Linear Response Theory: Finite Systems Lecture #1 M.T. Lusk
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Linear Response Theory: Finite Systems Lecture #1 M.T. Lusk 2 Linear_Response_Lusk_Lecture_1.nb Linear Response Theory The response to a weak, external perturbation is proportional to the perturbation itself. Therefore, we need only determine the proportionality factor. Examples: spring constant … displacement Ø internal force elastic constants … strain Ø internal stresses specific heat … temperature Ø internal energy thermal conductivity … temperature gradient Ø heat flux atomic diffusivity … density gradient Ø mass flux susceptibility … electric field Ø polarization of charge Linear_Response_Lusk_Lecture_1.nb 3 Kubo Formula ` Gives linear response in expected value of an observable, A, due to perturbation, H', of Hamiltonian: d XA HtL\ ª XA HtL\ - XA\0 = ‡ ¶ t0 „ t' CAH' Ht, t'L ` ` CAH' Ht, t'L :=  q Ht - t'L YAA HtL, H ' HtLE]0 Quantum statistical mechanical approach includes finite temperature ` ” ` ” Could have linear response, CA Hr”L B Hr”'L, between 2 field operators, A Hr L and B Hr 'L. We will come back to this later, but we will have to build up to it. 4 Linear_Response_Lusk_Lecture_1.nb Macroscopic Response Electric field … E ” Dipole operator … m = q r m = gè E … linear constitutive relation for dipole operator Polarizability operator … gè (tilde indicates tensor) Susceptibility operator, c è = N gè ” ” ” Polarization operator … P = N ” m = Ng cE èE= è # independent charges per unit volume … N Dielectric operator, eè = 1è + c è H4 p e0 = 1 in atomic units) All we need is gè. We want to estimate it without doing experiments. Linear_Response_Lusk_Lecture_1.nb 5 Dipole Contribution to Electron Hamiltonian F … external electrostatic potential r … electron density (electrons per volume) e = |e| … magnitude of charge of an electron (e = 1 in atomic units) Energy stored in electron configuration subjected to E-field, Eext = - Ÿ „3 r e r F = Ÿ „3 r Vext Assume F is slowly varying and expand in Taylor series: ” ” ” F Hr L = F H0L + r ÿ “F H0L + … = F H0L - r ÿ E H0L + … Then ” Eext = qe F H0L + ‡ „3 r e r r ÿ E H0L + … = q F H0L - ‡ „3 r r m ÿ E H0L + … ` qe … total electron charge. Energy contribution cancels with qion for neutral systetms. Then, to dipole order, Vext = - r m ÿ E = YY Vext Y] so ` Vext = - m ÿ E 6 Linear_Response_Lusk_Lecture_1.nb Polarizability: Static Field Ë Assume solution known to unperturbed, time-independent Kohn-Sham equation: ` H 0 j0n] = ¶ε0n j0n] Ë Want solution to perturbed KS equation: ` ` IH 0 + V M jn\ = ¶εn jn\ ` Here V = -m` ÿ E perturbs the LDA Hamiltonian (assumed small). ” Recall : V = r ÿ Iq EM = distance x force , q … charge, E … electric field Ë Expand jn \ and ¶εn in perturbation series. Ë Collect terms to find approximation of perturbed system energies (sum is over both occupied and unoccupied states): ` 0 Yj V j0n] 2 ` m 0 0 0 ¶εn = ¶εn + Yjn V jn] + ‚ ¶ε0n - ¶ε0m m=0, m ¹≠n Linear_Response_Lusk_Lecture_1.nb 7 Polarizability: Static Field ` Ë Put in V = -m` ÿ E and consider excitation from ground state: (superscript "0" denotes ground state value) ¶ε0 = ¶ε00 -E ÿ Yj00 Yj0m m` ÿ E j00] `m j0] + ‚ 0 ¶ε00 - ¶ε0m m>0 ¶ε00 - E ÿ Yj00 Yj00 m` jm\ Ä⊗ Yj0m m` j00] `m j0] + E ÿ E ‚ 0 0 0 ¶ε ¶ε 0 m m>0 2 = ¶ε00 -E ÿ Yj00 æ Xj0 Im` ÿ EM j0m] Ä⊗ Yj0m m` ÿ E j00] `m j0] + ‚ 0 ¶ε00 - ¶ε0m m>0 or ¶ε0 = Ë But the perturbed and induced energies are the same, and induced energy is Eind = - Ÿ „ E ÿ m = gè = 2 ‚ m>0 Note: Sum is over al (both occupied and unoccupied) states. 