Semiempirical Quantum-Chemical Methods with Orthogonalization and Dispersion Corrections

We present two new semiempirical quantum-chemical methods with orthogonalization and dispersion corrections: ODM2 and ODM3 (ODM x). They employ the same electronic structure model as the OM2 and OM3 (OM x) methods, respectively. In addition, they include Grimme's dispersion correction D3 with Becke-Johnson damping and three-body corrections E ABC for Axilrod-Teller-Muto dispersion interactions as integral parts. Heats of formation are determined by adding explicitly computed zero-point vibrational energy and thermal corrections, in contrast to standard MNDO-type and OM x methods. We report ODM x parameters for hydrogen, carbon, nitrogen, oxygen, and fluorine that are optimized with regard to a wide range of carefully chosen state-of-the-art reference data. Extensive benchmarks show that the ODM x methods generally perform better than the available MNDO-type and OM x methods for ground-state and excited-state properties, while they describe noncovalent interactions with similar accuracy as OM x methods with a posteriori dispersion corrections.

Second-Order Dispersion Energy Based on Multireference Description of Monomers

We propose a method for calculating a second-order dispersion energy for weakly interacting multireference systems in arbitrary electronic states. It is based on response properties obtained from extended random phase approximation equations. The introduced formalism is general and requires only one- and two-particle reduced density matrices of monomers. We combine the new method with either generalized valence bond perfect pairing (GVB) or complete active space (CAS) self-consistent field description of the interacting systems. In addition to a general scheme, three approximations, leading to significant reduction of the computational cost, are developed by exploiting Dyall partitioning of the monomer Hamiltonians. For model multireference systems (H₂···H₂ and Be···Be) the method is accurate, unlike its single-reference-based counterpart. Neither GVB nor CAS description of single-reference monomers improves the dispersion energy with respect to the Hartree-Fock-based results.

GFN2-xTB-An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions

An extended semiempirical tight-binding model is presented, which is primarily designed for the fast calculation of structures and noncovalent interaction energies for molecular systems with roughly 1000 atoms. The essential novelty in this so-called GFN2-xTB method is the inclusion of anisotropic second order density fluctuation effects via short-range damped interactions of cumulative atomic multipole moments. Without noticeable increase in the computational demands, this results in a less empirical and overall more physically sound method, which does not require any classical halogen or hydrogen bonding corrections and which relies solely on global and element-specific parameters (available up to radon, Z = 86). Moreover, the atomic partial charge dependent D4 London dispersion model is incorporated self-consistently, which can be naturally obtained in a tight-binding picture from second order density fluctuations. Fully analytical and numerically precise gradients (nuclear forces) are implemented. The accuracy of the method is benchmarked for a wide variety of systems and compared with other semiempirical methods. Along with excellent performance for the "target" properties, we also find lower errors for "off-target" properties such as barrier heights and molecular dipole moments. High computational efficiency along with the improved physics compared to its precursor GFN-xTB makes this method well-suited to explore the conformational space of molecular systems. Significant improvements are furthermore observed for various benchmark sets, which are prototypical for biomolecular systems in aqueous solution.

Big data analysis of ab Initio molecular integrals in the neglect of diatomic differential overlap approximation

Most modern semiempirical quantum-chemical (SQC) methods are based on the neglect of diatomic differential overlap (NDDO) approximation to ab initio molecular integrals. Here, we check the validity of this approximation by computing all relevant integrals for 32 typical organic molecules using Gaussian-type orbitals and various basis sets (from valence-only minimal to all-electron triple-ζ basis sets) covering in total more than 15.6 million one-electron (1-e) and 10.3 billion two-electron (2-e) integrals. The integrals are calculated in the nonorthogonal atomic basis and then transformed by symmetric orthogonalization to the Löwdin basis. In the case of the 1-e integrals, we find strong orthogonalization effects that need to be included in SQC models, for example, by strategies such as those adopted in the available OMx methods. For the valence-only minimal basis, we confirm that the 2-e Coulomb integrals in the Löwdin basis are quantitatively close to their counterparts in the atomic basis and that the 2-e exchange integrals can be safely neglected in line with the NDDO approximation. For larger all-electron basis sets, there are strong multishell orthogonalization effects that lead to more irregular patterns in the transformed 2-e integrals and thus cast doubt on the validity of the NDDO approximation for extended basis sets. Focusing on the valence-only minimal basis, we find that some of the NDDO-neglected integrals are reduced but remain sizable after the transformation to the Löwdin basis; this is true for the two-center 2-e hybrid integrals, the three-center 1-e nuclear attraction integrals, and the corresponding three-center 2-e hybrid integrals. We consider a scheme with a valence-only minimal basis that includes such terms as a possible strategy to go beyond the NDDO integral approximation in attempts to improve SQC methods. © 2018 Wiley Periodicals, Inc.

