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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
A comprehensive yet accessible exploration of quantum chemical methods for the determination of molecular properties of spectroscopic relevance
Molecular properties can be probed both through experiment and simulation. This book bridges these two worlds, connecting the experimentalist's macroscopic view of responses of the electromagnetic field to the theoretician’s microscopic description of the molecular responses. Comprehensive in scope, it also offers conceptual illustrations of molecular response theory by means of time-dependent simulations of simple systems.
This important resource in physical chemistry offers:
- A journey in electrodynamics from the molecular microscopic perspective to the conventional macroscopic viewpoint
- The construction of Hamiltonians that are appropriate for the quantum mechanical description of molecular properties
- Time- and frequency-domain perspectives of light–matter interactions and molecular responses of both electrons and nuclei
- An introduction to approximate state response theory that serves as an everyday tool for computational chemists
- A unified presentation of prominent molecular properties
Principles and Practices of Molecular Properties: Theory, Modeling and Simulations is written by noted experts in the field. It is a guide for graduate students, postdoctoral researchers and professionals in academia and industry alike, providing a set of keys to the research literature.
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Seitenzahl: 888
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Title Page
Copyright
Preface
Chapter 1: Introduction
Chapter 2: Quantum Mechanics
2.1 Fundamentals
2.2 Time Evolution of Wave Functions
2.3 Time Evolution of Expectation Values
2.4 Variational Principle
Further Reading
Chapter 3: Particles and Fields
3.1 Microscopic Maxwell's Equations
3.2 Particles in Electromagnetic Fields
3.3 Electric and Magnetic Multipoles
3.4 Macroscopic Maxwell's Equations
3.5 Linear Media
Further Reading
Chapter 4: Symmetry
4.1 Fundamentals
4.2 Time Symmetries
4.3 Spatial Symmetries
Further Reading
Chapter 5: Exact-State Response Theory
5.1 Responses in Two-Level System
5.2 Molecular Electric Properties
5.3 Reference-State Parameterizations
5.4 Equations of Motion
5.5 Response Functions
5.6 Dispersion
5.7 Oscillator Strength and Sum Rules
5.8 Absorption
5.9 Residue Analysis
5.10 Relaxation
Further Reading
Chapter 6: Electronic and Nuclear Contributions to Molecular Properties
6.1 Born–Oppenheimer Approximation
6.2 Separation of Response Functions
6.3 Molecular Vibrations and Normal Coordinates
6.4 Perturbation Theory for Vibrational Wave Functions
6.5 Zero-Point Vibrational Contributions to Properties
6.6 Pure Vibrational Contributions to Properties
6.7 Adiabatic Vibronic Theory for Electronic Excitation Processes
Further Reading
Chapter 7: Approximate Electronic State Response Theory
7.1 Reference State Parameterizations
7.2 Equations of Motion
7.3 Response Functions
7.4 Residue Analysis
7.5 Relaxation
Further Reading
Chapter 8: Response Functions and Spectroscopies
8.1 Nuclear Interactions
8.2 Zeeman Interaction and Electron Paramagnetic Resonance
8.3 Polarizabilities
8.4 Magnetizability
8.5 Electronic Absorption and Emission Spectroscopies
8.6 Birefringences and Dichroisms
8.7 Vibrational Spectroscopies
8.8 Nuclear Magnetic Resonance
Further Reading
Appendix A: Abbreviations
Appendix B: Units
Appendix C: Second Quantization
C.1 Creation and Annihilation Operators
C.2 Fock Space
C.3 The Number Operator
C.4 The Electronic Hamiltonian on Second-Quantized Form
C.5 Spin in Second Quantization
Appendix D: Fourier Transforms
Appendix E: Operator Algebra
Appendix F: Spin Matrix Algebra
Appendix G: Angular Momentum Algebra
Appendix H: Variational Perturbation Theory
Appendix I: Two-Level Atom
I.1 Rabi Oscillations
I.2 Time-Dependent Perturbation Theory
I.3 The Quasi-energy Approach
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: Introduction
Figure 1.1 Liquid benzene in a small volume corresponding to the focal point of a laser operating at 532 nm and releasing pulses with an energy of 1 mJ.
Figure 1.2 Electronic charge density of neon expanded in orders of the applied electric field . Light and dark gray regions indicate positive and negative values, respectively.
Figure 1.3 Hierarchy of quantum-chemical methods.
