A novel analytic potential function applied to neutral diatomic molecules and charged lons

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1 Vol., No., (00 Natual Scic A ovl aalytic pottial fuctio applid to utal diatomic molculs ad chad los Cha-F Yu, Cha-Ju Zhu, Cho-Hui Zha, Li-Xu So, Qiu-Pi Wa Dpatmt of physics, School of Scic, Xi a Polytchic Uivsity, Xi a, Chia; yuh55@6.com Rcivd 4 Novmb 009; visd 8 Jauay 00; accptd 0 Jauay 00. ABSTRACT I this pap, a w mthod o costucti aalytical pottial y fuctios is pstd, ad fom this a aalytical pottial y fuctio applid to both utal diatomic molculs ad chad diatomic molcula ios is obtaid. This pottial y fuctio icluds th dimsiolss udtmid paamts which ca b dtmid uiquly by solvi lia quatios with th xpimtal spctoscopic paamts of molculs. Th solutios of th dimsiolss udtmid paamts a al umbs ath tha complx umbs, this sus that th aalytical pottial y fuctio has xtsiv uivsality. Fially, th pottial y fuctio is xamid with fou kids of diatomic molculs o ios homoucla utal diatomic molcul H(X, K(B u ad Li (B u, homoucla chad diatomic molcula io H (X u, N (X ad O (X, htucla utal diatomic Molcul AlB(A, PuO(X ad NaLi(X, htucla ch- ad diatomic Molcula io BC (X, MH (X ad i,as a cosquc, ood sults a obtaid. Kywods: Diatomic Molculs Ad Ios; Pottial Ey Fuctio; Foc Costats; Spctoscopic Paamts; Phas Facto. INTRODUCTION Aalytical pottial y fuctios a of at siificac i th study of matial scic, molcula spctum, actio dyamics of atoms ad molculs, vibatioal ad otatioal y-lvl stuctus of molculs, itactios btw las ad matt, photoioizatio tc. [-] Du to th impotac ad xtsiv applicatios of th pottial y fuctio, th cospodi sach woks hav b caid o all alo [4-6]. So fa, th pstativ aalytical pottial y fuctio poposd hav Mos pottial [7], Rydb pottial [8], Mull-Sobi pottial (M-S [9] ad Huxly-Mull-Sobi pottial (HMS [0] tc. Rctly, Su Wiuo t al hav poposd a y cosistt mthod (ECM ad costuctd a w physically wll bhavd aalytical pottial fuctio of a diatomic systm calld ECM pottial []. Ths pottial fuctios abov hav mits ad dfcts spctivly, thy a valid i dscibi th bhavios of som idividual o classificatoy diatoms ad molculs. But o of thm ca dscib both utal diatomic molculs ad chad diatomic molcula ios ad dscib pcisly th bhavios of pottial y fuctio ov th whol a of itucla distac. S fom xpssioal foms, most of ths pottial y fuctios adopt th foms of ployomial ad xpotial. I this pap, a cosi fuctio with a phas facto is usd as basic pottial y fuctio ad, thouh omalizatio to th phas facto, a uivsal pottial y fuctio applid to fou kids of diatomic molculs o ios homoucla utal diatomic molculs, homoucla chad diatomic molcula ios, htucla utal diatomic molculs ad htucla chad diatomic molcula ios is iv. Fially, th pottial y fuctio is xamid with twlv difft kids of diatomic molculs ad ios tc., as a cosquc, ood sults a obtaid.. FUNDAMENTAL SUPPOSITIONS, AND DERIVATION OF A UNIVERSAL ANALYTIC POTENTIAL FUNCTION Suppos that th pottial fuctio of diatomic molcula satisfis th followi latio V( Acos ( B ( wh, wh ( accos( / ( A, B a udtmid costats, ( is a Copyiht 00 SciRs.