Development and application of retention time prediction models in the suspect and non-target screening of emerging contaminants

Hydrophilic interaction liquid chromatography (HILIC) and reversed phase LC (RPLC) coupled to high resolution mass spectrometry (HRMS) are widely used for the identification of suspects and unknown compounds in the environment. For the identification of unknowns, apart from mass accuracy and isotopic fitting, retention time (t_R) and MS/MS spectra evaluation is required. In this context, a novel comprehensive workflow was developed to study the t_R behavior of large groups of emerging contaminants using Quantitative Structure-Retention Relationships (QSRR). 682 compounds were analyzed by HILIC-HRMS in positive Electrospray Ionization mode (ESI). Moreover, an extensive dataset was built for RPLC-HRMS including 1830 and 308 compounds for positive and negative ESI, respectively. Support Vector Machines (SVM) was used to model the t_R data. The applicability domains of the models were studied by Monte Carlo Sampling (MCS) methods. The MCS method was also used to calculate the acceptable error windows for the predicted t_R from various LC conditions. This paper provides validated models for predicting t_R in HILIC/RPLC-HRMS platforms to facilitate identification of new emerging contaminants by suspect and non-target HRMS screening, and were applied for the identification of transformation products (TPs) of emerging contaminants and biocides in wastewater and sludge.

Tracing the Pathways of Waters and Protons in Photosystem II and Cytochrome c Oxidase

Photosystem II (PSII) uses water as the terminal electron donor, producing oxygen in the Mn4CaO5 oxygen evolving complex (OEC), while cytochrome c oxidase (CcO) reduces O2 to water in its heme–Cu binuclear center (BNC). Each protein is oriented in the membrane to add to the proton gradient. The OEC, which releases protons, is located near the P-side (positive, at low-pH) of the membrane. In contrast, the BNC is in the middle of CcO, so the protons needed for O2 reduction must be transferred from the N-side (negative, at high pH). In addition, CcO pumps protons from N- to P-side, coupled to the O2 reduction chemistry, to store additional energy. Thus, proton transfers are directly coupled to the OEC and BNC redox chemistry, as well as needed for CcO proton pumping. The simulations that study the changes in proton affinity of the redox active sites and the surrounding protein at different states of the reaction cycle, as well as the changes in hydration that modulate proton transfer paths, are described.

Semi-Empirical Born-Oppenheimer Molecular Dynamics (SEBOMD) within the Amber Biomolecular Package

Semi-empirical quantum methods from the neglect of differential diatomic overlap (NDDO) family such as MNDO, AM1, or PM3 are fast albeit approximate quantum methods. By combining them with linear scaling methods like the divide & conquer (D&C) method, it is possible to quickly evaluate the energy of systems containing hundreds to thousands of atoms. We here present our implementation in the Amber biomolecular package of a SEBOMD module that provides a way to run semi-empirical Born-Oppenheimer molecular dynamics. At each step of a SEBOMD, a fully converged self-consistent field (SCF) calculation is performed to obtain the semiempirical quantum potential energy of a molecular system encaged or not in periodic boundary conditions. We describe the implementation and the features of our SEBOMD implementation. We show the requirements to conserve the total energy in NVE simulations, and how to accelerate SCF convergence through density matrix extrapolation. Specific ways of handling periodic boundary conditions using mechanical embedding or electrostatic embedding through a tailored quantum Ewald summation is developed. The parallel performance of SEBOMD simulations using the D&C scheme are presented for liquid water systems of various sizes, and a comparison between the traditional full diagonalization scheme and the D&C approach for the reproduction of the structure of liquid water illustrates the potentiality of SEBOMD to simulate molecular systems containing several hundreds of atoms for hundreds of picoseconds with a quantum mechanical potential in a reasonable amount of CPU time.

