1 pubs.acs.org/jctc Quantum Chemistry for Solvated Molecules on Graphical Processing Units Using Polarizable Continuum Models Fang Liu,, Nathan Luehr,...

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Quantum Chemistry for Solvated Molecules on Graphical Processing Units Using Polarizable Continuum Models Fang Liu,†,‡ Nathan Luehr,†,‡ Heather J. Kulik,†,§ and Todd J. Martínez*,†,‡ †

Department of Chemistry and The PULSE Institute, Stanford University, Stanford, California 94305, United States SLAC National Accelerator Laboratory, Menlo Park, California 94025, United States § Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, United States ‡

S Supporting Information *

ABSTRACT: The conductor-like polarization model (C-PCM) with switching/Gaussian smooth discretization is a widely used implicit solvation model in chemical simulations. However, its application in quantum mechanical calculations of large-scale biomolecular systems can be limited by computational expense of both the gas phase electronic structure and the solvation interaction. We have previously used graphical processing units (GPUs) to accelerate the ﬁrst of these steps. Here, we extend the use of GPUs to accelerate electronic structure calculations including C-PCM solvation. Implementation on the GPU leads to signiﬁcant acceleration of the generation of the required integrals for C-PCM. We further propose two strategies to improve the solution of the required linear equations: a dynamic convergence threshold and a randomized block-Jacobi preconditioner. These strategies are not speciﬁc to GPUs and are expected to be beneﬁcial for both CPU and GPU implementations. We benchmark the performance of the new implementation using over 20 small proteins in solvent environment. Using a single GPU, our method evaluates the C-PCM related integrals and their derivatives more than 10× faster than that with a conventional CPU-based implementation. Our improvements to the linear solver provide a further 3× acceleration. The overall calculations including C-PCM solvation require, typically, 20−40% more eﬀort than that for their gas phase counterparts for a moderate basis set and molecule surface discretization level. The relative cost of the C-PCM solvation correction decreases as the basis sets and/or cavity radii increase. Therefore, description of solvation with this model should be routine. We also discuss applications to the study of the conformational landscape of an amyloid ﬁbril. as GCOSMO,8 and IEF-PCM9−11) are the most popular and accurate of these ASC algorithms. While PCM calculations are much more eﬃcient than their explicit solvent counterparts, their application in quantum mechanical calculations of large-scale biomolecular systems can be limited by CPU computational bottlenecks.4 Graphical processing units (GPUs), which are characterized as stream processors,12 are especially suitable for parallel computing involving massive data, and numerous groups have explored their use for electronic structure theory.13−24 Implementation of gas phase ab initio molecular calculations19−21 on GPUs led to greatly enhanced performance for large systems.25,26 Here, we harness the advances27 of stream processors to accelerate the computation of implicit solvent eﬀects, eﬀectively reducing the cost of PCM calculations. These improvements will enable simulations of large biomolecular systems in realistic environments.

1. INTRODUCTION Modeling the inﬂuence of solvent in quantum chemical calculations is of great importance to understanding solvation eﬀects on electronic properties, nuclear distributions, spectroscopic properties, acidity/basicity, and mechanisms of enzymatic and chemical reactions.1−4 Explicit inclusion of solvent molecules in quantum chemical calculations is computationally expensive and requires extensive conﬁgurational sampling to determine equilibrium properties. Implicit models based on a dielectric continuum approximation are much more eﬃcient and are an attractive conceptual framework to describe solvent eﬀects within a quantum mechanical (QM) approach.1 Among these implicit models, the apparent surface charge (ASC) methods are popular because they are easily implemented within QM algorithms and can provide excellent descriptions of the solvation of small- and medium-sized molecules when combined with empirical corrections for nonelectrostatic solvation eﬀects.4 ASC methods are based on the fact that the reaction potential generated by the presence of the solute charge distribution may be described in terms of an apparent charge distribution spread over the solute cavity surface. Methods such as the polarizable continuum models5 (PCM) and its variants such as conductor-like models (COSMO,6 C-PCM,7 also known © 2015 American Chemical Society

Received: April 20, 2015 Published: June 10, 2015 3131

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2. CONDUCTOR-LIKE POLARIZABLE CONTINUUM MODEL The original conductor-like screening model (COSMO) was introduced by Klamt and Schuurmann.6 In this approach, the molecule is embedded in a dielectric continuum with permittivity ε, and the solute forms a cavity within the dielectric with unit permittivity. In this electrostatic model, the continuum is polarized by the solute, and the solute responds to the electric ﬁeld of the polarized continuum. The electric ﬁeld of the polarized continuum can be described by a set of surface polarization charges on the cavity surface. Then, the electrostatic component of the solvation free energy can be represented by the interaction between the polarization charges and solute, in addition to the self-energy of the surface charges. For numerical convenience, the polarization charge is often described by a discretization in terms of M ﬁnite charges residing on the cavity surface. The locations of the surface charges are ﬁxed, and the values of the charges can be determined via a set of linear equations Aq = −f (Bz + c)

screened k Lμν = −(μ|Jk̂ |ν )

=−

erf(ζkl′ | rk⃗ − rl |⃗ ) | rk⃗ − rl |⃗ ζk −1 Sk 2π

Akk =

BJk =

ζk =

ck =

∑ PμνLμνk μν

(6)

ζ RI wk

(7)

where ζ is an optimized exponent for the speciﬁc Lebedev quadrature level being used (as tabulated28 by York and Karplus) and wk is the Lebedev quadrature weight for the kth point. The combined exponent is then given as ζkl′ =

(1)

ζkζl ζk2

+ ζl2

(8)

The atom-centered Gaussian basis functions used to describe the solute electronic wave function are denoted as ϕμ and ϕν and Pμν is the corresponding density matrix element. Finally, the switching function which smoothes the boundary of the van der Waals spheres corresponding to each atom (and thus makes the solvation energy continuous) is given by Sk. For ISWIG, this switching function is expressed as Sk =

atoms

∏

J ,k∉J

Swf ( rk⃗ ,RJ⃗ ) = 1 −

Swf ( rk⃗ ,RJ⃗ ) 1 {erf[ζk(RJ − | rk⃗ − RJ⃗ |)] 2

+ erf[ζk(RJ + | rk⃗ − RJ⃗ |)]}

(9)

Similar, but more involved, deﬁnitions are used in SWIG (which we have also implemented, but only ISWIG will be used in this paper). Once q is obtained by solving eq 1, the contribution of solvation eﬀects to the Fock matrix is given by ΔFS =

