phenomenology and calculation with tddft · 2018. 8. 21. · optical parametric amplifiers...
TRANSCRIPT
Linear and non-linear response:phenomenology and calculation with TDDFT
David A. StrubbeDepartment of Physics,
University of California, Merced, USA
Benasque, 21 August 2018
Linear and non-linear optics
Relation to other topics in school:
Background:Gross, fundamentals of TDDFT
Applications (mainly linear, could do nonlinear):Octopus tutorial
More details:Requist, quantum theory of polarizationTancogne-Dejean, Microscopic-macroscopic connectionJornet Somoza, linear response in chemistry
Linear and non-linear optics
Consider only perturbative processes via Taylor expansion of dipole moment. Not high-harmonic generation!
Polarizability (linear optics)
absorption, refraction
Hyperpolarizability: second-harmonic generation (SHG) etc.
Quantized picture
absorption two-photon absorption
second-harmonic generation
refraction
A zoo of nonlinear optical processes
sum-frequency generation (SFG)
difference-frequency generation (DFG)
second-harmonic generation (SHG)
optical rectification
Pockels (electrooptic) effect
third-harmonic generation (THG)
two-photon absorption
four-wave mixing
Energy conservation requires frequency arguments to sum to zero.
Applications of non-linear optics
Characterization in surface science and chemistry (very sensitive)
Optical parametric amplifiers
Pockels cells
Laser pointers
Tunable light sources
Optical logic
coherent anti-Stokes Raman spectroscopy (CARS) (kind of 4-wave mixing)
Two-photon fluorescencent labels in biology
Typically inorganic crystals are used in applications, but organic molecules have the potential to be cheaper and more efficient.
SHG and SFG at the National Ignition FacilityLawrence Livermore National Laboratory, Livermore, California
600 KH2PO4 crystals of 400 kg each, ultrapure to avoid absorption.
192 laser beams for inertial confinement fusion
Total power = 500 TW1053 nm1053 nm1053 nm
527 nm351 nm
The challenges of nonlinear optics: a cautionary tale
The original experimental report of SHG (quartz, 694 nm)
Lesson: check your proofs!
Symmetry properties of nonlinear susceptibility tensorsInversion symmetry: even orders are zero (µ, b, d, etc.).
Apply inversion:
Therefore
surface: breaks symmetry
e.g. Si crystal bulk: no SHG
Symmetry properties of nonlinear susceptibility tensors
Permutation symmetry:
“Kleinman symmetry,” only when off-resonance or
Also Kramers-Kronig relations.
Symmetry properties of susceptibility tensors
Spatial symmetries: e.g. chloroform (CHCl3)
Zincblende structure (e.g. GaAs) has only
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Conventions and units
A. Willetts, J. E. Rice, D. M. Burland, and D. P. Shelton, J. Chem. Phys. 97, 7590 (1992)
Many conventions for pre-factors! Multiple papers on just conventions…
Commonly used units: au, esu
1 au of β= 3.206361×10−53 C3 m3/J2 (SI)= 8.6392×10−33 cm4 statvolt−1 (esu of β)
Nonlinear optics in solidsSusceptibility tensors for solids
Now can have q-dependence as well.
β ~ 10000 au
KTiOPO4 c(2) ~ 10-8 esu
CHCl3 β ~ 1 au, c(2) ~ 10-8 esu
Phase-matching condition for constructive interference:
Adjust angle of incidence to satisfy and get significant conversion.
(NOT momentum conservation; it’s the wavevector in the material)
Molecules vs solids: orders of magnitude.
Other perturbations
Other perturbations: ionic displacement and strain (Raman tensors, Grüneisen parameters, phonon anharmonicity, pyroelectric tensor, piezoelectric tensor, second-order elastic coefficients, …)
S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Gianozzi, Rev. Mod. Phys. 73, 515 (2001)
Representation of electric field in finite and periodic systems:
Quantum theory of polarization
A Dal Corso, F Mauri, and A Rubio, Phys. Rev. B 53, 15638 (1996)
Related to theory of Wannier functions
Solution measurements of hyperpolarizability
(Time-averaged) inversion symmetry of solution makes ordinary measurement give zero.
