Natural Internal Coordinates Research Articles

A multiscale approach to coupled nuclear and electronic dynamics. II. Exact and approximated evaluation of nonradiative transition rates.

This work follows a companion article, which will be referred to as Paper I [Campeggio et al., J. Chem. Phys. 158, 244104 (2023)] in which a quantum-stochastic Liouville equation for the description of the quantum-classical dynamics of a molecule in a dissipative bath has been formulated in curvilinear internal coordinates. In such an approach, the coordinates of the system are separated into three subsets: the quantum coordinates, the classical relevant nuclear degrees of freedom, and the classical irrelevant (bath) coordinates. The equation has been derived in natural internal coordinates, which are bond lengths, bond angles, and dihedral angles. The resulting equation needs to be parameterized. In particular, one needs to compute the potential energy surfaces, the friction tensor, and the rate constants for the nonradiative jumps among the quantum states. While standard methods exist for the calculation of energy and dissipative properties, an efficient evaluation of the transition rates needs to be developed. In this paper, an approximated treatment is introduced, which leads to a simple explicit formula with a single adjustable parameter. Such an approximated expression is compared with the exact calculation of transition rates obtained via molecular dynamics simulations. To make such a comparison possible, a simple sandbox system has been used, with two quantum states and a single internal coordinate (together with its conjugate momentum). Results show that the adjustable parameter, which is an effective decoherence time, can be parameterized from the effective relaxation times of the autocorrelation functions of the conjugated momenta of the relevant nuclear coordinates.

A multiscale approach to coupled nuclear and electronic dynamics. I. Quantum-stochastic Liouville equation in natural internal coordinates.

Multiscale methods are powerful tools to describe large and complex systems. They are based on a hierarchical partitioning of the degrees of freedom (d.o.f.) of the system, allowing one to treat each set of d.o.f. in the most computationally efficient way. In the context of coupled nuclear and electronic dynamics, a multiscale approach would offer the opportunity to overcome the computational limits that, at present, do not allow one to treat a complex system (such as a biological macromolecule in explicit solvent) fully at the quantum mechanical level. Based on the pioneering work of Kapral and Ciccotti [R. Kapral and G. Ciccotti, J. Chem. Phys.110, 8919 (1999)], this work is intended to present a nonadiabatic theory that describes the evolution of electronic populations coupled with the dynamics of the nuclei of a molecule in a dissipative environment (condensed phases). The two elements of novelty that are here introduced are (i) the casting of the theory in the natural, internal coordinates, that are bond lengths, bond angles, and dihedral angles; (ii) the projection of those nuclear d.o.f. that can be considered at the level of a thermal bath, therefore leading to a quantum-stochastic Liouville equation. Using natural coordinates allows the description of structure and dynamics in the way chemists are used to describe molecular geometry and its changes. The projection of bath coordinates provides an important reduction of complexity and allows us to formulate the approach that can be used directly in the statistical thermodynamics description of chemical systems.

Molecular docking, spectroscopic studies and quantum calculations on nootropic drug

A systematic vibrational spectroscopic assignment and analysis of piracetam [(2-oxo-1-pyrrolidineacetamide)] have been carried out using FT-IR and FT-Raman spectral data. The vibrational analysis was aided by an electronic structure calculation based on the hybrid density functional method B3LYP using a 6-311G++(d,p) basis set. Molecular equilibrium geometries, electronic energies, IR and Raman intensities, and harmonic vibrational frequencies have been computed. The assignments are based on the experimental IR and Raman spectra, and a complete assignment of the observed spectra has been proposed. The UV–visible spectrum of the compound was recorded and the electronic properties, such as HOMO and LUMO energies and the maximum absorption wavelengths λmax were determined by the time-dependent DFT (TD-DFT) method. The geometrical parameters, vibrational frequencies and absorption wavelengths were compared with the experimental data. The complete vibrational assignments are performed on the basis of the potential energy distributions (PED) of the vibrational modes in terms of natural internal coordinates. The simulated FT-IR, FT-Raman, and UV spectra of the title compound have been constructed. Molecular docking studies have been carried out in the active site of piracetam by using Argus Lab. In addition, the potential energy surface, HOMO and LUMO energies, first-order hyperpolarizability and the molecular electrostatic potential have been computed.

