University of Groningen
On the use of the quasi-Gaussian entropy theory in noncanonical ensembles. I. Prediction of temperature dependence of thermodynamic properties Amadei, A.; Apol, M. E. F.; Berendsen, H. J. C.
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DOI: 10.1063/1.476893
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Citation for published version (APA): Amadei, A., Apol, M. E. F., & Berendsen, H. J. C. (1998). On the use of the quasi-Gaussian entropy theory in noncanonical ensembles. I. Prediction of temperature dependence of thermodynamic properties. Journal of Chemical Physics, 109(8), 3004 - 3016. https://doi.org/10.1063/1.476893
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Download date: 21-05-2019 On the use of the quasi-Gaussian entropy theory in noncanonical ensembles. I. Prediction of temperature dependence of thermodynamic properties A. Amadei, M. E. F. Apol, and H. J. C. Berendsen
Citation: J. Chem. Phys. 109, 3004 (1998); doi: 10.1063/1.476893 View online: https://doi.org/10.1063/1.476893 View Table of Contents: http://aip.scitation.org/toc/jcp/109/8 Published by the American Institute of Physics
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The quasi-Gaussian entropy theory: Free energy calculations based on the potential energy distribution function The Journal of Chemical Physics 104, 1560 (1996); 10.1063/1.470744 JOURNAL OF CHEMICAL PHYSICS VOLUME 109, NUMBER 8 22 AUGUST 1998
On the use of the quasi-Gaussian entropy theory in noncanonical ensembles. I. Prediction of temperature dependence of thermodynamic properties A. Amadei,a),b) M. E. F. Apol,b) and H. J. C. Berendsen Groningen Biomolecular Sciences and Biotechnology Institute (GBB), Department of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands ͑Received 16 January 1998; accepted 18 May 1998͒ In previous articles we derived and tested the quasi-Gaussian entropy theory, a description of the excess Helmholtz free energy in terms of the potential energy distribution, instead of the configurational partition function. We obtained in this way the temperature dependence of thermodynamic functions in the canonical ensemble assuming a Gaussian, Gamma or Inverse Gaussian distribution. In this article we extend the theory to describe the temperature dependence of thermodynamic properties in an exact way in the isothermal-isobaric and grand canonical ensemble, using the distribution of the appropriate heat function. For both ensembles restrictions on and implications of these distributions are discussed, and the thermodynamics assuming a Gaussian or ͑diverging͒ Gamma distribution is derived. These cases have been tested for water at constant pressure, and the results for the latter case are satisfactory. Also the distribution of the heat function of some theoretical model systems is considered. © 1998 American Institute of Physics. ͓S0021-9606͑98͒50232-2͔
I. INTRODUCTION rive the temperature dependence, an ordinary differential equation, the thermodynamic master equation ͑TME͒ was The prediction of the temperature and density behavior formulated. Assuming a Gamma or Inverse Gaussian distri- of realistic fluidlike molecular systems based on an exact bution, the resulting solutions of the TME provide the tem- statistical mechanical approach is both very challenging and perature dependence of all properties based on the knowl- important for practical applications and the prediction of edge of a limited set of data at one initial reference equations of state. For molecules in the ideal gas phase it is temperature and agree very well with experimental data ͑wa- well possible to derive the thermodynamic functions in this ter, methane, methanol͒ for all densities except in the vicinity way, see for example Frenkel et al.1 On the contrary, the of the critical point, that we did not investigate yet, and in evaluation of the partition function for systems with interact- multiphasic conditions. ing molecules is in general extremely difficult, and often However, the extension of the theory in this form using severe approximations have to be made. excess properties gives problems in other ensembles. Espe- However, for the evaluation of macroscopic thermody- namic properties of realistic systems most of the information cially the definition of a proper general reference turns out to which is present in the partition function is redundant. It is be difficult. Hence up to now this theory has been used only in an approximate way to describe temperature dependence sufficient to focus on the distribution of the appropriate fluc- 5,6 tuations in the system. in noncanonical ensemble conditions. Using this idea we have derived and applied in previous In this paper we will describe how to extend the quasi- papers2–4 the quasi-Gaussian entropy ͑QGE͒ theory to obtain Gaussian entropy theory to obtain the temperature depen- the temperature dependence of thermodynamic properties at dence of the thermodynamic functions in the isothermal- constant volume, based on the internal energy fluctuations of isobaric and grand canonical ensemble in an exact way. This the system. It is possible in the canonical ensemble to focus is accomplished by using a different reference state and dis- on excess ͑‘‘ideal reduced’’͒ properties with respect to a tributions of full thermodynamic properties ͑internal energy, proper reference. We showed that the ideal reduced Helm- enthalpy͒ instead of excess ones. Possible drawbacks can holtz free energy and entropy can be expressed in an exact arise from the fact that the distributions required to describe way in terms of the excess internal energy distribution, full thermodynamic properties with high accuracy may be which must be close to a Gaussian for macroscopic systems. more complex than the ones needed for excess properties. In Since the type of ͑model͒ distribution determines the free addition we will therefore also describe the use of a proper energy and all other derived thermodynamic functions, it excess enthalpy in the NpT ensemble. An advantage of the hence determines the statistical state of the system. To de- new reference states is the fact that we immediately obtain expressions for the thermodynamic functions without explic- itly solving the appropriate TME. a͒Author to whom correspondence should be addressed. b͒Present address: c/o Professor Di Nola, Department of Chemistry, Univer- In a similar way we can also obtain the density depen- sity of Rome, ͑La Sapienza͒, p.le A. Moro 5, 00185, Rome, Italy. dence of thermodynamic properties using the distribution of
0021-9606/98/109(8)/3004/13/$15.003004 © 1998 American Institute of Physics J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3005 the volume or number of particles within the QGE theory. ͗etX ͘. This will be briefly illustrated below. A more detailed 7 This will be described in a separate paper. treatment can be found in Sec. III A, in Sec. II A of Ref. 7 This article is organized as follows. and in previous papers.2,4 In order to avoid that the general idea gets obscured by In the NVT ensemble, for example, we can define a ref- ‘‘technical’’ details and specific applications in various en- erence state at the same temperature and density with the sembles, the basic principles of the QGE theory ͑both in the same Hamiltonian except for the classical inter and intramo- 2–6 previous form and the one described in this and the fol- lecular interactions which are switched off.4 The excess or 7 lowing article ͒ are summarized in Sec. II: The system and confined ideal reduced Helmholtz free energy A* can be the choice of the reference state͑s͒, the relation between free written as energy and the distribution of an extensive quantity ͑heat A*ϭ AϪA ϭln eUЈ ϭϪln eϪUЈ , ͑2͒ function, volume, number of particles͒ via the moment gen- ͑ *ref͒ ͗ ͘ ͗ ͘*ref erating or cumulant generating function of that distribution, where UЈ is the instantaneous ideal reduced internal energy the statistical state of a system, the relation between param- ͑basically the classical potential energy͒. Here X ϭUЈ and eters of the model distribution and thermodynamic input ͑ ͒ tϭ or Ϫ, depending on whether the expectation value is data and the derivation of related thermodynamic functions. evaluated in the actual system or reference ensemble (*ref). Moreover, a unified notation for thermodynamic properties In this paper ͑Sec. III A͒ we will show that in the NVT in the NVT, NpT and VT ensembles is introduced. ensemble we can write the Helmholtz free energy difference In Sec. III we will describe the temperature dependence between two ‘‘temperatures’’  and  as of thermodynamic functions at constant pressure ͑NpT͒ and 0 constant chemical potential ͑VT͒, using the unified nota- ⌬͑A͒ϭAϪ A ϭln e⌬U ϭϪln eϪ⌬U , ͑3͒ 0 0 ͗ ͘ ͗ ͘0 tion. We introduce the definitions of the system and the ref- erence state in Sec. III A and derive an expression of the free where X ϭU is the instantaneous internal energy and, tX energy difference in terms of the distribution of the heat depending on calculating ͗e ͘ in the  or 0 ensemble, function. In Sec. III B we focus on the characteristics of this tϭϪ0ϭ⌬ or Ϫ⌬. distribution and the way the parameters can be obtained. We In Ref. 7 we demonstrate that in the NpT ensemble we indicate in Sec. III C how other thermodynamic functions can define the Gibbs free energy difference between two 0 may be obtained from the free