The quasi-gaussian entropy theory: Free energy calculations based on the potential energy distribution function

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1 The quas-gaussan entopy theoy: Fee enegy calculatons based on the potental enegy dstbuton functon A. Amade, M. E. F. Apol, A. D Nola, a) and H. J. C. Beendsen Gonngen Bomolecula Scences and Botechnology Insttute (GBB), Depatment of Bophyscal Chemsty, Unvesty of Gonngen, Njenbogh 4, 9747 AG Gonngen, The Nethelands Receved 7 August 1995; accepted 18 Octobe 1995 A new theoy s pesented fo calculatng the Helmholtz fee enegy based on the potental enegy dstbuton functon. The usual expessons of fee enegy, ntenal enegy and entopy nvolvng the patton functon ae ephased n tems of the potental enegy dstbuton functon, whch must be a nea Gaussan one, accodng to the cental lmt theoem. We obtaned expessons fo the fee enegy and entopy wth espect to the deal gas, n tems of the potental enegy moments. These can be lnked to the aveage potental enegy and ts devatves n tempeatue. Usng themodynamcal elatonshps we also poduce a geneal dffeental equaton fo the fee enegy as a functon of tempeatue at fxed volume. In ths pape we nvestgate possble exact and appoxmated solutons. The method was tested on a theoetcal model fo a sold classcal hamonc sol and some expemental lquds. The hamonc sold has an enegy dstbuton, whch can be deved exactly fom the theoy. Expemental fee eneges of wate and methanol could be epoduced vey well ove a tempeatue ange of moe than 3 K. Fo wate, whee the appopate expemental data wee avalable, also the enegy and heat capacty could be epoduced vey well Amecan Insttute of Physcs. S I. INTRODUCTION Fee enegy s the quantty of cental nteest n physcs and chemsty, snce the fee enegy detemnes the equlbum physcs, and fee enegy dffeences detemne chemcal equlbum and dynamcs. Eale attempts to calculate the Helmholtz fee enegy and to elate t to well measuable basc popetes wee estcted to ethe dluted gases 1 o gases and monoatomc van de Waals lquds at elatvely hgh tempeatue, usng the so-called Hgh Tempeatue Expanson of Zwanzg. 2 Ths last expesson, appled to the fee enegy wth espect to the had sphee lqud, s gven by 3 A* kt 6 kt 2 1 wth the petubaton potental enegy, whee the ensemble aveages ae taken ove the efeence state. Often Eq. 1 s tuncated afte the second tem and n ths case the ght hand sde can be expessed n tems of g() o elated quanttes, usng ethe the Bake Hendeson theoy 4,5 o the Chandle Weeks Andesen theoy. 5,6 Late extensons to small molecula fluds ae evewed by Gay and Gubbns. 7 In ths pape we wll ephase the basc equatons of statstcal mechancs n tems of the one-dmensonal dstbuton functon of the total potental enegy, nstead of the mult-dmensonal patton functon. Snce we can defne the ntenal enegy and the Helmholtz fee enegy n tems of ths enegy dstbuton functon, all othe themodynamc popetes can be deved too. Fom the fact that at not too low tempeatue a macoscopc system can be egaded as an nfnte collecton of dentcal ndependent subsystems, t follows that the dstbuton must be close to a Gaussan we wll efe to these macoscopc systems as quas-gaussan systems. Hence fo evey quas-gaussan system t s possble to classfy dffeent statstcal states, accodng to the coespondng unnomal dstbuton, and to deve a geneal expesson of the educed entopy the quas-gaussan entopy n tems of the heat capacty and a dmensonless and ntensve quantty, the ntnsc entopy functon. The atcle s oganzed as follows. In Sec. II A we defne the educed and deal educed fee enegy, and n Sec. II B we gve the geneal shape of the potental enegy dstbuton functon and the physcal and mathematcal estctons, usng a genealzed Peason system of cuves. Then we wll wok out two cases, the deal Gaussan state Sec. II C and the Gamma state Sec. II D. In Sec. II E we defne the ntnsc entopy functon. In the same secton we also deve the geneal expesson fo the quas-gaussan entopy QGE and a geneal dffeental equaton maste equaton whch elates the heat capacty and the ntnsc entopy functon. In Sec. II F we solve the maste equaton fo the Gaussan and Gamma state whch yelds the the tempeatue dependence of the deal educed fee enegy, ntenal enegy and heat capacty. In Secs. II G and II H we descbe two appoxmated solutons whch can be appled to systems wth moe complex statstcal states. In Sec. II I we wll dscuss applcatons of these appoxmatons to small molecules. Fnally n Sec. III we pesent the esults obtaned fo thee dffeent systems: the classcal hamonc sold, fo whch we could deve an analytcal expesson fo the fee enegy and the enegy dstbuton, and lqud wate and lqa Depatment of Chemsty, Unvesty of Rome La Sapenza, p.lea. Moo 5, 185, Rome, Italy. 156 J. Chem. Phys. 14 (4), 22 Januay /96/14(4)/156/15/$ Amecan Insttute of Physcs

2 Amade et al.: The quas-gaussan entopy theoy 1561 ud methanol, whee we compaed the expemental esults wth the pedctons followng fom the QGE theoy, usng both appoxmatons. II. THEORY A. Reduced fee enegy The Helmholtz fee enegy of a system wth fxed tempeatue, volume and numbe of patcles, s gven by: AkT ln Q, 2 whee Q s the oveall patton functon. Fo a system of N dentcal molecules n the classcal lmt we can say that Q 1 N! Qe Q kn Q pot, whee Q e s the electonc patton functon, whch s a constant fo the majoty of systems of nteest, snce the fst electonc enegy gap s much lage than kt at usual tempeatue. Q kn s the knetc enegy patton functon and Q pot s the potental enegy patton functon. Q kn and Q pot ae patton functons whee all the bond lengths and angles and the tme devatves nvolved n the n b bond vbatons ae consdeed to be constaned fom a classcal pont of vew: Q kn 1 h R e K Q pot l 3 n b ḃ dp 4 e E l n b b b dx l e E l dx, 5 whee K s the classcal expesson fo the knetc enegy, s the ntemolecula potental functon and s the ntamolecula potental functon not ncludng bond vbatonal potental enegy. Futhe (b b ) and (ḃ ) ae Dac -functons epesentng the bond constants, whee b ae the equlbum bond lengths and angles of the quantum oscllatos, epesentng the ntenal vbatons of the molecules, and ḃ ae the bond constant tme devatves. x and p ae the atomc coodnates and momenta, h s Planck s constant, 1/kT and R s the numbe of the atomc degees of feedom of the system excludng the vbatonal degees of feedom. E l s the vbatonal enegy of the whole system, defned by the set of quantum numbes l of all quantum oscllatos pesent, and the sum l uns ove all possble sets. In geneal the enegy levels of the ndvdual oscllatos and theefoe E l ) may be a functon of the coodnates x, mplyng a tempeatue dependence. Only n the deal gas condton, we can consde the levels of the oscllatos to be ndependent fom the coodnates and theefoe fom the tempeatue. The pme on the ntegal sgn means an ntegaton wth all bond and angle constants. It must be noted that Q kn n Eq. 4 must be evaluated at one abtay confguaton, snce ts Dac functons (ḃ ) ae dependng on the coodnates, although the value of the ntegal s vtually ndependent of them snce the ntegaton s pefomed ove the whole ensemble of molecules. Theefoe the factozaton of Q s possble even wth constants. The fee enegy of an deal system at the same tempeatue and densty but wth no ntemolecula nteactons deal gas condton s: A kt ln Q, Q 1 N! Qe Q kn Q pot, Q pot l e E l n b b b dx 6 7 l e E l dx. 8 Agan, the pme on the ntegal sgn means an ntegaton wth all bond and angle constants. Now the only dffeence between ths deal state and the eal state of the system s n the potental enegy patton functon. It s convenent to calculate the dffeence between the eal fee enegy and the deal one. Ths dffeence s called the educed fee enegy: A AA kt ln Qpot Q pot. 9 In geneal we can ewte the constant confguatonal patton functon Q pot fo a system wth a potental enegy functon (x)e, wth E the total enegy of the quantum oscllatos, n the followng way Q pot l e dx e dx ldx l dx l le dx l dx l e e ldx dx e and smlaly 1 Q pot ldx e, 11 whee and denote the canoncal expectaton values n the eal and deal condton, and E. It s nteestng to note, that the expectaton values n Eqs coespond n statstcal tems to the moment geneatng functon 8 of the vaable n : G e ldx Q pot. 12 We wll use ths fact late n Sec. II E. Fom Eqs. 1 and 11 t follows that A kt ln e kt ln e. 13

