Density-Functional Theory and Chemistry in One Dimension
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1 Density-Functional Theory and Chemistry in One Dimension Pierre-François Loos Research School of Chemistry, Australian National University, Canberra, Australia 14ème Rencontre des Chimistes Théoriciens Francophones, École Nationale Supérieure de Chimie de Paris July 2nd 2014
2 Introduction Density-Functional Theory Chemistry of 1D Atoms Chemistry of 1D Molecules Conclusion My collaborators Quantum Chemistry at ANU Peter Gill Q-Chem president Andrew Gilbert IQmol Caleb Ball DFT Giuseppe Barca HF excited states Discovery Early Career Researcher Award Discovery Project 2014 PF Loos loos/ RCTF 2014 July 2nd 2014
3 Local-Density Approximation The local-density approximation (LDA)
4 Local-Density Approximation The local-density approximation (LDA) Find the correlation energy of the infinite uniform electron gas (UEG) for all densities ρ Ceperley & Alder, Phys Rev Lett 45 (1980) 566
5 Local-Density Approximation The local-density approximation (LDA) Find the correlation energy of the infinite uniform electron gas (UEG) for all densities ρ Ceperley & Alder, Phys Rev Lett 45 (1980) 566 Treat a molecular density as a collection of tiny bits of UEG Vosko, Wilk & Nusair, Can J Phys 58 (1980) 1200 Perdew & Zunger, Phys Rev B 23 (1981) 5048 Perdew & Wang, Phys Rev B 45 (1992) 13244
6 Local-Density Approximation The local-density approximation (LDA) Find the correlation energy of the infinite uniform electron gas (UEG) for all densities ρ Ceperley & Alder, Phys Rev Lett 45 (1980) 566 Treat a molecular density as a collection of tiny bits of UEG Vosko, Wilk & Nusair, Can J Phys 58 (1980) 1200 Perdew & Zunger, Phys Rev B 23 (1981) 5048 Perdew & Wang, Phys Rev B 45 (1992) The LDA is an ab initio model with no adjustable parameters This is an attractive approach to molecular electronic structure It also forms a foundation for more accurate approximations
7 Local-Density Approximation The local-density approximation (LDA) Find the correlation energy of the infinite uniform electron gas (UEG) for all densities ρ Ceperley & Alder, Phys Rev Lett 45 (1980) 566 Treat a molecular density as a collection of tiny bits of UEG Vosko, Wilk & Nusair, Can J Phys 58 (1980) 1200 Perdew & Zunger, Phys Rev B 23 (1981) 5048 Perdew & Wang, Phys Rev B 45 (1992) The LDA is an ab initio model with no adjustable parameters This is an attractive approach to molecular electronic structure It also forms a foundation for more accurate approximations Not very accurate for correlation energy (overestimated by roughly 100%)
8 Generalized Local-Density Approximation Jacob s ladder vs Generalized LDA idea The lowest rung (LDA) assumes that all UEGs of density ρ are equivalent
9 Generalized Local-Density Approximation Jacob s ladder vs Generalized LDA idea The lowest rung (LDA) assumes that all UEGs of density ρ are equivalent That assumption is not correct! Gill & Loos, Theor Chem Acc 131 (2012) 1069 Loos & Gill, J Chem Phys 138 (2013)
10 Generalized Local-Density Approximation Jacob s ladder vs Generalized LDA idea The lowest rung (LDA) assumes that all UEGs of density ρ are equivalent That assumption is not correct! Gill & Loos, Theor Chem Acc 131 (2012) 1069 Loos & Gill, J Chem Phys 138 (2013) We propose to follow an alternative route to heaven using finite-size UEGs!