energy function. Next a de- pressures p and p as scription of some important model distributions ͑Gaussian, 0 ⌬pV ⌬Gϭ͑G͑p͒ϪG͑p ͒͒ϭln͗e ͘p Gamma and diverging Gamma͒ and associated statistical Ϫ⌬pV states is presented in Sec. III D. Finally, some remarks are ϭϪln͗e ͘p0, ͑4͒ made on the use of the new and the previous form of the where X ϭV is the instantaneous volume and tϭ(p QGE theory in the NVT ensemble Sec. III E and a possible ͑ ͒ Ϫp0)ϭ⌬p or Ϫ⌬p. exact use of excess fluctuations in the NpT ensemble Sec. ͑ In all cases, the expectation value ͗etX ͘ and hence III F͒. the free energy difference and all other thermodynamic prop- Applications to water of the various statistical states are erties can be evaluated once we know the distribution ͑X ͒. presented in Sec. IV, along with some results on model sys- tX tX In fact, ͗e ͘ϭGX (t)ϭ͐e (X )dX is the moment gen- tems ͑harmonic oscillators, ideal gas͒. Finally, in Sec. V we 8,9 erating function ͑MGF͒ of the distribution ͑X ͒ and give some conclusions. t⌬X ϪtX G⌬X (t)ϭ͗e ͘ϭe GX(t) is the central moment gener- ating function where ⌬X ϭX ϪX and Xϭ͗X ͘. The loga- II. BASIC PRINCIPLES rithm of the MGF is called the cumulant generating function8–10 ͑CGF͒ of ͑X ͒. Since the precise distribution is As is well-known, because of the structure of statistical not known exactly for an arbitrary system, the CGF is often mechanics and especially of the semiclassical partition func- expanded in a Taylor series in t, obtaining a so-called cumu- tion, various free energy differences may be written in the lant expansion. Because the distribution is close to a Gauss- form ian for macroscopic systems this expansion is usually trun- cated after the second, third or fourth order. Zwanzig’s high 11 ln͗etX ͘ϭln etX ͑X ͒dX , ͑1͒ temperature expansion is basically such a cumulant expan- ͵ sion. Recently, cumulant expansions have also been used to where X is some instantaneous extensive property ͑e.g., the obtain free energy differences by Molecular Dynamics or internal or potential energy, enthalpy, volume, etc.͒, t is an Monte Carlo simulations.12–18 However, in the QGE theory intensive property ͑e.g., ϭ1/kT͒, ͗¯͘ denotes an expecta- we do not use such an expansion, but focus on the complete tion value in the appropriate ensemble and ͑X ͒ is the prob- distribution. ability distribution function ͑‘‘distribution’’͒ of X . As usual Based on physical and mathematical principles we can we denote instantaneous properties by calligraphic symbols, formulate several restrictions on the possible model distribu- whereas thermodynamic averages are denoted by usual capi- tions ͑X ͒, see Ref. 2, Sec. III B of this paper and Secs. II E tals. The precise nature of t and X depends on ͑a͒ the sta- and II F of Ref. 7. Acceptable model distributions are for tistical ensemble ͑NVT, NpT or VT͒, ͑b͒ the choice of the example the Gaussian, Gamma and Inverse Gaussian distri- reference state and ͑c͒ the choice in which state ͑the actual butions. The n unknown parameters of the distribution can system or reference͒ we evaluate the expectation value be obtained by the method of moments,9,19 i.e., equating the 3006 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen
TABLE I. Notation of different properties in various ensembles, where in general x, y, and z are the indepen- dent variables, is the appropriate label, F, W, S and C are the thermodynamic potential ͑free energy͒, heat function, entropy and heat capacity, Y and Z are ‘‘conjugated’’ properties and ⌼ is the partition function. JϭϪpV and DϭUϪN are the grand potential and grand canonical heat function, see Eqs. ͑11͒ and ͑13͒. For completeness we also included the canonical ensemble, see also Sec. III E.
Fץ Fץ Yϭ Zϭ zͪץͩ yͪץͩ Ensemble xyzFWS C T,z T,y ⌼
Isothermal-isobaric TpNpGHSCp V ⌬ Grand canonical T V JDSC ϪN Ϫp ⌶ Canonical TVNVAUSCV Ϫp Q
r III. THEORY first n theoretical moments M r͓X ͔ϭ͗(⌬X ) ͘ expressed in terms of the parameters and the first n sample moments, A. Definition of the system and reference state which can be related to thermodynamic input data at one The Gibbs free energy in the isothermal-isobaric en- state point via general statistical mechanical relations such as semble ͑NpT͒ is given by 2 ϭϪ p͒T , G kT ln ⌬͑5͒ץ/VץM 2͓U͔ϭkT CV , M2͓V ͔ϭϪkT͑ with etc. The Gaussian distribution, for example, is characterized ϱ dV by the first two moments, the Gamma and Inverse Gaussian ⌬ϭ eϪpV Q͑N,V ,T͒ by the first three. ͵0 v
With the appropriate thermodynamic input data at one Ϫd h N ϱ Ј Ј* state point we can obtain the free energy difference at that ϭ dV dpN͚ dxNeϪH ͑6͒ state point. However, we can also obtain the temperature vN! ͵0 ͵ l ͵V dependence of Eqs. ͑2͒ and ͑3͒ and the pressure dependence and of Eq. ͑4͒ in the following way. If we have used the reference state ensemble to evaluate HϭUϩpV , ͑7͒ the MGF, the distribution and its parameters are fixed at one the isothermal-isobaric partition function and the instanta- state point. Hence we immediately obtain the temperature or 2–6 pressure dependence of the free energy difference and all neous enthalpy, respectively. As in previous articles we other derived thermodynamic functions. will use calligraphic symbols for instantaneous properties If we have evaluated the MGF in the system ensemble, and usual roman symbols for thermodynamic averages. Fur- the distribution and hence its parameters are also temperature thermore, or pressure dependent, so we do not obtain an explicit ex- Ϫd h N Ј Ј* pression. However, it turns out that with the help of general Q͑N,V,T͒ϭ Qe͑N,T͒ dpN dxNeϪU N! ͵ ͚ ͵ thermodynamic relations we can formulate in each case an l V ͑8͒ ordinary differential equation, the thermodynamic master equation ͑TME͒, the solution of which provides the explicit is the canonical partition function, where h is Planck’s con- e temperature or pressure dependence. stant, dN and Q (N,T) are the number of degrees of freedom The route using the reference state ensemble seems ad- and the electronic partition function for N molecules, as- vantageous, although the thermodynamic input data may be sumed to be independent of the volume, U(x,p,l,N) the difficult to obtain in that ensemble, e.g. see Eq. ͑2͒, where ͑instantaneous͒ internal energy and xN and pN the coordi- the reference state is rather unphysical. In that case the other nates and conjugated momenta of N molecules. ϭ1/kT, route is more appropriate, at the expense of having to solve with k the Boltzmann constant and the summation runs over explicitly the TME ͑see Refs. 2 and 4͒. all accessible vibrational states ͕l͖. The prime and the star Since by either one of the two routes we obtain the free on the integrals denote the restrictions due to fixed bond energy difference and derived thermodynamic properties as a length and bond angle constraints and the restrictions due to function of an intensive parameter ͑e.g.,  or p͒, and the a possible confinement of the system within a part of con- relation between distribution and free energy is unique ͑Eq. figurational space, respectively.4 Furthermore V is the in- ͑1͒ is a Laplace transform of the distribution͒ the distribution stantaneous volume and p the pressure. The factor v in Eq. completely determines the thermodynamics of the system ͑6͒ is in fact a numerical volume differential, which arises and therefore the statistical state. from the definition of the entropy in the NpT ensemble with In order to facilitate the derivations we introduce a uni- continuous volume; it makes ⌬ dimensionless and assures fied notation for quantities in various ensembles, see Table I. that in the zero temperature limit the entropy correctly tends In this way we can derive the temperature dependence using to zero.20 In fact, v is a measure of the accuracy with which Eq. ͑3͒ for all three ensembles simultaneously ͑Sec. III͒. we want to describe our macroscopic properties. However, J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3007 from Eqs. ͑5͒ and ͑6͒ it follows that free energy differences So, denoting in general the free energy, heat function are independent of v. It is worth to note, since this was not and entropy by F, W and S ͑see Table I͒, we have using the explicitly mentioned in previous papers, that for polyatomic 0 ensemble molecules in the canonical partition function the correction ⌬͑F͒ϭW ⌬Ϫln eϪ⌬⌬W 0 , ͑16͒ for the phase-space positions which are permutations of 0 ͗ ͘0 identical particles, should not only involve molecular permu- tations (1/N!) but also the possible rotations and intramo- whence lecular displacements which are permutations of identical at- F͑T͒ϭW ϪTS ϪkT ln͗eϪ⌬⌬W 0͘ . ͑17͒ oms of single molecules. In fact, the correction factor in the 0 0 0 partition function should be (N!(1ϩ )N)Ϫ1, where 0is ␥ ␥у Other thermodynamic properties follow simply as tempera- a positive constant characteristic of the molecule, see the ture derivatives: Appendix. For sake of simplicity in this paper, as in the F͑T͒ץ previous ones, we have always included the factor (1 ϩ ϪN S͑T͒ϭϪ , ͑18͒ ͪ Tץ ͩ .into the electronic partition function (␥ The Gibbs free energy difference between two ‘‘tem- y,z peratures’’  and  can now be expressed as 0 W͑T͒ϭF͑T͒ϩTS͑T͒, ͑19͒ ⌬͑G͒ϭGϪ G W͑T͒ץ 0 0 ϱ N N ϪH C͑T͒ϭ , ͑20͒ ͪ Tץ ͩ dV ͐Јdp ͚l͐VЈ*dx e 0͐ ϭϪln y,z ϱ N N Ϫ0H ͐0 dV ͐Јdp ͚l͐VЈ*dx e with C, y and z the appropriate heat capacity and fixed prop- ϭH ⌬Ϫln eϪ⌬⌬H0 ͑9͒ 0 ͗ ͘0 erties, see Table I.
