Electronic correlation and Hubbard approaches

Electronic correlation and Hubbard approaches Matteo Cococcioni Department of Chemical Engineering and Materials Science University of Minnesota Notable failures of LDA/GGA: transition-metal oxides Introduction to correlation: the H 2 molecule DFT and correlation DFT+U: general formulation Examples and applications

Failures of LDA/GGA: transition metal oxides Cubic, rock-salt structure TM ion Oxygen Antiferromagnetic (AF) ground state possible structural distorsions (FeO) rhombohedral symmetry and Conduction properties (exp): insulators (Mott/charge-transfer kind)

Fe 2+ FeO: GGA results Antiferromagnetic ground state: NO (FM) crystal structure (cubic): OK but We obtain a metal!!!

NiO: GGA results Antiferromagnetic ground state: OK crystal structure (cubic): OK Crystal field produces a band gap, but The energy gap is too small

Two quantities are to be considered: Mott insulators: U vs W U W on-site electron-electron repulsion bandwidth (hopping amplitude, related to kinetic energy) Two different regimes: W/U >> 1: the energy is minimized making the kinetic term as small as possible through delocalization (little price is paid on the occupied atomic sites to overcome repulsion U) W/U << 1: the kinetic energy of electrons is not large enough to overcome the on-site repulsion. Electrons undergo a Mott localization LDA/GGA approximations to DFT always tend to over-delocalize electrons: U is not well accounted for I. G. Austin and N. F. Mott, Science 168, 71 (1970) electronic energy functionals are affected by self-interaction

H 2 molecule in DFT Let s consider the spherically and system-averaged pair electronic density: where N( N % 1) d$ f r d d d r 12 = & ' 12 3 N 3 N 2!...! 4# 2 12 ( ) " (r,r, r,..., r ) R r... r 1 r 12 = r1! r 2 R = (r 1+r 2) / 2 N P Gori-Giorgi et al. cond-mat/0605174 At very large internuclear distances each electron is still split between the two sites!

From He to 2H: abc of correlation Two protons a and b (as classical objects, no nuclear forces) and two (noninteracting) electrons 1 and 2: electronic ground state at increasing interproton distance.we use 1s single electron states (H) or bonding (σ) and antibonding (σ*) linear combinations (H 2+ ) 1) d = 0, a = b: He atom "(1,2) = N [# a ( 1)# a ( 2) ]$% ionic terms 2) d > 0: H molecule a) "(1,2) = N [# a ( 1)# b ( 2) + # b ( 1)# a ( 2) ]$% b) "(1,2) = N [# $ ( 1)# $ ( 2) ]%&= N { " a ( 1)" b ( 2) + " b ( 1)" a ( 2) ( )" a ( 2) + " b ( 1)" b ( 2) }#$ [ ] + [" a 1 ] 2) d >> 0: distinct H atoms each electron is on the 1s ground state of each H atom: "(i) = # 1s ( i)$

From He to 2H: continuous solution H 2 molecule: the importance of ionic terms should disappear with distance. The ground state wave function should be something like: [ ] + $(d) [# a 1 ] "(1,2) = N { # a ( 1)# b ( 2) + # b ( 1)# a ( 2) ( )# a ( 2) + # b ( 1)# b ( 2) }%& This can be achieved through a linear combination of bonding (σ) and antibonding (σ*) Slater determinants: "(1,2) = N{ a(d) [# $ ( 1)# $ ( 2) ] + b(d) [# $ * ( 1)# $ * ( 2) ]}%&= N { ( a(d) + b(d) ) " a ( 1)" a 2 ( ) # a 1 + a(d) " b(d) ( ) + " b 1 ( )" b 2 a(d) and b(d) can be treated and computed as variational parameters. This is beyond Hartree-Fock theory because we need multi-determinantal wave-functions (multi reference CI methods etc) ( ) ( ) + ( ( )# b ( 2) + # b ( 1)# a ( 2) )}$%

Unrestricted (open-shell) Hartree-Fock Let the molecular orbitals for different spin have different spacial parts: " # $ = cos%" # + sin%" # * " # $ = cos%" # & sin%" # * " $ # * = %sin&" # + cos&" # * " $ # * = sin%" # + cos%" # * Constructing a Slater determinant for the singlet state: " # $ " # $ " UHF (1,2) = # $ % (1)# $ & (2) " = 0 o we can use θ as a variational parameter to obtain the ground state energy and wave function at any inter-nuclear distance d 0 o < " < 45 o " = 45 o θ allows for on-site localization of the electrons. Note how the spin sticks to a particular atomic site. Pictures from A. Szabo and N. S. Ostlund, Modern Quantum Chemistry

