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1、VASP: Dielectric response Perturbation theory, linear response, and finite electric fieldsUniversity of Vienna, Faculty of Physics and Center for Computational Materials Science, Vienna, AustriaOutlineDielectric responseFrequency dependent dielectric propertiesThe static dielectric responseResponse
2、to electric fields from DFPTThe macroscopic polarizationSCF response to finite electric fieldsIonic contributions to dielectric propertiesExperiment: Static and frequency dependent dielectric functions: measurement of absorption, reflectance and energy loss spectra. (Optical properties of semiconduc
3、tors and metals.)The long-wavelength limit of the frequency dependent microscopic polarizability and dielectric matrices determine the optical properties in the regime accessible to optical and electronic probes.Theory: The frequency dependent polarizability matrix is needed in many post-DFT schemes
4、, e.g.:GW ) frequency dependent microscopic dielectric response needed to compute W. ) frequency dependent macroscopic dielectric tensor required for the analytical integration of the Coulomb singularity in the self-energy. Exact-exchange optimized-effective-potential method (EXX-OEP). Bethe-Salpete
5、r-Equation (BSE) ) dielectric screening of the interaction potential needed to properly include excitonic effects.Frequency dependentFrequency dependent microscopic dielectric matrix ) In the RPA, and including changes in the DFT xc-potential.Frequency dependent macroscopic dielectric tensor ) Imagi
6、nary and real part of the dielectric function. ) In- or excluding local field effects. ) In the RPA, and including changes in the DFT xc-potential:StaticStatic dielectric tensor, Born effective charges, and Piezo-electric tensor, in- or excluding local field effects. ) From density-functional-pertur
7、bation-theory (DFPT). Local field effects in RPA and DFT xc-potential. ) From the self-consistent response to a finite electric field (PEAD). Local field effects from changes in a HF/DFT hybrid xc-potential, as well.Macroscopic continuum considerationsThe macroscopic dielectric tensor couples the el
8、ectric field in a material to an applied external electric field: E = ?1Eextwhere is a 33 tensor For a longitudinal field, i.e., a field caused by stationary external charges, this can be reformulated as (in momentum space, in the long-wavelength limit): vtot= ?1vextvtot= vext+ vindwithThe induced p
9、otential is generated by the induced change in the charge density #$%. In the linear response regime (weak external fields):ind= ?vextind= Pvtotwhere is the reducible polarizabilitywhere P is the irreducible polarizabilityIt may be straightforwardly shown that:?1= 1 + ? = 1 ? P? = P + P?(a Dyson eq.
10、)where is the Coulomb kernel. In momentum space: = 4e2/q2Macroscopic and microscopic quantitiesThe macroscopic dielectric function can be formally written asE(r,!) =Z dr0?1mac(r ? r0,!)Eext(r0,!)or in momentum space E(q,!) = ?1mac(q,!)Eext(q,!)The microscopic dielectric function enters ase(r,!) =Z d
11、r0?1(r,r0,!)Eext(r0,!)and in momentum spacee(q + G,!) =XG0?1G,G0(q,!)Eext(q + G0,!)The microscopic dielectric function is accessible through ab-initio calculations. Macroscopic and microscopic quantities are linked through:E(R,!) =1 Z(R)e(r,!)drMacroscopic and microscopic quantitiesAssuming the exte
12、rnal field varies on a length scale much larger that the atomic distances, one may show thatE(q,!) = ?10,0(q,!)Eext(q,!)and?1mac(q,!) = ?1 0,0(q,!)mac(q,!) =?10,0(q,!)?1For materials that are homogeneous on the microscopic scale, the off-diagonal elements of ,*+,(,), (i.e., for ) are zero, andmac(q,
13、!) = 0,0(q,!)This called the “neglect of local field effects”The longitudinal microscopic dielectric functionThe microscopic (symmetric) dielectric function that links the longitudinal component of an external field (i.e., the part polarized along the propagation wave vector q) to the longitudinal c
14、omponent of the total electric field, is given by?1G,G0(q,!) := ?G,G0+4e2 |q + G|q + G0|ind(q + G,!) vext(q + G0,!)G,G0(q,!) := ?G,G0?4e2 |q + G|q + G0|ind(q + G,!) vtot(q + G0,!)and with?G,G0(q,!) :=ind(q + G,!) vext(q + G0,!)PG,G0(q,!) :=ind(q + G,!) vtot(q + G0,!)sG,G0(q) :=4e2 |q + G|q + G0|ando
15、ne obtains the Dyson equation linking P and ?G,G0(q,!) = PG,G0(q,!) +XG1,G2PG,G1(q,!)sG1,G2(q)?G2,G0(q,!)ApproximationsProblem: We know neither P and Problem: The quantity we can easily access in Kohn-Sham DFT is the: “irreducible polarizability in the independent particle picture” 3(or 45)?0G,G0(q,
16、!) :=ind(q + G,!) ve(q + G0,!)Adler and Wiser derived expressions for 3which, in terms of Bloch functions, can be written as (in reciprocal space):?0G,G0(q,!) =1 Xnn0k2wk(fn0k+q? fn0k)h n0k+q|ei(q+G)r| nkih nk|e?i(q+G0)r0| n0k+qi n0k+q? nk? ! ? iApproximations (cont.)For the Kohn-Sham system, the following relations can be shown to hold? = ?0+ ?0( + fxc)?P = ?0+ ?0fxcP ? = P + P?
