Achieving linear-scaling computational cost for the polarizable continuum model of solvation

This work describes a new and low-scaling implementation of the polarizable continuum model (PCM) for computing the self-consistent solvent reaction field. The PCM approach is both general and accurate. It is applicable in the framework of both quantum and classical calculations, and also to hybrid quantum/classical methods. In order to further extend the range of applicability of PCM we addressed the problem of its computational cost. The generation of the finite-elements molecular cavity has been reviewed and reimplemented, achieving linear scaling for systems containing up to 500 atoms. Linear scaling behavior has been achieved also for the iterative solution of the PCM equations, by exploiting the fast multipole method (FMM) for computing electrostatic interactions. Numerical results for large (both linear and globular) chemical systems are discussed.     

Acceleration of Correlation-corrected Vibrational Self-consistent Field Calculation Times for Large Polyatomic Molecules

Acceleration of the correlation-corrected Vibrational self-consistent field (CC-VSCF) method for anharmonic calculations of vibrational states of polyatomic molecules is described. The acceleration assumes pairwise additive interactions between different normal modes, and employs orthogonality of the single-mode vibrational wavefunctions. This greatly reduces the effort in computing correlation effects between different vibrational modes, which is treated by second order perturbation theory in CC-VSCF. The acceleration can improve the scaling of the overall computational effort from N ⁶ to N ⁴, where N is the number of vibrational modes. Sample calculation times, using semi-empirical potential surfaces (PM3), are given for a series of glycine peptides. Large computational acceleration, and significant reduction of the scaling of the effort with system size, is found and discussed.     

An integral direct, distributed-data, parallel MP2 algorithm

Summary A scalable integral direct, distributed-data parallel algorithm for four-index transformation is presented. The algorithm was implemented in the context of the second-order M?ller-Plesset (MP2) energy evaluation, yet it is easily adopted for other electron correlation methods, where only MO integrals with two indices in the virtual orbitals space are required. The major computational steps of the MP2 energy are the two-electron integral evaluationO(N ⁴) and transformation into the MO basisO(ON ⁴), whereN is the number of basis functions, andO the number of occupied orbitals, respectively. The associated maximal communication costs scale asO(n _Σ O ² V N), whereV andn _Σ denote the number of virtual orbitals, and the number of symmetry-unique shells. The largest local and global memory requirements areO(N ²) for the MO coefficients andO(OV N) for the three-quarter transformed integrals, respectively. Several aspects of the implementation such as symmetry-treatment, integral prescreening, and the distribution of data and computational tasks are discussed. The parallel efficiency of the algorithm is demonstrated by calculations on the phenanthrene molecule, with 762 primitive Gaussians, contracted to 412 basis functions. The calculations were performed on an IBM SP2 with 48 nodes. The measured wall clock time on 48 nodes is less than 15 min for this calculation, and the speedup relative to single-node execution is estimated to 527. This superlinear speedup is a result of exploiting both the compute power and the aggregate memory of the parallel computer. The latter reduces the number of passes through the AO integral list, and hence the operation count of the calculation. The test calculations also show that the evaluation of the two-electron integrals dominates the calculation, despite the higher scaling of the transformation step.     

A fast adaptive multigrid boundary element method for macromolecular electrostatic computations in a solvent

A fast multigrid boundary element (MBE) method for solving the Poisson equation for macromolecular electrostatic calculations in a solvent is developed. To convert the integral equation of the BE method into a numerical linear equation of low dimensions, the MBE method uses an adaptive tesselation of the molecular surface by BEs with nonregular size. The size of the BEs increases in three successive levels as the uniformity of the electrostatic field on the molecular surface increases. The MBE method provides a high degree of consistency, good accuracy, and stability when the sizes of the BEs are varied. The computational complexity of the unrestricted MBE method scales as O(N_at), where N_at is the number of atoms in the macromolecule. The MBE method is ideally suited for parallel computations and for an integrated algorithm for calculations of solvation free energy and free energy of ionization, which are coupled with the conformation of a solute molecule. The current version of the 3-level MBE method is used to calculate the free energy of transfer from a vacuum to an aqueous solution and the free energy of the equilibrium state of ionization of a 17-residue peptide in a given conformation at a given pH in ∼ 400 s of CPU time on one node of the IBM SP2 supercomputer. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 569–583, 1997     

