Recent advances in development of single-point orbital-free kinetic energy functionals

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PacifiChem 2010 p. 1/29 Recent advances in development of single-point orbital-free kinetic energy functionals Valentin V. Karasiev vkarasev@qtp.ufl.edu Quantum Theory Project, Departments of Physics and of Chemistry, University of Florida, Gainesville, FL 32611 Pacifichem 2010, December 15-20, 2010, Honolulu, Hawaii Support by the U.S. Dept. of Energy TMS program grant DE-SC0002139

PacifiChem 2010 p. 2/29 Goal: Forces on nuclei for MD. ab-initio wave function methods: Electronic Structure Methods Survey - Ĥ system of SCF Eqns. {ψ i } E[{ψ i }, {R I }] F I = I E. Slow, scales as O(N 4 ). Kohn-Sham density functional theory (DFT) methods: Exc approx [n] system of KS Eqns. {φ i }, n T s [{φ i }] E[{φ i }, n, {R I }] F I = I E. The exact, explicit density functional for non-interacting kinetic energy (KE), T s, is not known orbitals are used. Slow, scales as O(N 4 ) (order-n methods are approximate). Orbital-free DFT (OF-DFT) methods: T approx s [n] single Euler-Lagrange Eqn. n E[n, {R I }] F I = I E. Relatively fast, may scale approximately as O(N). Classical; semi-empirical parametrical Classical: {R I } E[{R I }] F I = I E. Semi-empirical parametrical: {R I } parametric Ĥ {ψ i } E[{ψ i }, {R I }] F I = I E. Fast, but not universal, not reliable especially for states far from equilibrium.

PacifiChem 2010 p. 3/29 OF-DFT: Basics - Kohn-Sham noninteracting kinetic energy (KE) is minimum KE of one-determinant wave function for density n N e /2 T s [{φ i } N i=1 ] = 2 X Z φ i (r)( 1 2 2 )φ i (r)d 3 r Z t orb (r)d 3 r. i=1 The difference between interacting and noninteracting KE T c = Ψ ˆT Ψ T s is included in XC functional E xc [n]. Kohn-Sham DFT orbital dependent total energy functional: E KS-DFT [{φ i } N i=1, n] = T s[{φ i } N i=1 ] + E ne[n] + E H [n] + E xc [n] + E ion. In OF-DFT, T s [{φ i } N i=1 ] T s[n], and tot. energy becomes explicit functional of n: E OF-DFT [n] = T s [n] + E ne [n] + E H [n] + E xc [n] + E ion.

PacifiChem 2010 p. 4/29 OF-DFT: Basics - Electron density n(r) is the sole variable. There is a single Euler equation: δt s [n] δn(r) + v KS([n];r) = µ ; where v KS ([n];r) δ δn(r) «E ne [n] + E H [n] + E xc [n]. Computational cost of OF-DFT does not depend on N e, but scales as the spatial extent of the system. The Pauli term: KE may be decomposed as: T s [n] = T W [n] + T θ [n], T θ 0. The non-negativity of T θ is an important advantage of this decomposition. Euler eqn. takes the Schrödinger-like form: j 1 2 2 + v θ ([n];r) + v KS ([n];r)ff pn(r) = µ p n(r), where v θ ([n];r) = δt θ[n] δn(r), v θ(r) 0 is the Pauli potential.

PacifiChem 2010 p. 5/29 OF-DFT: Basics - A formal expression for v θ in terms of the KS orbitals: v θ ([n];r) = t θ([n];r) n(r) + NX (ε N ε i ) φ i(r) 2, n(r) i=1 where t θ ([n];r) = t orb (r) n(r)v W ([n];r); v W δt W [n] δn(r) = 1 8 h n 2 n 2 i 2 2 n. n Born-Oppenheimer (B-O) interatomic forces, F I = RI E OF-DFT [n], assume the form: F OF DFT I = E ion R I Z» n(r) v W ([n];r) + v θ ([n];r) + v KS ([n];r) d 3 r. R I Here there are three sources of error from approximate quantities: (i) approximate XC in v KS (ii) approximate v θ (Pauli potential) (iii) approximate density n(r). If we use converged KS density as input, then the major sorce of error would be approximate Pauli potential.

