Molecular chains with fixed bond lengths and angles: a study based on quantum statistical mechanics

This work studies large three-dimensional open molecular chains at thermal equilibrium in which bond lengths and angles are fixed (hard variables), based upon quantum statistics. A model for a chain formed by N particles interacting through harmonic-like vibrational potentials is treated in the high-frequency limit in which all bond lengths and angles become constrained, while other N angles (soft variables) remain unconstrained. The associated quantum partition function is bounded rigorously, using a variational inequality (related to the Born-Oppenheimer approximation), by another quantum partition function, Z. The total vibrational zero-point energy is shown to be independent of the soft variables thereby solving for this model a generic difficulty in the elimination of hard variables. Z depends only on soft variables and, under certain conditions, it can be approximated by a classical partition function Z_c. The latter satisfies the equipartition principle and it differs from other classical partition functions for related molecular chains. The extension of the model when only part of the bond angles become fixed in the high-frequency limit is outlined. As another generalization, a systematic study of macromolecules, as composed of electrons and heavy particles with Coulomb interactions, is also presented. Its exact quantum partition function is bounded, supposing that the effective molecular potential also tends to constrain all bond lengths and angles, and under suitable assumptions, by another quantum partition function. The latter depends only on the remaining soft variables and it generalizes the one obtained for the first model.     

Batch tautomer generation with MolTPC

Besides all their conformational degrees of freedom, drug‐like molecules and natural products often also undergo tautomeric interconversions. Compared to the huge efforts made in experimental investigation of tautomerism, open and free algorithmic solutions for prototropic tautomer generation are surprisingly rare. The few freely available software packages limit their output to a subset of the possible configurational space by sometimes unwanted prior assumptions and complete neglection of ring‐chain tautomerism. Here, we describe an adjustable fully automatic tautomer enumeration approach, which is freely available and also incorporates the detection of ring‐chain variants. The algorithm is implemented in the MolTPC framework and accessible on SourceForge. © 2013 Wiley Periodicals, Inc.     

Calculation of rotational partition functions by an efficient Monte Carlo importance sampling technique

The evaluation of the classical rotational partition function represented by a configuration integral over all external and internal rotational degrees of freedom of nonrigid chain polyatomic molecules is described. The method of Pitzer and Gwinn is used to correct the classical partition function for quantum mechanical effects at low temperatures. The internal rotor hindrance and all coupling arising from the external and internal rotational degrees of freedom are explicitly taken into account. Importance sampling Monte Carlo based on the adaptive VEGAS algorithm to perform multidimensional integration is implemented within the TINKER program package. A multidimensional potential energy hypersurface is calculated with the MM3(2000) molecular mechanics force field. Numerical tests are performed on a number of small n-alkanes (from ethane to octane), for which the absolute entropies calculated at three different temperatures are compared both with the experimental values and with the previous theoretical results. The application of a more efficient importance sampling technique developed here results in a substantial reduction of statistical errors in the evaluation of the configuration integral for a given number of Monte Carlo steps. Error estimates for the calculated entropies are given, and possible sources of systematic errors, and their importance for a reliable prediction of the absolute entropy, are discussed.     

A quantum mechanical Hamiltonian for unhindered macromolecular chains: consistency with Gaussian models

A model for an open unhindered three-dimensional macromolecular chain, based upon quantum mechanics and proposed in previous works, is studied in order to investigate its physical properties and consistency. The chain is formed by N particles interacting through harmonic-like vibrational potentials in the high-frequency limit (in which all successive bond lengths become fixed). This formulation leads to a specific Hamiltonian for the chain: it constitutes an improvement in comparison with standard Gaussian models, which do not. The classical partition function Z_c resulting from the quantum formulation is represented through an integral, which exhibits explicitly rotational invariance in the integrand and provides the basis for further approximations for large N. Approximate formulae are obtained for correlations between pairs of bond vectors, the distribution function for the end-to-end vector, distribution functions for individual bond vectors, “rubber eleasticity” (when stretching forces act) and the structure factor for small wave vector. In all cases, the results which have arisen from the quantum mechanical formulation coincide with those obtained for the standard Gaussian chain. This agreement appears to confirm the physical consistency of the quantum Hamiltonian characterizing the model.     

