Computer Physics Communications 179 (2008) 784–790 Contents lists available at ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Force calculation for orbital-dependent potentials with FP-(L)APW + lo basis sets ∗ Fabien Tran a, , Jan Kuneš b,c, Pavel Novák c,PeterBlahaa,LaurenceD.Marksd, Karlheinz Schwarz a a Institute of Materials Chemistry, Vienna University of Technology, Getreidemarkt 9/165-TC, A-1060 Vienna, Austria b Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, Augsburg 86135, Germany c Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, CZ-162 53 Prague 6, Czech Republic d Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA article info abstract Article history: Within the linearized augmented plane-wave method for electronic structure calculations, a force Received 8 April 2008 expression was derived for such exchange–correlation energy functionals that lead to orbital-dependent Received in revised form 14 May 2008 potentials (e.g., LDA + U or hybrid methods). The forces were implemented into the WIEN2k code and Accepted 27 June 2008 were tested on systems containing strongly correlated d and f electrons. The results show that the Availableonline5July2008 expression leads to accurate atomic forces. © PACS: 2008 Elsevier B.V. All rights reserved. 61.50.-f 71.15.Mb 71.27.+a Keywords: Computational materials science Density functional theory Forces Structure optimization Strongly correlated materials 1. Introduction The majority of present day electronic structure calculations on periodic solids are performed with the Kohn–Sham (KS) formulation [1] of density functional theory [2] using the local density (LDA) or generalized gradient approximations (GGA) for the exchange–correlation energy. Among the different methods to solve the KS equations, the full-potential (linearized) augmented-plane-wave and local-orbitals (FP-(L)APW + lo) methods [3,4] are among the most accurate schemes. For a given crystal structure one is very often interested in finding the atomic positions corresponding to the lowest energy. For solids with a complicated unit cell a direct minimization of the total energy is impractical, but the knowledge of atomic forces simplifies the structure optimization and allows one to move the atoms until the forces vanish. The formalism of the force calculation for the original LAPW basis set [5] was independently developed by Soler and Williams [6,7] and Yu et al. [8], leading to two formulations which later on were shown to be equivalent both from the formal as well as practical points of view [9–11]. The formulation of Yu et al. [8] was adopted for the WIEN2k code [12–15] (the details of the implementation can be found in Ref. [16]). Then, the formalism was adapted for the APW + lo basis set [4,17,18]. Other works on forces for the LAPW basis set can be found in Refs. [19,20]. There are important classes of solids for which the LDA and GGA functionals are known to yield even qualitatively incorrect ground states. Notorious examples are the so-called strongly correlated systems, e.g., the transition-metal oxides or rare-earth compounds. The ground state of such systems can often be described significantly better by using the LDA + U [21,22] or hybrid functionals (see, e.g., Refs. [23,24]). The WIEN2k code, which is based on the FP-(L)APW + lo method, allows such calculations [24–26], but so far structural relaxation has not been possible due to the lack of an atomic force formalism for LDA + U and hybrid functionals in the FP-(L)APW + lo basis set. Therefore, it is of great importance to develop such a formalism, which is presented here. * Corresponding author. E-mail address: [email protected] (F. Tran). 0010-4655/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cpc.2008.06.015 F. Tran et al. / Computer Physics Communications 179 (2008) 784–790 785 The paper is organized as follows. The next section describes the theory, namely, the FP-(L)APW + lo basis set, the basic KS equations with orbital-dependent potentials, and the derivation of the atomic forces for such potentials. Then, numerical tests are presented in Section 3, and finally a summary is given in Section 4. 