Andersen’s LMTO method
S. Y. Savrasov New Jersey Institute of Technology
O. K. Andersen, PRB 12, 3060 (1975) O. K. Andersen, O. Jepsen, PRL 53, 2571 (1984)
NJIT Content q Solving Schroedinger’s equation for solids q Basis sets & variational principle q Envelope functions q Linear muffin-tin orbitals q Tight-binding LMTO representation q Advanced topics
NJIT Material Research
Computations of properties of materials Ground state: • Density, total energy, magnetization • Volume & crystal structure • Lattice dynamics, frozen magnons Excitations: • One-electron spectrum • Photoemission & Optics • Superconductivity • Transport
Ab initio Material Design
NJIT Solving Schroedinger equation for solids
Density Functional Theory (Hohenberg, Kohn, 1964 Kohn, Sham, 1965) 2 (-Ñ+VDFT(r)-=Erkj)y kj ()0 y kj ()r describe one-electron Bloch states ikR yykj(r+=R)erkj () due to periodicity of the potential
VDFT(r+=R)VrDFT ()
NJIT Solving Schroedingers equation for solids
Many-Body Theory (Hartree-Fock, GW, DMFT, etc.) -Ñ2y(r)+S=(r,r',w)yy(r')dr'Er() kjwò kjwkjwwkj where S ( rr , ' ,) w is a non-hermitian complex frequency-dependent self-energy operator.
Bloch property is retained ikR yykjww(r+=R)erkj () due to periodicity of the self-energy: S(r+R,r'+R,ww)=S(rr,',)
NJIT Solving Schroedingers equation for solids
Solution of differential equation is required 2 (-Ñ+V(r)-=Erkj)y kj ()0 Properties of the potential Vr()
NJIT Solving Schroedingers equation for solids
Properties of the solutions: Erkj,y kj ()
Energy Bands
Core Levels
Ze2 - r NJIT Basis Sets & Variational Principle
Solving differential equation using expansion kjk yckj (r)= å Araa() a k where c a () r is a basis set satisfying Bloch theorem kikRk ccaa(r+=R)er()
Two most popular examples:
ki()kGr+ • Plane waves, c a () re ® , a ® G
• Linear combinations of local orbitals kikR ccaa(r)=-åe()rR R NJIT Basis Sets & Variational Principle
Variational principle leads us to solve matrix eigenvalue problem k2 kkj åácca||-Ñ+V-EAkj bbñ= b kkkj å(Hab-=EkjOAabb)0 b where
kkk2 HVab=áccab||-Ñ+ñis hamiltonian matrix
kkk Oab=áñccab| is overlap matrix
NJIT Basis Sets & Variational Principle
Linear combinations of local orbitals will be considered. kikR ccaa(r)=-åe()rR R
NJIT Basis Sets & Variational Principle
Smart choice of c a () r is important. Linear Combination of Atomic Orbitals (LCAO) l cja (r)= l(r,)EnlmiYrlm ()ˆ
Enlm Atomic potential
kikR to be used in variational ccnlm(r)=-åenlm ()rR R principle NJIT Basis Sets & Variational Principle
Muffin-tin Construction: Space is partitioned into non-overlapping spheres and interstitial region. potential is assumed to be spherically symmetric inside the spheres, and constant in the interstitials.