3 1562 Amade et al.: The quas-gaussan entopy theoy The two tems on the ght hand sde ae dffcult to evaluate, and thee s no geneal way to calculate them. In ths atcle, we wll deve fo these knd of tems a geneal analytcal expesson. Hence we ntoduce anothe efeence state, wth fee enegy N 1, N N A kt ln Q, 14 Q 1 N! Qe Q kn l dx. 15 Note, that ths state has the same densty and tempeatue as the eal system, but no angle and bond vbatons, and no nte and ntamolecula potental, coespondng to an deal gas of gd molecules. The fee enegy of the eal system wth espect to ths state we call the deal educed fee enegy A, AAA A kt ln e kt ln e. 16 Equaton 16 can be modfed to expess the enegy wth espect to the aveage potental enegy U: whee and AUkT ln e y, UE, y, Skln e y 2 s the deal educed entopy of the system. Ths s the dffeence between the entopy of the eal and the new efeence system. It should be noted, that when, Eq. 16 educes to the second tem n Eq. 13. Theefoe we can use the expessons we wll obtan n the moe geneal case fo ths tem as well. In Sec. II I we wll see, that fo small molecules A can be obtaned qute staghtfowadly fom A. B. Geneal dstbuton and estctons It s mpotant to note, that snce the enegy dstbuton functon (y) s that of a macoscopc system, t must be unnomal and vey smla to a Gaussan dstbuton. We can pove ths as follows. Snce the system s macoscopc, we may teat t as a collecton of N dentcal subsystems. Although these subsystems contan a dffeental amount of matte fom a macoscopc pont of vew, they can be egaded as statstcally ndependent themodynamc systems snce they stll contan an nfnte numbe of molecules. Ths means that the nteacton enegy between subsystems s neglgble wth espect to the ntenal enegy of each subsystem. 9 If theefoe evey subsystem has a potental enegy and a fluctuaton, we can expess the potental enegy and fluctuaton of the total system as The cental lmt theoem 1 states, that as N a macoscopc system ()(y) wll tend to a Gaussan dstbuton, whateve s the dstbuton of. Hence fo a macoscopc system (y) wll be unnomal and vey smla to a Gaussan dstbuton. Evey macoscopc system whch can be descbed n ths way, we call a quas-gaussan system, and each type of coespondng enegy dstbuton defnes ts statstcal state. In Sec. II E we wll show that all quas- Gaussan systems shae the same geneal expesson fo the deal educed entopy. At vey low tempeatue though, the system must be descbed n a complete quantum mechancal way. Ths means, that the enegy and ts dstbuton (y) ae dscete nstead of contnuous, and ths dscete dstbuton need not be unnomal, snce at low tempeatue the degeneacy of enegy levels can be small. In geneal (y) s a poduct of an nceasng functon the degeneacy of levels and a deceasng functon the Boltzmann facto. So fo systems whch ae not at too low tempeatue we can fomulate seveal estctons on the shape of ths nea Gaussan enegy dstbuton functon: R1 The dstbuton functon (y) s defned on the enegy nteval y,. Fo physcal easons, y must be a fnte value, snce any eal system must have an oveall potental enegy mnmum. The functon theefoe cannot be pefectly symmetc. Snce fo molecula systems at usual tempeatue, the mode of the enegy dstbuton s vey fa fom the absolute mnmum, y can be egaded as a numecal enegy mnmum, much hghe than the eal physcal mnmum. Fo systems at low densty and/o hgh tempeatue the mode s so much hghe than the numecal mnmum that a symmetc cuve could be a good descpton of the eal dstbuton. R2 Snce we assume the exact pobablty densty angng fom the absolute mnmum to nfnty and at least ts fst devatve to be contnuous, (y) and ts devatve /dy must be contnuous and eal on the nteval y,. R3 The devatve /dy must be zeo whee equals zeo that s at y and ), as follows fom R1 and R2. R4 The functon (y) s unnomal. It has one extemum at y m on the nteval y,, and ths extemum has to be a maxmum the mode. The devatve /dy theefoe must change sgn aound y m n the followng way: dy dy yy m, yy m. 23 R5 Any acceptable soluton of (y) has to esult n a fnte deal educed fee enegy. Theefoe the ntegal