Unrestricted Hartree-Fock H 2 : results Let the molecular orbitals for different spin have different spacial parts: Small d: the standard RHF solution is obtained: " UHF (1,2) = # $ % (1)# $ & (2) Large d: electrons localize on either atom: lim R "# $ UHF (1,2) = % a & (1)% b ' (2) For d --> " we also obtain that E H 2 = 2E H However the wave function is wrong and so is also E for finite d. Why? actual wfc: [ ] & 1 lim R "# $(1,2) = N % a (1)% b (2) + % b (1)% a (2) [ ( )'( 2) ( '( 1)& ( 2) ]

DFT H 2 Despite the wave function is wrong, the charge density (in the long-distance limit) is right even within a minimal basis set representation: n(r) = " a ( r) 2 + " b ( r) 2 Thus DFT offers better chances than UHF to provide an accurate description DFT is a density method: don t try to get Kohn-Sham orbitals close to real one-electrons wfc; density and energy will be wrong.

DFT and UHF On a given basis of spin-orbitals UHF energy is: N E = # " " h i + 1 2 i" # " N " # i N # " ( J "" $ K "" ) ij + ij j 1 2 # " N " # i N # $" j ( " $" J ) ij One-body: kinetic energy + external potential Coulomb Exchange DFT looks similar to UHF (we let the two spin wave-functions have different spacial parts and solve separate Kohn-Sham equations). To see this: " kv % # # (r) = $ i (r) $ i " kv 1) 2) i 3) Insert in Hartree potential n(r) = $ k,v,# " # kv (r) 2 = $ & %i (r)% j (r) " # # $ kv % i % j " kv Difference: E xc. It contains exchange interactions and correlation. However it s not exact Idea: we can use the expansion in 2) to correct it. We will use the projections on atomic orbitals to optimize the weights of Coulomb and exchange integrals. ij k,v,#

Open systems in DFT: analyticity of E DFT In open systems exchanging electrons with a reservoir E should be piece-wise linear and show discontinuities n its first derivative 1. E DFT instead is analytical Needed correction 1 Perdew et al., PRL 49, 1691 (1982)

E Hub { n I mm' } [ ] = 1 2 E [ dc { n I" }] = { & { m},i,$ "m,m'' V ee m',m'''# n I$ I %$ mm' n m''m''' ( ) n mm' + "m,m'' V ee m',m'''# $ "m,m'' V ee m''',m'# n mm' The LDA+U energy functional The LDA+U method consists in a correction to the LDA (or GGA) energy functional to give a better description of electronic correlations. It is shaped on a Hubbard-like Hamiltonian including effective on-site interactions. It was introduced and developed by Anisimov and coworkers (1990-1995). E LDA +U Fully rotationally invariant formulation (Lichtenstein et al. PRB 1995) Occupations: Keep in mind: [{ n(r) }] = E LDA { n(r) } # U 2 n I n I $1 I # $% " I kv & m' I" = f kv k,v I" [ ] + E Hub { n m } ( ) $ '$& m I # J 2 n I" n I" $1 I," ( ) % " kv ' n I" = I" # n mm m { } [ ] # E [ dc n I" ] I% I% n m''m''' } n I = # n I" only occupations of localized orbitals included (e.g. d or f states) no cross-site terms: integer on-site occupation are favored against hybridization "

a k ( m,m',m'',m''' ) = 4" 2k +1 J = U = Electronic interactions Hartree-Fock formalism (for d states): from the expansion of 1/ r-r in spherical harmonics we get: % "mm'' V ee m'm'''#= $ dr $ * dr' m (r)% * m'' (r')% m' (r)% m''' (r') r & r' k & q=%k 1 n I" mm' = # mm' #lm Y kq lm'$#lm'' Y kq * I" Tr[ n ] 2l +1 1 $ "m,m' V ( 2l +1) 2 ee m,m'# = F 0 m,m' "m,m' V ee m',m# = F 2 + F 4 1 2l 2l +1 ( ) $ m,m' 14 lm'''$ The double counting term is evaluated as the Mean Field Approximation of the Hubbard one. So in the expression of E Hub we put and get Keep in mind: we want screened (effective) interactions; ' = a k (m,m',m'',m''')f k k k F are unscreened

