A Computatonal Vewpont on Classcal Densty Functonal Theory Matthew Knepley and Drk Gllespe Computaton Insttute Unversty of Chcago Department of Molecular Bology and Physology Rush Unversty Medcal Center Geometrc Modelng n Bomolecular Systems SIAM Lfe Scences 14, Charlotte, NC August 6, 2014 M. Knepley (UC) DFT LS 14 1 / 31
Outlne CDFT Intro 1 CDFT Intro 2 Model 3 Verfcaton M. Knepley (UC) DFT LS 14 3 / 31
CDFT Intro What s CDFT? A fast, accurate theoretcal tool to understand the fundamental physcs of nhomogeneous fluds M. Knepley (UC) DFT LS 14 4 / 31
CDFT Intro What s CDFT? For concentraton ρ ( x) of speces, solve mn ρ ( x) Ω[{ρ ( x)}] where Ω s the free energy. M. Knepley (UC) DFT LS 14 4 / 31
CDFT Intro What s CDFT? For concentraton ρ ( x) of speces, solve δω δρ ( x) = 0 whch are the Euler-Lagrange equatons. M. Knepley (UC) DFT LS 14 4 / 31
What s CDFT? DFT CDFT Intro Computes ensemble-averaged quanttes drectly Can have physcal resoluton n tme (µs) and space (Å) Requres an accurate Ω Requres sophstcated solver technology Can predct expermental results! For example, D. Gllespe, L. Xu, Y. Wang, and G. Messner, J. Phys. Chem. B 109, 15598, 2005 M. Knepley (UC) DFT LS 14 4 / 31
Outlne Model 1 CDFT Intro 2 Model Hard Sphere Repulson Bulk Flud Electrostatcs Reference Flud Densty Electrostatcs 3 Verfcaton M. Knepley (UC) DFT LS 14 5 / 31
Equlbrum Model ( µ bath ρ ( x) = exp µ ext ( x) µ ex ) ( x) kt where µ ex ( x) = µ HS ( x) + µ ES ( x) = µ HS ( x) + µ SC ( x) + z eφ( x) and ɛ φ( x) = e ρ ( x) M. Knepley (UC) DFT LS 14 6 / 31
Detals Model The theory and mplementaton are detaled n Knepley, Karpeev, Davdovts, Esenberg, Gllespe, An Effcent Algorthm for Classcal Densty Functonal Theory n Three Dmensons: Ionc Solutons, JCP, 2012. M. Knepley (UC) DFT LS 14 7 / 31
Outlne Model Hard Sphere Repulson 2 Model Hard Sphere Repulson Bulk Flud Electrostatcs Reference Flud Densty Electrostatcs M. Knepley (UC) DFT LS 14 8 / 31
Model Hard Spheres (Rosenfeld) Hard Sphere Repulson µ HS ( x) = kt α Φ HS n α (n α ( x ))ω α ( x x ) d 3 x where Φ HS (n α ( x )) = n 0 ln(1 n 3 ) + n 1n 2 n V 1 n V 2 1 n 3 ( n2 3 + 24π(1 n 3 ) 2 1 ) 3 n V 2 n V 2 n2 2 M. Knepley (UC) DFT LS 14 9 / 31
Hard Sphere Bass Model Hard Sphere Repulson n α ( x) = ρ ( x )ω α ( x x ) d 3 x where ω 0 ( r) = ω2 ( r) 4πR 2 ω 1 ( r) = ω2 ( r) 4πR ω 2 ( r) = δ( r R ) ω 3 ( r) = θ( r R ) ( r) = ωv 2 ( r) ω V 2 ( r) = r 4πR r δ( r R ) ω V 1 M. Knepley (UC) DFT LS 14 10 / 31
Hard Sphere Bass Model Hard Sphere Repulson All n α ntegrals may be cast as convolutons: n α ( x) = ρ ( x )ω α ( x x)d 3 x = = F 1 (F (ρ ) F (ω α )) F 1 (ˆρ ˆ ω α ) and smlarly µ HS ( x) = kt α ( ˆ F 1 Φ HS n α ˆ ω α ) M. Knepley (UC) DFT LS 14 11 / 31