Figure 1.4 Speed of -electrons relative to the speed of light (b) and the corresponding Lorentz factor (a) versus atomic number. Relativistic contractions (c), defined as ratio between relativistic and nonrelativistic HF radial expectation values, of valence -orbitals for elements of rows 4, 5, and 6 of the periodic table. Atomic numbers are given relative to atomic number of the coinage metal of each row (M=Cu, Ag, Au).
Figure 1.5 Hydrogen spin densities in the -plane for the (a) and (b) orbitals.
Figure 1.6 Anionic polythiophene acetic acid with sodium counterions in water solution.
Chapter 2: Quantum Mechanics
Figure 2.1 Time evolution of a wave packet in a harmonic oscillator potential. The black line indicates the amplitude of the generally complex wave function. The full wave packet (gray line) has been traced in the complex plane normal to the -axis; the orientation of the imaginary axis and real axis is indicated in the lower curve plot.
Figure 2.2 Time dependence of the probability current density for the harmonic oscillator.
Chapter 3: Particles and Fields
Figure 3.1 A classical point charge in motion.
Figure 3.2 Decomposition of vector field into irrotational and solenoidal parts.
Figure 3.3 Linearly polarized light oscillating with an amplitude at an angle with the -axis.
Figure 3.4 Left-handed circularly polarized wave with wave vector . The electric field vector is illustrated at a certain coordinate along the direction of propagation (at a certain instant of time) and the direction of rotation (here anticlockwise) of this vector as time passes is depicted.
Figure 3.5 Conducting electrons in a wire placed in a uniform, static, magnetic field with inward direction. Electrons travel at velocity , giving rise to a magnetic Lorentz force that in turn causes a displacement of the electronic charge density. The net charge inhomogeneity gives rise to an electric force on the electrons and a resulting counterforce on the wire.
Figure 3.6 Electric dipole formed by point charges and separated by the distance vector . An observer point is indicated at which the scalar potential is given by Eq. (3.192).
Figure 3.7 Magnetic dipole generated by current loop.
Figure 3.8 Field of an electric or magnetic dipole aligned with the -axis.
Figure 3.9 Electric field strength of a monochromatic plane wave along the direction of propagation and at two separate times corresponding to equal to 0 (solid black line) and (solid gray line). The upper and lower panels refer to wavelengths in the near ultraviolet and soft X-ray regions, respectively. The horizontal bar depicts the size of the C fullerene.
Figure 3.10 Visualization of magnetic field lines using iron fillings sprinkled on a sheet of paper held over a bar magnet.
Figure 3.11 Gaussian cylinder surface bisected by an infinite flat sheet with a constant and homogeneous surface charge density . The area of the cylinder cross-section is so that the enclosed charge is equal to . Because of symmetry, the electric field is perpendicular to the sheet and directed in opposite directions on separate sides of the sheet.
Figure 3.12 Electric field due to a parallel-plate capacitor. The electric field inside the capacitor is constant and homogeneous, whereas outside it is zero.
Figure 3.13 Rowland ring in terms of an iron toroid that is magnetized by a tightly wound coil. A sampling coil with a small number of windings measures the induced magnetic flux. G indicates a galvanometer. The direction of the magnetic field in the Figure corresponds to current entering/leaving the page in the inner/outer ring.
Figure 3.14 Spatial averaging of the distribution of bound charges is carried out over the dummy integration variable to obtain , and it introduces, for each molecule , a frame of reference with internal coordinates . The molecular charge density is . The weight function depends on the distance between the observer point and .
Figure 3.15 Magnetic hysteresis curve.
Figure 3.16 Surface boundary between media A and B. (a) A cylinder with area and height straddles the interface, with the cylinder axis being parallel with the normal of the boundary surface. (b) A rectangle with width and heigth is placed across the surface, oriented so that the surface tangent is a normal to the rectangle.
Figure 3.17 Dielectric sphere in dielectric medium.
Figure 3.18 The internal field can be calculated as the difference between the total field inside the dielectric and the field associated with the molecular volume.
Figure 3.19 The field of a spherical plug of frozen polarization can be calculated by considering two uniformly charged spheres.
Figure 3.20 The Langevin function, Eq. (3.385), and its linear approximation . Selected values are tabulated to the right.
Figure 3.21 Projectiles (photons) with flux are incident on a target medium slab of thickness and area containing particles with individual cross section . As a result of the interaction with the medium, the photon flux is reduced upon passage.
