An introduction to linear response and to phonon calculations - PowerPoint PPT Presentation

An introduction to linear response and to phonon calculations —– Reference: Phonons and related crystal properties from density-functional perturbation theory , S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev. Mod. Phys. 73 , 515-562 (2001). – Typeset by Foil T EX –

Electronic screening Consider a static perturbation δV 0 ( r ) to a system of electrons under an external (nuclear) potential V 0 ( r ) . At linear order, � χ ( r , r ′ ) δV 0 ( r ′ ) d r ′ δn ( r ) = where χ ( r , r ′ ) is the density response of the system. The polarization charge δn ( r ) produces an electrostatic field that screens the perturbing potential δV 0 ( r ) : � δn ( r ′ ) δV ( r ) = δV 0 ( r ) + e 2 | r − r ′ | d r ′ that is: � � � χ ( r ′′ , r ′ ) � � δV 0 ( r ′ ) d r ′ ≡ δ ( r − r ′ ) + e 2 ǫ − 1 ( r , r ′ ) δV 0 ( r ′ ) d r ′ . | r − r ′′ | d r ′′ δV ( r ) = ǫ − 1 ( r , r ′ ) is the dielectric response function as usually defined in electrostatics.

Linear Response functions • χ ( r , r ′ ) yields the charge response to a bare (external) perturbing potential via � χ ( r , r ′ ) δV 0 ( r ′ ) d r ′ δn ( r ) = • ǫ − 1 ( r , r ′ ) yields the screened potential from the bare one via � ǫ − 1 ( r , r ′ ) δV 0 ( r ′ ) d r ′ . δV ( r ) = and is related to χ ( r , r ′ ) via � χ ( r ′′ , r ′ ) ǫ − 1 ( r , r ′ ) ≡ δ ( r − r ′ ) + e 2 | r − r ′′ | d r ′′ These are the functions that determine electronic response. Their calculation is however a nontrivial many-body problem.

Density-Functional Linear Response We assume that the system obeys Kohn-Sham (KS) equations: h 2 H KS = − ¯ 2 m ∇ 2 + V KS ( r ) ( H KS − ǫ i ) ψ i ( r ) = 0 , where V KS ( r ) = V 0 ( r ) + V H ( r ) + V xc [ n ( r )] and the charge is given by � f i | ψ i ( r ) | 2 n ( r ) = i (general case of noninteger occupancy f i ). Let us add an external δV 0 ( r ) to V 0 ( r ) : the potential V KS will be modified by δV KS = δV 0 ( r )+ δV H ( r )+ δV xc [ n ( r )] . Let us consider first order (linear response). We introduce the independent-particle polarizability χ 0 ( r , r ′ ) via � χ 0 ( r , r ′ ) δV KS ( r ′ ) d r ′ . δn ( r ) = Unlike χ ( r , r ′ ) , this quantity can be easily calculated using perturbation theory.

Independent-particle polarizability The first-order variation of KS orbitals: � ψ j ( r ) � ψ j | δV KS | ψ i � δψ i ( r ) = ǫ i − ǫ j j � = i and of the charge density (after some manipulations): � � f i − f j f i δψ ∗ ψ ∗ δn ( r ) = i ( r ) ψ i ( r ) + c.c. = i ( r ) ψ j ( r ) � ψ j | δV KS | ψ i � ǫ i − ǫ j i i,j,i � = j Note that contribution from i, j states vanishes if both are fully occupied. For a closed-shell (insulating) system: � v ( r ) ψ c ( r ) � ψ c | δV KS | ψ v � ψ ∗ δn ( r ) = 4 Re ǫ v − ǫ c v,c v = filled (valence) states, c = empty (conduction) states, a factor 2 from spin.

Independent-particle polarizability II We can write the independent-particle polarizability χ 0 ( r , r ′ ) as � ψ ∗ v ( r ) ψ c ( r ) ψ ∗ c ( r ′ ) ψ v ( r ′ ) χ 0 ( r , r ′ ) = 4 Re . ǫ v − ǫ c v,c which can be recast into the form � 1 χ 0 ( r , r ′ ) = 4 Re ψ ∗ P c ψ v ( r ′ ) v ( r ) P c ǫ v − H KS v where P c is the projector operator over conduction states. Note that: • this expression is valid only if V KS ≡ V KS ( r ) , i.e. is a local potential: • χ 0 ( r , r ′ ) is a ground-state property : it yields the difference between two ground states, even if it seems to depend on excited-state energies ǫ c

Physical Response Operator ...but we need χ ( r , r ′ ) , not χ 0 ( r , r ′ ) ! How can we get from χ 0 to χ ? In operator notations: δn = ˆ χδV 0 = ˆ χ 0 δV KS , and δV KS = δV 0 + δV H + δV XC . Screening from Hartree potential: � e 2 δn ( r ′ ) | r − r ′ | d r ′ ≡ ˆ δV H ( r ) = e 2 v c ( r , r ′ ) = v c δn, where | r − r ′ | Screening from exchange-correlation: � f xc ( r , r ′ ) = δV xc ( r ) f xc ( r , r ′ ) δn ( r ′ ) d r ′ ≡ ˆ δV xc ( r ) = f xc δn, where δn ( r ′ ) After some little algebra (remember that these are operators!): v c + ˆ χ = ˆ ˆ χ 0 + ˆ χ 0 (ˆ f xc )ˆ χ and finally � � − 1 v c − ˆ χ − 1 χ = ˆ ˆ − ˆ f xc 0

