Machine Learning with and for Molecular Dynamics Simulations.

From simple clustering techniques to more sophisticated neural networks, the use of machine learning has become a valuable tool in many fields of chemistry in the past decades. Here, we describe two different ways in which we explore the combination of machine learning (ML) and molecular dynamics (MD) simulations. One topic focuses on how the information in MD simulations can be encoded such that it can be used as input to train ML models for the quantitative understanding of molecular systems. The second topic addresses the utilization of machine learning to improve the set-up, interpretation, as well as accuracy of MD simulations.

Beyond the Rosenfeld Equation: Computation of Vibrational Circular Dichroism Spectra for Anisotropic Solutions.

The difference in absorption of left and right circularly polarized light by chiral molecules can be described by the Rosenfeld equation for isotropic samples. It allows the assignment of the absolute stereochemistry by comparing experimental and computationally derived spectra. Despite the simple form of the Rosenfeld equation, its evaluation in the infrared regime remained challenging, as the contribution from the magnetic dipole operator is zero within the Born-Oppenheimer (BO) approximation. In order to resolve this issue, "beyond BO" theories had to be developed, among which Stephen's magnetic field perturbation (MFP) approach offers a computationally easily accessible form. In this work, optical activity is discussed for cylindrically symmetric solutions, which cannot be described anymore by Rosenfeld's equation due to broken spherical symmetry. Mathematical properties of natural and electric-field induced anisotropies are discussed on the basis of the gauge-independent theoretical framework of Buckingham and Dunn. The issue of achiral noise arising from external field perturbations is considered, and potential remedies are introduced. Natural anisotropic vibrational circular dichroism (VCD) equations are solved numerically by applying the MFP approach within the Hartree-Fock (HF) formalism. Properties of anisotropic VCD spectra are discussed for R-(+)-methyloxirane and (1 S,2 S)-cyclopropane-1,2-dicarbonitrile. In particular, by using a group theoretical argument, a gauge-independent lower bound for the quadrupole contribution of C2-symmetric molecules can be identified, which allows the importance of additional quadrupole terms in anisotropic VCD spectra calculation to be assessed.

A molecular mechanism for the enzymatic methylation of nitrogen atoms within peptide bonds.

The peptide bond, the defining feature of proteins, governs peptide chemistry by abolishing nucleophilicity of the nitrogen. This and the planarity of the peptide bond arise from the delocalization of the lone pair of electrons on the nitrogen atom into the adjacent carbonyl. While chemical methylation of an amide bond uses a strong base to generate the imidate, OphA, the precursor protein of the fungal peptide macrocycle omphalotin A, self-hypermethylates amides at pH 7 using S-adenosyl methionine (SAM) as cofactor. The structure of OphA reveals a complex catenane-like arrangement in which the peptide substrate is clamped with its amide nitrogen aligned for nucleophilic attack on the methyl group of SAM. Biochemical data and computational modeling suggest a base-catalyzed reaction with the protein stabilizing the reaction intermediate. Backbone N-methylation of peptides enhances their protease resistance and membrane permeability, a property that holds promise for applications to medicinal chemistry.

Machine Learning of Partial Charges Derived from High-Quality Quantum-Mechanical Calculations.

Parametrization of small organic molecules for classical molecular dynamics simulations is not trivial. The vastness of the chemical space makes approaches using building blocks challenging. The most common approach is therefore an individual parametrization of each compound by deriving partial charges from semiempirical or ab initio calculations and inheriting the bonded and van der Waals (Lennard-Jones) parameters from a (bio)molecular force field. The quality of the partial charges generated in this fashion depends on the level of the quantum-chemical calculation as well as on the extraction procedure used. Here, we present a machine learning (ML) based approach for predicting partial charges extracted from density functional theory (DFT) electron densities. The training set was chosen with the goal to provide a broad coverage of the known chemical space of druglike molecules. In addition to the speed of the approach, the partial charges predicted by ML are not dependent on the three-dimensional conformation in contrast to the ones obtained by fitting to the electrostatic potential (ESP). To assess the quality and compatibility with standard force fields, we performed benchmark calculations for the free energy of hydration and liquid properties such as density and heat of vaporization.

Power Series Approximation for the Correlation Kernel Leading to Kohn-Sham Methods Combining Accuracy, Computational Efficiency, and General Applicability.

A power series approximation for the correlation kernel of time-dependent density-functional theory is presented. Using this approximation in the adiabatic-connection fluctuation-dissipation (ACFD) theorem leads to a new family of Kohn-Sham methods. The new methods yield reaction energies and barriers of unprecedented accuracy and enable a treatment of static (strong) correlation with an accuracy of high-level multireference configuration interaction methods but are single-reference methods allowing for a black-box-like handling of static correlation. The new methods exhibit a better scaling of the computational effort with the system size than rivaling wave-function-based electronic structure methods. Moreover, the new methods do not suffer from the problem of singularities in response functions plaguing previous ACFD methods and therefore are applicable to any type of electronic system.

Self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation theorem and the exact-exchange kernel.

