Introduction to LMTO method

Transcription

1 1 Introduction to MTO method 24 February 2011; V172 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

2 Ab initio Eectronic Structure Cacuations in Condensed Matter 2 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

3 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method 3

4 DMo 3 : inear Combination of Atomic Orbita 4 i j c ij j ( r ) j m n ( r ) R ( r) Y (, ) Radia portion atomic DFT eqs. numericay m Anguar Portion R cut Periodic and a periodic systems Good for moecues, custers, zeoites, moecuar crystas, poymers "open structures" P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

5 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method 5

6 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method 6

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8 The muffin-tin approximation 8 Spherica atoms in a constant interstitia potentia P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

9 MTO Method 9 Andersen (1975) PRB, 12, 3060 Andersen and Jepsen (1984) PR, 53, 2571 Constant, r I V MT ( r ) V(r ), r MT d du 2 r 2 r dr dr r 2 u ( r ( 1) V( r Partitioning of the unit ce into atomic sphere (I) and interstitia regions (II) Inside the MT sphere, an eigen state is better described by the soutions of the Schrödinger equation for a spherica potentia: The function satisfies the radia equation: u 1, )Y ) u u 0 The ony boundary condition: be we defined at r ( rˆ u ) P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

10 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method 10 s r ), r N ( s r ), r ( ))J ( cot( ),r ( ) ( rˆ Y i ),r, ( MTO s r, )) E ( cot( d d ),r E ( ) r ( J s r, ) r / s ( s r, ) ( s ) / r ( ) P( ),r ( ) ( rˆ Y i ),r ( MTO The basis functions can now be constructed as Boch sums of MTO: An MTO basis function in terms of energy and the decay constant may be expressed as: Here and represent the Besse and Neumann functions respectivey. J N Since the energy derivative of vanishes at for it eads to: E, r s

11 In the atomic sphere approximation (ASA), the MTO s can be simpified as : MTO m ( r ) i Y ( rˆ )( (r) (D) (r)), 11 where D is given by : ( s )D ( D ) ( s )D D( ) ( r ) D( ) D( ) D og arithmic derivative is chosen such that and its energy derivative matches continuousy to the tai function at the muffin-tin sphere boundary. Disadvantages of MTO-ASA method : (1) It negects the symmetry breaking terms by discarding the non-spherica parts of the eectron density. (2) The interstitia region is not treated accuratey as MTO repaces the MT spheres by space fiing Wigner spheres. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

12 inear Augmented pane wave (APW) method 12 Augmented pane waves: re rea Y ry kg rs kg * kg(, ) 4 (, ) () ( ), i( kg) r * kg(, ) 4 ( ) () ( ), re e j kg ry ry kg r int become smooth inear augmented pane waves: r r E a r E b Y r Y k G rs kg kg * kg() 4 { (, ) (, ) } () ( ), i( kg) r * kg() 4 ( ) () ( ),, r e j k G r Y r Y k G rs r int P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

13 inear Muffin-Tin Orbita (MTO) method 13 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

14 KKR partia waves 14 Basic idea of KKR method is to construct a partia wave (, re) { (, re) aj(,)}, r r S MT (, re) bh(,), r r S MT Consider its Boch sum k ikr (, re) e ( rre, ) R k re j' r S' b ' a ' (, ) (,){ ( ) } And demand tai-canceation: k k k A (, r E) A (, r E) () r k P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

15 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method 15

16 Non-inear KKR Equations 16 { S ( E) P( E)} A 0 k k ' ' where potentia parameters function is P( E) h a( E) W( h, ) h( S, E)[ D ( E) D ( E)] j b( E) W( j, ) j ( S, E)[ D ( E) D ( E)] and where KKR structure constants are S ( E) e C h ( E, R) k ikr '' ' ' '' R0 '' P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

17 ogarithmic Derivatives 17 Behavior of ogarithmic Derivative D ( E) S '( S, E) ( SE, ) Consider s-wave: 1s has no nodes, 2s has 1 node, or nodes=n--1. From the point of view of ( SE, ) node appears when which means that og. derivative diverges! ( SE, ) 0 So ogartihmic derivatives behave as tan(e), they diverge each time a new node of radia wave function appears. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

18 Wavefunction as a Function of Energy 18 (, ) 0 SE E 1 E 2 E 3 S r Energy Window for 3s states E 3 Energy Window for 2s states E 2 VMT () r E 1 Energy Window for 1s states P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

19 ogarithmic Derivative as a Function of Energy (, ) 0 SE D ( ) 0 E 19 Centers of the n band E 1 E 2 E 3 r E D ( E) 1 New node of wave function appears! 1s 2s 3s 4s P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

20 inearized Soutions 20 If D (E) can be expanded in Taior series around some energy E ν, we obtain potentia function in a inearized form 2(2 1) D ( E) 1 1 E C D ( E) EV which soves the band structure probem E k ws C 1 S j kj k j C gives the center of the -band, w gives its width whie denominator 1-γS gives additiona distortion of the band. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

