terils Physics nd echnics (0 9- Received: rch 0 THEORY OF VIRTIONS OF TETR-TOIC SYETRIC ENT OLECLES lexnder I eler * ri Krupin Vitly Kotov Deprtment of Physics of Strength nd Plsticity of terils Deprtment of Physics nd themticl odeling in echnics Deprtment of Experimentl Physics StPetersurg Stte Polytechnic niversity Polytehnichesy 9 955 St Petersurg Russi *e-mil: newton@imopspsturu strct In this contriution we sumit theory of virtions of tetr-tomic symmetric ent molecule in cis-conformtion Compred to the previous clcultions where some pproximtions hve een mde now we gve more rigorous version For the molecule studied we hve clculted freuencies of ll five virtions mong them three virtions x y nd sy re norml or nerly norml The longitudinl virtions re of two types: ( nd ( Introduction Structure of molecules nd mcromolecules mnifest themselves in virtion spectr odern physics chemistry nd iology widely use the virtion spectr for solving numerous nd diverse prolems which refer to the study of molecule nd mcromolecule structure nd its chnging in physicl chemicl nd iologicl processes olecule virtions ply crucil role in relxtion phenomen in inetics of chemicl rections in self-orgniztion of polymers nd iopolymers with the resulting formtion of prticulr structure [] For this reson the theory of molecule virtions is of fundmentl importnce for condensed mtter physics moleculr physics iology nd even for nnotechnology Virtion freuencies of molecules nd mcromolecules fter the dissocition energy nd the energy of chemicl onds re the most importnt constnts of these sustnces [] One of the most relile nd precise methods for the experimentl determintion of eigen freuencies of molecules nd mcromolecules re infrred spectroscopy nd Rmn spectroscopy To nlyze the virtion spectr of complex molecules which re very complicted we must e le to estlish purely theoreticlly chrcteristic virtions of tomic groups incorported in one or nother molecule Such virtions depend very little on the presence of other tomic groups y the purely theoreticl pproch to molecule virtions we understnd ny nlyticl solution otined in the frmewor of mechnics or nother science without ug semiempiricl potentil functions with lot of prmeters usully elstic constnts which re necessry for numericl solution nd which numer exceeds usully significntly the numer of toms in molecule The ppernce of excess prmeters hd led to endless discussion out the correctness of prmeter choice insted of developing new pproches eg electronic theory of molecule virtions ws creted with the purpose to gin etter 0 Institute of Prolems of echnicl Engineering
0 lexnder I eler ri Krupin Vitly Kotov insight into the nture of those virtions [] Creful nlysis of this sitution hs een given in [] nfortuntely the mjority of reserchers prefer clculting insted of thining Proly for this reson the purely nlyticl theory ws creted t this moment only for tritomic molecules in the frmewor of mechnics [ 5] nd reltively recently y the comintion of mechnics nd the ond-chrge theory which tes into ccount electronic degrees of freedom [] The mechnicl theory hs ecome clssicl one nd ws incorported into text-oos long go [ 5] The new theory enhnces the possiilities for studying properties of molecules significntly In prticulr it gives unified pproch to virtions nd rottions of molecule nd mcromolecules [6] Nevertheless it is not worth leving the clssicl mechnicl theory out of scientific reconing s efore it remins the sis for clssifiction of virtions We hve developed the theory of molecule virtions in the frme wor of clssicl mechnicl pproch for tetr-tomic liner symmetric molecules [7] nd for tetr-tomic ent symmetric molecules [8] However in the lst cse due to some pproximtions one of virtions ws lost In this contriution we give revised nd more complete version of the clcultions done in [8] without the previous pproximtions Trget setting Consider tetr-tomic ent symmetric molecule Let in n euilirium position the molecule hs the cis form of symmetric ent chin with the vlence ngles θ the euilirium distnce etween toms nd eing eul to nd etween toms respectively (Fig v y u v v u v u O x Fig Displcements of toms in tetr-tomic symmetric ent molecule Denote y the msses of the toms nd y u u u u their displcements from euilirium positions Suppose s efore [7] tht intertomic forces oth stretching the ounds nd ending the ngle etween onds re of short-rnge chrcter The forces cting etween the nerest neighors re centrl Let e their elstic constnts The forces chnging the vlence ngles θ whose vertices re occupied y the second nd the third tom re of noncentrl chrcter Let θ e their elstic constnt The vlence ngle nd elstic constnts re input prmeters Hence we will study the virtions of this tetr-tomic molecule in the frmewor of four prmeter model Geometric constrints nd new coordintes In the sme wy s efore [7] we exclude trnsltion motion of the molecule long the xes x nd y putting the totl liner momentum of the molecule eul to zero It mens tht the inerti center of the molecule is immoile so the following reltion tes plce u
Theory of virtions of tetr-tomic symmetric ent molecules i u i 0 In our cse this leds to ppernce of the geometric constrints of tomic displcements wht loo lie ( u u ( u u 0 ( v v ( v v 0 Introduce the new coordintes x u u x u u sx u u sx u u y v v y v v s y v v s y v v t tht the geometric constrints te the form x x y y To exclude rottion of the molecule one needs to ssume tht its totl ngulr momentum is zero For smll virtions this condition tes the form [7] i u 0 r i0 i where r i0 is the rdius vector of the immoile euilirium loction of i tom Let us project the displcement of tom on the direction norml to the line O (Fig y n n v u O x Fig ngles in ent tetr-tomic symmetric molecule used in clcultion This gives the displcement producing the rottion of the molecule round the xis z going through the origin of coordintes
lexnder I eler ri Krupin Vitly Kotov ( v u Here tn ( / O The displcement producing the rottion of the molecule in opposite direction cn e otined y the ction of rottion C ( y on ( ( C ( y ( v u If the rottion does not te plce then or ( v ( v ( v v ( u u ( v v Therefore v ( v v ( u u v In the new coordintes x s y s y Thus three coordintes nmely x y s y re superfluous Kinetic energy The inetic energy of the molecule is E in ( u u u u u u u u u u v v u u v v
Theory of virtions of tetr-tomic symmetric ent molecules Write down the inetic energy in the new coordintes Since u v u ( x sx u u ( x sx v ( y s y v v ( y s y we hve E in x sx y sy x sx y sy Eliminte the superfluous coordintes x x y y ecuse of x x y y the second term tes the form x y sx sy s result we hve E in x y sx sy sx sy Now eliminte the superfluous coordinte x s y s y ecuse of sy x sy x sy the second term tes the form
lexnder I eler ri Krupin Vitly Kotov sx x sy x sy Finlly we otin E in sx x y x sx sy sy x sy Or in nother form E in y sx sx x sy ( x sy 5 Potentil energy The potentil energy of the molecule consists of three prts ( u u u u where ( l ( l ( ( ( l Here l l nd l re the length chnges of the intertomic onds nd respectively is the devition of the vlence ngle from its euilirium vlue θ during deformtion virtions We cn find the vlue l projecting the vector u u on the direction ( u u ( v v l where / From this it follows tht ( l ( u u ( v v ( u u ( v v cting y the rottion C ( y on ( l we find ( l
Theory of virtions of tetr-tomic symmetric ent molecules 5 ( l ( u u ( v v ( u u ( v v The sum of these two lst expressions is ( u u ( u u ( v v ( v v ( l ( l ( u u ( v v ( u u ( v v Since ( u u ( u u u u u u ( uu u u ( v v ( v v v v v v ( v v v v u x sx ( ( u x sx u x sx ( ( u x sx uu ( x sx ( x sx uu ( x sx ( x sx u u u u ( s s u u ( x nd nlogous formuls re vlid for the components v i we hve x y sx s y x y sx s y x y x y Let us eliminte the superfluous coordintes x x y y ecuse of sx sx s y s y sx x x y y we hve xsx x sx x sx sx
