log(ωτ) on the bottom plane, ϵ′′(ω) versus log(ωτ) on the right side plane, and ϵ′′(ω) versus ϵ′(ω) on the left side plane.

Figure 16 Basic layout of a time-domain measurement of permittivity, that is, polarization under constant field condition. The switch is closed for times t > 0. Measuring the displacement via a reference capacitor is sometimes referred to as modified Sawyer–Tower bridge [115]. Electrometer plus attached meter is meant to indicate an ideal voltmeter, with the voltage measured being V(t) = Q(t)/Cref. and thus proportional to ϵ(t).

Figure 17 Basic layout of a time-domain measurement of electric modulus, that is, polarization under constant charge condition [117]. The switch is closed shortly at time t = 0 in order to generate a charge step. The guard eliminates the cable capacitance. Electrometer plus attached meter is meant to indicate an ideal voltmeter, with the voltage measured being proportional to M(t).

Figure 18 Principle of a time-domain reflectometry measurement [120]. An incident signal is sent down a transmission line that is terminated by the sample. Both incident signal v0(t) and reflected signal rx(t) are recorded.

Figure 19 Basic schematic of a typical impedance experiment, with applied voltage V0, the current being measured with a transimpedance amplifier (I→V) with virtual ground input characteristic. The wiring to the sample capacitor includes the typical parasitic components Rs, Ls, Cp, Gp. A gain/phase analysis of current and voltage is then used to determine the impedance Z and then the permittivity of the sample.

Figure 20 Fringing fields as indicated in panel (a) generate stray capacitances and inhomogeneous fields when the electrode distance is not very small compared with the lateral dimensions. The panel (b) indicates how a Kelvin guard ring held at the same potential as the low-potential electrode eliminates fringing field errors if the gap is small compared with the electrode distance.

Figure 21 (a) Schematic illustration of a pair of interdigitated electrodes (IDEs) and the connections to the gain–phase analyzer equipped with two transimpedance amplifiers (I→V) as used in the simultaneous dual-channel dielectric measurements [138]. (b) Side view of an IDE on the substrate loaded with sample. The electric field lines between neighboring electrode fingers are meant to indicate that both substrate and sample contribute to the impedance.

Figure 22 The solid line represents synthetic dielectric loss data for a 100 μm thick dielectric with a dipole reorientation peak at ν ≈ 10⁵ Hz and dc conductivity. The dashed line indicates the apparent loss curve if the same sample is measured with an additional 15 μm thick Teflon sheet. Without proper correction, the peak amplitude and position as well as the appearance of dc conductivity appear altered [152].

Figure 23 Schematic representation of nonlinear D(E) behavior by a third-order polynomial that deviates symmetrically from the dashed linear counterpart. Nonlinear effects can be probed by a large amplitude sine with bias, Eac(t), or by a small oscillation around a large dc field, Edc(t). Only the latter case will also show higher-order susceptibilities of even order, for example, χ2.

Figure 24 Time resolved variant of a high-field impedance experiment using a field that changes amplitude for a given number of cycles [163]. Recording both field, E(t), and displacement, D(t), allows one to perform a period-by-period Fourier analysis. The lower panel indicates that transient effect and a slowly accumulated nonlinear effect could be seen.

Figure 25 (a) Schematic representation of the phase cycle used in DHB. The traces P1 and P2 are for the linear experiment without a preceding pump field. The traces P3–P6 are for large amplitude pump sine waves preceding the step response measurement. The aim is to eliminate the additional large responses that originate from the pump field. (b) Final response curves, the original P(t) and the field modified P*(t), that differ in a certain time range.

Figure 26 Temperature dependence of the Kirkwood correlation factor, gK, for four different octanol isomers, as reported by Dannhauser [180]. The compounds are 6-methyl-3-heptanol, 5-methyl-3-heptanol, 4-methyl-3-heptanol, and 3-methyl-3-heptanol, in the order of decreasing gK at low temperature.