1 2 E ÿ gè E . Therefore Yj00 m` j0m] Ä⊗ Yj0m m` j00] ¶ε0m - ¶ε00 8 Linear_Response_Lusk_Lecture_1.nb Polarizability: Oscillating Field ` Put in V = -m` ÿ E cos Hw tL q(t) and consider excitation from ground state. q(t) is unit step function. Use time-dependent perturbation theory, applied to the time-dependent Kohn-Sham equation with LDA Hamiltonian, to obtain 1 1 gè HwL = ‚ Yj00 m` jm\ Ä⊗ Yj0m m` j00] + wm - w -  h wm + w +  h m>0 H0L Here  h terms enforce causality and wm ª ¶εH0L m - ¶ε0 Note: Sum is over al (both occupied and unoccupied) states. Linear_Response_Lusk_Lecture_1.nb 9 Brief Note on Causality Enforcement First consider the Fourier transform of the unit step function 1 IntegrateAUnitStep@tD ‰Â w t, 8t, -¶, ¶<E 2p  2p w Suppose we want to take the inverse Fourier transform of the resulting quantity. We want to get the unit step function back! F0 HwL = Fmod HwL =  2p w  w + Âh 1 2p Take inverse Fourier transform of each of these and see what the role of h is. F0@w_D =  ; 2p w Fmod@w_D = 1  2p w + Âh ; 10 Linear_Response_Lusk_Lecture_1.nb Simplify@InverseFourierTransform@F0@wD, w, tDD fmod@t_, h_D = Simplify@InverseFourierTransform@Fmod@wD, w, tD, h > 0D Manipulate@Plot@8UnitStep@tD, fmod@t, hD<, 8t, -2, 5<, PlotRange Ø All, PlotStyle Ø 88Green, Thickness@.01D<, Thickness@.007D<, Frame Ø True, GridLines Ø Automatic, PlotLabel Ø "Causality: h = " <> ToString@hDD, 8h, 0.1, 0.001<D Sign@tD 2 ‰-t h HeavisideTheta@tD h Causality: h = 0.0577 1.0 0.8 0.6 0.4 0.2 0.0 -2 -1 0 1 2 3 4 5 Linear_Response_Lusk_Lecture_1.nb 11 Inverse Causality Let's see what happens if we shift the pole into the upper-half plane instead: Fmod2@w_D = 1  2p w - Âh ; 12 Linear_Response_Lusk_Lecture_1.nb fmod2@t_, h_D = Simplify@InverseFourierTransform@Fmod2@wD, w, tD, h > 0D Manipulate@Plot@8UnitStep@tD, fmod2@t, hD<, 8t, -5, 2<, PlotRange Ø All, PlotStyle Ø 88Green, Thickness@.01D<, Thickness@.007D<, Frame Ø True, GridLines Ø Automatic, PlotLabel Ø "Inverse Causality: h = " <> ToString@hDD, 8h, 0.1, 0.001<D -‰t h HeavisideTheta@-tD h Inverse Causality: h = 0.0052 1.0 0.8 0.6 0.4 0.2 0.0 -5 -4 -3 -2 -1 This is inverse causality. 0 1 2 Linear_Response_Lusk_Lecture_1.nb References Static Polarizability from time-independent perturbation theory: (1) R. W. Boyd, Nonlinear Optics, 3rd Ed., Academic Press (2008) (2) http://en.citizendium.org/wiki/Polarizability (3) http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics) Oscillating Field Polarizability from time-dependent perturbation theory: Mahan and Subbaswamy, Local Density Theory of Polarizability, p94 and on. Boyd Eq 3.5.22 13 Linear Response Theory: Finite Systems Lecture #2 M.T. Lusk 2 Linear_Response_Lusk_Lecture_2.nb Time-Dependent Density Functional Response Theory (TD-DFRT) Ë Start with time-dependent Kohn-Sham equation `  ¶∂t Yj] = H Yj] Ë Response functions can be constructed by perturbing Hamiltonian Ë Called Time-dependent Density Functional Response Theory (TD-DFRT) Ë Called TDLDA when exchange and correlation restricted to local density approximation Ë Current application: optical response of confined systems (electron screening, photon absorption, transition probabilities) Ë GW and BS approaches can use TDLDA screening