Adiabatic connections in DFT
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1 Adiabatic connections in DFT Andreas Savin Torino, LCC2004, September 9, 2004
2 Multi reference DFT Kohn Sham: single reference Variable reference space capability to approach the exact result
3 Overview The problem Adiabatic connections in DFT Modified Hamiltonian Approximations An exact energy/particle ε x c
4 Overview The problem
5 Correlation energies in atomic series E c in the He (N 2) and Be series (N 4) E c Z He series: lim Z E c Z 0 Be series: lim Z E c Z 1
6 Difficulties with DFAs in atomic series DFA: density functional approximation DFAs transfer correlation from one system to another: do not describe both behaviors Perdew, McMullen, Zunger : LDA gives log Z
7 Difficulties with DFAs in atomic series DFAs transfer correlation from one system to another: do not describe both behaviors Solution (?) MBA describes near degeneracy, if present DFA describes the rest (He series type of behavior) MBA: many body approximation DFA: density functional approximation
8 An analogy Quantum Monte Carlo experience for Be,... 2 s 2 and 2 p 2 configurations Jastrow factor
9 A problem with exact Kohn Sham Identifying states (KS peculiar with quasi degeneracy) Be series C 2 at R e... Gritsenko, Baerends
10 A problem with exact Kohn Sham Identifying states (KS peculiar with quasi degeneracy) Kohn Sham ε 2 p ε 2 s in the Be series Ε p Ε s F. Colonna; also Kohn, Ullrich or Morrison Z
11 A problem with exact Kohn Sham Identifying states (KS peculiar with quasi degeneracy) States are easily identified in MBAs
12 Combining MBAs with DFAs Status of standard approximations MBAs too heavy when accurate. DFAs don t work for strongly correlated systems Objective Compromise with simple many body treatment (MBAs), density functional approximations (DFAs)
13 Combining MBAs with DFAs MBA DFA near degeneracy dynamic correlation
14 Overview The problem Adiabatic connections in DFT
15 Adiabatic connections in DFT = connections between the Kohn Sham and the physical system Kohn Sham system non interacting fermions physical ground state density Physical system electronic Schrödinger equation
16 Adiabatic connections in DFT Connection Hamiltonian changes along the path: MB effects progressively switched on
17 Adiabatic connections in DFT Connection Hamiltonian changes along the path: MB effects progressively switched on Connections Multiplicity of paths
18 Adiabatic connections in DFT DFT= Density Functional Theory Density functional corrects model energy to yield the physical one
19 Adiabatic connections in DFT Adiabatic DFT: ground state does not change along the path State following?
20 Adiabatic connections in DFT Adiabatic DFT: ground state does not change along the path
21 Choice of the intermediate system Mathematically Choice of the Hamiltonian Physically What is transferable? Short range (?) High lying states (?)
22 Adiabatic connections in DFT Moving from a pure DFA to a pure MBA, systematically series of approximations understanding what DFT does
23 Adiabatic connections in DFT: History Understand E xc Harris,Jones Langreth,Perdew Gunnarsson, Lundqvist Construct E xc Becke (hybrid functionals) Seidl (interpolation) Multiplicity of paths Yang
24 Acknowledgement F. Colonna (Paris) H. J. Flad (Leipzig) P. Gori Giorgi (Paris) T. Leininger (Toulouse) R. Pollet (Paris) H. Stoll (Stuttgart) J. Toulouse (Paris)
25 Acknowledgement János Ángyán (Nancy) Iann Gerber (Nancy) H. J. Jensen (Odensee) J. Pedersen (Odensee) C. Gutle (Paris) J. Rey (Paris)
26 Acknowledgement MOLPRO (modifications by J. Ángyán, F. Colonna, I. Gerber, Th. Leininger, H. Stoll, J. Toulouse, H. J. Werner) CASDI (D. Maynau) Service programs (F. Colonna) Mathematica DALTON (J. Pedersen, H. J. Jensen)
27 Overview The problem Adiabatic connections in DFT Modified Hamiltonian
28 Modified Hamiltonian Ground state energy expression Traditional way E min T V ne V ee
29 Modified Hamiltonian Ground state energy expression Traditional way E min T V ne V ee Kohn Sham E min T V ne 0 n
30 Modified Hamiltonian Ground state energy expression Traditional way E min T V ne V ee Kohn Sham E min T V ne 0 n Kohn Sham extensions E min T V ne W W n
31 Modified Hamiltonian Ground state energy expression Traditional way E min T V ne V ee Kohn Sham E min T V ne 0 n Kohn Sham extensions E min T V ne W W n W i j w r i, r j 0, or V ee special cases
32 Modified Hamiltonian Ground state energy expression E min T V ne W W n
33 Parametric dependence of W E min T V ne W W n W: dependence on parameter Μ Infinity of choices of w W W Μ i j w r, r ; Μ Limiting cases Limit Μ W W n Kohn Sham 0 0 n physical V ee 0