ϭH⌬ϩln͗e⌬⌬H͘, ͑10͒ where H ϭ H , Hϭ H , ⌬H ϭHϪH and ⌬H B. Distribution of the heat function 0 ͗ ͘0 ͗ ͘ 0 0 Ϫ⌬⌬W ϭHϪH with ͗ ͘ and ͗ ͘ ensemble averages in the 0 The term ln e 0 can be written as ¯ 0 ¯ ͗ ͘0 and  ensemble. In the grand canonical ensemble ͑VT͒ the ‘‘grand po- ln͗eϪ⌬⌬W 0͘ ϭln eϪ⌬⌬W 0 ͑⌬W ͒d⌬W , tential’’ JϭϪpV is 0 ͵ 0 0 0 ͑21͒ JϭϪkT ln ⌶͑11͒ with 0(⌬W 0) the probability distribution of the fluctua- with tions ⌬W 0 in the 0-ensemble. In fact, 0 is a continuous probability density, since in the quasiclassical limit U, H ϱ and D are continuous. Since we can subdivide a macroscopic ⌶ϭ ͚ eN Q͑N ,V,T͒ N ϭ0 system into n identical, independent subsystems, each with linear dimension L larger than the typical correlation ϱ Ϫd 21,22 h N Ј Ј* length and n ϱ, we can apply the central limit e N N ϪD 8,23,24 → ϭ ͚ Q ͑N ,T͒ dp ͚ dx e , ͑12͒ theorem to show that (⌬W ) is unimodal and close N ϭ0 N ! ͵ l ͵V 0 0 to a Gaussian distribution ͑‘‘quasi-Gaussian’’͒. Such a con- DϭUϪN , ͑13͒ dition might not be fulfilled in the critical point region, where the correlation length tends to infinity, and hence the the grand canonical partition function and the instantaneous distribution is not necessarily well modeled by a unimodal grand canonical heat function ͑‘‘granthalpy’’͒, respectively. one. However, the fact that everywhere else in the phase N is the instantaneous number of molecules in the system diagram the system can be described by unimodal distribu- and the chemical potential. Also in this case we can ex- tions suggests that even at the critical point the distribution press the free energy difference as can be considered at least as a limit condition of a general ͑highly complex͒ unimodal distribution. The term Ϫ⌬⌬W 0 ⌬͑J͒ϭJϪ0J0 ͗e 0͘ ϵG (Ϫ⌬) is in fact the central moment 0 ⌬W 0 8,9 ͚ϱ ͑hϪdN / !͒͐ dpN ͚ ͐Ј*dxN eϪD generating function of the distribution 0(⌬W 0). The N ϭ0 N Ј l V 0 ϭϪln ϱ zero superscript on G denotes the reference state condi- ͚ ͑hϪdN /N !͒͐ЈdpN ͚ ͐Ј*dxN eϪ0D ⌬W 0 N ϭ0 l V tion. The distribution is statistical mechanically defined as ϭD ⌬Ϫln eϪ⌬⌬D0 ͑14͒ 0 ͗ ͘0 ⍀* eϪ0͑W0ϩ⌬W 0͒ ⌬W 0 ϭD ϩln e⌬⌬D , 15 0͑⌬W 0͒ϭ , ͑22͒ ⌬ ͗ ͘ ͑ ͒ ⌼0 where D ϭ D , Dϭ D , ⌬D ϭDϪD and ⌬DϭD 0 ͗ ͘0 ͗ ͘ 0 0 where ⌼0 is the appropriate partition function ͑Table I͒ ϪD. evaluated at 0 , and 3008 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen
Ϫd rϪ2 rϪ2 T )y,z . Hence for a specific distribution viaץ/ C0 ץ) h N ϱ e Ј N ⍀⌬*H ϭ Q ͑N͒ dV dp Eqs. ͑17͒–͑21͒ the temperature dependence of F, W, S and C 0 vN! ͵0 ͵ is completely determined by the knowledge of W0 , S0 and the heat capacity C and a limited set of temperature deriva- Ј* N 0 ϫ͚ dx ␦͑⌬H0͑x,p,l,V ͒Ϫ⌬H0͒, tives at one temperature T . Therefore, each different type of l ͵V 0 distribution defines a different statistical state of the system. ͑23͒ Note that since we have used Eqs. ͑9͒ and ͑14͒ to express the
ϱ Ϫd thermodynamic properties in terms of the distribution in the h N Ј * e N 0 ensemble, we do not explicitly have to solve a ‘‘thermo- ⍀⌬D ϭ ͚ Q ͑N ͒ dp 0 N ϭ0 N ! ͵ dynamic master equation,’’ like in Refs. 2–6. Secondly, from all the possible distributions arising from Ј* N the basic differential equation, Eq. ͑25͒, we have to select the ϫ͚ dx ␦͑⌬D0͑x,p,l,N ͒Ϫ⌬D0͒, ͑24͒ l ͵V ones which are compatible with physical-mathematical restrictions,2 like the fact that the distribution is unimodal are the appropriate ‘‘volumes’’ of the hypersurface in phase and should be defined on the interval ͓⌬W ,ϱ , where space of constant ⌬H or ⌬D with ␦͑ ͒ the Dirac delta 0,min ͘ 0 0 • ⌬ is the finite lower limit of ⌬ .33 This restriction function. We therefore find4 W 0,min W 0 is not strictly necessary for the granthalpy in some special ln ⍀* cases where Ͼ0 and so 0(⌬D0) could be defined on the ץ d0 ⌬W 0 ϭϪ0 0Ϫ interval ͗Ϫϱ,ϱ͘. In this paper we do not discuss these very -W 0 ͬ unusual cases. Possible unimodal distributions already inves⌬ץ ͫ d⌬W 0 m tigated are the Gaussian, Gamma and Inverse Gaussian ͑⌬W 0Ϫ⌬W 0m͒P ͑⌬W 0͒ 2–4,8 ϭϪ0 ͑25͒ distribution. From Eqs. 9 and 14 it is clear that the Gn͑⌬W ͒ ͑ ͒ ͑ ͒ 0 free energy diverges if  0, because the volume integral by expanding the term between square brackets around the and the summation over the→ number of particles, as well as 25,26 mode ͑maximum͒ of 0 in a Pade´ approximant. ⌬W 0m the kinetic part of the partition function, tend to infinity in is the position of the mode of the distribution and absence of the Boltzmann factor. For the same reason 0
m must be different from zero, too. In fact, the moment gener- ating function of the distribution , G0 (Ϫ⌬) Pm͑⌬W ͒ϭ a ͑⌬W ͒i, ͑26͒ 0 W 0 ͚ i,0 0 ϭ eϪ⌬W must be finite for any finite  except for iϭ0 ͗ ͘0 ϭ0, i.e., Ϫ⌬ϭ0 . Note that 0(⌬W 0)ϭ0(W ). n 2 n j As previously discussed, for all distributions ͑͒ aris- G ͑⌬W 0͒ϭ b j,0͑⌬W 0͒ , ͑27͒ j͚ϭ0 ing from the generalized Pearson system ͓Eqs. ͑25͒–͑27͔͒ t with mϩ1Ͼn, the MGF G(t)ϭ͗e ͘ is finite for all values are polynomials of order m and n in ⌬W 0 , where without of tϭϪ⌬. For distributions defined on the interval loss of generality we can set am,0ϭ1. Note that the zero ͓⌬W 0,min ,ϱ͘ where mϩ1Ͻn the MGF always diverges and subscript on the coefficients ͕ai,0͖ and ͕b j,0͖ reflects that when mϩ1ϭn the MGF is finite only for tϭϪ⌬ they are evaluated at 0 and therefore temperature indepen- Ͻ1/bn,0 . Therefore if dent. Equation ͑25͒ represents the generalized Pearson sys- tem of distributions,2,4,27–30 the solutions of which are distri- 1/bn,0ϭtdϭ lim Ϫ⌬ϭ0 ͑29͒  0 butions