Linear_Response_Lusk_Lecture_2.nb 3 Energies, ¶ε Ts (t) … kinetic energy of independent Kohn-Sham particles ” ” ” ” ” ” ¶εH HtL = ‡ „3 r 1 ‡ „3 r 2 V Hr 1, r 2L rHr 1, tL rHr 2, tL ” ” V Hr 1, r 2L = 1 ” ” r2 -r1 … Hartree energy … bare Coulomb potential ” ” ” ¶εxcHtL = ‡ „3 r 1 fxcHr Hr 1, tL L rHr 1, tL … exchange -correlation energy ” ” ¶εextHtL = ‡ „3 ”r1 VextHr 1, tL rHr 1, tL … external energy ¶εtot = Ts + ¶εeff ¶εeff = ¶εH + ¶εxc + ¶εext 4 Linear_Response_Lusk_Lecture_2.nb Potentials and Time-Dependent Kohn-Sham Equation Ë Use functional differentiation to get potentials for KS Hamiltonian … ” Notation: (1) ª Hr 1, t1) VH H1L = d¶εH H1L = ‡ „ 2 V H1, 2L rH2L dr VxcH1L = d¶εxc „ fxc H1L = fxcHrH1LL + rH1L H1L dr „r … Hartree potential … exchange - correlation potential Veff = VH + Vxc + Vext … effective potential ` H0 = ` Ts + Veff Time-dependent Kohn-Sham equation: `  ¶∂t Yj] = H Yj] Linear_Response_Lusk_Lecture_2.nb 5 Position/Time Dependent Response Functions Ë Want linear response at position/time 1 from perturbation at position/time 2: dA H1L = ‡ „ 2 CAB H1, 2L dB H2L ó CAB H1, 2L = ó Pf H1, 2L = dA H1L dB H2L Example: TDLDA Polarization dr H1L = ‡ „ 2 Pf H1, 2L dVext H2L dr H1L dVext H2L Parr and Yang, Appendix A for review of functional calculus. 6 Linear_Response_Lusk_Lecture_2.nb Examples: Polarizability: (non-interacting system) dr H1L = ‡ „ H2L P0 H1, 2L dVeff H2L ó P0 H1, 2L = dr H1L dVeff H2L TDLDA Polarizability: (interacting system) dr H1L = ‡ „ H2L Pf H1, 2L dVext H2L ó Pf H1, 2L = dr H1L dVext H2L Inverse Dielectric Response: dVeff H1L = ‡ „ H2L e-1 H1, 2L dVext H2L ó e-1 H1, 2L = dVeff H1L dVext H2L Screened Response, convolution of inverse dielectric response with bare Coulomb response: W H1, 2L = ‡ „ H3L e-1 H1, 3L V H3, 2L ¶εW = ‡ „ 1 ‡ „ 2 W H1, 2L rH1L rH2L … Screened Hartree energy Linear_Response_Lusk_Lecture_2.nb 7 Typically Want Response to Perturbation of External Potential Ë Perturbed Hamiltonian cannot be decomposed into base and perturbation terms. Recall VH H1L = d¶εH H1L = ‡ „ 2 V H1, 2L rH2L dr VxcH1L = d¶εxc „f H1L = fxcHrH1LL + rH1L xc H1L dr „r … Hartree potential … exchange - correlation potential Ë Therefore change in Vext changes electron density which changes … dVext Ø dr Ø d VH + dVxc so dVeff = d VH + dVxc + dVext 8 Linear_Response_Lusk_Lecture_2.nb Polarizability of Non-Interacting System, P0 Starting point: calculate polarizability for non-interacting system: P0 H1, 2L = dr H1L dVeff H2L ó dr H1L = ‡ „ H2L P0 H1, 2L dVeff H2L … polarizability, gives probability amplitude of a change in density at (1) due to a change in density at (2) caused by a perturbation to the potential. Step 1: Find perturbation to wave function Step 2: Use perturbed wave function to obtain perturbed electron density Step 3: Write perturbed electron density as dr H1L = Ÿ „ H2L P0 H1, 2L dVeff H2L Step 4: Extract expression for P0 from integral Linear_Response_Lusk_Lecture_2.nb 9 Polarizability of Non-Interacting System, P0 Step 1a: Set up two perturbation equations … Ë Ground state system: H0 y0j HrL = lj y0j HrL ` ` ` ` æ Âwt -Âwt Ë Perturbed Hamiltonian: H = H0 + Veff ‰ + Veff ‰ Ë Perturbed wave function (y-j and y+j to be determined): ” ” ” ” Yj Hr , tL = ‰-Âlj t Iy0j Hr L + y-j Hr , wL ‰-Âwt + y+j Hr , wL ‰ÂwtM Ë Substitute into TDKS eqn to get 2 first order eqs: IH0 - lj - wM y-j = -Veff y0j , IH0 - lj + wM