A self-consistent Kohn-Sham method based on the adiabatic-connection fluctuation-dissipation (ACFD) theorem, employing the frequency-dependent exact exchange kernel fx is presented. The resulting SC-exact-exchange-only (EXX)-ACFD method leads to even more accurate correlation potentials than those obtained within the direct random phase approximation (dRPA). In contrast to dRPA methods, not only the Coulomb kernel but also the exact exchange kernel fx is taken into account in the EXX-ACFD correlation which results in a method that, unlike dRPA methods, is free of self-correlations, i.e., a method that treats exactly all one-electron systems, like, e.g., the hydrogen atom. The self-consistent evaluation of EXX-ACFD total energies improves the accuracy compared to EXX-ACFD total energies evaluated non-self-consistently with EXX or dRPA orbitals and eigenvalues. Reaction energies of a set of small molecules, for which highly accurate experimental reference data are available, are calculated and compared to quantum chemistry methods like Møller-Plesset perturbation theory of second order (MP2) or coupled cluster methods [CCSD, coupled cluster singles, doubles, and perturbative triples (CCSD(T))]. Moreover, we compare our methods to other ACFD variants like dRPA combined with perturbative corrections such as the second order screened exchange corrections or a renormalized singles correction. Similarly, the performance of our EXX-ACFD methods is investigated for the non-covalently bonded dimers of the S22 reference set and for potential energy curves of noble gas, water, and benzene dimers. The computational effort of the SC-EXX-ACFD method exhibits the same scaling of N(5) with respect to the system size N as the non-self-consistent evaluation of only the EXX-ACFD correlation energy; however, the prefactor increases significantly. Reaction energies from the SC-EXX-ACFD method deviate quite little from EXX-ACFD energies obtained non-self-consistently with dRPA orbitals and eigenvalues, and the deviation reduces even further if the Coulomb kernel is scaled by a factor of 0.75 in the dRPA to reduce self-correlations in the dRPA correlation potential. For larger systems, such a non-self-consistent EXX-ACFD method is a competitive alternative to high-level wave-function-based methods, yielding higher accuracy than MP2 and CCSD methods while exhibiting a better scaling of the computational effort than CCSD or CCSD(T) methods. Moreover, EXX-ACFD methods were shown to be applicable in situation characterized by static correlation.

Stability conditions for exact-exchange Kohn-Sham methods and their relation to correlation energies from the adiabatic-connection fluctuation-dissipation theorem.

The occurrence of instabilities, in particular singlet-triplet and singlet-singlet instabilities, in the exact-exchange (EXX) Kohn-Sham method is investigated. Hessian matrices of the EXX electronic energy with respect to the expansion coefficients of the EXX effective Kohn-Sham potential in an auxiliary basis set are derived. The eigenvalues of these Hessian matrices determine whether or not instabilities are present. Similar as in the corresponding Hartree-Fock case instabilities in the EXX method are related to symmetry breaking of the Hamiltonian operator for the EXX orbitals. In the EXX methods symmetry breaking can easily be visualized by displaying the local multiplicative exchange potential. Examples (N2, O2, and the polyyne C10H2) for instabilities and symmetry breaking are discussed. The relation of the stability conditions for EXX methods to approaches calculating the Kohn-Sham correlation energy via the adiabatic-connection fluctuation-dissipation (ACFD) theorem is discussed. The existence or nonexistence of singlet-singlet instabilities in an EXX calculation is shown to indicate whether or not the frequency-integration in the evaluation of the correlation energy is singular in the EXX-ACFD method. This method calculates the Kohn-Sham correlation energy through the ACFD theorem theorem employing besides the Coulomb kernel also the full frequency-dependent exchange kernel and yields highly accurate electronic energies. For the case of singular frequency-integrands in the EXX-ACFD method a regularization is suggested. Finally, we present examples of molecular systems for which the self-consistent field procedure of the EXX as well as the Hartree-Fock method can converge to more than one local minimum depending on the initial conditions.

Hydrogen storage on metal oxide model clusters using density-functional methods and reliable van der Waals corrections.

We investigate the capability of low-coordinated sites on small model clusters to act as active centers for hydrogen storage. A set of small magic clusters with the formula (XY)6 (X = Mg, Ba, Be, Zn, Cd, Na, Li, B and Y = O, Se, S, F, I, N) and a "drumlike" hexagonal shape showing a low coordination number of three was screened. Oxide clusters turned out to be the most promising candidates for hydrogen storage. For these ionic compounds we explored the suitability of different van der Waals (vdW) corrections to density-functional calculations by comparing the respective H2 physisorption profile to highly accurate CCSD(T) (Coupled Cluster Singles Doubles with perturbative Triples) calculations. The Grimme D3 vdW correction in combination with the Perdew-Burke-Ernzerhof exchange-correlation functional was found to be the best approach compared to CCSD(T) hydrogen physisorption profiles and is, therefore, suited to study these and other light metal oxide systems. H2 adsorption on sites of oxide model clusters is found to meet the adsorption energy criteria for H2 storage, with bond strengths ranging from 0.15 to 0.21 eV. Energy profiles and estimates of kinetic constants for the H2 splitting reaction reveal that H2 is likely to be adsorbed molecularly on sites of (MgO)6, (BaO)6, and (BeO)6 clusters, suggesting a rapid H2 uptake/release at operating temperatures and moderate pressures. The small mass of beryllium and magnesium makes such systems appealing for meeting the gravimetric criterion for H2 storage.

