New Approaches to an Old Idea - Orbital-free Density Functional Theory

Transcription

1 New Approaches to an Old Idea - Orbital-free Density Functional Theory Sam Trickey Quantum Theory Project Physics, Chemistry - University of Florida trickey@qtp.ufl.edu Sept. 2010

2 A Bit of DFT History The seminal role of J.C. Slater should not be forgotten. X was the first parameterized E xc approximation. It had some exceedingly zealous advocates. Much of the progress (conceptual, numerical) on early DFT arose from Slater (average exchange, APW, X, spin orbitals) And there was much confusion over the entanglement of X and the multiple-scattering basis and muffin-tin potential.

3 Big Systems and Simulations Core Issues Molecular dynamics is the dominant simulation tool in materials physics and much of computational biomolecular research. MD is Newton s 2nd Principle: m R V( R, R,, R ) I I I 1 2 N The potential should be the Born-Oppenheimer energy surface V R E0 R E Nuc Nuc R E 0 ({R}) is the ground state electronic energy. V({R}) often is represented by an empirically calibrated force field. Clearly inadequate for predictive treatment of bond formation, re-arrangement, or breaking. Genuine B-O forces in the MD QM QM $ $ $ $ $ or Multiscale is the solution? (artificial boundaries?) Go to bigger machines! (Exascale solves everything?) Section of 102 Gold atoms surrounded by 44 p-mercaptabenzoic acid groups Kornberg et al., Science 318, 430(2007) Mn 25 single-molecule magnet; Mn II yellow, III is blue, IV is olive Stamatos et al., Polyhedron 26, 2095 (2007)

4 A Viewpoint Common in Materials Physics and Quantum Chemistry 4

5 Everyone knows Amdahl s Law but quickly forgets it T.Puzak, IBM (2007) 1 S P (1 P) N S = speedup; P fraction parallel; N number of processors 99.9 % parallel, 2048 processors, speedup 675 IEE Computer, 41, 33 (2008) Also see

6 B-O Forces from DFT So there s both urgency and opportunity for better methods. The KS eigenvalue form of the DFT ground state energy illustrates the computational barrier to using ordinary KS DFT for generating B-O forces at every MD step: 1 n( r) n( r) E ( R ) drdr E [ n] dr[ n( r) v ( r)] 0 n( r) n ( r) k k k k k xc xc 2 r r 2 ˆKS ˆ 1 h h v v v v v 2 2 k ( r) k k ( r); KS ( r); KS ( r) ext ( r) H ( r) xc ( r) This eigenvalue approach is slow. There are order-n approximate methods but they introduce additional assumptions (e.g. about basis locality, etc.). But in DFT, E[n] is fundamental, E[φ K-S [n]] is not. Challenge: to get the content of KS DFT without doing the KS eigenvalue problem, yet accurately enough to do physics and chemistry studies on realistically available computer clusters.

7 Another KS Orbital Problem (Jacob s Ladder) Biblical (Genesis 28: 11-19) DFT XC approximations (John Perdew) Heaven of chemical accuracy unocc. {φ i } x and/or 2 n n n Hartree world generalized RPA, etc. hyper-gga meta-gga GGA LSDA Landschaft mit der Darstellung von Jakobs Traum: Die Engelsleiter, Michael Willmann, c Also African-American Spiritual (folk hymn) Red = explicit orbital dependence J. P. Perdew and K. Schmidt, in Density Functional Theory and its Application to Materials, V. Van Doren, C. Van Alsenoy, and P. Geerlings, Eds., AIP, Melville, New York, 2001 Everybody is working on orbital-dependent XC functionals!