y+j = -Veff y0j 10 Linear_Response_Lusk_Lecture_2.nb Polarizability of Non-Interacting System, P0 Step 1b: Solve perturbation equations using Green functions … Ë Construct Green fns for each perturbation equation. ” G-j Hr , y0i HrL y0* ” i Hr'L r ', wL = ‚ li - l j - — w i¹≠j , ” G+j Hr , y0i HrL y0* ” i Hr'L r ', wL = ‚ li - l j + — w i¹≠j È Sums over all (both occupied and unoccupied) states. Use sum rule to help estimate missing contributions È Other ways of finding these Green functions which do not rely on an infinite sum over efns Ë Convolve Green fns with RHS terms to get perturbed wfs … ” ” ” ” ” y-j Hr , wL = - ‡ „3 r G-j Hr , r ', wL Veff y0j Hr 'L ” ” ” ” ” y+j Hr , wL = - ‡ „3 r G+j Hr , r ', wL V*eff y0j Hr 'L Linear_Response_Lusk_Lecture_2.nb 11 Polarizability of Non-Interacting System, P0 Step 2: Use perturbed wave function to obtain perturbed electron density … Ë Recall that ” ” ” ” Yj Hr , tL = ‰-Âlj t Iy0j Hr L + y-j Hr , wL ‰-Âwt + y+j Hr , wL ‰-ÂwtM Ë Substitute into electron density expression: Nocc ” ” r Hr , tL = ‚ Yj Hr , tL 2 j=1 Nocc = ” -* ” +* ” - ” 0* ” Âwt -Âwt 0 ” -Âwt + y+j Hr , wL ‰ÂwtM ‚ Iyj Hr L + yj Hr , wL ‰ + yj Hr , wL ‰ M Iyj Hr L + yj Hr , wL ‰ j=1 Ë Subtract out base density get density perturbation to first order: Nocc ” ” - ” ” +* ” -Âwt Âwt dr Hr , tL = ‚ Iy0* + y0* j Hr L yj Hr , wL ‰ j Hr L yj Hr , wL ‰ M j=1 12 Linear_Response_Lusk_Lecture_2.nb Polarizability of Non-Interacting System, P0 Step 3: Write perturbed electron density as dr H1L = Ÿ „ H2L P0 H1, 2L dVeff H2L Ë Substitute convolution integrals, ” ” ” ” ” y-j Hr , wL = - ‡ „3 r G-j Hr , r ', wL Veff y0j Hr 'L , ” ” ” ” ” y+j Hr , wL = - ‡ „3 r G+j Hr , r ', wL V*eff y0j Hr 'L , into Nocc ” ” - ” -Âwt 0* ” +* ” Âwt dr Hr , tL = ‚ Iy0* Hr L y Hr , wL ‰ + y Hr L y Hr , wL ‰ M j j j j j=1 Ë Stir vigorously … Linear_Response_Lusk_Lecture_2.nb 13 Polarizability of Non-Interacting System, P0 Step 4: Extract expression for P0 from integral … after massaging expression for dr, find and extract P0: P0 H1, 2L = 0 rij = y0* i yj … elements of density matrix fi = : 0, unoccupied 1, occupied ¶εij = li - lj  h … infinitessimal added to enforce causality Hh > 0L Ifi - fjM rji H1L rij H2L dr H1L = ‚ dVeff H2L —w - ¶εij + Âh i,j 14 Linear_Response_Lusk_Lecture_2.nb TDLDA Polarizability, Pf Ë Now have P0 H1, 2L= dr H1L , but cannot perturb Veff in real systems dVeff H2L Ë Recall dVext Ø dr Ø d VH + dVxc so dVeff = d VH + dVxc + dVext Ë Really want Pf H1, 2L= dr H1L dVext H2L … TDLDA polarizability response, gives probability amplitude of a change in density at (1) due to a change in density at (2) caused by a perturbation to the external potential. Linear_Response_Lusk_Lecture_2.nb Strategy for Getting Pf Step 1: Write perturbed electron density in two ways, in terms of P0 and in terms of Pf Step 2: Derive a Dyson series for Pf Step 3: Formally invert operators to obtain Pf 15 16 Linear_Response_Lusk_Lecture_2.nb TDLDA Polarizability, Pf Step 1: Write perturbed electron density in two ways, in terms of P0 and in terms of Pf dr H1L = ‡ „ H2L P0 H1, 2L dVeff H2L = ‡ „ H2L Pf H1, 2L dVext H2L Ë Equivalent expression: P0 dVeff = Pf dVext Ë Secret plan: re-write dVeff in terms of dVext dVeff = dVext + dVH + dVxc = dVext + dVind = dVext + dVind dr dr Ë Define a new response operator (can immediately calculate) K := dVind „ Vxc dVH dVxc = + = V+1 dr dr dr „r Linear_Response_Lusk_Lecture_2.nb 17 TDLDA Polarizability, Pf Step 2: Derive a Dyson series for Pf Ë Put K into last expression for dVeff and recall Pf = dr ê dVext: dVeff = dVext + K dr = dVext + K Pf dVext Ë Put this into the P0dVeff to obtain P0 dVeff = P0 dVext + P0 K Pf dVext Ë But this must equal Pf dVext, and dVext is arbitrary. Therefore Pf = P0 + P0 K Pf This is a Dyson series for P0. 