PacifiChem 2010 p. 6/29 Pauli term in the GGA form: T GGA θ [n] = Z t TF ([n];r)f θ (s(r))d 3 r, F θ (s) = F t (s) 5 3 s2, OF-DFT: Basics - where s n 2nk F. Few examples of the GGA KE functionals: The second-order gradient approximation (SGA). An ab-initio GGA proposed by Perdew (Phys. Lett. A 165, 79 (1992)). The PW91 functional (by Lacks and Gordon) conjoint to the PW91 exchange (J. Chem. Phys. 100, 4446 (1994)). The semi-empirical DePristo and Kress (DPK) functional: some parameters are fixed from the exact conditions and others by fitting to the KE of closed-shell atoms (Phys. Rev. A 35, 438 (1987)). Thakkar functional, F t is a heuristic combination of different GGA and with parameters fitted to the KE of 77 molecules (Phys. Rev. A 46, 6920 (1992)). The Tran and Wesolowski functional (PBE-TW) with parameters fitted to the KE of the He and Xe atoms (Int. J. Quantum Chem. 89, 441 (2002)).

Non-SCF results: F GGA θ and F GGA I - We use Kohn-Sham density as the input to the OF-DFT density functional and calculate corresponding energies and forces. The biggest source of error in these type of calculations is the error due to approximate Pauli functional, T θ. Pauli term enhancement factor F θ of various GGA KE functionals. Gradients of the total energy of the N 2 molecule. F θ 1.25 1 0.75 0.5 0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 1 0 1 2 SGA GGA Perdew PW91 DPK Thakkar PBE TW 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 s 2 de/dr, hartree/angstrom 2 3 4 5 6 7 exact SGA GGA Perdew PW91 DPK Thakkar PBE TW 8 0.6986 1.1986 1.6986 2.1986 R, angstrom PacifiChem 2010 p. 7/29

Non-SCF results: F GGA θ and F GGA I - GGA (=PBE-TW) Pauli potential (orange line) for the SiO molecule near Si atom. The GGA Pauli potential is negative and singular at the nucleus position. (a.u.) 300 200 100 0 100 10(4π( z R/2) 2 n) t θ PBE TW v θ PBE TW 100 F θ PBE TW 200 1.5 1 0.5 z (Å) Exact" KS Pauli potential (orange line) for the SiO molecule near Si atom. It is non-negative and finite at nucleus. a.u. 120 100 80 60 4π( z R/2) 2 n t θ KS v θ KS 10 F θ KS 40 20 0 1.5 1 0.5 z, angstrom PacifiChem 2010 p. 8/29

PacifiChem 2010 p. 9/29 OF-DFT: analysis of v GGA θ - SGA Pauli enhancement factor: Fθ SGA = 1 + a 2 s 2, with a 2 = 40/27. Near nucleus-cusp behavior defined by Kato s cusp condition: n(r) exp( 2Z r ), vθ GGA becomes: vθ GGA (r 0) = 3Z 5r F GGA θ (s 2 ) + nonsingular terms. At small s 2, F GGA θ 1 + as 2 and we obtain: vθ GGA (r 0) = 3aZ 5r + nonsingular terms. Important results: the GGA Pauli potential is (i) Singular at the nuclear position (ii) The singularity in published GGAs is negative, i.e. it violates the constraint. Solution: (i) It is possible to satisfy the non-negativeness constraint by changing the sign of the F GGA θ (s 2 ) derivative near nuclear positions ( at s 2 0.14). (ii) But positive GGAs have nuclear site divergences go beyond GGA.

PacifiChem 2010 p. 10/29 OF-DFT: modified conjoint GGA - We proposed modified conjoint GGA (mcgga) KE functionals [J. Comp.-Aided Mater. Des. 13, 111 (2006)] to remove the negative singularity of the Pauli potential. "Conjointness" hypothesis [Lee, Lee, Parr, 1991]: E x [n] = C x R n 4/3 F x (s)d 3 r, T s [n] = C TF R n 5/3 F t (s)d 3 r, F t (s) F x (s). Useful, not literally correct. Parameterization of KE functional using energetic objective function, ω E = P m i=1 EKS i Ei OF-DFT 2, for a set of nucleai arrangement of same system(s) failed. E KS The following E criterion was used: ω E = P M,i where E M,i = E M,i E M,e is the energy difference w.r.t equilibrium nuclear configuration. It is equivalent to parameterization to forces. M,i EOF-DFT M,i 2 Fitted functionals, modified conjoint GGA, reproduce qualitatively correct total energy gradients ( correct interatomic forces). mcgga Pauli potential is non-negative (satisfies the important constraint).