Figure 3.22 Complex relative permittivity of liquid water at 25 C as a function of frequency on a logarithmic scale.
Figure 3.23 A decomposition of a time-dependent electric field into a sum of static fields.
Figure 3.24 Imaginary part of the complex permittivity of liquid water at 25 C as a function of frequency on a logarithmic scale. The displayed frequencies are in the infrared and microwave regions of the spectrum.
Figure 3.25 Complex relative permittivity of liquid water at 25 C as a function of frequency on a logarithmic scale. Compared to that shown in Figure 3.22, the infrared to X-ray domain has been enlarged.
Figure 3.26 Complex permittivity of liquid water at 25 C as a function of frequency around the first electronic excitation, found in the UV region. Results from the Lorentz model are compared with experimental data.
Figure 3.27 Log–log plot of the absorption coefficient of liquid water at 25 C as a function of frequency.
Figure 3.28 Real part of the refractive index of liquid water at 25 C as a function of frequency.
Figure 3.29 The chiral CHFClBr molecule ( isomer).
Chapter 4: Symmetry
Figure 4.1 A sequence of symmetry operations transforms a function step by step. The original and resulting functions are illustrated by contour plots. The upper and lower sets of panels provide the
active
and
passive
points of view, respectively.
Figure 4.2 Flowchart for determining molecular point groups.
Figure 4.3 Group table (a) for a group containing four elements {A,B,C,D} and illustration (b) of two isomorphic groups.
Figure 4.4 Character table (a) and direct product table (b) for an Abelian group containing four elements {A,B,C,D} and irreducible representations , , , .
Figure 4.5 (a, b) Time-inversion symmetry.
Figure 4.6 (a, b) Space inversion, or parity, symmetry.
Figure 4.7 An infinitesimal rotation by an angle about an arbitrary laboratory axis . The length of the displacement is and it is directed along .
Figure 4.8 Infinitesimal rotation by an angle of a two-dimensional vector field.
Figure 4.9 Rotation of a system in a triplet state by an angle about the laboratory -axis.
Figure 4.10 Ethylene molecule in point group.
Chapter 5: Exact-State Response Theory
Figure 5.1 Energy levels and corresponding wave functions of a two-level atom together with matrix representations of the Hamiltonian and electric dipole moment operators.
Figure 5.2 Induced electric dipole moment in a two-level atom. The external field , induced dipole moment , and the time are given in a.u., and denotes the population of excited state . The vertical bar indicates the time of 10 fs. A time-integration step length of a.u. is used.
Figure 5.3 Quasi-energy in a two-level atom. The external field and quasi-energy are given in a.u. The vertical bar indicates the time of 10 fs. The horizontal bar in the upper panel indicates the time-averaged quasi-energy minus the ground-state energy, that is, , in the steady-state region. A time-integration step length of a.u. is used.
Figure 5.4 Field derivative of the quasi-energy in a two-level atom. The external field and the derivative of the quasi-energy are given in a.u. The vertical bar indicates the time of 10 fs. The horizontal bar indicates the time-averaged quasi-energy derivative in the steady-state region. A time-integration step length of a.u. is used and the field derivative is determined by finite differences using a.u.
Figure 5.5 Elastic scattering of incident photons of frequency .
Figure 5.6 Second-harmonic generation involving two incident photons of frequency and a sum-frequency-generated photon of frequency .
Figure 5.7 Frequency decomposition of two superimposed laser fields.
Figure 5.8 Frequency decomposition of the molecular polarization.
Figure 5.9 Difference-frequency generation involving photons of frequencies and .
Figure 5.10 Parametrization of the wave function by projections.
Figure 5.11 Parametrization of the wave function by rotations.
Figure 5.12 Energy as a function of rotation parameters.
Figure 5.13 Electronic first-order hyperpolarizability of hydrogen fluoride for the SHG and EOPE optical processes. Data refer to time-dependent Hartree–Fock calculations using Dunning's t-aug-cc-pVTZ basis set. The dipole moment is directed along the positive -axis and the experimental bond length of 1.733 a.u. is employed. All quantities are given in a.u.
Figure 5.15 Induced electric dipole moment in a two-level atom for a frequency of the perturbation that is in resonance with the electronic transition. The external field , induced dipole moment , and the time are given in a.u., and denotes the population of excited state . The gray curve in panel (b) indicates the excited-state population obtained by using perturbation theory, that is, Eq. (5.239). The vertical bar indicates a time of 10 fs after the start. A time-integration step length of a.u. is used.