M

∑ qkLμνk

(10)

k=1

where the Fock matrix of the solvated system is Fsolvated = F0 + ΔFS and F0 is the usual gas phase Fock operator. This modiﬁed Fock matrix is then used for the self-consistent ﬁeld (SCF) calculation. As usual, the atom-centered basis functions are contractions over a set of primitive atom-centered Gaussian functions

ϕμ( r ⃗) =

(2)

lμ

∑ cμi χi ( r ⃗)

(11)

i=1

Thus, the one electron integrals from eq 6 that are needed for the calculation of c and ΔFS are

(3)

erf(ζk| rk⃗ − RJ⃗ |) | rk⃗ − RJ⃗ |

erf(ζk| r ⃗ − rk⃗ |) ϕν( r ⃗) d r ⃗ | r ⃗ − rk⃗ |

where rk⃗ is the location of the kth Lebedev point and R⃗ J is the location of the Jth nucleus with atomic radius RJ. The Gaussian exponent for the kth point charge belonging to the Ith nucleus is given as

where q ∈ M is the discretized surface charge distribution, A ∈ M × M is the Coulomb interaction between unit polarization charges on two cavity surface segments, B ∈ M × N is the interaction between nuclei and a unit polarization charge on a surface segment, z ∈ N is the vector of nuclear charges for the N atoms in the solute molecule, and c ∈ M is the interaction between the unit polarization charge on one surface segment and the total solute electron density. The parameter f = (ε − 1)/(ε + k) is a correction factor for a polarizable continuum with ﬁnite dielectric constant. In the original COSMO paper, k was set to 0.5. Later work by Truong and Stefanovich8 (GCOSMO) and Cossi and Barone7 (C-PCM) suggested that k = 0 was more appropriate on the basis of an analogy with Gauss’ law. We use k = 0 throughout in this work, although both cases are implemented in our code. The precise form of the A, B, and c matrices/vectors depends on the speciﬁc techniques used in cavity discretization. In order to obtain continuous analytic gradients of solvation energy, York and Karplus28 proposed the switching-Gaussian formalism (SWIG), where the cavity surface van der Waal spheres are discretized by Lebedev quadrature points. Polarization charges are represented as spherical Gaussians centered at each quadrature point (and not as simple point charges). Lange and Herbert29 proposed another form of switching function, referred to here as improved Switching-Gaussian (ISWIG). Both SWIG and ISWIG formulations use the following deﬁnitions for the fundamental quantities A, B, and c

Akl =

∫ ϕμ( r ⃗)

(μ|Jk̂

screened

(4)

|ν ) =

lμ

lν

∑ ∑ cμi cνj[χi |Jk̂ i=1 j=1

screened

|χj ]

(12)

where we use brackets to denote one-electron integrals over primitive basis functions and parentheses to denote such integrals for contracted basis functions. In the following, we

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use the indices μ, ν for contracted basis functions, and the indices i, j, k, and l are used to refer to primitive Gaussian basis functions. Smooth analytical gradients are available for COSMO SWIG/ ISWIG calculations due to the use of a switching function that makes surface discretization points smoothly enter/exit the cavity deﬁnition. The total electrostatic solvation energy of COSMO is ΔGels = (Bz)† q + c†q +

1 † q Aq 2f

(13)

Thus, the PCM contribution to the solvated SCF energy gradient with respect to the nuclear coordinates RI of the Ith atom is given by 1 ∇*RI (ΔGels) = z†(∇RI B†)q + (∇*RI c†)q + q†(∇RI A)q 2f (14)

where ∇*RI denotes that the derivative with respect to the density matrix is not included. The contribution of changes in the density matrix to the gradient is readily obtained from the gradient subroutine in vacuo (see the Supporting Information for details). In the COSMO-SCF process described above, there are three computationally intensive steps: (1) building c and ΔFS from eqs 5 and 10; (2) solving the linear system in eq 1; (3) evaluating the PCM gradients from eq 14. We discuss our acceleration strategies for each of these steps in Section 4 below.

Figure 1. Molecular geometries used to benchmark the correlation between COSMO energy error and CG convergence threshold.

summary of the name, size, and preparation method for these molecules, together with coordinate ﬁles, is provided in the Supporting Information. In the performance section, we select a test set of 20 experimental protein structures identiﬁed by Kulik et al.,39 where inclusion of a solvent environment was essential to ﬁnd optimized structures in good agreement with experimental results. The molecules are listed in the Supporting Information and range in size from around 100 to 500 atoms. Most were obtained from aqueous solution NMR experiments. For these test molecules, we conduct a number of restricted Hartree−Fock (RHF) single-point energy and nuclear gradient evaluations with the 6-31G basis set.40 These calculations are carried out in both PCM environment and in the gas phase. For some of these test molecules, we also use basis sets of diﬀerent sizes, including STO-3G,41 3-21G,42 6-31G*, 6-31G**,43 6-31++G,44 6-31+G*, and 6-31++G*. We use these test molecules to identify optimum algorithm parameters and to study the performance of our approach as a function of basis set size. In the Applications section, we investigate how COSMO solvation inﬂuences the conformational landscape of a model protein by expansive geometry optimization with both RHF and the range corrected exchange-correlation functional ωPBEh.45 Both of these approximations include the full-strength longrange exact exchange interactions that are vital to avoid selfinteraction and delocalization errors. Such errors can lead to unrealistically small HOMO−LUMO gaps.46 We obtain seven diﬀerent types of stationary point structures for the protein in gas phase and in COSMO aqueous solution (ε = 78.39) with a number of diﬀerent basis sets (STO-3G, 3-21G, 6-31G). Grimme’s D3 dispersion correction47 is applied to some minimal basis set calculations, here referred to as RHF-D and ωPBEh-D.

