Reliability of the Phonon Density of States Determined by Real-Coded Genetic Algorithm from Heat Capacities of Benzoic Acid Crystals

Abstract

Constant-pressure heat capacities (C P s) of crystalline benzoic acid (BAh) and its deuterated analogue (C 6 H 5 COOD, BAd) were measured by adiabatic calorimetry, and the phonon density (g(ω)) of states was determined from their C P data using a real-coded genetic algorithm (RCGA) with just generation gap + real-coded ensemble cross-over. The distribution of g(ω) was in reasonable agreement with the spectroscopic one observed for molecular vibration modes, indicating sufficient reliability of g(ω) determined by the RCGA. Based on the fact that C P s reveals an inverse-isotope effect in the temperature range 30−130 K, the determined g(ω)s were used to investigate the molecular mechanism of the effect: g(ω) of BAd revealed blue shifts in the ranges of ω = 80−100 and 150− 230 K, as referred to that of BAh. It was suggested from the combined considerations on g(ω) and spectroscopic results that an anticooperative correlation exists between O−H•••O hydrogen bonds and interdimer interactions in BA.

Phase-Transition Behaviors of Crystalline Hypophosphorous Acid and a Feasible Incommensurate Wave-like Array of the Proton Displacements from Each Center of Hydrogen Bonds along the 1-Dimensional Chain

“…While a low-temperature heat-capacity tail of the second-order phase transition was represented by the mean-field approximation, the excess heat capacity near the critical temperature cannot be expressed by the model; thus, the g (ω) was optimized from the set of C P data without those in the temperature range of 130–210 K. Similarly, the C P data above 237 K were omitted from the set for optimization since they involved the contribution from a fusion phenomenon. The presentations of the genotype for g (ω), the C P – C V correction, and the mean-field approximation term were the same as the method reported previously. , …”

“…C P , cal was thus calculated as the sum of C V , the C P – C V correction term ( ATC P 2 ), and the excess heat capacity assumed by the mean-field approximation. The fitness ( u ) of the derived g (ω) was evaluated by the same function, given by eq using C P ,cal and C P ,exp , as reported previously. , u = 1 n ∑ false( C italicP , cal − C italicP , exp false) 2 false( μ C italicP , exp false) 2 where μ is the experimental inaccuracy and n is the total number of data points: μ is determined to be 0.003 and 0.002 below and above 35 K, respectively, by the previous heat-capacity measurement …”

“…The baseline heat capacity to a sequence of phase transitions (see Section 3.1) in the HPA crystal was determined based on the density (g(ω)) of states of phonons, which was derived, from a set of C P data, by using the RCGA 12−15 with the just generation gap + real-coded ensemble crossover method 16,17 in the same way as reported previously. 18,19 In this study, the molecular-vibration frequency (ω) was taken in K units.…”

“…The presentations of the genotype for g(ω), the C P −C V correction, and the mean-field approximation term were the same as the method reported previously. 18,19 C P, cal was thus calculated as the sum of C V , the C P −C V correction term (ATC P 2 ), and the excess heat capacity assumed by the mean-field approximation. The fitness (u) of the derived g(ω) was evaluated by the same function, given by eq 3 using C P,cal and C P,exp , as reported previously.…”

“…In this study, heat capacities of the crystalline HPA were measured in a temperature range of 20–306 K, temperature dependence of dielectric constants was examined as a supplemental means to characterize the phase transitions found, and a density ( g (ω), where ω is the vibration frequency) of states of phonons and thereby baseline heat capacities were determined, from the obtained heat-capacity data, by using the real-coded genetic algorithm (RCGA) − with the just generation gap + real-coded ensemble crossover , method in the same way as reported previously. , A sequence of phase transitions was found. The transitions occurred apparently as due to structural changes of a normal–incommensurate–(superlattice commensurate) type.…”