6 lexnder I eler ri Krupin Vitly Kotov ysy ysy ysysy Now eliminte the superfluous coordinte x s y s y ecuse of sy x sy x sy we otin x sx sx sx sx y x sy s y The next term is the potentil energy Here ( l ( u u ( v v sx sy nd we hve ( l ( sx sy gin let us eliminte the superfluous coordinte x s y s y ecuse of sy x sy x sy
Theory of virtions of tetr-tomic symmetric ent molecules 7 we otin ( sx x sy x sy Now find the chnge of the ngle for the ent molecule For this purpose project the vector u u on the direction norml to Then we hve ( v v ( u u The ngle chnge cn e found y cting the rottion C ( y on C ( y ( v v ( u u Therefore ( v v ( u u ( u u ( v v ( v v ( u u ( u u ( v v Compring these expressions with ( l ( u u ( v v ( u u ( v v ( l ( u u ( v v ( u u ( v v nd the potentil energies ( ( ( l ( l with ech other we cn write right wy x sx sx sx y x sy sx s y
8 lexnder I eler ri Krupin Vitly Kotov 6 Lgrnge function nd norml coordintes Comining ll the previous results otined we otin the Lgrnge function L y sx sx x sy ( x sy x sx sx sx sx y sy x sy ( sx x sy x sy x sx sx sx sx y x sy s y sully << s in hydrogen peroxide H O The Lgrnge function contins the terms of different order of smllness: of order nd of order / Eliminting smll untities of order / one is le to seprte out three functions which relte to norml nd nerly norml virtions L( x x x x x
y y ( y L sy sy sy sy L ( The freuencies of these virtions re eul to x y sy 7 Lgrnge function nd longitudinl virtions Consider the reminder of the Lgrnge function sx sx x s x s L sx sx sx sx sx ccording to the generl formul 0 E dt d n n in 9 Theory of virtions of tetr-tomic symmetric ent molecules
0 lexnder I eler ri Krupin Vitly Kotov find the eutions of motion sx sx sx 0 sx sx sx 0 Denote the comintions of elstic nd geometric constnts in the following wy s is customry we see the solution of this system of liner homogeneous differentil eutions with constnt coefficients in the form i t s x e s x i t e where re some constnts Sustitution of these functions in the eution system nd i t cncelltion y the term e gives the system of liner homogeneous lgeric eutions for the constnts ( 0 0 In order to hve solutions not eul to zero the system determinnt should e zero 0 Therefore 0 The eution hs two roots
Theory of virtions of tetr-tomic symmetric ent molecules which define the freuencies of longitudinl virtions Since t tht x 0 ie u u nd ( u u ( u u 0 we hve u u These conditions give the virtions of two types ( ( 8 Conclusion We hve developed the theory of molecule virtions in the frme wor of clssicl mechnicl pproch for tetr-tomic symmetric ent molecules Compred to the previous clcultions where some pproximtions hve een mde [8] now we gve revised nd more complete version s result we hve found the freuencies of ll five virtions In prticulr the new symmetricl virtion long the direction norml to the longitudinl xis of the molecule is found esides the freuencies of norml nd nerly norml virtions differ from the vlues erlier otined t the sme time the freuencies of longitudinl virtions did not chnge This mens tht the verticl displcement of internl usully more hevy toms do not influence on the longitudinl virtions References [] I eler // Proc SPIE 6597 (007 6597-0 [] VN Kondrtiev Structure of toms nd olecules (Fizmtgiz oscow 959 in Rus [] I eler Voroyev // Proc SPIE 65 (006 6505 [] I eler Voroyev // Reviews on dvnced terils Science 0 (009 [5] LD Lndu E Lifshitz echnis (Nu oscow 988 in Rus [6] I eler Krupin // terils Physics nd echnics 9 (00 0 [7] Voroyev I eler // Proc SPIE 65 (006 650 [8] I eler Voroyev // Proc SPIE 6597 (007 659708