Monte Carlo free energy calculations using electronic structure methods

The molecular mechanics-based importance sampling function (MMBIF) algorithm [R. Iftimie, D. Salahub, D. Wei, and J. Schofield, J. Chem. Phys. 113, 4852 (2000)] is extended to incorporate semiempirical electronic structure methods in the secondary Markov chain, creating a fully quantum mechanical Monte Carlo sampling method for simulations of reactive chemical systems which, unlike the MMBIF algorithm, does not require the generation of a system-specific force field. The algorithm is applied to calculating the potential of mean force for the isomerization reaction of HCN using thermodynamic integration. Constraints are implemented in the sampling using a modification of the SHAKE algorithm, including that of a fixed, arbitrary reaction coordinate. Simulation results show that sampling efficiency with the semiempirical secondary potential is often comparable in quality to force fields constructed using the methods suggested in the original MMBIF work. The semiempirical based importance sampling method presented here is a useful alternative to MMBIF sampling as it can be applied to systems for which no suitable MM force field can be constructed.     

Monte Carlo free energy estimates using non-Boltzmann sampling: Application to the sub-critical Lennard-Jones fluid

The paper reports a Monte Carlo technique for estimation of the free energies of fluids by sampling on distributions designed for this purpose, rather than on the usual Boltzmann distribution. As an illustration of its use, the free energy of a Lennard-Jones fluid in the liquid-vapour coexistence region has been estimated by relating it to that of the inverse-twelve (soft sphere) fluid, which itself shows no condensation.     

Quantum Monte Carlo calculations of the dimerization energy of borane

Accurate thermodynamic data are required to improve the performance of chemical hydrides that are potential hydrogen storage materials. Boron compounds are among the most interesting candidates. However, different experimental measurements of the borane dimerization energy resulted in a rather wide range (-34.3 to -39.1) ± 2 kcal/mol. Diffusion Monte Carlo (DMC) simulations usually recover more than 95% of the correlation energy, so energy differences rely less on error cancellation than other methods. DMC energies of BH(3), B(2)H(6), BH(3)CO, CO, and BH(2)(+) allowed us to predict the borane dimerization energy, both via the direct process and indirect processes such as the dissociation of BH(3)CO. Our D(e) = -43.12(8) kcal/mol, corrected for the zero point energy evaluated by considering the anharmonic contributions, results in a borane dimerization energy of -36.59(8) kcal/mol. The process via the dissociation of BH(3)CO gives -34.5(2) kcal/mol. Overall, our values suggest a slightly less D(e) than the most recent W4 estimate D(e) = -44.47 kcal/mol [A. Karton and J. M. L. Martin, J. Phys. Chem. A 111, 5936 (2007)]. Our results show that reliable thermochemical data for boranes can be predicted by fixed node (FN)-DMC calculations.     

Calculation of the entropy and free energy by the hypothetical scanning Monte Carlo method: application to peptides

A new approach, the hypothetical scanning Monte Carlo (HSMC), for calculating the absolute entropy, S, and free energy, F, has been introduced recently and applied first to fluids (argon and water) and later to peptides. In this paper the method is further developed for peptide chains in vacuum. S is calculated from a given MC sample by reconstructing each sample conformation i step-by-step, i.e., calculating transition probabilities (TPs) for the dihedral and bond angles and fixing the related atoms at their positions. At step k of the process the chain's coordinates that have already been determined are kept fixed (the "frozen past") and TP(k) is obtained from a MC simulation of the "future" part of the chain whose TPs as yet have not been determined; when the process is completed the contribution of conformation i to the entropy is, S(i) approximately -ln Pi(k) TP(k). In a recent paper we studied polyglycine chains, modeled by the AMBER force field with constant bond lengths and bond angles (the rigid model). Decaglycine [(Gly)(10)] was studied in the helical, extended, and hairpin microstates, while (Gly)(16) was treated only in the first two microstates. In this paper the samples are increased and restudied, (Gly)(16) is also investigated in the hairpin microstate, and for (Gly)(10) approximations are tested where only part of the future is considered for calculating the TPs. We calculate upper and lower bounds for F and demonstrate that like for fluids, F can be obtained from multiple reconstructions of a single conformation. We also test a more realistic model of (Gly)(10) where the bond angles are allowed to move (the flexible model). Very accurate results for S and F are obtained which are compared to results obtained by the quasiharmonic approximation and the local states method. Thus, differences in entropy and free energy between the three microstates are obtained within errors of 0.1-0.3 kcal/mol. The HSMC method can be applied to a macromolecule with any degree of flexibility, ranging from local fluctuations to a random coil. The present results demonstrate that the difference in stability, DeltaF(mn)=F(m)-F(n) between significantly different microstates m and n, can be obtained from two simulations only without the need to resort to thermodynamic integration. Our long-term goal is to extend this method to any peptide and apply it to a peptide immersed in a box with explicit water.     