The electronic mean-field configuration interaction method. I. Theory and integral formulas

In this article, we introduce a new method for solving the electronic Schrodinger equation. This new method follows the same idea followed by the mean-field configuration interaction method already developed for molecular vibrations; i.e., groups of electronic degrees of freedom are contracted together in the mean field of the other degrees. If the same partition of electronic degrees of freedom is iterated, a self-consistent field method is obtained. Making coarser partitions (i.e., including more degrees in the same groups) and discarding the high energy states, the full configuration interaction limit can be approached. In contrast with the usual group function theory, no strong orthogonality condition is enforced. We have made use of a generalized version of the fundamental formula defining a Hopf algebra structure to derive Hamiltonian and overlap matrix element expressions which respect the group structure of the wave function as well as its fermionic symmetry. These expressions are amenable to a recursive computation.     

Spectroscopic interpretation: the high vibrations of CDBrClF

We extract the dynamics implicit in an algebraic fitted model Hamiltonian for the deuterium chromophore's vibrational motion in the molecule CDBrClF. The original model has four degrees of freedom, three positions and one representing interbond couplings. A conserved polyad allows in a semiclassical approach the reduction to three degrees of freedom. For most quantum states we can identify the underlying motion that when quantized gives the said state. Most of the classifications, identifications, and assignments are done by visual inspection of the already available wave function semiclassically transformed from the number representation to a representation on the reduced dimension toroidal configuration space corresponding to the classical action and angle variables. The concentration of the wave function density to lower dimensional subsets centered on idealized simple lower dimensional organizing structures and the behavior of the phase along such organizing centers already reveals the atomic motion. Extremely little computational work is needed.     

Application of significant structures theory to amorphous polyethylene

The thermodynamic properties of amorphous polyethylene are calculated from a model based on the method of significant structures. The motion of a molecule as a whole is described by the motion of segments, each segment moving independently of all others. It is assumed that on melting, holes appear in the solid lattice and the segments can move into these vacancies, obtaining some gaslike degrees of freedom. The complete frequency distribution for polyethylene is used for the solidlike degrees of freedom, while a corrected classical partition function is used for the gaslike degrees of freedom. The calculated thermodynamic properties are in reasonable agreement with experimentally determined values, assuming each gaslike segment to consist of 20 CH₂ groups.     

A new approach to the exact and approximate anharmonic vibrational partition function of diatomic and polyatomic molecules utilizing Morse and Rosen–Morse oscillators

Exact closed forms of the equilibrium partition functions in terms Jacobi elliptic functions are derived for a particle in a box and Rosen–Morse (Poschl–Teller) oscillator (perfect for modeling bending vibrational modes). An exact form of the equilibrium partition function of Morse oscillator is reported. Three other approximate forms of Morse partition function are presented. Having an exact closed‐form for the vibrational partition function can be very helpful in evaluating thermodynamic state functions, e.g., entropy, internal energy, enthalpy, and heat capacity. Moreover, the herein presented closed forms of the vibrational partition function can be used for obtaining spectroscopic and dynamical information through evaluating the two‐ and four‐point dipole moment time correlation functions in anharmonic media. Finally, a closed exact form of the rotational partition function of a particle on a ring in terms of the first kind of complete elliptic integral is derived. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011     

Grassmann coherent states representation of the path integral: Evaluation of the generating function for spin systems

An interacting spin system is investigated within the scenario of the Feynman path integral representation of quantum mechanics. Short‐time propagator algorithms and a discrete time formalism are used in combination with a basis set involving Grassmann variables coherent states to get a many‐body analytic propagator. The generating function thus obtained leads, after an adequate tracing over Grassmann variables in the imaginary time domain, to the partition function. A spin 1/2 Hamiltonian involving the whole set of interactions is considered. Fermion operators satisfying the standard anticommutation relations are constructed from the raising and lowering spin operators via the Jordan–Wigner transformation. The partition function obtained is more general than the partition function of the traditional Ising model involving only first‐neighbor interactions. Computations were performed assuming that the coupling as a function of the distance can be reasonably well represented by an Airy function. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002     

Electronuclear multiconfiguration time‐dependent hartree calculations on the confined H atom with mobile electron and nucleus

A multiconfiguration time‐dependent Hartree method oriented toward calculations of a non‐Born‐Oppenheimer nature has been applied to the calculation of the dynamical properties of a confined H atom. The calculation is fully six‐dimensional and does not take into account constraints arising from linear or angular momentum conservation. The orbital evolution is monitored and the energy level spectrum of the system, as well as the dependence of the results on the decomposition of the Hamiltonian and on the correlation between radial degrees of freedom, is determined. © 2012 Wiley Periodicals, Inc.     