of increasing complexity. At this point several → remarks have to be made. ͑where td denotes the value of t at which the MGF diverges͒ 2 First, the parameters ͕ai,0͖ and ͕b j,0͖ can be expressed the free energy will fulfil the requirement to be always finite k in terms of a set of central moments M ϭ͗(⌬W ) ͘ , except when  0. In fact, this eliminates the parameter bn,0 k,0 0 0 → which in turn via statistical mechanics2,31,32 can be expressed and reduces the complexity of the solution. So for all the in terms of some thermodynamic derivatives at temperature ‘‘temperature lines’’ the first possible exact distributions T ϭ1/k . In fact, the central moments of W are related to are the Gamma distribution (mϭ0, nϭ1) and Inverse 0 0 4 the heat capacity and some temperature derivatives in the Gaussian distribution ͑a degenerate mϭ1, nϭ2 solution ͒ following way:2 with 1/bn,0ϭ0 , defining the diverging Gamma and diverg- ing Inverse Gaussian states. M 2,0ϭ͑kT0͓͒T0C0͔, C. Derivation of conjugated properties C 0 ץ 2 2 M3,0ϭ͑kT0͒ T ϩ2T0C0 , ͑28͒ Using the appropriate distribution from the generalized ͪ Tץ 0ͩ ͫ y,z ͬ Pearson system we obtain an explicit expression of rϪ2 rϪ2 -T ), where the paץ/ C0 ץ,...,Tץ/ C0ץ, F(T;S0 ,W0 ,C0 ,... rameters of the distribution are determined by the moments T)y,z are the values of C and up to M r,0 ͑rϭ2 for a Gaussian or diverging Gamma, rϭ3ץ/ C0ץ) where C0 and T)y,z at T0 .IfMr,0 is the highest order moment re- for a Gamma or Inverse Gaussian distribution͒. Subsequentץ/Cץ) quired to express the parameters ͕ai,0͖ and ͕b j,0͖, this corre- temperature derivatives of this expression yield S(T), W(T) sponds to the knowledge of C0 and derivatives up to and C(T), see Eqs. ͑18͒–͑20͒. However, further ‘‘conju- J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3009 gated’’ thermodynamic properties follow from derivatives in where we defined ⌬ϭϪ͗͘, the cumulant generating the other two independent variables y and z, see Table I: function is t⌬ 1 2 F ln G⌬͑t͒ϭln͗e ͘ϭ 2b0t ͑40͒ץ Yϭ , ͑30͒ ͪ yץ ͩ T,z with b0ϭM 2 . In this case substituting tϭϪ⌬, ⌬ ϭ⌬W , b ϭb ϭM and M given by Eq. ͑28͒ we ob- F 0 0 0,0 2,0 2,0ץ Zϭ . ͑31͒ tain ͪ zץ ͩ T,y 1 T0 rϪ2 rϪ2 F͑T͒ϭW0ϪT S0ϩ C0 ϩT0C0 1Ϫ , ͑41͒ T ͩ 2 ͪ ͩ 2Tͪץ/ C0 ץ,...,Tץ/ C0ץ, Since the parameters S0 ,W0 ,C0 are also functions of y and z, we can obtain Y(T) and Z(T) via T0 W͑T͒ϭW ϩT C 1Ϫ , ͑42͒ 0 0 0ͩ T ͪ rϪ2 f l,0 ץ Fץ S0ץ Fץ Y T ϭ ϩ , 32 2 ͑ ͒ ͚ ͑ ͒ 1 1 T0 ͪ yץ ͩ fץ y ͪ lϭϪ1ץ ͩ Sץ 0 T,z l,0 T,z S͑T͒ϭS ϩ C Ϫ C , ͑43͒ 0 2 0 2 0ͩ T ͪ rϪ2 f l,0 2 ץ Fץ S0ץ Fץ Z͑T͒ϭ ϩ ͚ , ͑33͒ T0 z ͪ C͑T͒ϭC0 , ͑44͒ץ ͩ fץ z ͪ lϭϪ1ץ ͩ Sץ 0 T,y l,0 T,y ͩ T ͪ where we have defined where W0 , S0 and C0 are the values of W, S and C at the reference temperature T ϭ1/k . Furthermore, using Eq. C0 0 0ץ f ϭW ; f ϭC ; f ϭ , ͑34͒ ͑32͒ we find for the density-related properties ͑see Table I͒ ͪ Tץ ͩ Ϫ1,0 0 0,0 0 1,0 y,z T T0 etc. Using the Maxwell relations4,32 we find Y͑T͒ϭY ϩB Ϫ1 ϪB 1Ϫ , ͑45͒ 0 1ͩ T ͪ 2ͩ T ͪ 0 lϩ1 lϩ2 Y0 ץ Y0 ץ f l,0 ץ ϭϪl ϪT , ͑35͒ where ͪ Tlϩ2ץ Tlϩ1 ͪ 0ͩץ ͩ ͪ yץ ͩ T,z y,z y,z 2 Y 0 ץ Y 0 1 2ץ lϩ1 lϩ2 B1ϭT0 ϩ T0 2 , ͑46͒ ͪ Tץ ͩ T ͪ 2ץ ͩ Z0 ץ Z0 ץ f l,0 ץ ϭϪl ϪT , ͑36͒ y,z y,z ͪ Tlϩ2ץ Tlϩ1 ͪ 0ͩץ ͩ ͪ zץ ͩ T,y y,z y,z 2 Y 0 ץ 2 1 B2ϭ T0 2 , ͑47͒ ͪ Tץ ͩ Y0 2ץ S0ץ ϭϪ , ͑37͒ y,z ͪ Tץ ͩ ͪ yץ ͩ T,z y,z and with Eq. ͑33͒ we obtain a similar expression for Z(T), replacing Y by Z in Eqs. ͑45͒–͑47͒. Note that the Gaussian Z0ץ S0ץ ϭϪ , ͑38͒ state, because of the very special properties of the Gaussian ͪ Tץ ͩ ͪ zץ ͩ T,y y,z distribution, is equivalent to a second order cumulant expan- 0 0 sion of ⌬(F)in⌬. T ϭY 0 . Hence Y(T) and Z(T) areץ/ Y 0 ץ where obviously For a Gamma state, we use the fact that in general for a / Y 0ץ, simply expressions of the form Y(T;Y 0 r r r r Gamma distribution2,4 .( Tץ/ Z0 ץ,...,Tץ/ Z0ץ, T ) and Z(T;Z0ץ/ Y0 ץ,...,Tץ 2 2 b0 /b1 b1͑1/b1͒ 2 b0 /b Ϫ1 ͑⌬͒ϭ 2 ͑b0ϩb1⌬͒ 1 ⌫͑b0 /b1͒ D. Statistical states b ϩb ⌬ Next we will derive expressions for F(T), W(T), S(T), 0 1 ϫexp Ϫ 2 ͑48͒ C(T), Y(T) and Z(T) for the Gaussian, Gamma and diverg- ͭ b1 ͮ ing Gamma state. The equations of the Inverse Gaussian and 34 with ⌫͑•͒ the Gamma function, the cumulant generating diverging Inverse Gaussian states can be obtained straight- function is forwardly, but we omitted them because in most cases their 1 1 behavior is almost indistinguishable from the corresponding t⌬ 4 ln G⌬͑t͒ϭln͗e ͘ϭϪb0 tϩ 2 ln͑1Ϫb1t͒ ͑49͒ Gamma expressions. The way to solve Eqs. ͑17͒–͑21͒ for ͫb1 b ͬ 1 these distributions is mathematically very similar to the case with b ϭM and b ϭM /2M . In this case substituting t previously described in the canonical ensemble using excess 0 2 1 3 2 ϭϪ⌬, ⌬ϭ⌬W , b ϭb , b ϭb , M ϭM and properties. Therefore we will only give here the final expres- 0 0 0,0 1 1,0 2 2,0 M ϭM we find with Eq. ͑28͒ sions and refer to Refs. 2 and 4 for further details. 3 3,0 For a Gaussian state, we use the fact that in general for T0C0 TC0 a Gaussian distribution2,4 F͑T͒ϭW0Ϫ ϪTS0ϩ 2 ͓␦0ϩln͑1Ϫ␦0͔͒ ␦0 ␦0 2 1 ͑⌬͒ TC0 ͑⌬͒ϭ exp Ϫ , ͑39͒ Ϫ 2 ln͑1Ϫ␦͑T͒͒, ͑50͒ ͭ 2b0 ͮ ͱ2b0 ␦0 3010 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen
␦͑T͒ T T T W͑T͒ϭW ϩ͑TϪT ͒C , ͑51͒ Y͑T͒ϭY ϩB Ϫ1 ϩB ln ͑64͒ 0 0 0ͩ ␦ ͪ 0 1ͩ T ͪ 2ͩ T ͪ ͩ T ͪ 0 0 0 0 with C0 S͑T͒ϭS0Ϫ 2 ͓␦0ϩln͑1Ϫ␦0͔͒ 2 Y 0 ץ Y 0ץ 0␦ B ϭT ϪT2 , ͑65͒ ͪ T2ץ T ͪ 0ͩץ 1 0ͩ C0 y,z y,z ϩ ͓␦ ͑T͒ϩln͑1Ϫ␦ ͑T͔͒͒, ͑52͒ ␦2 2 Y 0 ץ 0 2 B2ϭT0 2 , ͑66͒ ͪ Tץ ͩ 2 ␦͑T͒ y,z C͑T͒ϭC , ͑53͒ 0ͩ ␦ ͪ and using Eq. 33 a similar expression for Z(T), replacing Y 0 ͑ ͒ where we have defined by Z in Eqs. ͑64͒–͑66͒.
