Basics of density functional perturbation theory

Transcription

1 CECAM Tutoral: Dynamcal, delectrc and magnetc propertes of solds wth ABINIT Lyon, may 2014 Bascs of densty functonal perturbaton theory Gan-Marco Rgnanese European Theoretcal Spectroscopy Faclty Insttute of Condensed Matter and Nanoscences, B-1348 Louvan-la-Neuve, Belgque

2 Born-Oppenhemer approxmaton Quantum treatment for electrons Kohn-Sham equaton E el [r]=â y v T +V ext y v + E Hxc [r] v r(r)=âyv (r)y v (r) apple v V ext (r)+v Hxc (r) y (r)=e y (r) Classcal treatment for nucle Newton equaton unt cell F a ka = E tot R a ka {R} = M k d 2 R a ka dt 2 atom drecton (α=1,2,3) CECAM Tutoral, Lyon, may 2014 E tot = E on + E el 2

3 CECAM Tutoral, Lyon, may Harmonc approxmaton E tot R a κα = R a α + τ κα + u a κα cell equlbrum atomc poston dsplacement u E tot ({u})=e (0) tot + aκα 1 a κ α 2 ( 2 E tot u a κα u a κ α ) u a καu a κ α

4 CECAM Tutoral, Lyon, may Phonons The matrx of nteratomc force ( constants (IFCs) ) s defned as 2 E tot C κα,κ α (a,a )= u a κα u a κ α Its Fourer transform (usng translatonal nvarance) C κα,κ α (q)= a C κα,κ α (0,a )e q R a allows one to compute phonon frequences and egenvectors as solutons of the followng generalzed egenvalue problem: Â C ka,k 0 a 0(q)U mq(k 0 a 0 )=M k wmqu 2 mq (ka) k 0 a 0 phonon dsplacement pattern masses square of phonon frequences

5 CECAM Tutoral, Lyon, may Example: Damond Theory vs. Experment [X. Gonze, GMR, and R. Caracas, Z. Krstallogr. 220, 458 (2000)]

6 CECAM Tutoral, Lyon, may Total energy dervatves In fact, many physcal propertes are dervatves of the total energy (or a sutable thermodynamc potental) wth respect to external perturbatons. Possble perturbatons nclude: atomc dsplacements, expanson or contracton of the prmtve cell, homogeneous external feld (electrc or magnetc), alchemcal change...

7 CECAM Tutoral, Lyon, may Total energy dervatves Related dervatves of the total energy (E el + E on ) 1 st order: forces, stress, dpole moment,... 2 nd order: dynamcal matrx, elastc constants, delectrc susceptblty, Born effectve charge tensors, pezoelectrcty, nternal strans 3 rd order: non-lnear delectrc susceptblty, phonon-phonon nteracton, Grünesen parameters,... Further propertes can be obtaned by ntegraton over phononc degrees of freedom (e.g., entropy, thermal expanson,...)

8 CECAM Tutoral, Lyon, may Total energy dervatves These dervatves can be obtaned from drect approaches: fnte dfferences (e.g. frozen phonons) molecular-dynamcs spectral analyss methods perturbatve approaches The former have a seres of lmtatons (problems wth commensurablty, dffculty to decouple the responses to perturbatons of dfferent wavelength,...). On the other hand, the latter show a lot of flexblty whch makes t very attractve for practcal calculatons.

9 CECAM Tutoral, Lyon, may Outlne Perturbaton Theory Densty Functonal Perturbaton Theory Atomc dsplacements and homogeneous electrc feld

10 CECAM Tutoral, Lyon, may Outlne Perturbaton Theory Densty Functonal Perturbaton Theory Atomc dsplacements and homogeneous electrc feld

11 CECAM Tutoral, Lyon, may Reference and perturbed systems Let us assume that all the solutons are known for a reference system for whch the one-body Schrödnger equaton s: E E H (0) = e (0) wth the normalzaton condton: D E = 1 Let us now ntroduce a perturbaton of the external potental characterzed by a small parameter λ: V ext (λ)=v (0) ext + λv (1) ext + λ 2 V (2) ext + known at all orders.