Open three-dimensional unhindered molecular chains: Partition function

An open molecular chain, formed by N classical particles moving in three dimensions and having N − 1 bonds of constant lengths between successive neighbours, is considered. Hamiltonian methods with manifest rotational invariance allow to characterize all constraints. The classical partition function, Z₀, for a large open chain in thermal equilibrium is analyzed in general. Simpler integral representations for Z₀ are obtained, with the following advantages: (i) explicit rotational invariance in the integrands, (ii) only tridiagonal matrices appear and, moreover, approximations for their determinants can be obtained. For a two-dimensional open chain, an approximate factorized formula for Z₀ is presented, and its essentials are generalized to the three-dimensional case. In both cases, the features of the partition functions bear certain similarities to that for a classical ideal gas. The approximate partition functions lead to approximate analytical computations of the correlations between pairs of different bond vectors, of the squared end-to-end distance, of the probability distribution for the end-to-end vector and of the structure factor, which display some novel features. Some comparisons with the corresponding results for the Gaussian chain are made.     

Molecular modeling and simulation of conjugated polymer oligomers: ground and excited state chain dynamics of PPV in the gas phase

The ground and excited state dynamics of poly(p-phenylenevinylene) (PPV) chains is studied through an implementation of mixed quantum/classical molecular dynamics simulation. The model used in the simulations combines the semiempirical Pariser-Parr-Pople (PPP) Hamiltonian to treat the pi molecular electronic structure with a mechanical force field capturing all other aspects. Nuclear degrees of freedom are treated classically. We first validate the model by simulating PPV chains of various length, and evaluate the absorption spectra. The thermal disorder contribution to the breadth of the first absorption band is estimated to be 0.2 eV at T = 300 K. To investigate the relationship between the emission and chain conformation, we simulate an isolated ten unit chain of PPV in the ground and the lowest excited state. The emission spectrum, red-shifted with respect to absorption of about 0.2 eV as found in experiments, shows a structured line shape that we relate to the photoinduced CC bond distortions. In accord with earlier studies, the exciton self-traps in the middle of the chain. We introduce two collective variables that reflect geometrical distortion, and find these to be effective in describing the contribution of chain conformation to the emission spectrum. The collective variables are also shown to be effective in describing the bond relaxation dynamics after photoexcitation. Such a relaxation is found to occur in approximately 100 fs and is guided by a compensatory release of energy between the double and single bonds in the vinylene junctions and p-phenyl rings. Finally, we find that the chain has a very slight preference for a more planar conformation in the excited state, compared to the ground state. However, the thermal motions induce the chain to explore out-of-plane conformations in both the ground and the excited states with an amplitude significantly greater than this difference.     

The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

Discrete variable approaches to tetratomic molecules: part I: DVR(6) and DVR(3) + DGB methods

We present two discrete variable representation (DVR) based methods for the determination of the vibrational energy levels of tetratomic molecules. Both methods are designed for orthogonal internal coordinates in a body-fixed reference frame and make use of the DVR of three angular variables. The angular DVRs allow the construction of a fixed-angle three-mode Hamiltonian for the stretching vibrations. For each of the angular triples, the stretching eigenvalue problems are solved by employing 3D radial DVRs in the DVR(6) approach and real three-dimensional distributed Gaussian functions in the DVR(3) + DGB method. The angular degrees of freedom are taken sequentially into account in conjunction with a contraction scheme resulting from several diagonalization/truncation steps. Vibrationally averaged geometries, expectation values of rotational constants, and several adiabatic projection schemes developed in this work for tetratomic molecules are used to characterize the vibrational levels calculated by the DVR(6) and DVR(3) + DGB.     

Advancing the computational methodology of rigid rod and semiflexible polymer systems: A new solution to the wormlike chain model with rod‐coil copolymer calculations

A wormlike chain model for rod type blocks in a rod‐coil diblock copolymer is implemented in the self‐consistent field theory (SCFT) formalism. A pseudo‐spectral method is used to solve for the single‐chain partition function of this copolymer system. Orientation degrees of freedom are discretized using Lebedev sphere rules such that orientation integrations are carried out through a Lebedev quadrature, an approach not used previously in tandem with the pseudo‐spectral method. Phase behavior in the rigid‐rod limit as a function of rod segment volume fraction, Flory–Huggins interaction parameter χ , degree of polymerization N , and rod contour length ratio β are examined in detail in one and two dimensions. Examples extending to three dimensions are included. Semiflexible behavior via the rod bending rigidity κ is explored. An approximation is used for rigid‐rods that do not need spherical harmonics leading to increased speed in finding equilibrium morphologies. The results show that standing vertical structures may be more easily produced with rigid‐rod blocks compared to coil‐coil lamellae, an important feature in nanolithographic applications. Suggestions are made for using the model in future molecular orientation studies where the model can be used with inverse search methods to measure the values of the model parameters for the real systems. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 29–39     