V0 Muffin-tin potential
VrVMT()= VrVMT()=sph(SMT),=>V0 rSMT Muffin-tin sphere SMT NJIT Basis Sets & Variational Principle Solve radial Schroedinger equation inside the sphere l jl(rE,)iYrlm ()ˆ 2 (-Ñrl+Vsphl(r)-=E)j (rE,)0 Solve Helmholtz equation outside the sphere 2 l jkl(r,)iYrlm ()ˆ E 2 2 (-Ñrll+V0 -=Er)jk(,)0 2 k =-EV0 Muffin-tin sphere S NJIT MT Basis Sets & Variational Principle Solution of Helmholtz equation outside the sphere 2 jl(r,k)=+aljrl(kk)bllhr() where coefficients ab ll , provide smooth matching with jl (rE,) -SMT SMT al= Wj{ll,}j bl= Wh{ll,}j Wf{,g}=-f''ggf NJIT Basis Sets & Variational Principle Linear combinations of local orbitals should be considered. kikR ccLL(r,E)=-åe(rRE,) R l cjL(r,E)= Consider instead: l cL(r,E)={jkl(r,E)- Bloch sum: kikR ccLL(r,E)=åe(r-=RE,) R ikR jL(r,E)-aljL(kk,r)+åeblLh(,)rR-= R¹0 k jL(r,E)-aljL(k,r)+=å jL''(,kkr)SbLLl() L ' k jL(r,E)+-å jL'(,kr){SL''L(kd)}balLLl L' where structure constants are: kikRL'' SL'L(kk)= ååeCLLL'hR'' (,) RL¹0'' NJIT Basis Sets & Variational Principle A single L-partial wave kk cL(r,E)=jL(r,E)+-å jL'(k,r){SL''L(kd)}balLLl L ' is not a solution: 2 k (-Ñ+VrMTL()-¹E)c (rE,)0 However, a linear combination can be a solution kkk ååALcL(r,E)==ALjyLk(r,Er)() LL Tail cancellation is needed kk å{SL''L()(kdblE)-=LLalL(EA)}0 L kj which occurs at selected EAkjL, NJIT Basis Sets & Variational Principle is a good basis, basis of MUFFIN-TIN ORBITALS (MTOs), k c L (rE,)which solves Schroedinger equation for MT potential exactly! For general (or full) potential it can be used with variational principle kjk yckj(r)= å ALL(rE,)kj L k2 kkj åáccL' |-Ñ+VMT+VNMT-EAkj|0LLñ= L which solves the entire problem sufficiently accurate. Unfortunately, drawback: implicit E-dependence! NJIT Envelope Functions General idea to get rid of E-dependence: use Taylor series and get LINEAR MUFFIN-TIN ORBITALS (LMTOs) jl(r,E)=jjl(r,Enl)+-(EEnnl)&ll(rE,) -1 jl(r,D)=jjl(r,Enl)+-DD&nnnl(Dl)&ll(rE,) Dlll(E)= Sjj¢(S,E)/(SE,) Before doing that, consider one more useful construction: envelope function. In fact, concept of envelope functions is very general. By choosing appropriate envelope functions, such as plane waves, Gaussians, spherical waves (Hankel functions) we will generate various electronic structure methods (APW, LAPW, LCGO, LCMTO, LMTO, etc.) NJIT Envelope Functions Algorithm, in terms of which we came up with the MUFFIN-TIN ORBITAL construction: hrL(,)k Step 1. Take a Hankel function hLL(r,E-=Vhr0 )(,)k Step 2. Augment it inside the sphere cL (rE,) by linear combination: {jkL(r,E)- aljLl(,rb)}/ Step 3. Construct a Bloch sum c k (rE,) kikR L ccLL(r,E)=-åe(rRE,) R NJIT Envelope Functions Why take Hankel function as an envelope? QaL ()r Step 1. Take ANY function Qa L ()r which has one center expansion in terms of Xb L' ()r Step 2. Augment it inside the sphere cL (rE,) by linear combination: aalj L(r,E)-XbraalL() Step 3. Construct a Bloch sum k cL (rE,) kikR ccaaLL(r,Ee)=-å (rRE,) R NJIT Envelope Functions Envelope functions can be Gaussians or Slater-type orbitals. They can be plane waves which generates augmented plane wave method (APW) ei()kGr+ ei()k+Gr= * 4p å jkl(|++G|r)YLL(r)Y()kG L S S S S c kG+ (rE,)= kG+ * 4pjå l(rE,)alYLL(r)Y()kG+ L NJIT Envelope Functions Condition of augmentation – not necessarily smooth like with Hankel functions. APWs are not smooth but continuous. We can require that linear combination of APWs is smooth: kj å Ak++GcykG(r,Erkj)= kj () G which solves the problem and delivers spectrum Erkj,y kj () Another option – use APWs in the variational principle which takes into account discontinuity in the derivative