Efficient calculation of short-range Coulomb energies

An efficient algorithm for the calculation of short-range Coulomb energies is examined. The algorithm uses a boxing scheme and a prescreening for negligible integrals to evaluate the short-range Coulomb energy via computational work that scales only linearly with the size of the system. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 921–927, 1999     

Density functional theory for systems of very many atoms

The standard Kohn-Sham formulation of density functional theory (DFT ) is limited, for practical reasons, to systems of less than about 50-100 atoms. The computational effort scales as N, where N_at is the number of atoms and 2 < α > 3. (By comparison, conventional configuration interaction methods are limited to 5-10 atom systems.) This article deals with the prospect of practical methods that scale linearly in N_at and may thus allow calculations for systems of 10³-104 atoms. The physical reason (“near-sightedness”) for linear scaling is presented. Implementations of linear scaling DFT by the use of generalized Wannier functions or the one-particle density matrix are discussed. © 1995 John Wiley & Sons, Inc.     

Substitution effects in the 15N NMR chemical shifts of heterocyclic azines evaluated at the GIAO‐DFT level

A systematic study of the accuracy factors for the computation of ¹⁵N NMR chemical shifts in comparison with available experiment in the series of 72 diverse heterocyclic azines substituted with a classical series of substituents (CH₃, F, Cl, Br, NH₂, OCH₃, SCH₃, COCH₃, CONH₂, COOH, and CN) providing marked electronic σ‐ and π‐electronic effects and strongly affecting ¹⁵N NMR chemical shifts is performed. The best computational scheme for heterocyclic azines at the DFT level was found to be KT3/pcS‐3//pc‐2 (IEF‐PCM). A vast amount of unknown ¹⁵N NMR chemical shifts was predicted using the best computational protocol for substituted heterocyclic azines, especially for trizine, tetrazine, and pentazine where experimental ¹⁵N NMR chemical shifts are almost totally unknown throughout the series. It was found that substitution effects in the classical series of substituents providing typical σ‐ and π‐electronic effects followed the expected trends, as derived from the correlations of experimental and calculated ¹⁵N NMR chemical shifts with Swain–Lupton's F and R constants.     

Exploiting graphical processing units to enable quantum chemistry calculation of large solvated molecules with conductor-like polarizable continuum models

The conductor-like polarizable continuum model (C-PCM) with switching/Gaussian smooth discretization is a widely used implicit solvation model in quantum chemistry. We have previously implemented C-PCM solvation for Hartree-Fock (HF) and density functional theory (DFT) on graphical processing units (GPUs), enabling the quantum mechanical treatment of large solvated biomolecules. Here, we first propose a GPU-based algorithm for the PCM conjugate gradient linear solver that greatly improves the performance for very large molecules. The overhead for PCM-related evaluations now consumes less than 15% of the total runtime for DFT calculations on large molecules. Second, we demonstrate that our algorithms tailored for ground state C-PCM are transferable to excited state properties. Using a single GPU, our method evaluates the analytic gradient of the linear response PCM time-dependent density functional theory energy up to 80× faster than a conventional central processing unit (CPU)-based implementation. In addition, our C-PCM algorithms are transferable to other methods that require electrostatic potential (ESP) evaluations. For example, we achieve speed-ups of up to 130× for restricted ESP-based atomic charge evaluations, when compared to CPU-based codes. We also summarize and compare the different PCM cavity discretization schemes used in some popular quantum chemistry packages as a reference for both users and developers.     