PEPCONF, a diverse data set of peptide conformational energies

We present an extensive and diverse database of peptide conformational energies. Our database contains five different classes of model geometries: dipeptides, tripeptides, and disulfide-bridged, bioactive, and cyclic peptides. In total, the database consists of 3775 conformational energy data points and 4530 conformer geometries. All the reference energies have been calculated at the LC-ωPBE-XDM/aug-cc-pVTZ level of theory, which is shown to yield conformational energies with an accuracy in the order of tenths of a kcal/mol when compared to complete-basis-set coupled-cluster reference data. The peptide conformational data set (PEPCONF) is presented as a high-quality reference set for the development and benchmarking of molecular-mechanics and semi-empirical electronic structure methods, which are the most commonly used techniques in the modeling of medium to large proteins.

Quantitative Assessment of rPM6 for Fluorine- and Chlorine-Containing Metal Complexes: Comparison with Experimental, First-Principles, and Other Semiempirical Results

We report a reparameterization of PM6 parameters for fluorine and chlorine using our training set containing transition metal complexes. Spin unrestricted calculations with the resulting rPM6 (UrPM6) were examined quantitatively using two test sets: (i) the description of magnetic interactions in 25 dinuclear metal complexes and (ii) the prediction of barrier heights and reaction energies for epoxidation and fluorination reactions catalyzed by high-valent manganese-oxo species. The conventional UPM6 and UPM7 methods were also evaluated for comparison on the basis of either experimental or computational (the UB3LYP/SVP level) outcomes. The merits of UrPM6 are highlighted by both the test sets. As regards magnetic exchange coupling constants, the UrPM6 method had the smallest mean absolute errors from the experimental data (19 cm-1), followed by UPM7 (119 cm-1) and UPM6 (373 cm-1). For the epoxidation and fluorination reactions, all of the transition state searches were successful using UrPM6, while the success rates obtained by UPM6 and UPM7 were only 50%. The UrPM6-optimized stationary points also agreed well with the reference UB3LYP-optimized geometries. The accuracy for estimating reaction energies was also greatly remedied.

A Density Functional Tight Binding Layer for Deep Learning of Chemical Hamiltonians

Current neural networks for predictions of molecular properties use quantum chemistry only as a source of training data. This paper explores models that use quantum chemistry as an integral part of the prediction process. This is done by implementing self-consistent-charge Density-Functional-Tight-Binding (DFTB) theory as a layer for use in deep learning models. The DFTB layer takes, as input, Hamiltonian matrix elements generated from earlier layers and produces, as output, electronic properties from self-consistent field solutions of the corresponding DFTB Hamiltonian. Backpropagation enables efficient training of the model to target electronic properties. Two types of input to the DFTB layer are explored, splines and feed-forward neural networks. Because overfitting can cause models trained on smaller molecules to perform poorly on larger molecules, regularizations are applied that penalize nonmonotonic behavior and deviation of the Hamiltonian matrix elements from those of the published DFTB model used to initialize the model. The approach is evaluated on 15 700 hydrocarbons by comparing the root-mean-square error in energy and dipole moment, on test molecules with eight heavy atoms, to the error from the initial DFTB model. When trained on molecules with up to seven heavy atoms, the spline model reduces the test error in energy by 60% and in dipole moments by 42%. The neural network model performs somewhat better, with error reductions of 67% and 59%, respectively. Training on molecules with up to four heavy atoms reduces performance, with both the spline and neural net models reducing the test error in energy by about 53% and in dipole by about 25%.

Nonadiabatic dynamics: The SHARC approach

We review the Surface Hopping including ARbitrary Couplings (SHARC) approach for excited-state nonadiabatic dynamics simulations. As a generalization of the popular surface hopping method, SHARC allows simulating the full-dimensional dynamics of molecules including any type of coupling terms beyond nonadiabatic couplings. Examples of these arbitrary couplings include spin-orbit couplings or dipole moment-laser field couplings, such that SHARC can describe ultrafast internal conversion, intersystem crossing, and radiative processes. The key step of the SHARC approach consists of a diagonalization of the Hamiltonian including these couplings, such that the nuclear dynamics is carried out on potential energy surfaces including the effects of the couplings-this is critical in any applications considering, for example, transition metal complexes or strong laser fields. We also give an overview over the new SHARC2.0 dynamics software package, released under the GNU General Public License, which implements the SHARC approach and several analysis tools. The review closes with a brief survey of applications where SHARC was employed to study the nonadiabatic dynamics of a wide range of molecular systems. This article is categorized under: Theoretical and Physical Chemistry > Reaction Dynamics and KineticsSoftware > Simulation MethodsSoftware > Quantum Chemistry.