of the basis functions (Slater, 1960) In all cases so far implicit energy dependence is present! NJIT Envelope Functions We learned how envelope functions augmented inside the spheres generate good basis functions. The basis functions can be continuous and smooth like MTOs or simply continuous like APWs, in all cases either condition of tail cancellation or requirement of smoothness leads us to a set of equation which delivers the solution of the Schroedinger equation with MT potential. Variational principle can be used for a full potential case. If basis functions are not smooth additional terms in the functional have to be included. Implicit energy dependence complicates the problem! NJIT Linear Muffin-Tin Orbitals General idea to get rid of E-dependence: use Teilor series and get read off the energy dependence. jl(r,E)=jjl(r,Enl)+-(EEnnl)&ll(rE,) -1 jl(r,D)=jjl(r,Enl)+-DD&nnnl(Dl)&ll(rE,) Dlll(E)= Sjj¢(S,E)/(SE,) Introduction of phi-dot function gives us an idea that we can always generate smooth basis functions by augmenting inside every sphere a linear combinations of phi’s and phi-dot’s The resulting basis functions do not solve Schroedinger equation exactly but we got read of the energy dependence! The basis functions can be NJITused in the variational principle. Linear Muffin-Tin Orbitals Augmented plane waves: kG+ * ck+G(r,E)=4pjå l(rE,)alYLL(r)Y(k+ÎG),rS L i(k+Gr)* cpk+G(r,E)=e4å jkl(|+G|r)YLL(r)Yk(+Gr), ÎWint L become smooth linear augmented plane waves: k++GkG * ck+G(r)=4på{jjl(r,Euul)al+&l(r,El)bl}YLL(r)Y(k+ÎG),rS L i(k+Gr)* cpk+G(r)=e=4å jkl(|+G|r)YLL(r)Yk(+G),,rÎSrÎWint L NJIT Linear Muffin-Tin Orbitals Consider local orbitals. Energy-dependent muffin-tin orbital defined in all space: l cL(r,E)={jkl(r,E)- Bloch sum should be constructed and one center expansion used: ikR åec L ()rR-= R ikR aljL(r,Ennl)+bljk&L(r,ElL)+åeh(,)rR-= R¹0 k aljL(r,Ennl)++blj&L(r,El)å jL''(kk,rS)LL() L' Final augmentation of tails gives us LMTO: khh cL(r)=++aljjL(r,Ennl)bl&Ll(rE,) jjk å{al'jL'(r,Ennl')+ bl'jk&L'(r,ESl'')}LL() L' NJIT Linear Muffin-Tin Orbitals In more compact notations, LMTO is given by khjk ckL(r)=FL(r)+Få L''(rS)LL() L' where we introduced radial functions hhh FL(r)=+aljjL(r,Ennl)bl&Ll(rE,) jjj FL(r)=+aljjL(r,Ebnnl)l&Ll(rE,) which match smoothly to Hankel and Bessel functions. NJIT Linear Muffin-Tin Orbitals Another way of constructing LMTO. Consider envelope function as kikR ck% LL(r)=-åeh(,)rR R Inside every sphere perform smooth augmentation kk c% L(r)=+hL(k,r)å jL''(kk,rS)LL() L' ß khjk ckL(r)=FL(r)+Få L''(rS)LL() L' S S S S which gives again LMTO construction. NJIT Linear Muffin-Tin Orbitals We could do the same trick for a single Hankel function ck% LL(r)= hr(,) Inside every sphere perform one- Center expansion c% LLR(r)=+hr(,)kd0 (1)--dkR0å jL''(,rR)SRLL() L' and augmentation h cdL(rr)=F+LR() 0 S S S S j (1-d R0)åF-L''(rR)SRLL() Bloch summation is trivial. L' NJIT Linear Muffin-Tin Orbitals LMTO definition (k dependence is highlighted): khjk ckLk (r)=FL(r)+åFL''(r)SrLL(), ÎWMT L' kikR ckLLk (r)=åeh(,r-Rr), ÎWint R which should be used as a basis in expanding kjk yckj(r)= å ArLLkk() Lk Variational principle gives us matrix eigenvalue problem. k2 kkj åáccL''k|-Ñ+V-EAkj|0LLkkñ= Lk NJIT Linear Muffin-Tin Orbitals Accuracy and Atomic Sphere Approximation: LMTO is accurate to first order with respect to (E-En) within MT spheres. LMTO is accurate to zero order (k2 is fixed) in the interstitials. Atomic sphere approximation can be used: Blow up MT-spheres until total volume occupied by spheres is equal to cell volume. Take matrix elements only over the