We demonstrate how an iterative method for potential inversion from distribution functions developed for simple liquid systems can be generalized to polymer systems. It uses the differences in the potentials of mean force between the distribution functions generated from a guessed potential and the true distribution functions to improve the effective potential successively. The optimization algorithm is very powerful: convergence is reached for every trial function in few iterations. As an extensive test case we coarse-grained an atomistic all-atom model of polyisoprene (PI) using a 13:1 reduction of the degrees of freedom. This procedure was performed for PI solutions as well as for a PI melt. Comparisons of the obtained force fields are drawn. They prove that it is not possible to use a single force field for different concentration regimes. © 2003 Wiley Periodicals, Inc. J Comput Chem 13: 1624–1636, 2003

Deriving effective mesoscale potentials from atomistic simulations.

A practical polarization propagator method devised for the treatment of valence electron excitations in atoms and molecules is presented. This method, referred to as (second-order) algebraic-diagrammatic construction (ADC(2)), allows for a theoretical description of single and double excitations consistently through second and first order, respectively, of perturbation theory. The computational scheme is essentially an eigenvalue problem of a Hermitian secular matrix defined with respect to the space of singly and doubly excited configurations. The configuration space is smaller (more compact) than that of comparable configuration interaction (CI) expansions and the method leads to size-consistent results. The performance of the ADC(2) method is tested in exemplary applications to Ne, Ar and CO, where detailed comparison can be made with experiment and previous theoretical results. While the accuracy of the absolute excitation energies is only moderate, a very satisfactory description is obtained for the relative energies and, in particular, for the spectral intensities. Aspects related to the Thomas-Reiche-Kuhn sum rule and the equivalence of the dipole-length and dipole-velocity forms of the transition moments are discussed. Due to the relatively small computational expense and the possibility of a direct ADC(2) formulation this method should prove particularly useful in applications to large molecules.

https://iopscience.iop.org/article/10.1088/0953-4075/28/12/003/pdf

An efficient polarization propagator approach to valence electron excitation spectra

We have formulated a polarization propagator approach which yields excitation energies, transition moments, and dynamic polarizabilities which are consistent through second order in the electronic repulsion. Certain terms are proven to be missing in our previous second order calculations of transition moments and dynamic polarizabilities and in the equation‐of‐motion calculations of the same quantities. Numerical calculations on carbon monoxide are performed. The calculations show that the major difference between the polarizability (and some transition moments) in the RPA and in the second order polarization propagator approximation is due to these terms. The total effect of all correction terms has been to improve considerably the agreement between theoretical and experimental estimates of the excitation properties for carbon monoxide.

Transition moments and dynamic polarizabilities in a second order polarization propagator approach

We employ the inverse Boltzmann method to coarse-grain three commonly used three-site water models (TIP3P, SPC and SPC/E) where one molecule is replaced with one coarse-grained particle with isotropic two-body interactions only. The shape of the coarse-grained potentials is dominated by the ratio of two lengths, which can be rationalized by the geometric constraints of the water clusters. It is shown that for simple two-body potentials either the radial distribution function or the geometrical packing can be optimized. In a similar way, as needed for multiscale methods, either the pressure or the compressibility can be fitted to the all atom liquid. In total, a speed-up by a factor of about 50 in computational time can be reached by this coarse-graining procedure.

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Comparative atomistic and coarse-grained study of water: What do we lose by coarse-graining?

In this study we explore a systematic procedure to coarsen a microscopic model towards a mesoscopic model. The procedure is applied to a system of chains of ten beads, representing a low molecular weight polymer melt. Our method consists of defining coarse-grained sites in the microscopic system, and calculating their spatial distribution on the pair level. The effective interaction between the coarse-grained sites is then obtained by bringing the pair interaction in consistence with the pair density. We investigate both a dynamic and a stochastic method for this step. The so obtained mesoscopic interaction is used in a molecular dynamics simulation to investigate the pressure of the coarse-grained system. We found that the pair interaction that reproduces the pair density predicts a pressure that is significantly lower than the microscopic value, even if we take the state-dependency of the coarse-grained interactions into account. We therefore conclude that coarse-grained models lack thermodynamic consistency.

A structure-based coarse-grained model for polymer melts

The classical demonstrations that a unique single particle external field is associated with each equilibrium single particle density field (Hohenberg and Kohn, Mermin) are reinterpreted in the language of functional Legendre transformations. This picture is readily extended to the pair distribution function problem and the extension offers a context for understanding how the singlet and pair number densities fix the state of a system. It is shown that one can be sure that there are closure relations to integral equations in general and that in principle the correct closure relation fixes not only the distribution functions but also the complete thermodynamic state of a system. It also follows that a correctly closed integral equation possesses a unique solution. Integral equations for the radial distribution function alone, however, are typically produced by projecting out the singlet density field and for this reason they provide an incomplete characterization of the system. The failure to specify a unique state leads to the existence of multiple solutions.

When does a pair correlation function fix the state of an equilibrium system

An infinite set of ladder diagrams are summed to yield an augmented random phase approximation (ARPA) to the Bethe−Salpeter amplitude equation. The final equation is of the easily solved form of the RPA amplitude eigenvalue equation but with a more complete (augmented) vertex which is the product of a matrix and an inverse matrix. The diagonal ARPA vertex is obtained by taking only the diagonal terms of the inverse matrix, making inversion necessary. The diagonal ARPA should include the most important hole−hole and particle−particle terms which are left out of the RPA. The final ARPA equation is derived two ways, by direct diagram summation and from the integral equation for the ladder vertex.

Augmented random phase approximation

Expressions are derived for the transition moments and absorption resulting from the electric moment induced in one molecule of a pair by the field of the quadrupole moments of a neighbor with axial or near axial symmetry. Values of relevant matrix elements of squares and products of direction cosines are included. For axial polar molecules, the infrared absorption has a different frequency dependence than that by permanent dipoles, permitting separation of the effects. For asymmetric rotors with nearly axial inertial symmetry, the absorption transitions with ΔK=0 are associated with one of the two quadrupole moments and transitions with ΔK=±2 with the other, again permitting spectral resolution of the different contributions. These possibilities are analyzed in some detail for the polar molecules CH3F and CHF3 and for the asymmetric rotor C2H4.

Judith O. Zwicker

Papers

When does a pair correlation function fix the state of an equilibrium system

Augmented random phase approximation

Pressure induced spectra of axially symmetric and slightly asymmetric quadrupolar molecules