Density functional theory in solution: Implementing an implicit solvent model for CASTEP and ONETEP
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1 Density functional theory in solution: Implementing an implicit solvent model for CASTEP and ONETEP, November 2016 James C. Womack University of Southampton, Southampton, UK
2 Overview Background The model Density functional theory Theoretical design Why (implicit) solvation? Implementation details Essential components of an implicit solvation model Performance of the model Extending the model Project road map Software (re)engineering Where are we now? Boundary conditions What's next? Implementation in CASTEP 2
3 Density functional theory Incredibly successful and widely used method Relatively low computational cost Includes electron correlation in a SCF model conventional Facilitated by linear-scaling Modern parallel computer hardware Efficient theoretical methods / approximations 3
4 The need for solvation Theorists like to think matter exists in a vacuum No environment to consider Theoretical simplifications possible But many natural and artificial systems exist in solution The environment can significantly affect the behaviour of molecules and their reactions Simulations which account for solvent effects allow us to address new questions, e.g. How is the activity of my catalyst impacted by the solvent used? How does the cell environment affect the folding of this protein? How will this pharmaceutical behave when dissolved in the stomach? Image: "Acid & base test using vinegar", courtesy of Carolina Biological Supply Company ( 4
5 Approaches to solvation Explicit solvent Implicit solvent Solute interacts with individual solvent molecules Solute interacts with implicit representation of solvent + Detailed description of solute-solvent interactions + Do not increase system size, N, compared to solute. + Can describe solvent structural effects, e.g. H-bonding + Solvent degrees of freedom are implicitly averaged in model - Significant increase in system size, N, and thus computational cost + Explicit treatment of solvent-solvent interactions is avoided - Many solvent degrees of freedom which need to be statistically averaged - Solvent structural effects are generally not described - Solvent-solvent interactions must be treated - Specific interactions between solute and solvent not accounted for - Typically require empirical parameterization 5
6 Approaches to solvation Explicit solvent Implicit solvent Explicit and implicit approaches can be combined, e.g. Many possible approaches, e.g. QM Solvent-accessible surface area Solvent exclusion by solute volume Dielectric screening by solvent MM Continuum Reference interaction site model (RISM) Continuum dielectric Focus of this work Reviews on implicit solvation: 1 M. Feig and C.L. Brooks III, Curr. Opin. Struct. Biol. 14, 217 (2004). 2 J. Tomasi, B. Mennucci, and R. Cammi, Chem. Rev. 105, 2999 (2005). 6
7 Continuum dielectric model Without continuum dielectric Potential due to charge density For periodic boundary conditions, can be solved in reciprocal space: Solution to homogeneous Poisson equation 7
8 Continuum dielectric model With continuum dielectric Hartree potential Solvent reaction potential Potential due to solute charge Potential due to dielectric medium P is the polarization density Solution to homogeneous Poisson equation As described in: J.-L. Fattebert and F. Gygi, Int. J. Quantum Chem. 93, 139 (2003). 8
9 Free energy of solvation What do we mean by solvation? Change in internal energy due to the environment is not the full story: Physically meaningful results must account for entropy, temperature. These are included in the thermodynamic free energy. We are typically interested in the free energy of solvation the reversible work required to transfer the solute in a fixed configuration from vacuum to solution 1 Electrostatic Non-electrostatic Interaction of solute with dielectric medium Cost of creating cavity in solvent Solute/solvent dispersion-repulsion interactions 1 J. Chen, C.L. Brooks III, and J. Khandogin, Curr. Opin. Struct. Biol. 18, 140 (2008). 9
10 Building an implicit solvation model The essential components Solute cavity Solute charge representation Model for non-electrostatic components of free energy change Cavity Many possibilities! But we are interested in just one combination today... 10
11 Minimal parameter implicit solvent model The ONETEP *TEP solvent model Solute charge From density functional theory Cavity Density dependent Smoothly varying dielectric function Non-electrostatic part From cavity surface area Effective surface tension Self-consistently solve KS equations subject to solvent Electrostatic/non-electrostatic terms depend on density A refined Fattebert-Gygi-Scherlis (FGS) solvation model J.-L. Fattebert and F. Gygi, J. Comput. Chem. 23, 662 (2002). J.-L. Fattebert and F. Gygi, Int. J. Quantum Chem. 93, 139 (2003). D.A. Scherlis, J.-L. Fattebert, F. Gygi, M. Cococcioni, and N. Marzari, J. Chem. Phys. 124, (2006). J. Dziedzic, H.H. Helal, C.-K. Skylaris, A.A. Mostofi, and M.C. Payne, EPL 95, (2011). J. Dziedzic, S.J. Fox, T. Fox, C.S. Tautermann, and C.-K. Skylaris, Int. J. Quantum Chem. 113, 771 (2013). 11