3. COMPUTATIONAL METHODS We have implemented a GPU-accelerated COSMO formulation in a development version of the TeraChem package. All COSMO calculations use parameters stated as follows unless otherwise speciﬁed. The environment dielectric constant corresponds to aqueous solvation (ε = 78.39). The cavity uses an ISWIG29 discretization density of 110 points/atom and cavity radii that are 20% larger than the Bondi radii.30−32 An ISWIG screening threshold of 10−8 is used, meaning that molecular surface (MS) points with a switching function value less than this threshold are ignored. The conjugate gradient33 (CG) method is used to solve the PCM linear equations, with our newly proposed random Jacobi preconditioner (RBJ) with block size 100. The electrostatic potential matrix A is explicitly stored and used to calculate the necessary matrix−vector products during CG iterations. In order to verify correctness and also to assess performance, we compare our code with the CPU-based commercial package, Q-Chem34 4.2. For all of the comparison test cases, Q-Chem uses exactly the same PCM settings as those in TeraChem, except for the CG preconditioner. Q-Chem uses diagonal decomposition together with a block-Jacobi preconditioner based on an octree spatial partition. We use OpenMP paralellization in Q-Chem because we found this to be faster than its MPI35 parallelized version based on our tests on these systems. In order to use OpenMP parallelization in this version of Q-Chem, we use the fast multipole method36,37 (FMM) and the no matrix mode, which rebuilds the A matrix on the ﬂy. We use a test set of six molecules (Figure 1) to investigate the relationship of the threshold and resulting error in the CG linear solve. For each molecule, we used ﬁve diﬀerent structures: one optimized structure and four distorted structures obtained by performing classical molecular dynamics (MD) simulations on the ﬁrst structure with Amber ﬀ03 force ﬁelds38 at 500 K. A

4. ACCELERATION STRATEGIES 4.1. Integral Calculation on GPUs. Building c and ΔFS requires calculation of one-electron integrals and involves a signiﬁcant amount of data parallelism, making these well suited for calculation on GPUs. The ﬂowchart in Figure 2 summarizes our COSMO-SCF implementation. Following our gas phase 3133

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Figure 2. Flowchart for COSMO SCF implementation.

Figure 3. Algorithm for calculating ΔFS for ss integrals of a system composed of 3 s shells (the ﬁrst and the third s shells contain 3 primitive Gaussian function each; the second s shell has 2 primitive Gaussian functions). On top of the graph, the pale green array represents primitive pairs belonging to ss shell pairs. The GPU cores are represented by orange squares (threads) embedded in pale yellow rectangles (one-dimensional blocks with 16 threads/ block). The output is an array where each entry stores a primitive pair integral. Primitive pair integrals are ﬁnally added to the Fock matrix entry of the conrresponding contracted function pair. All red lines and text indicate contracted Gaussian integrals. Blue arrows and text indicate memory operations.

SCF implementation,19,48 the COSMO related integrals needed for c and ΔFS are calculated in a direct SCF manner using GPUs. Here, each GPU thread calculates integrals corresponding to one ﬁxed primitive pair. However, the rest of the calculation, most signiﬁcantly the solution of eq 1, is handled on the CPU. From eq 5 and 10, it follows that the calculations for c and ΔFS are very similar, so one might be tempted to evaluate Lkμv once and use it in both calculations. In practice, this approach is not eﬃcient. Because ΔFS depends on the surface charge distribution (qk) and therefore on c through eq 1, c and ΔFS cannot be computed simultaneously. As the storage requirements for Lkμv are excessive, it is ultimately more eﬃcient to calculate the integrals for c and ΔFS separately from scratch. The algorithm for evaluating ΔFS is shown schematically in Figure 3 for a system with three s shells and a GPU block size of 1

× 16 threads. The ﬁrst and the third s shells contain 3 primitive Gaussian functions each; the second s shell has 2 primitive Gaussian functions. A block of size 1 × 16 is used for illustrative purposes. In practice, a 1 × 128 block is used for optimal occupancy and memory coalescence. Primitive pairs, χiχj, that make negligible contributions are not calculated, and these are determined by using a Schwartz-like bound49 with a cutoﬀ, εscreen, of 10−12 atomic units [ij|Schwartz = [χi χj |χi χj ]1/2 < ε screen

(15)

The surviving pair quantities are preloaded to the GPU global memory, and each is fetched by a unique GPU thread at the beginning of the integral kernel. Quantities related to each molecular surface (MS) grid point (charge qk, coordinates rk, switching-Gaussian exponent ζk) are also preloaded in global 3134

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Figure 4. MS point-driven algorithm for building c for ss integrals of a system composed of 3 s shells (the ﬁrst and the third s shells contain 3 primitive Gaussian functions each; the second s shell has 2 primitive Gaussian functions). The pale green array at the top of the ﬁgure represents primitive pairs belonging to ss shell pairs. The GPU cores are represented by orange squares (threads) embedded in pale yellow rectangles (one-dimensional blocks with 16 threads/block). The output is an array where each entry stores a primitive pair integral. Primitive pair integrals are ﬁnally added to the Fock matrix entry of the corresponding contracted function pair. All red lines and text indicate contracted Gaussian integrals. Blue arrows and text indicate memory operations.

memory. Each thread loops over all MS grid points to accumulate the Coulomb interaction between its primitive pair and all grid points as follows. screened ΔFijS = −∑ qk cμicνj[χi |Jk̂ |χj ] k

This sum is then stored in an output array in global memory of size M × nb, where nb is the number of GPU thread blocks in use and M is the number of MS grid points. After looping over all MS grid points, the output array is copied to CPU, where we sum b cbk. across diﬀerent blocks and obtain the ﬁnal ck = ∑nb=1 Alternatively, the frequent block reductions can be eliminated from the kernel’s inner loop. Instead of mapping each primitive pair to a thread, each MS point is distributed to a separate thread. Each thread loops over primitive pairs to accumulate the Coulomb interaction between its MS point and all primitive pairs so that each entry of c is trivially accumulated within a single thread. This algorithm can be seen as a transpose of the ΔFS kernel and is referred to here as the pair-driven kernel. The reduction heavy algorithm is referred as the MS-driven kernel. Depending on the speciﬁcs of the hardware, one or the other of these might be optimal. We found little diﬀerence on the GPUs we used, and the results presented here use the MS-driven kernel. All algorithms discussed above can be easily generalized to situations with angular momenta higher than s functions. In each loop, each thread calculates the Coulomb interaction between a MS point and a batch of primitive pairs instead of a single primitive pair. For instance, for an sp integral, each GPU thread calculates integrals of 3 primitive pairs, [χs,χxp], [χs,χyp], and [χs,χzp], in each loop. We wrote six separate GPU kernels for the following momentum classes: ss, sp, sd, pp, pd, and dd. These kernels are launched sequentially. 4.2. Conjugate Gradient Linear Solver. The typical dimension of A in eq 1 is 103 × 103 or larger. Since eq 1 needs to be solved only for a few right-hand sides, iterative methods can be applied and are much preferred over direct methods based on matrix inversion. Because the Coulomb operator is positive deﬁnite, conjugate gradient (CG) methods are a good choice. At the kth step of CG, we search for an approximate solution xk in the kth Krylov subspace k(A ,b), and the distance between xk and the exact solution can be estimated by the residual vector

(16)

The result is stored to an output array in global memory. The last step is to form the solvation correction to the Fock matrix S ΔF μν =

