Advances in Chemical Physics, Volume 163 E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Advances in Chemical Physics
- Sprache: Englisch
The
Advances in Chemical Physics series provides the chemical physics field with a forum for critical, authoritative evaluations of advances in every area of the discipline.
• This is the only series of volumes available that presents the cutting edge of research in chemical physics
• Includes 10 contributions from leading experts in this field of research
• Contains a representative cross-section of research in chemical reaction dynamics and state of the art quantum description of intramolecular and intermolecular dynamics
• Structured with an editorial framework that makes the book an excellent supplement to an advanced graduate class in physical chemistry, chemical physics, or molecular physics
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Seitenzahl: 602
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Editorial Board
List of Contributors Volume 163
Foreword
Preface to the Series
Chapter 1: Applications of Quantum Statistical Methods to the Treatment of Collisions
I. Introduction
II. Quantum Statistical Theory
III. Fine-Structure Branching in Reactive O(
1
D) + H
2
Dynamics
IV. Inelastic OH + H Collisions
V. OH + O Reaction and Vibrational Relaxation
VI. Inelastic Collisions of the CH Radical
VII. H + O
2
Transport Properties
VIII. Conclusion
Acknowledgments
References
Chapter 2: Quantum Dynamics in Photodetachment of Polyatomic Anions
I. Introduction
II. Potential Energy Surfaces
III. Quantum Dynamics
IV. Systems
V. Conclusions
Acknowledgments
References
Chapter 3: Recent Advances in Quantum Dynamics Studies of Gas-Surface Reactions
I. Introduction
II. Theoretical Methods
III. Potential Energy Surface
IV. Site-Averaged Approximation
V. Applications
VI. Conclusions
Acknowledgments
References
Chapter 4: Quantum Scattering and Semiclassical Transition State Theory Calculations on Chemical Reactions of Polyatomic Molecules in Reduced Dimensions
I. Introduction
II. Quantum Scattering Calculations on Polyatomic Reactions in Reduced Dimensions
III. Semiclassical Transition State Theory
IV. Conclusions
Personal Note on John Light (David Clary)
Acknowledgements
References
Chapter 5: Adiabatic Switching Applied to the Vibrations of syn-CH3CHOO and Implications for “Zero-Point Leak” and Isomerization in Quasiclassical Trajectory Calculations
I. Introduction
II. Theory and Computational Details
III. Results and Discussion
IV. Summary and Conclusions
Acknowledgments
References
Chapter 6: Inelastic Charge-Transfer Dynamics in Donor–Bridge–Acceptor Systems Using Optimal Modes
I. Introduction
II. Theoretical Approach
III. Inelastic Electronic Coupling in Donor–Bridge–Acceptor Complexes
IV. Discussion
Acknowledgments
References
Chapter 7: Coupled Translation–Rotation Dynamics of H2 and H2O Inside C60: Rigorous Quantum Treatment
I. Introduction
II. H
2
@C
60
III. H
2
O@C
60
IV. Conclusions and future prospects
Acknowledgments
References
Chapter 8: Using Iterative Eigensolvers to Compute Vibrational Spectra
I. Introduction
II. Direct-Product Basis Sets
III. Using a Direct-Product Basis Set to Solve the Schroedinger Equation
IV. Using a DVR to Make a Contracted Basis
V. Using Pruning to Reduce Both Basis and Grid Size
VI. Conclusion
Acknowledgments
References
Chapter 9: Large Scale Exact Quantum Dynamics Calculations: Using Phase Space to Truncate the Basis Effectively
I. Introduction
II. Background and Theory
III. New Results and Discussion
IV. Summary and Conclusions
Acknowledgments
References
Chapter 10: Phase-Space Versus Coordinate-Space Methods: Prognosis for Large Quantum Calculations
I. Introduction
II. Pedagogical Aspects of the Discrete Variable Representation
III. The von Neumann Basis
IV. Multidimensional Considerations
V. Applications
VI. Conclusions
Acknowledgments
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Applications of Quantum Statistical Methods to the Treatment of Collisions
Figure 1 Schematic diagram of the potential energy surfaces of the OHH system. Only the lowest () PES was taken into account in the initial quantum statistical calculations on the O(D) + reaction [14, 15]. Note that O () well lies 59,000 below the O(D) + asymptote.
Figure 2 Energies of the lower rotational/fine-structure levels of the OH(, ) manifold. The -doublet splitting has been exaggerated for clarity.
Figure 3 Contour plots (in ) of the OH()−H PESs for the (top row) , , and (bottom row) , states determined at the vibrationally averaged OH bond length . The angle corresponds to linear OHH geometry.
Figure 4 Schematic diagram of the potential energy surfaces relevant to OH(, ) + H OH() + H inelastic scattering. Relaxation can occur through both noncapture (direct) inelastic scattering (Eq. (8)) as well as direct (no hydrogen exchange; Eq. (9)) and exchange processes (Eq. (10)), both of which can sample the deep O well.
Figure 5 Direct and complex-mediated initial state-selected total cross section for OH() + H OH() + H vibrational relaxation. The cross section is averaged and summed over the initial and final -doublet levels, respectively. The inset panel reveals the relative size of the direct (noncomplex mediated) contribution.
Figure 6 Cross sections for the vibrational relaxation of OH() initial rotational levels in the lower, fine-structure manifold to OH() in collision with H atoms. The cross section is averaged and summed over the initial and final -doublet levels, respectively.