A simplified approach First order approximation: let s neglect the exchange interaction J: We get: J = F 2 = F 4 = 0 U 2 E [{ U n I" }] I" mm' = E Hub n mm' % $ $ & n I" # $ n I" n I" mm ) = U ' mm' m'm I m," m' * 2 [ ] = [{ }] # E { dc n I" } Note: a) U is the only interaction parameter in the functional Note: b) the rotational invariance is preserved. This is the formula implemented in PWscf. We have ( $ I," Tr[ n I" ( 1 # n I" )] E DFT +U = E DFT " I# [ ] + E U n mm' [{ }] = E DFT " [ ] + U 2 % I,# Tr[ n I# ( 1 $ n I# )]

How does it work? Because of rotational invariance we can use a diagonal representation: where E U = U 2 % I,# % m [" I# ( I# m 1$ " )] m Potential: & k,v n I" v m = # I" m v m " I# m = f kv $ # I kv % m % m I # $ kv # V U " kv = $E U $" = U # * kv 2 " m I# > 1 2 $ V U < 0 " m I# < 1 2 $ V U > 0 ( I,# (( I# 1% 2& ) I I m ' m ' m m # " kv } Partial occupations are! discouraged Gap opening! E g! U

Potential discontinuities and energy gaps E U = U 2 % I,# % m [" I# ( I# m 1$ " )] m λ The +U correction is the one needed to recover the exact behavior of the energy. What is the physical meaning of U? Keep in mind: the +U correction is only applied to localized (atomic-like) states treated as impurities exchanging electrons with the crystal

Input file for LDA+U calculation with PWscf Only the namelist system is modified: &system... lda_plus_u =.true., Hubbard_U(1) = $U 1, Hubbard_U(2) = $U 2,.. Hubbard_U(ntype) = $U ntype,.. / There is a different U for each distinct type of Hubbard atom U is in ev Typical values: U is rarely larger than 7-8 ev (in most cases 0<U<5 ev) There might be need of using finer k-points grids for a better evaluation of on-site occupations.

GGA GGA+U FeO GGA+U M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

GGA+U FeO: the Broken Symmetry Phase (111) Plane of Fe old GS New frustrated electronic GS: correlation-stabilized orbital ordering new GS In the new electronic ground state the observed tetrahedral distorsion under pressure is reproduced. M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

GGA LDA+U NiO GGA+U M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

Mineral in the Earth s mantle: Fe 2 SiO 4 band structure structural parameters M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

Useful references A. Szabo and N. S. Ostlund, Modern quantum chemistry, Dover Publications Inc., 1996 Perdew et al., PRL 49, 1691 (1982) Anisimov, Zaanen, and Andersen, PRB 44, 943 (1991) Anisimov et al., PRB 48, 16929 (1993) Liechtenstein, Anisimov, and Zaanen, PRB 52, R5467 (1995) Anisimov, Aryasetiawan, and Liechtenstein, J. Phys.: Cond Matt 9, 767 (1997) Pickett, Erwin, and Ethridge, PRB 58, 1201 (1998)

Summary I Introduction to correlation: the case of H 2 H 2 : HF, UHF, CI DFT, exact exchange and correlation: between UHF and CI DFT and correlation: open systems and discontinuous potentials DFT+U: general formulation and main approximations Examples and applications

A consistent, linear-response approach To DFT+U Matteo Cococcioni Department of Chemical Engineering and Materials Science University of Minnesota DFT+U energy functional The meaning of U Linear-response calculation of U self-consistent U Examples and applications

The DFT+U energy functional [{ }] E [ DFT +U "] = E [ DFT "] I# + E U n mm' Where: E [{ U n I" }] I" mm' = E Hub n mm' [ ] = [{ }] # E { dc n I" } U 2 % $ $ & n I" # $ n I" n I" mm ) = U ' mm' m'm I m," m' * 2 ( $ I," Tr[ n I" ( 1 # n I" )] a) U is the only interaction parameter in the functional b) The functional is (on-site) rotationally invariant. This is the formula implemented in PWscf. We have E DFT +U = E DFT " I# [ ] + E U n mm' [{ }] = E DFT " [ ] + U 2 % I,# Tr[ n I# ( 1 $ n I# )]

What does U mean? E U = U 2 % I,# % m [" I# ( I# m 1$ " )] m λ The +U correction is the one needed to recover the exact behavior of the energy. What is the physical meaning of U?