PacifiChem 2010 p. 11/29 OF-DFT: modified conjoint GGA - The mcgga functional enhancement factors (parameterized to forces): FmcGGA θ (s 2 > 0 ) near nuclear positions (at the nucleus s 0.38, s 2 0.14) The Pauli potential near the nuclei position is positive. Ft PBEν (s) = 1 + P ν 1 i=1 C i 1.25 h s 2 1+a 1 s 2 i i, ν = 2,3,4. 1 0.75 0.5 GGA PBE2 PBE3 PBE4 0.25. 0 0 0.05 0.1 0.15 0.2 0.25 s 2

PacifiChem 2010 p. 12/29 Non-SCF results: mcgga - Left panel: Gradients of the total energy of the N 2 molecule for the PBE2 modified conjoint GGA potential, compared with results from the PBE-TW potential and the exact value. Right panel: Electronic density (scaled by the factor (4π( z R/2) 2, with R the internuclear distance), Pauli term, Pauli potential, and enhancement factor F θ.

OF-DFT: reduced derivative approximation - Both GGA and mcgga Pauli potentials are singular at nuclei. To remove singularities one must go beyond the GGA. Based on 4th order gradient expansion for the KE: h t (4) 8 θ ([n];r) = t TF([n];r) p 2 9 81 8 s2 p + 1 3 s4 i, where s n, p 2nk F (RDD) variable : 2 n (2k F ) 2, we introduce the reduced density derivative n κ 4 = a 4 s 4 + b 2 p 2 + c 21 s 2 p, For n exp( 2Zr) the near-nucleus behavior of the 4th order potential is:» v (4) θ (r) 16Z2 3r 2 (5b 2 + 3c 21 ) + 32Z3 9r (18a 4 + 17b 2 + 18c 21 ) + nonsingular terms. and define coefficients in κ 4 from requirement of cancelation of singular terms κ 4 = s 4 + 18 13 p2 30 13 s2 p. PacifiChem 2010 p. 13/29

PacifiChem 2010 p. 14/29 OF-DFT: reduced derivative approximation - However this cancelation of singularities is not "universal". Only holds for densities with exponential behavior up to 4th term in Taylor series n(r) 1 2Zr + 2Z 2 r 2 (4/3)Z 3 r 3 Observation: κ 4 O( 4 ) suggests that the effective or operational order of in such a candidate variable should be reduced to second order. Another RDD variable is defined as κ 4 = p a 4 s 4 + b 2 p 2 + c 21 s 2 p for density n(r) 1 + C 1 r + C 2 r 2 + C 3 r 3, the corresponding Pauli potential is v θ (r) c 21 b2 1 r + nonsingular terms. With the choice c 21 = 0 the cancelation of singularities would be universal (for arbitrary C i ), hence finally we define the 4 th order RDD variable as κ 4 = p s 4 + b 2 p 2, b 2 > 0.

PacifiChem 2010 p. 15/29 OF-DFT: reduced derivative approximation - The fourth order, κ 4 reduced density derivative for different values of b 2 along the internuclear axis z for the SiO diatomic molecule near the Si atom: Si at (0, 0, 0.963)Å, O at (0, 0,+0.963)Å. Variables s and p, are shown for comparison....

PacifiChem 2010 p. 16/29 OF-DFT: reduced derivative approximation - In parallel with definition of κ 4 we introduce a 2 d order RDD κ 2 = s 2 + b 1 p, Thus, we define a class of approx. KE functionals, the reduced derivative approximation (RDA) functionals: Z Ts RDA [n] T W [n] + t TF ([n];r)f θ (κ 2 (r), κ 4 (r)) d 3 r, Parameterization of a RDA functional: F m0 θ ({κ}) = A 0 + A 1 κ 4a «2 + A 2 κ 4b «4 + A 3 κ 2c 1 + β 1 κ 4a 1 + β 2 κ 4b 1 + β 3 κ 2c «. This form has properties: (i) the corresponding v θ is finite (ii) the divergences of κ 4a, κ 4b and κ 2 near the nucleus cancel out. We used the usual energy fitting criterion to minimize ω E = P m i=1 E i KS Ei OF-DFT 2, for M = {H 6 Si 2 O 7,H 4 SiO 4,Be,Ne}, with a set of six Si O bond lengths for each molecule.

