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Two Faces of the Two-Phase Thermodynamic Model Two Faces of the Two-Phase Thermodynamic Model Ad´amMadar´asz,´ ∗,y Andrea Hamza,y D´avidFerenc,y,z and Imre Bak´oy yResearch Centre for Natural Sciences, Magyar Tud´osokK¨or´utja2, H-1117 Budapest, Hungary zInstitute of Chemistry, ELTE, E¨otv¨osLor´andUniversity, P´azm´anyP´eters´et´any1/A, Budapest, H-1117, Hungary Received January 26, 2021; E-mail: madarasz:adam@ttk:hu ferent articles. The classical heat capacities were also taken Abstract: into account in the calculation of the 2PT heat capacity of The quantum harmonic model and the two-phase thermo- water, 9,10,17,33 but in the case of organic solvents classical dynamics method (2PT) are widely used to obtain quantum heat capacities were discarded. 19{21 According to refs 19{21 corrected properties such as isobaric heat capacities or molar we refer to 1PT and 2PT heat capacities that do not contain entropies. 2PT heat capacities were calculated inconsistently anharmonic corrections calculated from classical values. In in the literature, and the excellent correlations are due to error previous studies there was no systematic comparison of the cancellation for organic liquids. We reanalyzed the performance effect of this anharmonic correction. In this communication of different quantum corrections on the heat capacities of com- we fill this gap and analyse the 2PT and 1PT+AC methods mon organic solvents against experimental data. The accuracy in more details. of the computations was also assessed with the determination Here we focus on heat capacities to evaluate different of the self-diffusion coefficients. types of quantum corrections because it contains large nu- clear quantum effect and there are accurate experimen- tal data that can be used for the benchmark of force Accounting for nuclear quantum effects is essential to ob- fields. 6,17,19{21,25{27,29,49{57 In contrast to enthalpy or Gibbs tain meaningful thermodynamic properties that are compa- energy, heat capacity is an absolute quantity meaning that 1 rable to experimental observations. The most typical ex- there is no need to set the zero point. Additionally, the iso- ample is that zero point energies are indispensable in the baric heat capacity is a state function, so if we know the cp determination of reaction free energies. The quantum har- as a function of T and p, the other state functions such as monic oscillator model works quite well for small molecules the enthalpy and entropy can be calculated as well. Previ- and solid states, but the anharmonicity becomes significant ously, quantum corrected thermodynamic properties of or- in macromolecules, interfaces and liquids and the potential ganic solvents were investigated in two systematic studies by energy surfaces must be mapped using molecular dynamics Pascal, and Caleman. 19,20 For the same solvents they found or Monte Carlo simulation. Berens proposed to add quan- similar results: the 2PT heat capacities were in good agree- tum correction to the classically calculated properties us- ments with the experimental data. Both studies showed that 2 ing the harmonic oscillator model. Goddard improved this OPLS force field gave better results than other general force by the separation of different motions like translation rota- fields such as GAFF or CHARMM. We recalculated the heat tions and vibrations and using different partition functions capacities of 92 organic solvents from ref 20 to test further 3,4 for each of them. This was abbreviated as two-phase ther- the 1PT+AC and 2PT methods. modynamic (2PT) model referring to the gas phase and solid For the determination of the quantum corrected thermo- phase motions in contrast to the one-phase thermodynamic dynamic properties the velocity autocorrelation functions (1PT) method where only vibrations were considered. An (VACF) are computed from molecular dynamics simulations anharmonic correction was also included in Berens' origi- that can be defined as follows: nal idea, and thus we refer to that method as one-phase- 1 thermodynamics with anharmonic correction (1PT+AC). R mv(t + τ) · v(τ)dτ 2PT and 1PT+AC methods were successfully applied for 0 VACF(t) = 1 (1) the calculation of thermodynamic properties of several sys- R mv(τ) · v(τ)dτ tems such as Lennard-Jones fluids, 3,5 water, 2,4,6{15 aqueous 0 solutions, 16,17 molten salts, 18 organic liquids, 19{21 carbon where m is the atomic mass and v is the velocity as a function dioxide, 22 urea, 23 ionic liquids, 24{27 carbohydrates, 28 cellu- of time (t). The vibrational density of states (VDOS) is the lose, 29 mixtures 30 and interfaces. 