Figure 7 Comparison of thermally averaged vibrational removal rate constants for OH(, 2) in collision with H atoms. The experimental room-temperature value is from [53]. The filled circle and filled square designate rate constants derived from experiments in which O was excited, respectively, to the or stretch levels before photolysis.
Figure 8 Cross sections for the production of the energetically accessible OD rotational-fine-structure levels in the reaction D + OH() OD() + H at a collision energy of 6.7 K. The dashed and solid curves designate, respectively, final levels of nominal and symmetry, respectively. In the spin–orbit manifold, the and reflection symmetry labels correspond, respectively, to and [45]. This assignment is reversed for levels in the manifold.
Figure 9 Computed rate constant (solid curve) for the isotope exchange reaction OH + D OD + H. The open circles and open squares represent experimental data from Howard and Smith [57] obtained using O or , respectively, as the OH precursor. The dashed curve is the fit proposed by these authors. The “X” denotes the experimental rate constant of Kaufman and coworkers [58]. The filled data points represent earlier theoretical predictions: circles from [57] and squares from [59].
Figure 10 Hyperfine doubling of the lowest -doublet of OH (see Figure 1.2). The quantum number
F
is the total (molecular plus nuclear spin
I
;
F
=
j
+
I
) angular momentum, while designates the parity. The arrows indicate the OH maser transitions.
Figure 11 Schematic diagram of the potential energy surfaces of the OOH system. Note that the + H asymptote is endoergic for collision of O with OH in , but becomes energetically allowed when OH is vibrationally excited. The wells in the and states lie, respectively, 22,500 and 15,600 below the OH + O asymptote.
Figure 12 Cross sections for the OH(,
j
) + H + H reaction (top panel
j
= 0, bottom panel
j
= 1), as predicted by the quantum-statistical method [71], compared with predictions [69] of TID and time-dependent (wavepacket, WP) fully quantum calculations, as well as from application of the -shifting approximation [72], and quasi-classical trajectory calculations.
Figure 13 Comparison of the coupled-states, statistical rate constants for reactive and inelastic quenching of the OH(, ) state by collision with O(P) atoms as a function of temperature. The solid and dashed curves designate, respectively, collisions occurring on the lowest and PESs. The uppermost curve designates the rate constant for the total (inelastic plus reactive) removal. The room-temperature value of this OH() total removal from the experiments of Khachatrian and Dagdigian [75] is also shown. For comparison, the heavy black curve (the second highest curve) designates the calculated CS-ST rate constant for
reactive
removal of the OH(, ) state. At 298 K, the recommended value (Ref. 4) for this rate constant is 3.3 0.7 10/molecule/s.
Figure 14 Energies of the lower rotational/fine-structure levels of the CH(,) manifold. The -doublet splitting has been exaggerated for clarity.
Figure 15 Cross sections for direct and indirect collisions and total (the sum of the direct and indirect) integral cross sections for the (a) and (b) transitions in collision with .
Figure 16 Integral cross sections (sum of direct and indirect contributions) as a function of the collision energy for collisions between the CH -doublet levels and
ortho
- and
para
- for transitions to the and levels.
Figure 17 Schematic diagram of the potential energy surfaces of the CHH system. The triplet states are drawn with solid lines, while the singlet states with dashed lines. The lowest two states of methylene ( and ) have well depths of 34,000 and 31,800 , respectively, with respect to the CH + H asymptote. Reaction of CH with H is exoergic by 7900 to form C(P) + , but endoergic by 2000 to form C(D) + .
Figure 18 Contour plots (in ) of the CH−H PESs for the (top row) , , and (bottom row) , states for the equilibrium CH bond length . The angle corresponds to linear CHH geometry.
Figure 19 Integral cross sections as a function of the collision energy for collisions between the CH -doublet levels and H atoms for transitions to the and levels.
Figure 20 Diffusion coefficient for the H− system, computed from quantum scattering calculations (labeled as , PESs) and from a two-center (atom–atom) model based on a LJ 12-6 potential.
Figure 21 Cross sections for collision-induced transitions out of the lowest rotational level
n
=1 for H + and H + - at a collision energy of 300 . (a) Cross sections for H + scattering on the and PESs separately; (b) cross sections for H + and H + , averaged over the degeneracies of the and PESs.
Chapter 2: Quantum Dynamics in Photodetachment of Polyatomic Anions
Figure 1 The HCCH–CC isomerization path (a) and PES cuts of the excited electronic states of CC along the C–C distance in geometry (b). Configurations of various species at the stationary points (a) and conical intersection (b) are also depicted in the figures.
Figure 2 Experiment–theory comparison for the CC (and CC) photoelectron spectra for the ground (a) and excited states (b and c). In panels (b) and (c), the low and high-resolution experimental data are given in upper and lower lines, while the theoretical results in sticks.
Figure 3 Illustration of PESs for the O and species. (a) PES contours superimposed by the initial anionic wave packets. (b) Energetics of both the anion and neutral PESs along the reaction path for the OH H O reaction.
Figure 4 Comparison of the measured (rugged lines) and calculated (smooth lines) photoelectron spectra of the ground (a) and excited (b) vibrational states of O. The experimental spectra were measured at and , respectively. The narrow peak near eV is due to the signal and should be ignored.
Figure 5 Illustration of the PESs for the O and species, along with the adiabatic channels (dotted lines) for several low-lying vibrational states of HF. The wavefunctions of the anion and some Feshbach resonances are depicted on the right panels. The energetics of the photodetachment is depicted in the lower-left corner.