Molecular conformational analysis, vibrational spectra and normal coordinate analysis of trans-1,2-bis(3,5-dimethoxy phenyl)-ethene based on density functional theory calculations

The conformational behavior and structural stability of trans-1,2-bis(3,5-dimethoxy phenyl)-ethene (TDBE) were investigated by using density functional theory (DFT) method with the B3LYP/6-311++G(d,p) basis set combination. The vibrational wavenumbers of TDBE were computed at DFT level and complete vibrational assignments were made on the basis of normal coordinate analysis calculations (NCA). The DFT force field transformed to natural internal coordinates was corrected by a well-established set of scale factors that were found to be transferable to the title compound. The infrared and Raman spectra were also predicted from the calculated intensities. The observed Fourier transform infrared (FTIR) and Fourier transform (FT) Raman vibrational wavenumbers were analyzed and compared with the theoretically predicted vibrational spectra. Comparison of the simulated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibrational modes. Information about the size, shape, charge density distribution and site of chemical reactivity of the molecules has been obtained by mapping electron density isosurface with electrostatic potential surfaces (ESP).

Spectroscopic studies, potential energy surface and molecular orbital calculations of pramipexole

A systematic vibrational spectroscopic assignment and analysis of pramipexole [(S)-N6-propyl-4,5,6,7-tetrahydro-1,3-benzothiazole-2,6-diamine] has been carried out using FT-IR and FT-Raman spectral data. The vibrational analysis was aided by an electronic structure calculation based on the hybrid density functional method B3LYP using a 6-311G(d, p) and cc-pVTZ basis sets. Molecular equilibrium geometries, electronic energies, IR and Raman intensities, harmonic vibrational frequencies have been computed. The assignments are based on the experimental IR and Raman spectra, and a complete assignment of the observed spectra has been proposed. The UV–visible spectrum of the compound was recorded and the electronic properties, such as HOMO and LUMO energies and the maximum absorption λmax were determined by time-dependent DFT (TD-DFT) method. The geometrical parameters, vibrational frequencies and absorption wavelengths were compared with the experimental data. The complete vibrational assignments are performed on the basis of the potential energy distributions (PEDs) of the vibrational modes in terms of natural internal coordinates. The simulated FT-IR, FT-Raman, and UV spectra of the title compound have been constructed. In addition, the potential energy surface, HOMO and LUMO energies, the molecular electrostatic potential and the first-order hyperpolarizability have been computed. The magnitude of the first-order hyperpolarizability is 5 times larger than that of urea and the title compound may be a potential applicant for the development of NLO materials.

An effective scaling frequency factor method for scaling of harmonic vibrational frequencies: The use of redundant primitive coordinates

A modified effective scaling frequency factor (ESFF) method that takes advantage of the potential energy distribution (PED) coefficients calculated in the basis of redundant primitive internal coordinates is presented. This approach is simpler and more flexible than that based on the natural internal coordinates. The sets of optimal scaling factors for routine 9- and 11-parameter ESFF calculations based on B3LYP/6-311G∗∗ force fields are derived from Baker’s training set of 30 molecules (660 frequencies). The calculated root-mean-square (RMS) deviations for all frequencies are 11.42 and 11.44 cm −1 for 9- and 11-parameter scaling, respectively. They are somewhat lower than in the case of ordinary ESFF calculations. The new sets of factors seem to be particularly well suited for scaling of frequencies in the middle region of the vibrational spectra (1000–2500 cm −1) – the RMS values in this range are 8.37 for 9-, and 9.56 cm −1 for 11-parameter scaling. These values are to be compared with 9.20 and 10.29 cm −1, respectively, calculated within the natural coordinates based ESFF formalism.

Interpolation schemes for peptide rearrangements

A variety of methods (in total seven) comprising different combinations of internal and Cartesian coordinates are tested for interpolation and alignment in connection attempts for polypeptide rearrangements. We consider Cartesian coordinates, the internal coordinates used in CHARMM, and natural internal coordinates, each of which has been interfaced to the OPTIM code and compared with the corresponding results for united-atom force fields. We show that aligning the methylene hydrogens to preserve the sign of a local dihedral angle, rather than minimizing a distance metric, provides significant improvements with respect to connection times and failures. We also demonstrate the superiority of natural coordinate methods in conjunction with internal alignment. Checking the potential energy of the interpolated structures can act as a criterion for the choice of the interpolation coordinate system, which reduces failures and connection times significantly.

Density functional and experimental studies on the FT‐IR and FT‐Raman spectra and structure of benzoic acid and 3,5‐dichloro salicylic acid

AbstractFT‐IR and FT‐Raman spectra of benzoic acid (BA) and 3,5‐dichloro salicylic acid (SA) have been recorded in the regions of 4000–400 and 4000–50 cm−1 respectively. The spectra were interpreted with the aid of normal coordinate analysis following the full structure optimizations and force field calculations based on density functional theory (DFT) using standard B3LYP6‐31G** method and basis set combinations. The DFT force field transformed to natural internal coordinates was corrected by a well‐established set of scale factors that were found to be transferable to the title compounds. The infrared and Raman spectra were also predicted from the calculated intensities. Comparison of the simulated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibrational modes. Copyright © 2008 John Wiley & Sons, Ltd.