Diagnostics of Data-Driven Models: Uncertainty Quantification of PM7 Semi-Empirical Quantum Chemical Method

We report an evaluation of a semi-empirical quantum chemical method PM7 from the perspective of uncertainty quantification. Specifically, we apply Bound-to-Bound Data Collaboration, an uncertainty quantification framework, to characterize (a) variability of PM7 model parameter values consistent with the uncertainty in the training data and (b) uncertainty propagation from the training data to the model predictions. Experimental heats of formation of a homologous series of linear alkanes are used as the property of interest. The training data are chemically accurate, i.e., they have very low uncertainty by the standards of computational chemistry. The analysis does not find evidence of PM7 consistency with the entire data set considered as no single set of parameter values is found that captures the experimental uncertainties of all training data. A set of parameter values for PM7 was able to capture the training data within ±1 kcal/mol, but not to the smaller level of uncertainty in the reported data. Nevertheless, PM7 was found to be consistent for subsets of the training data. In such cases, uncertainty propagation from the chemically accurate training data to the predicted values preserves error within bounds of chemical accuracy if predictions are made for the molecules of comparable size. Otherwise, the error grows linearly with the relative size of the molecules.

Ultrafast action chemistry in slow motion: atomistic description of the excitation and fluorescence processes in an archetypal fluorescent protein

We report quantum mechanical/molecular mechanical non-adiabatic molecular dynamics simulations on the electronically excited state of green fluorescent protein mutant S65T/H148D. We examine the driving force of the ultrafast (τ < 50 fs) excited-state proton transfer unleashed by absorption in the A band at 415 nm and propose an atomistic description of the two dynamical regimes experimentally observed [Stoner Ma et al., J. Am. Chem. Soc., 2008, 130, 1227]. These regimes are explained in terms of two sets of successive dynamical events: first the proton transfers quickly from the chromophore to the acceptor Asp148. Thereafter, on a slower time scale, there are geometrical changes in the cavity of the chromophore that involve the distance between the chromophore and Asp148, the planarity of the excited-state chromophore, and the distance between the chromophore and Tyr145. We find two different non-radiative relaxation channels that are operative for structures in the reactant region and that can explain the mismatch between the decay of the emission of A* and the rise of the emission of I*, as well as the temperature dependence of the non-radiative decay rate.

Fragment-based quantum mechanical calculation of protein-protein binding affinities

The electrostatically embedded generalized molecular fractionation with conjugate caps (EE-GMFCC) method has been successfully utilized for efficient linear-scaling quantum mechanical (QM) calculation of protein energies. In this work, we applied the EE-GMFCC method for calculation of binding affinity of Endonuclease colicin-immunity protein complex. The binding free energy changes between the wild-type and mutants of the complex calculated by EE-GMFCC are in good agreement with experimental results. The correlation coefficient (R) between the predicted binding energy changes and experimental values is 0.906 at the B3LYP/6-31G*-D level, based on the snapshot whose binding affinity is closest to the average result from the molecular mechanics/Poisson-Boltzmann surface area (MM/PBSA) calculation. The inclusion of the QM effects is important for accurate prediction of protein-protein binding affinities. Moreover, the self-consistent calculation of PB solvation energy is required for accurate calculations of protein-protein binding free energies. This study demonstrates that the EE-GMFCC method is capable of providing reliable prediction of relative binding affinities for protein-protein complexes. © 2018 Wiley Periodicals, Inc.

Chemi- and Bioluminescence of Cyclic Peroxides

Bioluminescence is a phenomenon that has fascinated mankind for centuries. Today the phenomenon and its sibling, chemiluminescence, have impacted society with a number of useful applications in fields like analytical chemistry and medicine, just to mention two. In this review, a molecular-orbital perspective is adopted to explain the chemistry behind chemiexcitation in both chemi- and bioluminescence. First, the uncatalyzed thermal dissociation of 1,2-dioxetane is presented and analyzed to explain, for example, the preference for triplet excited product states and increased yield with larger nonreactive substituents. The catalyzed fragmentation reaction and related details are then exemplified with substituted 1,2-dioxetanone species. In particular, the preference for singlet excited product states in that case is explained. The review also examines the diversity of specific solutions both in Nature and in artificial systems and the difficulties in identifying the emitting species and unraveling the color modulation process. The related subject of excited-state chemistry without light absorption is finally discussed. The content of this review should be an inspiration to human design of new molecular systems expressing unique light-emitting properties. An appendix describing the state-of-the-art experimental and theoretical methods used to study the phenomena serves as a complement.