T͒y,zץ/ C0ץb1,0 M3,0 T0͑ ␦0ϭ ϭ ϭ ϩ1, ͑54͒ kT0 2kT0M2,0 2C0 E. Canonical ensemble
T0␦0 All derivations up to here are also valid for the canonical ␦͑T͒ϭ , ͑55͒ T͑1Ϫ␦0͒ϩT0␦0 ͑NVT͒ ensemble, see also Table I. In that case we have to use the distribution of the full instantaneous internal energy U. -T)y,z at T0 . Further- Any exact distribution in this ensemble must have a divergץ/Cץ) T)y,z the value ofץ/ C0ץ) with more, from Eq. ͑32͒ we find after straightforward algebra ing MGF for  0 as well, since the kinetic part of the → T T 1Ϫ␦͑T͒ Helmholtz free energy A tends to infinity for T ϱ͑note, Y͑T͒ϭY ϩB Ϫ1 ϩB Ϫ1 → 0 1ͩ T ͪ 2ͩ T ͪ 1Ϫ␦ however, that the ideal reduced Helmholtz free energy AЈ is 0 0 0 finite in this limit4͒. In previous papers2–4 we showed that a T 1Ϫ␦͑T͒ formulation of the theory using the excess ͑‘‘ideal reduced’’͒ ϩB ln ͑56͒ 3ͩ T ͪ ͩ 1Ϫ␦ ͪ internal energy UЈ provides an excellent description of 0 0 many systems at constant volume, already using a Gamma with distribution. This already suggests that the description using 2 the full internal energy U must be less successful than the Y 0 ץ Y 0 4␦0Ϫ3 2ץ B1ϭT0 Ϫ 2 T0 2 description using the ideal reduced internal energy UЈ for ͪ Tץ ͩ ␦T ͪ 2ץ ͩ y,z 0 y,z the same type of distribution ͑i.e., level of the theory͒. 3 In fact, there are two general reasons for this: first it is Y 0 ץ 3 1 ϩ T , ͑57͒ very difficult to model the distribution of the energy which ͪ T3ץ 0ͩ 2␦2 0 y,z contains intramolecular quantum vibrational energy ͑which 2 3 is not present in UЈ͒, and secondly the kinetic energy ͑also Y0 ץ Y0 1 3 ץ 0Ϫ3 2␦2 B 2ϭϪ T ϩ T , not included in UЈ͒ always requires as the simplest exact ͪ T3ץ T2 ͪ 2␦2 0ͩץ 2␦2 0ͩ 0 y,z 0 y,z statistical state for the full energy a diverging Gamma, while ͑58͒ for the ideal reduced energy a Gamma state of full complex- 2 3 .Y 0 ity is the simplest exact solution ץ Y 0 1 ץ 0Ϫ3␦3 B ϭ T2 Ϫ T3 , ͑59͒ . T3 ͪ Consider for example the isochoric heat capacity CVץ T2 ͪ ␦3 0ͩץ 3 ␦3 0ͩ 0 y,z 0 y,z The first exact expression using the full internal energy is a and using Eq. ͑33͒ we obtain a similar expression for Z(T), diverging Gamma state, where from Eq. ͑63͒ we have replacing Y by Z in Eqs. ͑56͒–͑59͒. CV(T)ϭCV0 . Restricting ourselves for simplicity to mon- Finally, for a diverging Gamma state the restriction due atomic molecules, we see that the ideal reduced heat capacity to the divergence of the MGF at ϭ0 reduces the complexity CVЈ , i.e., with respect to an ideal gas at the same density and of the solutions. In fact, combining the general expression of temperature, is simply a constant 2,4 b1,0 for a Gamma state, b1,0ϭM 3,0 /2M 2,0 , with Eq. ͑29͒, 3 CVЈ͑T͒ϭCV0Ϫ 2Nk. ͑67͒ b1,0ϭ1/0 , we obtain M 3,0 /2kT0M2,0ϵ␦0ϭ1 and thus via Eq. ͑55͒ also ␦(T)ϭ1. This gives However, using the ideal reduced internal energy distribu- tion, we obtain for a Gamma state2–4 T0 F͑T͒ϭW ϪT C ϩT͑C ϪS ͒ϩTC ln , ͑60͒ 2 0 0 0 0 0 0 T T0 ͩ ͪ CЈ͑T͒ϭCЈ ͑68͒ V V0ͩT͑1Ϫ␦ ͒ϩT ␦ ͪ 0 0 0 W͑T͒ϭW0ϩ͑TϪT0͒C0 , ͑61͒ with T0 T͒ץ/CVЈ 0ץS͑T͒ϭS ϪC ln , ͑62͒ T0͑ 0 0 ͩ T ͪ ϭ ϩ1 69 ␦0 ͑ ͒ 2CVЈ 0 C͑T͒ϭC , ͑63͒ 0 which is clearly different, and moreover proved to be an and, after careful inspection35 excellent model of water, methane and methanol. Therefore J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3011 in the canonical ensemble we use the description in terms of 1 vNv e kin v rot Ј* N the ideal reduced internal energy UЈ instead of the full en- G refϭϪkT ln Q Q QidZid dxint , ͑72͒ * N! v ͵id ergy U.
