University of Groningen Extensions of the quasi-Gaussian entropy theory Amadei, A.; Apol, M. E. F.; Berendsen, H. J. C. Published in: Journal of Chemical Physics DOI: 10.1063/1.473328 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1997 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Amadei, A., Apol, M. E. F., & Berendsen, H. J. C. (1997). Extensions of the quasi-Gaussian entropy theory. Journal of Chemical Physics, 106(5), 1893-1912. https://doi.org/10.1063/1.473328 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). 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Berendsen Citation: The Journal of Chemical Physics 106, 1893 (1997); doi: 10.1063/1.473328 View online: https://doi.org/10.1063/1.473328 View Table of Contents: http://aip.scitation.org/toc/jcp/106/5 Published by the American Institute of Physics Articles you may be interested in 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 On the use of the quasi-Gaussian entropy theory in noncanonical ensembles. II. Prediction of density dependence of thermodynamic properties The Journal of Chemical Physics 109, 3017 (1998); 10.1063/1.476894 Extensions of the quasi-Gaussian entropy theory A. Amadei,a) M. E. F. Apol, 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 26 August 1996; accepted 21 October 1996! In this paper we present the quasi-Gaussian entropy theory in a comprehensive and consistent way, introducing a new derivation of the theory very suited for applications to molecular systems, and addressing its use in the case of multi-phase systems. A general derivation of the possible confinement of the system within a part of phase space is given, and for water it is shown that for this a hard sphere excluded volume model can be used. To obtain the temperature dependence of the pressure, a new differential equation is derived, and besides the previously introduced Gaussian and Gamma states, in this paper we also describe a new statistical state, the Inverse Gaussian state. We discuss the properties of these different statistical states and for water compare their thermodynamics with experimental data, finding that both the Gamma and Inverse Gaussian states are excellent descriptions. © 1997 American Institute of Physics. @S0021-9606~97!50305-9# I. INTRODUCTION with the previous properties, of the excess enthalpy H8 and Gibbs free energy G8. The key point in statistical mechanics is the evaluation In Ref. 2 we proposed the idea of a phase-space confine- of the ~configurational! partition function, which is in general ment, i.e., a division of phase space into a ‘‘stable’’ and a high-dimensional integral over all coordinates. For a sys- ‘‘unstable’’ part, in a somewhat ad hoc way, based on a tem not at very low density with a realistic Hamiltonian, it is general double state model. In this article we will derive the virtually impossible to derive in this way rigorous but hand- phase-space confinement in a more natural way, and discuss able expressions for the various thermodynamic properties. its implications and the behaviour of the confinement for In recent articles1–4 however, we showed that in the canoni- water as a function of the density. Moreover, we will discuss cal ensemble it is possible to switch from the high- the properties of the statistical states described and we will dimensional partition function to the one-dimensional poten- compare their zero and infinite temperature limits with gen- tial energy distribution function of the system, which is in a eral statistical mechanical results. way a special projection of the Hamiltonian. The advantage In section II we will present the general theory in a com- of this approach is the fact that, because of the macroscopic prehensive and consistent way, introducing a new derivation character of usual systems, the potential energy distribution of the theory very suited for molecular systems, and in sec- function is almost Gaussian ~‘‘quasi-Gaussian’’!. Hence tion III we summarize the properties of the various statistical mathematically already relatively simple distributions can be states. In ‘‘Discussion and results’’ ~section IV! we will ad- dress aspects of multiphase systems, the temperature limits used to model the real distribution, yielding very compact of the expressions of the thermodynamic properties as dis- expressions for the excess thermodynamic properties ~i.e., cussed in section III, the phase-space confinement and its with respect to a proper reference state!. relation to hard-sphere excluded volume and present new We also showed that, using this approach, in a very natu- results of the excess pressure at constant density using the ral way a hidden differential equation, the thermodynamic conjugated pressure equation. master equation, arises. For every type of distribution, which defines a different statistical state of the system, the solution of this master equation yields the temperature dependence of II. GENERAL THEORY the excess heat capacity CV8 and hence of the excess entropy S8, internal energy U8 and Helmholtz free energy A8.Inthe A. Definition of the system previous articles we derived and tested the temperature de- The Helmholtz free energy of a system at fixed volume, pendence of these excess thermodynamic properties at con- temperature and number of molecules is stant volume for a Gaussian and Gamma distribution. In this article we will introduce a new distribution, the A52kT ln Q, ~1! Inverse Gaussian distribution, which is very related to the where Q is the overall partition function. For a system of N Gamma distribution, and derive its thermodynamics. We will identical molecules in the classical limit also derive a new differential equation, the conjugated pres- 1 sure equation, the solution of which yields the temperature e kin pot Q> Q Q Q , ~2! dependence of the excess pressure p8 and hence, combined N! with Qe the electronic partition function, a constant for most kin a!Author to whom correspondence should be addressed. of the systems, Q the kinetic energy partition function and J. Chem. Phys. 106 (5), 1 February 1997 0021-9606/97/106(5)/1893/20/$10.00 © 1997 American Institute of Physics 1893 1894 Amadei, Apol, and Berendsen: Quasi-Gaussian entropy theory Qpot the configurational partition function involving the po- with E0 the overall vibrational ground state energy of the tential energy of the system. The most general expression of system. Since the same argument also implies that Qpot is1 2 El2E0 2 El 2E0 ( e b~ !>( e b~ id id! ~7! 8 l l l Qpot5( E e2b~F1c1E !dx, ~3! l we finally obtain where x are the atomic coordinates, F is the intermolecular pot v 8 2bU8 potential energy, c is the intramolecular potential energy Q >Qid E e dx ~8! ~excluding bond length and bond angle vibrational energy!, l 0 0 0 v E is the overall lth quantum vibrational bond energy ~in with U85F1c1E 2Eid and where Eid Qid are the overall general a function of the coordinates!, b51/kT and the sum vibrational ground state energy and partition function of the runs over all accessible vibrational states $l% of the system in ideal gas ~with F50!, respectively. It is easy to see that Eq. the temperature range of interest. The prime on the integral 8 is always exact if the system is completely confined in the means that we integrate only over configurations where all vibrational ground state, or if no vibrations are present ~mon- the bond lengths and angles are fixed at their equilibrium atomic molecules! or when the vibrational energies can be values, see the Appendix. In a previous article1 we derived considered independent of the coordinates, and therefore the quasi-Gaussian entropy theory from the completely gen- from the temperature, hence being identical to the ideal gas eral expression given in Eq. 3. However, in many cases such ones. a derivation of the theory is not very suitable for applica- If moreover, at least in the whole temperature range of tions. In this paper we will therefore introduce a very general interest, only a part of the configurational space is energeti- approximation of Qpot, valid for many different types of mol- cally accessible ~i.e., the system is confined within a part of ecules, from which we will derive a very useful form of the the configurational space! we can rewrite Eq. 8 as theory for applications to molecular systems. * pot v 8 2bU8 In fact we can simplify Eq. 3, considering that in general Q >Qid E e dx, ~9! for systems where the vibrational energies have a significant dependence on the coordinates, such a dependence is ob- where now the star denotes an integration over the accessible served only up to temperatures where the molecules are part of the configurational space only.