Figure 6 Experiment–theory comparison of the PPC spectrum for .
Figure 7 Energetics of the HOCO/ (a) and (b) PESs, with the molecular configurations of the stationary points.
Figure 8 Experiment–theory comparison of the photoelectron spectrum of (a) and / (b).
Figure 9 Energetics of the and PESs with the molecular configurations of the stationary points.
Figure 10 Experiment–theory comparison of the photoelectron spectrum of .
Figure 11 Energetics of the and PESs with the molecular configurations of the stationary points.
Figure 12 Experiment–theory comparison of the photoelectron spectrum of .
Chapter 3: Recent Advances in Quantum Dynamics Studies of Gas-Surface Reactions
Figure 1 (a) Six-dimensional Jacobi coordinates for the HCl/Au(111) system; (b) nine-dimensional Jacobi coordinates for the O/Cu(111) system; (c) eight-dimensional Jacobi coordinates for the /Ni(111) system; (d) an irreducible triangle unit cell of a rigid flat (111) surface (solid grey lines) with the top, fcc, hcp, and bridge surface symmetry impact sites.
Figure 2 Sideview of the transition state configurations of: the (a) HCl/Au(111) system; (b) O/Cu(111) system; (c and d) /Ni(111) system.
Figure 8 (a) Schematic of the distribution of 9 and 15 sites considered in an irreducible triangle unit cell of a rigid flat Cu(111) surface; (b) sticking probability of GS O molecule dissociation from 9-, 15-, 25-sites, and 9D results
Figure 3 Comparisons of six-dimensional, four-dimensional, and site-averaged dissociation probabilities for the GS HCl molecule on a rigid flat Au(111). The site-averaged results are obtained by averaging the dissociation probabilities of four fixed sites (bridge, hcp, fcc, and top) with appropriate relative weights.
Figure 4 The evolution of the probability density of GS DCl scattering from a rigid flat Au(111) surface at the specified top impact site on the contour plots of the dynamical PES. These contours are relative to the DCl + Au(111) asymptote with an interval of 0.1 eV. The probability density is shown as a function of the and with the other coordinates integrated, and each contour plot is shown with other coordinates optimized. Different propagation times are indicated by the label “T” in panels (a), (b), (c), (d), (e), and (f), respectively.
Figure 5 (a) Schematic of the distribution of 4, 9, and 25 sites considered in an irreducible triangle unit cell of a rigid flat Au(111) surface; (b) sticking probability of GS HCl molecule dissociation from 4-, 9-, and 25-sites and 6D results.
Figure 6 Comparisons of 6D dissociation probabilities and the site-averaged dissociation probabilities of in (,) on Cu(111) obtained by averaging the 4D quantum results over 3, 6, and 15 sites with appropriate relative weights based on: (a) the DZ PES [63] and (b) the SRP PES.
Figure 7 Comparisons of 7D sticking probabilities and 6D results of GS O with the azimuthal angle fixed at the saddle point for: (a) the fixed TS; (b) top; (c) bridge; and (d) hcp sites.
Figure 9 The dissociation probabilities obtained by the site-averaged approximation and the site-sudden approximation, together with the nine-dimensional dissociation probability, for O initially in the first vibrationally excited state(001).
Figure 10 Sticking probability of GS impinging on: (a) Top site with =, and ; (b) fcc site with =, , and ; (c) hcp site with =, , and ; and (d) brg site with = , , and . The solid lines are from a -in-quantum approach, and solid points are from the -in-averaged approach.
Figure 11 Sticking probability of GS obtained from quantum dynamics simulations taking azimuthal DOF into account for top, fcc, hcp, and bridge surface impact sites.
Figure 12 (a) Schematic of the distribution of 9, 16, and 24 sites considered in an irreducible triangle unit cell of a rigid flat Ni(111) surface; (b) sticking probability of GS molecule dissociation from 9-, 16-, and 24-site results.
Chapter 4: Quantum Scattering and Semiclassical Transition State Theory Calculations on Chemical Reactions of Polyatomic Molecules in Reduced Dimensions
Figure 1 Schematic representation of the H-abstraction or H-exchange reaction.
Figure 2 Comparison of (a) the calculated quantum thermal rate constants with the literature values of the H
2
+ CF
3
→ H + HCF
3
reaction at 500–2000 K and (b) the rate constant ratio with experimental results at 333–1000 K.
Figure 3 Plot of state-to-state integral cross sections of the Cl(
2
P
J
) + HCD
3
(v) → HCl(v′) + CD
3
reaction xversus collision energy (panel a), and comparisons to experiment of the state-to-state integral cross sections relative to that of the ground state versus collision energy (panels b and c). R1, R2, and R3 refer to Cl()()(), Cl()()(), and Cl()()(), respectively.
Figure 4 Plots of the minimum energy paths of the H + cyc−C
3
H
6
→ H
2
+ cyc−C
3
H
5
reaction in hyperspherical coordinates converted from (a) conventional Jacobi coordinates and (b) Jacobi coordinates with defined relative to the C-atom from which the H-atom is abstracted. The calculated quantum and TST thermal rate constants are compared with results from the literature for the H + cyc−C
3
H
6
→ H
2
+ cyc−C
3
H
5
reaction at (c) 200–2000 K and (d) 333–1000 K.
Figure 5 Rate constants calculated from 2-D PESs using SCTST with and without deep tunneling corrections (dotted and solid curves, respectively) presented in comparison to results from TST (dashed–dotted curve) and QRS calculations on the same PES (dashed curve) for four reactions.