8 Affordable QM? - Density Functional Theory Without Orbitals? But in DFT, E[n] is fundamental, E[φ K-S [n]] is clever, highly insightful, but not fundamental. Reiterated Challenge: to get the content of KS DFT without doing the KS eigenvalue problem, yet accurately enough to do physics and chemistry studies on realistically available computer clusters. The ideal: orbital-free DFT The Beer Store

9 Orbital-free DFT: How Hard Could That Be?

10 The Ideal - Orbital-free DFT Were we to have the K-S KE density kernel, and a good density-dependent (NOT orbitally dependent) E XC, then DFT B-O forces would be simple: E[ n] T [ n] E [ n] E [ n] E [ n] T [ n] dr t [ n( r)] KS KE without KS orbitals s s S ee xc Ne Ts E[ n] d n( ) 0 VKS[ n( )] n r r r n VKS [ n( r)] Eee[ n] E xc[ n] E Ne[ n] n OF DFT F E [ n] E dr n( r) v ( r) I I I NN I Ne Ts dr VKS [ n( r)] I n(r) n Is this goal realistic? DFT in Hydrodynamic form

11 Orbital-free DFT Sobering History of a Long-time Ideal The pursuit of OF-DFT has a long history of big names and big difficulties - Thomas-Fermi-Dirac: T [ n] c dr n r, TF 5/3 3 (3 ) 10 TF TF c Teller s non-binding theorem (1962) Thomas-Fermi-von Weizäcker with scaling: n W TTFW [ n] T TF W TW TTF dr 8 n r 0 1 W When combined with E ne [n] + E ee [n] + E nn : binds neutrals, has negative ions. 2 2 / 3 2 L.H. Thomas, Sanibel, 1975.

12 Orbital-free KE Problems with Simple Approaches Observe: Scaled Thomas-Fermi-von Weiszäcker used in OF-DFT and Treated variationally, or even with the actual KS n 0 (for a given XC) as input, can lead to weirdness: E [ n] TTF W[ n] TF W E [ n] E [ n] E [ n] ee E [ n ] E [ n ] TF W 0, TF W KS 0, KS xc ne Functional (all simple LDA XC) E total (Hartree) TFW, λ W = TFW, λ W = 1/ TFW, λ W = 1/ KS (13s8p GTO) TFW from G. Chan, A. Cohen, and N. Handy, J. Chem. Phys. 114, 631 (2001) Paul Ayers (Sanibel 2007): An ab initio quantum chemist will wonder- Is an N-representability constraint missing? If so, should we surrender? N-representability problems are very difficult.

13 Orbital-free KE: Fermion N-representability (for guidance) Def n : A KS kinetic energy functional T s [n] is N-representable iff for each proper density n, there exists a proper N-fermion state (or ensemble) which has the same KS kinetic energy for that density. Remark there are infinitely many N-fermion states associated with each proper n. Recall the constrained search definition of the K-S KE: 2 * min,, j,, N T n r r r r dr d dr d s 1 1 N N N N 1 1 N N n j Implications: 1. An approximate KE functional which delivers a value below the K-S KE T s for any system is NOT N-representable. 2. A non-n-representable KE functional will give a value below the K-S KE for at least one system. Observation the risks of non-n-representability are serious. Credit: Paul Ayers

14 Orbital-free KE: TFvW anyway Nevertheless, calculations with Thomas-Fermi-von Weiszäcker in OF-DFT continue to appear. Some very clever finite element techniques for NONperiodic systems and some interesting results for electronic structure of defects in Aluminum are in, for example, V. Gavini et al. [J. Mech. Phys. Solids 55, (2007)]

15 Some Prior Orbital-free KE Approaches Several groups, e.g. Madden et al., Carter et al., Teeter, Alvarellos et al., and others, have focused on T s [n] before us, but with a different development strategy OF-KE approximations. Use Carter et al. as an example of much of the literature. [Y.A. Wang and E.A. Carter in Theoretical Methods in Condensed Phase Chemistry, S.D. Schwartz ed. (Kluwer, NY 2000) p. 117] They develop their KE density models in the context of linear response on grounds that 1. In atoms and molecules, shell structure is the barometer of a good OF-KE 2. short range [density] oscillations and Friedel oscillations are the corresponding physical standard in solid state physics, and 3. correct linear response behavior is the key to predicting such oscillations.

