Direct calculation of the functional inverse of realistic interatomic potentials in field-theoretic simulations

Abstract

We discuss the functional inverse problem in field-theoretic simulations for realistic pairwise potentials such as the Morse potential (widely used in particle simulations as an alternative to the 12-6 Lennard-Jones one) and we propose two solutions: a) a numerical one based on direct inversion on a regular grid or deconvolution, and b) an analytical one by expressing attractive and repulsive contributions to the Morse potential as higher-order derivatives of the Dirac delta function; the resulting system of ordinary differential equations in the saddle-point approximation is solved numerically with appropriate model-consistent boundary conditions, using a Newton-Raphson method. For the first time, exponential-like, physically realistic pair interactions are analytically treated and incorporated into a field-theoretic framework. Advantages and disadvantages of the two approaches are discussed in detail in connection with numerical findings from test simulations for the radial distribution function of a monatomic fluid at realistic densities providing direct evidence for the capability of the analytical method to resolve structural features down to the Angstrom scale.

Excluded-Volume Interactions in Field-Theoretic Simulations: Multiconvolutions and Model Equivalence

“…Nonetheless, the details of such a generalization can be demonstrated by means of a novel example. Motivated by the use of decaying exponential interactions as already discussed in ref , we consider in particular the effective potential u 1 ′ ( r ) = A 0.25em exp ( − B false| boldr false| ) , goodbreak0em1em⁣ B > 0 whose 3D-Fourier transform is û 1 ′ ( k ) = 8 π italicAB false( | k | 2 + B 2 false) 2 Using eq and comparing with the functional form of the Yukawa potential u normalY ( r ) = A normalY 0.25em exp ( − B Y false| boldr false| ) false| boldr false| , goodbreak0em1em⁣ B normalY > 0 whose Fourier transform is û normalY …”

“…For the Yukawa potential of eq or the Coulomb potential u normalc ( r ) = λ normalB ′ r (which can also be considered as the limit of the Yukawa potential when B Y → 0), it is interesting to note that such an increase of û –1 with k is captured by the Laplacian (acting on the delta function) with u –1 then being specified by u normalc − 1 ( r ) = prefix− 1 4 π λ normalB ′ ∇ 2 δ ( r ) . This nice property has already been exploited in a recent publication to directly calculate u –1 for advanced interatomic potentials.…”

“…Mathematically, the new mass density field ρ̆( r ) is defined by the convolution of the masking (or distribution) function with the monomer position density ρ̂( r ) as ρ̆( r ) = ∫Γ( r – r ′)ρ̂( r ′) d 3 r ′ or, equivalently, ρ̆( r ) = [Γ ∗ ρ̂] ( r ). Due to the convoluted microscopic density, the total interaction potential between n particles in volume V , , which is the starting point for developing the smeared version of the model, is expressed as where we have made use of Dirac’s bra–ket notation. , From eq , we understand that the self-interaction term also involves the convolution of the masking function Γ with u ( 0 ), thus possible model divergences arising from u ( 0 ) can be removed. From eq , we also get the impression that one could have considered the entire formulation in the context of an effective potential Γ ∗ u ∗ Γ acting on ρ̂; however, it is important to emphasize that the particle-to-field transformation (the next step in the field-theoretic simulations, FTS model-building hierarchy) is with respect to the original pair potential u , i.e., it is the quantity Γ ∗ ρ̂ (namely, ρ̆) that is subjected to the transformation and not ρ̂.…”

“…Due to the convoluted microscopic density, the total interaction potential between n particles in volume V , , which is the starting point for developing the smeared version of the model, is expressed as where we have made use of Dirac’s bra–ket notation. 14 , 15 From eq 2 , we understand that the self-interaction term also involves the convolution of the masking function Γ with u ( 0 ), thus possible model divergences arising from u ( 0 ) can be removed. From eq 2 , we also get the impression that one could have considered the entire formulation in the context of an effective potential Γ ∗ u ∗ Γ acting on ρ̂; however, it is important to emphasize that the particle-to-field transformation (the next step in the field-theoretic simulations, FTS model-building hierarchy) is with respect to the original pair potential u , i.e., it is the quantity Γ ∗ ρ̂ (namely, ρ̆) that is subjected to the transformation and not ρ̂.…”

“…. This nice property has already been exploited in a recent publication 15 to directly calculate u −1 for advanced interatomic potentials.…”