∑

χi χj ∈ μν

ΔFijS (17)

on the CPU by adding each entry of the output array to its corresponding Fock matrix entry. The algorithm for evaluating c is shown schematically in Figure 4. Although the same set of primitive integrals is evaluated as for the evaluation of ΔFS, there are several signiﬁcant diﬀerences. First, the surface charge density, qk, is replaced by the density matrix element corresponding to each contracted pair. The screening formula can then be augmented with the density as follows. [ij|Schwartz = |Pμν|[χi χj |χi χj ]1/2

(18)

The density matrix elements are loaded with the other pair quantities at the beginning of the kernel. Second, the reduction is now carried out over primitive pairs rather than MS points. For the ΔFS kernel, the sum over MS points was trivially achieved by accumulating the integral results evaluated within each thread. For c, however, the sum over pair quantities would include terms from many threads, assuming pair quantities are again distributed to separate threads as in the ΔFS kernel. In this case, each thread in the CUDA block must evaluate a single integral between its own primitive pair and a common kth grid point. The result can then be stored to shared memory, and a block reduction for the bth block produces the following partial sum ckb = −

∑

χi , χj ∈ block(b)

screened Pμνcμicνj[χi |Jk̂ |χj ]

rk = Ax k − b

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The CG process terminates when the norm of the residual vector, ||rk||, falls below a threshold δ. A wise choice of δ can reduce the number of CG steps while maintaining accuracy. The CG process converges more rapidly if A has small condition number, i.e., looks more like the identity. Preconditioning transforms one linear system to another that has the same solution, but it is easier to solve. One approach is to ﬁnd a preconditioner matrix, C, that approximates A−1. Then, the problem CAx = Cb has the same solution as the original system, but the matrix CA is better conditioned. The matrix A of eq 1 is often ill-conditioned because some of the diagonal elements, which represent the self-energy of surface segments partially buried in the switching area, are ∼7 to 8 orders larger in magnitude than other diagonal elements. In the following paragraphs, we discuss our strategies to choose the CG convergence threshold δ and to generate a preconditioner for the linear equation eq 1. 4.2.1. Dynamic Convergence Threshold for CG. We must solve eq 1 in each SCF step. The traditional strategy (referred to here as the ﬁxed threshold scheme) is to choose a CG residual threshold value (e.g., δ ≈ 10−6) and use this threshold for all SCF iterations. With this strategy, CG may require hundreds of iterations to converge in the ﬁrst few SCF iterations for the computation of medium-sized systems (∼500 atoms), making the linear solve cost as much time as one Fock build. However, in the early SCF iterations, the solute electronic structure is still far from the ﬁnal solution, so it is pointless to get an accurate solvent reaction ﬁeld consistent with the inaccurate electronic structure. In other words, we can use larger δ for eq 1 in the early stages of the SCF, allowing us to reduce the number of CG iterations (and thus the total cost of the linear solves over the entire SCF process). The simplest approach to leverage this observation uses a loose threshold δ1 for the early iterations of the SCF and switches to a tight threshold δ2 when close to SCF convergence. The maximum element of the DIIS error matrix XT(SPF-FPS)X, henceforth the DIIS error, was used as an indicator for SCF convergence, where S is the AO overlap matrix50 and X is the canonical orthogonalization matrix. When the DIIS error reached 10−3, we switched from the loose threshold δ1 to the tight threshold δ2 in the CG solver. We deﬁne the loose and tight thresholds according to the relation δ1 = s·δ2, where s > 1 is a scaling factor. We call this adaptive strategy the 2-δ switching threshold. Numerical experimentation on a variety of molecules showed that for reasonable values of δ2 (10−5 to 10−7), s = 104 was a good choice that minimized the total number of CG steps required for an SCF calculation. The eﬀect of the 2-δ switching threshold strategy is shown in Figure 5. The number of CG steps in the ﬁrst few SCF iterations is signiﬁcantly reduced, and the total number of CG steps over the entire SCF procedure is halved. However, there is an abrupt increase of CG steps at the switching point, making that particular SCF iteration expensive. In order to remove this artifact and potentially increase the eﬃciency, we investigated an alternative dynamic threshold strategy. Luehr et al.51 ﬁrst proposed a dynamic threshold for the precision (32-bit single vs 64-bit double) employed in evaluating two-electron integrals on GPUs. We extend this idea to the estimation of the appropriate CG convergence threshold for a given SCF energy error. We use a set of test molecules (shown in Figure 1) at both equilibrium and distorted nonequilibrium geometries (using RHF with diﬀerent basis sets and ε = 78.39) to empirically determine the relationship between the CG residual

Figure 5. Number of CG steps taken in each SCF iteration for diﬀerent CG residual convergence threshold schemes in COSMO RHF/6-31G calculation on a model protein (PDB ID: 2KJM, 516 atoms, shown in inset).

norm and the error it induces in the COSMO energy. We focus on the ﬁrst COSMO iteration (i.e., the ﬁrst formation of the solvated Fock matrix). The CG equations are ﬁrst solved with a very accurate threshold for the CG residual norm, δ = 10−10 atomic units. Then, the CG equations are solved with progressively less accurate values of δ, and the resulting error in the COSMO energy (compared to the calculation with δ = 10−10) is tabulated. The average error for the six tested molecules is plotted as a function of the CG threshold in Figure 6. We found

Figure 6. Average absolute error in ﬁrst COSMO energies versus the CG residual convergence threshold. Both minimized and distored nonequilibrium geometries for the test set are included in averages. Error bars represent 2 standard deviations above the mean. The black line represents the empirical error bound given by eq 21.

the resulting error to be insensitive to the basis set used. Therefore, we used the 6-31G results to generate an empirical equation relating the error and δ by a power-law ﬁt. We further shifted this equation above twice the standard deviation to provide a bound for the error. This ﬁt is plotted in Figure 6 and given by Err(δ) = 0.01 × δ1.07

(21)

where Err(δ) is the COSMO energy error. We use eq 21 to dynamically adjust the CG threshold for the current SCF iteration by picking the value of δ that is predicted to result in a DIIS error safely below (10−3 times smaller than) the DIIS error of the previous SCF step. This error threshold ensures that error in CG convergence does not dominate the total SCF error. For the ﬁrst SCF iteration, where there is no previous DIIS error as reference, we choose a loose threshold, δ = 1. As shown in Figure 5, the number of CG steps required for each SCF iteration is now rather uniform. This strategy eﬃciently reduces CG steps without inﬂuencing the accuracy of the result. As shown in Figure 7, this approach typically provides a speedup of 2× to 3× for systems with 100−500 atoms. 3136

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Figure 7. Speed-up for CG linear solve methods compared to ﬁxed δ + Jacobi preconditioner of TeraChem for COSMO RHF/6-31G singlepoint energy calculations. Calculations were carried out on 1 GPU (GeForce GTX TITAN).