Figure 5.16 Induced electric dipole moment in a two-level atom for a frequency of the perturbation that is in resonance with the electronic transition. The external field , induced dipole moment , and the time are given in a.u., and denotes the population of the excited state . The gray curve in panel (b) indicates the excited-state population obtained by using perturbation theory, that is, Eq. (5.239). The vertical bar indicates a time of 10 fs after the start. A time-integration step length of a.u. is used.
Figure 5.17 Excited-state population in a two-level atom for a frequency of the perturbation that is in resonance with the electronic transition, that is, a.u. The amplitude of the electric field is equal to a.u. The time are given in a.u. and denotes the population of the excited state . The gray curve indicates the result obtained by using perturbation theory, that is, Eq. (5.239). A time-integration step length of a.u. is used.
Figure 5.18 A resonance of the linear response function.
Figure 5.19 Resonances of the first-order nonlinear response function.
Figure 5.20 Resonances of the second-order nonlinear response function.
Figure 5.21 A selection of absorption and relaxation channels in molecular systems: one-photon absorption (OPA), two-photon absorption (TPA), excited-state absorption (ESA), internal conversion (IC), and inter-system crossing (ISC). Typical interaction times for the relaxation processes are indicated. Singlet (denoted by ) and triplet (denoted by ) electronic states are included in the diagram, which is known as a Jablonski diagram. For each electronic state, several ro-vibrational states are indicated. Wiggly and straight lines refer to nonradiative thermal and photon interactions, respectively.
Figure 5.22 Relaxation in a two-level atom where denotes the population of the excited state . The vertical bars indicate times of and 10 fs, respectively, and the horizontal bar indicates a population of . A time-integration step length of a.u. is used.
Figure 5.23 Induced electric dipole moment in a two-level atom for a frequency of the perturbation that is in resonance with the electronic transition. Relaxation from the upper to the lower level is described by the parameter a.u. The external field , induced dipole moment , and the time are given in a.u., and denotes the population of the excited state . The vertical bar indicates the time of 10 fs. A time-integration step length of a.u. is used. The lower illustration shows the same results in the zoomed-in time interval from 450 to 500 a.u.
Figure 5.24 Structures of zeroth-, first-, and second-order density matrices. The filled squares denote nonzero matrix elements.
Figure 5.25 The upper illustration shows the real and imaginary parts of the linear polarizability for the two-level system. The full width at half maximum of the imaginary part is equal to . The numerical value of is here chosen as 0.04. The lower illustration shows the same polarizability but plotted on a polar form.
Chapter 6: Electronic and Nuclear Contributions to Molecular Properties
Figure 6.1 Illustration of a seam intersection occurring when two potential energy surfaces cross, allowing electrons to pass from one electronic state to another at nuclear geometries giving rise to very similar (or identical) energies for two different electronic states.
Figure 6.2 One of the twofold degenerate bending modes in CO.
Figure 6.3 The three normal modes in water. Note that it is largely the light nuclei (hydrogen) that move in all of the three normal modes as a consequence of the fact that the center of mass is spatially fixed.
Figure 6.4 The curvilinear motion associated with the bending mode in water.
Figure 6.5 Morse and harmonic oscillator potentials.
Figure 6.6 Property dependencies with respect to the low-frequency rotational mode in hydrogen peroxide. The electronic ground-state potential energy curve (in kilocalories per mole) and oscillator strength for the transition to the second electronically excited state are plotted in black and grey color, respectively. Results are obtained at the B3LYP/aug-cc-pVTZ level of theory.
Figure 6.7 The two enantiomers of the CFHDT molecule.
Figure 6.8 Population of the ground vibrational state at a temperature of 298.15 K as a function of the harmonic vibrational frequency.
Figure 6.9 Dependence of the bond distance and dipole moment of the HF molecule as a function of the field strength of an external static electric field. The molecule is located along the axis with the direction of the dipole moment being in the positive direction. Calculations have been carried out using the taug-cc-pVTZ basis set at the Hartree–Fock level of theory.
Figure 6.10 Illustration of the Franck–Condon principle with a fixed equilibrium ground-state geometry during the excitation process (vertical transition).
Figure 6.11 Coordinate transformation between normal coordinates of electronic ground and excited states denoted by and , respectively, and with origins at the respective equilibrium molecular geometries. Point represents a particular nuclear configuration with Cartesian coordinates . Cartesian displacements of this point are given by and for the ground and excited states, respectively.