Molecular orientations at interfaces by extended polarizable continuum model

This study presents an investigation into orientation of molecular solutes at the interface of liquid water and other media. The calculation of electrostatic free energy of molecular solute is based on an extension of the polarizable continuum model (PCM) to interfacial system. The extended PCM computational scheme is incorporated with the self‐consistent field procedure which is necessary to obtain more accurate electrostatic free energy and charge density distribution. The computation of non‐electrostatic energy for interfacial system is also realized. Applying the numerical procedure to molecular systems, N,N′‐diethyl‐p‐nitroaniline (DEPNA) at air/water interface and p‐nitrophenol (PNP) at cyclohexane/water interface, the average orientational angles are in reasonable agreement with the experimental results. Taking both the electrostatic and the non‐electrostatic energies into account, the analysis on the energy profiles shows that the electrostatic solvation energy is the dominant factor in determining the orientation angle for PNP, whereas for DEPNA, the orientation angle mainly depends on the cavitation energy. This suggests that, in addition to the electrostatic energy, taking the cavitation energy into account may provide a more complete view when we survey the molecular orientation at interface. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Scaling the Coulomb interaction in calculations of electron spectra of transition metal complexes

A simple technique of scaling two-electron integrals in ab initio calculations of the electronically excited states of transition metal complexes is proposed. This technique uses the fact that one-center two-electron integrals depend linearly on the scaling factor when Slater type functions are subjected to scaling transformation. This leads to a linear dependence of the d—d transition energy on the “scale” of Coulomb interaction, which allows one to affect the calculation result by varying the Slater exponential. To test the technique, ab initio configuration interaction and full active space calculations of the low excited states of the CrF ₆ ^3- , MnF ₆ ^2- , and VF ₆ ^3- complexes are performed. For transition elements, a basis of Slater type effective functions chosen from the optical spectra of the atoms and ions of transition elements is used. It is shown that in the STO-6G basis with effective exponentials, experimental transitions are reproduced with an accuracy of about 2000 cm^-1 even with the use of small active space determined by the orbitals localized on the central atom of the complex.     

The fast multipole boundary element method for molecular electrostatics: An optimal approach for large systems

We propose a fast implementation of the boundary element method for solving the Poisson equation, which approximately determines the electrostatic field around solvated molecules of arbitrary shape. The method presented uses computational resources of order O(N) only, where N is the number of elements representing the dielectric boundary at the molecular surface. The method is based on the Fast Multipole Algorithm by Rokhlin and Greengard, which is used to calculate the Coulombic interaction between surface elements in linear time. We calculate the solvation energies of a sphere, a small polar molecule, and a moderately sized protein. The values obtained by the boundary element method agree well with results from finite difference calculations and show a higher degree of consistency due to the absence of grid dependencies. The boundary element method can be taken to a much higher accuracy than is possible with finite difference methods and can therefore be used to verify their validity. © 1995 by John Wiley & Sons, Inc.     

Parallel processing in the isoelectric focusing chip

Investigation of isoelectric focusing (IEF) kinetics has been performed to provide the theoretical basis for miniaturization of classical IEF in immobilized pH-gradients. Standard IEF demands colinearity of the electric field and pH-gradient directions (serial devices). It is shown that the IEF separation process based on a continuous, serial pH gradient is incompatible with miniaturization of separation devices. The new realization of the IEF device by a parallel IEF chip is suggested and analyzed. The main separation tool of the device is a dielectric membrane (chip) with conducting channels that are filled by Immobiline gels of varying pH. The membrane is held perpendicular to the applied electric field and proteins are collected (trapped) in the channels whose pH are equal to the pI of the proteins. The pH value of the surrounded aqueous solution is not equal to any channel's pH. The fast particle transport between different channels takes place due to convection in the aqueous solution. The new device geometry introduces two new spatial scales to be considered: the scale of transition region from a solution to the gel in a channel and a typical channel size. The corresponding time scales defining the IEF process kinetics are analyzed and scaling laws are obtained. It is shown both theoretically and experimentally that parallel IEF accelerates the fractionation of proteins by their pI down to several minutes and enables possible efficient sample collection and purification.     

