Ryo Maezono. Japan Advanced Institute of Science and Technology, Ishikawa, Japan. treated by CHAMP. School of Information Science,
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1 Sotto, Italy, 28.Jul.09 treated by CHAMP Ryo Maezono School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa, Japan.
2 - Masayoshi Shimomoto - Yasutami Takada - Cyrus. J. Umrigar
3 Inhomogeneity effects on XC potentials Inverse Kohn-Sham scheme Charge Density DMC evaluation As a simplest system... Immersed Atom into Jellium Sphere
4 Background (1)
5 (Kohn-Sham) In Principle, v XC! r v XC ( r ) XC potential ( ) can be constructed so that it can reproduce! ( r! ) (Kohn-Sham) (Inverse Kohn-Sham)! ( r! )!! r ( ) can be obtained Charge Density by reliable treatment about electronic correlation
6 2-elec. Singlet systems He atom Only the lowest orbital occupied... v XC ([!]; r! ) = " KS # 2 $ lowest $ lowest % v ext! r! r! & ( ) % ( ) '! r % r! & d r! & " r! ( ) = ( )!! lowest r 2! KS = E G + Z 2 2 ; (Closed-shell in a lowest orbital) ; (KS-level equals to ionization energy) v XC! r ( ) is obtained analytically, directory from! ( r! ) & E G evaluated by Hylleraas-type Variational calc. Umrigar and Gonze, PRA50, 3827 (1994)
7 Relying on such special relations (analytically feasible inverse KS) - A.C. Pedroza, PRA33, 804 (1986). - Umrigar and Gonze, PRA50, 3827 (1994)
8 E. S. Kadantsev and M. J. Stott, Phys. Rev. A 69, (2004) Exploiting Haydock-Foulkes functional V 0 V Trial for Inverse Kohn-Sham! ( r )! ; true potential to reproduce given!0 ( r )! r ( ) General/numerical feasibility ; trial potential!! I " # V Trial ( r );!0 ( r ) $% := & '! ( " i # V Trial ( r ) $%!! + d 3 r V Trial ( r )!0 r Haydock-Foulkes Functional iocc. ) ( ) Upper bound property!!! I # $ V 0 ( r );"0 ( r ) %& W.M.C. Foulkes and R. Haydock, PRB39, (1989). How to get XC potential to reproduce a given! 0!! 0 ( r )! r ( )! r ( ) numerically?!! Optimize V Trial so that it may minimize I " # V Trial ( r );!0 ( r ) $% evaluated by CI methods for Ne atom and methane molecule
9 Provide DMC density! 0! r ( ) Initial Guess of V ( r! ) by LDA KS-SCF to get KS-level {! j }! r! & ( ) Evaluate!( r!! ) "! 0 ( r ) < # Finish Update V ( r! ) by Conjugate Gradient method
10 Background (2)
11 Firstly modelled as Homogeneous Electron Gas Inhomogeneity effects due to ionic cores How it dominates for the origin of FCC/BCC structures? Energy gain by immersing atom into Jellium (Embedding Energy) Practical application of Embedding Energy --> Effective Medium Theory M.J. Puska et.al., PRB24, 3037 (1981). Inhomogeneity effects on XC potential
12 for immersed system D.C. Thompson and Ali Alavi, PRB 66, (2002) 2-electrons in a sphere with and w/o background. Exact diagonalization to get accurate densities Then obtain XC potentials by Inverse Kohn-Sham. Comparison with LYP, P91, PBE, PZ Difference investigated
13
14 - DMC calculation of a Immersed Atom into HEG to get Charge densities - Inverse Kohn-Sham scheme using Haydock-Foulkes functional minimization to get XC potential V XC Immersed Atom into HEG ( ) r;z,r s Localized basis Delocalized basis Immersed Atom into Jellium Sphere.
15 Specified by Infinite Potential Wall ( r S,Z ) electrons (!N ) Jellium Sphere Background ( +N! Z ) nucleus ( +Z ) Infinite Potential Wall is introduced for convergence reason
16 Generate Trial Node by LDA QMC by CHAMP Trial/Guiding WaveFunction DMC charge density Inverse Kohn-Sham scheme Exchange Correlation Potential
17 without impurity Several reference QMC works available upto N=106 - P. Ballone, C. J. Umrigar, and P. Delaly, Phys. Rev. B 45, 6293 (1992) - F. Sottile and P. Ballone, Phys. Rev. B 64, (2001) ; VMC ; DMC
18 to be prepared Implementation of DFT for shperically symmetric systems (LDA part of PBE/Numerov method) - High angular momentum (upto any L by recursive generation) - Matrix operation with large size (General treatment for Multi-det. Sometimes fails) Implemented by Conjugate Gradient method.... Other extensions are quite straight forward.
19 upto Higher angular momentum (e.g., Z=2, rs=5,25, N=60) 1s(2) 2s(2) 1p(6) 3s(2) 2p(6) 1d(10) ** ** ** 1f(14) ** ** ** ** 1g(18) We follow the unfamiliar convention such as 1p or 1d as in - F. Sottile and P. Ballone, Phys. Rev. B 64, (2001)
20
21 (Simple Jellium sphere without impurity) Z = 0 r s N
22 (Z=0, N=106, rs=1.0/ without Infinite pot. wall) Energy (ev/electron.) Std.err LDA,S&B LDA,present * VMC,S&B VMC,present DMC,S&B DMC,present - F. Sottile and P. Ballone, Phys. Rev. B 64, (2001)
23 (with Infinite pot. wall) r s 5.25 N = 58 N = Z
24 (Z=2, rs=5.25, N=60) !r 2 " r ( )!( r) (qmc)4pi*r^2*rho_r 4pi*r^2*rho_r(lda) fitted_rho_r rho_r(lda) raw_rho_r r a.u ( ) r( a.u. )
25 Z=2, rs=1.0, N=60 4!r 2 "( r) r( a.u. )
26 Z=2, rs=3.0, N=60 4!r 2 "( r) r( a.u. )
27 Z=2, rs=5.25, N=60 4!r 2 "( r) r( a.u. )
28
29 (Benchmark)
30 (Benchmark)
31 Z=0, rs=1, N=106
32 rs=5.25, N=60
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