2 C. F. Yu t al. / Natual Scic ( phas facto latd to, ad th itucla distac, h is quivalt phas diffc btw two itacti atoms, is quilibium itucla distac. Substituti Eq. ito Eq., yilds V( Acos accos( / B A ( / cos / si B ( Eq. is a basic aalytical pottial y fuctio. I od to obtai th uivsal aalytical pottial fuctio of diatomic molculs ad ios, omalizatio should b dd fo th tm i Eq., so as to su that th divativs of ach od of th Eq. a cotiuous ad fiit at quilibium distac. Thus w ca xpad th tm ito biomial sis (i! i ( (4 i i0 4 ( i! (i H, Eq.4 is a ifiit sis, it d to b tucatd ito fiit tms ad its followi ifiit tms should b absobd ito th udtmid cofficits a, b, c, so fom Eq.4, w hav i0 H( i( wh i i0 (i! ( i 4 ( i! (i a( b( 4 (i! H ( i i 4 ( i! (i i a( c( 6 (5 (6 Gally, th pottial y fuctio satisfis asymptotic coditio lim V( 0,so fom Eq. w hav B Asi (7 Substituti Eq.5 ad Eq.7 ito Eq., ad otic H ( 0, yilds A i V ( cos Asi H ( i( i (8 4 6 a( b( c( I Eq.8, th udtmid costat A ca b dtmid accodi to th poptis of pottial y fuctio. At th quilibium distac, th pottial valu is qual to th ativ valu of dissociatio y D, i.. V ( D, ad th fist divativs of V ( with spct to is zo. So fom Eq.8, w obtai V ( Acos Asi H i a b c ( D (9 i dv d A A cos si{ H ( i(i i a ( b( 4 c( 6 0 (0 Fom Eq.9 ad Eq.0, th solutios of A ad cos ca b iv as follows D A si i H( i(i a( b( c( 5 ( cos si{ i H ( i(i a( b ( 4 c( 6} ( Substituti Eq. ad Eq. ito Eq.8, yilds V ( i 4 D H ( i( / a( / b( / i c( 6 c( 6 H ( i(i a( b( 4 i i H ( i(i a( b ( c( 5 ( Eq. is th uivsal aalytical pottial y fuctio that is quid. Th udtmid paamts a, b, c ca b dtmid with th xpimtal spctoscopic paamts (,, B of molculs o fitti mthod usi silpoit pottial y scai. Wh,,, fom Eq.,w hav 4 D ( V a 6a 0b 4c 6 8 b c 4a6b8 c, ( 4 8D ( V 7 40a 56b 7c 8 ( a b c 6 a 8 b 0 c ( (5 Copyiht 00 SciRs.

3 86 C. F. Yu t al. / Natual Scic ( D V ( 9 a 44b 76c a b c 6 5 /8 8a 0b c ( ( a4b60c f X 740a56b7c D 4 400a896b680c f Y 740a56b7c 6D a576b5640c f4 Z 740a56b7c 4D (, ad 98X Y 4Z 80 0 (0. USING EXPERIMENTAL SPECTROSCOPIC PARAMETERS TO DETERMINE a,b,c Th udtmid paamts a, b, c ca b dtmid with th xpimtal spctoscopic paamts (,, B of diatomic molculs o ios. Th picipl of this mthod is, accodi to th latioship btw udtmid paamts ad foc costats, to obtai a, b, c by solvi lia quatios. Fom Eq., th al xpssio of foc costats at th quilibium itucla distac ca b iv as follows [ m m m d V ( V 0 fm [ ( ( ] m m Hi i j d i j 0 ( m! ( m! (5 m! a b c (! (! (5! i wh V 0 H ( i(i a( b( 4 c( 6] i m! ( m,, 4 (7 D H ( i(i a( b( c( 5 (8 Fom Eq.7 ad Eq.8, wh,,, th followi lia quatios ca b obtaid a0b56c f X 6a 0b 4c D a 00b 4c f Y 6a0b4c 6D 4 6a 40b 644c f4 Z 6a0b4c 4D (, ad 6X 5Y 4Z 0 ( a70b056c f X 9 a44b76c D 98 79a60b56c f Y 9 a44b76c 6D a80b648c f4 Z 9 a44b76c 4D (, ad 4X 7Y 4Z 60 0 ( I Eqs.9-, th latioships btw foc costats ad spctoscopic paamts a as follows f f 4 c ( f ( ( 6B f 8 f4 5( (4 6B B Th Eqs.9- abov a all lia quatios, wh th coditios of 6X 5Y 4Z 0, 98X Y 4Z 800ad4X 7Y 4Z 60 0 a satisfid with spct to Eqs.9-, thy hav uiqu al umb solutios fo th udtmid paamts a, b, c. Calculatios show that th coditios abov a always tabl i al. This sus that th aalytical pottial fuctio Eq. has xtsiv uivsality, which ca dscib ay of diatomic molculs ad ios spcially th bhavios of molculs a quilibium itucla distac. So fa, th most xtsivly usd aalytical pottial y fuctio is Mul-Sobi (M-S pottial. Th udtmid paamts i Mul-sobi pottial which a dtmid by xpimtal spctoscopic paamts hav o uiqu solutios ad cotai complx umb solutios. Thus, th M-S pottial is xtmly limitd i applicatios to som diatomic molculs ad ios. [] 4. APPLIED EXAMPLES OF THE UNIVERSAL ANALYTICAL POTENTIAL ENERGY FUNCTION Fo xamii pottial y fuctio Eq., fifty kids of utal diatomic molculs ad chad diatomic Copyiht 00 SciRs.