16 Orbital-free KE Approaches Response Function Here, in somewhat sloppy translationally invariant notation, is the basic theme of the response function approach n(r) dr (r - r ) V (r ) n(q) (q) V (q) KS n(r) V (r KS ) VKS (r - r ) dr (q - q ) (q) (q,q ) V (r ) n(r ) n KS 2 Ts KS Ts V [ n(r)] (q - q ) (q) (q,q ) n n n KS Result is a set of non-local (two-point) approximations based related to Average Density Approximation, Weighted Density approximation, etc. Various versions work moderately well for metals. Another recent version works moderately well for insulators. The non-locality is too complicated (computationally costly, introduces local pseudo-potentials) for our purposes

17 Orbital-free KE Approaches Response Function FIG. 1. Static structure factors of liquid Al at 943 K. Full circles: experimental x-ray diffraction data ~14 and 15!. Open circles: experimental neutron diffraction data ~16!. Continuous line: OF-AIMD simulations. s T n T T vw 3 T d n k 10 5/32 2 r (r) (r) 0 2 1/3 F F F k (r) : 3 2k dr n (r ) w (2k r r ) k w (3 n ) 0.51 weight function from linear response Density-independent kernel González, González,, López, and Stott, J. Chem. Phys. 115, 2373 (2001) FIG. 3. Dynamic structure factor, S(q,v), for several q values, for liquid aluminum at T5943 K. Continuous line: OF-AIMD simulations. Full circles: experimental data ~Ref. 22!. Dashed line: best fit to the experimental data.

18 Orbital-free KE Approaches Response Function Density-dependent kernel for metals Wang, Govind, and Carter, Phys. Rev. B 60, (1999), Erratum, Phys. Rev. B 64, (2001). The surface actually is a slab and the axis should be atomic units not arb units. T n T T T s TF vw,, T c0 d d n n w 2 1/3 r r (r) (r ) (r-r ), r-r kf(r) kf(r) (r-r ) : 2 k F (3 n) 1/ 5, 5 /

19 Orbital-free KE Approaches Response Function Density-dependent kernel for semiconductors C. Huang and E.A. Carter, Phys. Rev. B 81, (2010) T n T T T s TF vw, r r (r) (r ) r,r, T c0 d d n n w w w k s 2 (r,r ) : F (r)(1 ) r-r n 2 1/3 k F (3 n) s 4/3 n

20 Two-point OFKE Functional Problems Density-independent kernels suffer from nonlinear instability in the sense that the corresponding kinetic-energy functionals are not bounded from below. X. Blanc and E. Cancès, J. Chem. Phys. 122, (2005) Back to the N- representability issue mentioned earlier. Introducing density-dependence in the kernel comes at the expense of greatly complicating the derivation and making a straightforward numerical implementation computationally expensive Y.A. Wang, N.Govind, and E.A.Carter, Phys. Rev. B 60, (1999) Different kernels for metals and semi-conductors: but what about pressureinduced metal-insulator transitions? Reference density is non-unique for bulk systems and undefined for finite ones. Multiple parameters without physical constraints.

21 Orbital-free Density Functional Theory: Strategy Our goal is a workable recipe for t s [n] purely for driving MD E[ n] T [ n] E [ n] E [ n] E [ n] s ee xc ne 1 * 2 Ts[ n] dr ts[ n(r)] ni dr i (r) i ( r) 2 Ts FI, electronic I E[ n] dr VKS [ n(r)] I n(r) dr n(r) IVne n Observe: we do NOT seek a KE density kernel that will do everything that is in the basic DFT theorems Desired: t s [n(r)] be no more complicated than GGA E XC (depends on gradient of the density) or meta-gga (depends on Laplacian of the density also). Assumption: continued progress on pure E XC approximations, i.e., not hybrids or OEP.