A fragment multipole approach to long-range Coulomb interactions in Hartree–Fock calculations on large systems

An efficient ab initio method for electronic structure calculations on extended molecular systems is presented, along with some illustrative applications. A division of the system into subunits allows the interactions to be separated into short- and long-range contributions, leading to a reduction of the computational effort from the original fourth-power size-dependence to one that is approximately quadratic. The short-range contributions to the Fock matrix are obtained in an essentially conventional fashion, while the long-range interactions are evaluated using a two-center multipole expansion formalism. The number of short-range contributions grows only linearly with the number of subunits, while the long-range contributions grow as N². Systematic studies of the computational efforts for systems of up to 99 water molecules organized as one-stranded chains, three-stranded chains, and three-dimensional clusters, as well as alkane chains with up to 69 carbon atoms, have been performed. In these model systems, the overall computational effort grows as N^K where 1 < K < 2.     

Reduced variable molecular dynamics

This article describes an extension to previously developed constraint techniques. These enhanced constraint methods will enable the study of large computational chemistry problems that cannot be easily handled with current constrained molecular dynamics (MD) methods. These methods are based on an O(N) solution to the constrained equations of motion. The benefits of this approach are that (1) the system constraints are solved exactly at each time step, (2) the solution algorithm is noniterative, (3) the algorithm is recursive and scales as O(N), (4) the algorithm is numerically stable, (5) the algorithm is highly amenable to parallel processing, and (6) potentially greater integration step sizes are possible. It is anticipated that application of this methodology will provide a 10- to 100-improvement in the speed of a large molecular trajectory as compared with the time required to run a conventional atomistic unconstrained simulation. It is, therefore, anticipated that this methodology will provide an enabling capacity for pursuing the drug discovery process for large molecular systems. © 1995 John Wiley & Sons, Inc.     

Solvent effects in the GIAO‐DFT calculations of the 15N NMR chemical shifts of azoles and azines

The calculation of ¹⁵N NMR chemical shifts of 27 azoles and azines in 10 different solvents each has been carried out at the gauge including atomic orbitals density functional theory level in gas phase and applying the integral equation formalism polarizable continuum model (IEF‐PCM) and supermolecule solvation models to account for solvent effects. In the calculation of ¹⁵N NMR, chemical shifts of the nitrogen‐containing heterocycles dissolved in nonpolar and polar aprotic solvents, taking into account solvent effect is sufficient within the IEF‐PCM scheme, whereas for polar protic solvents with large dielectric constants, the use of supermolecule solvation model is recommended. A good agreement between calculated 460 values of ¹⁵N NMR chemical shifts and experiment is found with the IEF‐PCM scheme characterized by MAE of 7.1 ppm in the range of more than 300 ppm (about 2%). The best result is achieved with the supermolecule solvation model performing slightly better (MAE 6.5 ppm). Copyright © 2014 John Wiley & Sons, Ltd.     

The Gaussian and augmented-plane-wave density functional method for ab initio molecular dynamics simulations

A new algorithm for density-functional-theory-based ab initio molecular dynamics simulations is presented. The Kohn–Sham orbitals are expanded in Gaussian-type functions and an augmented-plane-wave-type approach is used to represent the electronic density. This extends previous work of ours where the density was expanded only in plane waves. We describe the total density in a smooth extended part which we represent in plane waves as in our previous work and parts localised close to the nuclei which are expanded in Gaussians. Using this representation of the charge we show how the localised and extended part can be treated separately, achieving a computational cost for the calculation of the Kohn–Sham matrix that scales with the system size N as O(NlogN). Furthermore, we are able to reduce drastically the size of the plane-wave basis. In addition, we introduce a multiple-cutoff method that improves considerably the performance of this approach. Finally, we demonstrate with a series of numerical examples the accuracy and efficiency of the new algorithm, both for electronic structure calculations and for ab initio molecular dynamics simulations. Received: 15 December 1998 /Accepted: 18 February 1999 /Published online: 14 July 1999     

Improving performance of polarizable continuum model for study of large molecules in solution