G refץ S ϭϪ * , ͑73͒ ͪ Tץ ͩ ref* p,N F. Excess properties in the NpT ensemble H ϭG ϩTS , ͑74͒ Compared to the canonical ensemble, it is much more *ref *ref *ref
H refץ difficult to define and use excess properties to set up the C ϭ * , ͑75͒ ͪ Tץ ͩ theory in the isothermal-isobaric ensemble. So far we were p*ref only able to derive the theory in that ensemble using excess p,N properties in an approximated way.5 Here we will illustrate G refץ how to define a proper reference state that can be used to V ϭ * ϭ0, ͑76͒ ͪ pץ ͩ ref* derive the theory for fluctuations of an excess instantaneous T,N c enthalpy H , where we have removed the intramolecular etc. We define the corrected thermodynamic properties as quantum vibrational and kinetic parts of the energy. Such a ϱ ϪpV N ϪUЈ formally equivalent and exact derivation of the theory could ͐0 dV e ͐VЈ*dx e GcϭGϪG ϭϪkT ln , ͑77͒ be preferable from a practical point of view, since for a given *ref ͐Ј*dxN ZrotvNv level of the theory ͑defined by the type of statistical state we id int id Gcץ -want to use to model the system͒ the excesslike heat func ScϭSϪS ϭϪ , ͑78͒ ͪ Tץ ͩ tions might be described with higher accuracy as for the *ref canonical ensemble, Sec. III E. p,N We can use the previously introduced general approxi- c c c H ϭHϪH refϭG ϩTS , ͑79͒ mation for the canonical partition function4 * Hcץ 1 c e kin v Ј* N ϪUЈ C pϭC pϪC p refϭ , ͑80͒ ͪ Tץ ͩ * QХ Q Q Qid dx e , ͑70͒ N! ͵ p,N with Qkin the semiclassical kinetic partition function includ- G ץ c ͑ ϪdN 0 0 V ϭVϭ . ͑81͒ ͪ pץ ͩ ing the factor h ͒, UЈϭ⌽ϩ⌿ϩE ϪEid where ⌽ and ⌿ are the inter and intramolecular classical potential energies, T,N E0 is the overall vibrational ground state energy of the sys- Defining the instantaneous corrected enthalpy H cϭUЈ 0 v tem and where Eid and Qid are the overall vibrational ground ϩpV , from the definition of the corrected properties we can state energy and partition function of the ideal gas ͑⌽ϭ0͒, write the corrected free energy difference in the same way as respectively. Eqs. ͑9͒ and ͑10͒:
We define a reference state as an ideal gas with no clas- c ͐ϱdV ͐ *dxNeϪH sical inter and intramolecular potential energy ͑⌽ϭ⌿ϭ0, so c 0 Ј 0 0 ⌬͑G ͒ϭϪln c E ϭE ͒ where by an infinite attractive potential among the ϱ N Ϫ0H id ͐0 dV ͐Ј*dx e molecules the centers of mass of the molecules are confined c Ϫ c within a small volume ‘‘differential’’ v, as well as the full ϭH ⌬Ϫln e ⌬⌬H 0 ͑82͒ 0 ͗ ͘0 volume of the system. The corresponding reference isothermal-isobaric partition function is c ⌬⌬H c ϭH ⌬ϩln͗e ͘ ͑83͒ N 1 v v c c c c c c c e kin v rot Ј* N with ⌬H 0ϭH ϪH0 , ⌬H ϭH ϪH with H0 ⌬ refϭ Q Q QidZid dxint , ͑71͒ c c c * N! v ͵id ϭ H and H ϭ H . From this follows ͗ ͘0 ͗ ͘ where the translational configurational volume is given by c c c c Ϫ⌬⌬H 0 N rot rot N G ϭH0ϪTS0ϪkT ln͗e ͘ . ͑84͒ v , Zid ϭ(zid ) is the ideal gas rotational configurational 0 N 1 N volume and ͐idЈ*dxintϭ(͐idЈ*dxint) is the ideal gas configu- From these equations it is evident that using the distribution rational volume of the classical intramolecular coordinates c N of the instantaneous corrected enthalpy H in the 0 en- xint , for a given definition of the internal and rotational co- semble, ordinates of the single molecule. The factor v/v arises from 4 c the integration over the volume. As usual the prime and the eϪ0H ͐ϱdV ͐Ј*dxN␦͑H c͑x,V ͒ϪH c͒ c 0 V star on the integrals denote the possible integration restric- 0͑H ͒ϭ c ͐ϱdV ͐Ј*dxNeϪ0H tions due to fixed bond lengths and angles and to a confine- 0 V ͑85͒ ment of the system within a part of configurational space, here in the ideal gas condition. Note that the use of transla- also fully defined by a set of moments of H c or temperature tional, rotational and intramolecular coordinates in Eq. ͑71͒ derivatives of Hc, in the NpT ensemble all the considerations implies that also the kinetic partition function Qkin is ex- and derivations of Sec. III B–III D can be applied to the pressed using the conjugated momenta of these coordinates. corrected thermodynamic properties by simply exchanging Hence the reference thermodynamic properties are in every equation the full properties with the corresponding 3012 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen corrected ones. For instance, for a diverging Gamma state range of interest the vibrations are largely confined to the ͑the formally exact Gamma state for both the corrected and ground state, if the corrected enthalpy fluctuations are prop- full enthalpy fluctuations͒ we have erly modeled by a diverging Gamma state, also the full en- thalpy fluctuations can be described by a diverging Gamma T c c c c c c 0 state. On the contrary, for any other discussed statistical state G ͑T͒ϭH0ϪT0C p0ϩT͑C p0ϪS0͒ϩTCp0 ln , ͑86͒ ͩ T ͪ such correspondence is lost. c c c Finally it must be mentioned that in the grand canonical H ͑T͒ϭH0ϩ͑TϪT0͒C p0 , ͑87͒ ensemble Eq. ͑70͒ cannot provide any derivation based on an T0 excesslike granthalpy and so in that ensemble we can only Sc͑T͒ϭSc ϪCc ln , ͑88͒ 0 p0 ͩ T ͪ use the full property. c c C p͑T͒ϭC p0 , ͑89͒ and IV. RESULTS T T T c c Before presenting a comparison between the Gamma V͑T͒ϭV0ϩB1 Ϫ1 ϩB2 ln , ͑90͒ ͩ T0 ͪ ͩ T0 ͪ ͩ T0 ͪ state expressions and experimental data of water, we will with first show that the Gamma distribution is the exact distribu- tion for the heat function of two simple systems, i.e., the 2 V0 internal, kinetic and potential energy of a set of classical ץ V0 2ץ c B ϭT ϪT , ͑91͒ T2 ͪ harmonic oscillators in the NVT ensemble and the enthalpyץ T ͪ 0ͩץ 1 0ͩ p,N p,N ͑internal energy͒ of an ideal gas in the NpT ͑NVT͒ ensemble. 2 V0 ץ Bc ϭT2 , ͑92͒ First consider a set of N classical harmonic oscillators T2 ͪ ͑e.g. normal modes͒ in the canonical ensemble. It is alwaysץ 2 0ͩ p,N possible to define a set of generalized coordinates and and using Eq. ͑33͒ a similar expression for c(T), replacing i momenta ͑not necessarily conjugated͒ such that the in- V by c in Eqs. 90 – 92 . It is very useful to link the ref- i ͑ ͒ ͑ ͒ stantaneous internal energy can be written as erence properties to ideal gas ones. From the ideal gas free energy N 1 N 1 1 Uϭ 2ϩ 2 e kin v rot Ј* N Ϫ⌿ ͚ 2 i i ͚ 2m i GidϭϪkT ln Q Q QidZid dxinte iϭ1 iϭ1 i N! ͵id N N kT 2 kT 2 ϱ d i i V ϪpV N ϭ ϩ ϫ e V 2 i͚ϭ1 2 i͚ϭ1 ͩ iͪ ͩ iͪ ͵0 v 2N kT 2 kT Ј* 1 i 2 ϭϪkT ln QeQkinQv Qrot dxN eϪ⌿ ϭ ͚ ϭ ͑98͒ id id ͵ int Nϩ1 2 iϭ1 2 v p ͩ iͪ id ͑ ͒ ͑93͒ with i and mi force constants and reduced masses and we obtain after a few steps where we used the fact that the variance of the position is 2 ϭkT/ and that of the momenta is 2 ϭm kT. Further- pv i i i i G ϭG ϪNkT ln͗e͘ ϪNkT ln ϪNkT, *ref id id ͩ NkTͪ more, i denotes a general coordinate. Since U is the sum of squares of an independent standard normal variable / ,it ͑94͒ i i is proportional to a 2 variable,36 which follows a N͗͘ pv 2  id -distribution with in this case 2N degrees of freedom. S refϭSidϩNk ln͗e ͘idϪ ϩNk ln , * T ͩ NkTͪ Hence the distribution of Uϭ(kT/2)2 is a 2-distribution ͑95͒ as well,2,8 H ϭH ϪN ϪNkT, ͑96͒ *ref id ͗ ͘id ͑1/kT͒N NϪ1 ϪU/kT ͘id ͑U͒ϭ U e ͑99͒͗ץ C ϭC ϪN ϪNk, ͑97͒ ⌫͑N͒ ͪ Tץ ͩ p*ref p id p,N where is the molecular intramolecular potential with which is a special subfamily of Gamma distributions. In fact,  1 Ϫ 1 ͗⌿͘idϭN͗͘id and ͗e ͘idϭ͐idЈ*dxint/͐idЈ*e dxint is the it is a diverging Gamma distribution since b1ϭkT ͓cf. Eq. moment generating function of the molecular intramolecular ͑48͔͒. potential energy fluctuation. In the case of small molecules Of course, the same argument can also be applied to the N 2 where ϭ0 ͑e.g., water͒ the previous equations simplify potential energy Upotϭ͚iϭ1(i/2) or the kinetic energy N 2 even further. K ϭ͚iϭ1(1/2mi)pi only, giving in both cases a It should be noted that for systems where no intramo- 2-distribution with N degrees of freedom, i.e., a diverging lecular potential is present and in the whole temperature Gamma distribution of the form J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3013
