An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method

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1 Fakultät für Mathematik und Naturwissenschaften - Lehrstuhl für Physikalische Chemie I / Theoretische Chemie An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method Jan-Ole Joswig Physikalische Chemie Technische Universität Dresden

2 An Introduction to Density-Functional Theory Dresden, /34

3 Density-Functional Theory The Hohenberg-Kohn theorems (1964): 1. Once we know the ground-state electron density in position space any ground-state property is uniquely defined. Any ground-state property is a functional of the electron density in position space. We do not need the full wavefunction. Instead: we calculate the electron density directly. ρ ρ δρ 2. A variation of the ground-state electron density results in a positive change of the total ground-state energy: E ρ r E r e ( ) ρ( ) with ρ( ) ρ( ) e ( r) = ( r) + ( r) r dr = r dr = N Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 3/34

4 Density-Functional Theory The Kohn-Sham method (1965): The Hohenberg-Kohn theorems provide formalistic proof for the correctness of the Thomas-Fermi approach. They do not provide any practical scheme. From the 2. Hohenberg-Kohn Theorem: [ ] [ ] [ ] δe ρ E ρ δρ E ρ e e + e = 0 Number of electrons N stays constant for all δρ: ρ ( r ) dr = Minimization: introducing a Lagrange multiplier μ: δ ρ μ ρ N { E ( r) ( r) dr N } e = 0 Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 4/34

5 Density-Functional Theory The Kohn-Sham method (1965): What does E e [ρ] contain? Ee ρ( r) = T ρ( r) + Vext( r) ρ( r) dr + VC( r) ρ( r) dr + E xc ρ( r) ρ r ρ r + δρ r For the functional derivative, we change and calculate μ: δt δ E xc μ = + V ext ( r ) + V C ( r ) + δρ δρ ( ) ( ) ( ) Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 5/34

6 Density-Functional Theory The Kohn-Sham method (1965): The Trick: We take a fictitious system of non-interacting particles with the same electron density as the real system and the same energy as the real system. Therefore, the particles move in an effective potential V eff and the total energy is and we get Ee ρ( r ) = T0 ρ( r ) + Veff ( r ) ρ( r ) dr δt0 Veff ( r) δρ + = μ NB: T0 ρ( r ) T ρ( r) Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 6/34

7 Density-Functional Theory The Kohn-Sham method (1965): Comparing both Lagrange multipliers δt δ E xc μ = + V ext ( r ) + V C ( r ) + δρ δρ we arrive at The Hamiltonian of the model system is i.e., there are only single-particle operators and μ δt 0 = + δρ δt δt0 δ E xc Veff ( r) = + Vext ( r) + VC ( r) + δρ δρ δρ N N 1 2 ˆ ri eff i eff i= 1 2 i= 1 ( ) ( ) Hˆ = + V r h i V eff ( r) Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 7/34

8 Density-Functional Theory The Kohn-Sham method (1965): Therefore, the Schrödinger equation can be written as a single Slater determinant where Ψ= φ1, φ2,, φn hˆ φ = εφ eff i i i is the single-particle equation that determines the single-particle orbital. Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 8/34

9 Density-Functional Theory The Kohn-Sham method (1965): Furthermore, ρ N ( r) φ ( r) 2 = i= 1 i summing over the N orbitals with the lowest eigenvalues ε i. But: We do not know the effective potential V eff, i.e., the terms δt δt δe δe δρ δρ δρ δρ 0 xc xc + ( r) the so-called exchange-correlation energy and potential, which have to be approximated. V xc Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 9/34

10 Density-Functional Theory Extension: Spin. So far, the system was supposed to be spin-unpolarized, i.e. and In the spin-polarized case, ρ ρ ρ ( r) = ( r) + ( r) ρ α And we have to extend the Hohenberg-Kohn theorems: α β ( r) = ρ ( r) m r r r ( ) = ρ ( ) ρ ( ) 0 α Ee = Ee ρ ( r), m( r) β β Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 10/34