12 CECAM Tutoral, Lyon, may Reference and perturbed systems We now want to solve the perturbed Schrödnger equaton: H(l) y (l) = e (l) y (l) wth the normalzaton condton: y (l) y (l) = 1 Idea: all the quanttes (X=H, ε, ψ ) are wrtten as a perturbaton seres wth respect to the parameter λ : X(λ)=X (0) + λx (1) + λ 2 X (2) + λ 3 X (3) + wth X (n) = 1 n! d n X dl n l=0

13 CECAM Tutoral, Lyon, may Expanson of the Schrödnger equaton Startng from: and nsertng: H(l) y (l) = e (l) y (l) H(!)=H (0) +!H (1) +! 2 H (2) +... l we get: H (0) +!H (1) +! 2 H (2) +... # (0) +!# (1) +! 2 # (2) +... y (l)= + ly (1) + l 2 y (2) +... e (l)=e (0) + le (1) + l 2 e (2) +... " (0) +! " (1) +! 2 " (2) +... " (0) +! " (1) +! 2 " (2) +... =

14 CECAM Tutoral, Lyon, may Expanson of the Schrödnger equaton Ths can be rewrtten as: H (0) +l H (0) y (1) + H (1) +l 2 H (0) y (2) + H (1) y (1) + H (2) +... = e (0) +l e (0) y (1) + e (1) +l 2 e (0) y (2) + e (1) y (1) + e (2) +... settng λ=0 dervng w.r.t. λ and settng λ=0 dervng twce w.r.t. λ and settng λ=0

15 CECAM Tutoral, Lyon, may Expanson of the Schrödnger equaton Fnally, we have that: H (0)! (0) = " (0)! (0) 0 th order H (0)! (1) + H (1)! (0) = " (0)! (1) + " (1)! (0) 1 st order H (0)! (2) + H (1)! (1) + H (2)! (0) = " (0)! (2) + " (1)! (1) + " (2)! (0) 2 nd order

16 CECAM Tutoral, Lyon, may Expanson of the normalzaton condton Startng from: and nsertng: we get: +l +l 2 y (l) y (l) = 1 y (l)= + ly (1) + l 2 y (2) = 1 y (1) + y (1) y (2) + y (1) y (1) + y (2) l

17 CECAM Tutoral, Lyon, may Expanson of the normalzaton condton Fnally, we have that = 1 0 th order y (1) + y (1) = 0 1 st order y (2) + y (1) y (1) + y (2) = 0 2 nd order

18 CECAM Tutoral, Lyon, may st order correctons to the energes Startng from the 1 st order of the Schrödnger equaton H (0) y (1) + H (1) = e (0) y (1) + e (1) and premultplyng by, we get: H (0) y (1) + H (1) = e (0) y (1) + e (1) {z } e (0) y (1) and thus, fnally, we have the Hellman-Feynman theorem: e (1) = H (1) The 0 th order wavefunctons are thus the only requred ngredent to obtan the 1 st order correctons to the energes. {z } 1

19 CECAM Tutoral, Lyon, may nd order correctons to the energes Startng from the 2 nd order of the Schrödnger equaton H (0)! (2) + H (1)! (1) + H (2)! (0) = " (0)! (2) + " (1) and premultplyng by! (1) + " (2)! (0), we get: e (0) y (2) {z }! (0) H (0)! (2) +! (0) H (1)! (1) +! (0) H (2)! (0) = " (0)! (0)! (2) + " (1)! (0)! (1) + " (2)! (0)! (0) {z } and thus, fnally, we have: 1! (2) = " (0) H (2) " (0) + " (0) H (1)! (1) " (1)

20 2 nd order correctons to the energes Snce the energes are real, we can wrte that:! (2) = " (0) H (2) " (0) + " (0) H (1)! (1) = " (0) H (2) " (0) + " (1) H (1)! (1) or, combnng both equaltes:! (2) = " (0) Usng the expanson of the normalzaton condton at 1 st order, we can fnally wrte that:! (2) = " (0) H (2) " (0) + 1 " (0) H (1) " (1) + " (1) H (1) " (0) 2 To obtan the 2 nd order correctons to the energes, the only requred ngredents are the 0 th and 1 st order wavefunctons. CECAM Tutoral, Lyon, may 2014 " (1) " (0) H (2) " (0) " (0) H (1)! (1) " (1) + " (1) H (1)! (1) " (0) 20

