Alignment of Buckingham Paametes to Genealized Lennad-Jones Potential Functions Teik-Cheng Lim School of Science and Technology SIM Univesity 535A Clementi oad S 599490 epublic of Singapoe epint equests to T.-C. L.; E-mail: alan tc lim@yahoo.com Z. Natufosch. 64a 200 204 (2009); eceived July 8 2008 / evised August 21 2008 The Lennad-Jones(12-6) and the Exponential-6 potential functions ae commonly used in computational softwaes fo descibing the van de Waals inteaction enegy. Some softwaes allow switching between these two potentials unde pescibed condition(s) that attempt to connect the paamete elationship between the two functions. Hee we popose a technique by which the paamete elationship between both potentials is extacted by simultaneously imposing an equal foce constant at the well depth s minimum and an equal mean inteatomic enegy fom the point of equilibium to the point of total sepaation. The fome imposition induces good ageement fo the inteatomic compession and a small change in the inteatomic distance nea the equilibium while the latte enables good ageement fo lage inteatomic sepaation. The excellent ageement exhibited by the plots validates the technique of combined citeia poposed heein. Key wods: Paamete Convesion; Potential Function; van de Waals. PACS numbes: 33.15.Dj 33.15.Fm 34.20.Cf 1. Intoduction Aising fom the intoduction of the Lennad-Jones potential enegy function [1 3 U LJ = A m B (0 < n < m) (1) n a numbe of molecula mechanics foce fields apply the Lennad-Jones function fo descibing the inteaction enegy between non-bonded neutal atoms. A majoity of these foce fields adopt the Lennad- Jones(12-6) function (e. g. [4 12) followed by the Lennad-Jones(9-6) function (e. g. [13 14). Less common ae the Lennad-Jones(12-10) which is available in AMBE [11 as an option in addition to the usual Lennad-Jones(12-6) function and the buffeed Lennad-Jones(14-7) function [15. Othe foce fields adopt the Exponential-6 potential function (e. g. [16 21) which is a special case of the Buckingham potential function [22 U B = aexp(b) c η (2) with η = 6. In DEIDING [8 and UFF [10 both the Lennad-Jones(12-6) and the Exponential-6 functions ae available as options. It has been appeciated that the Exponential-6 function is moe stable than the Lennad-Jones function [10. As a esult a loose fom of the Exponential-6 function was intoduced as { 6 [ ( U X6 = D ξ 6 exp ξ 1 ) ξ } 6 ξ 6 (3) whee substitution by ξ = 13.772 and ξ = 12 fulfills the foce constant ( 2 ) U LJ(12-6) 2 U X6 k = 2 = 2 (4) = = and the long ange elationship lim U LJ(12-6) = lim U X6 (5) espectively with efeence to the Lennad-Jones(12-6) function [8. A dawback in the use of these two values is obvious duing the switch of this facto in the intemediate ange. ecently an intemediate scaling facto of ξ = 12.6533 was obtained by imposing the following equal enegy integal fom equilibium to dissociation: U LJ(12-6)d = U X6 d. (6) 0932 0784 / 09 / 0300 0200 $ 06.00 c 2009 Velag de Zeitschift fü Natufoschung Tübingen http://znatufosch.com
T.-C. Lim Alignment of Buckingham Paametes 201 With 3 scaling factos instead of 2 the Exponential-6 function is split into 3 pats with 2 switches in the scaling facto. An additional switch educes the abuptness in the van de Waals enegy desciption hence esulting in a compaatively smoothe Exponential-6 potential enegy cuve [23. In the pesent pape an attempt is made to popose the use of only one scaling facto to ensue a pefectly smooth potential enegy cuve while maintaining an almost exact ageement not only with the commonly used Lennad-Jones(12-6) function but also to any Lennad-Jones-type of function. 