We present a new implementation of the polarizable continuum model (PCM) that significantly improves its performance, especially for large solutes. This approach avoids the separation between electronic and nuclear sources in the calculation of the solvation charges, allowing the extension of iterative procedures to all the PCM versions [dielectric (D), conductor (C), and integral equation formalism (IEF)], so that the best method and/or algorithm can be selected, depending on the system at hand. In particular, the new balanced two-term iterative procedure with total charges avoids any nonlinear computational step and memory occupation. Furthermore, it also shows a good convergence for the C-PCM and IEF-PCM versions, which were quite problematic for the conventional separate charges approach. Also, first and second analytical derivatives are available in this context for Hartree–Fock and Kohn–Sham models. A number of examples are analyzed; they show that the new algorithms couple fully satisfactory numerical accuracy with remarkable computational efficiency. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 1186–1198, 1999     

Solving the low dimensional Smoluchowski equation with a singular value basis set

Reaction kinetics on free energy surfaces with small activation barriers can be computed directly with the Smoluchowski equation. The procedure is computationally expensive even in a few dimensions. We present a propagation method that considerably reduces computational time for a particular class of problems: when the free energy surface suddenly switches by a small amount, and the probability distribution relaxes to a new equilibrium value. This case describes relaxation experiments. To achieve efficient solution, we expand the density matrix in a basis set obtained by singular value decomposition of equilibrium density matrices. Grid size during propagation is reduced from (100–1000)^N to (2–4)^N in N dimensions. Although the scaling with N is not improved, the smaller basis set nonetheless yields a significant speed up for low‐dimensional calculations. To demonstrate the practicality of our method, we couple Smoluchowsi dynamics with a genetic algorithm to search for free energy surfaces compatible with the multiprobe thermodynamics and temperature jump experiment reported for the protein α₃D. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Evolution of the theoretical description of the isoelectric focusing experiment: II. An open system isoelectric focusing experiment is a transient,bidirectional isotachophoretic experiment

The carrier ampholytes-based (CA-based) isoelectric focusing (IEF) experiment evolved from Svensson's closed system IEF (constant spatial current density, absence of convective mixing, counter-balancing electrophoretic and diffusive fluxes yielding a steady state pH gradient) to the contemporary open system IEF (absence of convective mixing, large cross-sectional area electrode vessels, lack of counter-balancing electrophoretic- and diffusive fluxes leading to transient pH gradients). Open system IEF currently is described by a two-stage model: In the first stage, a rapid IEF process forms the pH gradient which, in the second stage, is slowly degraded by isotachophoretic processes that move the most acidic and most basic CAs into the electrode vessels. An analysis of the effective mobilities and the effective mobility to conductivity ratios of the anolyte, catholyte, and the CAs indicates that in open system IEF experiments a single process, transient bidirectional isotachophoresis (tbdITP) operates from the moment current is turned on until it is turned off. In tbdITP, the anolyte and catholyte provide the leading ions and the pI 7 CA or the reactive boundary of the counter-migrating H₃O⁺ and OH⁻ ions serves as the shared terminator. The outcome of the tbdITP process is determined by the ionic mobilities, pK_a values, and loaded amounts of all ionic and ionizable components: It is constrained by both the transmitted amount of charge and the migration space available for the leading ions. tbdITP and the resulting pH gradient can never reach steady state with respect to the spatial coordinate of the separation channel.     

Crosslinked polymer chains with excluded volume: A new class of branched polymers?

In this note microgels with and without excluded volume interactions are considered. Based on earlier exact computations on Gaussian mircogels, which are formed by self-crosslinking (with M crosslinks) of polymer chains with chainlength N, Flory type approximations are used to get first insight to their behavior in solution. It is shown that two different types of microgels exist: A special type of branched polymer whose size scales as R ∝ N^2/5/M^−1/5, instead of R ∝ N^1/2. The second type are c^*-microgels whose average mesh sizes r are swollen and form self avoiding walks with a scaling law of the form r = a(N/M)^3/5.