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Sternheimer equation: 2n+1 Theorem Solving n th -order perturbation theory gives 2n+1 derivative of total energy. n = 0: Hellman-Feyman Theorem. No wavefunction derivatives at all. n = 1: Linear variation of wavefunctions gives quadratic response. X. Gonze and J.-P. Vigneron, Phys. Rev. B 39, 13120 (1989) Quantum ESPRESSO and Octopus use equations on previous slide. ABINIT formulation: find β by minimizing with respect to ψ (1) (variational).
Perturbations in Sternheimer equation Electric field in finite system: (Can also study response to magnetic fields, ionic displacements, strain, etc.) Dipole reformulation with quantum theory of polarization in extended system A Dal Corso, F Mauri, and A Rubio, Phys. Rev. B 53, 15638 (1996) k·p perturbation theory (non-SCF):
Page 1 and 2: Non-linear optics, k·p perturbatio
Page 3 and 4: Quantized picture of non-linear opt
Page 5 and 6: Question: How does the liquid envir
Page 7 and 8: Solution measurements of hyperpolar
Page 9 and 10: 1. Effect of electronic confinement
Page 11 and 12: Finite differences Apply static fie
Page 13: Sternheimer equation: projectors an
Page 17 and 18: Si k.p perturbation
Page 19 and 20: Vibrational / rotational contributi
Page 21 and 22: Example: H 2 O molecule O -0.47 0.0
Page 23 and 24: Linear solvers All have failure con
Page 25 and 26: Performance on non-Hermitian system
Page 27: Acknowledgments Steven Louie, Angel

Sternheimer equation - TDDFT.org

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