Hard Sphere Bass Spectral Quadrature Model Hard Sphere Repulson There s a fly n the ontment: standard quadrature for ω α s very naccurate (O(1) errors), and destroys conservaton propertes, e.g. total mass We can use spectral quadrature for accurate evaluaton, combnng FFT of densty, ˆρ, wth analytc FT of weght functons. M. Knepley (UC) DFT LS 14 12 / 31
Hard Sphere Bass Spectral Quadrature Model Hard Sphere Repulson There s a fly n the ontment: standard quadrature for ω α s very naccurate (O(1) errors), and destroys conservaton propertes, e.g. total mass We can use spectral quadrature for accurate evaluaton, combnng FFT of densty, ˆρ, wth analytc FT of weght functons. M. Knepley (UC) DFT LS 14 12 / 31
Hard Sphere Bass Spectral Quadrature Model Hard Sphere Repulson ˆω 0 ( k) = ˆω2 ( k) 4πR 2 ˆω 2 ( k) = 4πR sn(r k ) k ˆω V 1 ˆω 1 ( k) = ˆω2 ( k) 4πR ( k) = ˆωV 2 ( k) ˆω V 2 ( 4πR k) = 4πı ( k 2 ˆω 3 ( k) = 4π k 3 ( sn(r k ) R k cos(r k ) sn(r k ) R k cos(r ) k ) ) M. Knepley (UC) DFT LS 14 13 / 31
Hard Sphere Bass Numercal Stablty Model Hard Sphere Repulson Recall that Φ HS (n α ( x )) =... + and note that we have analytcally n 3 2 24π(1 n 3 ) 2 n V 2 (x) 2 n 2 (x) 2 1. ( 1 n V 2 n V 2 n 2 2 However, dscretzaton errors n ρ near sharp geometrc features can produce large values for ths term, whch prevent convergence of the nonlnear solver. Thus we explctly enforce ths bound. ) 3 M. Knepley (UC) DFT LS 14 14 / 31
Outlne Model Bulk Flud Electrostatcs 2 Model Hard Sphere Repulson Bulk Flud Electrostatcs Reference Flud Densty Electrostatcs M. Knepley (UC) DFT LS 14 15 / 31
Model Bulk Flud (BF) Electrostatcs Bulk Flud Electrostatcs µ SC = µ ES,bath j x x R j ( ) c (2) j ( x, x ) + ψ j ( x, x ) ρ j ( x ) d 3 x Usng λ k = R k + 1 2Γ, where Γ s the MSA screenng parameter, we have c (2) j ( x, x ) + ψ j ( x, x ) = z z j e 2 8πɛ ( x x λ + λ j + 2λ λ j λ λ j (( ) 2 λ λ j 1 x x 2λ λ j + 2 ) ) M. Knepley (UC) DFT LS 14 16 / 31
Model Bulk Flud (BF) Electrostatcs Bulk Flud Electrostatcs µ SC = µ ES,bath j x x R j ( ) c (2) j ( x, x ) + ψ j ( x, x ) ρ j ( x ) d 3 x It s a convoluton too! M. Knepley (UC) DFT LS 14 16 / 31
Model Bulk Flud (BF) Electrostatcs Bulk Flud Electrostatcs µ SC = µ ES,bath j x x R j ( ) c (2) j ( x, x ) + ψ j ( x, x ) ρ j ( x ) d 3 x F ( ρ j ) = F ( ρj ρ bath ) = F ( ρj ) F (ρbath ) F ( ρ j ) was already calculated F (ρ bath ) s constant ( ( F x, x ) ( + ψ j x, x )) s constant c (2) j so we only calculate the nverse transform on each teraton. M. Knepley (UC) DFT LS 14 16 / 31
Model Bulk Flud (BF) Electrostatcs Bulk Flud Electrostatcs µ SC = µ ES,bath j x x R j ( ) c (2) j ( x, x ) + ψ j ( x, x ) ρ j ( x ) d 3 x FFT s also naccurate! M. Knepley (UC) DFT LS 14 16 / 31