Figure 6 Rate constants calculated using 1-D SCTST methods presented in comparison to 2-D rate constants.
Figure 7 Rate constants for the H + CH
4
→ H
2
+ CH
3
and H + C
2
H
6
→ H
2
+ C
2
H
5
reactions calculated using the FD SCTST-W method from PES derivatives obtained from
ab initio
calculations, presented in comparison to results from other theoretical and experimental studies.
Figure 8 Rate constants for the H + CH
4
and H + C
2
H
6
reactions calculated using the 1-D SCTST-W method presented in comparison to the FD results. The results labeled “Hessians” were calculated using force constants obtained by differentiating Hessian matrix elements, and those labeled “energies” were obtained by differentiating single-point energies.
Chapter 5: Adiabatic Switching Applied to the Vibrations of syn-CH3CHOO and Implications for “Zero-Point Leak” and Isomerization in Quasiclassical Trajectory Calculations
Figure 1 Schematic of the isomerization of syn-CHOO to VHP.
Figure 2 Zero-point vibrational energy of syn-CH
3
CHOO using adiabatic switching for three switching times.
Figure 3 Time dependence of for zero-point adiabatically switched trajectory.
Figure 4 Vibrational energy of syn-CH
3
CHOO with mode 9 excited, using adiabatic switching with one switching time.
Figure 5 Time dependence of for excited adiabatically switched trajectory.
Chapter 6: Inelastic Charge-Transfer Dynamics in Donor–Bridge–Acceptor Systems Using Optimal Modes
Figure 1 Sketch of Marcus parabolas for a model energy or charge-transfer system. Labeled are the key parameters used to compute the Marcus rate constant (Eq. (3)).
Figure 2 Sketch of adiabatic and diabatic representations for a two-state system. Compared to adiabatic representations, the diabatic representation has smoother energy surfaces and couplings.
Figure 3 (a) Chemical structures of the donor (P), bridge (–Pt–), and acceptor (NAP) complexes considered here. (b) Triplet energy along a linear interpolation coordinate connecting the NAP minimum energy geometry and the CT minimum energy geometry.
Figure 4 Energy level diagram for the triplet states of PTZ at the NAP and CT state geometries. The electron/hole distributions for the CT and CSS are shown to the right (light gray = electron, dark gray = hole).
Figure 5 Correlation functions of various numbers of projected modes, compared to the exact correlation, for (a) CSS NAP at NAP geometry, (b) CT NAP at NAP geometry, (c) CT NAP at CT geometry, and (d) CT CSS at CT geometry.
Figure 6 Component projection of the primary mode onto the normal modes for the following transitions: (a) CSS
3
NAP, (b) CT
3
NAP calculated at
3
NAP geometry. (c)CT
3
NAP, and (d) CT CSS calculated at CT geometry. The embedded molecule shows the atomic displacement vectors of primary mode.
Chapter 7: Coupled Translation–Rotation Dynamics of H2 and H2O Inside C60: Rigorous Quantum Treatment
Figure 1 Lower-lying translation–rotation (TR) energy levels of -H and -H molecules inside C from the quantum 5D calculations [30]. They are labeled by the quantum numbers defined in the text, and are arranged in columns according to their values.
Figure 2 Lower-lying translation–rotation (TR) energy levels of -HO and -HO molecules inside C from the quantum 6D calculations. They are labeled by the quantum numbers defined in the text. The and rotational levels are shown with thicker lines, to indicate their splitting into a near-degenerate pair a levels, due to the “crystal field” of C. For additional explanation see the text.
Chapter 9: Large Scale Exact Quantum Dynamics Calculations: Using Phase Space to Truncate the Basis Effectively
Figure 1 PS Wigner function, , for the projection operator corresponding to the lowest states of the 1D harmonic oscillator system, : (a) (quasi)classical approximation from the right-hand-side of Eq. (4); (b) exact quantum result. The latter oscillates about the former (constant) value, within the classically allowed region of PS. Outside this region, , and decays to zero very quickly (typically as a Gaussian).
Figure 2 Schematic representation of doubly-dense weylet/SG RPSL in 1D, for the fourth normal mode of CN. The gray shaded area is the PS region representing the (eight) basis functions retained via PS truncation – for a calculation of energy states in the dynamically relevant range up to (outer contour, denoted by thick curve).
Figure 3 Contour plots of the electron density for the ground state of the He atom, as computed in Section III.B, and integrated over all but two phase space variables: (a) ; (b) ; (c) . Contours correspond to exponentially-decreasing values of the electron density.
Chapter 10: Phase-Space Versus Coordinate-Space Methods: Prognosis for Large Quantum Calculations
Figure 1 Graphical illustration of the difference between basis orthogonality relations and grid orthogonality relations for the harmonic oscillator wavefunctions.
Figure 2 Depiction of the pseudospectral basis functions, in the second variant of the Fourier method (see Eq. (21)), for . These functions display the characteristic properties of all pseudospectral bases: the basis functions are orthonormal and each basis function vanishes at all grid points except the one where it is centered.
Figure 3 (a) coordinate grid points and von Neumann unit cells cover the same area in phase space, . Shown also is a typical von Neumann function. Note that its boundary conditions are not appropriate for the rectangular area in phase space. (b) Coverage of the same phase space by a discrete coordinate basis. (c) Coverage of the same phase space by a discrete momentum basis [44].