Figure 8. Number of CG steps taken in each SCF iteration for diﬀerent choices of CG preconditioner in COSMO RHF/6-31G calculation on a model protein (PDB ID: 2KJM, 516 atoms, shown in inset). RBJ-100 and RBJ-800 represent the randomized block-Jacobi preconditioner with block size of 100 and 800, respectively. The block-Jacobi preconditioner based on an octree partition of surface points (denoted octree-800) is also shown, where the maximum number of points in a box is 800.

4.2.2. Randomized Block-Jacobi Preconditioner for CG. York and Karplus28 proposed a symmetric factorization, which is equivalent to Jacobi preconditioning. Lange and Herbert52 later used a block-Jacobi preconditioner, which accelerated the calculation by about 20% for a large molecule. Their partitioning scheme (referred to as octree in our later discussion) of the matrix blocks is based on the spatial partition of MS points in the fast multipole method (FMM),36,37 implemented with an octree data structure. Here, we propose a new randomized algorithm, which we refer to as RBJ, to eﬃciently generate the block diagonal preconditioner without detailed knowledge of the spatial distribution of surface charges. The primary advantage of the RBJ approach is that it is very simple to generate the preconditioner, although it may also have other beneﬁts associated with randomized algorithms.53 As we will show, the performance of the RBJ preconditioner is at least as good as the more complicated octree preconditioner. Since A ∈ m × m is symmetric, there exists some permutation matrix P such that the permuted matrix PAP is block-diagonal dominant. The block-diagonal matrix, M, is then constructed from l × l diagonal blocks of PAP and can be easily inverted to obtain C = PM−1P ≈ A−1 as a preconditioner of A. We generate the permutation matrix P in the following way: at the beginning of the CG solver, we randomly select a pivot Akk, sort the elements of the kth row by descending magnitude, pick the ﬁrst l column indices, and form the ﬁrst diagonal block of M with the corresponding elements, repeating the procedure for the remaining indices until all rows of A have been accounted for. The inverse M−1 is then calculated, and its nonzero entries (diagonal blocks) are stored and used throughout the blockJacobi preconditioned CG algorithm.54 The eﬃciency of the RBJ preconditioner depends on the block size. As block size increases, more information about the original matrix A is kept in M, and the preconditioner C becomes a better approximation to A−1. Thus, larger block sizes will lead to faster convergence of the CG procedure, at the cost of expending more eﬀort to build C. In the limit where the block size is equal to the dimension of A, C is an exact inverse of A and CG will converge in 1 step. However, in this case, building C is as computationally intensive as that from inverting A. We ﬁnd that a block size of 100 is usually large enough to get signiﬁcant reduction in the number of CG steps required for molecules with 100−500 atoms at a moderate discretization level 110 points/atom (Figures S1 and S2). The performance of the randomized block-Jacobi preconditioner is shown in Figure 8, using as an example a single-point COSMO RHF/6-31G calculation on a model protein (PDB ID:

2KJM, 516 atoms). Because RBJ is a randomized algorithm, each data point stands for the averaged results of 50 runs with diﬀerent random seeds (error bars corresponding to the variance are also shown). For this test case, RBJ with a block size of 100 reduces the total number of CG steps (matrix−vector products) by 40% compared to that with ﬁxed threshold CG. Increasing the block size to 800 only slightly enhances the performance. As a reference, we also implemented the block-Jacobi preconditioner based on the octree algorithm. In Figure 8, octree-800 denotes the octree preconditioner with at most 800 points in each octree leaf box. Unlike RBJ, the number of points in each block of the octree is not ﬁxed. For octree-800, the mean block size is 289. RBJ-100 already outperforms octree-800 in the number of CG steps, despite the smaller size of blocks, because RBJ provides better control of the block size and is less sensitive to the shape of the molecular surface. For RBJ and octree preconditioners with the same average blocksize l ̅, if the molecular shape is irregular (which is common for large asymmetric biomolecules), then the octree will contain both very small and large blocks for which l ≪ l ̅ or l ≫ l ̅, respectively. This eﬀect reduces the eﬃciency of the octree algorithm in two ways: (1) the small blocks tend to be poor at preconditioning and (2) the large blocks are less eﬃciently stored and inverted. Another important aspect of the preconditioner is the overhead. For a system with a small number of MS points (e.g., less than 1000), the time saved by reducing CG steps cannot compensate the overhead of building blocks for RBJ. Thus, a standard Jacobi preconditioner is faster. For a system with a large number of MS points, the RBJ preconditioner is signiﬁcantly faster than Jacobi, despite some overhead for building and inverting the blocks. As shown in Figure 7, compared with the ﬁxed δ + Jacobi method, ﬁxed δ + RBJ provides a 1.5× speedup, and dynamic δ + RBJ provides a 3× speedup. 4.3. PCM Gradient Evaluation. To eﬃciently evaluate eq 14, we note that ∇RIA, ∇RIB, and ∇RIc are all sparse and do not need to be calculated explicitly for all nuclear coordinates. This is a direct result of the fact that each MS point moves only with the atom on which it is centered, which is also true for the basis functions. Therefore, the strategy here is to evaluate only the nonzero terms and add them to the corresponding gradients. Speciﬁcally, 3137

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we focus on the evaluation of the second term (∇*RIc†)q in eq 14, which involves one-electron integrals and is the most demanding. For each interaction between an MS point and a primitive pair, ̂ there are three nonzero derivatives: [∇RIχi|Jkscreened |χj], [χi| screened screened Jk̂ |∇RJχj], and [χi|∇RKJk̂ |χj], where χi, χj, and MS point k are located on atoms I, J, and K, respectively. Therefore, (∇R*Ic†) q is composed of three parts (∇*RI c†)q =

∑

ij , i ∈ I

ga[ij] +

∑

gb[ij] +

ij , j ∈ I

ratio is achieved for the calculation of analytic gradients (Figure S3). Of course, this ratio will change with the level of quantum chemistry method and MS discretization. For a medium-sized molecule, the ratio decreases as the basis set size increases (Figure S4) because the COSMO-speciﬁc evaluations only involve one-electron integrals, whose computational cost grows more slowly than that of the gas phase Fock build. Speciﬁcally, for basis sets with diﬀuse functions, the PCM calculation can be faster since the SCF often converges in fewer iterations for PCM compared to vacuum. The COSMO overhead also decreases as larger cavity radii are used (Figure S5) because the number of MS points decreases with increasing cavity radii (more points are buried in the surface). This trend is expected to apply to molecules in a wide range of sizes (ca. 80−1500 atoms), as they share a general trend of decreasing the number of MS points with increasing radii (Figure S6). As a speciﬁc example, we turn to Photoactive Yellow Protein (PYP, 1537 atoms). When the most popular choice55 of cavity radii (choosing atomic radii to be 20% larger than Bondi radii, i.e., 1.2*Bondi) is used (76 577 MS points in total), the computational eﬀort associated with COSMO takes approximately 25% of the total runtime for COSMO RHF/6-31G* single-point calculation (Figure 10).