4 C. F. Yu t al. / Natual Scic ( Tabl. Expimtal spctoscopic paamts of diatomic molculs ad ios. stats / cm / cm B / cm / cm / m D / V Rfs. H(X [] K(B u [4] Li(B u [5] H (X u [6] N (X [4] O (X [7] AlB(A [4] PuO(X [8] NaLi(X [] BC (X [9] MH (X [4] [4] i Tabl. Pottial paamts ad foc costats of diatomic molculs ad ios. stats 4 4 D / V / m. a b c f /0 aj m f /0 aj m f4 /0 aj m H(X K(B u Li(B u H ( X u N (X O (X AlB(A PuO(X NaLi(X BC (X MH (X i Tabl. Pottial paamts of Mul-Sobi pottial of diatomic molculs ad ios. stats a / m a / m a / m / m D / V H(X K(B u Li(B u H ( X u N (X O (X AlB(A PuO(X NaLi(X BC (X MH (X i Copyiht 00 SciRs.

5 88 C. F. Yu t al. / Natual Scic ( Fiu. Pottial cuv of H X. Fiu 4. Pottial cuv of BC (X. u Fiu. Pottial cuv of H ( X Fiu. Pottial cuv of AlB A. molcula ios hav v b ivstiatd ad ood sults a obtaid. Calculatios show that two commo pottial y cuvs, i.. stadystat ad mtastabl stat ca b iv by usi th pottial y fuctio dtmid with xpimtal spctoscopic paamts. Th xpimtal spctoscopic paamts of H(X, H (X u, AlB(A ad BC (X tc. a listd i Tabl. Accodi to Eqs.-4, th cospodi foc costats ca b obtaid by usi th xpimtal spctoscopic paamts abov, ad substituti ths foc costats ito Eq.9 o Eq., th th udtmid paamts a, b, c ca b calculatd by solvi th lia quatios. Th calculatio valus a listd i Tabl. Th pottial y cuvs (to b calculatd ad plottd by usi Eq.4 ad Eq.6 dictly with Oii 7.0 softwa plottd by Eq.4 ad Eq.6 of H(X, H (X u, AlB(A ad BC (X a illustatd i Fius -4. As compaiso, i th Fis., th dot lis a th pottial cuvs which a plottd by usi th most xtsivly usd Mul-Sobi Pottial. Th M-S pottial xpssio is as follows V( D a a xp a a (5 Th latioships btw udtmid paamts of M-S pottial ad foc costats a as follows ( a a f (6 D D D (a a a a 4 a a a 4aa f (7 ( f (8 5.CONCLUSIONS I this pap, w fist itoduc th phas cocpt to th studis of aalytical pottial y fuctios ad t 4 Copyiht 00 SciRs.