22 OF-DFT Functional Construction Key Ingredient - Pauli KE, Pauli potential, and square root of density [M. Levy and H. Ou-Yang, Phys. Rev. B 38, 625 (1988); A. Holas and N.H. March, Phys. Rev. A 44, 5521 (1991); E.V. Ludeña, V.V. Karasiev, R. López-Boada, E. Valderama, and J. Maldonado, J. Comp. Chem. 20, 155 (1999) and references in these] An exact expression is T [ n] T [ n] T [ n], T [ n] 0 s W Useful (but incorrect) clue: Conjointness conjecture [H. Lee, C. Lee, and R.G. Parr, Phys. Rev. A 44, 768 (1991)]. The GGA E X is GGA 5 / 3 Conjointness: T [ n] dr n (r) F s(r) s t (r) (r) F s F s x E n c d n F s GGA 4 / 3 x [ ] x r (r) x (r) s(r) = n 2(3 ) n t 2 1/ 3 4 / 3

23 OF-DFT: Test of existing KE functionals Tests of 6 existing functionals, three conjoint. PW91: Lacks and Gordon, J.Chem. Phys. 100, 4446 (1994) [conjoint] PBE-TW: Tran and Wesolowski, Internat. J. Quantum Chem. 89, 441 (2002) [conjoint] GGA-Perdew: Perdew, Phys. Lett. A 165, 79 (1992) [conjoint] DPK: DePristo and Kress, Phys. Rev. A 35, 438 (1987) Thakkar: Thakkar, Phys. Rev. A 46, 6920 (1992) SGA: Second order Gradient Approx. T S = T TF + (1/9) T W SiO Stretch; Total energy vs. Bond length All six T s approximations fail to bind! V (not shown) violates positivity. J. Comput. Aided Matl. Design 13, 111 (2006)

24 OF-DFT Functional Construction (continued) Write T W in GGA-like form and define a shifted enhancement factor (so as to separate the Pauli term) 5 5 T n c d n s F F s 3 3 Weakly inhomogeneous limit: 5/3 2 2 W [ ] 0 r (r) (r) ; t t (r) 40 Ft ( s(r)) 1 s s 0 27 Conjoint form Pauli KE and potential for GGA models: T n T n d t n c d n F s GGA GGA 5/3 [ ] [ ] r [ (r)] 0 r (r) t ( (r)) 2 5 2/3 5/3 Ft s 5 n s s 5/3 Ft s V c0n Ft c0n c0n s 2 3 s n 3 n n n s n 2 F t form but not parameters from F x

25 OF-DFT: Modified conjoint KE functionals Parameterizations Tried two simple forms for modified enhancement factors. These forms would be convenient for use in MD. No guarantee that any of these is optimal. N 1 2 PBE N s Ft ( s) 1 c j 2 j1 1 as N=2 is typical PBE-like form as also used by Tran & Weslowski N=3 is the form used by Adamo and Barone [J. Chem. Phys. 116, 5933 (2002)] N=4 highest tried j exp 4 t a2s F C (1 e ) C (1 e ) a s 1 2 Constrain parameters to v 0 Initial parameterization used (a) single SiO or (b) SiO, H 4 SiO 4, and H 6 Si 2 O 7. Stretched single Si-O bond in all cases. Used known KS density.

26 OF-DFT - Performance of Modified Conjoint, Positive-definite Functionals Single bond stretching gradient in H 2 O. OF-KE parameters from 3-member training set (SiO, H 4 SiO 4, and H 6 Si 2 O 7 ) except PBE2. NO information about H 2 O in the set. All the functionals give too large an equilibrium bond length.