Electric-field-induced second-harmonic generation (EFISH) is coherent third-order process, based on field lining up molecules.
Hyper-Rayleigh scattering (HRS) is incoherent second-order process, based on orientational fluctuations.
Measurements are usually in solution for molecules. Solvent effects can be strong and complicate comparison between experiment and theory.(In theory, handle via polarizable continuum models or explicit solvent in small clusters or periodic system.)
What is measured directly and often reported:
Solution measurements of hyperpolarizability
Consider projection along dipole moment.
Vertical-vertical and horizontal-vertical polarizations for experiment.
Vibrational / rotational contributions
Many measurements are at l = 1064 nm. For organic molecules, typically:- above vibrational frequencies, so can neglect vibrations- below electronic resonances so little dispersion
D. Bishop, Rev. Mod. Phys. 62, 343 (1990)
Rotational and vibrational contributions can be estimated from simple harmonic models, for low-frequency response (e.g. THz).
E. Roman, J. R. Yates, M. Veithen, D. Vanderbilt, and I. Souza, Phys. Rev. B 74, 245204 (2006)
Z* = Born effective charge
Local-field factors for solvent
+
+
+
+
+
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+
Lorentz:
Clausius-Mossotti relations,relate bulk to molecular susceptibilities.
Also generalizations for ellipsoidal cavity, polar solvents.
Theoretical methods for nonlinear response
1. Finite differences (static)
2. Explicit time-propagation
3. Sternheimer equation (2n+1 Theorem)
4. Sum over states (Casida equation)
5. Dyson equation
Finite differencesApply static fields and calculate dipole moment.No need for special capabilities in code.Probably most common method used.Hope static and IR are similar!
F. Vila, D.A. Strubbe, Y. Takimoto, X. Andrade, A. Rubio, S. G. Louie, and J. J. Rehr, J. Chem. Phys. 133, 034111 (2010)
CHCl3
fit or stencil
Convergence is more demanding for nonlinear response
Response extends far from molecule (> 15 a0)
Compare: only 12 a0 converges ground stateall atoms contained within 3 a0 radius
R
F. Vila, D.A. Strubbe, Y. Takimoto, X. Andrade, A. Rubio, S. G. Louie, and J. J. Rehr, J. Chem. Phys. 133, 034111 (2010)
Comparison of Gaussian-type orbitals, SIESTA numerical basis sets, real-space grid: need five-zeta basis sets with diffuse functions, or very long-range real-space grid.
(SIESTA)
(aug-cc-pV5Z)
(h =0.25, r=22)
Explicit time-propagationCannot use “kick” for all frequencies at once (as for linear optics).Must calculate separately for each frequency combination.Scaling is like one ground-state calculation per time step.Incident laser pulses (duration ~ imaginary broadening).
Takimoto, Vila, and Rehr, J. Chem. Phys. 127, 154114 (2007)
p-nitroaniline
Explicit time-propagation
Takimoto, Vila, and Rehr, J. Chem. Phys. 127, 154114 (2007)
Sternheimer equation
aka density-functional perturbation theory (DFPT)Calculate variation of wavefunctions in linear response.No need for unoccupied states.SCF cycle for one frequency at a time
Need small imaginary broadening ih near resonances for numerical stability.
Sternheimer equation: 2n+1 Theorem
Solving nth-order perturbation theory gives 2n+1 derivative of total energy.
X. Gonze and J.-P. Vigneron, Phys. Rev. B 39, 13120 (1989)
n = 0: Hellman-Feynman Theorem. No wavefunction derivatives at all.n = 1: Linear variation of wavefunctions gives quadratic response.
Quantum ESPRESSO and Octopus use equations on previous slide.ABINIT formulation: find b by minimizing with respect to y(1) (variational).
Sternheimer equation: examples from Octopus code
CO SHG
X Andrade, S Botti, MAL Marques, and A Rubio, J. Chem. Phys. 126, 184106 (2007)
H2O optical rectification
CO linear spectrum
Sternheimer equation: examples from Octopus codeEfficient scaling with system size: same as ground state.e.g. comparison of TDLDA vs. CIS with Gaussian basis for vibrating N@C60
G. P. Zhang, D.A. Strubbe, S. G. Louie, and T. F. George, Phys. Rev. A 84, 023837 (2011)
Applicable to any method giving excited-state energies and matrix elements, in particular Casida equation for TDDFT.