spheres. ASA is a fast, accurate method which eliminates interstitial region and increases the accuracy. Works well for close packed structures, for open structures needs empty spheres. NJIT Tight-Binding LMTO Tight-Binding LMTO representation LMTO decays in real space as Hankel function which 2 depends on k =E-V0 and can be slow. Can we construct a faster decaying envelope? Advantage would be an access to the real space hoppings: kikR cckkLL(re)=-å ()rR R kikR Hk'L'kL= åeHRkk''LL() R NJIT Tight-Binding LMTO Any linear combination of Hankel functions can be the envelope which is accurate for MT-potential ()a hL(kk,r)=-å ALLL''(R)h(,)rR RL' where A matrix is completely arbitrary. Can we choose A-matrix so that screened Hankel function is localized? Electrostatic analogy in case k2=0 l '1+ MrL' / l+1 Vrscr ()~0 ZrL / Outside the cluster, the potential may indeed be screened out. The trick is to find appropriate screening charges (multipoles) NJIT Tight-Binding LMTO Once problem of screening is solved (Andersen, Jepsen, 1984) screened Hankel functions can be used as envelope functions and this leads us to so called: Tight-Binding LMTO Representation. Since mathematically it is just a transformation of the basis set, the obtained one-electron spectra are identical with original (long-range) LMTO representation. However we gain access to short-range representation and access to hopping integrals, and building low-energy tight-binding models. NJIT Advanced Topics Exact LMTOs LMTOs are linear combinations of phi’s and phi-dot’s inside the spheres, but only phi’s (Hankel functions at fixed k) in the interstitials. Can we construct the LMTOs so that they will be linear in energy both inside the spheres and inside the interstitials (Hankels and Hankel-dots)? Yes, Exact LMTOs are these functions! NJIT Advanced Topics Let us revise the procedure of designing LMTO: Step 1. Take Hankel function (possibly screened) as an envelope. Step 2. Replace inside all spheres, the Hankel function by linear combinations of phi’s and phi-dot’s with the condition of smooth matching at the sphere boundaries. S S S S Step 3. Perform Bloch summation. NJIT Advanced Topics Design of exact LMTO (EMTO): Step 1. Take Hankel function (possibly screened) as an envelope. Step 2. Replace inside all spheres, the Hankel function by only phi’s with the continuity condition at the sphere boundaries. The resulting function K kk L () r -= RK LR S S S S is no longer smooth! NJIT Advanced Topics Step 3. Take energy-derivative of the partial wave K&k LR So that it involves phi-dot’s inside the spheres and Hankel-dot’s in the interstitials. Step 4. Consider a linear combination c kLR(rK)=+kLR(r)å MKrLRL'R'&k LR''() LR'' where matrix M is chosen so that the whole construction becomes smooth in all space (kink-cancellation condition) This results in designing NJITExact Linear Muffin-Tin Orbital. Advanced Topics Non-linear MTOs (NMTOs) Do not restrict ourselves by phi’s and phi-dot’s, continue Tailor expansion to phi-double-dot’s, etc. In fact, more useful to consider just phi’s at a set of additional energies, instead of dealing with energy derivatives. This results in designing NMTOs which solve Schroedinger’s equation in a given energy window even more accurately. NJIT Advanced Topics: FP-LMTO Method Problem: Representation of density, potential, solution of Poisson equation, and accurate determination of matrix elements k2 kkj åáccL''k|-Ñ+V-EAkj|0LLkkñ= Lk with LMTOs defined in whole space as follows khjk ckLk (r)=FL(r)+åFL''(r)SrLL(), ÎWMT L' kikR ckLLk (r)=åeh(,r-Rr), ÎWint R NJIT Advanced Topics: FP-LMTO Method