21 CECAM Tutoral, Lyon, may st order correctons to the wavefunctons The 1 st order of the Schrödnger equaton H (0) y (1) + H (1) = e (0) y (1) + e (1) can be rewrtten gatherng the terms contanng y (1) : H (0) e (0) y (1) = H (1) e (1) producng the so-called Sternhemer equaton. In order to get y (1) one would lke to nvert the (H (0) e (0) ) operator, but t cannot be done as such snce s an egenvalue of H (0) e (0). The problem can be solved by expressng the 1 st order wavefuncton as a lnear combnaton of the 0 th order ones: y (1) = Â c (1)

22 1 st order correctons to the wavefunctons We separate the 0 th order wavefunctons nto two subsets: those assocated to e (0) : I f H (0) = e (0) (ust f the energy s non-degenerate) those that belong to the subspace that s orthogonal: I y (1) = Âc (1) = Â c (1) + Â c (1) I I The Sternhemer equaton can thus be rewrtten as: H (0) e (0) CECAM Tutoral, Lyon, may 2014 y (1) = Âc (1) = Â = c (1) I H (0) e (0) e (0) e (0) H (1) e (1) 22

23 CECAM Tutoral, Lyon, may st order correctons to the wavefunctons Premultplyng by  c (1) I and, thus: e (0) e (0) c (1) k k wth k I, we get: k = k H (1) e (1) {z } δk e (0) k e (0) = k H (1) k = 0 snce k I c (1) = 1 e (0) e (0) H (1) for I

24 CECAM Tutoral, Lyon, may st order correctons to the wavefunctons Premultplyng by  c (1) I e (0) e (0) k wth k I, we get: k = k H (1) e (1) {z } 0 e (1) k d k = k H (1) k For k =, t s nothng but the Hellmann-Feynman theorem. But, t does not provde any nformaton on the c (1) for 2 I. In fact, there s a gauge freedom that allows to choose them equal to zero. = d k snce k I

25 CECAM Tutoral, Lyon, may st order correctons to the wavefunctons Fnally, we can wrte the so-called sum over states expresson: y (1) = Â 1 I e (0) e (0) H (1) whch requres the knowledge of all the 0 th order wavefunctons and energes. Instead, f we defne the proector P I? onto the subspace I? by: P I = Â I we can rewrte the Sterhemer equaton n that subspace: P I H (0) e (0) P I y (1) = P I H (1)

26 CECAM Tutoral, Lyon, may st order correctons to the wavefunctons In ths form, the sngularty has dsappeared and t can thus be nverted: h P I y (1) = P I H (0) e (0) 1 P I H (1) and defnng the Green s functon n the subspace h G I?(e)= P I? e H (0) 1 P I? we can wrte: P I y (1) = G I (e (0) )H (1) Ths s the Green's functon technque for dealng wth the Sternhemer equaton. I? as:

27 CECAM Tutoral, Lyon, may nd order correctons to the energes The sum over states expresson for the 1 st order wavefunctons: y (1) = Â 1 I e (0) can be nserted n the 2 nd e (0) H (1) order correctons to the energes:! (2) = " (0) H (2) " (0) + 1 " (0) H (1) " (1) + " (1) H (1) " (0) 2 leadng to:! (2) = " (0) H (2) " (0) + # " (0) H (1) " (0) I 1! (0)! (0) " (0) H (1) " (0)

28 CECAM Tutoral, Lyon, may nd order correctons to the energes Alternatvely, we can wrte that:! (") H(") # (")! (") = 0 " and the perturbaton expanson at the 2 nd order gves:! (0) H (0) " (0)! (2) +! (0) H (1) " (1)! (1) +! (1) H (0) " (0) +! (0) H (2) " (2)! (0) +! (1) H (1) " (1)! (0) +! (2) H (0) " (0)! (0) = 0! (1) It can be shown that the sum of the terms n a row or n column vanshes! Gettng rd of the frst row and the last column, we get another expresson for the 2 nd order correctons to the energes:! (2) = " (0) H (2) " (0) + " (1) H (0)! (0) " (1) + " (0) H (1)! (1) " (1) + " (1) H (1)! (1) " (0)