4 e y ydy 24 has to convege. Accodng to Eq. 12 ths coesponds to the equement, that a sutable enegy dstbuton functon should have a convegng moment geneatng functon at the coespondng.) So we have the necessay condton that lm e y. 25 y It should be mentoned hee, that ths physcal estcton s not vald fo a classcal deal hamonc sold a set of classcal oscllatos wth no lmts on the coodnates. We wll show n Sec. III, that n such a case the system stll can be descbed as a specal case of the theoy. The most geneal dffeental equaton, descbng such a unnomal functon, s dy yy m Pm y G n 26 y wth P m (y) and G n (y) some abtay polynomals n y of ode m and n: m P m y n G n y j a y, b j y j Eq. 26 s just a genealzaton of the nomal Peason system of fequency cuves, developed by Kal Peason between 189 and It s based on the dffeental equaton dy yy m b yb 2 y 2 29 coespondng to m, n2 n ou notaton. The choce b 2 ob 2 s equal to n1 on. The esultng famly ncludes some well-known types, lke the nomal, Gamma, Beta and t-dstbuton. 14,15 Futhe genealzatons of Eq. 29 ae descbed by Od, 14 fo the case n2. The use of a Padé expanson as n Eq. 26 was fst descbed by Dunnng and Hanson. 16 We wll use the genealzed Peason equaton to classfy dffeent unnomal dstbutons, close to a Gaussan, whch satsfy the physcal and mathematcal estctons. Followng Peason s wok we can moeove lnk the paametes y m, a, and b j to the cental moments of the potental enegy M n, defned as M n n y n. 3 If we have n p ndependent paametes, we have to solve the followng set of k max equatons y k G n y dy dy y k yy m P m ydy k...k max 1 Amade et al.: The quas-gaussan entopy theoy 31 wth k max n e n p, 32 whee n e s the numbe of equatons needed to solve the paametes. n e can be lage than n p because the fst n p equatons not necessaly ae ndependent. Integatng by pats we obtan y k G n y d dy yk G n ydy y k yy m P m ydy. 33 Snce all moments of fnte ode exst, as t follows fom the fact that we can expess evey moment M n1 as a sum of poducts of and ts fst n devatves n tempeatue, we can conclude that lm y y k G n (y). Hence, snce s zeo at y d dy yk G ydy n y k yy m P m ydy. 34 Usng the defntons of G n (y) and P m (y), we get n j jkb j y jk1 dy m a y k1 y m y k dy, 35 whch esults n the followng set of elatons between the paametes and moments n j m jkb j M jk1 a M k1 y m M k k...k max wth M 1, and M 1. Late on we wll see, that m1n Eqs. 41 and 42; theefoe the hghest moment that s equed s M kmax m. Fom the same equatons as Eq. 36 wth kk max 1, we get the moments fom M kmax m1 on, expessed n tems of the lowe ones. We wll now nvestgate the effect of the estctons R1 to R5 on the possble solutons of the dffeental equaton of (y), Eq. 26. Restcton R2 contnuty demands that G n (y) has no eal oots on the nteval y,. G n (y) theefoe can have eal oots only at y o some value ȳ smalle than y. Snce (y) and ts devatve must be eal, also pas of conjugated complex oots at ˆ ae possble. So G n (y) can be bult up of factos lke (yy ) n, (yȳ ) n, and n ((y ) 2 ˆ 2 ) wth nnn/2 n. None of these factos can change sgn n the nteval y,. Smlaly, estcton R4 demands that P m (y) has no eal oots on the nteval y, othe than y m. P m (y) can have othe eal oots smalle than o equal to y, o has pas of conjugated complex oots at ˆ. Theefoe P m (y) can

5 1564 Amade et al.: The quas-gaussan entopy theoy consst of factos lke (yy m ) m, (yy ) m, (yȳ ) m, and m v ((y ) 2 ˆ 2 ). Of these factos, only (yy m ) m can change sgn n the nteval. Snce (y) must have a maxmum, estcton R4 demands that P m (y)/g n (y), so m even. Restcton R5 convegence eques that Eq. 25 s fulflled. We can use the fomal soluton of Eq. 26 fo (y): y exp yy m Pm y G n y dy. If we combne ths wth Eq. 25 we obtan lm exp y yy m Pm y y G n y dy, whch s equvalent to lm yy y m Pm y G n y dyy, y lm y lm y lm y yy m Pm y G n y dy y P m y yym G n y dy y 1. Usng l Hôptal s ule we fnally obtan G n y lm yy m P m y 1 kt. y Ths nequalty s always fulflled f m1n. 1, It s neve fulflled when m1n. In the case that m1 n, we use the defntons of the polynomals P m (y) and G n (y) Eqs. 27 and 28 and fnd that the fee enegy can convege only f b n a m kt mn1. 42 We can ode the possble solutons of Eq. 26 accodng to the complexty. Ths s defned by the numbe of paametes mn2 and the numbe of equed moments (k max m). The smplest soluton of Eq. 26 m, n s a Gaussan dstbuton defnng the deal Gaussan state and the fst completely acceptable soluton m, n1 s a Gamma dstbuton, defnng the Gamma state. In ths way, a state s defned by the type of potental enegy dstbuton, because t descbes all the equlbum popetes of the system. In the followng sectons we wll focus on these two solutons. C. Ideal Gaussan state Fom the cental lmt theoem we saw that (y) tends to a Gaussan dstbuton as the system becomes moe and moe macoscopc. Theefoe we can consde as deal efeence a system whee (y) s exactly a Gaussan. We call ths the deal Gaussan state. Ths case coesponds to m, n. It s the smplest soluton, wth two paametes snce we can put a to 1 and two moments, wth the followng dffeental equaton fo (y): dy yy m. 43 b Eq. 36 yelds the followng set of elatons between paametes and moments: y m. 44 b M 2 The dffeental equaton thus becomes dy y, M 2 whch has the tval soluton y e y2 /2M wth 1/2M 2. It should be noted, that a eal Gaussan state s not possble snce t would mply a pefectly symmetc dstbuton wth no enegy lmts. Ths s physcally mpossble because evey system must have an oveall enegy mnmum estcton R1. Unde cetan condtons though, t mght be a good numecal appoxmaton. The deal educed Helmholtz fee enegy s calculated as follows, accodng to Eq AUkT ln 2M 2 e y e y2 /2M 2 dy. Usng the substtuton z 1/2M 2 (y M 2 ), we obtan AU M 2 2kT, S M 2 2kT Eq. 49 shows that n an deal Gaussan state the educed entopy of the system s completely defned by T and the second cental moment of the potental enegy. It s clea then, that n such a condton A s a functon only of T and the fst two potental enegy moments. We can lnk these moments M n to the devatves of n T: kt e 2 kt T 2 C dx e dx 2 2 M

6 Amade et al.: The quas-gaussan entopy theoy 1565 Hee agan the pme on the ntegaton sgn means ntegatng wth all bond and angle constants, and the summaton uns ove all possble sets of quantum vbatonal states. So M 2 kt 2 C, 52 AU TC 2, 53 S C Hee C s the deal educed heat capacty of the total system at fxed volume, excludng the knetc enegy pat. D. Gamma state The fst physcally acceptable soluton of just beyond the Gaussan dstbuton Eq. 46 s gven by m, n1. In ths case we have 3 ndependent paametes puttng a to 1 agan and 3 potental enegy moments. The dffeental equaton of s dy yy m b y yy m yy, 55 whee y b /. Usng Eq. 36, the thee paametes y m, b, and ae lnked to the enegy moments n the followng way: y m b M 2 3 M 2 M 3 y m M 2. Hence we have b M 2, y m M M 2 Fom Eq. 36 fo kk max 1 we obtan the ecusve elatons that lnk the moments hghe than M 3 to M 2 and M 3. It tuns out that we can expess M n as M n n1 M 2 M n2 M 3 M 2M 2 n1. 59 Note the smlaty between ths expesson and the one obtaned fo the Gaussan dstbuton, M n n1m 2 M n2, 6 M odd. The dstbuton functon follows fom dy y b y 61 and s only defned n y,. Fo convenence, we defne zb y, z, and obtan dz 1 b 2 z. 62 By ntegatng ths equaton, we fnd z z b /b e z/b 2 1, 63 whee s the nomalzaton constant, defned as 1 2 z b /1 e z/b 2 1 dz 2 b / 2 x b / 2 1 e x dx 2 b / 2 b b 2, 1 64 whee we used the substtuton xz/ 2. In the last step we used the defnton of the Gamma functon. 17 Now the dstbuton functon can be expessed as ethe o z y 1 2 b 2 / b b 2 / b 2 z b /b e z /b 2 1 b y b /b e b y/b 2 1, whee we used (z)dz(y)dy. Eqs. 65 o 66 epesent the Gamma dstbuton, 8 o Peason s type III cuve, whch has the geneal fom a a a1 e,. 67 In ths case, zb y, ab / 2 and 1/ 2. The deal educed Helmholtz fee enegy can be calculated fom Eq. 17; theefoe we need to evaluate e y e y ydy e zb / zdz e b / e b / 1 2 b 2 / b 2 2 z b /1 e 1/b 2 1 /b1 z dz 1 b /b x b /1 e x dx b 2 e b /1 b /b 2 1, 68 whee we used x(1 )z/ 2. Hence the fee enegy s