11 Generalized Local-Density Approximation Jacob s ladder vs Generalized LDA idea The lowest rung (LDA) assumes that all UEGs of density ρ are equivalent That assumption is not correct! Gill & Loos, Theor Chem Acc 131 (2012) 1069 Loos & Gill, J Chem Phys 138 (2013) We propose to follow an alternative route to heaven using finite-size UEGs! We add a new local two-electron parameter occ hole curvature: η(r) 2 ψ i 2 ρ 2 2ρ i Loos, Ball & Gill, J Chem Phys 140 (2014) 18A524
12 GLDA in 1D GLDA correlation functional for 1D systems [ Ec GLDA (ρ, η) = Υ 0(η) F 1, 3 ] Υ0(η)(1 Υ(η)), Υ(η), ρ 1 2 Υ (η)
13 GLDA in 1D GLDA correlation functional for 1D systems [ Ec GLDA (ρ, η) = Υ 0(η) F 1, 3 ] Υ0(η)(1 Υ(η)), Υ(η), ρ 1 2 Υ (η) F (a, b, c, x) = Hypergeometric function exact for small and large ρ Υ 0 (η) = electrons are close to each other perturbation theory Υ(η) = intermediate densities quantum Monte Carlo Υ (η) = electrons are far apart perturbation theory
14 GLDA in 1D GLDA correlation functional for 1D systems [ Ec GLDA (ρ, η) = Υ 0(η) F 1, 3 ] Υ0(η)(1 Υ(η)), Υ(η), ρ 1 2 Υ (η) F (a, b, c, x) = Hypergeometric function exact for small and large ρ Υ 0 (η) = electrons are close to each other perturbation theory Υ(η) = intermediate densities quantum Monte Carlo Υ (η) = electrons are far apart perturbation theory By construction, E GLDA c (ρ, η = 1) = Ec LDA (ρ)
15 GLDA in 1D GLDA correlation functional for 1D systems [ Ec GLDA (ρ, η) = Υ 0(η) F 1, 3 ] Υ0(η)(1 Υ(η)), Υ(η), ρ 1 2 Υ (η) F (a, b, c, x) = Hypergeometric function exact for small and large ρ Υ 0 (η) = electrons are close to each other perturbation theory Υ(η) = intermediate densities quantum Monte Carlo Υ (η) = electrons are far apart perturbation theory By construction, E GLDA c (ρ, η = 1) = Ec LDA (ρ) Electrons in a box (L = π) Electrons in a harmonic well (k = 1) n = 2 n = 3 n = 4 n = 5 n = 6 n = 2 n = 3 n = 4 n = 5 n = 6 LDA GLDA FCI Loos, Phys Rev A 89 (2014)
16 The Coulomb Operator Rules the 1D World! Chemistry in 1D with the Coulomb operator x 1 Loos, Ball & Gill (submitted)
17 Can a Wavepacket Go Through the Coulomb Potential? Impenetrability of the 1D Coulomb potential: H atom Newton, J Phys A 27 (1994) 4717; Nunez-Yepez et al., Phys Rev A 83 (2011)
18 Hydrogen Atom Hydrogen atom in 1D Left-handed ground state: 1H Right-handed ground state: H Loudon, Am J Phys 27 (1959) 649
19 Hydrogen Atom Hydrogen atom in 1D Left-handed ground state: 1H Right-handed ground state: H 1 1 ψ(x) = x exp(+x) 1 ψ(x) = x exp( x) E = 0.5 µ = +1.5 R = x 2 = 1.8 E = 0.5 µ = 1.5 R = x 2 = 1.8 Loudon, Am J Phys 27 (1959) 649
20 Introduction Density-Functional Theory Chemistry of 1D Atoms Chemistry of 1D Molecules Conclusion Helium Atom Helium atom in 1D One-sided helium: +2 PF Loos loos/ He1,2 Two-sided helium: 1 He1 +2 RCTF 2014 July 2nd 2014
21 Introduction Density-Functional Theory Chemistry of 1D Atoms Chemistry of 1D Molecules Conclusion Helium Atom Helium atom in 1D One-sided helium: Two-sided helium: E = E = µ = +3.5 R = 4.8 PF Loos loos/ He1,2 µ=0 1 He1 R = 1.0 RCTF 2014 July 2nd 2014
22 1D Atoms More 1D atoms... Lithium: µ = 1.5 and R = 2.8 Beryllium: µ = 0 and R = Boron: µ = 1.9 and R = 4.7 Carbon: µ = 0 and R =
23 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides
24 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons
25 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell
26 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell
27 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Odd electron reactive
28 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Odd electron reactive Even electron unreactive
29 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Odd electron reactive Even electron unreactive Odd electron alkali metals
30 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Odd electron reactive Even electron unreactive Odd electron alkali metals Even electron noble gases
31 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Odd electron reactive Even electron unreactive Odd electron alkali metals Even electron noble gases
32 Mendeleev s Periodic Table The periodic table in a 1D world... Periodic trends in 1D atoms 1D atoms have only two sides Shells hold only two electrons Odd electron unfilled shell Even electron filled shell Group! Period! 1! 2! 1!! Alkali! metals! 1! H! 3! Li! 2!! Noble! Gases! 2! He! 4! Be! Odd electron reactive Even electron unreactive Odd electron alkali metals Even electron noble gases 3! 4! 5! 5! B! 7! N! 9! F! 6! C! 8! O! 10! Ne!
33 One-Electron Diatomics The H + 2 molecule in 1D The H 1H + state: µ = 0 The HH 1 + state: µ 0
34 One-Electron Diatomics The H + 2 molecule in 1D The H 1H + state: µ = 0 Potential energy curves for H + 2 The HH 1 + state: µ 0 EHH 2 + L-EHHL-EHH + L Ha.u.L HH H 1 H R Ha.u.L
35 One-Electron Diatomics One-electron diatomic molecules in 1D Potential energy curves for H + 2 -like molecules Electron densities for one-electron diatomics EHABL-EHAL-EHBL Ha.u.L R Ha.u.L + H 2 3+ He 2 5+ Li 2 HeH 2+ LiH 3+ LiHe 4+ rhxl Ha.u.L He 1 H 2+ H 1 H xêreq Ha.u.L
36 Two-Electron Diatomics The H 2 molecule in 1D The H 1,2H state The H 1H 1 state The 1HH 1 state The HH 1,2 state
37 Two-Electron Diatomics Two-electron diatomic molecules in 1D Potential energy curves for the H 2 molecule Electron densities for two-electron diatomics E H2 2E H a.u. 1.0 H 1,2 H HH 1, HH H 1 H rhxl Ha.u.L 4 + He 1 H H 1 H 1 He 1 He 1 2+ H 1 He xêr eq Ha.u.L
38 Two-Electron Diatomics Two-electron diatomic molecules in 1D Potential energy curves for the H 2 molecule Electron densities for two-electron diatomics E H2 2E H a.u. 1.0 H 1,2 H HH 1, HH H 1 H rhxl Ha.u.L 4 + He 1 H H 1 H 1 He 1 He 1 2+ H 1 He xêr eq Ha.u.L 1D atoms are bound by one-electron bonds!!
39 Hydrogen nanowire Lego-style formation of 1D polymers A single H 1 atom has a dipole moment Two H 1 atoms will feel dipole-dipole attraction The resulting H 1 H 1 molecule also has a dipole moment H 1 H 1 and H 1 will feel dipole-dipole attraction The resulting H 1 H 1 H 1 molecule also has a dipole moment H 1 H 1 H 1 and H 1 will feel dipole-dipole attraction H 1 + H 1 H 1 H 1 H 1 H 1 + H 1 H 1 H 1 H 1 H 1 H 1 H 1 + H 1 H 1 H 1 H 1 H 1
40 Final Remarks Take-home messages All uniform electron gases are equal, but some are more equal than others! GLDA improves LDA (a lot) 1D chemistry is very different from 3D chemistry Electrons cannot penetrate the nuclei Periodic Table has only two groups: alkali metals and noble gases Dipole-dipole contribution to bonding is important 1D atoms are bound by one-electron bonds!
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