Non-SCF results: RDA for molecules - Table 1: KS kinetic energy T s, and differences (T OF DFT s T KS s ). All in Hartrees. KS Thakkar MGGA RDA H 2 1.080 0.022 0.103 0.036 LiH 7.784 0.021 0.296 0.161 H 2 O 75.502 0.285 0.318 0.188 HF 99.390 0.353 0.329 0.171 N 2 108.062 0.340 0.300 0.182 LiF 106.183 0.261 0.566 0.175 CO 111.832 0.333 0.300 0.179 BF 123.117 0.273 0.456 0.080 NaF 260.097 0.348 1.295 0.711 SiH 4 290.282 0.084 3.112 0.574 SiO 362.441 0.262 2.825 0.370 H 4 SiO 364.672 0.163 3.338 0.421 H 4 SiO 4 587.801 0.860 4.133 0.081 H 6 Si 2 O 7 1100.227 1.408 7.968 0.064 MAE a 0.358 1.810 0.242 PacifiChem 2010 p. 17/29

PacifiChem 2010 p. 18/29 Non-SCF results: RDA for molecules - Table 2: Energy gradient (Hartree/Å) calculated at extremum point R m R m, Å KS Thakkar MGGA RDA H 2 1.2671 0.164 0.029 0.112 0.017 LiH 2.455 0.046 0.016 0.037 0.028 H 2 O(1R) 1.3714 0.216 0.050 0.073 0.261 H 2 O(2R) 1.3714 0.416 0.127 0.163 0.454 HF 1.3334 0.232 0.071 0.003 0.192 N 2 1.3986 0.576 0.349 0.819 0.212 LiF 2.0405 0.079 0.019 0.032 0.041 CO 1.4318 0.474 0.248 0.659 0.472 BF 1.6687 0.207 0.037 0.118 0.244 NaF 2.4284 0.067 0.007 1.169 0.157 SiH 4 1.9974 0.447 0.102 0.189 0.088 SiO 1.9261 0.278 0.098 0.281 0.125 H 4 SiO 2.057 0.162 0.027 0.086 0.113 H 4 SiO 4 2.037 0.712 0.278 0.714 0.600 H 6 Si 2 O 7 2.010 0.194 0.022 0.173 0.117

PacifiChem 2010 p. 19/29 All-electron SCF results: Atoms - Table 3: OF-DFT self-consistent x-only total energies of atoms obtained from various kinetic energy functionals (in Hartrees). 1/9 vw+tf 1/5 vw+tf vw+tf GGA mcgga KS H -0.6664-0.5666-0.2618-0.719-0.321-0.4065 He -3.2228-2.8184-1.4775-3.39-1.53-2.7236 Li -8.2515-7.3227-4.1054-8.58-4.10-7.1749 Be -16.1631-14.4841-8.4922-16.69-8.36-14.2233 B -27.2876-24.6284-14.9259-28.05-14.59-24.5275 C -41.9053-38.0332-23.6569-42.94-22.98-37.6863 N -60.2623-54.9429-34.9084-59.53-34.07-54.3977 O -82.5799-75.5765-48.8832-84.26-47.53-74.8076 F -109.0594-100.13xx -65.7676-111.11-63.76-99.4072 Ne -139.8867-128.8015-85.7344-142.34-82.95-127.4907

PacifiChem 2010 p. 20/29 All-electron SCF results: SiO - 400 300 KS mcgga GGA TF+vW TF+vW/9 200 v θ (a.u.) 100 Figure 1: Pauli potential v θ obtained from selfconsistent all-electron 0 Kohn-Sham and OFDFT calculations for SiO 100 with mcgga (PBE2), 1.3 1.2 1.1 1 0.9 0.8GGA(PBETW) and TF z (angstrom) kinetic energy functionals.