Efficient self-consistent treatment of electron correlation within the random phase approximation.

A self-consistent Kohn-Sham (KS) method is presented that treats correlation on the basis of the adiabatic-connection dissipation-fluctuation theorem employing the direct random phase approximation (dRPA), i.e., taking into account only the Coulomb kernel while neglecting the exchange-correlation kernel in the calculation of the Kohn-Sham correlation energy and potential. The method, denoted self-consistent dRPA method, furthermore treats exactly the exchange energy and the local multiplicative KS exchange potential. It uses Gaussian basis sets, is reasonably efficient, exhibiting a scaling of the computational effort with the forth power of the system size, and thus is generally applicable to molecules. The resulting dRPA correlation potentials in contrast to common approximate correlation potentials are in good agreement with exact reference potentials. The negatives of the eigenvalues of the highest occupied molecular orbitals are found to be in good agreement with experimental ionization potentials. Total energies from self-consistent dRPA calculations, as expected, are even poorer than non-self-consistent dRPA total energies and dRPA reaction and non-covalent binding energies do not significantly benefit from self-consistency. On the other hand, energies obtained with a recently introduced adiabatic-connection dissipation-fluctuation approach (EXXRPA+, exact-exchange random phase approximation) that takes into account, besides the Coulomb kernel, also the exact frequency-dependent exchange kernel are significantly improved if evaluated with orbitals obtained from a self-consistent dRPA calculation instead of an exact exchange-only calculation. Total energies, reaction energies, and noncovalent binding energies obtained in this way are of the same quality as those of high-level quantum chemistry methods, like the coupled cluster singles doubles method which is computationally more demanding.

Directional Noncovalent Interactions: Repulsion and Dispersion.

The interaction energies between an argon atom and the dihalogens Br2, BrCl, and BrF have been investigated using frozen core CCSD(T)(fc)/aug-cc-pVQZ calculations as reference values for other levels of theory. The potential-energy hypersurfaces show two types of minima: (1) collinear with the dihalogen bond and (2) in a bridging position. The former represent the most stable minima for these systems, and their binding energies decrease in the order Br > Cl > F. Isotropic atom-atom potentials cannot reproduce this binding pattern. Of the other levels of theory, CCSD(T)(fc)/aug-cc-pVTZ reproduces the reference data very well, as does MP2(fc)/aug-cc-pVDZ, which performs better than MP2 with the larger basis sets (aug-cc-pVQZ and aug-cc-pvTZ). B3LYP-D3 and M06-2X reproduce the binding patterns moderately well despite the former using an isotropic dispersion potential correction. B3LYP-D3(bj) performs even better. The success of the B3LYP-D3 methods is because polar flattening of the halogens allows the argon atom to approach more closely in the direction collinear with the bond, so that the sum of dispersion potential and repulsion is still negative at shorter distances than normally possible and the minimum is deeper at the van der Waals distance. Core polarization functions in the basis set and including the core orbitals in the CCSD(T)(full) calculations lead to a uniform decrease of approximately 20% in the magnitudes of the calculated interaction energies. The EXXRPA+@EXX (exact exchange random phase approximation) orbital-dependent density functional also gives interaction energies that correlate well with the highest level of theory but are approximately 10% low. The newly developed EXXRPA+@dRPA functional represents a systematic improvement on EXXRPA+@EXX.

Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation.

Two related methods to calculate the Kohn-Sham correlation energy within the framework of the adiabatic-connection fluctuation-dissipation theorem are presented. The required coupling-strength-dependent density-density response functions are calculated within exact-exchange time-dependent density-functional theory, i.e., within time-dependent density-functional response theory using the full frequency-dependent exchange kernel in addition to the Coulomb kernel. The resulting resolution-of-identity exact-exchange random-phase approximation (RI-EXXRPA) methods in contrast to previous EXXRPA methods employ an auxiliary basis set (RI basis set) to improve the computational efficiency, in particular, to reduce the formal scaling of the computational effort with respect to the system size N from N(6) to N(5). Moreover, the presented RI-EXXRPA methods, in contrast to previous ones, do not treat products of occupied times unoccupied orbitals as if they were linearly independent. Finally, terms neglected in previous EXXRPA methods can be included, which leads to a method designated RI-EXXRPA+, while the method without these extra terms is simply referred to as RI-EXXRPA. Both EXXRPA methods are shown to yield total energies, reaction energies of small molecules, and binding energies of noncovalently bonded dimers of a quality that is similar and in some cases even better than that obtained with quantum chemistry methods such as Mo̸ller-Plesset perturbation theory of second order (MP2) or with the coupled cluster singles doubles method. In contrast to MP2 and to conventional density-functional methods, the presented RI-EXXRPA methods are able to treat static correlation.