Many-body dispersion interactions between semiconducting wires

Transcription

1 Many-body dispersion interactions between semiconducting wires Alston J. Misquitta TCM Group, Cavendish Laboratory University of Cambridge 26 th July, 2010

2 Collaborators James Spencer (Imperial College, London) Ali Alavi (Cambridge) Tim Totton (Chem. Engg., Cambridge) Anthony Stone (Cambridge)

3 Hückel chains Interactions between two parallel infinite wires. Two-band Hamiltonian: H = n (βa 2ia 2i 1 + β a 2i+1a 2i + h.c.) (1) i Models interactions between (H 2 ) n chains or π-conjugated polyenes. Band-gap is given by E g = 2(β β ) (2)

4 Dispersion energy: disp = i A,j B a A,b B ij r 1 12 ab 2 ε i + ε j ε a ε b. (3) Large finite-size effects: k-point sampling equivalent to a crystal cell with sites.

5 Infinite, parallel chains with gap: E g = 2β disp / β β /β = 0 z z / d

6 Infinite, parallel chains with gap: E g = z disp / β β /β = 1 β /β = 0 z z / d

7 Dobson, White & Rubio, PRL 96, (2006). System Metals Insulators 1-D R 2 (ln(kr)) 3/2 R 5 2-D R 5/2 R 4 Verified using DMC calculations on the 1-D and 2-D HEG by Drummond & Needs (PRL 99, (2007)). 1-D result probably known much earlier: Coulson & Davies (Trans. Faraday Soc. 1952) and Longuet-Higgins & Salem (Proc. R. Soc. A. 1961). The latter studied the non-additivity of the dispersion energy.

8 Infinite, parallel chains with gap: E g = 2(β β ) 10 3 z 2.16 z 5.00 z 2.02 disp / β z 2.65 β /β = 1 β /β = 0.99 β /β = 0.90 β /β = 0 z 4.99 z z / d

9 SAPT(DFT) Interaction of (H 2 ) n chains using SAPT(DFT) (symmetry-adapted perturbation theory based on DFT). Finite chains: n = 2, 4, 8, 16, 32 Control HOMO LUMO gap using bond alternation. η = y/x. Gap goes from 10 ev (smallest chain) to 1 ev.

10 SAPT(DFT) dispersion is calculated using the generalized Casimir-Polder expression (Longuet-Higgins (1965)): disp = 1 2π 0 αa (r 1, r 1; iw)α B (r 2, r 2; iw) dw r 1 r 2 r 1 r dr 1 dr 1dr 2 dr 2 2 Where α(r, r ; ω) is the frequency-dependent density susceptibility function (FDDS). Advantages of perturbation theory: separate out the dispersion from the long-range electrostatics.

11 Bond alternation: η = y/x = 2, E g = a.u z 4.84 disp / n / a.u z η = z / a.u.

12 Bond alternation: η = y/x = 1, E g = a.u z 3.17 disp / n / a.u z 4.84 η = 1.00 η = 2.00 z z / a.u.

13 Bond alternation: η = y/x 10 3 disp / n / a.u η = 1.00 η = 1.25 η = 1.50 η = z / a.u.

14 No single power law. Two regions (distances in a.u.): z 6 : for z L z x : x < 5 for 6 < z < 20. The anomalous power law occurs at physically important separations. Enhanced dispersion: At z = 40 a.u. (roughly half chain length), two orders of magnitude between chains with η = 1 and η = 2. Enhancement orders of magnitude less than for the Hu ckel chains: importance of screening.

15 Non-additivity of total polarizabilities can explain three order of magnitude enhancement at very long range. Longitudinal static polarizabilities for (H 2 ) 32 chains: α(η = 1.0) α(η = 2.0) = = 28 (4) 415 Dispersion energy depends quadratically on the polarizability. But extrapolation to intermediate distances overestimates dispersion by orders of magnitude. Failure of additivity. Not damping or retardation. Severe finite-size effects...