29 CECAM Tutoral, Lyon, may nd order correctons to the energes Actually, a number of other expressons exst for the 2 nd order correctons to the energes. However, t can be demonstrated that ths expresson s varatonal n the sense that the 2 nd order correctons to the! (2) = mn " (1) under the constrant that:! (1) energes can be obtaned by mnmzng t wth respect to : n " (0) H (2) " (0) + " (1) H (0)! (0) " (1) + " (0) H (1)! (1) " (1) + " (1) H (1)! (1)! (0)! (1) +! (1)! (0) = 0 " (0) o

30 3 rd order correctons to the energes Startng from the 3 rd order of the Schrödnger equaton H (0)! (3) + H (1)! (2) + H (2)! (1) + H (3)! (0) = and premultplyng by! (0) " (0) " (0)! (3) + " (1)! (2) + " (2)! (1) + " (3)! (0) " (3), we get: {z }! (0) H (0)! (3) +! (0) H (1)! (2) +! (0) H (2)! (1) +! (0) H (3)! (0) = " (0)! (0)! (3) + " (1)! (0)! (2) + " (2)! (0)! (1) + " (3)! (0)! (0) {z } 1 CECAM Tutoral, Lyon, may

31 CECAM Tutoral, Lyon, may rd order correctons to the energes Fnally, we we can wrte:! (3) = " (0) H (3) " (0) + " (0) H (2)! (2) + " (0) H (1)! (1) " (1) " (2) Ths expresson of the 3 rd order correctons to the energes requres to know the wavefunctons up to the 2 nd order.

32 3 rd order correctons to the energes Alternatvely, we can wrte that:! (") H(") # (")! (") = 0 " and the perturbaton expanson at the 3 rd order gves:! (0) H (0) " (0)! (3) +! (0) H (1) " (1)! (2) +! (1) H (0) " (0) +! (0) H (2) " (2)! (1) +! (1) H (1) " (1)! (1) +! (2) H (0) " (0) Agan, the sum of the terms n a row or n column vanshes. So, gettng rd of the frst two rows and the last two columns, we get another expresson that does not requre to know the 2 nd order wavefunctons : CECAM Tutoral, Lyon, may 2014! (2) +! (0) H (3) " (3)! (0) +! (1) H (2) " (2)! (0) +! (2) H (1) " (1)! (0) +! (3) H (0) " (0)! (0) = 0! (1)! (3) = " (0) H (3) " (0) + " (1) H (1)! (1) " (1) + " (0) H (2) " (1) + " (1) H (2) " (0) 32

33 Summary There are 4 dfferent methods to get the 1 st order wavefunctons: solvng the Sternhemer equaton drectly, complemented by a condton derved from the normalzaton requrement usng the Green s functon technque explotng the sum over states expresson mnmzng the constraned functonal for the 2 nd order correctons to the energes Wth these 1 st order wavefunctons, both the 2 nd and 3 rd order correctons to the energes can be obtaned. More generally, the n th order wavefunctons gve access to the (2n) th and (2n+1) th order energy [ 2n+1 theorem]. CECAM Tutoral, Lyon, may

34 CECAM Tutoral, Lyon, may Outlne Perturbaton Theory Densty Functonal Perturbaton Theory Atomc dsplacements and homogeneous electrc feld

35 Reference system In DFT, one needs to mnmze the electronc energy functonal: D E E el [r (0) ]= T +V (0) ext + E (0) Hxc [r(0) ] N e  =1 n whch the electronc densty s gven by: h r (0) (0) (r)= (r) y (r) CECAM Tutoral, Lyon, may 2014 N e  =1 under the constrant that: D E = d Alternatvely, one can solve the reference Shrödnger equaton: E apple H (0) 1 E E = 2 2 +V (0) ext +V (0) Hxc = e (0) where the Hartree and exchange correlaton potental s: V (0) (r)=de(0) Hxc [r(0) ] Hxc dr(r) 35

