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1 Information-Theoretic Construction of an Orbital-Free Kinetic-Energy Functional Ian P Hamilton 1 Ricardo M Mosna 2 Luigi Delle Site 3 Luca M. Ghiringhelli 3 1 Wilfrid Laurier University, Canada 2 Universidade Estadual de Campinas, Brazil 3 Max-Planck-Institute for Polymer Research, Mainz, Germany
2 First steps tow ards the Information-Theoretic Construction of an Orbital-Free Kinetic-Energy Functional Ian P Hamilton 1 Ricardo M Mosna 2 Luigi Delle Site 3 Luca M. Ghiringhelli 3 1 Wilfrid Laurier University, Canada 2 Universidade Estadual de Campinas, Brazil 3 Max-Planck-Institute for Polymer Research, Mainz, Germany
3 Recent Publications Interacting electrons, spin statistics and, and information theory, L. Ghiringhelli, I.P. Hamilton and L. Delle Site, J. Chem. Phys. 132, (2010). Information-theoretic approach to kinetic-energy functionals: The nearly uniform electron gas, L. Ghiringhelli, L. Delle Site, R.A. Mosna, I.P. Hamilton, J. Math. Chem. 48, 78 (2010). Fisher information and kinetic-energy functionals: A dequantization approach, I.P. Hamilton, R.A. Mosna, J Comp. Appl. Math. 233, 1542 (2010). Quantum Fluctuations, Dequantization, Information Theory and Kinetic-Energy Functionals, I.P.Hamilton, R.A. Mosna and L. Delle Site, book chapter in Recent Advances in Orbital-Free Density Functional Theory edited by Tomasz Wesolowski and Alex Wang, (World Scientific, Singapore, 2010).
4 Goals Formulation of both quantum and classical mechanics in which the electron density,,, is the fundamental variable (not the wavefunction, ψ,, or the N-electron density, N ). 4
5 Goals Formulation of both quantum and classical mechanics in which the electron density,,, is the fundamental variable (not the wavefunction, ψ,, or the N-electron density, N ). Decomposition of the kinetic energy: - Classical and purely quantum kinetic energy. - Noninteracting and interacting kinetic energy. - Stronger connection with information theory. - Construction of orbital-free kinetic-energy energy functional. 5
6 Outline Decomposition into classical and purely quantum kinetic energy via dequantization process that leaves unchanged. Decomposition of the purely quantum kinetic energy: A) purely quantum noninteracting kinetic energy - functional of a Fisher information expression (or Weizsäcker term) B) purely quantum interacting kinetic energy - functional of (N-1) conditional electron density Monte-Carlo procedure functional of a Shannon information expression 6
7 Quantization Altering or augmenting equations of classical mechanics to obtain quantum mechanics. 7
8 Quantization Altering or augmenting equations of classical mechanics to obtain quantum mechanics. In Nelson s approach quantum fluctuations are added to the classical momentum resulting in the quantum momentum. 8
9 Quantization Altering or augmenting equations of classical mechanics to obtain quantum mechanics. In Nelson s approach quantum fluctuations are added to the classical momentum resulting in the quantum momentum. In Bohm s approach a quantum potential is added to the classical potential. 9
10 Dequantization Set of rules for turning quantum mechanics into classical mechanics Not the 0 limit of quantum mechanics 10
11 Dequantization Set of rules for turning quantum mechanics into classical mechanics Not the 0 limit of quantum mechanics In our approach dequantization strips the quantum fluctuations from the quantum momentum operator resulting in the classical momentum operator. 11
12 Dequantization Set of rules for turning quantum mechanics into classical mechanics Not the 0 limit of quantum mechanics In our approach dequantization strips the quantum fluctuations from the quantum momentum operator resulting in the classical momentum operator. In our approach the electron density is a fundamental variable as in Madelung s hydrodynamic f ormulation of quantum mechanics. 12
13 Witten deformation approach Mosna, Hamilton and Delle Site, J. Phys. A 38,, 3869 (2005) 13
14 Witten deformation approach continued (1) 14
15 Witten deformation approach continued (2) 15
16 Witten deformation approach continued (2) 16
17 Witten deformation approach continued (2) 17
18 Witten deformation approach continued (2) 18
19 Witten deformation approach continued (3) 19
20 Variational approach Mosna, Hamilton and Delle Site, J. Phys. A 39,, L229 (2006) For an N-electron system, P N and T N are the usual (quantum) N-electron momentum and kinetic energy operators while P N and T N are the momentum and kinetic energy of the s ystem. 20
21 Variational approach continued (1) 21
22 Variational approach continued (2) 22
23 N-electron classical kinetic energy Hamilton, Mosna and Delle Site, Theo. Chem. Acct. 118, 407 (2007). 23
24 In 1D (with I N = I): 24
25 1D Example 25
26 1D Example 26
27 Kinetic energy decomposition Our expression for the N-electron kinetic energy is T N = T C,N + T W,N The N-electron Weizsäcker term can be written as T W + T corr W where T W is the (one-electron) electron) Weizsäcker term and T corr W is the purely quantum correlation term. - T W results from local quant um fluctuations. - T corr W results from nonlocal quant um fluctuations. 27