27 OF-DFT - Pauli Potentials Compared Pauli potential v θ, Pauli KE energy density t θ, enhancement factor F θ SiO at R = Å. Exact potential is from the KS solution. PBE-TW is the Tran -Wesolowski potential. PBE2 is our modified conjoint-form (positivity enforced). For comparison the density also is displayed. PROBLEM: Pauli potential v θ singularities at the nuclei. TOO positive! Phys. Rev. B 80, (2009)

28 Interlude Solving the OFKE Euler equation with a KS code? T [ n] T [ n] T [ n], T [ n] 0 s W Stationarity of variation with respect to density n yields 1 V (r) V (r) n(r) n(r) 2 KS 2 V (r) T n 0 r Levy, Perdew, and Sahni, Phys. Rev. A 30, 2745 (1984): Chan, Cohen, and Handy J. Chem. Phys. 114, 631 (2001)

29 Interlude Solving the OFKE Euler equation with a KS code? TFDvW, λ W = 1 E total (Hartree) Chan, Cohen & Handy Numerical KS code GTO KS code H atom Li atom Ne atom However, with simple linear mixing of densities and starting from the pure von Weizsäcker KE, the iterative convergence is very slow and unpredictable. SUCCESS! For mcgga PBE2, the all-numerical OFKE Euler (density) equation has a minimum for SiO at almost the right bond length. (KS result is shifted up in energy by Hartree.). So far, we cannot find a minimum with the Tran- Wesolowski KE functional. CHALLENGE Again convergence is slow and doesn t go below 1 mhartree.

30 OF-DFT - Removing Pauli Potential Singularities Gradient Expansion for T (0) (2) (4) T n dr t [ n(r)] t [ n(r)] t [ n(r)] (0) 5/ (0) 2 t [ n(r)] c0n (r) 1 s 3 t0[ n(r)] 1 s t 3 0[ n(r)] F ( s ) (2) 5 2 (2) 2 t [ n(r)] t0[ n(r)] s t 27 0[ n(r)] F ( s ) t [ n(r)] t [ n(r)] p s p s (4) p : n 4(3 ) n 2 2 4/3 5/3 F F F 1 a s (0) (2) ( sga ) 2 2 F b p c s p a s (4) t n F s p (4) 2 0[ (r)] (, ) Instead of using the grad expansion values for the coefficients, write and choose the parameter values to eliminate the singularities at finite order

31 OF-DFT - Removing Pauli Potential Singularities Functional differentiation gives ( sga ) 2/3 5 2 v c0n 3 a2 s 2 p 2 nd order: only one constant go to 4 th order Near nucleus (Kato cusp condition): n(r) exp 2Zr (4) v ( r 0) (5b c21) (18a 4 17b2 18 c21) non sing. n(r) r r, constants 1 2 Non-singular if c 5b 13b ; a

32 OF-DFT - Removing Pauli Potential Singularities Define the fourth order reduced density derivative: : s p s p Then candidate enhancement factors are F a or F or F 1 C (4) (4) 4 (4), j (4) j 1 j 4 1 j 4 The first form is manifestly non-singular. The others are in the spirit of GGAs each such form must be examined for its behavior. But these also will exhibit singularities for second- and third-order non-hydrogenic densities. So define an effective second-order form F a s b p c s p (42) If c 21 =0, this is non-singular for a density with near-nuclear form n r 1 C r C r C r j s : n 2(3 ) n 2 1/3 4/3 p : n 4(3 ) n 2 2 4/3 5/3 j So use : s b p, b

33 OF-DFT - Removing Pauli Potential Singularities One candidate RDD enhancement factor studied (a little) so far i j k F A A A A m 0 4 a 4 b 2 c a 1 2 4b 1 3 2c The parameters A i, β i, and the a b c, in the κ themselves all are to be determined. Some exploration led to studying i=2, j=4, k=1. Fit to total energies for six bond lengths in H 6 Si 2 O 7, H 4 SiO 4, plus Be, Ne atoms,. H 2 O molecular total energy as function of one bond KS = reference calc (source of input density) DPK = DePristo-Kress OFKE functional reparameterized for this case MGGA = meta-gga functional of Perdew and Constantin Phys. Rev. B 75, (2007). Thakkar functional RDA = This work Phys. Rev. B 80, (2009)