Figure 27 Dielectric loss data of water at 19°C, as compiled by Kaatze et al. [192]. The curve is described by a main Debye process with Δϵ = 72, τ1 = 10.2 ps, and another faster and smaller process with Δϵ = 2.3, τ2 = 0.13 ps. For comparison, hexagonal ice at −10.8°C [196] shows a somewhat larger and much slower loss peak with Δϵ = 92, τ2 = 60 μs.

Figure 28 Cole–Cole representation, ϵ′′(ω) versus ϵ′(ω), of the prominent loss component of the dielectric response of 2-methyl-1-butanol at T = 142 K [200]. The symbols represent experimental results in the frequency range 40 mHz to 2 MHz. The inset enlarges the deviations from the Debye fit at high frequencies, and the dashed line shows the Debye component of the fit.

Figure 29 Experimental decay results of the normalized dielectric modulus M(t) for poly(vinylacetate) for temperatures from 291 K to 323 K in steps of 2 K and in the order from slowest to fastest decay [39, 111]. The polarization decay reflects segmental reorientation. The time scale is shifted by log10(τϵ/τM) = 1.36 to reflect the experimental time it would have required if the polarization had been measured at constant field. The average dielectric retardation times, τϵ, derived from these data range from 2 seconds to 3.4 × 10⁷ seconds, which exceeds 1 year at T = 291 K.

Figure 30 Dielectric permittivity, ϵ′ and ϵ′′, of glycerol for frequencies from below 10−5 Hz to above 10¹² Hz [210]. Different curves are for different temperatures as indicated.

Figure 31 (a) The dielectric peak loss time constant τmax versus reciprocal temperature for 1-propanol together with the failure of a single VFT fit. (b) Identification of three distinct temperature regimes via a derivative analysis that converts a VFT type dependence into a straight line. TB marks the transition from a lower temperature VFT law to another VFT regime with different parameters, TA indicates the transition to Arrhenius behavior at high temperatures [216].

Figure 32 Frequency-dependent dielectric loss, ϵ′′(ω)/ϵ′′(ωmax) versus log10(ω/ωmax), of deuterated ααβ-tris-naphthylbenzene (C36H10D14, DTNB, symbols), obtained by normalizing the 345 K to 417 K loss data in order to generate a master curve [232]. At the bottom end of the ordinate scale, the loss factor is as low as tan δ = 3 × 10−5. Within this temperature range, the dynamics follows time–temperature superposition. The lines represent the HN slopes with α = 0.93 and γ = 0.40.

Figure 33 (a) Dielectric loss of a 1 wt% mixture of di-n-butylether in 3-methylpentane. The 27 curves are for different temperatures between 81.5 K and 107.0 K, measured in steps of approximately 1 K. Each curve displays two retardation peaks separated by one decade on the frequency scale. (b) Master plot of the dielectric loss of a 1 wt% mixture of di-n-butylether (DBE) in 3-methylpentane (3MP). The solid line represents a fit using the sum of two HN processes, the dashed lines represent the individual contributions with fit parameters: Δϵh = 0.028, αh = 0.88, γh = 0.46, τh = 1.03 × 10−4 seconds for 3MP (host) and Δϵg = 0.026, αg = 0.98, γg = 1.00, τg = 8.61 × 10−4 seconds for DBE (guest) [259].

Figure 34 Dielectric retardation data for glycerol at a temperature T = 238.0 K obtained using a parallel disc cell with standard filling procedure [85]. Solid diamonds and dots show ϵ′(ω) and ϵ′′(ω), respectively. Open diamonds are ϵ′′(ω) values as estimated via the derivative of ϵ′(ω), Eq. (52). The dashed curve represents the conductivity peak of the modulus, M′′(ω), scaled arbitrarily. The arrow indicates the frequency at which ϵ′ = ϵ′′ and ωτσ = 1, with τσ being the conductivity relaxation time.

Figure 35 Experimental dielectric data for ortho-terphenyl (OTP) plotted as tanδ measured at ν = 1 kHz as a function of temperature T [271]. The solid curve below Tg = 246 K is for the sample after a quench to T = 25 K. The dashed curve below Tg is obtained after annealing the sample for 24 hours at T ≈ Tg − 10 K, the portion above Tg is recorded afterward.