21 Energy inearization 21 Andersen proposed to spit energy dependence coming from inside the spheres and from interstitias. Since interstitia region is sma, Andersen proposed to fix this energy kappa to some vaue (originay to zero) Energy Bands 2 =E-V MT-zero V 0 0 Average kinetic energy of eectron in the interstitia region P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

22 Partia waves of fixed energy tais 22 (, re) { (, re) aj(,)} r { (, re) ary()}, r r S 0 MT 1 (, re) bh(,) r by (), r r S r 0 1 MT Consider as before Boch sum and demand tai-canceation: k ikr (, re) e ( rre, ) R ' k (, re) ry' (){ r S' ( ) b ' a} ' k k k A(, r E) A(, r E) k() r P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

23 KKR equations become 23 k 2 k { S ' ( 0) ' P( E)} A 0 where potentia parameters function is [ D ( E) 1] P ( E) 2(21) [ D ( E) ] and where the fixed energy structure constants are 1 S ( 0) e C Y ( r) k 2 ikr '' ' ' '' 1 '' R0 '' r To minimize the error of fixing the energy, Andersen proposed to enarge MT spheres to atomic spheres. This method has the name KKR-ASA. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

24 Canonica band structures (Andersen, 1973) 24 At the absence of hybridization, a remarkabe consequence of KKR ASA equations is canonica energy bands: [ D ( E) 1] k 2 det{ Sm' m( 0) m' m2(21) } 0 [ D ( E) ] For a given bock, one can diagonaize the structure constants and obtain (2+1) non-inear equations S k j 2 [ D ( E) 1] ( 0) 2(21) [ D ( E) ] whose soutions give rise to band structures E(kj), so caed canonica band structures. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

25 Canonica d-band for fcc materia 25 w C P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

26 Comparison with bands of Cu 26 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

27 Energy inearization (Andersen, 1973) 27 Genera idea to get rid of E-dependence: use Taior series and get INEAR MUFFIN-TIN ORBITAS (MTOs) (, re) (, re ) ( EE ) (, re ) 1 (, rd) (, re) D ( DD) (, re) D ( E) S( S, E)/ ( S, E) Before doing that, consider one more usefu construction: enveope function. In fact, concept of enveope functions is very genera. By choosing appropriate enveope functions, such as pane waves, Gaussians, spherica waves (Hanke functions) we wi generate various eectronic structure methods (APW, APW, CGO, CMTO, MTO, etc.) P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

28 Enveope Functions 28 Enveope functions can be Gaussians or Sater-type orbitas. They can be pane waves which generates augmented pan wave method (APW) e i( k G) r e i( kg) r j k G r Y r Y k G * 4 ( ) ( ) ( ) kg (, re) kg * 4 ( rea, ) Y( ry ) ( k G) S S S S P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

29 Construction of Augmented Spherica Wave (, re) (, reiy ) (), rˆ r S MT ( re, ) { aj( r) bh( r)} iy( rˆ ), r S MT 29 inear combinations of oca orbitas shoud be considered. k ikr (, re) e ( rre, ) R However, it ooks bad since Besse does not fa off sufficienty fast! Consider instead: (, re) { (, re) aj( r)} iy(), rˆ r S MT (, re) bh( riy ) (), rˆ r S MT P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

30 Enveope Functions 30 Agorithm, in terms of which we came up with the augmented spherica wave (MUFFIN-TIN ORBITA) construction: h (, r ) Step 1. Take a Hanke function h (, r EV ) h (,) r Step 2. Augment it inside the sphere by inear combination: Step 3. Construct a Boch sum 0 { ( re, ) aj(, r)}/ b k ikr (, re) e ( rre, ) R (, re ) k (, re) P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

31 inearization over Energy 31 Genera idea to get rid of E-dependence: use Taior series and get read off the energy dependence. (, re) (, re ) ( EE ) (, re ) 1 (, rd) (, re) D ( DD) (, re) D ( E) S( S, E)/ ( S, E) Introduction of phi-dot function gives us an idea that we can aways generate smooth basis functions by augmenting inside every sphere a inear combinations of phi s and phi-dot s The resuting basis functions do not sove Schroedinger equation exacty but we resoved the energy dependence! The basis functions can be used in the variationa principe. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

32 inear Muffin-Tin Orbitas 32 Consider oca orbitas. Energy-dependent muffin-tin orbita defined in a space: ( re, ) { ( re, ) aj( r)}/ biy( rˆ ), r S MT (, re) h( riy ) (), rˆ r S MT becomes energy-independent (, re) { a (, re ) b (, re )} iy(), rˆ r S MT (, re) h( riy ) (), rˆ r S MT provided we aso fix E V 0 to some number (say 0) P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