Figure 6.12 Absorption and fluorescence spectrum of a diatomic molecule.
Chapter 7: Approximate Electronic State Response Theory
Figure 7.1 Complete set of Slater determinants for a system of two electrons in four orbitals.
Figure 7.2 Structures of the matrix representations of the operators and . The filled squares denote nonzero matrix elements.
Figure 7.3 Parameterization of reference state determinant.
Figure 7.4 Hartree–Fock orbital energy diagram of helium using Dunning's aug-cc-pVDZ basis set. Energies are given in units of .
Figure 7.5 Active space for helium based on the Hartree–Fock orbitals using Dunning's aug-cc-pVDZ basis set. Energies are given in units of .
Figure 7.6 Complete active space.
Figure 7.7 A system divided into subsystems A and B.
Figure 7.8 Hartree–Fock (left value) and DFT/B3LYP (right value) orbital energy diagram of beryllium using Pople's 6-31G basis set. Energies are given in units of .
Figure 7.9 Electric dipole oscillator strength distribution for helium in the time-dependent Hartree–Fock approximation employing Dunning's (a) t-aug-cc-pVTZ and (b) q-aug-cc-pV5Z basis sets. Summed oscillator strengths are given for transitions below and above the experimental ionization potential, respectively. For visual clarity, the bars representing the oscillator strengths are broadened by Lorentzian profiles. The experimental transition energy is 21.22 eV.
Figure 7.10 (a) Linear absorption cross section and (b) real part of electric dipole polarizability for helium in the time-dependent Hartree–Fock approximation employing Dunning's t-aug-cc-pVTZ basis set. A common -parameter equal to 0.1 eV is employed. The data points marked with triangles are interpolated with cubic splines.
Chapter 8: Response Functions and Spectroscopies
Figure 8.1 Illustration of different models for the nuclear charge distributions for Fe (bottom) and Au (top). The skin regions ( fm) of the Fermi distribution is between the dotted lines.
Figure 8.2 Schematic setup of a Mössbauer spectrometer.
Figure 8.3 Classical model of a magnetic dipole.
Figure 8.4 Oblate (a) and prolate (b) ellipsoids.
Figure 8.5 Zeeman splitting of the -lines in sodium.
Figure 8.6 Zeeman splitting of the red line in cadmium.
Figure 8.7 Polarizability per carbon atom for the -alkanes CH. Results are obtained at the time-dependent Hartree–Fock level of theory with use of the aug-cc-pVDZ basis set.
Figure 8.8 Polarizability per carbon atom for the
trans
-polyenes CH. Results are obtained at the time-dependent Hartree–Fock level of theory with use of the aug-cc-pVDZ basis set.
Figure 8.9 Convergence of the Gauss–Legendre quadrature scheme with respect to the number of quadrature points used for obtaining the coefficients (in a.u.) of benzene from Eq. (8.145). Results are obtained at the Kohn–Sham/B3LYP level of theory using the polarized basis set of triple zeta quality by Sadlej.
Figure 8.10 Magnetic-field-induced currents in the (a) hydrogen molecule and (b) hydrogen fluoride.
Figure 8.11 Magnetizabilities for a number of hydrocarbons as obtained at the Hartree–Fock/aug-cc-pVDZ level of theory and with the use of Pascal's rule. All results are reported in units of J T.
Figure 8.12 Magnetic-field-induced ring currents in cyclobutadiene (a) and benzene (b) as due to a magnetic field applied perpendicular to the molecular plane. The currents shown are at a height of 1.0 above the molecular plane.
Figure 8.13 Photoisomerization of retinal.
Figure 8.14 The Lyman, Balmer, and Paschen series of hydrogen with transitions from levels with principal quantum numbers , respectively.
Figure 8.15 Illustration of vertical, adiabatic, and 0–0 excitation energies.
Figure 8.16 Linear absorption spectrum of l-alanyl-l-tryptophan obtained at the CAM-B3LYP/6-31G level of theory. Oscillator strengths of electronic transitions are represented as bars and the corresponding absorption spectra are obtained with Lorentzian and Gaussian line broadenings employing 0.2 and 0.4 eV, respectively.
Figure 8.17 Illustration of Gaussian, Lorentzian, and Voigt lineshape functions. The and broadening parameters are so chosen as to give a common HWHM value of 0.1 for the Gaussian and Lorentzian functions, respectively.