7 1566 Amade et al.: The quas-gaussan entopy theoy AUkT ln e y UkT b b 2 ln 1 U b b 2 kt ln 1 kt. 69 Snce m1n, we have the addtonal estcton that b n /a m kt see Eq. 42. Ths also follows fom Eq. 69, snce the agument of the logathm cannot be negatve o zeo. As aleady mentoned n Sec. II B a classcal deal hamonc sold has exactly the enegy dstbuton of the Gamma state Eq. 65, but wth kt, esultng n a dvegng deal educed fee enegy, see Sec. III. Remembeng the expessons of b and Eqs. 57 and 58, t s clea that n the Gamma state the fee enegy s gven by U, M 2 and M 3. Just as we dd pevously, we can expess M 2 and M 3 as tempeatue devatves of. Usng Eqs. 5 and 51 we see that kt2 C kt 2 C 2k T 2 T 3 C 7 and 2 2 e dx e dx 2 2 e dx e dx so M 3 2 M 3 kt 2 C 2k T 2 T 3 C In the Gamma state theefoe, the deal educed fee enegy s gven by the aveage potental enegy and ts fst two devatves n tempeatue. Now we can use the expessons fo M 2 Eq. 52 and M 3 n the defnton of b and. It follows that AUTC kt kt 73 wth UTC 2 T kt 1 T 2C C T ln 1. ln 1 kt 74, Snce kt, t follows that 1 and fom Eq. 75 t s clea that (C /T), mplyng that C ast. The functon (T) plays a vey mpotant ole n the theoy. Among othes t defnes the deal educed entopy as SC T. 77 Inspectng Eqs. 75 and 76 we see, that s actually a functon of T(C /T) /2C. It s vey mpotant to note that (T) s a dmensonless and ntensve quantty, ndependent of the sze of the system. Compang Eqs. 48 and 74 we see that the deal Gaussan state mples that (T)1/2. Such a condton can be egaded as a lmt of Eq. 76 when. The paamete s theefoe a measue of the asymmety of the enegy dstbuton. E. The ntnsc entopy functon In the pevous sectons we studed the fst two possble statstcal states of a quas-gaussan system. Although t s possble to study hghe ode solutons of the genealzed Peason equaton, such solutons wll be vey complcated, nvolvng hghe ode moments and devatves of n T. It tuns out that solutons just beyond the Gamma dstbuton nvolve at least M 5. In ths secton we wll show that fo evey quas- Gaussan system can be expessed n tems of the moment geneatng functon of the subsystems. We wll also deve a geneal dffeental equaton themodynamc maste equaton whch lnks (T) toc (T). In Sec. II B we defned a quas-gaussan system as a collecton of N dentcal ndependent dffeental themodynamc subsystems elementay systems, wth N. We can defne the moment geneatng functon of the enegy fluctuaton fo each elementay system as 2 g ! k k 78 k! wth k () k the kth potental enegy moment of the elementay system. Fo the whole system the moment geneatng functon beng the poduct of the moment geneatng functons of the ndependent subsystems 18 and the deal educed entopy see Eq. 2 ae G lm g N, N 79 SNkln g. Snce C N T T, 8 t follows that S C kln g /T ln ! T. 81 It s evdent that s a dmensonless and ntensve popety, snce t s completely defned by the elementay system. In geneal s a functon of the tempeatue, densty and com-

8 Amade et al.: The quas-gaussan entopy theoy 1567 poston of the system. We can call ths quantty the ntnsc entopy functon. It s nteestng to note that snce S and C ) and n the lmt of nfnte tempeatue we have lm T T lm ln ! as t follows fom the fact that evey k conveges to a fnte value as T beng equal to () k ). Fom Eq. 81 we can also defne the deal educed fee enegy as AUkT ln G UTC T, 83 whee C s the geneal expesson fo the quas-gaussan entopy. (T) s gven by 1/2 fo a Gaussan state and by Eq. 76 fo the Gamma state. Evey type of dstbuton wll have ts own functonal fom of dependng on a lmted numbe of moments. Ths numbe nceases wth nceasng ode of the soluton of the genealzed Peason equaton. It s nteestng to note that solutons of lowe ode can be egaded as lmts of hghe ode solutons. Wth nceasng tempeatue, the system can be descbed by a lowe ode soluton and n the lmt of nfnte tempeatue evey system wll tend to an deal Gaussan state. Usng Eq. 83, we can deve a new dffeental equaton, snce we know fom themodynamcs that A T SC T. Dffeentatng Eq. 83 we obtan on the othe hand A C T C TTT C T 84 TC T 85 T and hence we ave at the themodynamc maste equaton TT C C T TC T. 86 T It s a emakable fact that f the exact statstcal state of a quas-gaussan system s known at least at one tempeatue, Eq. 86 wll povde the complete behavo of C and hence of all othe themodynamcal popetes at evey possble tempeatue at fxed volume, f of couse thee ae no sngula ponts, whee (C /T) o not contnuous. It should be mentoned also that the devatons of ths secton ae vald even when the pobablty dstbuton functon of the potental enegy s not contnuous and/o unnomal. F. Exact solutons of the themodynamc maste equaton In ths secton we wll solve the maste equaton, Eq. 86, fo the two cases we dscussed befoe n the pevous sectons. Fo a Gaussan state, snce 1/2 and hence (/T) the themodynamc maste equaton s smply dc C 2 T dt 87 wth soluton of C (T) C TC T T 2 T. 88 The potental enegy s UTUT T T C TdT UT T C T TT T and fee enegy ATUTTC TT 89 UT T C T 1 T 9 2T. In the case of the Gamma state, we can ewte the maste equaton n tems of Eq. 75 as T T211, 91 whee (/T)(/)(/T). Snce fom Eq we get (/T)(1)/T, so TdT T T T 93 T 1 wth soluton T T, 94 T1 T whee (T ). Fom Eq. 75 t follows: C TC T T 2 T1 T UTUT T C T, 95 TT T1 T, ATUT T C T TC T 2 T1 ln T1 T Fom the solutons t s clea that n the Gaussan state the knowledge of U and C at one tempeatue s equed to know the complete behavo of the system. In the Gamma state we need to know (C /T) at one tempeatue as well.

