An exact a posteriori correction for hydrogen bond population correlation functions and other reversible geminate recombinations obtained from simulations with periodic boundary conditions. Liquid water as a test case

2021 ◽  
Vol 154 (21) ◽  
pp. 214501
Author(s):  
Johanna Busch ◽  
Jan Neumann ◽  
Dietmar Paschek
Keyword(s):  
Hydrogen Bond ◽  
Boundary Conditions ◽  
Liquid Water ◽  
Correlation Functions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Test Case ◽  
A Posteriori ◽  
Bond Population ◽  
Population Correlation

scholarly journals Correlation functions by separation of variables: the XXX spin chain

2021 ◽  
Vol 10 (1) ◽  
Author(s):  
Giuliano Niccoli ◽  
Hao Pei ◽  
Véronique Terras
Keyword(s):  
Integral Representation ◽  
Boundary Conditions ◽  
Thermodynamic Limit ◽  
Correlation Functions ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Periodic Boundary Conditions ◽  
Multiple Integral ◽  
Periodic Case ◽  
Zero Temperature

We explain how to compute correlation functions at zero temperature within the framework of the quantum version of the Separation of Variables (SoV) in the case of a simple model: the XXX Heisenberg chain of spin 1/2 with twisted (quasi-periodic) boundary conditions. We first detail all steps of our method in the case of anti-periodic boundary conditions. The model can be solved in the SoV framework by introducing inhomogeneity parameters. The action of local operators on the eigenstates are then naturally expressed in terms of multiple sums over these inhomogeneity parameters. We explain how to transform these sums over inhomogeneity parameters into multiple contour integrals. Evaluating these multiple integrals by the residues of the poles outside the integration contours, we rewrite this action as a sum involving the roots of the Baxter polynomial plus a contribution of the poles at infinity. We show that the contribution of the poles at infinity vanishes in the thermodynamic limit, and that we recover in this limit for the zero-temperature correlation functions the multiple integral representation that had been previously obtained through the study of the periodic case by Bethe Ansatz or through the study of the infinite volume model by the q-vertex operator approach. We finally show that the method can easily be generalized to the case of a more general non-diagonal twist: the corresponding weights of the different terms for the correlation functions in finite volume are then modified, but we recover in the thermodynamic limit the same multiple integral representation than in the periodic or anti-periodic case, hence proving the independence of the thermodynamic limit of the correlation functions with respect to the particular form of the boundary twist.

Simulation of electrostatic systems in periodic boundary conditions. II. Equivalence of boundary conditions

1980 ◽  
Vol 373 (1752) ◽  
pp. 57-66 ◽  
Keyword(s):  
Dielectric Constant ◽  
Boundary Conditions ◽  
Correlation Functions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Perturbation Series ◽  
Large Sphere ◽  
Dipolar Systems ◽  
Simulation Cell ◽  
Number Of Particles

We consider simulations of dipolar systems under periodic boundary conditions in which a large sphere consisting of periodic replications of a central simulation cell is surrounded by a continuum of dielectric constant ε'. We develop a perturbation theory expressing correlation functions with ε" in terms of correlation functions with ε' exactly to order N* 1 , N being the number of particles in the sample. In the thermodynamic limit, the correlation functions and internal energy density are independent of e The Kirkwood g -factor is strongly dependent on ε ' but in such a way as to make the dielectric constant independent of ε'. The dependence upon ε' of h A {r) at large r, described in paper I, is explained in terms of the perturbation series.

scholarly journals SemiEmpirical Born-Oppenheimer Molecular Dynamics (SEBOMD) Within the Amber Biomolecular Package

2018 ◽  
Author(s):  
Antoine Marion ◽  
Hatice Gokcan ◽  
Gerald Monard
Keyword(s):  
Molecular Dynamics ◽  
Boundary Conditions ◽  
Liquid Water ◽  
Periodic Boundary ◽  
Molecular System ◽  
Periodic Boundary Conditions ◽  
Ewald Summation ◽  
Molecular Systems ◽  
Parallel Performance ◽  
Scaling Methods

<div>Semiempirical 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 SemiEmpirical Born-Oppenheimer Molecular Dynamics. At each step of a SEBOMD molecular dynamics, a fully converged 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 equirements 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.</div>

scholarly journals SemiEmpirical Born-Oppenheimer Molecular Dynamics (SEBOMD) Within the Amber Biomolecular Package

2018 ◽  
Author(s):  
Antoine Marion ◽  
Hatice Gokcan ◽  
Gerald Monard
Keyword(s):  
Molecular Dynamics ◽  
Boundary Conditions ◽  
Liquid Water ◽  
Periodic Boundary ◽  
Molecular System ◽  
Periodic Boundary Conditions ◽  
Ewald Summation ◽  
Molecular Systems ◽  
Parallel Performance ◽  
Scaling Methods

<div>Semiempirical 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 SemiEmpirical Born-Oppenheimer Molecular Dynamics. At each step of a SEBOMD molecular dynamics, a fully converged 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 equirements 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.</div>

Molecular Dynamics Using Non-Variational Polarizable Force Fields: Theory, Periodic Boundary Conditions Implementation and Application to the Bond Capacity Model

2019 ◽  
Author(s):  
Pier Paolo Poier ◽  
Louis Lagardere ◽  
Jean-Philip Piquemal ◽  
Frank Jensen
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Force Fields ◽  
Periodic Boundary Conditions ◽  
Computational Effort ◽  
Polarizable Force Fields ◽  
Energy Models ◽  
Capacity Model ◽  
Energy Gradient ◽  
Polarization Models

<div> <div> <div> <p>We extend the framework for polarizable force fields to include the case where the electrostatic multipoles are not determined by a variational minimization of the electrostatic energy. Such models formally require that the polarization response is calculated for all possible geometrical perturbations in order to obtain the energy gradient required for performing molecular dynamics simulations. </p><div> <div> <div> <p>By making use of a Lagrange formalism, however, this computational demanding task can be re- placed by solving a single equation similar to that for determining the electrostatic variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p><div><div><div> </div> </div> </div> <p> </p><div> <div> <div> <p>variables themselves. Using the recently proposed bond capacity model that describes molecular polarization at the charge-only level, we show that the energy gradient for non-variational energy models with periodic boundary conditions can be calculated with a computational effort similar to that for variational polarization models. The possibility of separating the equation for calculating the electrostatic variables from the energy expression depending on these variables without a large computational penalty provides flexibility in the design of new force fields. </p> </div> </div> </div> </div> </div> </div> </div> </div> </div>

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

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