Figure 4 Depiction of the modified Gaussians associated with the reduced basis. The phase space spanned by the reduced basis is the non-gray area in both plots. On the left, the modified Gaussian , is in the interior of the phase space; is almost identical to the original Gaussian . On the right, we see a heavily deformed Gaussian, , whose center is close to the reduced subspace boundary; it is significantly different from the original Gaussian to which it corresponds. The states are plotted as heat maps, where the value of each cell of the von Neumann lattice is the absolute value of the overlap of the state plotted (here the modified Gaussians), with the Gaussian centered at that cell of the lattice, .
Figure 5 (a) Comparison of the classically allowed phase space for the harmonic oscillator at energy (circular boundary) versus the phase space spanned by a (momentum-symmetrized) Gaussian basis showed schematically as the union of black squares. Note that the Gaussian basis protrudes from the classical phase space in some regions and leaves part of the classical phase space uncovered. (b) The same (momentum-symmetrized) Gaussians modified by the quantum projector corresponding to . Notice how the quantum projector distorts the Gaussians near the classical phase-space boundary to conform to the shape of the boundary while leaving the Gaussians in the interior unchanged.
Figure 6 (a) The matrix product that appears in the orthogonal projector onto the subspace (Eq. (69)). Note that elements of outside of the subspace contribute. This structure is characteristic of orthogonal projection in a nonorthogonal basis. (b) The matrix product that appears in the nonorthogonal projector onto the subspace (Eq. (70)). Counterintuitively, the simpler diagonal structure of this object is the signature of a nonorthogonal projection.
Figure 7 Comparison of pruned pW, weylets, PvB and FGH for the simple harmonic oscillator. The number of phase-space-basis functions is determined by phase-space pruning based on classical arguments. The FGH basis covers a square area in phase space that is enlarged as the number of basis functions is increased. The error is defined by the norm of the difference between the vector of the exact and the numerical energies.
Figure 8 Representation of , Eq. (109), (panels a–c) and , Eq. (110), (panels d–f) in PvB (panels a and d), pW (panels b and e) and biorthogonal projected symmetrized Gaussians (panels c and f) in phase space [18]. Note the different ordinates for the representations.
Figure 9 Comparison of the absolute value of the inverse of (panel a) and its approximations (panels b and c). The difference between the correct and the approximate inverse is shown in panels d and e. The pruned basis represents states for a two-dimensional coupled harmonic oscillator.
Figure 10 (a) Accuracy of the dynamics for the 2D double well as a function of the percentage ratio of reduced and unreduced basis sizes. The more basis functions are used, the larger the wave amplitude threshold. The full basis size is 13365. The accuracy is determined by the infidelity of the autocorrelation and shown for projected weylets (pW, filled circles), pruned FGH (squares), PvB with the orthogonal projector (rings) and PvB with the nonorthogonal projector (triangles). Adapted from Machnes
et al
. 2006 [13] and Larsson
et al
. 2016 [18]. (b) Computing time against accuracy. The black horizontal line denotes the computing time of the unpruned FGH method. Due to the inferior behavior in panel a and the lack of an optimized implementation, values for PvB with the nonorthogonal projection are not shown.
Figure 11 Absolute value of the autocorrelation for pW and FGH dynamics compared to the exact dynamics (black line) for the six-dimensional model of pyrazine. The full basis size is . The given times are the runtimes.
List of Tables
Chapter 3: Recent Advances in Quantum Dynamics Studies of Gas-Surface Reactions
Table I Numerical parameters used in the HCl/Au(111), H
2
O/Cu(111), and CH
4
/Ni(111) systems
Table II Geometries and activation barriers of the TS configurations of HCl/Au(111), H
2
O/Cu(111), and CH
4
/Ni(111) from: the DFT calculations and the NN (PES)
Chapter 5: Adiabatic Switching Applied to the Vibrations of syn-CH3CHOO and Implications for “Zero-Point Leak” and Isomerization in Quasiclassical Trajectory Calculations
Table I The Projection of the syn-CH
3
CHOO Normal Modes on the Imaginary-Frequency Mode of the Saddle Point
Chapter 6: Inelastic Charge-Transfer Dynamics in Donor–Bridge–Acceptor Systems Using Optimal Modes
Table I Comparison between Experimental and Computed State-to-State Transition Rates for PTZ
Table II Driving force , reorganization energy , diabatic coupling , mean diabatic coupling , and (driving force calculated with ), for different transitions
Chapter 9: Large Scale Exact Quantum Dynamics Calculations: Using Phase Space to Truncate the Basis Effectively
Table I Number of Converged Energy Levels for CN at Varying Levels of Accuracy, . Convergence was Determined by Comparing the Second Largest Calculation () to the Largest ()
Table II Gaussian Expansion Coefficients, , for Normalized (but not Momentum-Symmetrized) “Fiducial” Weylet, , on Doubly-Dense Rectilinear PS Lattice
Table III Gaussian Expansion Coefficients, , for Normalized (but not Momentum-Symmetrized) “Fiducial” Weylet, , on Doubly-Dense Rectilinear PS Lattice
Table IV Ground State Energy Level and Excited State Frequencies of the He Atom, in Hartrees, as Computed Using the Method Described in Section III.B (Column II), and Compared with Experiment (Column III)
Chapter 10: Phase-Space Versus Coordinate-Space Methods: Prognosis for Large Quantum Calculations
Table I Summary of the most important definitions and relations of the biorthogonal bases in the reduced subspaces
Advances in Chemical Physics
Volume 163
Edited by
K. Birgitta Whaley
University of California, Berkeley Palatine, United States
This edition first published 2018
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Library of Congress Catalog Number: 58-9935
ISBN: 9781119374992
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Cover image: Courtesy of Hua Guo. Potential energy surfaces and wavefunctions involved in the reaction dynamics of fluorine with water.