Ideas: Use of plane wave Fourier transforms Weirich, 1984, Wills, 1987, Bloechl, 1986, Savrasov, 1996 Use of atomic cells and once-center spherical harmonics expansions Savrasov & Savrasov, 1992 Use of interpolation in interstitial region by Hankel functions Methfessel, 1987 NJIT Advanced Topics: FP-LMTO Method At present, use of plane wave expansions is most accurate l ˆ rr(r)=<å L(r)iYrL(), rSMT L iGr rr(r),=å GerÎWint G To design this method we need representation for LMTOs kklˆ cckkL(r)=<å LL''(r)iYL(r),rSMT L' ki()kGr+ cckkLL(r)=å (k+G),erÎWint G NJIT Advanced Topics: FP-LMTO Method Problem: Fourier transform of LMTOs is not easy since kikR ckLLk (r)=åeh(,r-Rr), ÎWint R Solution: Construct psuedoLMTO which is regualt everywhere k c% Lk (r),= NJIT Advanced Topics: FP-LMTO Method The idea is simple – replace the divergent part inside the spheres hrL(,)k by some regular function which matches continuously and differentiably. What is the best choice of these regular functions? c%k L ()r The best choice would be the one when the Fourier transform is fastly convergent. The smoother the function the faster Fourier transform. NJIT Advanced Topics: FP-LMTO Method c ()r Weirich proposed to use linear %k L combinations aljrl(kk)+ bll&jr() This gives 4 c%k L (k+ GG)~1/ Wills proposed to match up to nth order aljrl(kkkk)++++bl&jrl()bl&j&l(r)cll&&&jr()... 3+n This gives c%k L (k+ G)~nG!!/ with optimum n found near 10 to 12 NJIT Advanced Topics: FP-LMTO Method Another idea (Savrasov 1996, Methfessel 1996) Smooth Hankel functions 22 ll-hrl2 (-Ñ-=kk)h%l(r)iYLL(r)reiYr() Parameter h is chosen so that the right-hand side is nearly zero when r is outside the sphere. Solution of the equation is a generalized error-like function which can be found by some recurrent relationships. It is smooth in all orders and gives Fourier transform 22 decaying exponentially -+|kG|/4h c%k L (k+ Ge)~ NJIT Advanced Topics: FP-LMTO Method Finally, we developed all necessary techniques to evaluate matrix elements áck|-Ñ22+VV|ckñ=ácckk||-Ñ+ñ+ L'k'LkVLL''kkWMT áck|-Ñ22+VV|ckñ-ácckk||-Ñ+ñ %L'k'%%%LkV%%LL''kkWMT where we have also introduced pseudopotential V%(r),= NJIT Advanced Topics: FP-LMTO Method Computer Programs available: LmtART (ASA-LMTO & FP LMTO methods) http://physics.njit.edu/~savrasov NJIT Advanced Topics: FP-LMTO Method LmtART features: q Multiple-kappa LMTO basis sets and multi-panel technique. q LSDA together with GGA91 and GGA96. q Total energy and force calculations q LDA+U method for strongly correlated systems. q Spin-orbit coupling for heavy elements. q Finite temperatures q Full 3D treatment of magnetization in relativistic calculations. q Non-collinear magnetizm. q Tight-binding regime. q Hopping integrals extraction regime. q Optical Properties (e1,e2, reflectivity, electron energy loss spectra) NJIT Advanced Topics: FP-LMTO Method Computer Programs available: MindLab (Material Information & Design Laboratory) Educational edition MS Windows based software freely available on CDs during our Workshop, http://physics.njit.edu/~savrasov NJIT Conclusion • LMTO, TB-LMTO, EMTO, NMTO is economical localized orbital basis set to solve the Schroedinger equation. • It is physically transparent and allows readily to analyze the nature of bonding in solid-state and molecular systems. • It is one of the most popular basis sets and widely used in density functional total energy calculations • It provides a minimal basis for building correct models described by many-body Hamiltonians and further developments of combining electronic structure techniques with many-body dynamical mean-field methods. NJIT