PacifiChem 2010 p. 21/29 All-electron SCF results: SiO - 3000 KS mcgga GGA vw+tf TF+vW/9 2000 n 1000 Figure 2: All-electron selfconsistent Kohn-Sham and OFDFT electron density profiles calculated for point on the 0 internuclear axis for the SiO 1.2 1.1 1 0.9molecule; Si at (0,0,-1.05), O z (angstrom) at (0,0,+1.05).

PacifiChem 2010 p. 22/29 All-electron SCF results: SiO - 4πr 2 n 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 KS mcgga GGA TF+vW TF+vW/9 0 3 2.5 2 1.5 z (angstrom) Figure 3: All-electron selfconsistent Kohn-Sham and OFDFT electron densities scaled by the factor 4π( z R/2) 2.

PacifiChem 2010 p. 23/29 All-electron SCF results: SiO - 1 0.5 E tot (Hartrees) 0 KS TF+1/9vW TF+vW GGA Figure 4: Total energy of the mcgga SiO molecule as a function of 0.5 bond length obtained from self- all-electron Kohn- 0.5 1 1.5 2 2.5 3 3.5consistent R (angstrom) Sham and OFDFT calculations.

PacifiChem 2010 p. 24/29 OF-DFT results: GGA and mcgga F θ - F θ 1.25 1 0.75 0.5 0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 TF+vW/9 TF+vW GGA mcgga Figure 5: Pauli term enhancement factors F θ of kinetic en- 0 0.25 0.5 0.75 1 1.25 1.5 1.75 functionals as a function of 2ergy s 2 s 2.

PacifiChem 2010 p. 25/29 1 Local pseudo-potentials for OF-DFT - 0 v, Ry 1 2 3 n NLPP n LPP v spd1 v s v p v d 4 0.01 0.1 1 10 r, Bohr Figure 6: Real space potentials for Li: local v spd1, and different l-components obtained with Troullier-Martins (TM) pseudopotential generation procedure. 4 3 v spd1 v mod1 v (ev) 2 1 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 q (angtsrom 1 ) Figure 7: Reciprocal space local potentials for Li: v spd1 (c 0 = 0.69, c 1 = 0.24, c 2 = 0.1) and v mod1 (A = 0.45499, r c = 2.2261, g c = 2.86).

PacifiChem 2010 p. 26/29 Pseudo-potential OF-DFT results - Energy per atom (ev) 0.2 0.15 0.1 0.05 TF TF+vW/9 TF+vW GGA mcgga KS (Vasp) KS (Siesta) 0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4 4.1 Lattice constant (angstom) lattice constant for bulk bcc-li. Figure 8: Energy per atom vs.

PacifiChem 2010 p. 27/29 Pseudo-potential OF-DFT results - Energy per atom (ev) 2 1.5 1 0.5 TF TF+vW/9 TF+vW GGA mcgga KS (Vasp) 0 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 Lattice constant (angstom) Figure 9: Energy per atom vs. lattice constant for bulk Al.

PacifiChem 2010 p. 28/29 SUMMARY REMARKS - Negative singularity of the GGA Pauli potential is responsible for the poor performance of the GGA KE functionals (no attraction predicted). mcgga Pauli potential is positive. The functional gives better results (attraction region), but it still is singular at nucleus kinetic energy is overestimated. To eliminate the singularity of the Pauli potential at nucleus requires going beyond the GGA. New variable, reduced density derivative (RDD) provides regular behavior of the Pauli potential at nuclear positions. RDA is a promising approximation for development functionals capable to predict correctly both, energy and energy gradients (forces). References: Phys. Rev. B 80, 245120 (2009) V.V. Karasiev, R.S. Jones, S.B. Trickey, and F.E. Harris, in New Developments in Quantum Chemistry, J.L. Paz and A.J. Hernández eds. (Research Signposts) (2010); J. Comp.-Aided Mater. Des. 13, 111 (2006).

Thanks to - Laboratory of Theoretical Physical Chemistry, Center of Chemistry, Venezuelan Institute for Scientific Research (IVIC), Caracas, VENEZUELA Collaborators: Prof. E. V. Ludeña Quantum Theory Project, University of Florida. Collaborators: Prof. S. B. Trickey and Prof. F.E. Harris PacifiChem 2010 p. 29/29

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