6 C. F. Yu t al. / Natual Scic ( ood sults. This shows that th mthod of costucti aalytical pottial y fuctio by mas of phas is ffctiv ad liabl. Compad with oth pottial y fuctios, th pottial y fuctio iv i this pap has two mits: Th udtmid paamt quatios dtmid by xpimtal spctoscopic paamts a lia quatios. Bcaus ths lia quatios hav uiqu al umb solutios, so this pottial y fuctio has a xtsiv uivsality; This pottial y fuctio ca dscib fou difft kids of diatomic molculs o ios homoucla utal diatomic molculs, homoucla chad diatomic molcula ios, htucla utal diatomic molculs ad htucla chad diatomic molcula ios; I additio, This pottial y fuctio ca also dscib accuatly th bhavios of pottial cuvs ov a faily wid a of itucla distac. Pottial y fuctios of diatomic molculs a th basis to th studis of multi-atomic molculs, ios ad clusts, which hav xtmly siificacs ad applid valus i th study of matial scic, molcula spctum chmical actio tc. Chmical actio, molcula collisio ad may oth poblms d pcis aalytical pottial y fuctios. Thus, th studis of aalytical pottial y fuctio will still b impotat subjct i atomic ad molcula physics. REFERENCES [] Yiaopoulou, A., Ju, G.-H., Su, J.P., t al. (999 Pottial-y cuvs fo hihly xcitd lctoic stats i diatomic molculs latd to th atomic obital udulatios [J]. Physical Rviw A, 59(, [] Liu, G.Y., Su, W.G. ad F, H. (004 Studis o th aalytical pottial y fuctio of diatomic molcula Io XY + usi vaiatioal mthod [J]. Scic i Chia (Sis G, 47(, [] Maio, M. ad Acioli, P.H. Full cofiuatio itactio psudopottial dtmiatio of th oud-stat pottial y cuvs of Li ad LiH [J]. Itatioal Joual of Quatum Chmisty, 0(5, [4] Xi, R.H. ad Go, J.B. (005 A Simpl Th- paamt modl pottial fo diatomic systms: Fom wakly ad sto to mtastabl molcula ios [J]. Physical Rviw Ltts, 95, 60. [5] Yu, C.f., Ya, K. ad Liu, D.Z. (006 A uivsal aalytic pottial-y fuctio basd o a phas facto [J]. Acta Mtalluica Sica (Elish Ltts, 9(6, [6] Estvs, C.S., d Olivia, H.C.B., Ribio, L., t al. (006 Modli diatomic pottial y cuvs thouh th alizd xpotial fuctio [J]. Chmical Physics Ltts, 47(, 0-. [7] Mos, P.M. (99 Diatomic molculs accodi to th wav mchaics. Ⅱ. Vibatioal lvls [J]. Physical Rviw Ltts, 4: [8] Rydb, R. (9 Gaphisch Dastllu ii Badsp-ktoskopisch Ebiss [J]. Z Physics, 7: [9] Mul, J.N. ad Sobi, K.S. (974 Nw aalytic fom fo th pottial y cuvs of stabl diatomic stat [J]. Joual of th Chmical Socity, Faaday Tas, II, 70, [0] Huxly, P. ad Mul, J.N. (98 Goud-stat diatomic pottial [J]. Joual of th Chmical Socity, Faaday Tas Ⅱ, 79, -8. [] Su, W.G. ad F, H. (999 A y-cosistt mthod fo pottial y cuvs of diatomic molculs [J]. Joual of Physics B: Atomic, Molcula ad Optical Physics, (, [] Zhu, Z.H. ad Yu, H.G. (997 Molcula stuctu ad molcula pottial y fuctio [M]. Scic Pss, Biji, 8-. [] G, Z.D., Fa, X.W. ad Zha, Y.S. (006 Stuctu ad pottial y fuctio of th oud stat of XY (H, Li, Na.[J] Acta Physica Siica, 55(05, [4] Hzb, G. (98 Molcula spcta ad molcula stuctu (i. spcta of diatomic molculs [M]. Scic Pss, Biji, [5] Yu, B.H., Shi, D.H., Su, J.F., Zhu, Z.L., Liu, Y.F. ad Ya, X.D. (007 Ab iitio Calculatio o Accuat Aalytic Pottial Ey Fuctios ad Hamoic Fqucis of c ad B u Stats of Dim 7Li [J]. Chis Physics, 6(8, [6] Hablad, H., Issdoff, B.V., Fochticht, R., t al. (995 Absoptio Spctoscopy ad Photodissociatio Dyamics of Small Hlium Clust Ios [J] Joual of Chmical Physics, 0(, [7] Savpt, K. ad Mahaja, C.G. (999 Wi hua s fou-paamt pottial commts ad computatio of molcula costats_ ad [J]. Pamaa joual of physics, 5(4, [8] Gao, T., Wa, H.Y., Yi, Y.-G., Ta, M.-L., Zhu, Z.-H., Su, Y., Wa, X.-L. ad Fu, Y.-B. (999 Ab itio calculatio of th pottial y fuctio ad thmodyamic fuctios fo oud stat X of PuO [J]. 5 Acta. Physica Siica, 48(, -7. [9] Tzli, D. ad Mavidis, A. (00 Cotiui ou study o th lctoic stuctu of th cabids BC ad AlC [J]. Joual of Physical Chmisty, A05, Copyiht 00 SciRs.

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