Dissociation energy of the water dimer from quantum Monte Carlo calculations

We report a study of the electronic dissociation energy of the water dimer using quantum Monte Carlo techniques. We have performed variational quantum Monte Carlo and diffusion quantum Monte Carlo (DMC) calculations of the electronic ground state of the water monomer and dimer using all-electron and pseudopotential approaches. We have used Slater-Jastrow trial wave functions with B3LYP type single-particle orbitals, into which we have incorporated backflow correlations. When backflow correlations are introduced, the total energy of the water monomer decreases by about 4-5 mhartree, yielding a DMC energy of -76.428 30(5) hartree, which is only 10 mhartree above the experimental value. In our pseudopotential DMC calculations, we have compared the total energies of the water monomer and dimer obtained using the locality approximation with those from the variational scheme recently proposed by Casula [Phys. Rev. B 74, 161102(R) (2006)]. The time step errors in the Casula scheme are larger, and the extrapolation of the energy to zero time step always lies above the result obtained with the locality approximation. However, the errors cancel when energy differences are taken, yielding electronic dissociation energies within error bars of each other. The dissociation energies obtained in our various all-electron and pseudopotential calculations range between 5.03(7) and 5.47(9) kcalmol and are in good agreement with experiment. Our calculations give monomer dipole moments which range between 1.897(2) and 1.909(4) D and dimer dipole moments which range between 2.628(6) and 2.672(5) D.     

Gamma-point lattice free energy estimates from O1 force calculations

We present a new method for estimating the vibrational free energy of crystal (and molecular) structures employing only a single force calculation, for a particularly displaced configuration, in addition to the calculation of the ground state configuration. This displacement vector is the sum of the phonon eigenvectors obtained from a fast-relative to, e.g., density-functional theory (DFT)-Hessian calculation using interatomic potentials. These potentials are based here on effective charges obtained from a DFT calculation of the ground state electronic charge density but could also be based on other, e.g., empiric approaches.     

Quantum Monte Carlo calculations of the dissociation energy of the water dimer

We report diffusion quantum Monte Carlo (DMC) calculations of the equilibrium dissociation energy D(e) of the water dimer. The dissociation energy measured experimentally, D(0), can be estimated from D(e) by adding a correction for vibrational effects. Using the measured dissociation energy and the modern value of the vibrational energy Mas et al., [J. Chem. Phys. 113, 6687 (2000)] leads to D(e)=5.00+/-0.7 kcal mol(-1), although the result Curtiss et al., [J. Chem. Phys. 71, 2703 (1979)] D(e)=5.44+/-0.7 kcal mol(-1), which uses an earlier estimate of the vibrational energy, has been widely quoted. High-level coupled cluster calculations Klopper et al., [Phys. Chem. Chem. Phys. 2, 2227 (2000)] have yielded D(e)=5.02+/-0.05 kcal mol(-1). In an attempt to shed new light on this old problem, we have performed all-electron DMC calculations on the water monomer and dimer using Slater-Jastrow wave functions with both Hartree-Fock approximation (HF) and B3LYP density functional theory single-particle orbitals. We obtain equilibrium dissociation energies for the dimer of 5.02+/-0.18 kcal mol(-1) (HF orbitals) and 5.21+/-0.18 kcal mol(-1) (B3LYP orbitals), in good agreement with the coupled cluster results.     

Internal and free energy in a pair of like-charged colloids: Monte Carlo simulations

The effective interaction between two colloidal particles in a bath of monovalent co- and counterions is studied by means of lattice Monte Carlo simulations with the primitive model. The internal electrostatic energy as a function of the colloid distance is studied fixing the position of the colloids. The free energy of the whole system is obtained introducing a bias parabolic potential, that allows us to sample efficiently small separations between the colloidal particles. For small charges, both the internal and free energy increase when the colloids approach each other, resulting in an effective repulsion driven by the electrostatic repulsion. When the colloidal charge is large enough, on the other hand, the colloid-ion coupling is strong enough to form double layers. The internal energy in this case decreases upon approaching the colloids because more ions enter the double layer. This attractive contribution to the interaction between the colloids is stronger for larger charges and larger ionic concentrations. However, the total free energy increases due to the loss of ionic entropy, and resulting finally in a repulsive interaction potential driven by the entropic contributions. The loss of ionic entropy can be almost quantitatively reproduced with the ideal contribution, the same level of approximation as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The overall behavior is captured by the DLVO theory qualitatively, and a comparison is made with the functional form predicted by the theory, showing moderate agreement.     