36 Perturbed system The electronc energy functonal to be mnmzed s: E el [r(l)] = N e  =1 n whch the electronc densty s gven by: N e r(r;l)=  y (r;l)y (r;l) =1 under the constrant that: y (l) y (l) = d Alternatvely, applethe Shrödnger equaton to be solved s: 1 H(l) y (l) = 2 2 +V ext (l)+v Hxc (l) y (l) = e (l) y (l) where the Hartree and exchange correlaton potental s: V Hxc (r;l)= de Hxc[r(l)] dr(r) CECAM Tutoral, Lyon, may 2014 y (l) T +V ext (l) y (l) + E Hxc [r(l)] 36

37 CECAM Tutoral, Lyon, may st order of perturbaton theory n DFT For the energy, t can be shown that: D (T +V ext ) (1) E (1) el = N e  =1 Ths s the equvalent of the Hellman-Feynman theorem for densty-functonal formalsm. For the wavefunctons, the constrant leads to: D E D E y (1) + y (1) = 0 E + d dl E Hxc[r (0) ] l=0

38 2 nd order energy n DFPT For the energy, t can be shown that: hd E E (2) el = y (1) (T +V ext ) (1) + wth N e  =1 + N e  = Z D E (T +V ext ) (1) y (1) hd E D E (T +V ext ) (2) + y (1) (H e ) (0) y (1) Z Z d 2 E Hxc [r (0) ] dr(r)dr(r 0 ) r(1) (r)r (1) (r 0 )drdr 0 d dl E Hxc [r (0) ] dr(r) h r (1) N e (r)=  y (1) (r) =1 CECAM Tutoral, Lyon, may 2014 l=0 r (1) (r)dr h (0) y (r)+ d 2 dl 2 E Hxc[r (0) ] l=0 (1) (r) y (r) 38

39 CECAM Tutoral, Lyon, may st order wavefunctons n DFPT The 1 st order wavefunctons can be obtaned by mnmzng: hn o n o E (2) el = E (2) el, y (1) n o wth respect to y (1) D E under the constrant: y (1) = 0 These can also be obtaned by solvng the Sternhemer equaton: Ĥ(0) e (0) y (1) Ĥ(1) = e (1) H (1) =(T +V ext ) (1) + Z d 2 E Hxc [r (0) ] dr(r)dr(r 0 ) r(1) (r 0 )dr 0 e (1) = H (1)

40 CECAM Tutoral, Lyon, may Hgher orders n DFPT More generally, t easy to show that snce there s a varatonal prncple for the 0 th order energy: hn o E (0) el = E (0) el non-varatonal expresson can be obtaned for hgher orders: hn o E (1) el = E (1) el hn o n o E (2) el = E (2) el, y (1) hn o n o n o E (3) el = E (3) el, y (1), y (2) But, ths s not the best that can be done!

41 CECAM Tutoral, Lyon, may Hgher orders n DFPT We assume that all wavefunctons are known up to (n-1) th order: ȳ = y <n + O(l n )= + ly (1) + + l n 1 y + O(l n ) The varatonal property of the energy functonal mples that: E el [{y tral + O(h)}]=E el [{y tral }]+O(h 2 ) Takng {y tral } = y <n and h = l n, we see that: f the wavefunctons are known up to (n-1) th order, the energy s know up to (2n-1) th order; f the wavefunctons are known up to n th order, the energy s know up to (2n+1) th order. Ths s the 2n+1 theorem. (n 1)

42 Hgher orders n DFPT Snce the varatonal prncple s also an extremal prncple [the error s ether > 0 mnmal prncple, or CECAM Tutoral, Lyon, may 2014 < 0 maxmal prncple], the leadng mssng term s also of defnte sgn (t s also an extremal prncple): E (0) el = E (0) el E (1) el = E (1) el E (2) el = E (2) el E (3) el = E (3) el E (4) el = E (4) el E (5) el = E (5) el hn hn hn hn hn hn o o o o n, n, n o, o n, y (1) y (1) y (1) y (1) o o o o, n, n y (2) y (2) o o n varatonal w.r.t. n varatonal w.r.t. n varatonal w.r.t. y (1) y (2) o o o 42

43 CECAM Tutoral, Lyon, may Mxed dervatves n DFPT Smlar expressons exst for mxed dervatves (related to two dfferent perturbatons 1 and 2 ): n o E 1 2 el = E 1 2 el ;y 1 ;y 2 The extremal prncple s lost but the expresson s statonary: the error s proportonal to the product of errors made n the 1 st order quanttes for the frst and second perturbatons; f these errors are small, ther product wll be much smaller; however, the sgn of the error s undetermned, unlke for the varatonal expressons.