33 inear Muffin-Tin Orbitas 33 Boch sum shoud be constructed and one center expansion used: R ikr e ( rr) a (, r E ) b (, r E ) e h (, rr) ikr R0 a (, r E ) b (, r E ) j (,) r S ( ) k ' ' ' Fina augmentation of tais gives us MTO: k h h () r a (, r E ) b (, r E ) ' { a ( r, E ) b ( r, E )} S ( ) j j k ' ' ' ' ' ' ' P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

34 inear Muffin-Tin Orbitas 34 In more compact notations, MTO is given by () r () r () r S ( ) k h j k ' ' ' where we introduced radia functions h h h () r a (, r E ) b (, r E ) j j j () r a (, r E ) b (, r E ) which match smoothy to Hanke and Besse functions. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

35 Summary of MTO method 35 P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

36 inear Muffin-Tin Orbitas 36 Accuracy and Atomic Sphere Approximation: MTO is accurate to first order with respect to (E-E ν ) within MT spheres. MTO is accurate to zero order (k 2 is fixed) in the interstitias. Atomic sphere approximation can be used: Bow up MT-spheres unti tota voume occupied by spheres is equa to ce voume. Take matrix eements ony over the spheres. ASA is accurate method which eiminates interstitia region and increases the accuracy. Works we for cose packed structures, for open structures needs empty spheres. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

37 Variationa Equations 37 MTO definition (k dependence is highighted): () r () r () r S ( ), r k h j k ' ' MT ' k ikr () r e h (, rr), r R which shoud be used as a basis in expanding Variationa principe gives us matrix eigenvaue probem. kj k () r A () r kj k 2 ' ' k kj V Ekj A 0 int P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

38 Tight-Binding MTO 38 Tight-Binding MTO representation (Andersen, Jepsen 1984) MTO decays in rea space as Hanke function which depends on 2 =E-V 0 and can be sow. Can we construct a faster decaying enveope? Advantage woud be an access to the rea space hoppings, perform cacuations with disorder, etc: k ikr () r e ( rr) R H e H ( R) k ikr ' ' ' ' R P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

39 Tight-Binding MTO Any inear combination of Hanke functions can be the enveope which is accurate for MT-potentia h (, r) A ( R) h (, r R) ( ) ' ' R' where A matrix is competey arbitrary. Can we choose A-matrix so that screened Hanke function is ocaized? 39 Eectrostatic anaogy in case 2 =0 M ' / ' 1 r Z / 1 r Vscr ()~0 r Outside the custer, the potentia may indeed be screened out. The trick is to find appropriate screening charges (mutipoes) P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

40 Screening MTO orbitas: 40 Unscreened (bare) enveopes (Hanke functions) h ( rr) j ( r) S ( R) ' ' ' Screening is introduced by matrix A h ( rr) A h ( rr') ( ) R' R' ' R ' ' Consider it in the form A ( ) R' R ' ' RR ' ' S ' R ' R where apha and S α coefficients are to be determined. P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

41 Demand now that 41 R' R'' ' S ( S ) S ( ) ( ) '' R '' ' R ' R' R ' ' R' R ' '' R '' R we obtain one-center ike expansion for screened Hanke functions h ( rr) [ h ( rr'') j ( rr'')] S ( ) ( ) ' ' '' R' R'' ' ' j ( rr'') S ( ) ( ) ' R' R'' where S α pays a roe of (screened) structure constants and we introduced screened Besse functions j () r h () r j () r ( ) ' P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

42 Screened structure constants are short ranged: SI ( S ) S ( ) ( ) ( ) S S I S /( ) 42 For s-eectrons, transforming to the k-space Sk ( ) 1/ k 2 ( ) 2 S ( k) S( k)/( I S( k)) 1/( k ) Choosing apha to be negative constant, we see that it pays the roe of Debye screening radius. Therefore in the rea space screened structure constants decay exponentiay ( S ) ( R) exp( / R) whie bare structure constants decay as SR ( ) 1/ R P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

43 43 Screening parameters apha have to be chosen from the condition of maximum ocaization of the structure constants in the rea space. They are in principe unique for any given structure. However, it has been found that in many cases there exist canonica screened constants apha (detais can be found in the iterature). Since, in principe, the condition to choose apha is arbitrary we can aso try to choose such apha s so that the resuting MTO becomes (amost) orthogona! This woud ead to first principe oca-orbita orthogona basis. In the iterature, the screened, mosty ocaized, representation is known as apha-representation of TB-MTOs. The representaiton eading to amost orthogona MTOs is known as gamma-representation of TB-MTOs. If screening constants =0, we return back to origina (bare/unscreened) MTOs P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

44 Tight-Binding MTO 44 Since mathematicay it is just a transformation of the basis set, the obtained one-eectron spectra in a representations (apha, gamma) are identica with origina (ong-range) MTO representation. However we gain access to short-range representation and access to hopping integras, and buiding ow-energy tight-binding modes because the Hamitonian becomes short-ranged: k ikr H' ' e H' ' ( R) R P.Ravindran, FME-course on Ab initio Modeing of soar ce Materias 24 February 2011 Introduction to MTO method

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