Monte Carlo simulation of electrodeposition of copper: a multistep free energy calculation

Electrodeposition of copper (Cu) involves length scales of a micrometer or even less. Several theoretical techniques such as continuum Monte Carlo, kinetic Monte Carlo (KMC), and molecular dynamics have been used for simulating this problem. However the multiphenomena characteristics of the problem pose a challenge for an efficient simulation algorithm. Traditional KMC methods are slow, especially when modeling surface diffusion with large number of particles and frequent particle jumps. Parameter estimation involving thousands of KMC runs is very time-consuming. Thus a less time-consuming and novel multistep continuum Monte Carlo simulation is carried out to evaluate the step wise free energy change in the process of electrochemical copper deposition. The procedure involves separate Monte Carlo codes employing different random number criterion (using hydrated radii, bare radii, hydration number of the species, redox potentials, etc.) to obtain the number of species (CuCl(2) or CuSO(4) or Cu as the case may be) and in turn the free energy. The effect of concentration of electrolyte, influence of electric field and presence of chloride ions on the free energy change for the processes is studied. The rate determining step for the process of electrodeposition of copper from CuCl(2) and CuSO(4) is also determined.     

Absolute Helmholtz free energy of highly anharmonic crystals: theory vs Monte Carlo

We discuss the problem of the quantitative theoretical prediction of the absolute free energy for classical highly anharmonic solids. Helmholtz free energy of the Lennard-Jones (LJ) crystal is calculated accurately while accounting for both the anharmonicity of atomic vibrations and the pair and triple correlations in displacements of the atoms from their lattice sites. The comparison with most precise computer simulation data on sublimation and melting lines revealed that theoretical predictions are in excellent agreement with Monte Carlo simulation data in the whole range of temperatures and densities studied.     

Accurate vibrational-rotational partition functions and standard-state free energy values for H2O2 from Monte Carlo path-integral calculations

Accurate quantum mechanical partition functions and absolute free energies of H(2)O(2) are determined using a realistic potential energy surface [J. Koput, S. Carter, and N. C. Handy, J. Phys. Chem. A 102, 6325 (1998)] for temperatures ranging from 300 to 2,400 K by using Monte Carlo path integral calculations with new, efficient polyatomic importance sampling methods. The path centroids are sampled in Jacobi coordinates via a set of independent ziggurat schemes. The calculations employed enhanced-same-path extrapolation of trapezoidal Trotter Fourier path integrals, and the paths were constructed using fast Fourier sine transforms. Importance sampling was also used in Fourier coefficient space, and adaptively optimized stratified sampling was used in configuration space. The free energy values obtained from the path-integral calculations are compared to separable-mode approximations, to the Pitzer-Gwinn approximation, and to values in thermodynamic tables. Our calculations support the recently proposed revisions to the JANAF tables.     

Valence energy in variational Monte Carlo: CuH dissociation energy

We show how to estimate the dissociation energy of CuH using the variational Monte Carlo method. The techniques involved are (i) an all-electron approach, (ii) a diffusion-only Metroplis algorithm which is well-suited for sampling the nodal regions properly, and (iii) a core-valence partitioning scheme such that the dissociation energy is estimated from the valence energies of CuH and Cu only. This approach avoids several of the approximations inherent in pseudopotential methods. Using relatively crude wave functions, we obtain an estimate of the dissociation energy and dipole moment with an accuracy on par with much more elaborate calculations in the literature. © 1996 John Wiley & Sons, Inc.     

Computation of binding free energy with molecular dynamics and grand canonical Monte Carlo simulations

The binding of a ligand to a receptor is often associated with the displacement of a number of bound water molecules. When the binding site is exposed to the bulk region, this process may be sampled adequately by standard unbiased molecular dynamics trajectories. However, when the binding site is deeply buried and the exchange of water molecules with the bulk region may be difficult to sample, the convergence and accuracy in free energy perturbation (FEP) calculations can be severely compromised. These problems are further compounded when a reduced system including only the region surrounding the binding site is simulated. To address these issues, we couple molecular dynamics (MD) with grand canonical Monte Carlo (GCMC) simulations to allow the number of water to fluctuate during an alchemical FEP calculation. The atoms in a spherical inner region around the binding pocket are treated explicitly while the influence of the outer region is approximated using the generalized solvent boundary potential (GSBP). At each step during thermodynamic integration, the number of water in the inner region is equilibrated with GCMC and energy data generated with MD is collected. Free energy calculations on camphor binding to a deeply buried pocket in cytochrome P450cam, which causes about seven water molecules to be expelled, are used to test the method. It concluded that solvation free energy calculations with the GCMC/MD method can greatly improve the accuracy of the computed binding free energy compared to simulations with fixed number of water.     