12 Minimal parameter implicit solvent model Solute charge obtained from DFT Cavity defined by smoothly varying density-dependent dielectric function Non-polar free energy contribution based on cavity surface area 12
13 Solving the NPE How to determine the reaction potential? Solve the non-homogeneous Poisson equation (NPE) Yields total electrostatic potential Subject to boundary conditions Current model uses approximate open BCs Assumes dielectric function is homogeneous 13
14 SCRF method Self-consistent reaction field Self-consistent solution for the solute charge and reaction potential Solve for the solute charge and reaction potential due to the interaction of the solute with the dielectric medium. Initial guess Solve KS equations Determine reaction potential Determine cavity surface area Converged result 14
15 Implementation highlights Smeared ionic cores1,2 Represent ionic core charge with Gaussians Solve NPE for total molecular charge density Avoids numerical issues with point-charges Electrostatic energy is evaluated for total charge Actual functional represents all electrostatic interactions Functional is modified to correct for smeared cores Coarse-grained boundary conditions2 Evaluation of BC integral is computationally costly Replace integral with sum over point charges Point charges represent summed charge of block of space 1 D.A. Scherlis, J.-L. Fattebert, F. Gygi, M. Cococcioni, and N. Marzari, J. Chem. Phys. 124, (2006). 2 J. Dziedzic, S.J. Fox, T. Fox, C.S. Tautermann, and C.-K. Skylaris, Int. J. Quantum Chem. 113, 771 (2013). 15
16 Implementation highlights Solve NPE using a multigrid solver (DL_MG)1 Efficient real-space solution to second-order Highly parallelised: MPI + OpenMP Higher-order accuracy obtained by defect correction Input and output on real-space Cartesian grid Defect correction loop DL_MG Model currently available in ONETEP2 Linear-scaling DFT code Strictly localized orbitals (NGWFs) Direct energy minimization approach Capable of calculations on 1000s of atoms 1 L. Anton, J. Dziedzic, C.-K. Skylaris, and M. Probert, Multigrid Solver Module for ONETEP, CASTEP and Other Codes (dcse, 2013). 2 C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, and M.C. Payne, J. Chem. Phys. 122, (2005). 16
17 How accurate is the model? Testing the model in ONETEP for 71 neutral molecules* Approach XC functional Error (RMS) Error (Max) *TEPa R PBE *TEPb PBE PCM PBE SMD M05-2X AMBER (classical) Free energies of solvation in kcal/mol a Self-consistent cavity, bfixed cavity PCM, SMD and AMBER force field PoissonBoltzmann are competing implicit solvent models. See Ref 3 for more calculation details. *Taken from blind tests in 1 J. P. Guthrie, J. Phys. Chem. B 113, 4501 (2009). 2 A. Nicholls et al., J. Med. Chem. 51, 769 (2008). Experimental energies of solvation from Minnesota solvation database. Results provided by J. Dziedzic, also appear in: 3 J. Dziedzic et al., EPL 95, (2011). 17
18 Extending the model ARCHER ecse project Existing implementation of minimal parameter solvent model in ONETEP Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP J. C. Womack, J. Dziedzic, C.-K. Skylaris, M. I. J. Probert J. Dziedzic, H. H. Helal, C.-K. Skylaris, L. Anton, A. A. Mostofi, M. C. Payne. Import defect correction code from ONETEP into DL_MG Alternative iterate change criteria for multigrid procedure Add support for periodic and mixed boundary conditions (currently only open BCs are supported) Periodic and mixed BCs for saline solvent model Improvements to multigrid solver (DL_MG) Extensions to solvent model in ONETEP Atomic forces terms for saline solvation energy terms Poisson-Boltzmann equation Implementation of solvent model in CASTEP 18
19 Extending the model ARCHER ecse project Existing implementation of minimal parameter solvent model in ONETEP Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP J. C. Womack, J. Dziedzic, C.-K. Skylaris, M. I. J. Probert J. Dziedzic, H. H. Helal, C.-K. Skylaris, L. Anton, A. A. Mostofi, M. C. Payne. DL_MG interface DL_MG-compatible parallel decomposition of charge density Evaluation of real-space electrostatic potential using multigrid (rather than in reciprocal space) Supporting functionality Real-space representation of ionic cores as Gaussian charge distributions ( smeared ion representation ) Support for open boundary conditions Local pseudopotentials in real-space Solvent model Evaluation of dielectric function Electrostatic energy terms Non-electrostatic (cavitation) energy terms Integration of solvation terms into self-consistency scheme Support for atomic forces evaluation with solvent terms Steric potentials for saline solvent model Improvements to multigrid solver (DL_MG) Extensions to solvent model in ONETEP Implementation of solvent model in CASTEP Software development Modification of CASTEP build process to include DL_MG New keywords for running simulations with solvent model Documentation (source and user manual) 19
