van der Waals corrections to approximative DFT functionals
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1 van der Waals corrections to approximative DFT functionals Ari P Seitsonen Physikalisch Chemisches Institut, Universität Zürich February 8, 2011
2 van der Waals interactions van der Waals interactions important in several cases Rare gases Non-polar molecular systems with closed-shell consituents Biology... Even in bulk Au the van der Waals interactions significant! [Pyykkö, 2004] Typical (semi-)local approximations like LDA, GGA cannot reproduce it
3 van der Waals interactions Example: Benzene crystal
4 Outline Theory of van der Waals interactions Popular/recently introduced approximations Example Liquid water
5 van der Waals interactions Arise from instantaneous fluctations in electron densities Fluctuation-dissipation theory: E c = 1 2π 1 α=0 ω=0 r [αχ 0 (r 1, r 3, ω) W (r 3, r 4, ω) χ 0 (r 4, r 2, ω) +α 2 χ 0 (r 1, r 3, ω) W (r 3, r 4, ω) χ 0 (r 4, r 5, ω) W (r 5, r 6, ω) χ 0 (r 6, r 2, ω) + O(α 3 ) +... ] dr i After some approximations: E asymp disp = 3 π 1 R 6 0 α A αβ αb γδ dω i
6 van der Waals interactions After some approximations: E asymp disp = 3 π 1 R 6 0 α A αβ αb γδ dω An idea: E asymp disp 1 R 6 Divergence has to be cut off at short radii
7 vdw semi-empirical treatments Most popular now: Grimme 2006 DFT-D2 Grimme et alia 2010 DFT-D3 General properties: Do not depend on electron densities, only on atomic coordinates Usually pair-wise Energy of electronic system no longer minimimised in equilibrium structure
8 vdw XC-D2 Additional term into energy: Pair-wise interaction: Damping function: C 6 parametrised E DFT D2 = E KS DFT + E disp N at E disp = s 6 N at i=1 j=i+1 C ij 6 R 6 ij f damp ( Rij ) f damp ( Rij ) = e d(r ij /R r 1)
9 vdw XC-D3 Like XC-D2, but includes R 8 term three-body term dependency on local coordination
10 vdw XC-D3; pair-wise terms Coefficients C AB 6 = 3 π α A (iω) α B (iω) dω ω=0 C8 AB = 3C6 AB Q A Q B r 4 A Q A = s 42 Z A r 2 A
11 vdw XC-D3; three-body term From perturbation theory E ABC = CABC 9 (3 cos θ a cos θ b cos θ c + 1) (r AB r BC r CA ) 3 C ABC 9 = 3 π C9 AB ω=0 α A (iω) α B (iω) α C (iω) dω C6 ABCBC 6 CCA 6 Energy E (3) = ABC f d,(3) ( r ABC ) E ABC
12 vdw XC-D3; coordination number Dependency on local environment: Eg sp 2 versus sp 3 carbon atoms CN A = N at B A e k 1(k 2 (R A,cov +R B,cov )/r AB 1) C AB 6 ( CN A, CN B) = Z W N A Z = W = i N A N B i j N B L ij j C AB 6,ref (CNA, CN B )L ij
13 Comparison vdw XC-D3; performance
14 vdw (e)dcacp Additional term in external (in practise, pseudo) potential: V (e)dcacp ( l r, r ) = l m= l p l (r) r l exp r 2 /(2σ 2 2 ) Y lm (ˆr) p l (r)σ 1 p l (r )Y lm (ˆr ) Parametres sigma 1, σ 2 optimised in reference system: ( min E ref R ref) E (R ref) 2 N at ( + w F I R ref) 2 I=1 Parametrised values used in applications Does modify electronic structure, minimise total energy Does not reproduce asymptotic R 6 tail
15 vdw (e)dcacp Example: Benzene/bi-layer graphite
16 vdw real functional Langreth-Lundqvist et alia, vdw-df or vdw-ll Truly non-local: Ec nl [n] = n(r)φ(r, r )n(r ) dr dr r r Usually accompanied with revpbe-gga functional (reproduces best Hartree-Fock-exchange) Newest edition:
17 vdw XC-D3; performance Left: Adenine-adening; right: bi-layer graphite
18 van der Waals and CP2K Approximations currently implemented: Grimme, 2006: XC-D2 Grimme et alia, 2010: XC-D3 (e)dcacp
19 CP2K Input Example: XC-D2 1 &FORCE_EVAL &DFT &XC 10 &XC_FUNCTIONAL BLYP 11 &END XC_FUNCTIONAL &vdw_potential 14 DISPERSION_FUNCTIONAL PAIR_POTENTIAL 15 &PAIR_POTENTIAL 16 TYPE DFTD2 17 REFERENCE_FUNCTIONAL BLYP 18 R_CUTOFF [angstrom] &END PAIR_POTENTIAL 20 &END vdw_potential &END XC &END DFT &END FORCE_EVAL
20 CP2K Input Example: XC-D3 1 &FORCE_EVAL &DFT &XC 6 &XC_FUNCTIONAL BLYP 7 &END XC_FUNCTIONAL 8 9 &vdw_potential 10 DISPERSION_FUNCTIONAL PAIR_POTENTIAL 11 &PAIR_POTENTIAL 12 TYPE DFTD3 13 CALCULATE_C9_TERM.TRUE. 14 REFERENCE_C9_TERM.TRUE. 15 LONG_RANGE_CORRECTION.TRUE. 16 PARAMETER_FILE_NAME ${DATA_PATH}/dftd3.dat 17 VERBOSE_OUTPUT.FALSE. 18 REFERENCE_FUNCTIONAL BLYP 19 R_CUTOFF [angstrom] EPS_CN 1.0E-6 21 # COORDINATION_NUMBERS # COORDINATION_NUMBERS &END PAIR_POTENTIAL 24 &END vdw_potential &END XC &END DFT &END FORCE_EVAL
21 Example
22 Summary van der Waals interactions can be crucial (biology etc) Currently implemented in CP2K: XC-D2, XC-D3, (e)dcacp Water is not always water
23 Literature XC-D2: Stefan Grimme, Journal of Computational Chemistry 27, 1787 (2006) XC-D3: Stefan Grimme, Jens Antony, Stephan Ehrlich and Helge Krieg, A consistent and accurate ab initio parametrisation of density functional dispersion correction (DFT-D) for the 94 elements H-Pu, Journal of Chemical Physics 132, (2010), DOI: / (e)dcacp: O Anatole von Lilienfeld, Ivano Tavernelli, Ursula Röthlisberger, and Daniel Sebastiani, Optimisation of effective atom centred potentials for London dispersion forces in density functional theory, Physical Review Letters 93, (2004); DOI: /PhysRevLett
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