Figure 36 Dielectric loss curves ϵ′′(ω) of D-sorbitol as a function of temperature in the range 132 K ≤ T ≤ 302 K [129, 274]. The curves are plotted in steps of 4 K for 132 K ≤ T ≤ 264 K and in steps of 2 K for 266 K ≤ T ≤ 302 K. The low-frequency wings due to dc conductivity are drawn as dashed lines.

Figure 37 (a) Equilibrium loss profile of D-sorbitol at T = 261 K indicating the frequency position of the aging experiment. (b) Symbols represent the experimental results for the dielectric loss, ϵ′′, versus aging time, tage, for sorbitol (Tage = 261 K) at various test frequencies between ν = 10 mHz and 100 kHz, as indicated in (a). Lines are fits based upon the LWSL model based on a common να(tage) curve, that is, the lines differ only in their limiting values ϵ′′st and ϵ′′eq [281].

Figure 38 Solid symbols represent the experimental results for the equilibrium dielectric retardation times, τα, as a function of temperature for D-sorbitol (SOR), xylitol (XYL), glycerol (GLY), propylene glycol (PG), and propylene carbonate (PC). Lines are VFT fits to equilibrium data. Open diamonds represent the values of τα(Tage) obtained from analyzing the aging data, stars identify the retardation times τα of the VFT fits if extrapolated to T = Tage [281].

Figure 39 Dielectric loss spectra of a 600 nm ultrastable IMC film at different times during annealing at Tann = 325.0 K, with loss amplitudes rising with increasing annealing time. The lines are best HN fits using the same retardation time (τHN) and shape parameters (αHN, γHN) that describe the conventional IMC supercooled liquid. The inset shows the retardation intensity obtained by HN fitting as a function of annealing time, indicating a linear rise of Δϵ(tann) until transformation is complete [290].

Figure 40 Impedance data shown as σ′(ω) for the ionic liquid butyl-methyl-immidazolium-PF6 for temperatures from T = 186 K to 210 K in steps of 2 K, and from T = 214 K to 230 K in steps of 4 K. The dashed line indicates the level of σdc = 1.7 × 10−11 S cm−1 for the T = 200 K case (full symbols), where σ′ is practically independent of frequency and, accordingly, σ′′ ≪ σ′ [305].

Figure 41 Impedance data shown as M′′(ω) for the ionic liquid butyl-methyl-immidazolium-PF6 for temperatures from T = 186 K to 210 K in steps of 2 K, and from T = 214 K to 230 K in steps of 4 K. The dashed line indicates the largest possible dc-conductivity contribution consistent with σdc = 1.7 × 10−11 S cm−1 and ϵs = 10 for the T = 200 K case (full symbols) [305].

Figure 42 Comparison of a dielectric measurement (tan δϵ) of bulk poly(vinylacetate) at T = 323.1 K and a Kelvin force microscope measurement (tan δv) of two films of different thickness. The 45 nm and 55 nm films were actually measured at T = 324.0 K and T = 322.5 K, respectively, but the data shown are shifted by 0.148/K along the log10(ν/Hz) scale to correspond to the temperature of the bulk curve [314, 315].

Figure 43 Plot of the optical variance, σ²(t), reduced by its steady state value and normalized to the square of the dynamical Stokes shift, versus the Stokes shift correlation function, C(t), for quinoxaline in 2-methyltetrahydrofuran. Symbols are for different temperatures as indicated, and stretched exponential fits to C(t) at each temperature yielded β = 0.5 and the time constants τ0 listed in the legend. The line is based on Eq. (108) and assumes purely exponential local dynamics, χ(t,τ) = exp(−t/τ) [328].

Figure 44 Two simple ways of arranging a two-component heterogeneous dielectric with permittivities ϵ1 and ϵ2 and volume fraction ϕ = 0.5. The parallel