Lecture on First-principles Computations (14): The Linear Combination of Atomic Orbitals (LCAO) Method
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1 Lecture on First-principles Computations (14): The Linear Combination of Atomic Orbitals (LCAO) Method 任新国 (Xinguo Ren) 中国科学技术大学量子信息重点实验室 Key Laboratory of Quantum Information, USTC Hefei,
2 Recall: the plane wave basis set V r = V r + R I Periodic crystal potential 2 2m + V r ψ nk r = ε nk r Ritz variational principle: expanding ψ nk r in terms of basis functions: ψ nk r = c μ,n k χ μk r μ H μν k c υn k = ε nk c υn k v Basis-type I -- Plane waves : χ μk r = χ G k r = 1 Ω μ G ei k+g r
3 Linear combination of atomic orbitals (LCAO) V r = V r + R I Periodic crystal potential 2 2m + V r ψ nk r = ε nk r Expanding ψ nk r in terms of basis functions: ψ nk r = c μ,n μ k χ μk r Basis-type II localized atom-centered orbitals: The LCAO method χ μk r = 1 N R χ a,j,l,m r R r a e ik R μ = a, j, l, m Atomic position within the unit cell
4 General features of local orbitals χ a,j,l,m r r a = R a,j,l r r a Y lm r r a Centered at atomic positions. The angular part Y lm r are spherical harmonics, while the radial functions R a,j,l r are flexible -- they can be analytical functions (Gaussians, Slater-type orbitals), or completely numerical. j denotes different radial functions with the same angular momentum, and hence controls the size of the basis set (from tight-binding to full quantitative calculations). Single-ζ ( SZ ): 1s1p for O, C, N, etc. (minimal basis) Double-ζ ( DZ ): 2s2p for O, C, N, etc. Double- ζ plus polarization function ( DZP ): 2s2p1d for O, C, N, etc. Compact in size (Efficient), and suitable for local symmetry analysis Constructing systematically converging atomic basis set is a highly nontrivial task, and needs special attention.
5 The Hamiltonian and overlap matrices Hψ nk r = ε nk ψ nk r ψ nk r = c μ,n k χ μk r μ H μν k c ν,n k = ε nk S μν k c ν,n k ν ν H μν k = χ μk H χ νk S μν k = χ μk χ νk
6 The Hamiltonian and overlap matrices in the LCAO method H μν k c ν,n k = ε nk S μν k c ν,n k ν ν H μν k = χ μk H χ νk, S μν k = χ μk χ νk Now χ μk r = χ a,j,l,m r R r a e ik R R H μν k = 1 N R H μν R e ik R, S μν k = 1 N R S μν R e ik R H μν R = χ a,j,l,m r r a S μν R = χ a,j,l,m r r a H χ a,j,l,m r r a R χ a,j,l,m r r a R
7 Two-center integrals: Multi-center integrals S μν R = χ a,j,l,m r r a T μν R = χ a,j,l,m r r a χ a,j,l,m r r a R 2 2m χ a,j,l,m r r a R Can be computed efficiently in Fourier space. Three-center integrals: V μν R = χ a,j,l,m r r a V χ a,j,l,m r r a R V r = a,r V a r r a R Three centers For numerical orbitals, this is often evaluated by grid integration.