Benchmarking density functional tight binding models for barrier heights and reaction energetics of organic molecules

Density Functional Tight Binding (DFTB) models are two to three orders of magnitude faster than ab initio and Density Functional Theory (DFT) methods and therefore are particularly attractive in applications to large molecules and condensed phase systems. To establish the applicability of DFTB models to general chemical reactions, we conduct benchmark calculations for barrier heights and reaction energetics of organic molecules using existing databases and several new ones compiled in this study. Structures for the transition states and stable species have been fully optimized at the DFTB level, making it possible to characterize the reliability of DFTB models in a more thorough fashion compared to conducting single point energy calculations as done in previous benchmark studies. The encouraging results for the diverse sets of reactions studied here suggest that DFTB models, especially the most recent third-order version (DFTB3/3OB augmented with dispersion correction), in most cases provide satisfactory description of organic chemical reactions with accuracy almost comparable to popular DFT methods with large basis sets, although larger errors are also seen for certain cases. Therefore, DFTB models can be effective for mechanistic analysis (e.g., transition state search) of large (bio)molecules, especially when coupled with single point energy calculations at higher levels of theory. © 2017 Wiley Periodicals, Inc.

QTAIM and Stress Tensor Characterization of Intramolecular Interactions Along Dynamics Trajectories of a Light-Driven Rotary Molecular Motor

A quantum theory of atoms in molecules (QTAIM) and stress tensor analysis was applied to analyze intramolecular interactions influencing the photoisomerization dynamics of a light-driven rotary molecular motor. For selected nonadiabatic molecular dynamics trajectories characterized by markedly different S₁ state lifetimes, the electron densities were obtained using the ensemble density functional theory method. The analysis revealed that torsional motion of the molecular motor blades from the Franck-Condon point to the S₁ energy minimum and the S₁/S₀ conical intersection is controlled by two factors: greater numbers of intramolecular bonds before the hop-time and unusually strongly coupled bonds between the atoms of the rotor and the stator blades. This results in the effective stalling of the progress along the torsional path for an extended period of time. This finding suggests a possibility of chemical tuning of the speed of photoisomerization of molecular motors and related molecular switches by reshaping their molecular backbones to decrease or increase the degree of coupling and numbers of intramolecular bond critical points as revealed by the QTAIM/stress tensor analysis of the electron density. Additionally, the stress tensor scalar and vector analysis was found to provide new methods to follow the trajectories, and from this, new insight was gained into the behavior of the S₁ state in the vicinity of the conical intersection.

Reliable Estimation of Prediction Uncertainty for Physicochemical Property Models

One of the major challenges in computational science is to determine the uncertainty of a virtual measurement, that is the prediction of an observable based on calculations. As highly accurate first-principles calculations are in general unfeasible for most physical systems, one usually resorts to parameteric property models of observables, which require calibration by incorporating reference data. The resulting predictions and their uncertainties are sensitive to systematic errors such as inconsistent reference data, parametric model assumptions, or inadequate computational methods. Here, we discuss the calibration of property models in the light of bootstrapping, a sampling method that can be employed for identifying systematic errors and for reliable estimation of the prediction uncertainty. We apply bootstrapping to assess a linear property model linking the ⁵⁷Fe Mössbauer isomer shift to the contact electron density at the iron nucleus for a diverse set of 44 molecular iron compounds. The contact electron density is calculated with 12 density functionals across Jacob's ladder (PWLDA, BP86, BLYP, PW91, PBE, M06-L, TPSS, B3LYP, B3PW91, PBE0, M06, TPSSh). We provide systematic-error diagnostics and reliable, locally resolved uncertainties for isomer-shift predictions. Pure and hybrid density functionals yield average prediction uncertainties of 0.06-0.08 mm s^-1 and 0.04-0.05 mm s^-1, respectively, the latter being close to the average experimental uncertainty of 0.02 mm s^-1. Furthermore, we show that both model parameters and prediction uncertainty depend significantly on the composition and number of reference data points. Accordingly, we suggest that rankings of density functionals based on performance measures (e.g., the squared coefficient of correlation, r², or the root-mean-square error, RMSE) should not be inferred from a single data set. This study presents the first statistically rigorous calibration analysis for theoretical Mössbauer spectroscopy, which is of general applicability for physicochemical property models and not restricted to isomer-shift predictions. We provide the statistically meaningful reference data set MIS39 and a new calibration of the isomer shift based on the PBE0 functional.

