The Nuts and Bolts of FirstPrinciples Simulation 22

The Nuts and Bolts of First-Principles Simulation 22: Linear response theory Durham, 6 th-13 th December 2001 Nuts and Bolts 2001 CASTEP Developers’ Group with support from the ESF k Network Lecture 22: Linear response theory

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

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

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

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

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

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

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

Expression for E(2) l l l For order n, the “ 2 n+1 theorem” allows us to write a constrained variational expression for E: n E(2 n) depends on of order n or below only n The terms in E(2 n) are the set having order 2 n 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 9

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

Off-diagonal elements The E(2) just considered is stationary r There are non-stationary expressions that can be used to find off-diagonal elements r l l Combine the y(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) r Can check diagonal elements this way r Nuts and Bolts 2001 Lecture 22: Linear response theory 11

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

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

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

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

Other perturbations r 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 16

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