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, IL, USA
Daan Frenkel, Department of Chemistry, Trinity College, University of Cambridge, Cambridge, UK
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, IL, USA
Gerhard Hummer, Theoretical Biophysics Section, NIDDK-National Institutes of Health, Bethesda, MD, USA
Ronnie Kosloff, Department of Physical Chemistry, Institute of Chemistry, Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem, Israel
Ka Yee Lee, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA
Todd J. Martinez, Department of Chemistry, Photon Science, Stanford University, Stanford, CA, USA
Shaul Mukamel, Department of Chemistry, School of Physical Sciences, University of California, Irvine, CA, USA
Jose N. Onuchic, Department of Physics, Center for Theoretical Biological Physics, Rice University, Houston, TX, USA
Stephen Quake, Department of Bioengineering, Stanford University, Palo Alto, CA, USA
Mark Ratner, Department of Chemistry, Northwestern University, Evanston, IL, USA
David Reichman, Department of Chemistry, Columbia University, New York City, NY, USA
George Schatz, Department of Chemistry, Northwestern University, Evanston, IL, USA
Steven J. Sibener, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA
Andrei Tokmakoff, Department of Chemistry, James Franck Institute, University of Chicago, Chicago, IL, USA
Donald G. Truhlar, Department of Chemistry, University of Minnesota, Minneapolis, MN, USA
John C. Tully, Department of Chemistry, Yale University, New Haven, CT, USA
List of Contributors Volume 163
Millard H. Alexander, Department of Chemistry and Biochemistry, Institute for Physical Science and Technology, University of Maryland, College Park, MD 20741-2021, USA
Elie Assémat, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel;
Theoretical Physics, Saarland University, D-66123 Saarbrücken, Germany
Zlatko Bačić, Department of Chemistry, New York University, New York, NY 10003, USA;
NYU-ECNU Center for Computational Chemistry, New York University Shanghai, Shanghai 200062, China
Eric R. Bittner, Department of Chemistry, University of Houston, Houston, TX 77004, USA
Joel M. Bowman, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA
Tucker Carrington Jr., Chemistry Department, Queen's University, Kingston, Ontario K7L 3N6, Canada
David C. Clary, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK
Paul J. Dagdigian, Department of Chemistry, The Johns Hopkins University, Baltimore, MD 21218-2685, USA
Peter M. Felker, Department of Chemistry and Biochemistry, University of California, Los Angeles, CA 90095-1569, USA
Samuel M. Greene, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK
Hua Guo, Department of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, NM 87131, USA
Henrik R. Larsson, Institut für Physikalische Chemie, Christian-Albrechts-Universität zu Kiel, Olshausenstraße 40, D-24098 Kiel, Germany;
Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
Jianyi Ma, Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, China
Shai Machnes, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel;
Department of Theoretical Physics, Saarland University, D-66123 Saarbrücken, Germany
Apurba Nandi, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA
Andrey Pereverzev, Department of Chemistry, University of Missouri-Columbia, Columbia, MO 65211, USA
Bill Poirier, Department of Chemistry and Biochemistry, and Department of Physics, Texas Tech University, Lubbock TX 79409-1061, USA
Chen Qu, Department of Chemistry and Cherry L. Emerson Center for Scientific Computations, Emory University, Atlanta, GA 30322, USA
Xiao Shan, Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, Oxford OX1 3QZ, UK
Xiangjian Shen, Research Center of Heterogeneous Catalysis and Engineering Science, School of Chemical Engineering and Energy, Zhengzhou University, Zhengzhou 450001, People's Republic of China;
State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China
David Tannor, Department of Chemical Physics, Weizmann Institute of Science, 76100 Rehovot, Israel
Minzhong Xu, Department of Chemistry, New York University, New York, NY 10003, USA
Xunmo Yang, Department of Chemistry, University of Houston, Houston, TX 77004, USA
Dong H. Zhang, State Key Laboratory of Molecular Reaction Dynamics and Center for Theoretical Computational Chemistry, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian 116023, People's Republic of China
Foreword
This 163rd volume of Advances in Chemical Physics is dedicated to John C. Light, late Professor of Chemistry at the James Franck Institute and Department of Chemistry at the University of Chicago, who passed away in Denver, Colorado, on January 18, 2016. This memorial volume provides 10 contributions by former students and colleagues that focus on some of the core areas in quantum dynamics of molecular systems that fueled many years of pioneering research by John.
Born in Mt. Vernon in 1934, John studied at Oberlin College, Harvard, and then Brussels, before joining the faculty of the University of Chicago in 1961. He remained at the University of Chicago until his retirement in 2006, where he also provided long-term service to The Journal of Chemical Physics as an Editor (1983–1997) and an Associate Editor (1997–2007).
John's research interests were broadly focused on quantum dynamics of chemical systems. Within this area, he addressed a diverse and constantly evolving set of chemical and physical problems, with an emphasis on developing groundbreaking analytical and numerical formulations that took advantage of the rapidly growing power of computers during his career. John's legacy includes his pioneering work in developing the modern quantum theory of reactive molecular collisions, which laid the foundation for the high-precision quantum scattering calculations of chemical reactions being made today. This work led to his introduction of the highly efficient discrete variable representation (DVR) for scattering problems (with Jim Lill and Greg Parker in 1982). Recognizing the potential of the sparsity and flexibility provided by this dual representation, John subsequently extended the DVR to the analysis of intramolecular dynamics where it revolutionized the study of multidimensional bound states of molecular systems, allowing for a numerically exact quantum treatment of highly excited states, floppy molecules, and molecular clusters that was previously inaccessible.