9 1568 Amade et al.: The quas-gaussan entopy theoy The Gaussan state and ts soluton, as we mentoned befoe, can be egaded as an deal soluton. The Gamma state soluton on the contay, although n geneal t wll not be an exact descpton, can be consdeed as an acceptable appoxmaton fo a lage vaety of systems. In the next secton we wll descbe an appoxmated soluton of the themodynamc maste equaton, based on the assumpton that the state of a eal system s vey close to a Gamma state. G. Effectve Gamma appoxmaton In the pevous secton we deved the soluton fo an exact Gamma state. In ths secton we descbe an appoxmaton fo eal systems whch can not be descbed by the exact Gamma soluton, but whch can be consdeed to be n a petubed Gamma state. In geneal the heat capacty of the eal system can be wtten as C TC TT, 98 whee C s the heat capacty of a possble Gamma state and (T) the coespondng eo. Ou am s to fnd a specfc effectve Gamma state whch wll epoduce the entopy of the eal system as close as possble. If all the nfomaton on the system s confned to one tempeatue, one smple choce to defne unquely the Gamma state s to equate S (T ) wth S(T ) and C (T ) wth C (T ). Ths effectve Gamma state s thus defned as havng the same entopy and heat capacty as the eal system at the efeence tempeatue T. Theefoe we can use Eqs eplacng and by the effectve * and *, whee * follows fom S(T )/C (T )(T )( *) 1 ( *) 2 ln (1 *). Ths effectve Gamma state could povde a good appoxmaton fo the entopy especally fo TT ) and theefoe the fee enegy, snce A(T)A(T ) T T S(T)dT T T S (T)dT. It should be noted that the aveage enegy wll not be epoduced equally well, snce T T (T)dT s n geneal not neglgble. H. Constant alpha appoxmaton When the statstcal state of a quas-gaussan system s known and hence the functonal fom of (T,C,(C /T),...)) thethemodynamc maste equaton Eq. 86 can be solved exactly, as we dd fo the Gaussan and Gamma state. In the pevous secton we showed that an appoxmated soluton can be obtaned f the system s close to a Gamma state petubed Gamma state. In ths secton we wll deve an appoxmated local soluton of the maste equaton n some tempeatue nteval based on a tempeatue expanson of the ntnsc entopy functon. Ths appoxmaton does not eque any knowledge o assumpton on the statstcal state of the system. We can expand (T) ntaound a efeence tempeatue T T n 1 n! n T T n T TT n 99 and then use ths to ntegate Eq. 86. Wth Eq. 99 we can poduce a set of appoxmated analytcal solutons, vald fo an nceasng tempeatue ange. We stated testng the zeo ode soluton, whee (T)(T ) and ((T)/T). We obtaned aleady wth ths smple soluton vey accuate esults fo a lage tempeatue ange fo two eal lquds. Wthn the zeo ode appoxmaton, we can ewte Eq. 86 as C C T T T, 1 whch has the soluton C TC T T T, 11 1 T. 12 We can use Eq. 11 to calculate U as a functon of T: UTUT T T C TdT UT C T T T T T UT C TT 1 C T T Eq. 13 can also be used to calculate A fom Eq. 74 AUT C TT 1 C T T C TT 1 UT C T 1 T T dt T T T. 14 Eqs. 11, 13 and 14 gve C, U and A as a functon of tempeatue n the zeo ode appoxmaton of Eq. 86. I. Applcaton to small molecules In ths secton we wll apply the esults of the quas- Gaussan Entopy theoy to small molecules, lke wate and methanol. We can ewte Eq. 13, expessng the fst two tems on the ght hand sde accodng to Eq. 83 as: A UU TC TC T, 15 whee U, 16 C U. 17 T Fo an deal gas, Q pot can be factozed as Q pot Q v e dx. 18

10 Amade et al.: The quas-gaussan entopy theoy 1569 Now Q v s the poduct of all molecula vbatonal patton functons q v, whee we can appoxmate q v n tems of quantum oscllatos 5 m N Q v q v N v q, 19 q v ee 1e 2e wth e the zeo-pont enegy of the th hamonc mode, and m the numbe of modes pe molecule. The possblty to factoze Q pot mples, that e s ndependent of x and theefoe of T. As explaned befoe, the levels n the eal system may be tempeatue dependent. Fo wate fo example, thee s a shft of one of the thee modes of about 1 cm 1 n 4 K. 19 It s possble to calculate Q v usng spectoscopcal data, obtanng that fo most molecules up to at least 6 K vtually no othe vbatonal states othe than the gound state ae populated, snce e kt Ref. 2 except fo backbone vbatons n lage molecules 21. Fo wate 22 and methanol, 23 ths s fulflled even up to 1 K. Theefoe, the aveage vbatonal enegy s just the gound state enegy, and the vbatonal contbuton to the heat capacty s vtually zeo. Moeove, fo small molecules, lke wate and methanol fo the latte consdeng the dhedal angle to be feely otatng, we have, and theefoe and so m U N C, UU UE U, C C, A U TC T, e E, whee U s the vapozaton enegy and C (U /T). Hence fo pue systems, consstng of small molecules we can expess the equatons of the effectve Gamma appoxmaton as T * *T T1 *T *, c Tc T T 2 T1 *T * U m TU m T T c T 115, 116 TT T1 *T *, 117 A m TU m T T c T Tc T * * 2 T1 ln *, 118 T1 *T * whee * follows fom (T )( *) 1 ( *) 2 ln (1 *), and modfy the constant alpha equatons Eqs to obtan c Tc T T T, 119 U m TU m T c TT 1 c T T 1 A m U m T c T 1 T T T T T,, T wth c, U m and A m the mola educed heat capacty, potental enegy and fee enegy. III. RESULTS In ths secton we wll pesent the esults of the quas- Gaussan Entopy theoy QGE on thee systems. Fst we wll dscuss the hamonc sold a genealzaton of the classcal Ensten sol. Then we pesent data on lqud wate and methanol, usng both the effectve Gamma and constant alpha appoxmatons fo these lquds an exact Gamma soluton s not accuate enough. The classcal Ensten sold s a collecton of onedmensonal dentcal oscllatos, whch ae coupled to a themal bath. 24 Hee we wll nvestgate a collecton of nondentcal classcal hamonc oscllatos. The potental enegy of the whole system,, s a smple squae potental, 1 2 x 2, whee s the numbe of oscllatos, s the foce constant of the th oscllato, and x the coespondng coodnate. The confguatonal pat of the fee enegy s theefoe AkT ln Q kt ln e /kt dx 2 x kt ln 1 exp 2kT/ dx kt ln 1 2kT 1/ It s clea that x 2 2 kt/. Now the sum of ndependent squaes of a andom vaable wth a standad Gaussan dstbuton s a 2 -vaable, 25 so