2. Theory 2.1. FP-L(A)PW + lo basis sets In the FP-(L)APW + lo method, the unit cell volume is divided into two types of regions (see Ref. [3] for details): the interstitial I and the non-overlapping muffin-tin (MT) spheres Sα centered at the positions Rα of the nuclei. The basis functions, which are used for the expansion of the crystal orbitals (n is the band index, k is a vector in the first Brillouin zone, and σ is the spin index) ψσ (r) = dσ φσ (r) + dLOiσ φLOiσ (r), (1) nk nk+K k+K nk+Ki k+Ki K i are augmented plane waves (where V is the unit cell volume and rα = r − Rα is the position inside sphere α): + √1 ei(k K)r r ∈ I, σ V φ + (r) = (2) k K αmσ ασ + αmσ ˙ ασ ˆ ∈ ,m(ak+K u1, (rα) bk+K u1, (rα))Ym(rα) r Sα, ασ ασ ˙ ασ where u1, is a radial function evaluated at the energy 1, and u1, is its energy derivative, and local orbital (LO) basis functions αi imi σ αi σ + αi imi σ ˙ αi σ + αi imi σ αi σ ˆ ∈ LOiσ (aLOk+K u1, (rαi ) bLOk+K u1, (rαi ) cLOk+K u2, (rαi ))Yimi (rαi ) r Sαi , φ + (r) = i i i i i i (3) k Ki ∈ 0 r / Sαi . α m σ Eqs. (2) and (3) are for the LAPW + LO basis set [3], while for the APW + lo basis set [4] the coefficients bαmσ in Eq. (2) and c i i i in k+K LOk+Ki Eq. (3) are zero. 2.2. DFT + U and hybrid functionals In the WIEN2k code, the DFT + U and hybrid methods are implemented only inside the MT spheres [24–26], which is justified provided these methods are applied to electrons that are well localized inside the corresponding sphere, e.g., 3d and 4 f electrons in transition- metal and rare-earth atoms, respectively. In essence, an orbital-dependent contribution to exchange and correlation is added within the MT spheres of the atoms which contain strongly correlated electrons and the corresponding LDA or GGA contribution is subtracted. The DFT + U and hybrid total-energy functionals are given by (all equations are expressed in Hartree atomic units) = + + loc + E Ts ECoul Exc Eorb, (4) loc where Ts, ECoul, Exc , and Eorb are the KS kinetic, Coulomb (electron–electron, electron–nucleus, and nucleus–nucleus electrostatic inter- actions), (semi-)local (i.e., LDA or GGA) exchange–correlation, and orbital (Hubbard or hybrid) terms, respectively. The corresponding KS equations for the valence electrons are ˆ σ σ = ˆ + KS + ˆ σ σ = σ σ H ψnk(r) T veff,σ (r) U ψnk(r) nkψnk(r), (5) ˆ =−1 ∇2 where T 2 is the kinetic-energy operator, KS = + loc veff,σ (r) vCoul(r) vxc,σ (r) (6) ˆ σ is the multiplicative effective KS potential [1], and U is the non-multiplicative (i.e., orbital-dependent) operator associated with Eorb (see Refs. [27] and [26] for DFT + U and hybrid functionals, respectively): βσ δEorb ∂ Eorb δnm m βσ β Uˆ σ ψσ (r) = = 1 2 = v Pˆ ψσ (r). (7) nk σ ∗ βσ σ ∗ m1m2 m1m2 nk δψ (r) ∂n δψ (r) nk β,m1,m2 m1m2 nk β,m1,m2 In Eq. (7), mi =−,...,, where is the angular momentum of the electrons of atom β, for which the Hubbard or hybrid correction is βσ = βσ ˆ β applied, vm1m2 ∂ Eorb/∂nm1m2 , Pm1m2 is the projector which is defined such that β RMT ˆ β = | | 2 f Pm1m2 g f Ym1 Ω,β Ym2 g Ω,βr dr, (8) 0 βσ and nm1m2 is the occupation matrix which is calculated in the following way (see Eq. (11) of Ref. [27]): β RMT βσ β 2 n = wσ ψσ Pˆ ψσ = wσ ψσ Y Y ψσ r dr m1m2 nk nk m1m2 nk nk nk m1 Ω,β m2 nk Ω,β n,k n,k 0 σ σ ∗ σ βm1σ ∗ βm2σ βm1σ ∗ βm2σ βσ βσ βm1σ ∗ βm2σ = w d d a a + b b u˙ u˙ + c c nk nk+K nk+K k+K k+K k+K k+K 1, 1, r,β k+K k+K n,k,K,K βm1σ ∗ βm2σ βm1σ ∗ βm2σ βσ βσ βm1σ ∗ βm2σ βm1σ ∗ βm2σ βσ βσ + a c + c a u u + b c + c b u˙ u (9) k+K k+K k+K k+K 1, 2, r,β k+K k+K k+K k+K 1, 2, r,β , 786 F. Tran et al. / Computer Physics Communications 179 (2008) 784–790 σ | | where wnk is the weight for the integration in the Brillouin zone. In Eqs. (8) and (9), f g r,β and f g Ω,β denote radial and angular integrations in the sphere Sβ , respectively. Using the sum of eigenenergies, the KS kinetic energy is given by βσ βσ T = σ + wσ σ − vKS r σ r d3r − v n (10) s i nknk eff,σ ( )ρ ( ) m1m2 m1m2 , σ m m σ ,i σ ,n,k cell β,σ , 1, 2 where the first two sums are over the core and valence orbitals, respectively. 2.3. Forces for DFT + U and hybrid functionals =−∇ The force Fα Rα E which acts on the nucleus α is composed of two terms: = ES + IBS Fα Fα Fα , (11) ES IBS where Fα is the Hellmann–Feynman electrostatic (ES) force and Fα is the so-called Pulay force [28] which arises due to the use of an atomic-position-dependent incomplete basis set (IBS).