Evaluation of U U is the unphysical curvature of the DFT total energy We want effective interactions: we evaluate U from the DFT ground state A free-electron contribution (due to re-hybridization) is to be subtracted in crystals: U = d 2 E GGA d(n I ) " d 2 GGA E 0 2 d(n I ) 2 From self-consistent ground state (screened response) From fixed-potential diagonalization (Kohn-Sham response)

Second derivatives Second derivatives of the energy are not directly accessible We apply a potential shift α to the d states of each Hubbard atom I and use a Legendre transform: Legendre transform E [{" I }] = min n(r ) $ % & [ ] + " I E n( r) # n I I ' ( ) E { n I } [ ] = min d E { n J } [ ] % & E " I " I ' { } ( ) dn I = "# I n J $ n I [{ }] # " I I d 2 E ( ) * = E " I { n J } [ ] ( ) 2 = " d n I I $ n min [{ }] # " I I d# ( I { n J }) dn I

d n I " 0 0 IJ = d! J Linear response Using α I as perturbation parameters we can easily evaluate the response matrices: " = IJ I d n d! χ 0 is the bare response of the system, χ the fully interacting (screened) one Run a self-consistent (unperturbed) calculation. Starting from saved potential and wavefunction add the perturbation The response χ 0 is evaluated at the first iteration (at fixed potential) The response χ is evaluated at self consistency The effective interaction is finally obtained as: J U = " d# I dn I ( ) II + d# I = $ "1 I 0 " $ "1 dn 0 M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005)

What screening? U = " 0 #1 # " #1 is equivalent to U = " 0 #1 "" 0 #1 # " 0 #1 " = " 0 + " 0 U" " = " 0 + " 0 U" 0 diagrammatic representation:!! 0! 0 = + U!!! 0! 0 0 = + U! U is a bare interaction; U is the dressed (effective) one. We use U and let the electrons perform the screening through the relaxation to their self-consistent ground state. But weren t we looking for screened effective interactions? What screening are we including?

Linear response: what is inside U? Let s apply a perturbation to the external potential and study the response of the electronic charge density "n(r) = "V (r) = "V ext + "V ext (r) "n(r) we can introduce response functions and write: where: Inverting (1) we get: $ #(r,r')"v ext (r')dr' = $ # 0 (r,r')"v (r')dr' "n(r') $ dr'+ %v xc(r) $ "n(r')dr' r # r' %n(r') (1) (2) "V ext (r) = % # $1 (r,r')"n(r')dr' "V (r) = % # $1 0 (r,r')"n(r')dr' (3) From (2) and (3) we easily obtain: " 0 #1 (r,r') # " #1 (r,r') = 1 r # r' + $v xc(r) $n(r') The Coulomb interaction is screened by the xc kernel xc kernel See S. Baroni et al., Rev. Mod. Phys. 73, 515 (2001) and refs quoted therein

Input file for computing U with PWscf unperturbed run &system... lda_plus_u =.true., Hubbard_U(1) = 1.d-20, Hubbard_U(2) = 1.d-20,.. Hubbard_U(ntype) = 1.d-20,.. / Hubbard_alpha values are symmetrically distributed around 0 (typically from 0.1 to 0.1 ev) the perturbed atom has to be treated as of different kind the series of perturbed runs at different Hubbard_alpha is repeated for every different kind of Hubbard atom diago_thr_init is chosen close to the last ethr of the unperturbed run (ethr_conv) perturbed run foreach a (0.d0 0.01 0.01 ). &control. restart_mode = from_scratch. / &system. lda_plus_u =.true., Hubbard_U(1) = 1.d-20,. Hubbard_U(ntype) = 1.d-20, Hubbard_alpha(1) = $a, Hubbard_alpha(2) = 0.d0,. / &electrons. startingwfc = file, startingpot = file, diago_thr_init = $ethr_conv. /. end

Constructing the response matrices The perturbation must be fully accomodated in the considered cell (to avoid interactions with its periodic replicas): we need to run calculations in supercells Supercell: N Hubbard atoms of M distinct types. Response matrices: NxN Perturbed calculations: to be repeated for each of the M types of Hubbard atoms First M columns of the response matrices are calculated straightforwardly. The other N-M columns are obtained attributing the same response to equivalent shells of neighbors when perturbation is applied on equivalent atoms Background term (perturbation must be neutral) and other technical details: please refer to M. C. and S. d. G. PRB 71, 035105 (2005) Converge the computed U values with the size of the supercell