∑ gc[k] k∈I

screened ga[ij] = Pμνcμicνj ∑ qk [∇I χi |Jk̂ |χj ] k

gb[ij] = Pμνcμicνj ∑ qk [χi |Jk̂

screened

k

|∇J χj ]

screened gc[k] = qk ∑ Pμνcμicνj[χi |∇K Jk̂ |χj ] ij

(22)

The calculation of ga and gb requires reduction over MS points, whereas gc requires reduction over primitive pairs. Therefore, the GPU algorithm for evaluation of (∇*RIc†)q is a hybrid of the pairdriven ΔFS kernel and the MS-driven c kernel. Primitive pairs are prescreened with the density-weighted Schwartz bound of eq 18. Each thread is assigned a single primitive pair, and it loops over all MS points. Integrals ga[ij] and gb[ij] are accumulated within each thread. Finally, gc[k] is formed by a reduction sum within each block at the end of the kth loop, and the host CPU performs the cross-block reduction.

5. PERFORMANCE A primary concern is the eﬃciency of a COSMO implementation compared with that of its gas phase counterpart at the same level of ab initio theory. For our set of 20 proteins, Figure 9 shows the ratio of time required for COSMO compared to that for gas phase for RHF/6-31G (110 points/atom) and RHF/6-31++G* (590 points/atom) single-point energy calculations. The COSMO calculations introduce at most 60 and 30% overhead for 6-31G and 6-31++G* calculations, respectively. A similar

Figure 10. Breakdown of timings by SCF iteration for components of COSMO RHF/6-31G* calculation on Photoactive Yellow Protein (PYP) with cavity radii chosen as Bondi radii scaled by 1.2.

When larger cavity radii (2.0*Bondi) are used (17 266 MS points), the overhead for COSMO falls to 5% (Figure S7). Overall, our COSMO implementation typically requires at most 30% more time than that for gas phase energy or gradient calculations when a moderate basis set (6-31++G*) and ﬁne cavity discretization level (radii = 1.2*Bondi, 590 points/atom) are used. When a larger basis set or larger cavity radii is used, COSMO will be an even more insigniﬁcant part of the total computational cost relative to that for a gas phase calculation. To demonstrate the advantage of a GPU-based implementation, we compare our performance to that of a commercially available, CPU-based quantum chemistry code, Q-Chem.34 We compare runtimes for RHF COSMO-ISWIG gradients using the 6-31G and 6-31++G* basis sets for the smallest (PDB ID: 1Y49, 122 atoms) and largest (PDB ID: 2KJM, 516 atoms) molecules in our test set of proteins. TeraChem calculations were run on nVidia GTX TITAN GPUs and Intel Xeon [email protected] GHz CPUs. Q-Chem calculations were run on faster Intel Xeon [email protected] GHz CPUs. The number of GPUs/CPUs was varied in the tests to assess parallelization eﬃciency across multiple CPU/GPUs. Timing results are summarized in Tables 1−3. The PCM gradient calculation consists of four major parts: gas phase SCF (SCF steps in common with gas phase calculations), PCM SCF (including building the c vector, building ΔFS, and the CG linear

Figure 9. Ratio of time for COSMO versus gas phase single-point energy calculation for 20 small proteins using RHF/6-31G and RHF/631++G*. Dynamic precision for two-electron integrals is used with COSMO cavity radii chosen as 1.2*Bondi radii. An ISWIG discretization scheme is used with 110/590 Lebedev points/atom for 6-31G and 6-31++G* calculations, respectively. 3138

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417 (23%) 116 (19%) 59 (14%)

16 12 19

7789 (22%) 6043 (45%) 5768 (51%)

445 (25%) 181 (29%) 141 (34%)

18 33 41

18756 (53%) 3948 (29%) 2339 (21%)

885 (50%) 299 (48%) 196 (47%)

21 13 12

solve), gas phase gradients, and PCM gradients. For each portion of the calculation, the runtime is annotated in parentheses with the percentage of the runtime for that step relative to total runtime. As explained above, Q-Chem uses OpenMP with no matrix mode and FMM. Comparisons with the MPI parallelized version of Q-Chem are provided in the Supporting Information. The MPI version of Q-Chem does not use FMM and stores the A matrix explicitly. First, we focus on the single CPU/GPU performance, and we compare the absolute runtime values. For both the small and large systems, the GPU implementation provides a 16× reduction in the total runtime relative to that with Q-Chem at the RHF/6-31G level. As shown in Table 3, the speedup is even larger (up to 32×) when a larger basis set and Lebedev grid are used (6-31++G*, 590 points/atom). This is in spite of the fact that Q-Chem is using a linear scaling FMM method. The speedup for diﬀerent sections varies. The PCM gradient calculation has a speedup of over 40×, which is much higher than the overall speedup and the speedup for gas phase gradient. The FMM-based CG procedure in Q-Chem is slower than the version that explicitly stores the A matrix. Even compared to the latter, our CG implementation is about 3× faster (see the Supporting Information). We attribute this to the preconditioning and dynamic threshold strategies described above. On the other hand, it is interesting to note that Q-Chem and TeraChem both spend a similar percentage of their time on PCM SCF and gradient evalutions, regardless of the diﬀerence in absolute runtime. When we use multiple GPUs/CPUs, the total runtime decreases as a result of parallelization for both Q-Chem and TeraChem. However, for both programs, the percentage of time spent on PCM increases, showing that the parallel eﬃciency of the PCM related evaluations is lower than that of other parts of the calculation. Table 4 shows the parallel eﬃciency of TeraChem PCM calculation. The parallel eﬃciency is deﬁned here as usual56

49 81 91 40 (2%) 26 (4%) 23 (5%)

efficiency =

1787 622 419 35 345 13 506 11 339 1 4 8 2KJM (516, 26 025)

1 T1 P TP

(23)

where P is the number of GPUs/CPUs in use and T1/TP are the total runtime in serial/parallel, respectively. We compare the parallel eﬃciency of the four components of the PCM SCF calculation: building c, building ΔFS, solving CG, and building the other terms in common with gas phase SCF. The parallel eﬃciencies of building c and ΔFS are both higher than those of gas phase SCF. However, for our CG implementation, the matrix−vector product is calculated on the CPU, which hampers the overall PCM SCF parallel eﬃciency. Similarly, parallel eﬃciency of the PCM gradient evaluation is limited by our serial computation of ∇A, ∇B. Overall, the GPU implementation of PCM calculations in TeraChem demonstrates signiﬁcant speedups compared to those with Q-Chem, which serves as an example of the type of performance expected from a mature and eﬃcient CPU-based COSMO implementation. However, our current implementations of CG and ∇A, ∇B are conducted in serial on the CPU and do not beneﬁt from parallelization. This is a direction for future improvement.