8 Slater-type orbitals (STO) χ STO abc x, y, z = Nx a y b z c e ζr Normalization factor a, b, c control the angular momentum: l = a + b + c. ζcontrols the spread of the orbital. These basis functions don't have radial nodes, and not pure spherical harmonics. For 1s, it is hydrogen-like orbital. Long-range and short-range behaviors are correct. Hydrogen 1s orbital: ψ 1s r = 1 π ( 1 a 0 ) 3/2 e r/a 0 a 0 : Bohr radius
9 Gaussian-type orbitals (GTO) χ GTO lmn x, y, z = Nx l y m z n e ζr2 Normalization factor N = 2l 1!! 2m 1!! 2n 1!! π3/2 (4α) l+m+n (2α) 3/2 l, m, n control the angular momentum. ζ controls the width of the orbital. They are not hydrogen-like orbitals for 1s. Long-range and short-range behaviors are incorrect. The Coulomb integrals involving GTOs can be easily computed
10 Comparison between STOs and GTOs φ STO r = e αr i a i e β ir 2
11 The computation integrals among GTOs Gaussian product theorem: χ GTO l A = N A x 1 m A y 1 n A z 1 A e ζ 1 r R A 2, χ GTO l B = N B x 2 m B y 2 n B z 2 B e ζ 2 r R B 2 r A = r R A = (x A, y A, z A ) e ζ 1 r R A 2 e ζ 2 r R B 2 = e ζ 1ζ 2 ζ 1 +ζ 2 R A R B 2 e ζ 1+ζ 2 r ζ 1R A +ζ 2 R B ζ 1 +ζ 2 χ A GTO χ B GTO = e ζ 1ζ 2 ζ 1 +ζ 2 R A R B 2 linear combination of Gaussians centered at ζ 1R A + ζ 2 R B ζ 1 + ζ 2 Multi-centered integrals can be reduced to linear combination of twocentered integrals, that can be computed analytically.
12 STOs, GTOs, and NAOs GTOs STOs NAOs Accuracy of the basis functions Number of basis functions at a given level of accuracy Number of integrals to be evaluated Difficulty of evaluating the integrals
13 Symmetry analysis for two-center integrals χ a,j,l,m χ a,j,l,m = d 3 rr a,j,l r r a Y lm r r a R a,j,l r r a R Y l m r r a R = S μ,ν δ m.m S ll m δ m.m l = s, p, d, f, m = σ, π, δ, D = R + r a r a z ssσ spσ ppσ ppπ pdπ
14 From tight-binding to full calculations J. M. Soler et al, J. Phys: Condens. Matter 14, 2745 (2002)
15 Spanish Initiative for Electronic Simulations with Thousands of Atoms
16 FHI-aims (
17 ABACUS (
18 Other computer codes based on NAOs OpenMX (Open source package Material explorer) Dmol (Bernard Delley, Swiss, all-electron) FPLO (Full-potential local orbital, all-electron, Dresden) ADF (Amsterdam, Slater-type orbitals ) Crystal (Gaussian orbitals for solids)
19 Quantum chemistry codes based on GTOs Gaussian (commercial code) ( NWChem (free code) ( Q-CHEM (commercial code) ( Turbomole (
20 -value project in materials science: reproducibility of density-functional theory calculations for solids. i a, b = 1.06V 0,i (Eb,i V E a,i (V)) 2 dv 0.12V 0,i 0.94V 0,i A set of 71 elemental solids. K. Lejaeghere, S. Cottenier, et al. Science 351, 1415 (2016). 71 Test accuracies of methods/codes (basis sets, pseudopotentials, relativisitic treatment etc. )
21 Comparing solid-state DFT codes, basis sets, and potentials Code Version Basis Electron treatment Δ-value WIEN2k 13.1 LAPW/APW+lo all-electron 0 mev/atom FHI-aims tier2 numerical orbitals all-electron (relativistic atomic_zora scalar) 0.2 mev/atom Exciting development version LAPW+xlo all-electron 0.2 mev/atom VASP plane waves PAW 2015 GW-ready (5.4) 0.3 mev/atom FHI-aims tier2 numerical orbitals all-electron (relativistic zora scalar 1e-12) 0.3 mev/atom Quantum ESPRESSO 5.1 plane waves SSSP Accuracy (link is external) (mixed NC/US/PAW potential library) 0.3 mev/atom Elk APW+lo all-electron 0.3 mev/atom ABINIT plane waves PAW JTH v1.0 (link is external) 0.4 mev/atom FLEUR 0.26 LAPW (+lo) all-electron 0.4 mev/atom Quantum ESPRESSO 5.1 plane waves SSSP Efficiency (link is external) (mixed NC/US/PAW potential library) 0.4 mev/atom CASTEP 9.0 plane waves OTFG CASTEP mev/atom
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