18 Linear_Response_Lusk_Lecture_2.nb TDLDA Polarizability, Pf Step 3: Formally invert operators to obtain Pf Ë Found Dyson series: Pf = P0 + P0 K Pf Ë Therefore H1 - P0 KL Pf = P0 or Pf = H1 - P0 KL-1 P0 P0 known so we now have Pf = dr dVext Linear_Response_Lusk_Lecture_2.nb 19 Dielectric Response and Screening Ë Want inverse dielectric response: dVeff H1L = ‡ „ H2L e-1 H1, 2L dVext H2L ó e-1 H1, 2L = dVeff H1L dVext H2L Ë Recall dVeff = dVext + dVH + dVxc = dVext + dVind = dVext + Ë Therefore e-1 = dVeff = dVext 1+K or e-1 = 1 + K Pf Ë Screened Potential can the replace bare potential in Coulomb integrals: W H1, 2L = -1 ‡ „ H3L e H1, 3L V H3, 2L ó W= e-1 V dr dVext dVind dr = dr dVext + K dr 20 Linear_Response_Lusk_Lecture_2.nb Closing Remarks Ë Have formally determined polarizability within TDLDA framework Ë Must solve Dyson equation--i.e. inversion of H1 - P0 KL Ë Mark Casida has developed solution to Dyson equation based on a generalized EVP Ë We will talk about this in the next lecture Linear_Response_Lusk_Lecture_2.nb References Static Polarizability from time-independent perturbation theory: (1) R. W. Boyd, Nonlinear Optics, 3rd Ed., Academic Press (2008) (2) http://en.citizendium.org/wiki/Polarizability (3) http://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics) Oscillating Field Polarizability from time-dependent perturbation theory: (1) Mahan and Subbaswamy, Local Density Theory of Polarizability, p94 and on. (2) Boyd Eq 3.5.22 Functional Differentiation: (1) R. G. Parr and W. Yang, Density-functional theory of atoms and molecules, Oxford University Press (1994) Response Functions: (1) R.A. Broglia, G. Colo, G. Onida, H.E. Roman, Solid State Physics of Finite Systems, Springer (2004) --Ch 4 and especially the first appendix at the end of CH 4 (2) Hedin and Lundquist, p1, Solid State Physics Vol 23 (1968) Density Functional Response Theory: (1) M. E. Casida, Time dependent density functional response theory for molecules, Recent Advances in DFT Methods, (Ed. Chong) p155 (1995). [This paper has been published in many guises and can be downloaded for free from http://dcm.ujfgrenoble.fr/PERSONNEL/CT/casida/research/chong.ps] (2) M.E. Casida, Recent Developments and Applications of Modern Density Functional Theory, Theoretical and 21 22 (1) M. E. Casida, Time dependent density functional response theory for molecules, Recent Advances in DFT Methods, (Ed. Chong) p155 (1995). [This paper has been published in many guises and can be downloaded for free from http://dcm.ujfgrenoble.fr/PERSONNEL/CT/casida/research/chong.ps] (2) M.E. Casida, Recent Developments and Applications of Modern Density Functional Theory, Theoretical and Computational Chemistry, 4, 391-439 (1996) (3) C. Jamorski, M. E. Casida, and D. R. Salahub, Dynamic polarizabilities and excitation spectra from a molecular implementation of time-dependent density-functional response theory: N2 as a case study, J. Chem. Phys. 104 5134-5147 (1996) Linear_Response_Lusk_Lecture_2.nb