20 Defect correction DL_MG solves the NPE to second order Definitions Typically yields insufficient accuracy in this context The defect correction can greatly improve accuracy Iteratively improve upon second order solution Uses higher-order finite differences outside of solver DL_MG ONETEP DL_MG See J. Dziedzic et al, Int. J. Quantum Chem. 113, 771 (2013), and references therein for further details. 20
21 Defect correction Why is it important? FD order Hartree energy in vacuum MT CC Phenol vacuum Hartree energy of phenol in kcal/mol (electron density only) MT: Martyna-Tuckerman CC: Cutoff-Coulomb Reference values for vacuum calculations with open BCs Coulombic component is for total charge density: Binding energy is for phenol/protein complex* solution Results provided by J. Dziedzic, also appear in: 1 J. Dziedzic et al, Int. J. Quantum Chem. 113, 771 (2013). * Protein = T4 Lysozyme L99A/M102Q 21
22 Defect correction Migrate defect correction code into DL_MG Avoids need to re-implement in CASTEP or other codes DL_MG becomes more a capable package More complicated than it appears! Different data representations Different parallel data distributions (1-D vs. 3-D) DL_MG ONETEP 22
23 Porting the model to CASTEP ONETEP Linear-scaling density-matrix DFT code Plane-wave pseudopotential DFT code Key relevant similarities: + Solute charge density on real space grid + Hartree (electrostatic) potential on real space grid + Nuclei & core electrons represented by pseudopotentials + Written in Fortran 2003 with modular design + MPI and OpenMP parallelism C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, and M.C. Payne, J. Chem. Phys. 122, (2005). S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.I.J. Probert, K. Refson, and M.C. Payne, Z. Kristallogr. 220, 567 (2005). 23
24 Porting the model to CASTEP ONETEP Linear-scaling density-matrix DFT code Plane-wave pseudopotential DFT code Key challenges: - Currently no support for open BCs in CASTEP - Need real-space local pseudopotentials - Need to calculate BCs for electrostatic potential - Currently no support for smeared ions in CASTEP - Need smeared ions to calculate total electrostatic potential - Different parallel data distributions (1-D vs. 3-D) C.-K. Skylaris, P.D. Haynes, A.A. Mostofi, and M.C. Payne, J. Chem. Phys. 122, (2005). S.J. Clark, M.D. Segall, C.J. Pickard, P.J. Hasnip, M.I.J. Probert, K. Refson, and M.C. Payne, Z. Kristallogr. 220, 567 (2005). 24
25 Boundary conditions Adding periodic and mixed boundary conditions to model Currently only supports open BCs, i.e. We plan to implement periodic/mixed BCs, i.e. Mixed: periodic along some directions, open along others This capability is useful for some types of material, e.g. Surfaces (periodic in 2-dimensions) Polymers (periodic in 1-dimension) Periodic/mixed BCs will affect electrostatics Pseudopotentials Pseudoion core interactions Will require careful consideration... Images: Polymer from G. Boschetto, Pt on graphene from L. Verga. 25
26 Project roadmap ARCHER ecse project Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP J. C. Womack, J. Dziedzic, C.-K. Skylaris, M. I. J. Probert Improvements to multigrid solver (DL_MG) Extensions to solvent model in ONETEP Implementation of solvent model in CASTEP Imported defect correction code from ONETEP into DL_MG NB Only 1-D parallel decomposition, open BCs currently Upcoming tasks Periodic and mixed BCs for defect correction Generalized 3-D parallel data decomposition Already supported DL_MG's core second-order solver code, but need to be extended to the new defect correction component 26
27 Project roadmap ARCHER ecse project Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP J. C. Womack, J. Dziedzic, C.-K. Skylaris, M. I. J. Probert Zero BCs: Open BCs: Improvements to multigrid solver (DL_MG) interfaced with CASTEP DL_MG Integrated into CASTEP build system Evaluation of Hartree potential using multigrid Extensions to solvent model in ONETEP NB. Only zero BCs, single MPI process (with OpenMP) Upcoming tasks Implement open BCs in CASTEP Implementation of solvent model in CASTEP Calculation of potential on cell faces (boundary conditions) Real-space local pseudopotentials DL_MG-compatible parallel decomposition of charge Should be contiguous in x, y and z directions Open question: How best to achieve this? Gather data to 1 MPI process and run DL_MG here? Gather data to 1 MPI process, rearrange and redistribute? 27
28 Project roadmap ARCHER ecse project Implementation and optimisation of advanced solvent modelling functionality in CASTEP and ONETEP J. C. Womack, J. Dziedzic, C.-K. Skylaris, M. I. J. Probert Improvements to multigrid solver (DL_MG) support for periodic and mixed boundary conditions AddRequires periodic and mixed BCs in DL_MG Extensions to solvent model in ONETEP Currently not implemented in defect correction code Could implement in CASTEP & ONETEP simultaneously More progress on CASTEP implementation necessary Implementation of solvent model in CASTEP 28
29 Acknowledgements ecse project collaborators: Jacek Dziedzic (Southampton) Chris-Kriton Skylaris (Southampton) Lucian Anton (Cray Inc.) Matt Probert (York) Previous work on solvent model: Hatem H. Helal Arash A. Mostofi Mike C. Payne Jacek Dziedzic Chris-Kriton Skylaris Lucian Anton 29
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