Geometry optimization for peptides and proteins: Comparison of Cartesian and internal coordinates

We present the results of several benchmarks comparing the relative efficiency of different coordinate systems in optimizing polypeptide geometries. Cartesian, natural internal, and primitive internal coordinates are employed in quasi-Newton minimization for a variety of biomolecules. The peptides and proteins used in these benchmarks range in size from 16 to 999 residues. They vary in complexity from polyalanine helices to a beta-barrel enzyme. We find that the relative performance of the different coordinate systems depends on the parameters of the optimization method, the starting point for the optimization, and the size of the system studied. In general, internal coordinates were found to be advantageous for small peptides. For larger structures, Cartesians appear to be more efficient for empirical potentials where the energy and gradient can be evaluated relatively quickly compared to the cost of the coordinate transformations.

Vibrational spectra and structure of 5,6-diamino uracil and 5,6-dihydro-5-methyl uracil by density functional theory calculations

The molecular vibrations of 5,6-diamino uracil and 5,6-dihydro-5-methyl uracil were investigated in polycrystalline sample, at room temperature, by FT-IR and FT-Raman spectroscopies. The spectra were interpreted with the aid of normal coordinate analysis following a full structure optimization and force field calculations based on the density functional theory (DFT) using standard B3LYP/6-31G * and B3LYP/6-311+G ** methods and basis set combinations. The DFT force field transformed to natural internal coordinates was corrected by a well-established set of scale factors that were found to be transferable to the title compounds. The infrared and Raman spectra were also predicted from the calculated intensities. Comparison of the simulated spectra with the experimental spectra provides important information about the ability of the computational method to describe the vibrational modes.

Structural and vibrational study of isochroman

The fully optimized equilibrium structure of isochroman has been obtained from DFT calculations by using the hybrid B3LYP and B3PW91 functionals in combination with different basis sets (6-31G∗, 6-311G∗∗ and 6-311++G∗∗). Due to the absence of any experimental geometry data for isochroman, the optimized molecular structure has been compared to those of other compounds like naphthalene, benzene, cyclohexane, tetralin, 2H-pyran and chroman. The vapor, liquid, and solution (CCl 4 and CS 2) infrared (IR) and liquid Raman spectra have been recorded. The harmonic force field generated with B3LYP/6-31G∗, given in terms of the natural internal coordinates, has been scaled by the recommended scale factors by Rauhut and Pulay, using their multiple scaling (SQM) force field procedure. In order to get the best possible fitting between the calculated and experimental frequencies, the scale factors were refined by a linear least squares procedure obtaining a final rms of 7.5 cm −1. Furthermore, supposing a Lorentzian band contour, the infrared spectrum of isochroman has been simulated using the theoretical and scaled force fields. On the basis of the frequencies and infrared intensities calculated from the refined SQM force field, the first vibrational assignment of the normal modes of the individual isochroman molecule is presented, having resulted the infrared intensity calculation decisive in this case for assigning the fundamentals in the low energy region where the most of the uncertainties appeared. In addition, a new method has been developed to characterize the similarity between the experimental and calculated spectra: the inspection of their scalar product, yielding the best result for isochroman in the case of scaled force fields.

Density functional theory study on the structure and vibrational spectra for cyanuric chloride

The molecular structure and vibrational spectra of cyanuric chloride have been investigated by density functional theory (DFT) using standard B3LYP/6-31G* and B3LYP/6-311+G** method and basis set combinations. The DFT force field transformed to natural internal coordinates was corrected by a well-established set of scale factors that were found to be transferable to the title compound. Both the calculated structural parameters and vibrational frequencies are in good agreement with the available experimental data.

Vibrational spectra and potential energy distributions for 4,5-dichloro-3-hydroxypyridazine by density functional theory and normal coordinate calculations

The solid phase FT-IR and FT-Raman spectra of 4,5-dichloro-3-hydroxypyridazine have been recorded in the regions 4000–400 cm −1 and 3500–100 cm −1, respectively. The spectra were interpreted with the aid of normal coordinate analysis following a full structure optimization and force field calculations based on the density functional theory (DFT) using the standard B3LYP/6-31G* and B3LYP/6-311+G** method and basis set combinations. The DFT force field transformed to natural internal coordinates was corrected by a well-established set of scale factors that were found to be transferable to the title compound. The IR and Raman spectra were predicted theoretically and compared with the experimental spectra.

Transferable Scaling Factors for Vibrational Force Fields of Halogenated Molecules Based on Energy-Consistent Pseudopotentials

Within the framework of the scaled quantum mechanical procedure (SQM), transferable scaling factors (TSF) have been determined for a set of natural internal coordinates involving bromine and iodine atoms. Thereby, energy-consistent relativistic pseudopotentials adjusted to multiconfiguration Dirac−Hartree−Fock atomic reference data were used for simulating the Br7+ and I7+ cores, and a density-functional (DFT) treatment was applied to the valence-electron system. The TSFs were used to compute the vibrational spectra of dibromofuroxan and diiodofuroxan. Excellent agreement with available experimental data has been achieved.