11 157 Amade et al.: The quas-gaussan entopy theoy ktx 2 x and theefoe /2 /2 2 /2 1 e 2 / Clealy, the 2 -dstbuton s a Gamma dstbuton see Eq. 67 wth 2, a/2 and 1/2. Now we can expess the dstbuton of the potental enegy, snce 2 2/kT,as 1/kT/2 /2 /2 1 e /kt, 127 whch s agan a Gamma dstbuton wth, a/2 and 1/kT. Theefoe the hamonc sold has exactly the enegy dstbuton of the Gamma state and usng Eq. 65 n Eq. 1 we obtan exactly the expesson of the fee enegy, Eq The state of ths deal hamonc sold coesponds to the smplest possble soluton of a quas-gaussan system. Howeve, ths Gamma state does not fulfll estcton R5, and theefoe all devatons statng fom Eq. 68 ae not vald. Such a system can be egaded to be n a Gamma state wth a non convegng deal educed fee enegy. In fact, n ths case C 1 2 k, so (C /T), kt and hence. Ths s a consequence of the fact that we ae consdeng an deal classcal hamonc sold, whee the coodnates x ae not bound, and theefoe both the confguatonal volume dx and e dvege. It s clea then, that the ato /e conveges to (2kT/ ) 1/2, see Eq The classcal hamonc sold s a good appoxmaton of a eal monatomc sold above ts Debye tempeatue D, say, whee c 3R, the classcal lmt. 5 Fo many solds ths Debye tempeatue s n the ange of 15 4 K, so n the ode of oom tempeatue. Below D, the pevous appoach cannot be used, because of the quantum chaacte of the vbatons. In that case the Ensten o Debye appoxmaton can be used. 5 We also tested the effectve Gamma appoxmaton Eqs and the constant alpha appoxmaton Eqs n the case of small molecules, on lqud wate and methanol. We obtaned expemental educed fee eneges A m usng equlbum lqud-gas vapo pessues see Appendx. We used the values of c, heat of vapozaton v H m wth U m v H m RT), and A m at the efeence tempeatue to calculate (T ) and * and to pedct the tempeatue behavo of A m, U m and c usng ths (T )o *. It should be noted that the efeence tempeatue T and the coespondng equlbum pessue must be chosen n such a way, that the vapo at T can be egaded as an deal gas. Fo wate, the efeence condton was defned at T 3 K and the coespondng equlbum pessue. We calculated the efeence c fom c P, 17 usng the elaton c c P 2 T m l T 128 FIG. 1. The fee enegy A m of lqud wate as a functon of tempeatue. wth the sobac volume expansvty and the sothemal compessblty, and c P c P 3R, coectng fo the knetcal pat whch s 3R fo a gd body. We obtaned c.4942 kj/mol K at 3 K. The expemental heat of vapozaton, v H m at 3 K was obtaned fom Atkns, 26 gvng U m kj/mol. alues fo the equlbum vapopessue wee obtaned fom Schmdt, 27 whch ae n close ageement wth the Handbook of Chemsty and Physcs. 17 Above 373 K, say, coectons fo the non-deal behavo of l the vapo and the changng of the mola lqud volume m ae not completely neglgble. Theefoe we coected all data, as explaned n the Appendx, assumng that the devaton fom deal gas s popely descbed by the second val coeffcent B, and the devaton fom an ncompessble lqud s descbed by a lnea elaton between pessue and mola volume. The values of B( p*,t) and the sothemal compessblty (p*,t) wee calculated fom the tables n Schmdt. 27 The values of B(T) wee n easonable ageement wth those gven by othe authos. 28 At 573 K, the hghest tempeatue, the coecton due to the non-deal vapo s 1.15 kj/mol, and compessng the lqud back to the same ntal densty gves.32 kj/mol. Up to 473 K though, both tems almost cancel. Usng A m, U m and c at 3 K, we obtaned (T )1.186 and *.772. In Fg. 1 we plot the tempeatue pedcton of the QGE theoy usng both appoxmatons togethe wth the expemental data. The ageement s vey good, even ove a tempeatue ange of 3 K. We can see, that thee s a slght dscepancy of only.5 kj/mol fo the constant alpha and.2 kj/mol fo the effectve Gamma appoxmaton at the hghest tempeatue. Clealy, the constant alpha appoxmaton stats to become less accuate fo such a lage tempeatue nteval. In geneal, we expect to ncease towads the deal Gaussan state value of 1/2 wth nceasng tempeatue. An ncease of ndeed esults n the fact, that the eal A m s somewhat lowe than the one, pedcted wth constant. The pedcton of the effectve Gamma appoxmaton seems to be moe accuate fo such a lage tempeatue nteval. The ageement of ths pedcton wth expemental data n-

12 Amade et al.: The quas-gaussan entopy theoy 1571 of lqud wate as a functon of tempea- FIG. 2. The potental enegy U m tue. FIG. 3. The heat capacty c of lqud wate as a functon of tempeatue. dcates that lqud wate can be consdeed as a petubed Gamma state. Fo compason, we also calculated a second ode Taylo expanson of the fee enegy, usng the same amount of expemental data. We can expand A m aound T as follows: A m TA m T A m T T TT A m T 2 T TT wth A m T T S m T A m T U m T, 13 T 2 A m T 2 T S m T T c T, 131 T whee we used ds m dq/tc dt/t. We see, that the ft of ths Taylo expanson s much wose at hghe tempeatues. Ths s n a way obvous, snce the Taylo expanson does not contan futhe physcal nfomaton. It s meely a numecal ft. Ths s especally evdent, fom the compason of the expemental U m and c wth the pedcton of the second ode Taylo expanson of the fee enegy. In the Taylo expanson c s constant and U m s lnea n T. Fom the specfc heat tables at hgh pessue, 27 we wee able to calculate U m at thee dffeent tempeatues, apat fom 3 K, at constant densty. Note, that to keep the same densty as the equlbum densty at T 3 K, aleady at 373 K we need to apply a pessue of 98 ba. Results ae gven n Fg. 2. Fom the fgue t s evdent that /T s an excellent appoxmaton. As aleady mentoned n Sec. II G the effectve Gamma appoxmaton can epoduce wth a hgh accuacy the fee enegy, but less accuate U m and c,ast s clea fom the fgue. Fom Schmdt 27 we also obtaned values of c P, (T) and (T) at the fxed efeence lqud densty.e. at hgh pessue n the same tempeatue ange as Fg. 2. Usng Eq. 128 we obtaned values of c (T), see Fg. 3. Fom these last two fgues t s clea, that fo U m and c the constant alpha appoxmaton s excellent and the effectve Gamma appoxmaton s stll acceptable even fo these two popetes. Ths suggests that fo lqud wate the constant alpha appoxmaton can be consdeed as an excellent local descpton, vald fo a tempeatue ange of at least 1 K, whle the effectve Gamma appoxmaton can be consdeed as a moe geneal descpton, less accuate fo U m and c, but acceptable ove a lage tempeatue ange. Note, that n the ange of tempeatues between 3 and 4 K the second ode Taylo expanson gves a ft to the expemental fee enegy, whch s almost as good as the QGE appoxmatons, whle n ths ange fo the othe two popetes the Taylo expanson s clealy off. Ths s futhe evdence, that such an expanson s a pue numecal ft of the fee enegy only, unable to epoduce othe physcal popetes. We also appled the theoy to lqud methanol. Expemental densty and vapo-pessues wee obtaned fom Lley. 29 To calculate c, we used heat of vapozaton values fom Lley, 29 but, snce we need v H m at constant densty, we wee foced to calculate c at low tempeatue (T 18 K, whee the changng n the mola volume s least, obtanng c.4 kj/mol K and U m 4.32 kj/mol. Fo the coecton tems at hgh tempeatues, we used values of B fom Smth 3 and values of at one atmosphee n the ange fom K. 17 Fo hghe tempeatues, we used extapolated data. In ths case (T )1.96 and *.926. The esults ae gven n Fg. 4. Also hee we have an excellent ageement fo both appoxmatons ove a lage tempeatue ange 3 K. Just as n the case of wate, the Taylo expanson becomes qute wose at hghe tempeatue. Besdes the effect of less accuate expemental data and the assumptons, that ethe (T) s almost constant o the