(Also other theories such as configuration interaction, coupled cluster, Bethe-Salpeter equation, etc.)
Most commonly used as RPA though: just use Kohn-Sham energies and matrix elements.
Sum over states
Sum over states
Arbitrary (or physical but unknown) imaginary broadenings G.
Convergence in two sums on states is difficult!
Applied occasionally to two-photon absorption in TDDFT.
Y-R Shen, The Principles of Nonlinear Optics
Dyson-like equation
Efficient scheme for solids (for linear and non-linear optics): k-points.Need q->0 limit, equivalent to k.p perturbation theory.
G. Senatore and K. R. Subbaswamy, Phys. Rev. A 35, 2440 (1987)E. Luppi, H. Hübener, and V. Véniard, J. Chem. Phys.132, 241104 (2010)
Complicated macroscopic/microscopic relations…
GaAs SHG
Dyson-like equation
Must converge with respect to unoccupied states in a triple sum.
E. Luppi, H. Hübener, and V. Véniard, Phys. Rev. B 82, 235201 (2010)
Magnetic perturbations
Term in Hamiltonian:
Daniele Varsano, Leonardo A. Espinosa-Leal, Xavier Andrade, Miguel A. L. Marques, Rosa di Felice and Angel Rubio, Phys. Chem. Chem. Phys. 11, 4481 (2009)
~µ⇥ ~B<latexit sha1_base64="UXwNkqFwwcW6O3BpAqw+3i/d2LI=">AAACAXicjVBNS8NAEJ3Ur1q/ol4EL4tF8FQSEfRY6sVjBdsKTSib7aZdursJu5tCCfXiX/HiQRGv/gtv/hu3aQ8qCj4YeLw3w8y8KOVMG8/7cEpLyyura+X1ysbm1vaOu7vX1kmmCG2RhCfqNsKaciZpyzDD6W2qKBYRp51odDnzO2OqNEvkjZmkNBR4IFnMCDZW6rkHwZiSPBDZFAWGCapRITSmPbfq17wC6G9ShQWaPfc96CckE1QawrHWXd9LTZhjZRjhdFoJMk1TTEZ4QLuWSmyXhXnxwRQdW6WP4kTZkgYV6teJHAutJyKynQKbof7pzcTfvG5m4oswZzLNDJVkvijOODIJmsWB+kxRYvjEEkwUs7ciMsQKE2NDq/wvhPZpzfdq/vVZtd5YxFGGQziCE/DhHOpwBU1oAYE7eIAneHbunUfnxXmdt5acxcw+fIPz9gmmjZcA</latexit><latexit sha1_base64="UXwNkqFwwcW6O3BpAqw+3i/d2LI=">AAACAXicjVBNS8NAEJ3Ur1q/ol4EL4tF8FQSEfRY6sVjBdsKTSib7aZdursJu5tCCfXiX/HiQRGv/gtv/hu3aQ8qCj4YeLw3w8y8KOVMG8/7cEpLyyura+X1ysbm1vaOu7vX1kmmCG2RhCfqNsKaciZpyzDD6W2qKBYRp51odDnzO2OqNEvkjZmkNBR4IFnMCDZW6rkHwZiSPBDZFAWGCapRITSmPbfq17wC6G9ShQWaPfc96CckE1QawrHWXd9LTZhjZRjhdFoJMk1TTEZ4QLuWSmyXhXnxwRQdW6WP4kTZkgYV6teJHAutJyKynQKbof7pzcTfvG5m4oswZzLNDJVkvijOODIJmsWB+kxRYvjEEkwUs7ciMsQKE2NDq/wvhPZpzfdq/vVZtd5YxFGGQziCE/DhHOpwBU1oAYE7eIAneHbunUfnxXmdt5acxcw+fIPz9gmmjZcA</latexit><latexit