Photophysics and Photochemistry of Canonical Nucleobases’ Thioanalogs: From Quantum Mechanical Studies to Time Resolved Experiments

Interest in understanding the photophysics and photochemistry of thiated nucleobases has been awakened because of their possible involvement in primordial RNA or their potential use as photosensitizers in medicinal chemistry. The interpretation of the photodynamics of these systems, conditioned by their intricate potential energy surfaces, requires the powerful interplay between experimental measurements and state of the art molecular simulations. In this review, we provide an overview on the photophysics of natural nucleobases’ thioanalogs, which covers the last 30 years and both experimental and computational contributions. For all the canonical nucleobase’s thioanalogs, we have compiled the main steady state absorption and emission features and their interpretation in terms of theoretical calculations. Then, we revise the main topographical features, including stationary points and interstate crossings, of their potential energy surfaces based on quantum mechanical calculations and we conclude, by combining the outcome of different spectroscopic techniques and molecular dynamics simulations, with the mechanism by which these nucleobase analogs populate their triplet excited states, which are at the origin of their photosensitizing properties.

Ultrafast Electronic Energy Transfer in an Orthogonal Molecular Dyad

Understanding electronic energy transfer (EET) is an important ingredient in the development of artificial photosynthetic systems and photovoltaic technologies. Although EET is at the heart of these applications and crucially influences their light-harvesting efficiency, the nature of EET over short distances for covalently bound donor and acceptor units is often not well understood. Here we investigate EET in an orthogonal molecular dyad (BODT4), in which simple models fail to explain the very origin of EET. On the basis of nonadiabatic ab initio molecular dynamics calculations and ultrafast fluorescence experiments, we gain detailed microscopic insights into the ultrafast electrovibrational dynamics following photoexcitation. Our analysis offers molecular-level insights into these processes and reveals that it takes place on time scales ≲100 fs and occurs through an intermediate charge-transfer state.

Metal Ion Modeling Using Classical Mechanics

Metal ions play significant roles in numerous fields including chemistry, geochemistry, biochemistry, and materials science. With computational tools increasingly becoming important in chemical research, methods have emerged to effectively face the challenge of modeling metal ions in the gas, aqueous, and solid phases. Herein, we review both quantum and classical modeling strategies for metal ion-containing systems that have been developed over the past few decades. This Review focuses on classical metal ion modeling based on unpolarized models (including the nonbonded, bonded, cationic dummy atom, and combined models), polarizable models (e.g., the fluctuating charge, Drude oscillator, and the induced dipole models), the angular overlap model, and valence bond-based models. Quantum mechanical studies of metal ion-containing systems at the semiempirical, ab initio, and density functional levels of theory are reviewed as well with a particular focus on how these methods inform classical modeling efforts. Finally, conclusions and future prospects and directions are offered that will further enhance the classical modeling of metal ion-containing systems.

CADEE: Computer-Aided Directed Evolution of Enzymes

The tremendous interest in enzymes as biocatalysts has led to extensive work in enzyme engineering, as well as associated methodology development. Here, a new framework for computer-aided directed evolution of enzymes (CADEE) is presented which allows a drastic reduction in the time necessary to prepare and analyze in silico semi-automated directed evolution of enzymes. A pedagogical example of the application of CADEE to a real biological system is also presented in order to illustrate the CADEE workflow.

Non-Hermitian surface hopping

We present a generalized non-Hermitian equation of motion (nH-EOM) to go beyond standard trajectory surface hopping dynamics. The derivation is based on the Born-Huang expansion of the total wave function and the polar representation of the nuclear factor. The nH-EOM contains two additional terms, a skew symmetry term iΓ with dissipation operator Γ to account for decoherence, and a kinetic-energy renormalization term to account for phase shifts, without destroying the invariance to the choice of representation. Numerically, the nH-EOM can still be solved efficiently using a semiclassical approximation in the framework of Tully's fewest-switches surface hopping (FSSH) algorithm. Applications to model Hamiltonians demonstrate improved performance over the standard FSSH approach, through comparison to exact quantum results.