The Nuts and Bolts of First-Principles Simulation Durham, 6th-13th December : Linear response theory CASTEP Developers’ Group with support from the ESF  k Network

Nuts and Bolts 2001 Lecture 22: Linear response theory 2 Outline  Objectives  Perturbed potentials and E (2)  Which perturbations?  Phonons  Calculating E (2) for phonon perturbations  Cross derivatives and the force-constant matrix  The zone centre  Summary

Nuts and Bolts 2001 Lecture 22: Linear response theory 3 Objectives  To give an idea of what linear response theory is and what can be calculated with it  To outline the theory for phonons  To show the scheme of calculation

Nuts and Bolts 2001 Lecture 22: Linear response theory 4 Perturbed potentials  The central idea is to compute how the total energy responds to a perturbation, usually of the DFT external potential v  Expand quantities (E, n, , v)  Taylor series  Properties related to the derivatives

Nuts and Bolts 2001 Lecture 22: Linear response theory 5 Which perturbations?  External potential: arises from the ionic cores and any external fields Ionic positions  phonons Cell vectors  elastic constants Electric fields  dielectric response Magnetic fields  NMR  Not just the potential, any Hamiltonian perturbation d/dk  Born effective charges d/d(PSP)  alchemical perturbation

Nuts and Bolts 2001 Lecture 22: Linear response theory 6 Phonons: basics  For a periodic system the displacement pattern for each atom  Frequency  depends on q - dispersion  For N atoms in a (super)cell there are 3N phonon modes at each wavevector q

Nuts and Bolts 2001 Lecture 22: Linear response theory 7 Phonons: general expressions  The 3N eigenstates of D  Relation to energy second derivative

Nuts and Bolts 2001 Lecture 22: Linear response theory 8 Phonon perturbation for LR  For each atom i at a time, in direction   The potential becomes a function of  Take derivatives of the potential wrt  Hartree, xc: derivatives of potentials done by chain rule wrt n and  Then E (2) looks like this...

Nuts and Bolts 2001 Lecture 22: Linear response theory 9 Expression for E (2) For order n, the “2n+1 theorem” allows us to write a constrained variational expression for E: E (2n) depends on  of order n or below only The terms in E (2n) are the set having order 2n This is a variational quantity - more later This expression gives the electronic contribution to the diagonal elements

Nuts and Bolts 2001 Lecture 22: Linear response theory 10 Variational principle  E (2) is variational wrt  The plane-wave coefficients of are varied to find the minimum E (2) under a perturbation of a given ion i in a given direction  and for a given q  Similar to standard total energy calculation  Based on a ground state (E (0) ) calculation  Choice of q related to {k }

Nuts and Bolts 2001 Lecture 22: Linear response theory 11 Off-diagonal elements  The E (2) just considered is stationary  There are non-stationary expressions that can be used to find off-diagonal elements Combine the  (1 ) found for one perturbation with the potential perturbed for another ion or direction …but at the same q of course  We have a row of the matrix (electronic part)  Can check diagonal elements this way

Nuts and Bolts 2001 Lecture 22: Linear response theory 12 Whole calculation  Use  Find electronic force constant matrix  Add in Ewald part  Repeat for a mesh of q  Fourier transform to get F(R)  Fit and interpolate  Fourier transform and mass weight to get D at arbitrary q

Nuts and Bolts 2001 Lecture 22: Linear response theory 13 Pros and cons  Pros Fast, each wavevector less than a total energy calculation Arbitrary q General formalism  Cons Details of implementation considerable May not be ideal for  -point calculation of second derivatives in large systems (transition states)

Nuts and Bolts 2001 Lecture 22: Linear response theory 14 The zone centre  Need Born effective charges to get LO-TO splitting Found from d/dk calculation and similar cross- derivative expressions to the forgoing  Technical note: all expressions for perturbed potentials different at zone centre

Nuts and Bolts 2001 Lecture 22: Linear response theory 15 LR in CASTEP  What Dynamical matrix for arbitrary q “Back end”: dispersion, DOS, free energy... Born effective charges  With Insulators (metals) LDA and GGA Norm-conserving potentials  When Beta in spring 2002

Nuts and Bolts 2001 Lecture 22: Linear response theory 16 Other perturbations  Beware that the approach is general, but the major work is in the detail of implementation

Nuts and Bolts 2001 Lecture 22: Linear response theory 17 Summary  LR: powerful, general, efficient  Phonon calculations in CASTEP soon