sha1_base64="GASambTtqNrE+veB5VCwy2OmqjU=">AAACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5KIoMtSNy4r2Ac0oUymk3boZBJmbgolxI2/4saFIm79C3f+jdO0BZ8HLhzOuZd77wkSwTU4zoe1tLyyurZe2ihvbm3v7Np7+y0dp4qyJo1FrDoB0UxwyZrAQbBOohiJAsHawehq6rfHTGkey1uYJMyPyEDykFMCRurZh96Y0cyL0hx7wCOmcSHU855dcatOAfw/qaA5Gj373evHNI2YBCqI1l3XScDPiAJOBcvLXqpZQuiIDFjXUEnMMj8rPsjxiVH6OIyVKQm4UL9OZCTSehIFpjMiMNQ/van4l9dNIbz0My6TFJiks0VhKjDEeBoH7nPFKIiJIYQqbm7FdEgUoWBCK5sQnMXLv8kihNZZ1XWq7s15pVafx1FCR+gYnSIXXaAaukYN1EQU3aEH9ISerXvr0XqxXmetS9Z85gB9g/X2CaU3lv0=</latexit><latexit sha1_base64="GASambTtqNrE+veB5VCwy2OmqjU=">AAACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5KIoMtSNy4r2Ac0oUymk3boZBJmbgolxI2/4saFIm79C3f+jdO0BZ8HLhzOuZd77wkSwTU4zoe1tLyyurZe2ihvbm3v7Np7+y0dp4qyJo1FrDoB0UxwyZrAQbBOohiJAsHawehq6rfHTGkey1uYJMyPyEDykFMCRurZh96Y0cyL0hx7wCOmcSHU855dcatOAfw/qaA5Gj373evHNI2YBCqI1l3XScDPiAJOBcvLXqpZQuiIDFjXUEnMMj8rPsjxiVH6OIyVKQm4UL9OZCTSehIFpjMiMNQ/van4l9dNIbz0My6TFJiks0VhKjDEeBoH7nPFKIiJIYQqbm7FdEgUoWBCK5sQnMXLv8kihNZZ1XWq7s15pVafx1FCR+gYnSIXXaAaukYN1EQU3aEH9ISerXvr0XqxXmetS9Z85gB9g/X2CaU3lv0=</latexit><latexit sha1_base64="GASambTtqNrE+veB5VCwy2OmqjU=">AAACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5KIoMtSNy4r2Ac0oUymk3boZBJmbgolxI2/4saFIm79C3f+jdO0BZ8HLhzOuZd77wkSwTU4zoe1tLyyurZe2ihvbm3v7Np7+y0dp4qyJo1FrDoB0UxwyZrAQbBOohiJAsHawehq6rfHTGkey1uYJMyPyEDykFMCRurZh96Y0cyL0hx7wCOmcSHU855dcatOAfw/qaA5Gj373evHNI2YBCqI1l3XScDPiAJOBcvLXqpZQuiIDFjXUEnMMj8rPsjxiVH6OIyVKQm4UL9OZCTSehIFpjMiMNQ/van4l9dNIbz0My6TFJiks0VhKjDEeBoH7nPFKIiJIYQqbm7FdEgUoWBCK5sQnMXLv8kihNZZ1XWq7s15pVafx1FCR+gYnSIXXaAaukYN1EQU3aEH9ISerXvr0XqxXmetS9Z85gB9g/X2CaU3lv0=</latexit><latexit sha1_base64="GASambTtqNrE+veB5VCwy2OmqjU=">AAACAXicdVDLSsNAFJ34rPUVdSO4GSyCq5KIoMtSNy4r2Ac0oUymk3boZBJmbgolxI2/4saFIm79C3f+jdO0BZ8HLhzOuZd77wkSwTU4zoe1tLyyurZe2ihvbm3v7Np7+y0dp4qyJo1FrDoB0UxwyZrAQbBOohiJAsHawehq6rfHTGkey1uYJMyPyEDykFMCRurZh96Y0cyL0hx7wCOmcSHU855dcatOAfw/qaA5Gj373evHNI2YBCqI1l3XScDPiAJOBcvLXqpZQuiIDFjXUEnMMj8rPsjxiVH6OIyVKQm4UL9OZCTSehIFpjMiMNQ/van4l9dNIbz0My6TFJiks0VhKjDEeBoH7nPFKIiJIYQqbm7FdEgUoWBCK5sQnMXLv8kihNZZ1XWq7s15pVafx1FCR+gYnSIXXaAaukYN1EQU3aEH9ISerXvr0XqxXmetS9Z85gB9g/X2CaU3lv0=</latexit>