11 Density-Functional Theory The xc potential: Local-density approximation (LDA): xc functional depends only on the electron density LDA Exc [ ρ] = ρ( r) εxc ( ρ) dr E x : homogeneous electron gas E c : analytical expression not known various approaches: Vosko-Wilk-Nusair (VWN) Perdew-Zunger (PZ81) Perdew-Wang (PW92) E = E + E xc x c LDA x [ ρ] ρ( ) E = r dr 4 π Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 11/34

12 Density-Functional Theory The xc potential: Local-density approximation (LDA): r V xc at position as it would be for a homogeneous electron gas, i.e. ignoring spatial variations in ρ (and m). Generalized gradient approximation (GGA): Spatial variations are included (V xc may depend on,, ). ρ 2 ρ Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 12/34

13 Density-Functional Theory Comparison: Dresden, /34

14 Approximations Frozen-core approximation: Ee ρ( r) = Ee ρcore( r) + ρvalence( r) Pseudopotentials: Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 14/34

15 Approximations Linear combination of atomic orbitals (LCAO): LCAO ansatz: ψ = ( r) c ϕ ( r R ) Gauss-type atomic orbitals (GTOs): Slater-type atomic orbitals (STOs): (Linearized)-augmented-wave methods: LMTO and LAPW. LMTO: Non-overlapping atom-centered muffin-tin spheres + interstitial region i im m j m e ζ r 2 e ζ r LAPW: Using plane waves in the interstitial region: ikr e ( ) R ( ) V r R V r = V 0 Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 15/34

16 Properties The density of states (DOS): The number of states (orbital energy eigenvalues) in an intervall [E+dE]. E Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 16/34

17 Properties The electronic band structure: The Schrödinger equation of a solid has Bloch waves as solutions: ψ = nk ( r) e ikr u ( r) The band structure shows the dependence of the energtic states on the direction of motion of the electron (wave vector k ). nk Silicon direct gap: no momentum change required indirect gap: momentum change required Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 17/34

18 The Approximate Density-Functional Tight-Binding (DFTB) Method Dresden, /34

19 Density-Functional Tight-Binding Theory Linear combination of atomic orbitals (LCAO): The single-particle KS eigenfunctions are expanded ina set of atomcentered basis functions: ψ ( r) c ϕ ( r) i im m m Obtained from SCF-DFT calculations of the isolated atom (large Slater-type basis set) = Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 19/34

20 Density-Functional Tight-Binding Theory The effective potential: The DFT Hamiltonian is ˆ 1 h i = + V r = tˆ + V Effective potential in DFT Effective potential in DFTB 2 () ( ) eff r eff i eff 2 i δt δt0 δ E xc Veff ( r) = + Vext ( r) + VC ( r) + δρ δρ δρ V r V r R 0 ( ) = ( ) eff j j j i.e. superpostion of potentials of neutral atoms Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 20/34

21 Density-Functional Tight-Binding Theory The total energy: 1 E r V r r dr V r r dr occ. tot ρ( ) ε = i eff ( ) ρ( ) ext ( ) ρ( ) i E V ( r) ρ ( r) dr + E 2 Double counting xc xc N occ. occ. ε ( ) ˆ i = ϕi r heff ( i) ρ( r ) ϕi( r occ. ) i i i ε = n ε i i i i Optimizing the KS orbitals (non-self-consistent). Everything else is constant. Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 21/34

22 Density-Functional Tight-Binding Theory The tight-binding approximation: The Hamiltonian matrix elements are h = ϕ hˆ ϕ = ϕ tˆ+ V ϕ = ϕ tˆ+ V ϕ h 0 mn m n m eff n m j n j mn ( ) = ϕ tˆ + V + 1 δ V ϕ 0 0 m jm jn, jm jn n 0 ϕ ˆ m t + Vjm ϕm m= n = 0 0 ˆ ϕm t + Vjm + Vjn ϕn m n Through this approximation only two-center terms are considered h = ϕ hˆ ϕ S = ϕ ϕ mn m n mn m n Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 22/34