Figure 8.18 Phosphorescence parameters for the excited triplet state in formaldehyde. Molecular structures are optimized at the B3LYP/cc-pVTZ level of theory and property calculations are carried out at the four-component Hartree–Fock/taug-cc-pVTZ level of theory using the Dirac–Coulomb Hamiltonian.
Figure 8.19 Dominant OPA, TPA, three-, four-, and five-photon absorption cross sections for the lowest 15 singlet states in
para
-dinitrobenzene. Results are obtained at the CAM-B3LYP/aug-cc-pVDZ level of theory.
Figure 8.20 X-ray absorption and radiative and nonradiative core hole decay.
Figure 8.21 Fluorescence and Auger electron yields in -edge NEXAFS as a function of atomic number.
Figure 8.22 Radial densities for the carbon core -orbitals in CH in the ground- and core-ionized states at a frozen molecular geometry. The results refer to the nonrelativistic Hartree–Fock level of theory using an uncontracted basis set.
Figure 8.23 Valence electron density for CH in (a) the ground state and (b) a core-ionized state with the core hole localized to the rightmost carbon atom. The plots correspond to isodensity surfaces of the valence electron densities as determined at the Hartree–Fock level of theory.
Figure 8.24 X-ray radiation beam illuminating domains of self-assembled molecules in a monolayer sample. Detection is made of particles (fluorescence photons or Auger electrons) passing through a slit. The axis is aligned with the surface normal.
Figure 8.25 Orientational average of the absorption dipole for a monolayer sample. The unit vectors of the electric field and absorption dipole are denoted by and , respectively.
Figure 8.26 Relative intensities of a near -edge transition at normal and grazing incidence of the X-ray beam. The vertical line is located at the magic angle.
Figure 8.27 Near carbon -edge X-ray absorption spectra for 1,1-difluoroethylene. Theoretical spectra are obtained with configuration interaction singles (CIS), STEX, and coupled cluster singles and doubles with perturbative treatment of triples excitations [CCSDR(3)]. A HWHM broadening parameter eV is used.
Figure 8.28 Near carbon -edge X-ray absorption spectra for 1,1-difluoroethylene obtained with the complex polarization propagator method in conjunction with a description of the electronic structure at the levels of DFT and CCSD. The CCSD and DFT spectra are shifted 3.0 and 10.2 eV, respectively, so as to align the first peak with the corresponding experimental transition energy. A HWHM broadening parameter eV is used.
Figure 8.29 Illustration of classes of birefringences: (a) A circular birefringence known as the Faraday effect causes a rotation of the plane of polarization that is linearly proportional to the component of the uniform magnetic field in the direction of light propagation. (b) A linear birefringence known as the Kerr effect causes a linearly polarized field to become elliptically polarized in quadratic proportion to the externally applied static electric field.
Figure 8.30 Optical rotatory dispersion [in units of 10 deg/(dm g cm)] and oscillator strengths in the region of the lowest singlet excited states of ()-methyloxirane. Results are obtained at the B3LYP/aug-cc-pVDZ level of theory and with use of a damping parameter eV.
Figure 8.31 Specific optical rotation [deg cm dm g] for different conformers of ()-()-paraconic acid together with the Boltzmann-weighted average amounting to 85.2 (same units).
Figure 8.32 Optical rotation at 589 nm of propane as a function of the dihedral H–C–C–C angle. Results are obtained at the DFT/B3LYP level of theory with use of different basis sets and London orbitals. Optical rotatory dispersion in units of deg/(dm g ).
Figure 8.33 Ground- and excited-state structures of (1,4)-bicyclo[2.2.1]hept-5-en-2-one, where chirality may be induced in the excited state through pyramidalization of the ketone carbon, thus giving rise to circularly polarized luminescence.
Figure 8.34 Electronic circular dichroism [in units of L mol cm] and oscillator strengths in the region of the lowest singlet-excited states of ()-methyloxirane. Results are obtained at the B3LYP/aug-cc-pVDZ level of theory and with use of a damping parameter eV.
Figure 8.35 Schematic representation of the , , and bands of MCD spectra. The term is temperature dependent and will become progressively more absorptive in character the higher the temperature, as this would suggest a more equal population of the different sublevels.