~µ = ~r ⇥ i~~r<latexit sha1_base64="PfEffYIpKXOE1qCz+zraxC3DXPg=">AAACGHicjVBNS8NAEN34WetX1KOXxSJ4qokIehGKXjxWsB/QhDLZbtulu5uwuymUkJ/hxb/ixYMiXnvz37hNe1BR8MHA470ZZuZFCWfaeN6Hs7S8srq2Xtoob25t7+y6e/tNHaeK0AaJeazaEWjKmaQNwwyn7URREBGnrWh0M/NbY6o0i+W9mSQ0FDCQrM8IGCt13dNgTEkWiDTHV7jgKseBYYJqzHAwjEDN5UBCxCHvuhW/6hXAf5MKWqDedadBLyapoNIQDlp3fC8xYQbKMMJpXg5STRMgIxjQjqUS7OYwKx7L8bFVergfK1vS4EL9OpGB0HoiItspwAz1T28m/uZ1UtO/DDMmk9RQSeaL+inHJsazlHCPKUoMn1gCRDF7KyZDUECMzbL8vxCaZ1Xfq/p355Xa9SKOEjpER+gE+egC1dAtqqMGIugBPaEX9Oo8Os/Om/M+b11yFjMH6Buc6Sefo6Ah</latexit><latexit sha1_base64="PfEffYIpKXOE1qCz+zraxC3DXPg=">AAACGHicjVBNS8NAEN34WetX1KOXxSJ4qokIehGKXjxWsB/QhDLZbtulu5uwuymUkJ/hxb/ixYMiXnvz37hNe1BR8MHA470ZZuZFCWfaeN6Hs7S8srq2Xtoob25t7+y6e/tNHaeK0AaJeazaEWjKmaQNwwyn7URREBGnrWh0M/NbY6o0i+W9mSQ0FDCQrM8IGCt13dNgTEkWiDTHV7jgKseBYYJqzHAwjEDN5UBCxCHvuhW/6hXAf5MKWqDedadBLyapoNIQDlp3fC8xYQbKMMJpXg5STRMgIxjQjqUS7OYwKx7L8bFVergfK1vS4EL9OpGB0HoiItspwAz1T28m/uZ1UtO/DDMmk9RQSeaL+inHJsazlHCPKUoMn1gCRDF7KyZDUECMzbL8vxCaZ1Xfq/p355Xa9SKOEjpER+gE+egC1dAtqqMGIugBPaEX9Oo8Os/Om/M+b11yFjMH6Buc6Sefo6Ah</latexit><latexit sha1_base64="SAAYVGuvLpDpCYnNOeGWP7pzCrk=">AAACGHicdVDLSsNAFJ34rPUVdelmsAiuaiKCboSiG5cV7AOaUG6mk3boZBJmJoUS8hlu/BU3LhRx251/4zRtweeBC4dz7uXee4KEM6Ud58NaWl5ZXVsvbZQ3t7Z3du29/aaKU0log8Q8lu0AFOVM0IZmmtN2IilEAaetYHgz9VsjKhWLxb0eJ9SPoC9YyAhoI3XtU29ESeZFaY6vcMFljj3NIqoww94gADmTPQEBh7xrV9yqUwD/TypojnrXnni9mKQRFZpwUKrjOon2M5CaEU7zspcqmgAZQp92DBVgNvtZ8ViOj43Sw2EsTQmNC/XrRAaRUuMoMJ0R6IH66U3Fv7xOqsNLP2MiSTUVZLYoTDnWMZ6mhHtMUqL52BAgkplbMRmABKJNlmUTgrN4+TdZhNA8q7pO1b07r9Su53GU0CE6QifIRReohm5RHTUQQQ/oCb2gV+vRerberPdZ65I1nzlA32BNPgGeTaAe</latexit><latexit