13 1572 Amade et al.: The quas-gaussan entopy theoy FIG. 4. The fee enegy A m of lqud methanol as a functon of tempeatue. statstcal state can be consdeed a petubed Gamma state, n methanol we have the addtonal complcaton of the pesence of the dhedal angle H C O H, whch we assume to be feely otatng. Anyway, ths appoxmaton s wthn expemental eos, snce the expemental c of the gas at 18 Ks3.6R, 29 nstead of 3.5 R gd body wth one dhedal. I. DISCUSSION AND CONCLUSIONS In ths atcle we have shown that ewtng the equlbum statstcal mechancs n tems of the potental enegy dstbuton functon t s possble to defne dffeent statstcal states, accodng to the type of dstbuton. We also showed that fo a quas-gaussan system such states follow fom a genealzed Peason dffeental equaton fo unnomal dstbuton cuves. The solutons can be odeed by the numbe of cental potental enegy moments, that defne the cuve, whee such moments can be wtten n tems of tempeatue devatves of the aveage potental enegy. Snce n geneal moment M n s a functon only of the fst n1 devatves of the aveage potental enegy, t s clea that such solutons can be odeed accodng to the numbe of devatves as well. Ths mples that fo each soluton, the equlbum physcs s gven only by the aveage potental enegy and a lmted numbe of devatves, dependng on the type of dstbuton. We nvestgated the popetes of the fst two possble solutons, whch ae the Gaussan and Gamma dstbuton, defnng the fst two possble statstcal states, namely the deal Gaussan and Gamma state. The fst one can be egaded as an deal o lmt condton, but the second one s completely physcally acceptable. It s a emakable fact that the equlbum physcs of systems n ths Gamma state can be descbed completely by the aveage potental enegy and ts fst two tempeatue devatves only. We deved fo both states expessons fo the deal educed entopy and fee enegy. We also deved a geneal expesson fo the deal educed entopy of evey quas-gaussan system n tems of the heat capacty and the quantty, the ntnsc entopy functon. Fom ths expesson, we wee also able to obtan a geneal dffeental equaton themodynamc maste equaton, the soluton of whch descbes the tempeatue dependence of the fee enegy, aveage enegy and heat capacty of the system. When the exact statstcal state of the system s known, t s possble to solve ths equaton, as we showed fo the Gaussan and Gamma state. Even when such a condton s not pesent t s stll possble to obtan appoxmated solutons. In ths pape we descbed two possble appoxmatons, the effectve Gamma and constant alpha appoxmatons. They both gave vey accuate esults fo the fee enegy ove a lage ange of tempeatue, when appled to lqud wate and methanol fo these lquds the exact Gamma state s not accuate enough, showng that n such systems the ntnsc entopy functon s vey tempeatue nsenstve and the statstcal states of these lquds can be egaded as petubed Gamma states. The constant alpha appoxmaton was also able to epoduce vey accuately the aveage enegy and heat capacty, at least ove a ange of 1 K, whle the effectve Gamma appoxmaton poved to be less accuate n the same ange of tempeatue fo the latte popetes, although the eos neve exceeded 7%. We also poved that the enegy dstbuton of a classcal hamonc sold s exactly a Gamma dstbuton, showng that aleady the smplest statstcal state of a quas-gaussan system can exst. Moeove, aleady at oom tempeatue many eal solds ae well descbed by ths hamonc model. The quas-gaussan Entopy theoy can also be genealzed to mxtues of dffeent components. The expesson fo the deal educed fee enegy of the total system wll be gven agan by AUTC T 132 and the patal mola Helmholtz fee enegy and chemcal potental fo each component follow fom A m A, n p,t,n j A n,t,n j, wth n the numbe of moles of component. In the case of mxtues at nfnte dluton the functon s completely detemned by the solvent. Cuently we ae developng the theoy fo mxtues, accodng to these equatons. It s also nteestng to note that the same basc mathematcal appoach can be used fo dect calculatons of the educe chemcal potental, snce the latte can be expessed 31 as kt ln e kt ln e 135 whee s the ntemolecula potental enegy of one patcle nteactng wth the est of the system, and. Hee we have assumed, fo smplcty, that the