23 Density-Functional Tight-Binding Theory Secular equations: Related to other tight-binding schemes or the extended Hückel method. m ( ε ) = 0 c h S im mn i mn But all matrix elements are calculated exactly within DFT; none is determined through fitting to experimental results. Off-diagonal matrix elements depend only on the interatomic distance. They are tabulated. Pairwise interaction of the remaining terms (repulsive energy) 1 E = U R R ( ) rep jj' j j' 2 j j' Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 23/34

24 Density-Functional Tight-Binding Theory Binding energy: occ. 1 E ε ε + U R R ( ) B i jm jj' j j' i j m 2 j j' E rep is calculated once for every pair and fitted or tabulated E rep C 2 dimer Frozen core approximation: Ee ρ( r ) = Ee ρcore( r ) + ρvalence( r ) SCF-LDA occ. εi i Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 24/34

25 Density-Functional Tight-Binding Theory Self-Consistent Charge (SCC): Critical DFTB assumption: charge density is a superposition of unperturbed atomic charge densities E ρ r δρ r E ρ r ( ) + ( ) = ( ) δ E xc δρ ( r ) δρ ( r ) + + drdr 2 r1 r2 δρ ( r1) δρ ( r2) ρ0 atoms δρ ( r) = ΔqA Decomposition into atomic contributions A Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 25/34

26 Density-Functional Tight-Binding Theory Self-Consistent Charge (SCC): Critical DFTB assumption: charge density is a superposition of unperturbed atomic charge densities 1 E ρ r + δρ r = E ρ r + Δq Δq γ atoms 0( ) ( ) 0( ) 2 AB, Mulliken populations Partial atomic charges depend on the KS orbitals Self-consistency requirement A B AB Self-Consistent Charge DFTB (SCC-DFTB) Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 26/34

27 Density-Functional Tight-Binding Theory Summary: ( r ) c ϕ ( r ) = LCAO ansatz for the molecular orbitals: i im m m Basis: minimal set of atomic valence orbitals obtained from selfconsistent DFT calculations of the isolated atoms ψ Only two-centre terms: h mn = ϕ hˆ ϕ, S = m n mn ϕ m ϕ n Repulsion: short-range, 2-body potential, parametrized (diatomics, solids) ( ) R occ Binding energy: E B ε i ε jm + U jj R j i j m j j Efficient method for large systems ( atoms) 1 2 j Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 27/34

28 Density-Functional Tight-Binding Theory Dresden, /34

29 DFTB vs. DFT The H 3 PO 3 molecule Labels MP2 PBE0 PBE SVWN DFTB Exp* P1 H P1 O P1 O P1 O O3 H O4 H H2 P1 O H2 P1 O H2 P1 O Bond lengths [Å] and angles [deg.] of H 3 PO 3 J. Mol. Struct.: THEOCHEM 816 (2007), 119. *Neutron diffraction: Becker et al., Z. Anorg. Allg. Chem. 591 (1990), 17 Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 29/34

30 DFTB vs. DFT The proton transfer in the PA dimer DFTB asymmetric PA dimer J. Mol. Struct.: THEOCHEM 816 (2007), 119. Dresden, /34

31 DFTB vs. DFT The proton transfer in the PA dimer Performance of DFTB in comparison to the PBE functional in Gaussian J. Mol. Struct.: THEOCHEM 816 (2007), 119. Dresden, /34

32 DFTB vs. DFT Band structure of Cu: SCF-DFT vs. DFTB calculation J. Phys. Chem. A 111 (2007), Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 32/34

33 DFTB vs. DFT SCC-DFTB vs. DFTB: Phys. Rev. B 58 (1998), Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 33/34

34 DFTB vs. DFT Computational time: Method Real time [s] Single-point calculation DFTB 0.14 DFTB-SCC 0.24 DFT-LDA DFT-GGA Molecular-dynamics simulation of a single H 2 O molecule (100 time steps of 0.3 fs) Method Real time [s] DFTB-SCC 1.44 DFT-LDA Dresden, Jan-Ole.Joswig@chemie.tu-dresden.de 34/34