Figure 8.36 MCD spectra of Zn-porphyrin for various degrees of bending about the mesocarbon -metal–mesocarbon axis. Results are obtained at the CAM-B3LYP/aug-cc-pVDZ level of theory. A lifetime broadening of 1000 has been used.
Figure 8.37 Illustration of the Fermi resonance between the first overtone of the two degenerate bending modes in CO and the fundamental mode arising from the symmetric stretching motion. In Raman spectroscopy, the bending mode is symmetry forbidden, but Fermi observed in 1931 that the overtone, which is Raman allowed, could be observed due to intensity-borrowing from the fundamental mode.
Figure 8.38 Vibrational circular dichroism of ()-methyloxirane. Rotational strengths are given in units of esu cm and broadened by Gaussian line profiles. A spectral line profile is also provided for the ()-enantiomer in grey color.
Figure 8.39 Potential energy curve and isotropic polarizability for N. The polarizability is determined at the B3LYP/taug-cc-pVTZ level of theory for rad s. The zero-point vibrational energy as determined at the B3LYP/cc-pVTZ level of theory is indicated in the figure.
Figure 8.40 The molecular dynamics of N as given in terms of the internuclear distance and which gives rise to the time-dependent polarizability (here is equal to rad). The application of an electric field with strength and frequency 1.0 PHz results in a time-dependent induced dipole moment .
Figure 8.41 The Raman signal of N as given by together with its normalized autocorrelation function and the Fourier transform . The Fourier transform shows the Stokes and anti-Stokes Raman scattering signals.
Figure 8.42 Illustration of the processes of nonresonant Raman scattering and two-photon absorption. States and refer to the initial (ground) and final states, respectively, and the dashed line represents an intermediate virtual state.
Figure 8.43 Illustration of optical processes: (i) coherent two-photon absorption, (ii) normal instantaneous Raman scattering, (iii) resonant Raman scattering with interaction time smaller than excited-state lifetime , and (iv) fluorescence with and nonradiative energy transfer to the environment in the excited state.
Figure 8.44 Potential energy curves (PECs) for electronic ground and excited states and electronic transition moment for CH. Data are determined at the B3LYP/taug-cc-pVTZ level of theory. Fundamental vibrational energies are indicated by horizontal dashed lines. The harmonic ground-state PEC is drawn both in the ground and excited states; in the excited state, the gradient (slanted dashed line) of the harmonic potential matches that of the true PEC at the ground-state equilibrium distance. The linear approximation of the transition moment is shown as a slanted dashed line.
Figure 8.45 Absolute value of transition polarizability for CH in the region of electronic resonance. Results are obtained with use of Kramers–Heisenberg–Dirac theory, Eq. (8.250) (upper panel), the Franck–Condon approximation, Eq. (8.254) (mid panel), and the IMDHO model, Eq. (8.255) (lower panel) assuming a lifetime of 0.1 ps for the excited vibronic states. Vertical dashed lines indicate the vibronic resonances.
Figure 8.46 Absolute value of transition polarizability for CH in the region of electronic resonance. Result curves labeled by KHD and CPP are obtained with use of Kramers–Heisenberg–Dirac theory, Eq. (8.250), and the short-time approximation, Eq. (8.252), respectively, assuming a lifetime of 10 fs for the excited vibronic states. The vertical dashed lines indicate the vertical and 0–0 electronic transition energies.
Figure 8.47 Setups for ROA measurements based on incident, scattered, and dual circular polarization (ICP, SCP, and DCP) with light incident along the -axis and scattering intensities detected at an angle in the -plane. Scattering angles of 0, 90, and 180 correspond to forward, right-angle, and backward scattering geometries, respectively.
Figure 8.48 Raman scattering intensities and differential Raman optical activity scattering intensities for ()-methyloxirane for a backscattering experiment. The intensities and optical activities in the low-frequency region are scaled by factors of 10.0 and 5.0, respectively.
Figure 8.49 The power of nuclear magnetic resonance is illustrated by magnetic resonance imaging, here of the brain of one of the authors (TS).
Figure 8.50 Larmor precession with a radiofrequency probe adding nutation.
Figure 8.51 Hydrogen shielding constant (ppm) and indirect iodine–hydrogen spin–spin coupling constant ( kg m s A) for iodoethane as a function of dihedral angle (the hydrogen and iodine atoms are pointed out with arrows): (a) calculated with and without inclusion of spin–orbit interactions and (b) comparison of with the spin–orbit contribution to the total shielding , that is, the difference of curve plots in panel (a).