34 But, what about avoiding the orbital rungs of the XC ladder? Everybody is working on orbitaldependent XC functionals! Heaven of chemical accuracy unocc. {φ i } x and/or 2 n n n Hartree world generalized RPA, etc. hyper-gga meta-gga GGA LSDA Red = explicit orbital dependence J. P. Perdew and K. Schmidt, in Density Functional Theory and its Application to Materials, V. Van Doren, C. Van Alsenoy, and P. Geerlings, Eds., AIP, Melville, New York, 2001 But the most popular non-empirical XC functional is the PBE GGA

35 PBE X and the Lieb-Oxford Bound The PBE exchange functional is E [ n] c n ( r) F ( s( r)) dr F PBE 4/3 x x xpbe xpbe 1 n : 1 ( ) : 1 s / 2(3 ) n ; s r ; c 2 2 1/3 4 /3 x : / 3 Lieb-Oxford bound as constraint on construction of functionals E. Lieb and S. Oxford, Int. J. Quantum Chem. 19, 427 (1981); J.P. Perdew in Electronic Struct. Of Solids 91, 11 (1991); G. Chan and N. Handy, Phys. Rev. A 59, 3075 (1999) E E E ; 2.275, LDA xc LO x LO LO, CH LDA x /3 4/3 d n r r ( ) Satisfaction of the L-O bound for all possible spin densities n (r), and all inhomogeneity values s(r) is imposed as a sufficient condition - LO Fx, PBE[ n, s] /3 2 lim F [ n, s] s x PBE

36 LO Bounds in Real Systems Natural systems appear to respect a much tighter value than either λ LO or λ CH. [credit: Odashima & Capelle, J. Chem. Phys. 127, (2007); also Internat. J. Quantum Chem. 108, 2428 (2008)] But simply tightening the bound in PBE doesn t help much, if any. See Odashima, Capelle, and Trickey, J. Chem. Th. Comput. 5, 798 (2009).

37 Vela-Medel-Trickey X Enhancement Factor (VMT) - Lieb-Oxford bound as in PBE or Odashima-Capelle 1.804, Large s limit homogeneous electron gas Non-empirical (constraint-based) F X ( s) 1 where GEA (PBEsol) or (PBE) and F X ( s ) / 2 MAX s 2 e s 1 s 1/3 2 2 s MAX F xpbe : s / : J. Chem. Phys. 130, (2009).

38 VMT X Enhancement Factors Two Flavors VMT(μ GE ) = VMTGE VMT(μ PBE ) = VMT

39 VMT and PBE X Enhancement Factors - Variation of α parameter in VMT to satisfy LO bound Fx s Fx s s 1.5 PBE 96 PBEsol (µ from GEA) VMT = orig. LO Fx s s 1 2 s

40 Comparison of X Enhancement Factors - Only PW 91 satisfies the large-s constraint lim s 1/2 F XC s M. Levy & J.P. Perdew, Phys. Rev. B 48, (1993)

41 VT{mn} Exchange VMT recovers HEG behavior for large-s while staying below the LO bound for all but one value of s s 2 s2 e F VMT X (s) 1 1 s 2 VT{mn} changes to satisfy the large-s constraint m/2 s n F ( s) F ( s) 1 e s 1 VT { mn} VMT /2 X X 2 s m/2 VT { mn} s e s n /2 FX ( s) 1 1 e s s m and n are fixed by: must be integers. 2 The leading term in the small-s expansion of F x must be quadratic. To preserve rotational invariance only even powers of s are allowed. The limit s0 must = 1 to recover HEG behavior. m n; m / 2 and ( m n) / 2 are even

42 VT{84} Exchange As with VMT, μ can be μ PBE VT{mn} or μ GEA VT{mn} sol For each μ, α is set from satisfying the local LO bound at the maximum of F X. In both cases, the quadratic coefficient in the correlation is set to recover the linear response of the HEG. For mn = 84 α PBE = α GEA = Why choose mn -= 84?