The articles contributed to this volume in memory of John Light address topics in quantum molecular scattering dynamics, phase-space theory, intramolecular dynamics, and electron transfer dynamics. These areas reflect the breadth and enthusiasm of John's interest in both chemical reaction dynamics and the broader science that this connects to. John was a great scientist, a leader in his field, and a wonderful and highly respected colleague for many in the Chemical Physics community. He was also an inspiring mentor and scientific role model for generations of students and postdocs. John's vision and gracious persona will be missed by us all.
K. Birgitta Whaley Department of Chemistry The University of California, Berkeley
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 past 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. DinnerJuly 2017, Chicago, IL, USA
Chapter 1Applications of Quantum Statistical Methods to the Treatment of Collisions
Paul J. Dagdigian1 and Millard H. Alexander2
1Department of Chemistry, The Johns Hopkins University, Baltimore, MD, 21218-2685, USA
2Department of Chemistry and Biochemistry, Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20741-2021, USA
Contents
I. Introduction
II. Quantum Statistical Theory
III. Fine-Structure Branching in Reactive O(
1
D) + H
2
Dynamics
IV. Inelastic OH + H Collisions
A. OH + H Vibrational Relaxation
B. OH + D Isotope Exchange
C. OH + H Rotationally Inelastic Collisions
V. OH + O Reaction and Vibrational Relaxation
VI. Inelastic Collisions of the CH Radical
A. CH + H
2
B. CH + H
VII. H + O
2
Transport Properties
VIII. Conclusion
Acknowledgments
References
I. Introduction
In chemical kinetics, statistical theories were first developed to understand unimolecular reactions and predict their rates (see, for example, [1, 2]). In the field of molecular reaction dynamics, we would expect statistical models to be well suited to reactions proceeding through formation and decay of a strongly bound collision Examples would be the reaction of electronically excited atoms with [C(D), N(D), O(D), and S(D), for instance]. Here, atom M inserts into the H−H bond with the subsequent formation of a transient HMH complex, which then decays to form MH + H products.
Statistical models for reactions involving the formation and decay of a complex were first proposed in the 1950s to describe nuclear collisions [3]. These models were then applied to molecular collisions [4]. Molecular statistical theories were put on a firm theoretical footing by Miller [5], who used as justification the formal theory of resonant collisions [6–8].
Pechukas and Light [9, 10] pioneered a statistical theory to predict the rate and product internal state distribution of the reaction of an atom with a diatomic molecule. This theory was based on Light's work on the phase space theory of chemical kinetics [11, 12] but, in addition, imposed detailed balance. This work has formed the basis of modern quantum mechanical treatments of complex-forming chemical reactions [13–17]. Here, once the complex is formed, it can fall apart to yield any accessible reactant or product subject to conservation of the total energy and angular momentum.
Pechukas and Light [9, 10] made some additional simplifications: First, they assumed that the capture probability was zero or one, depending on whether the reactants had sufficient energy to surmount the centrifugal barrier for each partial wave (related to the total angular momentum J of the collision complex). Second, they assumed that the long-range potential could be described by an inverse power law, –. These assumptions, particularly the latter, were reasonable in an era where calculation of a potential energy surface (PES) was a major undertaking.
Subsequently, Clary and Henshaw [13] showed how to apply time-independent (TID) coupled-states and close-coupling methods to the determination of capture probabilities for systems with anisotropic long-range interactions. More recently, Manolopoulos and coworkers [14, 15] combined the statistical considerations of Pechukas and Light with the Clary–Henshaw TID quantum capture probabilities, using, in addition, accurate ab initio potential energy surfaces. This so-called “quantum statistical” method has been applied to a number of atom–diatom reactions that proceed through the formation and decay of a deeply bound complex. Guo has demonstrated how a time-dependent (wavepacket, WP) determination of the scattering wave function can be used in an equivalent quantum-statistical investigation of reactions proceeding through deep wells [18].
In related work, Quack and Troe [19] developed an adiabatic channel model to describe the unimolecular decay of activated complexes. This theory has been applied to a variety of processes, including the OH + O reaction [20].
González-Lezana [21] has written a comprehensive review of the use and applicability of quantum statistical models to treat atom–diatom insertion reactions. A good agreement with full quantum reactive scattering calculations has been found for properties such as the differential cross sections for the reactions of C(D) and S(D) with , while less satisfactory agreement was found for the O(D) and N(D) + reactions [15]. This comparison illustrates a limitation of the statistical theory: how to assess the accuracy of the approach without recourse to more onerous calculations. The differential cross-section of the product of a statistical reaction should have forward–backward symmetry. This is often not quite the case because the quenching of interferences between partial waves is not complete, particularly in the forward and backward directions [22].
Typically, fully quantum scattering calculations involve expansion of the scattering wavefunction in terms of all the triatomic states that are energetically accessible during the collision. The computational difficulty scales poorly with the number of these internal states. Both the large number of accessible vibrational states of a triatomic as well as the rotational degeneracy of the states corresponding to the A + BC orbital motion contribute to this bottleneck. Deep wells in any transient complexes are particularly problematic. An example is the O(D) + OH + H reaction, for which the PESs are illustrated schematically in Fig. 1. Even without taking anharmonicity into account, there are >1900 O vibrational levels with energy below the O(D)+ asymptote. And this does not include rotational levels. Thus, full quantum reactive scattering calculations for complex-mediated reactions are a heroic task [23].