Model Bulk Flud (BF) Electrostatcs Bulk Flud Electrostatcs µ SC = µ ES,bath j x x R j ( ) c (2) j ( x, x ) + ψ j ( x, x ) ρ j ( x ) d 3 x ĉ (2) j + ˆψ j = z ( z j e 2 1 ɛ I 1 λ (( ) 2 ) ) + λ j λ λ j I 0 + + 2 I 1 k 2λ λ j λ λ j 2λ λ j where I 1 = 1 k ( 1 cos( ) k R) I 0 = R k cos( k R) + 1 k 2 sn( k R) I 1 = R2 k cos( k R) + 2 R k 2 sn( k R) 2 k 3 ( 1 cos( k R) M. Knepley (UC) DFT LS 14 16 / 31 )
Outlne Model Reference Flud Densty Electrostatcs 2 Model Hard Sphere Repulson Bulk Flud Electrostatcs Reference Flud Densty Electrostatcs M. Knepley (UC) DFT LS 14 17 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs Expand around ρ ref ( x ), an nhomogeneous reference densty profle: [{ ( ρk y )}] µ SC µ SC kt kt 2,j c (1) c (2) j [{ ( ρ ref k y )}] [{ ρ ref k ( y )} ; x ] ρ ( x ) d 3 x [{ ( ρ ref k y )} ; x, x ] ( ρ x ) ( ρ j x ) d 3 x d 3 x wth ( ρ x ) ( = ρ x ) ρ ref ( x ) M. Knepley (UC) DFT LS 14 18 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs ρ ref [{ ( ρk x )} ; x ] = 3 4πR 3 SC ( x x ) α ( x ) ( ρ x ) d 3 x x R SC( x) Choose α so that the reference densty s charge neutral, and has the same onc strength as ρ Ths can model gradent flow M. Knepley (UC) DFT LS 14 19 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs ρ ref [{ ( ρk x )} ; x ] = 3 4πR 3 SC ( x x ) α ( x ) ( ρ x ) d 3 x x R SC( x) Choose α so that the reference densty s charge neutral, and has the same onc strength as ρ Ths can model gradent flow M. Knepley (UC) DFT LS 14 19 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs ρ ref [{ ( ρk x )} ; x ] = 3 4πR 3 SC ( x x ) α ( x ) ( ρ x ) d 3 x x R SC( x) Choose α so that the reference densty s charge neutral, and has the same onc strength as ρ Ths can model gradent flow M. Knepley (UC) DFT LS 14 19 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs We can rewrte ths expresson as an averagng operaton: ρ ref ( x) = ρ( x ) θ ( x x R SC ( x) ) 4π 3 R3 SC ( dx x) where R SC ( x) = We close the system usng ρ ( x)r ρ ( x) + 1 2Γ( x) Γ SC [ρ] ( x) = Γ MSA [ ρ ref (ρ) ] ( x). M. Knepley (UC) DFT LS 14 20 / 31
Model Reference Flud Densty Electrostatcs Reference Flud Densty (RFD) Electrostatcs We can rewrte ths expresson as an averagng operaton: ρ ref ( x) = ρ( x ) θ ( x x R SC ( x) ) 4π 3 R3 SC ( dx x) where R SC ( x) = We close the system usng ρ ( x)r ρ ( x) + 1 2Γ( x) Γ SC [ρ] ( x) = Γ MSA [ ρ ref (ρ) ] ( x). M. Knepley (UC) DFT LS 14 20 / 31
Outlne Verfcaton 1 CDFT Intro 2 Model 3 Verfcaton M. Knepley (UC) DFT LS 14 21 / 31
Consstency checks Verfcaton Check n α of constant densty aganst analytcs Check that n 3 s the combned volume fracton Check that wall soluton has only 1D varaton M. Knepley (UC) DFT LS 14 22 / 31
Verfcaton Sum Rule Verfcaton Hard Spheres βp HS bath = ρ (R ) where P HS bath = 6kT π ( ) ξ 0 + 3ξ 1ξ 2 2 + 3ξ3 2 3 usng auxlary varables ξ n = π 6 = 1 ξ 3 j ρ bath j σ n j n {0,..., 3} M. Knepley (UC) DFT LS 14 23 / 31
Verfcaton Sum Rule Verfcaton Hard Spheres Relatve accuracy and Smulaton tme for R = 0.1nm M. Knepley (UC) DFT LS 14 24 / 31
