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Harmonic-to-anharmonic thermodynamic integration made simple using REG TI

Venkat Kapil

Abstract

Standard harmonic-to-anharmonic thermodynamic integration (TI) is known to develop a near singularity in the integrand for solids exhibiting diffusive degrees of freedom, such as rotating functional groups or migrating defects. This pathology results in numerical challenges for estimating absolute free energies within a single thermodynamic cycle. In this work, we introduce a simple regularization that removes this singularity and yields a well-behaved integrand that can be accurately evaluated on a uniform grid. The approach -- termed Regularized End-point Gradient (REG) TI -- is demonstrated on a model system and on predicting the relative stability of paracetamol polymorphs for which quasi-free methyl rotations lead to a near singularity in standard TI. We expect REG TI to simplify anharmonic free energy calculations for solids and to potentially enable their automation.

Harmonic-to-anharmonic thermodynamic integration made simple using REG TI

Abstract

Standard harmonic-to-anharmonic thermodynamic integration (TI) is known to develop a near singularity in the integrand for solids exhibiting diffusive degrees of freedom, such as rotating functional groups or migrating defects. This pathology results in numerical challenges for estimating absolute free energies within a single thermodynamic cycle. In this work, we introduce a simple regularization that removes this singularity and yields a well-behaved integrand that can be accurately evaluated on a uniform grid. The approach -- termed Regularized End-point Gradient (REG) TI -- is demonstrated on a model system and on predicting the relative stability of paracetamol polymorphs for which quasi-free methyl rotations lead to a near singularity in standard TI. We expect REG TI to simplify anharmonic free energy calculations for solids and to potentially enable their automation.
Paper Structure (1 section, 17 equations, 2 figures)

This paper contains 1 section, 17 equations, 2 figures.

Table of Contents

  1. harmonic to ideal gas TI

Figures (2)

  • Figure 1: Free-energy analysis for the two-dimensional model system of methyl rotation from Ref. kapil_assessment_2019. Panels a–d show the spatial probability densities at 25 K and 300 K for the physical potential and its harmonic reference, with darker colors indicating higher probability density. Panel e shows the integrand for standard thermodynamic integration (blue) and for the regularized variant with $m=2,4,6,8,10$ at 300 K, where darker colors correspond to larger values of $m$. The inset magnifies the y-axis to compare the integrand of REG TI for various $m$ with that of standard TI. Panel f shows the temperature-dependent anharmonic free energies estimated from standard TI (blue) and its regularized variant for $m=2,4,6$. Round markers indicate numerical integration using the trapezoidal rule applied to Eq. \ref{['eq:H_alchemical_eval']}, while triangular markers indicate integration via Padé interpolation as detailed in Ref. rossi_anharmonic_2016. The exact anharmonic free energies obtained by direct numerical evaluation of the partition function are shown in black. Dashed lines are guide for eyes.
  • Figure 2: Standard and regularized thermodynamic integration integrands for paracetamol polymorphs. Panels a and b show the integrands corresponding to standard (blue) and regularized thermodynamic integration with $m=6$ (red) at 300 K, using the experimental lattice parameters for forms I and II, respectively. Error bars indicate a 2$\sigma$ confidence interval estimated via block averaging using five blocks. Dashed lines are guide for eyes.