sha1_base64="SAAYVGuvLpDpCYnNOeGWP7pzCrk=">AAACGHicdVDLSsNAFJ34rPUVdelmsAiuaiKCboSiG5cV7AOaUG6mk3boZBJmJoUS8hlu/BU3LhRx251/4zRtweeBC4dz7uXee4KEM6Ud58NaWl5ZXVsvbZQ3t7Z3du29/aaKU0log8Q8lu0AFOVM0IZmmtN2IilEAaetYHgz9VsjKhWLxb0eJ9SPoC9YyAhoI3XtU29ESeZFaY6vcMFljj3NIqoww94gADmTPQEBh7xrV9yqUwD/TypojnrXnni9mKQRFZpwUKrjOon2M5CaEU7zspcqmgAZQp92DBVgNvtZ8ViOj43Sw2EsTQmNC/XrRAaRUuMoMJ0R6IH66U3Fv7xOqsNLP2MiSTUVZLYoTDnWMZ6mhHtMUqL52BAgkplbMRmABKJNlmUTgrN4+TdZhNA8q7pO1b07r9Su53GU0CE6QifIRReohm5RHTUQQQ/oCb2gV+vRerberPdZ65I1nzlA32BNPgGeTaAe</latexit><latexit sha1_base64="SAAYVGuvLpDpCYnNOeGWP7pzCrk=">AAACGHicdVDLSsNAFJ34rPUVdelmsAiuaiKCboSiG5cV7AOaUG6mk3boZBJmJoUS8hlu/BU3LhRx251/4zRtweeBC4dz7uXee4KEM6Ud58NaWl5ZXVsvbZQ3t7Z3du29/aaKU0log8Q8lu0AFOVM0IZmmtN2IilEAaetYHgz9VsjKhWLxb0eJ9SPoC9YyAhoI3XtU29ESeZFaY6vcMFljj3NIqoww94gADmTPQEBh7xrV9yqUwD/TypojnrXnni9mKQRFZpwUKrjOon2M5CaEU7zspcqmgAZQp92DBVgNvtZ8ViOj43Sw2EsTQmNC/XrRAaRUuMoMJ0R6IH66U3Fv7xOqsNLP2MiSTUVZLYoTDnWMZ6mhHtMUqL52BAgkplbMRmABKJNlmUTgrN4+TdZhNA8q7pO1b07r9Su53GU0CE6QifIRReohm5RHTUQQQ/oCb2gV+vRerberPdZ65I1nzlA32BNPgGeTaAe</latexit><latexit sha1_base64="SAAYVGuvLpDpCYnNOeGWP7pzCrk=">AAACGHicdVDLSsNAFJ34rPUVdelmsAiuaiKCboSiG5cV7AOaUG6mk3boZBJmJoUS8hlu/BU3LhRx251/4zRtweeBC4dz7uXee4KEM6Ud58NaWl5ZXVsvbZQ3t7Z3du29/aaKU0log8Q8lu0AFOVM0IZmmtN2IilEAaetYHgz9VsjKhWLxb0eJ9SPoC9YyAhoI3XtU29ESeZFaY6vcMFljj3NIqoww94gADmTPQEBh7xrV9yqUwD/TypojnrXnni9mKQRFZpwUKrjOon2M5CaEU7zspcqmgAZQp92DBVgNvtZ8ViOj43Sw2EsTQmNC/XrRAaRUuMoMJ0R6IH66U3Fv7xOqsNLP2MiSTUVZLYoTDnWMZ6mhHtMUqL52BAgkplbMRmABKJNlmUTgrN4+TdZhNA8q7pO1b07r9Su53GU0CE6QifIRReohm5RHTUQQQ/oCb2gV+vRerberPdZ65I1nzlA32BNPgGeTaAe</latexit>
Perturbation:
↵⌫µ,� =@↵⌫µ
@B�<latexit sha1_base64="l3v5IvhTOt5Ueo8dq+2pTHlvLQY=">AAACN3icjVBNS8MwGE79nPNr6tFLcAgeZLQi6EUY8+JJJjgV1lHeZukWTNKSpMIo/Vde/Bve9OJBEa/+A9Ot4AcKPhB4eT7eJE+YcKaN6z44U9Mzs3PzlYXq4tLyymptbf1Cx6kitENiHqurEDTlTNKOYYbTq0RRECGnl+H1caFf3lClWSzPzSihPQEDySJGwFgqqJ36wJMhBJkvU+yLdBf7AxACcnyE/UgByfwElGHA8Q9nnn9KraBMBbW613DHwH8PdVSiHdTu/X5MUkGlIRy07npuYnpZsZdwmlf9VNMEyDUMaNeOEgTVvWz87xxvW6aPo1jZIw0es18TGQitRyK0TgFmqH9qBfmb1k1NdNjLmExSQyWZXBSlHJsYFyXiPlOUGD6yAxDF7FsxGYJty9iqq/8r4WKv4bkN72y/3myVdVTQJtpCO8hDB6iJTlAbdRBBt+gRPaMX5855cl6dt4l1yikzG+gbnPcPEcStSA==</latexit><latexit