N/2 rot int ͑1/kT͒ with qid and qid the ideal gas rotational and internal parti- ͑U ͒ϭ UN/2Ϫ1eϪUpot /kT ͑100͒ pot ⌫͑N/2͒ pot tion function per molecule. The rotational partition function is in general given by38 N/2 ͑1/kT͒ 1/2 3 1/2 1/2 3/2 N/2Ϫ1 ϪK /kT T 1 1 ͑K ͒ϭ K e . ͑101͒ qrotϭ ϭ ⌫͑N/2͒ id ͩ ⌰ ⌰ ⌰ ͪ ͩ k3⌰ ⌰ ⌰ ͪ ͩ ͪ A B C A B C The MGF of these distributions follows from Eq. ͑49͒, giv- ͑109͒ ing with ⌰A ,⌰B ,⌰C the principal characteristic rotational tem- peratures and the symmetry number.  N/2  N/2 G ϭG ͑t͒ϭ ϭ , ͑102͒ If the s internal degrees of freedom can be described Upot K ͩ 0 ͪ ͩ Ϫtͪ classically within a harmonic approximation ͑e.g., harmonic bond and angle potentials͒, the distribution of the total en- where we defined tϭ⌬. Note that Eq. ͑100͒ is also in thalpy is still a diverging Gamma distribution, since in that general the distribution of the kinetic energy in the canonical case and isothermal-isobaric ensemble. The second system is an ideal gas of N particles in the s s 1/2 int 2 i NpT ensemble. For simplicity we first consider a monatomic qid ϭ ͟ ͑110͒ ͩ h ͪ iϭ1 ͩ ͪ gas, and discuss polyatomic molecules afterwards. The i isothermal-isobaric partition function for a monatomic ideal with i and i the appropriate reduced mass and force con- gas is given by20 stant, and hence ͑5/2͒Nϩ1 ͑3/2͒N sN ͑qe͒N⌳Ϫ3N ⌬id,0    G ͑⌬͒ϭ ϭ ⌬idϭ , ͑103͒ H v͑p͒Nϩ1 ⌬id ͩ 0 ͪ ͩ 0 ͪ ͩ 0 ͪ ͑4ϩs͒Nϩ1 where ϭ h2 /2 m is the thermal wavelength with m the  ⌳ ͱ  ϭ . ͑111͒ mass of the atom. Since ⌬(G)ϭln(⌬ /⌬ ), we have ͩϪtͪ id,0 id from Eq. ͑10͒ Note that in all these cases also the distribution of the cor- rected enthalpy fluctuations ͑Sec. III F͒ is exactly a diverging ⌬id,0 G ͑⌬͒ϭ͗e⌬H͘ϭ Gamma distribution. Even when quantum intramolecular vi- H ⌬ id brations are present and if the classical intramolecular inter-  ͑5/2͒Nϩ1  ͑5/2͒Nϩ1 actions are absent or harmonic the corrected enthalpy fluc- ϭ ϭ ͑104͒ ͩ  ͪ ͩ Ϫtͪ tuations are still exactly described by a diverging Gamma 0 state. This moreover suggests that a diverging Gamma state defining tϭ⌬. Note that from ⌬id follows the ideal gas for the full or corrected enthalpy fluctuations might also be a 37 5 law pVϭ(Nϩ1)kT,soHϭ(2Nϩ1)kT, and using the good description for the temperature dependence of thermo- 9 ϪtX dynamic properties at constant pressure of real dilute general relation G⌬X (t)ϭe GX(t), we find ͑ ͒ gases. Note that the distribution of the internal energy U of 5/2 Nϩ1  ͑ ͒ a monatomic ideal gas in the NVT ensemble is a Gamma G ͑t͒ϭeϪt͓͑5/2͒Nϩ1͔/ . ͑105͒ ⌬H ͩ Ϫtͪ distribution too. For polyatomic molecules the same require- ments as described above for the NpT ensemble are valid. For a Gamma distribution ͑͒ we have from Eq. ͑49͒ a We tested the validity of the Gamma state description on similar expression water, using both full and corrected enthalpy fluctuations 2 ͓Eqs. ͑60͒–͑66͒ and ͑86͒–͑92͔. Experimental data at fixed b0 /b1 1/b1 39 Ϫt͑b0 /b1͒ pressure were taken from Schmidt at 1, 50, 400 and 1000 G⌬͑t͒ϭe . ͑106͒ ͩ 1/b1Ϫtͪ bar and ideal gas properties from Frenkel et al.1 The critical pressure is 221.2 bar. For the two lowest pressures a phase Using Hϭ( 5Nϩ1)kT,soCϭ(5Nϩ1)k and C / Tϭ0, -transition occurs, so adopting the usual thermodynamic ap ץ p ץ p 2 2 we obtain with b0ϭM 2 , b1ϭM 3 /(2M2) and Eq. ͑28͒ proach we have in that case on both sides of the singularity two independent solutions, a gas and a liquid branch. At 400 1 b0 5 b0 5 ϭ; ϭ Nϩ1 /; ϭ Nϩ1. ͑107͒ bar the behavior in the vicinity of the critical temperature b b ͩ 2 ͪ b 2 2 1 1 1 ͑647.3 K͒ is rather complex, so also there we used a ‘‘liq- Since the moment generating function, being a Laplace uid’’ and a ‘‘gas’’ branch. At 1000 bar we only used the 9 transform, is uniquely related to the distribution, this proves ‘‘liquid’’ branch. For the liquid solutions we used T0 that the distribution of the enthalpy fluctuations for an ideal ϭ313 K, for the gas branches we used T0ϭ613 K ͑1 bar͒, monatomic gas in the NpT ensemble is a diverging Gamma 813 K ͑50 bar͒ and 1013 K ͑400 bar͒. distribution (b1ϭkT). In Figs. 1–4 the predictions of the full enthalpy H and When we are dealing with a polyatomic ideal gas, heat capacity C p are given using the full enthalpy fluctua- tions ͓Eqs. ͑61͒ and ͑63͔͒. Since for the liquid side there is qe N Ϫ3N ͑ ͒ ⌳ rot int N no divergence restriction, the parameters were calculated for ⌬idϭ Nϩ1 ͑qid qid ͒ ͑108͒ v͑p͒ a general Gamma state, obtaining in all cases ␦ p0Х1. We 3014 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen
FIG. 1. Enthalpy H and heat capacity C p along a water isobar at pϭ1.0 bar. FIG. 3. Enthalpy H and heat capacity C p along a water isobar at p Legend: experimental values ͑ࡗ͒, diverging Gamma states ͓Eqs. ͑61͒ and ϭ400.0 bar. Legend: see Fig. 1. ͑63͔͒ on the liquid and gas side ͑ ͒, and corresponding Gaussian states ͓Eqs. ͑42͒ and ͑44͔͒ ͑ ͒. The critical pressure is 221.2 bar ͑Ref. 39͒. The values of T0 for the liquid and gas side are indicated by छ.
second order cumulant expansion of ⌬(G)in⌬͑see Sec. therefore used a diverging Gamma state, which according to II͒, using the same input data as the diverging Gamma state. the figures describes the liquid behavior for the two subcriti- From the figures it is clear that the Gaussian state, which has cal isobars very well up to the phase transition, or for the the same complexity as a diverging Gamma state ͑i.e., the supercritical isobars roughly up to the critical temperature. same number of input data at T0͒, is much worse, especially For the gas side a diverging Gamma state is clearly a good evident from the heat capacity. The use of a physically ac- description for temperatures starting somewhat above the ceptable Gamma distribution function hence gives a signifi- phase transition ͑or ϳ150 K above the critical temperature cant improvement, and the second order cumulant expansion for pϭ400 bar͒, as already suggested by the fact that the can only be used for very local extrapolations. ͑corrected͒ enthalpy fluctuations of an ideal gas are exactly a The predictions of the enthalpy and heat capacity using diverging Gamma distribution, see Eq. ͑104͒. Obviously the the corrected enthalpy fluctuations ͑not shown͒ and the full complexity of a diverging Gamma state is not high enough to enthalpy fluctuations are of comparable quality. However, properly describe the behavior in the vicinity of the critical the accuracy of the Gamma states for the excess heat func- point at 400 bar, although from Fig. 4 it is clear that at a tion in the NVT ensemble is clearly higher ͑see Refs. 3 and higher isobar ͑1000 bar͒ the behavior is much closer to a 4͒, especially for a sensitive property like the heat capacity, single diverging Gamma state in the whole temperature and in contrast to the canonical ensemble the use of excess range. For comparison we also included the predictions of fluctuations in the NpT ensemble does not really improve the the Gaussian state ͓Eqs. ͑41͒–͑47͔͒, which is equivalent to a accuracy.