43 VT{84} and PW91 Exchange Enhancement F x VT{84(}μ PBE ) = VT{84} VT{84}(μ GEA ) = VT{84}sol PW91 s

44 VT{84} and VMT: Molecular Tests Test Sets: Raghavachari, Curtiss, Perdew, Scuseria, Truhlar, Hobza, Cheeseman G1 for atomization energies (fixed geometries). G3 for standard heats of formation at 298 K (geometry optimizations + harmonic analysis). Ionization potentials, electron and proton affinities with optimized geometries. Weak interactions. Hydrogen and non-hydrogen transfer barrier heights (fixed geometries). Transition metals (Not completed). Chemical shifts VT{mn} and VMT implemented in development versions of demon2k and NWChem.

45 VT{84} and VMT: Molecular Tests G3: Standard heats of formation at 298 K / kcal-mol molecules VT{84} is 53% better than PBE and 42% with LYP using a TZVPP basis set TPSS PBEsol PBE VMT VT84 VMTsol VT84sol C-PBE except TPSS C-LYP except TPSS DZVP def2- TZVPP DZVP def2- TZVPP

46 VT{84} and VMT: Molecular Tests Non-hydrogen transfer Barrier heights / Forward reactions / kcal-mol reactions TPSS PBEsol PBE VMT VT{84} VMTsol VT{84}sol C-PBE C-LYP DZVP def2- TZVPP DZVP def2- TZVPP

47 VT{84} and VMT: Molecular Tests The best bond distances are obtained with PBEsol. VT{84} is 28% above by Ǻ Bond Distances def2-tzvpp/ Ǻ 96 bonds TPSS PBEsol PBE VMT VT{84} VMTsol VT{84}sol C-PBE C-LYP

48 VT{84} and VMT: Molecular Tests 13 C chemical shifts: MADs / ppm for 17 C atoms LSDA PBE PBEso l PW91 VMTsol- PBE VMT- PBE VMTsol- LYP VMT- LYP VT84sol- PBE VT84- PBE VT84sol- LYP VT84- LYP DZVP Def2- TZVPP IGLO- II IGLO- III

49 Crystalline Phase Stability - Cohesive energy differences between phases can be a tougher test than cohesive energies and bond lengths. Example: crystalline Lithium (ground state and high pressure) 9R is the T=0K ground state (ABCBCACAB stacking; 9 atoms per hex cell). A more recent experimental claim has FCC as the ground state. Use HCP as a proxy: order of ground state phases then is HCP < FCC < BCC. The energy differences are very small: estimated as about 3 mev/atom, 10 mev/atom respectively. The first phase transition is believed to be HCP FCC at volume ratio Ω/Ω But see above about FCC. Simple LSDA is believed to get it about right.

50 Crystalline Li Calculations for Eight XC Approximations Functional KSG PZ HL RSK PBE VMT1PBE PBEsol VMT1GE Efcc Ehcp (E_H) Ebcc Ehcp (E_H) hcp Equilib vol (au**3) fcc Equilib vol (au**3) bcc Equilib vol (au**3) P hcp->fcc (Mbar) V/V_0 hcp->fcc GTOFF code: 10s6p3d KS basis, 11s fitting basis. KSG = Kohn-Sham-Gaspar PZ = Perdew-Zunger HL = Hedin-Lundqvist (Moruzzi, Janak, & Williams) RSK = Rajagopal-Singhal-Kimball (von Barth-Hedin) PBE = Perdew-Burke-Ernzerhof PBEsol = gradient expansion coefficient in PBE KSG, HL, and RSK results match 1985, 1989, 1992 (GTO, LMTO, FLAPW) calculations It is a little difficult to find the hcp fcc transition in PBEsol, VMT1PBE, and VMT1GE. The transition is very weak (1 μhartree) for VMT1GE

51 Summary - I Construction of simple OF-KE functionals for FORCES ALONE is promising (an example of the KISS principle). For OF-KE, conjointness alone does not work. In fact, we have shown the conjointness hypothesis is false. But it is a useful guide. Pauli potential criterion is an essential constraint on OF-KE. Simple generalized versions of conjoint GGA KE functionals are singular at nuclei. Order-by-order truncation of gradient expansion allows elimination of nuclear singularities. At present, there is no systematic way of picking among the many possible reduced density derivative functionals. Constraints are needed.

