Band calculations: Theory and Applications Lecture 2: Different approximations for the exchange-correlation correlation functional in DFT Local density approximation () Generalized gradient approximation () +U + http://alps.comp-phys.org/mediawiki-1.9.3/index.php/dft-short-course 1
DFT: Short summary from previous lecture Thomas-Fermi-Dirac: Hohenberg-Kohn: For non-interacting electron gas: Kinetic energy Exchange interaction + For any electronic system: Coulomb repulsion Kohn-Sham: Real, interacting system Reference, noninteracting system But with some strange term 2
DFT: Kohn-Sham equations (1965) Kohn-Sham equations XC energy: XC potential: It acts locally, but may know about density distribution in other points 3
: Local density approximation for E xc [n] +U + The key point in DFT is an explicit form E xc [n]. : exchange-correlation energy density equals to exchange-correlation energy density of homogeneous electronic gas in given point It is local in sense that it knows only density in given point due to Dirac (see TFD theory) 4
: Local density approximation for E xc [n] +U +, was obtained from QMC simulation for varying densities [PRL 45, 566 (1980)] And latter interpolated by Vosko, Wilk and Nusair Can. J. Phys. 58, 1200 (1980): 5
Extension of : Local Spin Density Approximation () +U + 6
Advantages and disadvantages of L(S)DA +U + Structural properties of solids are often good Ł usually underestimates bulk lattice constants by a small amount: 7
Advantages and disadvantages of L(S)DA +U + Structural properties of solids are often good Ł usually underestimates bulk lattice constants by a small amount Ł phonons too stiff EXP 8
Advantages and disadvantages of L(S)DA +U + Structural properties of solids are often good Ł usually underestimates bulk lattice constants by a small amount Ł phonons too stiff This is due to the fact, that L(S)DA favors electronic densities that are more homogeneous than they should be; As a result binding energies are too large 9
and : Where they should and should not work? Generally one may expect that will work good for isotropic and homogeneous system such as metals. will be problematic for the description of inhomogeneous systems such as e.g. +U + a) Isolated molecules b) Polarized insulators c) Strongly correlated materials + + + - 10
: Generalized Gradient Approximation +U + Idea: Taylor expansion of the density near homogeneous gas point Total energy calculations for Fe Experiment: FM, bcc J. Phys.: Cond. Matter 10, 5081 (1998) generally improves magnetic energies (with respect to ) one of contrary instances: PRB 65, 132104 (2002) 11
: Generalized Gradient Approximation +U + Lattice constants: sometimes gives better agreement in structural constants, than 12
Ab-initio H O treatment 2 +U + meta For the description of isolated molecules nor L(S)DA neither cannot be used! 13
L(S)DA or : band gap problem +U + Important: Kohn-Sham orbital energies Band gap for correlated oxides have NO explicit physical meaning! In general they are no more than orbital energies of some auxiliary Kohn-Sham system. Thus are even not owed to give correct gap! But, this is not only uncomfortable... 14
Example of inappropriate use of : +U + Analysis of results: Ti 3+ : d 1 Ti-Ti dimers has triplet ground state, i.e. FM!? 15
L(S)DA or : band gap problem +U + Experiment: NiO: CT insulator Band gap: ~ 4 ev Crystal field splitting 16
+U: Different treatment of physically different electrons: s,p electrons are considered on level d electron part of functional is corrected, ( Hubbard ) as it is done in model approaches +U + double counting term occupied states : unoccupied states : Anisimov et al., J. Phys.: Condens. Matter 9 (1997) 767 17
Very simple +U example: +U + Ionization energy of Hydrogen atom: E exact (H) = 1.0 Ry e (H) = 0.538 Ry << E exact (H) E (H) = 0.957 Ry ~ E exact (H) Calculated U = 0.9448 Ry Occupied (H) state: e +U = e (H) + U/2 = 1.0104 Ry = 1 Ry Unoccupied (H + ) state: e +U (H + ) = e (H) - U/2 = 0.0656 Ry ~0 Ry 18
+U results for TMO +U + +U for NiO: U=8 ev, J=0.85 ev Correct gap, NiO: CT - insulator Anisimov et al., J. Phys.: Condens. Matter 9 (1997) 767 Problems of +U: How to chose U? How to chose DC term? 19
+U: How to chose U? Calculate it! +U + We want to compute U, from, but how much Coulomb interaction is in the? Let s s think that: In atomic limit: DC term orbital energy So in : center of gravity occupancy This derivative can be estimated numerically in super-cell calculation: 20
+U: How to chose U? Calculate it! +U + e d n-1 d n+1 e e e e e e e e e Conduction band Important: this type of calculation takes into account screening Thus, since U can be calculated, +U can be considered as fully ab-initio initio. Drawback: unfortunately calculated values of U strongly depends on the details of calculation (method, RMT, allowed screening channels)! 21
+U: How to chose DC? No definite answer. +U + In simple : Due to hybridization and non-liniarity Ec, it's not easy to extract the part of e-e Coulomb repulsion taken into account on level for TM-d orbitals. Let's think that correlated electrons can be described, as, (1) Fully localized limit: works for strongly correlated materials NiO in +U with different DC terms: (2) Near MF limit: works for materials with intermediate correlation strength PRB 67, 153106 (2003) PRB 48, 16929 (1993) 22
+: The way to include dynamical correlations on the top real band structure. +U + : Reality: T 0 Pay attention: most of the + calculations presently available are not self-consistent! There are very few exceptions where authors repeat part after scf solution: n start n V FULL SCF loop,, loop loop Imp. solver H(k) H scf (k) G(τ) cond-mat/0106308 Phys. Rev. B 71, 125119 (2005) Phys. Rev. Lett. 101, 096405 (2008) 23
+: The way to include dynamical correlations on the top real band structure. +U + Q: When full self consistent + needed? A: When the number of electrons is changed significantly Ce-alpha : n=1.19 +: n=1.06 Other problems of +: 1) Inherited from +U, DC problem and choice of U,J 2) How to retrieve correlated only part of hamiltonian from? n start n V FULL SCF loop,, loop loop Imp. solver H(k) H scf (k) G(τ) 3) Specific problems like choice of solver, temperature etc. 24
+ calculation example: gamma-alpha alpha Ce +U + + Experiments: PRB 28, 7354 (1983) PRB 55, 2056 (1997) PES: occ. part IPES: unocc. part 25
What is the difference between different band structure calculations? Treatment of potential: MT-potential, Atomic sphere approximation (ASA) Full potential Pseudopotential Choice of wave functions (Method): Plane waves (PW) Augmented plane waves (APW, LAPW) MT-orbitals (MTO, LMTO) Lin. Comb. of atomic orbitals (LCAO) Gaussians Kohn-Sham equations Approximations for exchange correlation part: meta +U + Relativistic treatment of electrons. Semi-relativistic treatment of electrons (SO only for core level electrons). 26