44 CECAM Tutoral, Lyon, may Order of calculaton n DFPT 1. Ground state calculaton: 2. FOR EACH pertubaton 1 DO use ENDDO 3. FOR EACH pertubaton par { 1, 2 } DO determne E 1 2 usng both and y 2 (statonary expresson) usng ust y 1 ENDDO and r (0) V 1 ext! y 1 and r 1 V (0) ext! and r (0) mnmze 2 nd order energy solve Sternhemer equaton y 1 4. Post-processng to get the physcal propertes from E 1 2

45 CECAM Tutoral, Lyon, may Outlne Perturbaton Theory Densty Functonal Perturbaton Theory Atomc dsplacements and homogeneous electrc feld

46 CECAM Tutoral, Lyon, may Perturbatons of the perodc sold Let us consder the case where the reference system s perodc: V (0) ext (r + R a )=V (0) ext (r) It can be shown that f the perturbaton s characterzed by a wavevector q such that: V (1) ext (r + R a )=e q R a V (1) ext (r) all the responses, at lnear order, wll also be characterzed by q: r (1) (r + R a )=e q R a r (1) (r) y (1),k,q (r + R a)=e q R a y (1),k,q (r)

47 CECAM Tutoral, Lyon, may Perturbatons of the perodc sold We defne related perodc quanttes: r (1) (r)=e q r r (1) (r) u (1),k,q (r)=(n e W 0 ) 1/2 e (k+q) R y (1),k,q (r) In the equatons of DFPT, only these perodc quanttes appear: the phases e q r and e (k+q) R can be factorzed. The treatment of perturbatons ncommensurate wth the unperturbed system perodcty s mapped onto the orgnal perodc system. Ths s nterestng for atomc dsplacements but more mportantly for electrc felds.

48 Electronc delectrc permttvty tensor The delectrc permttvty tensor s the coeffcent of proportonalty between the macroscopc dsplacement feld and the macroscopc electrc feld, n the lnear regme: At hgh frequences of the appled feld, the delectrc permttvty tensor only ncludes a contrbuton from the electronc polarzaton: ε αα = δ αα 4π 2E E α E α Ω el 0 1 unt vector CECAM Tutoral, Lyon, may 2014 D mac,a = Â a 0 e aa 0E mac,a 0 e aa 0 = D mac,a E mac,a 0 αα ˆq α ε αα ˆq α = = d aa 0 + 4p P mac,a E mac,a 0 ε 1 G=0,G =0 (q 0) 48

49 CECAM Tutoral, Lyon, may Treatment of homogeneous electrc felds When the perturbaton s an electrc feld, we have: V (1) ext (r)=e r whch breaks the perodc boundary condtons. To obtan the 2nd order dervatve of the energy: 2E E a E a 0 el = Z r E a 0 (r)r a dr the followng matrx elements need to be computed: D E D E u (0) c,k r a 0 ue a v,k and u E a c,k r a 0 u(0) v,k conducton valence

50 CECAM Tutoral, Lyon, may Treatment of homogeneous electrc felds These matrx elements can be determned wrtng that: D E D e (0),k e (0),k u (0),k r a u (0),k = u (0),k H(0) kk r a r a H (0) kk = D u (0),k H(0) E kk u (0),k k a whch leads to the Sternhemer equaton: E P c H (0) kk e (0),k P c r a u (0),k = P c H(0) E kk u (0),k k a P I H (0) e (0) P I y (1) = P I H (1) u(0),k E DDK perturbaton

51 Born effectve charge tensor It s defned as the proportonalty coeffcent relatng at lnear order, the polarzaton per unt cell, created along the drecton α, and the dsplacement along the drecton α of the atoms belongng to the sublattce κ: It also descrbes the lnear relaton between the force n the drecton α on an atom κ and the macroscopc electrc feld Both can be connected to the mxed 2 nd order dervatve of the energy wth respect to u κα and E α Sum rule: Z καα = Ω 0 κ Z καα = 0 CECAM Tutoral, Lyon, may 2014 P mac,α u κα (q = 0) Eα =0 = F κα E α uκα =0 51