sha1_base64="l3v5IvhTOt5Ueo8dq+2pTHlvLQY=">AAACN3icjVBNS8MwGE79nPNr6tFLcAgeZLQi6EUY8+JJJjgV1lHeZukWTNKSpMIo/Vde/Bve9OJBEa/+A9Ot4AcKPhB4eT7eJE+YcKaN6z44U9Mzs3PzlYXq4tLyymptbf1Cx6kitENiHqurEDTlTNKOYYbTq0RRECGnl+H1caFf3lClWSzPzSihPQEDySJGwFgqqJ36wJMhBJkvU+yLdBf7AxACcnyE/UgByfwElGHA8Q9nnn9KraBMBbW613DHwH8PdVSiHdTu/X5MUkGlIRy07npuYnpZsZdwmlf9VNMEyDUMaNeOEgTVvWz87xxvW6aPo1jZIw0es18TGQitRyK0TgFmqH9qBfmb1k1NdNjLmExSQyWZXBSlHJsYFyXiPlOUGD6yAxDF7FsxGYJty9iqq/8r4WKv4bkN72y/3myVdVTQJtpCO8hDB6iJTlAbdRBBt+gRPaMX5855cl6dt4l1yikzG+gbnPcPEcStSA==</latexit><latexit sha1_base64="xwjSxuvKIymxATeD4V//jl+ttpU=">AAACN3icdVDLSgMxFM34rPVVdekmWAQXUmZE0I0gunElFWwVOsNwJ820wSQzJBmhDPNXbvwNd7pxoYhb/8BMW/F9IHA459wk90QpZ9q47r0zMTk1PTNbmavOLywuLddWVts6yRShLZLwRF1GoClnkrYMM5xepoqCiDi9iK6OS//imirNEnluBikNBPQkixkBY6WwduoDT/sQ5r7MsC+ybez3QAgo8AH2YwUk91NQhgHHP5JF8WkdheOpsFb3Gu4Q+H9SR2M0w9qd301IJqg0hIPWHc9NTZCX9xJOi6qfaZoCuYIe7VgqQVAd5MO9C7xplS6OE2WPNHiofp3IQWg9EJFNCjB9/dMrxb+8Tmbi/SBnMs0MlWT0UJxxbBJcloi7TFFi+MASIIrZv2LSB9uWsVVXbQnux8q/yUcJ7Z2G5za8s9364dG4jgpaRxtoC3loDx2iE9RELUTQDXpAT+jZuXUenRfndRSdcMYza+gbnLd3EG6tRQ==</latexit><latexit sha1_base64="xwjSxuvKIymxATeD4V//jl+ttpU=">AAACN3icdVDLSgMxFM34rPVVdekmWAQXUmZE0I0gunElFWwVOsNwJ820wSQzJBmhDPNXbvwNd7pxoYhb/8BMW/F9IHA459wk90QpZ9q47r0zMTk1PTNbmavOLywuLddWVts6yRShLZLwRF1GoClnkrYMM5xepoqCiDi9iK6OS//imirNEnluBikNBPQkixkBY6WwduoDT/sQ5r7MsC+ybez3QAgo8AH2YwUk91NQhgHHP5JF8WkdheOpsFb3Gu4Q+H9SR2M0w9qd301IJqg0hIPWHc9NTZCX9xJOi6qfaZoCuYIe7VgqQVAd5MO9C7xplS6OE2WPNHiofp3IQWg9EJFNCjB9/dMrxb+8Tmbi/SBnMs0MlWT0UJxxbBJcloi7TFFi+MASIIrZv2LSB9uWsVVXbQnux8q/yUcJ7Z2G5za8s9364dG4jgpaRxtoC3loDx2iE9RELUTQDXpAT+jZuXUenRfndRSdcMYza+gbnLd3EG6tRQ==</latexit><latexit sha1_base64="xwjSxuvKIymxATeD4V//jl+ttpU=">AAACN3icdVDLSgMxFM34rPVVdekmWAQXUmZE0I0gunElFWwVOsNwJ820wSQzJBmhDPNXbvwNd7pxoYhb/8BMW/F9IHA459wk90QpZ9q47r0zMTk1PTNbmavOLywuLddWVts6yRShLZLwRF1GoClnkrYMM5xepoqCiDi9iK6OS//imirNEnluBikNBPQkixkBY6WwduoDT/sQ5r7MsC+ybez3QAgo8AH2YwUk91NQhgHHP5JF8WkdheOpsFb3Gu4Q+H9SR2M0w9qd301IJqg0hIPWHc9NTZCX9xJOi6qfaZoCuYIe7VgqQVAd5MO9C7xplS6OE2WPNHiofp3IQWg9EJFNCjB9/dMrxb+8Tmbi/SBnMs0MlWT0UJxxbBJcloi7TFFi+MASIIrZv2LSB9uWsVVXbQnux8q/yUcJ7Z2G5za8s9364dG4jgpaRxtoC3loDx2iE9RELUTQDXpAT+jZuXUenRfndRSdcMYza+gbnLd3EG6tRQ==</latexit><latexit