Newtonian molecular dynamics in general curvilinear internal coordinates

A simple formulation of Newtonian molecular dynamics in general internal coordinates is given. Our equation for the internal acceleration does not require the transformation matrix from internal to Cartesian coordinates, only the much simpler first and second-order transformation from Cartesians to internals. The formulation has a simple, common general expression for the centrifugal/Coriolis contribution to the internal acceleration. The method has been implemented using natural internal coordinates, and can be easily applied to cyclic molecules. Applications for the internal dynamics of cyclohexane are shown.

The conformational origin of the barrier to the formation of neighboring group assistance in glycosylation reactions: a dynamical density functional theory study.

Static and dynamical Density Functional Theory studies of 2,6-di-O-acetyl-3,4-O-isopropylidene-D-galactopyranosyl cation have shown that this cation can exist in two conformers characterized as (2)S(O) and B(2,5), respectively. The (2)S(O) conformer has the O-2 acyl group equatorial with the carbonyl syn to H-2 and is populated by monocyclic oxocarbenium ions. These conformational features are present in the structurally related glycosyl donor ethyl 2,6-di-O-benzoyl-3,4-O-isopropylidene-beta-D-galactothiopyranoside as determined by X-ray diffraction studies. The B(2,5) conformer has O-2 axial and allows the carbonyl to rotate and close the five-membered ring to form a bicyclic dioxolenium ion. Constraints based on natural internal coordinates were implemented to study this conformational transition. In this way the barrier to interconversion has been determined to be 34 kJ mol(-)(1) with a transition state characterized as (O)S(2) and a pathway involving pseudorotation. Thus, for the first time the structures and energetics of the key ions postulated to be involved in neighboring group assisted glycosylation reactions have been determined.

Geometry optimization in generalized natural internal coordinates

The automatic generation of natural internal coordinates is extended to all cases except multiply fused ring systems, i.e., “cages.” This is achieved by a relatively simple code; most of the geometrical cases can now be treated by a single subroutine. To handle molecules containing cages we combine the natural internal coordinates with the delocalized coordinates of Baker, Kessi, and Delley. We have also modified the delocalized coordinates to separate modes with strongly different force constants. The new methods show very good convergence properties in geometry optimizations. Our test set of molecules covers a wide range of atoms and of different bonding situations. For such molecules good internal coordinates are particularly important because no reliable initial force constant matrix can be obtained to accelerate convergence.

Graphical approach for defining natural internal coordinates

A simple, customizable connectivity scheme is rigorously defined in which pairs of atoms are classified into three categories. The tools of graph theory are used to analyze the molecular graph and to efficiently find rings and ring assemblies through a combination of pruning and homeomorphic reduction. The definition of natural internal coordinates is extended in a nonredundant fashion for the various cases of weakly interacting components and for fused ring systems. The ring system coordinates were tested and found to be superior to Z-matrix coordinates. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 504–511, 1998

Ab initio geometry optimization for large molecules

Various geometry optimization techniques are systematically investigated. The rational function (RF) and direct inversion in the iterative subspace (DIIS) methods are compared and optimized for the purpose of geometry optimization. Various step restriction and line search procedures are tested. The model Hessian recently proposed by Lindh et al. has been used in conjunction with different Hessian update procedures. Optimization for over 30 molecules have been performed in Z-matrix coordinates, local normal coordinates, and curvilinear natural internal coordinates, using the same approximations for the Hessian in all cases. The most effective and stable procedure for optimization of equilibrium structures was found to be the DIIS minimization in natural internal coordinates using the BFGS update of the model Hessian. Our method shows faster overall convergence than all previously published methods for the same test suite of molecules. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1473–1483, 1997

Ab initio geometry optimization for large molecules

Various geometry optimization techniques are systematically investigated. The rational function (RF) and direct inversion in the iterative subspace (DIIS) methods are compared and optimized for the purpose of geometry optimization. Various step restriction and line search procedures are tested. The model Hessian recently proposed by Lindh et al. has been used in conjunction with different Hessian update procedures. Optimization for over 30 molecules have been performed in Z-matrix coordinates, local normal coordinates, and curvilinear natural internal coordinates, using the same approximations for the Hessian in all cases. The most effective and stable procedure for optimization of equilibrium structures was found to be the DIIS minimization in natural internal coordinates using the BFGS update of the model Hessian. Our method shows faster overall convergence than all previously published methods for the same test suite of molecules. © 1997 John Wiley & Sons, Inc. J Comput Chem 18: 1473–1483, 1997