Physical Response Operator in practice Major problem: how to invert the operators! In solids, the response function χ 0 ( r , r ′ ) can be expressed in reciprocal space as a matrix, the dielectric matrix : χ 0 ( q + G , q + G ′ ) , for the response to an external perturbation of wavevector q . Operators become infinite matrix. By truncating them at an appropriate G cut one has a practical scheme for calculating response operators. Local-field effects : those due to the presence of G � = 0 terms. Random Phase Approximation (RPA): neglect the f xc term. Note that the addition of LDA exchange-correlation is straightforward: f xc is a local operator � � f xc ( r , r ′ ) = δ ( r − r ′ ) dV xc ( n ) � . � dn n = n ( r ) The dielectric matrix approach yields the response to all possible perturbations, but only local ones (i.e. δV local), and is computationally heavy. However we are often interested to the response to a specific and/or nonlocal perturbation.

Self-consistent Linear Response We consider the basic equations, to be self-consistently solved: v c δn + ˆ δV KS = δV 0 + ˆ f xc δn and � � v ( r ) ψ c ( r ) � ψ c | δV KS | ψ v � 1 ψ ∗ ψ ∗ δn ( r ) = 4 Re = 4 Re v ( r ) P c P c δV KS ψ v . ǫ v − ǫ c ǫ v − H KS v,c v The variation of the charge density can be recast into the form � 1 ψ ∗ δn ( r ) = 4 Re v ( r )∆ ψ v ( r ) , where ∆ ψ v = P c P c δV KS ψ v ǫ v − H KS v ∆ ψ v can be obtained from the solution of a linear equation: ( ǫ v − H KS ) P c ∆ ψ v = P c δV KS ψ v . The above equations define a self-consistent procedure that can be solved by iteration, much in the same way as in the solution of KS equations.

Linear Response to an Electric Field If the perturbing potential represents a macroscopic electric field δ E: δV 0 = − eδ E 0 · r it is ill-defined in a crystal, because r is not a lattice-periodic operator! it can however be recast into a well-defined expression using the following trick: � ψ c | r | ψ v � = � ψ c | [ H KS , r ] | ψ v � for c � = v. ǫ c − ǫ v We can rewrite | ¯ ψ α v � = P c r α | ψ v � as the solution of a linear system: ( H KS − ǫ v ) | ¯ ψ α v � = P c [ H KS , r α ] | ψ v � , where the commutator can be calculated from the following expression: � � h 2 [ H KS , r ] = − ¯ ∂ ˆ ∂ r + V NL , r . m ( V NL is the nonlocal term of the potential if present).

Macroscopic Polarization The bare macroscopic electric field will be screened according to electrostatic: 0 = � E α β ǫ α,β ∞ E β , where ǫ ∞ is the electronic (clamped-nuclei) contribution to the dielectric tensor. This is related to the induced polarization P via E 0 = E + 4 π P so that ∞ = δ α,β + 4 πδ P α ǫ α,β δ E β The macroscopic induced polarization can be calculated as � � � ¯ � δ P α = − e e ψ α r α δn ( r ) d r = v | ∆ ψ v . N c Ω N c Ω v ( N c is the number of cells of volume Ω c , N c Ω is the crystal volume) using the same trick as shown before. In practical calculations, the (screened) electric field E is kept fixed, iteration is performed on the microscopic terms of the potential: � � � e 2 | r − r ′ | + δv xc ( r ) δn ( r ′ ) . δV KS ( r ) = − eδ E α r α + δn ( r ′ )

Linear Response and Phonons An important advantage of the self-consistent approach to Linear Response: the typical PW-PP technology can be straightforwardly applied. Note that the projector over empty states can be written as � P c = 1 − P v = 1 − | ψ v �� ψ v | v so that conduction bands are never explicitly required. Typical application: calculation of normal vibrational modes, and especially phonons in crystals. The ”perturbing potential” is in this case the displacement of a nuclear potential (or of a group of them). Once δn ( r ) is (are) calculated, the dynamical matrix can be easily derived, along with phonon modes and frequencies. To this end, we need to know the form of the second-order expansion term of the energy. Such procedure is often called Density-Functional Perturbation Theory (DFPT). (in the following, notations change: derivatives replace infinitesimal increments)

Density-Functional Perturbation Theory Let us assume that the external potential depends on some parameter λ 2 λ 2 ∂ 2 V ( r ) V λ ( r ) ≃ V ( r ) + λ∂V ( r ) + 1 + ... ∂λ 2 ∂λ (all derivatives calculated at λ = 0 ) and expand the charge density 2 λ 2 ∂ 2 n ( r ) n λ ( r ) ≃ n ( r ) + λ∂n ( r ) + 1 + ... ∂λ 2 ∂λ and the energy functional into powers of λ : 2 λ 2 ∂ 2 E E λ ≃ E + λ∂E ∂λ + 1 ∂λ 2 + ... The first-order derivative ∂E/∂λ does not depend on any derivative of n ( r ) ( Hellmann-Feynman theorem ): � ∂E n ( r ) ∂V ( r ) ∂λ = ∂λ d r