sha1_base64="xwjSxuvKIymxATeD4V//jl+ttpU=">AAACN3icdVDLSgMxFM34rPVVdekmWAQXUmZE0I0gunElFWwVOsNwJ820wSQzJBmhDPNXbvwNd7pxoYhb/8BMW/F9IHA459wk90QpZ9q47r0zMTk1PTNbmavOLywuLddWVts6yRShLZLwRF1GoClnkrYMM5xepoqCiDi9iK6OS//imirNEnluBikNBPQkixkBY6WwduoDT/sQ5r7MsC+ybez3QAgo8AH2YwUk91NQhgHHP5JF8WkdheOpsFb3Gu4Q+H9SR2M0w9qd301IJqg0hIPWHc9NTZCX9xJOi6qfaZoCuYIe7VgqQVAd5MO9C7xplS6OE2WPNHiofp3IQWg9EJFNCjB9/dMrxb+8Tmbi/SBnMs0MlWT0UJxxbBJcloi7TFFi+MASIIrZv2LSB9uWsVVXbQnux8q/yUcJ7Z2G5za8s9364dG4jgpaRxtoC3loDx2iE9RELUTQDXpAT+jZuXUenRfndRSdcMYza+gbnLd3EG6tRQ==</latexit>
Quantity of interest (circular dichroism):
Gauge issue with A and non-local pseudopotentials.
Can use Sternheimer or real-time
Magnetic perturbations in solids
Vector potential can be expressed in different gauges.
Separate wavefunction into gauge-dependent and gauge-independent parts.E and B fields as in quantum theory of polarization.Sternheimer equation for response.Carefully combine parts into gauge-independent quantity.
IV Lebedeva, DA Strubbe, IV Tokatly, and A Rubio, arxiV:1806.09886 (2018)
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~E = �1
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@ ~A
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Ramanspectroscopy
34
phononcreation
Lightcreates(ordestroys)vibrationsthroughexcitedstates
Ramantensorfrompolarizabilityderivativewithrespecttoatomicpositions
staticlinear-responsecalculationofRamantensorsM.Lazzeri andF.Mauri,Phys.Rev.Lett. 90,036401(2003)
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resonant Raman: need optical method, e.g. TDDFT or Bethe-Salpeter eqn.non-resonant: can make static approximation
Some references on linear and nonlinear response
DA Strubbe, L Lehtovaara, A Rubio, MAL Marques, and SG Louie, “Response functions in TDDFT: concepts and implementation,” in Fundamentals of Time-dependent density-functional theory (ch. 7) and references therein.
Yuen-Ron Shen, The Principles of Nonlinear Optics (Wiley 1984 / 2002).
P. N. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).