Pseudopotentials for hybrid density functionals and SCAN
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1 Pseudopotentials for hybrid density functionals and SCAN Jing Yang, Liang Z. Tan, Julian Gebhardt, and Andrew M. Rappe Department of Chemistry University of Pennsylvania
2 Why do we need pseudopotentials? N e - n e - Valence electrons - Dominate bonding - Smooth exponential decay Core electrons - Insensitive to chemical environment - Oscillation near nucleus Pseudopotential Replace core region with a finite shallow potential Reduce the computational cost Preserve the accuracy Frozen-core approximation - Combine core electron and nucleus to effective core (pseudo ion) - Explicitly keep valence electrons (N > n) ψ ps ψ AE V ps -Z eff /r 2
3 Pseudopotential developments Hamann-Schlüter-Chiang norm conserving pseudopotential (1979) Generalized norm conserving pseudopotential (1989) RRKJ optimized pseudopotential (1990) Troullier-Martine soft pseudopotential (1990) Ultrasoft pseudopotential (1990) Projector-augmented waves potentials (1994) Links to pseudopotential databases
4 Density functional calculation vs. pseudopotential development Unoccupied {Φ i } ε x " n n n Generalized RPA Hybrid-GGA Meta-GGA GGA LSD Pseudopotentials still at GGA level Hartree World J. Chem. Phys. 123, (2005) 4
5 Hybrid density functional pseudopotentials The exchange-correlation functional must agree between pseudopotential and plane-wave calculation Non self-consistent PBE0 pseudopotential did not show improved accuracy [Phys. Rev. B 80, (2009)] First self-consistent construction of PBE0 pseudopotentials 5
6 Hartree-Fock formalism The nonlocal Hamiltonian ˆT + ˆVion + ˆV H [{ n 0 l 0 }]+ nl ˆV x [{ n 0 l 0 }] nl(r) = nl nl (r) Hartree potential ˆV H [{ n0 l 0 }] nl i = X n 0l 0 Z d 3 r 0 n 0 l 0 (r 0 ) 2 r r 0 nl(r) 6
7 Hartree-Fock formalism The nonlocal Hamiltonian ˆT + ˆVion + ˆV H [{ n 0 l 0 }]+ nl ˆV x [{ n 0 l 0 }] nl(r) = nl nl (r) Exchange potential operator ˆV x nl [{ n0 l 0 }] nl i = X Z n 0 l 0 d 3 r 0 n 0 l 0 (r 0 ) n 0 l 0 (r) r r 0 nl(r 0 ) 7
8 Hartree-Fock formalism The nonlocal Hamiltonian ˆT + ˆVion + ˆV H [{ n 0 l 0 }]+ nl ˆV x [{ n 0 l 0 }] nl(r) = nl nl (r) Hartree and exchange potentials are orbital dependent Treated as spherical average over all orbital configurations via Slater integral X E av = 1 N 2S+1 L J E[{ }2S+1 L J ] E.g., for C atom, valence configurations N=15 with 5 allowed term symbols: 3 P 2 (5), 3 P 1 (3), 3 P 0 (1), 1 D 2 (5), 1 S 0 (1) E av = 1 15 [5E{ }3 P 2 +3E{ }3 P 1 + E{ }3 P 0 +5E{ }1 D 2 + E{ }1 S 0 ] 8
9 Hartree-Fock formalism The nonlocal Hamiltonian ˆT + ˆVion + ˆV H [{ n 0 l 0 }]+ nl ˆV x [{ n 0 l 0 }] nl(r) = nl nl (r) Slater integral is used to obtain the average energy of Hartree and exchange terms E av = I[{ nl (r)}]+f [{ nl (r)}]+g[{ nl (r)}] [Slater integral J. C. Slater (1960)] I[{φ nl (r)}], F[{φ nl (r)}], G[{φ nl (r)}] are slater integrals containing radial wavefunctions with different n,l quantum numbers 9
10 Hybrid density functional formalism PBE0 formalism Mix the HF exchange with PBE exchange E PBE0 xc Combine HF formalism and Kohn-Sham equation apple ˆT + ˆV ion + ˆV H [{ n 0 l 0 }]+ 1 4 = aex HF +(1 a)ex PBE nl ˆV x [{ n 0 l 0 }] E PBE c a =0.25 ˆV PBE x Invert the radial equation to obtain a screened pseudopotential + [JCP 105, 9982 (1996)] ˆV PBE c nl (r) = nl nl (r) ˆV nl scr(r) = nl (r) l(l + 1) 2r val (r) d 2 dr 2 [ val(r)] 10
11 Hybrid functional all-electron solver Solve apple ˆT + ˆV ion + ˆV H [{ n 0 l 0}]+1 4 nl ˆV x [{ n 0 l 0}]+3 4 ˆV PBE x + ˆV PBE c nl (r) = nl nl (r) iteratively Start with an initial guess for the wavefunctions ψ (e.g. hydrogenic wavefunctions) Construct Hartree, exact-exchange potential, and DFT exchange-correlation potentials V H, V X, V XC ψ ps Solve the modified HF differential equations to obtain updated wavefunctions ψ, and eigenvalues ε ψ AE V ps -Z eff /r Orthogonalize the wavefunctions ψ 11
12 Pseudopotential construction Constructing the pseudowavefunctions using the RRKJ method Use spherical Bessel functions to construct Pseudowavefunctions [Phys. Rev. B 44, (1990)] ψ ps ψ AE Inverting the Schrödinger equation to obtain V PSP V ps -Z eff /r Design non-local transformation on V PSP [Phys. Rev. B 59, 1247 (1999)] Descreening to obtain V ion 12
![B 44, 13175 (1990)] ψ ps ψ AE Inverting the Schrödinger equation to obtain V PSP V ps -Z eff /r Design](/docs-images/83/88162542/images/13-1.jpg "non-local transformation on V PSP [Phys. Rev.")