14 Amade et al.: The quas-gaussan entopy theoy 1573 ntamolecula potental s constant. Fo the dstbuton functon of, (), we can set up a smla scheme, as we dd fo the dstbuton of the total potental enegy. Thee s a majo pont of dffeence though between (y) and (); snce (y) apples to a macoscopc system, we can use the cental lmt theoem to show that (y) must be close to a Gaussan dstbuton. But the nteacton enegy s detemned manly by a vey lmted numbe of local nteactons. Theefoe () s lkely to be moe asymmetc than (y). Pelmnay esults wth Molecula Dynamcs smulatons show ndeed that () can ange fom almost symmetc to vey asymmetc, dependng on the type of nteactons. Shot ange van de Waals nteactons n apola fluds fo example lqud agon, modeled by a Lennad-Jones 12-6 potental 32 poduce a vey asymmetc dstbuton, wheas a pola lqud lke wate modeled by a Coulombc and Lennad-Jones potental n the SPC model 33 esults n an almost Gaussan dstbuton. The slght asymmety though s vey mpotant, snce the ght tal of the dstbuton () s multpled by exp() to calculate exp(). A smple Gaussan dstbuton was used by Levy and cowokes 34 fo calculatng the fee enegy of hydaton of ons n soluton. Such a dstbuton coesponds to the smplest soluton of the genealzed Peason system. Ou futue wok wll concen both development of the theoy and connectons wth expemental and accuate smulaton data. ACKNOWLEDGMENTS We lke to thank D. J. Mav fo caefully eadng the manuscpt and many stmulatng dscussons. Ths wok was suppoted by the Nethelands Foundaton fo Chemcal Reseach SON wth fnancal ad fom the Nethelands Oganzaton fo Scentfc Reseach NWO and by the Tanng and Moblty of Reseaches TMR Pogam of the Euopean Communty. APPENDIX: EXPERIMENTAL HELMHOLTZ FREE ENERGY In ths appendx we show how to calculate the Helmholtz fee enegy fo a homogeneous lqud system at fxed volume, fom the equlbum vapo pessue at dffeent tempeatues. Ths method s based on two assumptons. Fst, snce the lqud densty has to be fxed at l, we need to change the pessue fom the equlbum pessue p* to p, to foce the densty fom the equlbum one l (p*,t) back to the efeence densty l (p *,T ) l. We assume, n such a pocess the pessue to be a lnea functon of the mola lqud volume fst ode appoxmaton. Second, we assume that the behavo of the vapo wth densty g can be descbed adequately by a val expanson, tuncated afte the second val coeffcent B: p g kt 1B g. A1 Wth these two assumptons, the educed fee enegy can be calculated well above the bolng pont of the lqud, but wll be less accuate at vey hgh tempeatues o gas denstes. We can expess the chemcal potental of a pue lqud at a tempeatue T and a pessue p as: l p,t l p*,t p* p m l p,tdp, A2 whee p* s the lqud-gas equlbum pessue at tempeatue T and m s the mola volume of the lqud. We can l dentfy p wth the pessue to constan the densty. Snce the vapo s n equlbum wth the lqud, we may eplace the fst tem on the ght hand sde by the gas chemcal potental: l p,t p* g T g p m p p,tdp l m p,tdp p* A3 wth g m the mola volume of the gas, p some efeence pessue, chosen n such a way that the vapo at p behaves lke an deal gas, and g (T) g (p,t). Fo the second tem on the ght hand sde of Eq. A3, we can ewte Eq. A1 wth g m 1/ g to get an explct expesson fo g m (p,t). Integatng we fnd p* g m p,tdprt ln p p* p RT ln 1 4B RT p* B RT p RT1 4B 4B p*1 RT RT p. A4 The thd tem of Eq. A3 can be appoxmated by p l m p,tdp l m p,tpp* 1 p* 2 m l pp* l m p,tpp* 1 l l m m 2 l m p*,tp* A5 wth m l m l (p*,t) m l (p,t) and (1/ m l ) ( m l /p) T the sothemal compessblty. The fee enegy can be calculated fom

15 1574 Amade et al.: The quas-gaussan entopy theoy A m p,t l p,tp m l p,t g TRT ln p* 4B 1 RT p*1 RT ln 4B 1 1 RT p 4B RT 1 RT p* 4B 1 p RT p 1 l l m m 2 l m p*,tp* p* m l p,t. A6 Fo lquds we can safely neglect the tem p* l m (p,t). Fo the deal gas at the same tempeatue and densty as the lqud, we fnd: A m g TRT ln l RT p RT A7 wth g (T) the same as n Eq. A3. Fom the pevous equatons t follows that the educed fee enegy Eq. 9 s A p* m p,trt ln l RTRT 4B 1 RT p*1 RTln 4B 1 1 RT p 1 4B RT p* 1 4B RT p 1 l l m m 2 l m p*,tp*. A8 The last equaton povdes a vey accuate method to evaluate the Helmholtz fee enegy of a lqud wthn the ange of tempeatues whee the lqud-gas equlbum s pesent. Note, that the fst two tems on the ght hand sde epesent the fee enegy, assumng the vapo to be deal and the lqud to be ncompessble. The thd one s the coecton tem due to the non-dealty of the vapo, and the last tem the coecton due to the compesson of the lqud. Also note, that the last two tems have opposte sgn. 1 L. D. Landau and E. M. Lfshtz, Couse of Theoetcal Physcs, 3d ed. Pegamon, Oxfod, 198, ol R. W. Zwanzg, J. Chem. Phys. 22, J. P. Hansen and I. R. McDonald, Theoy of Smple Lquds, 2nd ed. Academc, London, J. A. Bake and D. Hendeson, J. Chem. Phys. 47, D. A. McQuae, Statstcal Mechancs Hape & Row, New Yok, J. D. Weeks, D. Chandle, and H. C. Andesen, J. Chem. Phys. 54, C. G. Gay and K. E. Gubbns, Theoy of Molecula Fluds Oxfod Unvesty, Oxfod, J. K. Patel, C. H. Kapada, and D. B. Owen, Statstcs: Textbooks and Monogaphs Macel Dekke, New Yok, 1976, ol Ths condton s not necessaly fulflled n the case of Molecula Dynamcs smulatons. In ths stuaton an nvestgaton on the dependence of the enegy moments on the sze of the peodc box s equed. 1 H. Camé, Mathematcal Methods of Statstcs Pnceton Unvesty, Pnceton, K. Peason, Phlos. Tans. R. Soc. A 185, K. Peason, Phlos. Tans. R. Soc. A 186, W. P. Eldeton, Fequency-Cuves and Coelaton Layton, London, J. K. Od, Famles of Fequency Dstbutons Gffn, New Yok, N. I. Johnson and S. Kotz, Contnuous Unvaate Dstbutons - 1 Houghton Mffln, New Yok, K. A. Dunnng and J. N. Hanson, J. Stat. Comp. Smul. 6, CRC Handbook of Chemsty and Physcs, 75th ed., edted by D. R. Lde Chemcal Rubbe, Cleveland, N. I. Johnson and S. Kotz, Dscete Dstbutons Houghton Mffln, New Yok, K. Tödhede, n Wate: A Compehensve Teatse, edted by F. Fanks Plenum, New Yok, 1972, ol. 1, Chap. 13, pp J. P. Phllps, Specta-Stuctue Coelaton Academc, New Yok, F. R. Dollsh, W. G. Fateley, and F. F. Bentley, Chaactestc Raman Fequences of Oganc Compounds Wley, New Yok, J. P. M. Postma, Ph.D. thess, Unvesty of Gonngen, The Nethelands, C. J. Pouchet, The Aldch Lbay of Infaed Specta, 3d ed. The Aldch Chemcal Company, Mlwaukee, C. Kttel, Intoducton to Sold State Physcs Wley, New Yok, H. O. Lancaste, The Ch-Squaed Dstbuton Wley, New Yok, P. W. Atkns, Physcal Chemsty, 3d ed. Oxfod Unvesty, Oxfod, E. Schmdt, Popetes of Wate and Steam n SI-Unts Spnge, Beln, G. S. Kell, n Wate: A Compehensve Teatse, edted by F. Fanks Plenum, New Yok, 1972, ol. 1, Chap. 1, pp P. E. Lley, Chem. Eng. 89, D. S. Smth and R. Svastava, Themodynamc Data fo Pue Compounds Elseve, Amstedam, 1986, ol. B. 31 K. S. Shng and K. E. Gubbns, Mol. Phys. 46, M. P. Allen and D. J. Tldesly, Compute Smulaton of Lquds Oxfod Unvesty, Oxfod, H. J. C. Beendsen, J. P. M. Postma, W. F. an Gunsteen, and J. Hemans, n Intemolecula Foces, edted by B. Pullman Redel, Dodecht, 1981, p R. M. Levy, M. Belhadj, and D. B. Ktchen, J. Chem. Phys. 95,