A stereographic projection path integral study of the coupling between the orientation and the bending degrees of freedom of water

A Monte Carlo path integral method to study the coupling between the rotation and bending degrees of freedom for water is developed. It is demonstrated that soft internal degrees of freedom that are not stretching in nature can be mapped with stereographic projection coordinates. For water, the bending coordinate is orthogonal to the stereographic projection coordinates used to map its orientation. Methods are developed to compute the classical and quantum Jacobian terms so that the proper infinitely stiff spring constant limit is recovered in the classical limit, and so that the nonconstant nature of the Riemann Cartan curvature scalar is properly accounted in the quantum simulations. The theory is used to investigate the effects of the geometric coupling between the bending and the rotating degrees of freedom for the water monomer in an external field in the 250 to 500 K range. We detect no evidence of geometric coupling between the bending degree of freedom and the orientations.     

Frame transformation relations for fluxional symmetric rotor dimers

The theory of frame transformation relation connecting body oriented angular momentum states and lab weakly coupled momentum states have been extended from rotor-electron to rotor-dimer systems. Coupling schemes are analyzed for weak and strong cases of correlation between lab and two different rotor body frames. It is shown that the frame transformation relation is a purely quantum effect at low angular momentum but an approach to a classical limit for high J. Symmetry analysis of frame transformation is compared to eigensolutions of model coupling Hamiltonian.     

Simulating quantum dynamics in classical environments

Methods for simulating the dynamics of composite systems, where part of the system is treated quantum mechanically and its environment is treated classically, are discussed. Such quantum–classical systems arise in many physical contexts where certain degrees of freedom have an essential quantum character while the other degrees of freedom to which they are coupled may be treated classically to a good approximation. The dynamics of these composite systems are governed by a quantum–classical Liouville equation for either the density matrix or the dynamical variables which are operators in the Hilbert space of the quantum subsystem and functions of the classical phase space variables of the classical environment. Solutions of the evolution equations may be formulated in terms of surface-hopping dynamics involving ensembles of trajectory segments interspersed with quantum transitions. The surface-hopping schemes incorporate quantum coherence and account for energy exchanges between the quantum and classical degrees of freedom. Various simulation algorithms are discussed and illustrated with calculations on simple spin-boson models but the methods described here are applicable to realistic many-body environments.     

Rate constants from the reaction path Hamiltonian. I. Reactive flux simulations for dynamically correct rates

As ab initio electronic structure calculations become more accurate, inherent sources of error in classical transition state theory such as barrier recrossing and tunneling may become major sources of error in calculating rate constants. This paper introduces a general method for diabatically constructing the transverse eigensystem of a reaction path Hamiltonian in systems with many degenerate transverse frequencies. The diabatically constructed reaction path Hamiltonian yields smoothly varying coupling constants that, in turn, facilitate reactive flux calculations. As an example we compute the dynamically corrected rate constant for the chair to boat interconversion of cyclohexane, a system with 48 degrees of freedom and a number of degenerate frequencies. The transmission coefficients obtained from the reactive flux simulations agree with previous results that have been calculated using an empirical potential. Furthermore, the calculated rate constants agree with experimental values. Comparison to variational transition state theory shows that, despite finding the true bottleneck along the reaction pathway, variational transition state theory only accounts for half of the rate constant reduction due to recrossing trajectories.     

On the Car–Parrinello computational scheme: Rigorous treatment

Given a manifold of parameters associated with a trial many-electron wave function to evolve in a fictitious time, the Lagrangian of the corresponding classical system of the parameters' degrees of freedom is constructed explicitly for the case of the closed-shell Hartree–Fock approximation. The relevant equations of motion of a non-Newtonian type are derived. The formulae for the masses of the parameters' degrees of freedom are obtained in the limit of small velocities. The Newtonian dynamics described by the Car–Parrinello Lagrangian is discussed. © 1993 John Wiley & Sons, Inc.