20 22 27

1960 (6%) 2100 (16%) 2088 (18%)

6840 (19%) 1415 (10%) 1144 (10%)

13 9 7 66 (57%) 23 (56%) 17 (57%) 878 (47%) 200 (28%) 111 (19%) 20 34 39 25 (22%) 10 (24%) 8 (27%) 502 (27%) 337 (48%) 309 (53%) 43 14 18 22 (19%) 6 (15%) 4 (13%) 44 43 89 2 (2%) 2 (5%) 1 (3%) 1 4 8 1Y49 (122, 5922)

1878 706 581

115 41 30

16 17 19

88 (5%) 85 (12%) 89 (15%)

410 (22%) 84 (12%) 72 (12%)

speedup TC speedup molecule (no. atoms, no. MS points)

GPU/ CPU cores

QC

TC

speedup

QC

TC

QC

TC

speedup

QC

TC

speedup

QC

gas phase SCF PCM SCF gas phase gradient PCM gradient total runtime

Table 1. Timing Data (Seconds) for COSMO RHF/6-31G Gradient Calculation of TeraChem (TC) on GTX TITAN GPUs and Q-Chem (QC) on Intel Xeon CPUs [email protected] GHz

Journal of Chemical Theory and Computation

6. APPLICATIONS As a representative application, we studied the structure of a protein ﬁbril57 (protein sequence SSTVNG, PDB ID: 3FTR) 3139

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Table 2. As in Table 1 but with Detailed Timing Information for the PCM SCF Portion of the Calculation CG GPU/ CPU Cores

QC

TC

speedup

QC

1Y49 (122, 5922)

1 4 8

221 (12%) 56 (8%) 28 (5%)

6 (4%) 4 (9%) 4 (12%)

37 14 7

150 (8%) 150 (21%) 150 (26%)

9 (8%) 3 (7%) 2 (7%)

2KJM (516, 26 025)

1 4 8

2335 (7%) 582 (4%) 311 (3%)

124 (7%) 81 (16%) 81 (19%)

19 7 4

2914 (8%) 2919 (22%) 2918 (26%)

131 (7%) 39 (6%) 20 (5%)

molecule (no. atoms, no. MS points)

total runtime molecule (no. atoms, no. MS points)

GPU/CPU cores

QC

TC

speedup

1Y49 (122, 22 430)

1 4 8 8

12.2 3.8 2.8 82.9

0.60 0.19 0.12 2.55

20 21 23 32

TC

speedup

QC

TC

speedup

17 50 75

131 (7%) 132 (19%) 132 (23%)

10 (9%) 3 (7%) 2 (7%)

22 75 146

2539 (7%) 2542 (19%) 2539 (22%)

176 (10%) 48 (8%) 25 (6%)

13 44 66 14 53 102

all levels of theory. Through this procedure, seven diﬀerent types of stationary point structures were found (Figures 11 and 12 and Table S3), characterized by diﬀering protonation states and backbone structures. We characterize the backbone structure by the end-to-end distance of the protein, computed as the distance between the Cα atoms of the ﬁrst and last residues. We describe the protonation state of the amide N and O with a protonation score, deﬁned as follows

Table 3. Timing Data (Hours) for COSMO (590 Points/ Atom) RHF/6-31++G* Gradient Calculation of TeraChem (TC) on GTX TITAN GPUs and Q-Chem (QC) on Intel Xeon CPUs [email protected] GHz

2JKM (516, 97 923)

build ΔFS

build c

n

protonation score =

∑i =r 1 dOi − Hi /d Ni − Hi nr

(24)

where nr is the number of residues; Oi, Hi, and Ni represent the amide O, H, N belonging to the ith residue (for the ﬁrst residue, Hi represents the hydrogen atom at the N-terminus of the peptide closest to O). The higher the score is (e.g., >1.5), the more closely hydrogens are bonded with amide nitrogens, indicating a correct protonation state. The 3FTR crystal structure is zwitterionic with charged groups at both ends, and geometry optimized structures of isolated 3FTR peptides will ﬁnd minima that stabilize those charges. In gas phase, the zwitterionic state’s energy is lowered during geometry minimizations in two ways. In one case, the C-terminus carboxylate is neutralized by a proximal amide H, resulting in unusually protonated local minima. In the other case, the energy is minimized by backbone folding that brings the charged ends close to each other. Both rearrangements result in unexpected structures inconsistent with experiments in solution. We note, however, that such structural rearrangements are known to occur in gas phase polypeptides.58 COSMO solvation largely corrects the protonation artifact observed in gas phase. Two types of severely unusually protonated (protonation score < 1.5) local minima are observed. One (labeled min1u in Figures 11 and 12) has been previously reported with the straight backbone structure as crystal structure. The other unusually protonated local minimum is min2u, which has very similar protonation state as min1u but a slightly bent backbone (backbone length < 17 Å). The normally protonated counterparts of min1u and min2u are min1n and min2n, which are the two minima most resembling the crystal structure. In gas

with our COSMO code. This ﬁbril is known to be able to form dimers called steric zippers that can pack and form amyloids, insoluble ﬁbrous protein aggregates. In each zipper pair, the two segments are tightly interdigitated β-sheets with no water molecules in the interface. The experimental structure of SSTVNG is a piece of the zipper from a ﬁbril crystal. Kulik et al.39 found that minimal basis set ab initio, gas phase, geometry optimizations of a zwitterionic 3FTR monomer resulted in a structure with an unusual deprotonation of amide nitrogen atoms. In that structure, the majority of the amide protons are shared between peptide bond nitrogen atoms and oxygen atoms, forming a covalent bond with the oxygen and a weaker hydrogen bond with the nitrogen. This phenomenon was explained as an artifact caused by both the absence of surrounding solvent and the minimal basis set. We were interested to quantify the degree to which these two approximations aﬀected the outcome. Thus, we conducted more expansive geometry optimizations of 3FTR with and without COSMO to investigate how solvation inﬂuences the conformational landscape of the protein. Stationary point structures of 3FTR were obtained as follows: starting from the two featured structures found previously (an unusually protonated structure and a normally protonated stationary point structure close to experiment), geometry optimizations were conducted in gas phase and with COSMO to describe aqueous solvation (ε = 78.39). Whenever a qualitatively diﬀerent structure was encountered, that structure was set as a new starting point for geometry optimization under