Figure 8.52 Property surfaces of the indirect spin–spin coupling constant (a) and the shielding constant (b) of iodoethane at an dihedral angle of 120. Dark and light gray surfaces refer to positive and negative property values, respectively.
List of Tables
Chapter 2: Quantum Mechanics
Table 2.1 Postulates in classical and quantum mechanics
Table 2.2 A selection of quantum mechanical operators
Chapter 3: Particles and Fields
Table 3.1 Calculated (B3LYP/aug-cc-pVDZ) and tensors, corresponding to the electric dipole and magnetic dipole polarizabilities, respectively, (in atomic units) of the -CHFClBr molecule at = 589 nm, corresponding to the Fraunhofer D-line in sodium
Chapter 4: Symmetry
Table 4.1 Character Table for the point group
Table 4.2 Character Table for the point group
Chapter 5: Exact-State Response Theory
Table 5.1 Common nonlinear optical processes
Table 5.2 Permutations of terms A1 and B1 in Eq. (5.203) needed to obtain terms A2–A4, B2', B3, and B4'
Table 5.3 A selection of electric molecular properties described by the first-, second-, and third-order response functions
Chapter 6: Electronic and Nuclear Contributions to Molecular Properties
Table 6.1 Expressions for the different contributions to the pure vibrational contributions to , , and for exact nuclear–electronic wave functions within the Born–Oppenheimer approximation
Table 6.2 Dependence of electronic property with respect to nuclear displacements along a normal coordinate belonging to the irreducible representation
Table 6.3 Formulas for the pure vibrational contributions to dynamic vibrational polarizabilities and hyperpolarizabilities.
a
Table 6.4 Conversion of formulas for contributions given in Table 6.3 to those for , , and
Chapter 7: Approximate Electronic State Response Theory
Table 7.1 A selection of nonrelativistic two-electron integrals in the canonical Hartree–Fock molecular orbital basis for helium using Dunning's aug-cc-pVDZ basis set
Table 7.2 Nonrelativistic time-dependent Hartree–Fock results for neon of the equivalent dipole expression in Eqs. (7.66) and (7.67)
Table 7.3 Excitation energies (eV) for the lowest triplet and singlet states in beryllium
Table 7.4 Parts of the electronic Hessian for beryllium referring to valence spin-adapted excitations
Chapter 8: Response Functions and Spectroscopies
Table 8.1 Total ground-state Hartree–Fock energy (in Hartrees) using orbitals optimized for the ground-state configuration and different nuclear charge distribution models
Table 8.2 Data for selected isotopes
Table 8.3 Orbital contributions, multiplied with orbital occupation numbers of two, to the field gradient (in ) at the fluorine nucleus of fluoride and hydrogen fluoride
Table 8.4 Landé -factors for states in sodium related to the -line transitions
Table 8.5 Basis set convergence of the Hartree–Fock magnetizability of PF calculated with and without London atomic orbitals and Dunning's augmented correlation-consistent basis sets
Table 8.6 The magnetizability of fluorine determined using, respectively, experimental data (Experiment estimated) and theoretical data (Theory estimated) for calibration
Table 8.7 Isotropic averages for the lowest order multiphoton absorption processes
Table 8.8 Diagonal elements of the mixed electric dipole–magnetic dipole polarizability and the trace of the tensor for a few selected molecules
Table 8.9 Electric field gradient-induced birefringence of H, N, and CH at temperatures 297.15 K (H), 298.15 K (N), and 294.0 K (CH)
Table 8.10 Infrared absorption properties of HO in terms of vibrational wave numbers (cm) and dipole moment gradients with respect to normal coordinates ( a.u.)
Table 8.11 Harmonic and anharmonic vibrational frequencies (cm) for HO.
a
Table 8.12 Isotropic transition polarizabilities (in a.u.) for N
Table 8.13 Vibronic transition energies (eV), oscillator strengths , Stokes Raman shifts (cm), and absolute values of complex transition polarizabilities (a.u.) for vibronic transitions in CH
Table 8.14 Explicit expressions for the Raman scattering intensity and the difference in scattering of right and left circularly polarized light observed in Raman optical activity for the most commonly used experimental setups of backward-, right-angle- and forward-scattering in scattered circular polarization experiments
Table 8.15 Contributions of individual ROA tensor invariants in ()-methyloxirane for a backscattering SCP experiment