Advantages of the method Fully ab-initio estimation of the effective interaction (no guess or semiempirical evaluation is needed) Consistency of the effective interaction with the definition of the energy functional and of the on-site occupations; - other localized basis sets can be equivalently used: gaussians, Wannier functions etc Consistency with the DFT approximation Easy implementation in different computational schemes

Self-consistent U DFT and DFT+U can lead to very different ground states The response of the system can be different in DFT and DFT+U U must be computed self-consistently from a DFT+U ground state

The energy model We need a model for the electronic interactions in the GGA+U ground state E int = U scf 2 # I + -,- % (. # $1 * 0 & )/ 0 + # I," I," ' n i ' n j i," j," ' Double counting functional +U correction functional + U in 2 ( ) ## I," I," n i 1$ n i I," i We need the second derivatives of the total energy with respect to the total on-site occupation n I = " i n i I

Evaluation of U scf For small shifts in the on-site potential we have: "n i I We obtain: U out = d 2 E GGA +U = a i I "n I and ( ) 2 " d d n I 2 E 0 GGA +U = d 2 E int ( ) 2 d( n I ) = U 2 scf " U in m d n I d 2 d n I " i, j ( ) 2 = a i I a j I d 2 dn i I dn j I where m =1/ " i ( I a ) 2 i is an effective degeneracy. We want the value of U scf that can be obtained from a GGA+U ground state with U in = U scf

Evaluation of U scf in practice Study U out vs U in : identify the linear region Extrapolate to U in =0 U scf is the one from the extrapolation of the linear region that contains it.

Importance of computing U Consistency with the expression of the +U functional and with the choice of localized basis set: the computed U is exactly the one needed Sensitiveness to spin states, chemical/physical environments, structural changes (important in studying e.g. chemical reactions, phase transitions) (Mg x Fe 1-x )O HS to LS transition under pressure FeO rhombohedral distorsion under pressure M. Cococcioni and S. de Gironcoli PRB 71, 035105 (2005) P (GPa) T. Tsuchiya, R. M. Wentzcovitch, C. R. S. da Silva and S. De Gironcoli, Phys. Rev. Lett. 96, 198501 (2006)

Cathode material for Li ion batteries: Li x FePO 4 High voltage (3.5 V) and high theoretical capacity Electrochemical stability Thermal safety Non toxicity Low cost and easy synthesis Issue: low electronic conductivity Properties Exp GGA GGA+U Crystal structure olivine olivine olivine Magnetic structure AF AF AF Similar agreement In other compounds! Exp GGA+U Formation Energy Li x FePO 4 >0 <0 >0 Voltage 3.5 V 2.9 V 3.5 V TM valence states (0< x <1) Fe 2+ /Fe 3+ Fe 2.5+ Fe 2+ /Fe 3+ V = E( Li FePO )! E( Li FePO )! ( x! x ) E( Li) x 4 x 4 2 1 2 1 ( x! x ) F 2 1 PRB 69, 201101 (2004) PRB 70, 235121 (2004)

Spin states in the Heme group Fe site O 2 Penta-coordinated iron Magnetic ground state Esa-coordinated iron (O 2, CO, etc) Exp quintuplet (S=2) singlet (S=0) GGA triplet singlet B3LYP triplet singlet HF quintuplet quintuplet GGA+U quintuplet singlet D. Scherlis, H. L. Sit, M. Cococcioni and N. Marzari, submitted to Journ. of Phys. Chem.

H 2 addition-elimination to FeO + GGA GGA+U H. J. Kulik, M. Cococcioni, D. Scherlis and N. Marzari, PRL 97, 103001 (2006)

H. J. Kulik, M. Cococcioni, D. A. Scherlis, and N. Marzari, PRL 97,103001 (2006). The Fe dimer

Conclusions Linear response approach to compute U: the LDA+U as a fully ab-initio computational scheme Self-consistent evaluation of U Correct electronic ground state and structural properties of FeO Correct chemical behavior and voltage of Transition Metal compounds for cathodes of Li batteries Improved description of molecules and chemical reactions

Work in progress Better +U functionals (for better energetics) auto-consistent calculation of U (from a LDA+U ground state) inter-site interactions phonon+u covariant formulation of U compute J