Verfcaton Sum Rule Verfcaton Hard Spheres Volume fracton ranges from 10 5 to 0.4 (very dffcult for MC/MD) M. Knepley (UC) DFT LS 14 24 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres R caton R anon 0.1nm 0.2125nm 1M 2 2 6 nm 3 and perodc Concentraton Doman Uncharged hard wall z = 0 Grd 21 21 161 M. Knepley (UC) DFT LS 14 25 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres R caton R anon 0.1nm 0.2125nm 1M 2 2 6 nm 3 and perodc Concentraton Doman Uncharged hard wall z = 0 Grd 21 21 161 M. Knepley (UC) DFT LS 14 25 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres Caton Concentratons for 1M concentraton M. Knepley (UC) DFT LS 14 26 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres Anon Concentratons for 1M concentraton M. Knepley (UC) DFT LS 14 27 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres Mean Electrostatc Potental for 1M concentraton M. Knepley (UC) DFT LS 14 28 / 31
Verfcaton Ionc Flud Verfcaton Charged Hard Spheres These results were frst reported n 1D n Densty functonal theory of the electrcal double layer: the RFD functonal, J. Phys.: Condens. Matter 17, 6609, 2005. M. Knepley (UC) DFT LS 14 29 / 31
Man Ponts Verfcaton Real Space vs. Fourer Space O(N 2 ) vs. O(N lg N) Accurate quadrature only avalable n Fourer space Electrostatcs Bulk Flud (BF) model can be qualtatvely wrong Reference Flud Densty (RFD) model demands complex algorthm Solver convergence Pcard was more robust Newton rarely entered the quadratc regme Stll no multlevel alternatve (nterpolaton?) M. Knepley (UC) DFT LS 14 30 / 31
Man Ponts Verfcaton Real Space vs. Fourer Space O(N 2 ) vs. O(N lg N) Accurate quadrature only avalable n Fourer space Electrostatcs Bulk Flud (BF) model can be qualtatvely wrong Reference Flud Densty (RFD) model demands complex algorthm Solver convergence Pcard was more robust Newton rarely entered the quadratc regme Stll no multlevel alternatve (nterpolaton?) M. Knepley (UC) DFT LS 14 30 / 31
Man Ponts Verfcaton Real Space vs. Fourer Space O(N 2 ) vs. O(N lg N) Accurate quadrature only avalable n Fourer space Electrostatcs Bulk Flud (BF) model can be qualtatvely wrong Reference Flud Densty (RFD) model demands complex algorthm Solver convergence Pcard was more robust Newton rarely entered the quadratc regme Stll no multlevel alternatve (nterpolaton?) M. Knepley (UC) DFT LS 14 30 / 31
Concluson Verfcaton The theory and mplementaton are detaled n An Effcent Algorthm for Classcal Densty Functonal Theory n Three Dmensons: Ionc Solutons, JCP, 2012. M. Knepley (UC) DFT LS 14 31 / 31
Concluson Verfcaton The theory and mplementaton are detaled n An Effcent Algorthm for Classcal Densty Functonal Theory n Three Dmensons: Ionc Solutons, JCP, 2012. M. Knepley (UC) DFT LS 14 31 / 31