Table 4. Parallel Eﬃciency of TeraChem PCM RHF/6-31G Calculation PCM SCF

PCM gradient

molecule

no. GPU

CG

build c

build ΔFs

total

gas phase SCF

∇c

total

gas phase gradient

1y49

4 8 4 8

0.39 0.19 0.39 0.19

0.84 0.53 0.85 0.81

0.81 0.68 0.91 0.87

0.66 0.40 0.64 0.43

0.72 0.47 0.74 0.56

0.75 0.61 0.85 0.79

0.37 0.21 0.38 0.21

0.93 0.78 0.90 0.88

2kjm

3140

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Figure 11. Diﬀerent minima (min1n, min1u, min2n, min2u, min3, min4) of 3FTR found with RHF/6-31G geometry optimizations in COSMO and in gas phase. The x-axis is the collective variable that characterizes the backbone folding. The y-axis is the total energy including solvation energy of the geometries. Each optimized structure is represented by a symbol in the graph and labeled by name with the backbone structure (C, O, N, and H are colored gray, red, blue, and white, respectively). Side chains are omitted for clarity.

Figure 12. Same as that in Figure 11 but using ωPBEh/6-31G.

COSMO also plays an important role in stabilizing an extended backbone structure. In gas phase calculations, the larger the end-to-end distance is, the less stable the structure tends to be. For both RHF/6-31G and ωPBEh calculations (Figures 11 and 12, respectively), all unfolded structures (min1n, min1u, min2n, min2u, min2t) are very unstable in the gas phase with respect to the folded structure, min4. Among them, min1n and min2n have the largest charges separated by the largest distances (Table S6). COSMO stabilizes the terminal charges, thus signiﬁcantly lowering the energy of min1n and min2n. For COSMO RHF/6-31G, min2n is as stable as the folded min4. At the same time, the half-folded and twisted structure, min3, is destabilized by COSMO. For the most part, the local minima in the gas phase and solution are similar for this polypeptide, even across a range of basis sets including minimal sets. However, the relative energies of these minima are strongly aﬀected by solvation and basis set. Solvation is especially important in this case because of the zwitterionic character of the polypeptide. This is expected on physical grounds (and the structures of gas phase polypeptides and proteins likely reﬂect this) and strongly suggests that solvation eﬀects need to be modeled when using ab initio methods to describe protein structures.

phase calculations with 3-21G and 6-31G, these four minima are all over 50 kcal/mol higher in energy than a folded structure (min4). COSMO solvation stabilizes min1n and min2n by about 50 kcal/mol, while leaving the anomalous min1u and min2u as high-energy structures (Table 5 and Figures 11 and 12). Table 5. Energy Diﬀerence (kcal/mol) between the Normally and Unusually Protonated 3FTR Minima energy diﬀerence (kcal/mol) ΔE(min1u − min1n)a

ΔE(min2u − min2n)b

method/basis set

COSMO

Gas Phase

COSMO

gas phase

RHF-D/STO-3G RHF/STO-3G RHF/3-21G RHF/6-31G

−101 −106 77 90

−178 −179 13 29

−31 −27 83 102

−77 −76 6 13

a

min1n and min1u are minima with an extended backbone structure (as in the 3FTR crystal structure), where n stands for normal protonation state and u stands for unusual protonation state. bmin2n and min2u are minima with slightly bent backbone structure.

Moreover, this COSMO stabilization eﬀect is already quite large for the smallest basis set (COSMO stabilization for diﬀerent basis sets is summarized in Table 5). Although min1u and min2u are still preferred over the normally protonated structures in both gas phase and COSMO STO-3G calculations, this is perhaps expected since the basis set is so small.

7. CONCLUSIONS We have demonstrated that by implementing COSMO-related electronic integrals on GPUs, dynamically adjusting the CG 3141

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threshold for COSMO equations, and applying a new strategy for generating the block-Jacobi preconditioner we can signiﬁcantly decrease the computational eﬀort required for COSMO calculations of large biomolecular systems. We achieve speedups compared to CPU-based codes of more than 15−60×. The computational overhead introduced by the COSMO calculation (relative to gas phase calculations) is quite small, typically less than 30%. Finally, we showed an example where COSMO solvation inﬂuences the geometry optimization of proteins qualitatively. Our eﬃcient implementation of COSMO will be useful for the study of protein structures. Our approach for COSMO electron integral evaluation on GPU can be adapted for other variants of PCMs, such as the integral equation formalism (IEF-PCM or SS(V)PE).59 Since generation of the randomized block-Jacobi preconditioner depends only on the matrix itself (not the speciﬁc physical model used), the strategy can be applied to the preconditioning of CG in a variety of ﬁelds. For instance, for linear scaling SCF, an alternative to diagonalization is the direct minimization of the energy functional60 with preconditioned CG. Another example is the solution of a large linear system with CG to obtain the perturbative correction to the wave function in CASPT2.61 In the future, we will extend our acceleration strategies to nonequilibrium solvation, where the optical (electronic) dielectric constant is equilibrated with the solute while the orientational dielectric constant is not.62−64 This will allow modeling of biomolecules in solution during photon absorption, ﬂuorescence, and phosphorescence processes. Our accelerated PCM code will also facilitate calculation of redox potential of metal complexes65 in solutes and pKa values for large biomolecules.66

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ASSOCIATED CONTENT

S Supporting Information *

Implementation details for derivative density matrix contributions to PCM gradients, coordinates for benchmark molecules, details of the protein data set used for performance benchmarking, performance details for PCM with varying parameters (RBJ-preconditioner blocksize, basis set size, cavity radii), and additional PCM performance tests. The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b00370.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] Funding

This work was supported by the Oﬃce of Naval Research (N00014-14-1-0590). T.J.M. is grateful to the Department of Defense (Oﬃce of the Assistant Secretary of Defense for Research and Engineering) for a National Security Science and Engineering Faculty Fellowship (NSSEFF). Notes

The authors declare the following competing ﬁnancial interest(s): T.J.M. is a founder of PetaChem, LLC.

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REFERENCES

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