52 Born effectve charge tensor Model system: -q(r) P(r) +q(r) datomc molecule: r dpole moment related to statc charge q(r): P(r)= q(r) r Atomc polar charge Z * (r) such that P(r)= Z * (r) r purely covalent case: purely onc case: CECAM Tutoral, Lyon, may 2014 q(r)=0=z * (r) q(r)=q 0 Z * (r)=q mxed onc-covalent: P(r)= q(r) r + q(r) r q(r) Z * (r) r r Q Q 52

53 CECAM Tutoral, Lyon, may Statc delectrc permttvty tensor The mode oscllator strength tensor s defned as ( )( ) S m,αα = Z καβ U mq=0(κβ) Z κα β U mq=0 (κβ ) κβ κβ The macroscopc statc (low-frequency) delectrc permttvty tensor s calculated by addng the onc contrbuton to the electronc delectrc permttvty tensor: ε αα (ω)=ε αα + 4π Ω 0 m ε 0 αα = ε αα + 4π Ω 0 m S m,αα ω 2 m ω 2 S m,αα ω 2 m

54 CECAM Tutoral, Lyon, may LO-TO splttng The macroscopc electrc feld that accompanes the collectve atomc dsplacements at q 0 can be treated separately: C κα,κ α (q 0)= C κα,κ α (q = 0)+ C NA κα,κ α (q 0) where the nonanalytcal, drecton-dependent term s: ( ) C NA κα,κ α (q 0)= 4π β q γ Z κβα )( β q β Z κ β α Ω 0 ββ q β ε ββ q β The transverse modes are common to both C matrces but the longtudnal ones may be dfferent, the frequences are related by ω 2 m(q 0)=ω 2 m(q = 0)+ 4π Ω 0 αα q α S m,αα q α αα q α ε αα q α

55 CECAM Tutoral, Lyon, may Example 1: Zrcon (phonons) Theoretcal frequency (cm 1 ) LDA PBE PBEsol AM05 WC HTBS Expermental frequency (cm 1 ) Relatve error (%) Expermental frequency (cm 1 ) LDA PBE PBEsol AM05 WC HTBS

56 Example 1: Zrcon (Born effectve charges) Nomnal: M +4 S +4 O -2 M : anomalously large (esp. Z ) PbZrO 3, ZrO 2 S : smaller devatons ( and ) SO 2 (α-quartz or stshovte) O : strong ansotropy SO 2 stshovte or TO 2 rutle n the y-z plane (plane of the M-O bonds) (rem: from SO 2 α-quartz n the x drecton Interpretaton: mxed onc-covalent bondng CECAM Tutoral, Lyon, may 2014 where 2 components) closer to stshovte than α-quartz n agreement wth nave bond countng for O atoms 56

57 Example 1: Zrcon (delectrc propertes) S m and Z* m are the largest for the lowest and hghest frequency modes due to the frequency factor, t s the lowest frequency mode that contrbutes the most to ε 0 S m and Z* m are smaller for HfSO 4 mass dfference and Born effectve charges ths effect can be compensated by a lower frequency for hafnon [e.g. A 2u (1)] CECAM Tutoral, Lyon, may

58 CECAM Tutoral, Lyon, may Example 2: Copper (thermodynamcs) (b) T (K) 3 (c) LDA PBE PBEsol AM05 WC HTBS Expt. 1.6 LDA PBE PBEsol AM05 WC HTBS Expt. ( ) 1.5 B0 (MBar) T (K) T (K) 250 Frequency (cm -1 ) Γ X W L Γ K X

59 !! S. Baron, P. Gannozz & A. Testa, Phys. Rev. Lett. 58, 1861 (1987) X. Gonze & J.-P. Vgneron, Phys. Rev. B 39, (1989) X. Gonze, Phys. Rev. A 52, 1096 (1995) S. de Groncol, Phys. Rev. B 51, 6773 (1995) X. Gonze, Phys. Rev. B. 55, (1997) X. Gonze & C. Lee, Phys. Rev. B. 55, (1997) S. Baron, S. de Groncol, A. Dal Corso, P. Gannozz, Rev. Mod. Phys. 73, 515 (2001) CECAM Tutoral, Lyon, may