2. Analysis The enegy integal appoach was found to be useful fo obtaining the shape paamete of a potential function fom a goup of paametes in anothe potential function. Unlike the limit appoach e. g. (5) the use of an enegy integal fom equilibium inteatomic distance to dissociation povides a shape paamete that ensues minimal discepancies [24 26. Futhemoe the limit appoach is of little pactical value as most computational softwaes impose enegy cut-off beyond cetain ange of inteatomic distance. Howeve the enegy integal appoach does not ensue good ageement nea the minimum well depth. The imposition of an equal second-ode deivative o a elated appoach ensues good coelation nea the minimum well depth but not fo a longe ange [27 30. To achieve good coelation ove the whole ange i. e. both in the nea ange and the fa ange thee is need to implement both the imposition of an equal enegy cuvatue and an equal enegy integal. This will of couse lead to two diffeent values of the scaling facto. As such we heein conside the Buckingham function in its loose fom U B = D { η exp [ ξ ( 1 ) ξ η } (7) which contains two shape paametes ξ and η instead of just one. The Lennad-Jones function consideed heein is of the geneal fom [ n m U LJ = D m n (8) m n m n with m and n as the two shape paametes. Hee the paametes ξ and m can be viewed as the epulsive indices while the paametes η and n ae the attactive indices of the potential enegy functions. The imposition of an equal second-ode deivative at the minimum well depth 2 U LJ 2 U B 2 = 2 (9) = = gives the poduct of the Lennad-Jones indices as ξη( 1) mn = (10) while the imposition of an equal enegy integal fom the well depth s minimum to infinite inteatomic distance leads to U LJ d = U B d (11) 1 m n (m 1)(n 1) = 1 ( η ξ ξ ). (12) η 1 The two independent elations descibed by (10) and (12) ae useful fo obtaining the Lennad-Jones shape paametes (mn) fom those of the Buckingham potential (ξ η) and vice vesa. The coesponding Buckingham paametes to the conventional Lennad-Jones(12-6) can be obtained by substituting m = 2n = 12 into (10) and (12); solving these equations gives ξ = 14.3863 and η = 5.6518. In the same way by substituting 2m = 3n = 18 into (10) and (12) fo the case of the Lennad-Jones(9-6) function enables one to solve the Buckingham paametes numeically as ξ = 11.9507 and η = 5.3212. In ode to povide a pope measue between the two potentials we conside the Hilbet space fo these functions. The distance between the two functions U B and U LJ in an inne poduct space is witten as d(u B U LJ )= U B U LJ (13) whee the nom of the function U B U LJ is given in tems of the inne poducts as U B U LJ = U B U LJ U B U LJ 1 2. (14) The inne poduct U B U LJ U B U LJ is defined in the space of eal-valued functions with domain to the eal line as U B U LJ U B U LJ = (U B U LJ ) ( U B U LJ ) d (15)
202 T.-C. Lim Alignment of Buckingham Paametes m n ξ η α β γ d(u B U LJ ) ef. 12 6 13.772 6 0.168715 0.33962 0.171821 0.030210 D [8 12 6 12 6 0.183081 0.35309 0.171821 0.042563 D [8 12 6 14.3863 5.6518 0.171468 0.34277 0.171821 0.022717 D This pape Table 1. The distance d(u B U LJ ) between the Buckingham and the Lennad-Jones(12-6) functions based on the Hilbet space. whee the ba denotes the conjugate. In this analysis both potential functions do not consist of imaginay pats hence (U B U LJ ) ( U B U LJ ) =(UB U LJ ) 2. (16) Substituting (16) into (15) and taking the integal leads to the distance between both potential functions as d(u B U LJ )=D (α + β + γ) (17) whee α = ξη [ η () 2 2ξ 2 2 ξ + η + ξ (18) η(2η 1) 2 β = ()(m n) [ ηn ξ + m ηm ξ + n ξ n η + m 1 + ξ m η + n 1 (19) γ = mn [ n (m n) 2 m(2m 1) 2 m + n 1 + m. n(2n 1) (20) Table 1 compaes the distance between both potential functions in the inne poduct space. It can be seen that the use of a combined equal foce constant and an equal enegy integal gives the lowest distance compaed to a pevious appoach [8. The pesent appoach gives the distance between the two potentials as thee quate and half of the distances based on DEIDING s [8 nea ange indices ξ = 13.772 and η = 6 and fa anges indices ξ = 12 and η = 6 espectively. 