FIG. 2. Enthalpy H and heat capacity C p along a water isobar at pϭ50.0 FIG. 4. Enthalpy H and heat capacity C p along a water isobar at p bar. Legend: see Fig. 1. ϭ1000.0 bar. Legend: see Fig. 1. J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen 3015
tistical state. In the subsequent paper we will present a dis- cussion on the implications of the existence of thermody- namic master equations and phase transitions. It is interesting to note that in the canonical ensemble2–4 the use of fluctuations of the excess heat function ͑ideal re- duced internal energy͒ provides a more accurate description of the thermodynamics. In fact, the Gamma state solutions provide in that case a model along the isochores which can be used over a very large temperature range, even in the critical point region. On the contrary the Gamma state solu- tions in the NpT and VT ensembles cannot be used as a general model for the temperature dependence of fluid ther- modynamics, as they are too simple to describe the fluid behavior at phase transitions or close to the critical tempera- ture. However, when compared to a usual second order cu- FIG. 5. Entropy S of water along different isobars. Legend: experimental mulant expansion, equivalent to a Gaussian state, these di- values ͑ࡗ͒, diverging Gamma states ͓Eq. ͑62͔͒ on the liquid ͑pϭ400 bar͒ verging Gamma states clearly show that the use of a and gas side ͑ ͒ and corresponding Gaussian states ͓Eq. ͑43͔͒ ͑ ͒. physically acceptable Gamma distribution function improves The critical pressure is 221.2 bar ͑Ref. 39͒. The values of T0 for the liquid and gas side are indicated by छ. the quality of the model considerably. In fact, the Gaussian state has the same problems concerning phase transitions and the critical point region, but in addition can be used only for Finally, in Fig. 5 the predictions for the entropy using very local extrapolations. The diverging Gamma state on the full enthalpy fluctuations show the same trend as the en- other hand seems to describe the fluid behavior properly in thalpy and heat capacity predictions ͑for clarity only the first the ‘‘stable’’ regions, as shown by the fact that in the ‘‘liq- three isobars are shown͒. Also here the predictions using the uid’’ and the ‘‘gas’’ ranges of the solution a very sensitive corrected enthalpy fluctuations ͑not shown͒ and the full en- property like the heat capacity is rather constant, as predicted thalpy fluctuations have the same accuracy. by the diverging Gamma state. Finally, it should be noted that limitation to the semiclas- V. CONCLUSION sical limit is not necessary: this theory could be derived from In this article we showed how to extend the quasi- a full quantum description of the partition function as well. Gaussian entropy theory in an exact way to noncanonical However, in the case of an exact quantum description no ensembles. We derived general expressions for the solutions fluctuations of excess heat functions can be used. of thermodynamic master equations for the temperature de- In the subsequent article we will show how to use the pendence in different ensembles, and described specific as- QGE theory to derive the density dependence in the NpT and VT ensembles using the volume and particle number distri- pects of the NpT, VT and NVT conditions. Gaussian and Gamma statistical states were derived in detail in the new butions. ensembles ͑NpT and VT͒ and were applied to describe the thermodynamics of water in gas and liquid conditions. ACKNOWLEDGMENTS We first showed that in the NpT ensemble an ideal gas This work was supported by the Netherlands Foundation can be described in general by a diverging Gamma state for for Chemical Research ͑SON͒ with financial aid from the the corrected and full enthalpy fluctuations, suggesting that a Netherlands Organization for Scientific Research ͑NWO͒ diverging Gamma state could be a good description at least and by the Training and Mobility of Researchers ͑TMR͒ Pro- of the gas behavior. In contrast to the situation in the canoni- gram of the European Community. cal ensemble using the energy fluctuations in the NpT and VT ensembles we encounter phase transitions. Following APPENDIX the usual thermodynamic approach ͑i.e., regarding the phase transitions as singularities͒ we have for the subcritical iso- From quantum mechanics we know that any set of bars two distinct solutions: a gas and a liquid one. phase-space positions which are permutations of identical Applying Gaussian and Gamma states, the results clearly particles should be considered as a unique physical state and show that we can describe a considerable part of the thermo- so counted as a single phase-space position in the partition dynamics of water using two Gamma state solutions, a low- function. Since for a system in the classical limit the canoni- density ͑gas͒ diverging Gamma state, and a high-density ͑liq- cal partition function can be expressed as the product of two uid͒ one, but no single Gamma state can be used as a unique independent integrals, one only on the classical momenta and statistical state of the system for all densities, even at super- the other on the classical coordinates,4 we can obtain a cor- critical isobars. This implies that for a real system its unique rect evaluation of the partition function by counting any set exact solution, able to describe both gas and liquid condi- of configurations which are permutations of identical par- tions including phase transitions and the critical point region, ticles ͑indistinguishable configurations͒ as a single configu- is in the NpT and VT ensembles beyond the Gamma level ration. To accomplish this correction in the configurational of the theory and hence requires a more complex model sta- integral we can decompose the classical degrees of freedom 3016 J. Chem. Phys., Vol. 109, No. 8, 22 August 1998 Amadei, Apol, and Berendsen of the molecule ͑excluding bond lengths and angles which 9 N. I. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distribu- are considered completely constrained4͒ into translational, tions 2nd ed. ͑Wiley, New York, 1992͒. 10 rotational and intramolecular ones. Hence for any given con- R. Kubo, J. Phys. Soc. Jpn. 17, 1100 ͑1962͒. 11 R. W. Zwanzig, J. Chem. Phys. 22, 1420 ͑1954͒. figuration of the system we can evaluate the corresponding 12 S. R. Phillpot and J. M. Rickman, J. Chem. Phys. 94, 1454 ͑1991͒. total number of translations, rotations and intramolecular dis- 13 J. M. Rickman and S. R. Phillpot, Phys. Rev. Lett. 66, 349 ͑1991͒. placements which are permutations of identical particles as 14 S. R. Phillpot and J. M. Rickman, Mol. Phys. 75, 189 ͑1992͒. 15 B. Guillot and Y. Guissani, Mol. Phys. 79,53͑1993͒. N 16 G. Hummer, L. R. Pratt, and A. E. Garcı´a, J. Phys. Chem. 99, 14188 n͑xint͒ϭN!͟ ͑1ϩ␥i͑xi,int͒͒ ␥iϭ0,1,2,..., ͑A1͒ ͑1995͒. iϭ1 17 G. Hummer, L. R. Pratt, and A. E. Garcı´a, J. Phys. Chem. 100, 1206 where x are the classical intramolecular coordinates, x ͑1996͒. int i,int 18 G. Hummer and A. Szabo, J. Chem. Phys. 105, 2004 ͑1996͒. the ones of the ith molecule and 1ϩ␥i the total number of 19 A. Stuart and J. K. Ord, Kendall’s Advanced Theory of Statistics, 5th ed. indistinguishable configurations only due to rotations and in- ͑Griffin, London, 1987͒. tramolecular displacements of the molecule. For macro- 20 T. Hill, Statistical Mechanics ͑McGraw–Hill, New York, 1956͒. scopic systems we can safely assume that although each ␥ is 21 K. Binder, Z. Phys. B 43, 119 ͑1981͒. i 22 in principle a function of the corresponding intramolecular A. Milchev, K. Binder, and D. W. Heermann, Z. Phys. B 63, 521 ͑1986͒. 23 H. Crame´r, Mathematical Methods of Statistics ͑Princeton University coordinates the product of all the factors 1ϩ␥i is virtually Press, Princeton, 1946͒. identical for every configuration of the system, so 24 J. K. Patel and C. B. Read, Handbook of the Normal Distribution ͑Marcel N Dekker, New York, 1982͒. 25 ´ N The Pade Approximant in Theoretical Physics, edited by G. A. Baker, Jr. ͑1ϩ␥i͒Х͑1ϩ␥͒ ␥у0, ͑A2͒ and J. L. Gammel ͑Academic, New York, 1970͒. i͟ϭ1 26 G. A. Baker, Jr., Essentials of Pade´ Approximants ͑Academic, New York, where ␥ can be considered a density and temperature inde- 1975͒. 27 K. Pearson, Philos. Trans. R. Soc. London, Ser. A 185, 719 ͑1894͒. pendent constant, characteristic of the molecules. Hence, di- 28 N K. Pearson, Philos. Trans. R. Soc. London, Ser. A 186, 343 ͑1895͒. viding the configurational integral by N!(1ϩ␥) provides 29 J. K. Ord, Families of Frequency Distributions ͑Griffin, London, 1972͒. the required correction. 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McQuarrie, Statistical Mechanics ͑Harper & Row, New York, 3017 ͑1998͒, following paper. 1976͒. 8 J. K. Patel, C. H. Kapadia, and D. B. Owen, Handbook of Statistical 39 E. Schmidt, Properties of Water and Steam in SI-Units ͑Springer, Berlin, Distributions ͑Marcel Dekker, New York, 1976͒. 1969͒.