13 Pseudopotential construction Constructing the pseudowavefunctions using the RRKJ method Use spherical Bessel functions to construct Pseudowavefunctions [Phys. Rev. B 44, (1990)] ψ ps ψ AE Inverting the Schrödinger equation to obtain V PSP V ps -Z eff /r Design non-local transformation on V PSP [Phys. Rev. B 59, 1247 (1999)] V PS ion = V PS l (r) Y [{ val }](r) 1 4 X[{ val }](r) l(r) 3 4 V x PBE (r) Vc PBE (r) 13
![07 10 7.07 10 [Configs] 2 # 200 2.00-210 5.](/docs-images/83/88162542/images/14-1.jpg "20 - # 200 2.00-210 5.")
14 Use of OPIUM Only one input file: *.param How to run?./opium f f.log ae ps nl ke rpt upf plot vi [Atom] F [Pseudo] opt [Optinfo] [Configs] 2 # # [XC] hy Atomic reference configuration Pseudopotential radial cut off Pseudowavefunction settings Configurations for transferability test Exchangecorrelation method opium.sourceforge.net/index.html 14
15 PBE0 pseudopotential performance bond lengths Relative Error (%) 2 1 PBE0@PBE PBE0 Relative Error (%) 0 H 2 LiH FH Li 2 LiF CO N 0 2 F 2 BeH CH NH OH CN NO O PBE0@PBE PBE0 err% = l l AE l AE 100% Closed shell Open shell Overall small (but systematic) improvements for bond lengths without further tuning the potential parameters Influence tends to be large in open-shell systems All-electron (AE) calculations are performed using FHI-aims 15
16 PBE0 pseudopotential performance HOMO-LUMO gaps Relative Error (%) H 2 LiH FH Li 2 LiF CO N 2 F 2 Closed shell %err = PBE0@PBE PBE0 AE AE 100 Relative Error (%) PBE0@PBE PBE0 0 BeH CH NH OH NO O 2 Open shell The error in HOMO-LUMO gaps is reduced using PBE0 pseudopotentials by 2% on average 16
17 PBE0 pseudopotential performance bond formation energies Relative error (%) E AE err% = EPW E AE 100% PBE0@PBE PBE0-20 H 2 LiH BeH CH NH OH FH Li 2 LiF CN CO N 2 NO O 2 F 2 Formation energies are improved by up to 4 % using PBE0 pseudopotentials 17
18 PBE0 pseudopotential performance - solids Crystal PBE-PBE0 (err%) PBE0 (err%) All Electron Calc. Lattice constant (Å) PBE0 Si (0.073) (-0.037) GaN (0.066) (0.022) Band Gap (ev) Si 1.79 (9.82) 1.78 (9.20) 1.63 GaN 3.58 (1.13) 3.56 (0.56) 3.54 Lattice constants and band gaps are improved for solids 18
19 Implementation of SCAN SCAN (Strongly constrained and Appropriately Normed) density functional is a meta-gga functional that obeys all currently known constraints Achieves remarkable accuracy for lattice constants of simple solids and weak interactions The computational time is similar to GGA Implemented in FHI-AIMS, VASP, (and pw.x) Phys. Rev. Lett. 115, (2015) 19
20 Implementation of SCAN Start with an initial guess for the wavefunctions ψ (e.g. hydogenic wavefunctions) Initialize the spin-dependent charge density by ρ i = % " ψ i 2 Construct Hartree, and SCAN exchange-correlation potentials V H, V XC Solve spin-dependent Kohn-Sham equation to obtain updated wavefunctions ψ and eigenvalues ε Implement SCAN exchange-correlation Develop a new solver to take care of the spinpolarization Integrate the wavefunctions ψ to obtain charge density ρ 20
21 Apply to real systems Convergence test G2 dataset with bond length and HOMO-LUMO gaps Solids lattice constants, bulk moduli and band gaps Comparison between all electron and plane-wave calculations 21
22 First stage benchmarking G2 data set bond lengths PBE PBEPBE0 BPE0 AE-PBE AE-PBE0 AE-SCAN H LiH BeH CH NH OH FH Li LiF CN CO N NO O F MARE PBE0 pseudopotential plane-wave calculations still show smaller errors on bond lengths using SCAN all-electron data as a reference. 22
23 Conclusion First implementation of fully sefl-consistent PBE0 pseudopotentials Open source OPIUM v4.0 will be published soon on sourceforge ( Systematic improvement of accuracy for PBE0 calculations on molecules and solids Including relativity for spin-orbit coupling studies To implement SCAN functional in OPIUM 23
24 Acknowledgement Dr. Andrew M. Rappe Dr. Liang Z. Tan Dr. Julian Gebhardt Department of Chemistry University of Pennsylvania 24
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