3. esults and Discussion To test the validity of the elations descibed in (10) and (12) in a pactical sense the dimensionless inteaction enegy (U/D) vesus the dimensionless inteatomic distance (/) is plotted fo the Lennad-Jones (12-6) potential U LJ(12-6) D = 12 6 2 (21) Fig. 1. Nomalized Lennad-Jones(12-6) and Lennad- Jones(9-6) potentials compaed with the Buckingham foms descibed by (21) (24). and the Lennad-Jones(9-6) potential U LJ(9-6) 9 6 = 2 3 (22) D with thei coesponding countepats in the Buckingham foms U B1 D = 0.6471e14.3863(1 ) 5.6518 1.6471 (23) and U B2 D = 0.8027e11.9507(1 ) 5.3212 1.8027 (24) espectively in Figue 1. This figue shows the plots of the Lennad-Jones(12-6) and Lennad-Jones(9-6) potentials in tiangles and cicles espectively while the Buckingham potentials with (ξ η) =(14.3863 5.6518) and (11.95075.3212) ae denoted by thin and bold lines espectively. Since the inteatomic foce is defined as F = U (25) we intoduce the nomalized inteatomic foce defined heein as F = D U (26)
T.-C. Lim Alignment of Buckingham Paametes 203 coesponding to the Buckingham and the Lennad- Jones potentials espectively. See Fig. 2 fo the coesponding nomalized plots of the inteatomic foce. The excellent ageement of the inteatomic foce obseved between both pais of coesponding potential functions attest the validity of the paamete elationships of (10) and (12). 4. Conclusions Fig. 2. Nomalized inteatomic foce of the Lennad- Jones(12-6) and Lennad-Jones(9-6) potentials compaed with the Buckingham foms. to give and F B = ξη F LJ = { ( mn m n ) η+1 ( exp[ ξ 1 ) } (27) [ n+1 m+1 (28) It was shown that any Lennad-Jones-type potential enegy function can be expessed in tems of an Exponential-6-type function by modifying the latte s epulsive and attactive indices. The adjusted epulsive and attactive indices can be obtained by equating the foce constant and the enegy integal. The existence of the attactive index η emovesthe equiementfo the scaling facto ξ to be adjustable. The fixed value of the scaling facto ensues that the Exponential-6-type potential function is pefectly smooth. The two sets of paamete elationships of (10) and (12) also allow any given Exponential-6-type paametes to be conveted into those of the Lennad-Jones-type potential fo application in computational softwaes adopting the latte function fo quantifying the van de Waals inteaction enegy. [1 J. E. Lennad-Jones Poc.. Soc. London A 106 463 (1924). [2. Mecke Z. Phys. 42 390 (1927). [3 G. B. B. M. Sutheland Poc. Indian Acad. Sci. 8 341 (1938). [4 S. Lifson A. T. Hagle and P. Daube J. Am. Chem. Soc. 101 5111 (1979). [5. Books. E. Buccolei B. D. Olafson D. L. States S. Swaminathan and M. Kaplus J. Comput. Chem. 4 187 (1983). [6 W. F. van Gunsteenm and H. J. C. Beendsen Goningen Molecula Simulation (GOMOS) Libay Manual (1987). [7 M. Clak. D. Came III and N. van Opdenbosch J. Comput. Chem. 10 982 (1989). [8 S. L. Mayo B. D. Olafson and W. A. Goddad III J. Phys. Chem. 94 8897 (1990). [9 V. S. Allued C. M. Kelly and C.. Landis J. Am. Chem. Soc. 113 1 (1991). [10 A. K. appe C. J. Casewit K. S. Colwell W. A. Goddad III and W. M. Skiff J. Am. Chem. Soc. 114 10024 (1992). [11 W. D. Conell P. Cieplak C. I. Bayly I.. Gould K. M. Mez J. G. M. Feguson D. C. Spellmeye T. Fox J. W. Caldwell and P. A. Collman J. Am. Chem. Soc. 117 5179 (1995). [12 W. Damm A. Fontea J. Tiado-ives and W. L. Jogensen J. Comput. Chem. 18 1955 (1997). [13 M. J. Hwang T. P. Stockfisch and A. T. Hagle J. Am. Chem. Soc. 116 2515 (1994). [14 S. Balow A. A. ohl S. Shi C. M. Feeman and D. O Hae J. Am. Chem. Soc. 118 7578 (1996). [15 T. A. Halgen J. Comput. Chem. 17 490 (1996). [16 E. M. Engle J. D. Andose and P. v.. Schleye J. Am. Chem. Soc. 95 8005 (1973). [17 N. L. Allinge J. Am. Chem. Soc. 99 8127 (1977). [18 N. L. Allinge Y. H. Yuh and J. H. Lii J. Am. Chem. Soc. 111 8551 (1989) [19 P. Comba and T. W. Hambley Molecula Modeling of Inoganic Compounds 1st ed. VCH Weinheim 1995. [20 J. M. L. Dillen J. Comput. Chem. 16 595 (1995). [21 N. L. Allinge K. Chen and J. H. Lii J. Comput. Chem. 17 642 (1996).
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