Figure 1 Schematic diagram of the potential energy surfaces of the OHH system. Only the lowest () PES was taken into account in the initial quantum statistical calculations on the O(D) + reaction [14, 15]. Note that O () well lies 59,000 below the O(D) + asymptote.
Adapted from Rackham et al. 2001 [14] and Rackham et al. 2003 [15].
In the quantum statistical method, the close-coupled scattering equations are solved outside of a minimum approach distance, the “capture radius” , at which point the number of energetically accessible states (open channels) is much less than at the minimum of the complex. Also, because the point of capture occurs well out in the reactant and/or product arrangement, one does not need to consider the mathematical and coding complexities associated with the transformation from the reactant to product states [24]. It is these simplifications that make the quantum statistical method so attractive.
Formation of a transient complex does not always lead to chemical reaction. The complex may decay to the reactant arrangement, resulting in an inelastic collision. The present review outlines the application of quantum statistical theory to nominally nonreactive collisions that access PESs having one or more deep wells. Consider, the generic A + BC AB + C collision. The inelastic event A + BC(v, ) A + BC() can occur in a direct (non-complex-forming) collision, either through an encounter in which the partners approach in a repulsive geometry or at a larger impact parameter, where the centrifugal barrier prevents access to the complex. In addition, complex formation () and subsequent decay into the reactant arrangement will also contribute to inelasticity.
In general, weak, glancing collisions contribute substantially to rotational inelasticity. Thus, one might naively expect that both direct and complex-forming processes will contribute to rotational inelasticity. By contrast, vibrational inelasticity in collisions on basically repulsive PESs is very inefficient [25, 26], because the variation of repulsive PESs with the vibrational modes of the molecular moiety is weak. Thus, one might anticipate that the formation of a transient complex, in which substantial change in the bond distances might occur, could make a major contribution to vibrational relaxation.
Also, for A + BC collision systems where one of the reactants is an open-shell species, typically (as shown schematically in Fig. 1), there are a number of electronic states that correlate with the A + BC (or AB + C) asymptote. Of these states, one (or, only a few) leads to strongly bound intermediates, while the others are repulsive. The branching between the energetically accessible fine-structure levels of the products (in the case of OH, the spin–orbit and -doublet levels) will be controlled by the coupling between the various electronic states as they coalesce in the product arrangement, as the complex decays. We might predict that this branching, which can often be measured experimentally [27, 28], would be insensitive to any couplings within the complex, where the excited electronic states lie high in energy, and hence be an ideal candidate for prediction by a quantum-statistical calculation.
Reactions involving isotopologs of the same atom,
where designates an isotopolog of B, are an additional example where the quantum statistical method can provide predictions and useful insight into potential experiments.
The next section contains a formal review of the quantum-statistical method, followed, in the remainder of this review, by a discussion of applications of this method to the problems introduced earlier in this section. In addition, and related to our discussion of inelastic collisions in the presence of a collision complex, we will use the quantum statistical method to calculate transport cross sections, which are weighted averages of differential cross sections. Here, the goal will be the determination of transport cross sections for the A + BC collision pair, in the presence of a deep well.
II. Quantum Statistical Theory
Here, we describe the extension of the quantum statistical method to inelastic scattering, in the TID formulation due originally to Manolopoulos and coworkers [14, 15, 21]. Guo and coworkers have described an equivalent time-dependent formulation [29, 30], which they have applied to a number of reactive collisions. In principle, this time-dependent methodology could also be applied to inelastic scattering.
Consider the collision of two particles with internal structure, for example an open-shell molecule or atom, with total angular momenta and , respectively. We suppress any other labels, for example the fine-structure manifold for an open-shell molecule, required to designate fully the levels. The integral cross section for a transition between the initial level pair = (, ) and a final level pair = (, ) at total energy E is given by the expression
In Eq. (2), is the internal energy of the initial level pair, is the collision reduced mass, J is the total angular momentum, and are the initial and final orbital angular momenta, and . The angular momenta and are vector sums of , and , respectively. Note that the cross section for transition from pair to pair involves, implicitly, a multiple summation over the projection quantum numbers of both angular momenta, as well as that of the orbital angular momentum of the collision partners. For the collision of a molecule with a structureless atom, we have , , and .
The thermal rate constant as a function of temperature is given by [31]:
In Eq. (3), is the Boltzmann constant and is the collision energy ( = E – E).
In a quantum description, the probability of a transition between the initial and final scattering states (channels) for total angular momentum is given by the square modulus of the (or ) matrix element between these states:
The matrix can be obtained by the imposition of scattering boundary conditions with a TID close-coupling determination of the scattering wave function. Several extensive reviews of the general equations for inelastic scattering (with application to rotationally inelastic scattering) are available [32–34].
In the quantum statistical theory [14, 15], the probability in Eq. (2) is computed as
where is the capture probability, namely the probability of forming the collision complex from the initial level pair in the scattering channel for total angular momentum . In Eq. (5), is the fraction of collision complexes with total angular momentum , which dissociates into the final level pair in the scattering channel and equals
As discussed by Rackham et al. [14], the quantum statistical cross sections obey detailed balance.
Since the scattering event can lead to the formation of a collision complex, the matrix element