52 Summary - II Nature does not seem to make use of the entire energy interval permitted by the Lieb-Oxford bound Lieb-Oxford estimate: 2.27; Chan-Handy reduction: 2.21 Largest value by Odashima & Capelle: (dilute electron gas); Largest value for real systems: < 1.25 Changing the LO bound in the PBE X functional doesn t help. VMT1 X enhancement factor instead generalizes PBE X to give a pointwise enforcement of the LO bound that is tighter than any of the general values listed. The VMT1 X enhancement factor gives 20-25% gain in atomization energy accuracy, at no loss in bond-length accuracy, without empirical parameterization, relative to PBE or PBEsol, for small molecules The VMT1 X enhancement factor with LYP correlation gives useful improvement over VMT1 with PBE or PBEsol correlation for atomization energies but not bond lengths. Predictions of the first pressure-induced phase transition in Li are rather sensitive to the details of a GGA, more so than with LDA.

53 What s Next? Orbital-free Free Energy Density Functional Theory (Mermin- Hohenberg-Kohn finite temperature DFT).

54 Collaborators (XC functionals): Klaus Capelle (IFSC, Univ. do ABC, Brazil) José Luis Gázquez (UAM- I, México D.F.) Alberto Vela (Cinvestav, México D.F.) Victór Medel (Cinvestav, México D.F.) Mariana Odashima (IFSC, Univ. São Paulo, Brazil) Juan Pacheco Kato (Univ. Guanajuato, México) Jorge Martin del Campo Ramirez (UAM- I, México D.F.) Collaborators (Orbital-free KE functionals): Frank Harris (Univ. Utah and Univ. Florida) Jim Dufty (Univ. Florida) Valentin Karasiev (IVIC, Caracas) Keith Runge (BWD Associates) Travis Sjostrom (Univ. Florida) Collaborators (Materials and molecular calculations): Jon Boettger (Los Alamos National Laboratory) PUPIL (Joan Torras, Erik Deumens, Ben Roberts, Gustavo Seabra, Oscar Bertran demon2k (Andreas Köster, Gerald Geudtner, Patrizia Calaminici, Carlos Quintanar, ) Funding Acknowledgments: U.S. DoE DE-SC ; CONACyT (México)

55 References for this Work ``Variable Lieb-Oxford Bound Satisfaction in a Generalized Gradient Exchange-Correlation Functional'', A. Vela, V. Medel, and S.B. Trickey, J. Chem. Phys. 130, (2009) Tightened Lieb-Oxford Bound for Systems of Fixed Particle Number, M. M. Odashima, K. Capelle, and S.B. Trickey, J. Chem. Theory and Comput. 5, 798 (2009). Empirical Analysis of the Lieb Oxford Bound in Ions and Molecules, M. M. Odashima and K. Capelle, Internat. J. Quantum Chem. 108, 2428 (2008) How Tight is the Lieb-Oxford bound? M.M. Odashima and K. Capelle, J. Chem. Phys. 127, (2007) Constraint-based, Single-point Approximate Kinetic Energy Functionals, V.V. Karasiev, R.S. Jones, S.B. Trickey, and F.E. Harris, Phys. Rev. B (2009) Conditions on the Kohn-Sham Kinetic Energy and Associated Density'', S.B. Trickey, V.V. Karasiev, and R.S. Jones, Internat. J. Quantum Chem.109, 2943 (2009). Recent Advances in Developing Orbital-free Kinetic Energy Functionals, 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, 2009] Born-Oppenheimer Interatomic Forces from Simple, Local Kinetic Energy Density Functionals'', V.V. Karasiev, S.B. Trickey, and F.E. Harris, J. Computer-Aided Mat. Design, 13, (2006).