Monte Carlo calculations on unimolecular reactions,energy transfer,and IR-multiphoton decomposition

Monte Carlo techniques are described for stochastic calculations involving energy transfer, unimolecular reactions, and IR-multiphoton decomposition; all of these phenomena can be simulated using the same computer code. The details of the model are presented along with the following example calculations: (1) weak collision energy-transfer following photoactivation, (2) chemical activation unimolecular reactions with multiple reaction pathways, (3) incubation times in shock-tube experiments on systems with multiple reaction pathways, and (4) IR-multiphoton absorption and decomposition in collisional environments. The model uses a modular approach that permits application to numerous experimental systems.     

Quantum Monte Carlo calculations of the potential energy curve of the helium dimer

We report results of two quantum Monte Carlo methods -- variational Monte Carlo and diffusion Monte Carlo -- on the potential energy curve of the helium dimer. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all.     

Energy-consistent pseudopotentials for quantum Monte Carlo calculations

The authors present scalar-relativistic energy-consistent Hartree-Fock pseudopotentials for the main-group elements. The pseudopotentials do not exhibit a singularity at the nucleus and are therefore suitable for quantum Monte Carlo (QMC) calculations. They demonstrate their transferability through extensive benchmark calculations of atomic excitation spectra as well as molecular properties. In particular, they compute the vibrational frequencies and binding energies of 26 first- and second-row diatomic molecules using post-Hartree-Fock methods, finding excellent agreement with the corresponding all-electron values. They also show their pseudopotentials give superior accuracy than other existing pseudopotentials constructed specifically for QMC. Finally, valence basis sets of different sizes (VnZ with n=D,T,Q,5 for first and second rows, and n=D,T for third to fifth rows) optimized for our pseudopotentials are also presented.     

Excited electronic state calculations by the transcorrelated variational Monte Carlo method: application to a helium atom

We have implemented the excited electronic state calculations for a helium atom by the transcorrelated variational Monte Carlo (TC-VMC) method. In this method, Jastrow-Slater-type wave function is efficiently optimized not only for the Jastrow factor but also for the Slater determinant. Since the formalism for the TC-VMC method is based on the variance minimization, excited states as well as the ground state calculations are feasible. It is found that both the first and the second excitation energies given by TC-VMC are much closer to the experimental data than those given by the variational Monte Carlo method with using the Hartree-Fock orbitals. The successful results in the TC-VMC method are considered to be due to the nodal optimization of the wave functions.     

Lower and upper bounds for the absolute free energy by the hypothetical scanning Monte Carlo method: application to liquid argon and water

The hypothetical scanning (HS) method is a general approach for calculating the absolute entropy S and free energy F by analyzing Boltzmann samples obtained by Monte Carlo or molecular dynamics techniques. With HS applied to a fluid, each configuration i of the sample is reconstructed by gradually placing the molecules in their positions at i using transition probabilities (TPs). At each step of the process the system is divided into two parts, the already treated molecules (the "past"), which are fixed, and the as yet unspecified (mobile) "future" molecules. Obtaining the TP exactly requires calculating partition functions over all positions of the future molecules in the presence of the frozen past, thus it is customary to invoke various approximations to best represent these quantities. In a recent publication [Proc. Natl. Acad. Sci. USA 101, 9235 (2004)] we developed a version of HS called complete HSMC, where each TP is calculated from an MC simulation involving all of the future molecules (the complete future); the method was applied very successfully to Lennard-Jones systems (liquid argon) and a box of TIP3P water molecules. In its basic implementation the method provides lower and upper bounds for F, where the latter can be evaluated only for relatively small systems. Here we introduce a new expression for an upper bound, which can be evaluated for larger systems. We also propose a new exact expression for F and verify its effectiveness. These free energy functionals lead to significantly improved accuracy (as applied to the liquid systems above) which is comparable to our thermodynamic integration results. We formalize and discuss theoretical aspects of HSMC that have not been addressed in previous studies. Additionally, several functionals are developed and shown to provide the free energy through the analysis of a single configuration.