34 A choice for the e e interaction w r, r ; Μ w erf Μ r r r r Limits: w Μ 0 0 w Μ w Μ 1 r r w erf Μ r r Μ Μ 1 Μ r
35 A choice for the e e interaction w r, r ; Μ w erf Μ r r r r Remaining effects ("1 r r w") via W n Better with DFA?
36 Short range/long range separation Long history... Uniform electron gas (e.g., Nozières, Pines) Hanke, Kohn (unpublished) + H. Stoll (correlation) + H. J. Flad
37 Choosing W w erf Μ r r r r connects KS with physical system easy to implement uniform electron gas molecular integrals (GTOs) systematic improvement
38 Overview The problem Adiabatic connections in DFT Modified Hamiltonian Approximations
39 Approximations? E min T V ne W W n Exact: accurate, but very heavy, calculations possible. W n : approximations needed MBA: can be accurate, for certain W, without considerable effort
40 MBA: expansion of Errors due to expansion in terms of, as a function of Μ, for Be E s2s 1s2s2p 1s2s2p3s 1s2s3p3s3p Μ Errors increase with Μ (no error for Μ=0)
41 DFAs W n W n must be approximated Same type of approximations as in Kohn Sham
42 Splitting W n E min T V ne W W n Coulomb (classical) 1 2 n r n r 1 r r w Exchange correlation W Exc n
43 Splitting W n Coulomb (classical) Exchange correlation W Exc n Exchange W Ex n Correlation W Ec n
44 Splitting W n : Relative contributions No systematic approximations for n in Kohn Sham Μ 0 W n 0 as Μ E. n W n : accurate results (Be) U n W n E c n E x n 5 10 Μ
45 Errors due to Μ LDA (Be) Exchange energy, E W x n Correlation energy, E W c n E x LDA Μ E c LDA Μ Errors decrease with Μ
46 Why is Μ LDA good at large Μ? E W x a local functional (exact result, no double counting, no self interaction) E W x n Μ 2 Π 8 n 2 r d 3 r Μ 4 3 Π 48 n n 2 Τ... 8 n (closed shell)
47 Why is Μ LDA good at large Μ? E W x a local functional (exact result, no double counting, no self interaction) E W x n Μ 2 Π 8 n 2 r d 3 r Μ 4 3 Π 48 n n 2 Τ... 8 n E W c depends on the on top pair density, P 2 r, r (well approximated by LDA, Perdew, et al.) E W c n Μ 2 Π 2 P 2 r, r n r 2 2 d 3 r Μ 3 2 Π 3 P 2 r, r d 3 r
48 W Ec n Μ 2 Π 2 P 2 r, r 1 2 n r 2 d 3 r +... E W c n, Μ 2 Π 2 P 2 r, r 1 2 n r 2 d 3 r (CI,.., LDA, very accurate) Ec He Μ
49 E x : expansion in Μ, validity of Μ LDA 0 He -0.2 Ex Μ Μ LDA accurate
50 E x : expansion in Μ, validity of Μ LDA 0 Be Ex Μ Μ LDA accurate
51 E c : expansion in Μ, validity of Μ LDA 0 He Ec Μ Μ LDA accurate
52 E c : expansion in Μ, validity of Μ LDA 0 Be Ec Μ Μ LDA accurate
53 River in Stølsheimen
54 Error in E xc n, Be E xc LDA GEA PBE Μ
55 Total energy, in Μ LDA: E(Μ)
56 E(Μ): different expansions Etot s2s 1s2s2p 1s2s2p3s 1s2s2p3s3p Μ
57 E(Μ): different expansions Μ LDA Μ PBE Etot Etot s2s 1s2s2p 1s2s2p3s 1s2s2p3s3p s2s 1s2s2p 1s2s2p3s 1s2s2p3s3p Μ Μ
58 Overview The problem Adiabatic connections in DFT Modified Hamiltonian Approximations An exact energy/particle ε x c
59 An exact energy/particle ε x c No unique definition Local Μ
60 ε c with global Μ=0.2 Be ε c Μ r
61 ε c with global Μ=0.2 and Μ dependent on n r and n r Be ε c Μ r
62 Summary
63 Summary W Μ : Kohn Sham physical system w erf Μ r r Μ Μ 1 Μ r
64 Summary W Μ : Kohn Sham physical system Effect: MBA and a better transferable DF Etot 1s2s2p Μ Large error in n compromise Large error in expansion
65 Summary W Μ : Kohn Sham physical system Effect: MBA and a better transferable DF ε: a tool for locally checking DFAs ε c Μ r
66 E(R), scaled: Μ LDA van der Waals dimers János Ángyán, Iann Gerber
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