31{36 Lately, 2PT was used Fourier transform of the autocorrelation function (VACF) 37{40 for the definition of the Frenkel line. Both 1PT/2PT +1 methods are still in continuous development in respect of Z VDOS(ν) = F fVACF(t)g (ν) = 2 VACF(t)·cos(2πνt)dt accuracy and applicability. 41{48 t The 2PT method is the most excellent in the calculation of 0 (2) absolute entropy even from short trajectories. Although the where ν is the frequency. heat capacity is strictly determined from the temperature Originally Berens proposed that the quantum corrected dependence of entropy according to the laws of thermody- density of states can be determined by the multiplication of namics, the calculation of the 2PT heat capacity is not as VDOS with an appropriate weight function w: 2 consistent in the literature as the computation of the 2PT entropy. The 2PT abbreviation refers to two conceptually VDOSq(ν) = VDOS(ν) · w(ν) (3) different calculation procedures of the heat capacity in dif- In the 1PT method there is no separation of motions, all 1 are considered as vibrations. The weight function for the ism allows a much more effective calculation, because there heat capacity is 58 is no need to calculate the VDOS. The isobaric heat capacity can be determined from the βhν 2 wcV (ν) = exp (βhν) ; (4) isochoric heat capacity by employing the relation vib 1 − exp (βhν) 2 −1 T Mαp where β = (k T ) , k is the Boltzmann constant, T is the cp = cV + ; (11) B B ρκ temperature, h is the Planck constant. Thus the isochoric T heat capacity can be calculated as where αp denotes the thermal expansion coefficient, M is the 1 relative molar mass, ρ is the density and κT is the isother- Z mal compressibility. The isobaric 1PT+AC heat capacity c1PT = 2fR VDOS(ν) · wcV (ν)dν ; (5) V vib is computed as a sum of the classical isobaric heat capacity 0 and the quantum correction from eq 8 and the latter can be where R is the universal gas constant, and f = 3N is the determined from VACF or VDOS according to eqs 5 and 9: number of degrees of freedom of an N-atomic molecule. 1PT+AC cl QC Gaseous motions like translation and rotation are sepa- cp = cp + c (12) rated from vibrations in the 2PT method. The total VDOS We performed 10.6 ns long NpT simulations by using the is decomposed into two terms, solid and gaseous compo- GROMACS simulation software. 61 The settings and inputs nents: were taken from the ref 20 (more details can be found in VDOS(ν) = VDOSsol(ν) + VDOSgas(ν) (6) the Supporting Information). 2PT heat capacities were cal- culated with the "dos" analysis tool of GROMACS. The Different weight functions are used for the different mo- classical heat capacity was determined from the fluctuation tions in the calculation of 2PT heat capacity: 19,20 of enthalpy 1 h@H2i c2PT = 2fR R [VDOS (ν)wcV (ν)+ cl V sol vib cp = 2 (13) 0 (7) RT p cV VDOSgaswgas(ν)] dν According to the correspondence principle the quantum The weight function of the gaseous component is 1/2 for the calculations should agree with the classical results as the h heat capacity. One limitation of 2PT is that the molec- Planck constant formally approaches zero. The 1PT model ular topology needs to remain the same, so bond break- gives fR for the heat capacity in the classical limit. Ap- ing/formation, thus chemical reactions, cannot be modeled. plying the classical weight functions of 1/2 and 1 in eq 7 it In the 1PT+AC method a quantum correction (cQC) is is easy to see, that the 2PT model can give values between cl 2 added to the classical isochoric heat capacity (cV ): fR=2 and fR for the isochoric heat capacities in the classical limit. The 1PT+AC model always satisfies the correspon- c1PT+AC = ccl + cQC = fR + cAC + cQC = c1PT + cAC (8) V V V dence principle in contrast with the 1PT or 2PT methods: AC where c is the anharmonic correction. From eq 8 it can 1PT+AC cl lim cp = cp (14) be seen that the 1PT+AC heat capacity is actually a sum h!0 of three terms: the heat capacity of f classical harmonic This also implies that the technique is able to describe the oscillators plus an anharmonic- and a quantum correction. effects of anharmonic motions. The 1PT and 2PT isochoric Jorgensen proposed to correct the classical heat capacity heat capacities for a rigid water model with 3 translational by the estimation of the intramolecular component using the and 3 rotational degrees of freedom cannot be higher than ideal gas value taken from experiments or ab initio calcula- 6R = 49.9 J/mol/K. The fact that in ref 9, 10, 17, 33 the 49,50 tions. If a given force field reproduces the experimen- calculated heat capacities are in the range of 57 and 81 tal heat capacity of the gas accurately then Jorgensen's ap- J/K/mol which is significantly larger than the theoretical proach should give a similar value to the 1PT+AC method.
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