28 Kinetic energy decomposition continued The N-electron classical kinetic energy, T C,N, can be written as T C + T corr C where T C is the (one-electron) electron) classical kinetic energy and T corr C is the classical correlation term. Our expression for the N-electron kinetic energy is T N = T C + T W + kinetic correlation terms Our expression for the one-electron electron kinetic energy is T = T C + T W 28
29 Results for 1D systems particle in a box, harmonic oscillator For a stationary state with no nonzero angular momentum component, T C = 0 and T = T W. This is the case for ground or excited states of - a 1D particle in a box - a 1D harmonic os cillator. For a nonstationary state, T C is not equal to zero even if there is no nonzero angular momentum component. 29
30 Kinetic energy decomposition: Superposition of stationary states Sum of ground state and first excited state f or 1D particle in a box. densities of T, T C and T W integrands 30
31 Kinetic energy decomposition: Spreading Gaussian Ground state for 1D harmonic oscillator with potential energy set equal to zero (kinetic energy is therefore the total energy). densities of T, T C and T W 31
32 Charge distribution and kinetic energy decomposition: particle in a box Hamilton and Mosna, J. Comp. Appl. Math. 233,, 1542 (2010) 32
33 Charge distribution and kinetic energy decomposition: spreading Gaus sian Hamilton and Mosna, J. Comp. Appl. Math. 233,, 1542 (2010) 33
34 T s in the orbital approximation 34
35 T s in the orbital approximation 35
36 T s in the orbital approximation 36
37 T C and orbital angular momentum Englert and Schwinger (Phys. Rev. A 29, 2339 (1984); 32, 47 (1985)) included angular momentum effects for the express purpose of correcting the kinetic-energy energy functional for large r. 37
38 Hydrogenic orbitals 38
39 T C for hydrogenic orbitals 39
40 Results for hydrogenic orbitals Hamilton, Mosna and Delle Site, Theo. Chem. Acct. 118,, 407 (2007) n=2, l=1, m=1 solid black; density of T density of kinetic energy and its components dotted black; density of T W dashed black; density of T C T C = 1/16 T W = 1/16 T = 1/8 r 40
41 Results for hydrogenic orbitals n=5, l=1, m=1 Dashed, integrand of T C Dotted, integrand of T W Solid, integrand of T T C = 1/250 T W = 4/250 T = 5/250 41
42 Results for hydrogenic orbitals n=5, l=2, m=1 Dashed, integrand of T C Dotted, integrand of T W Solid, integrand of T T C = 1/250 T W = 4/250 T = 5/250 42
43 Results for hydrogenic orbitals n=5, l=3, m=1 Dashed, integrand of T C Dotted, integrand of T W Solid, integrand of T T C = 1/250 T W = 4/250 T = 5/250 43
44 Results for hydrogenic orbitals n=5, l=4, m=1 Dashed, integrand of T C Dotted, integrand of T W Solid, integrand of T T C = 1/250 T W = 4/250 T = 5/250 44
45 Tomas-Fermi Term 45
46 Kinetic Energy Densities Following García-Aldea and Alvarellos (J. Chem. Phys. 127, (2007)) we compare the T TF and T TF + (1/9)T W kinetic energy densities to calculated kinetic energy densities for the ground states of Ne and Ar. We use Hartree-Fock in Becke s NUMOL code (basis set free) and a positive definite kinetic energy density: 46
47 Kinetic Energy Densities Following García-Aldea and Alvarellos (J. Chem. Phys. 127, (2007)) we compare the T TF and T TF + (1/9)T W kinetic energy densities to calculated kinetic energy densities for the ground states of Ne and Ar. T W corr = 0 We use Hartree-Fock in Becke s NUMOL code (basis set free) and a positive definite kinetic energy density: 47
48 Ne: Hartree-Fock density of kinetic energy and related expressions black; density of T green; density of T TF 400 red; density of T TF + (1/9)T W r Thanks to Erin Johnson for calculations. 48
49 Ar: Hartree-Fock density of kinetic energy and related expressions black; density of T 3000 green; density of T TF red; density of T TF + (1/9)T W r Thanks to Erin Johnson for calculations. 49
50 Ne: Hartree-Fock density of kinetic energy and related expressions black; density of T dotted; density of T W 200 dashed; density of T - T W r Thanks to Erin Johnson for calculations. 50
51 Ar: Hartree-Fock density of kinetic energy and related expressions black; density of T dotted; density of T W dashed; density of T - T W r Thanks to Erin Johnson for calculations. 51
52 Ne: Hartree-Fock density of kinetic energy and related expressions dashed; density of T -T W black; density of (x10) r Thanks to Erin Johnson for calculations. 52
53 Ar: Hartree-Fock density of kinetic energy and related expressions dashed; density of T -T W black; density of (x50) r Thanks to Erin Johnson for calculations. 53
54 Ne: Hartree-Fock density of kinetic energy and related expressions dashed; density of T -T W blue; density of exchange energy (x2) r Thanks to Erin Johnson for calculations. 54
55 Ar: Hartree-Fock density of kinetic energy and related expressions dashed; density of T -T W blue; density of exchange energy (x4) r Thanks to Erin Johnson for calculations. 55
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