16 Multipole Expansion At the time, I had been working on very accurate multipole expansions for the dispersion energy using WSM method (available in the CamCASP program): disp = a,b ( C ab 6 R 6 ab + C 7 ab Rab 7 + C 8 ab Rab 8 ) + Based on density-response functions calculated using linear-response TD-DFT. Consistent with SAPT(DFT) dispersion energies. Very accurate models. R.m.s. errors of 0.5 kj mol 1 across large energy ranges (-40 to 0 kj mol 1 ). (5)

17 Pyridine dimer: WSM dispersion models (damped) disp,d (model) / kj mol-1-30 C 10,d C 6,iso,d / kj moldisp,tot

18 Multipole Expansion For z > 6 a.u.: Density overlap is negligible. So multipole expansion should be valid. Usually we take this to mean that disp = a,b C6 ab Rab 6. (6) But this cannot yield an effective power law with exponent less than 5. Nor can the more general expression: disp = a,b ( C ab 6 R 6 ab + C 7 ab Rab 7 + C 8 ab Rab 8 where each coefficient is angular dependent. ) + (7)

19 Longuet-Higgins (1965) expression for the dispersion energy: disp = 1 2π 0 αa (r 1, r 1; iw)α B (r 2, r 2; iw) dw r 1 r 2 r 1 r dr 1 dr 1dr 2 dr 2 2 Taylor expansion (systems separated by R): Leading order term is usually written as the dipole-dipole interaction: 1 r r = ˆµ 3R α R β R 2 δ αβ α R 5 ˆµ β (8) But this is insufficient. It leads to the usual C 6 R 6 form. Generalize: 1 r r = ˆQ A t T AB tu ˆQ B u (9) where t = 00, 10, 11c, 11s,... label the rank of the multipole moment operators.

20 A T-function of ranks l and l behaves like R l l 1. Distribute: For extended systems use multiple centres: 1 r r = a b ˆQ a t T ab tu ˆQ b u. (10) Inserting this in the Longuet-Higgins expression gives: disp = 1 Ttu ab T a b t 2π u a,a b,b α aa 0 tt (iw)αbb uu (iw)dw (11) where the non-local polarizabilities are defined as αtt aa (ω) = ˆQ t a (r)α(r, r a ; ω) ˆQ t (r )drdr (12) This is the correct multipole expansion for the dispersion. For details see Stone The Theory of Intermolecular Forces (1996).

21 To obtain the usual expansion we must localize the non-local polarizabilities. This is usually done by another multipole expansion, but there are other methods. The result is that only terms involving the same site remain. This is an approximation that assumes that where γ 1. α aa e γ R aa, (13) When this is true, the localization is valid and we get disp = 1 2π = a a b b ( C ab 6 R 6 ab T ab tu T ab t u + C 7 ab Rab 7 0 α a tt (iw)αb uu (iw)dw + C 8 ab Rab 8 ) +

22 Pyridine dimer: WSM dispersion models (damped) disp,d (model) / kj mol-1-30 C 10,d C 6,iso,d / kj moldisp,tot

23 What are the non-local polarizabilities? The lowest rank polarizability is α aa 00,00. If V a is the potential at site a, the change in charge at site a is given by ˆQ a 00 = a α aa 00,00(V a V a ). (14) So such terms describe the flow of charge in response to a potential. They are therefore termed charge-flow polarizabilities. These are the analogue of the low (zero) frequency plasmon modes. They contribute R 2 terms to the dispersion energy.

24 Sum rule: Charge conservation requires that α(r, r ; ω)dr = 0, (15) and this results in the charge-flow sum rule αt,00 aa = 0. (16) a

25 Charge-flow contribution to the dispersion energy: 1 disp (00, 00) = 2π 1 R aa,bb ab R a b z L c : Large R 2 contribution. 0 α aa 00,00(iw)α bb 00,00(iw)dw L c z < L: Charge-fluctuations small compared to R ab and only R a b close to R ab contribute...(see fig)... (17)

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27 L c z < L: 1 disp (00, 00) = 2π 1 2π ( 0 1 R aa,bb ab R a b 1 a ab R ab R ab α aa 00,00(iw) 0 ) ( b α aa 00,00(iw)α bb 00,00(iw)dw α bb 00,00(iw) ) dw = 0 (18) But higher-order terms are non-zero.

28 L z: In this limit both R ab and R a b can be expanded in a multipole expansion about R = (0, 0, z): R 1 ab = R (r a r b ) 1 = R r ab 1 z r ab 2 z 3 Use sum-rule. 1 1 disp (00, 00) 2π 4z 6 rab 2 r a 2 b aa bb 0 α aa 00,00(iw)α bb 00,00(iw)dw C 6(00, 00) z 6. (19) How do the charge-flow terms behave for the chains?

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33 10 3 z 3.17 disp / n / a.u z 4.84 η = 1.00 η = 2.00 z z / a.u.

34 disp / n / a.u η = 2.0 Charge-flow only Up to dipole-dipole SAPT(DFT) η = z / a.u.

35 Angyan (2007,2009) has shown that the charge-flow polarizabilities are related to the the XC-hole by relating the density autocorrelation function S(r, r ) = ρ(r)δ(r rp) + ρ(r)h xc (r, r ) (20) to the FDDS using the Unsold approximation: Therefore, α aa 00,00 = Ω a = π ω α(r, r ; 0) = π ω S(r, r ). (21) Ω a α(r, r ; 0)drdr ( N a δ aa + ρ(r) Ω a Ω a h xc (r, r )drdr )

36 Analysis Failure of additivity (and the pair-wise C 6 /R 6 model) as the gap decreases and system length increases. Require explicit non-local terms. This non-locality is contained in the full density response function α(r, r ; ω). The non-local charge-flow terms do two things: (1) change the power-law for the interaction, and (2) enhance the dispersion interaction. Is this really important? What about carbon systems...? What about 2-D systems like graphene?

37 PAHs: SAPT(DFT) Pair Potential [ ( ) ] U ab = G exp α ab R ab ρ ab (Ω ab ) f 6 (R ab ) C 6,iso R }{{} 6 + E elst (model) ab }{{} short range long range Benzene dimer interaction energies (500 geometries). Larger PAH dimers: naphthalene, anthracene, pyrene. (a) (b) (c) (d) Validated on coronene dimers (not included in the fit).

38 Graphene & graphite Estimate of additive exfoliation energy of graphite by extrapolation: E bind (m) m = am b + m (22) 200 Binding Energy (kj mol -1 molecule -1 ) a (mev carbon -1 ) Number of Molecules in Stack n

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40 3-body dispersion For 3 atoms: Axilrod Teller Muto: disp (3 body) = C cos  cos ˆB cos Ĉ 9 RAB 3 R3 AC R3 BC C 9 R 9 (23)

41 Generalization for non-local, anisotropic, polarizabilities based on expression given by Strogryn (1971): E asymp disp [3] = π 0 α aa αβ (iω)αbb γδ (iω)αcc ɛφ ba (iω)dωtγβ T ɛδ cb T αφ ac (24)

42 Infinite, parallel chains with finite gap Length scale: R

43 Infinite, parallel chains with finite gap Length scale: R Select section of order R.

44 Infinite, parallel chains with finite gap Interaction of blue atom with others is: C 9 (R R) (25) R9

45 Infinite, parallel chains with finite gap But there are order R blue atoms, so interaction between the bits in the box is: C 9 (R R) R (26) R9

46 Infinite, parallel chains with finite gap And interaction per unit length is: C 9 R 9 (R R) R/R = C 9 R 7 (27)

47 z 4.48 η = 2.0 η = 1.0 (3-body) / n / a.u. disp z 6.80 z z z / a.u.

48 References A. J. Misquitta, J. Spencer, A. J. Stone and A. Alavi Dispersion interactions between semiconducting wires, to appear in Phys. Rev. B (I hope!) (online at All calculations were performed with the CamCASP suite of programs. Download: Non-local polarizabilities are calculated using the constrained density-fitting method described in AJM & Stone, J. Chem. Phys. 124, (2006). The WSM method for local polarizability models is described in AJM, Stone and Price, J. Comput. Theor. Chem. 4, 7 (2008), J. Comput. Theor. Chem. 4, 19 (2008). Dispersion models in AJM & Stone, Mol. Phys. 106, 1631 (2008).