Vibrational Relaxation of HF (v=3) + CO

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1 Journl of the Koren Chemicl Society 26, Vol. 6, No. 6 Printed in the Republic of Kore Notes Vibrtionl Relxtion of HF (v3) + CO Chng Soon Lee Deprtment of Chemistry, Chngwon Ntionl University, Chngwon 54, Kore. E-mil: cslee@chngwon.c.kr Key words: V R mechnism, HF vibrtionl relxtion, Vibrtionl energy trnsfer probbility INTRODUCTION In the vibrtionl energy trnsfer of molecules with excited vibrtionl levels, the strong ttrction between the molecules increses the rection rte of vibrtionl relxtion. In generl, the vibrtionl energy trnsfer rection between molecules with wek interction occurs t short distnce, nd thus the collision is ffected by the repulsive potentil nd hs short rection time. However, perturbtion occurs in molecules with strong ttrction when the molecules re reltively further prt, nd thus the longer collision time llows energy trnsfer rection. For exmple, hydrogen hlides such s HF nd HCl hve high hydrogen bond energy, nd their vibrtionl energy trnsfer mechnism differs from tht of molecules with wek interction. Hydrogen hlides exhibit negtive temperture dependence; the probbility of energy trnsfer t low temperture decreses with n increse in temperture. Owing to such strong hydrogen bond (6 kcl/mole) t lower temperture, HF molecules cn form nonrigid dimer with longer lifetime, nd thus the vibrtionl energy trnsfer between HF molecules occurs t reltively long seprtion distnce. Shin reported tht in the ner equilibrium configurtion the HF molecules in the dimer undergo the hindered rottionl motion nd bcknd-forth trnsltionl motion. This shows tht the HF vibrtionl energy trnsfer from n excited vibrtionl level cn occur vi hindered rottionl nd/or trnsltionl motion. 2,3 Further, Shin suggested long-lived collision model to clculte the vibrtionl energy trnsfer probbility for HCl; this model ssumes tht the two collision rection molecules mintin the rection for sufficiently long time t room temperture owing the hydrogen bond energy (2. kcl/mole) between the HCl molecules. 4 Lee et l. used Shin s long-lived collision model to explin the experimentl results of the vibrtion vibrtion energy trnsfer rection in HF (v n) +H 2 (v ) nd DF (v n) + D 2 (v ) rection systems. 5 Becuse the hydrogen bond energy between HF nd CO is significntly lrge (3. kcl/mole), 6 the vibrtionl energy relxtion in HF (v) + CO rection system hs ttrcted much interest in numerous experimentl nd theoreticl studies. Sirkin nd Pimentel 7 predicted tht the HF vibrtionl relxtion will minly occur vi the V R mechnism in the vibrtionl relxtion experiment of HF + CO system. Rybone et l. predicted tht in the IR chemiluminescence experiment for the vibrtionl relxtion of HF (v 3) + CO, the primry rection pthwy for relxtion rection is V R,T, which is single vibrtionl quntum trnsition. 8 However, Dzelzkln nd Kufmn predicted tht for the high vibrtionl relxtion energy levels of HF (v 5,6,7) + CO, V V process would be the min mechnism for energy trnsfer. 9 Further, Wllis nd Thompson used the qusiclssicl trjectory method to clculte the vibrtionl relxtion rte for HF + CO rection. In this study, Shin s long-lived collision model ws used becuse the energy of hydrogen bond between HF nd CO is 3. kcl/mole, which is sufficiently lrge. The rte constnt for the vibrtionl energy relxtion of HF (v 3) + CO rection system ws clculted using the V R mechnism. To obtin the rte constnt for vibrtionl relxtion trnsfer, the interction potentil energy between two collision molecules ws clculted using Morse-type eqution. A two-dimensionl collision model ws used in the clcultion, nd the collision trjectory bsed on rection time ws clculted in terms of clssicl mechnics using Newton s equtions of motion. The rte for V R energy trnsfer of HF molecules ws clculted using the quntum mechnicl method. Further, the clculted result ws compred with the existing experimentl nd theoreticl clcultion dt to demonstrte V R process s the mechnism for the vibrtionl relxtion in HF (v 3) + CO rection system. INTERACTION POTENTIAL The equilibrium orienttion of HF nd CO in the dimer intercting vi hydrogen bond energy ws ssumed to be -462-

2 Vibrtionl Relxtion of HF (v3) + CO 463 liner, nd the interction between the H in HF molecule nd the C in CO molecule ws considered. Then, the Morse potentil eqution stted below ws used for the clcultion. Vr () D exp r e r exp r e r () 2 where, r is the distnce between the H in HF nd the C in CO, nd r e is the equilibrium distnce between the two molecules. D nd re the Morse potentil constnts to be determined. In the current model, D is the energy of hydrogen bond between the two rectnt molecules. If the reltive distnce between the mss centers of the two collision molecules, R, is much lrger thn the equilibrium bond distnces of HF nd CO, d nd d 2, distnce r between the two toms H nd C cn be written s follows: r~ R γ ( d + x )cosθ + γ 2 ( d 2 + x 2 )cosθ 2 where, γ m F /(m H + m F ) nd γ 2 m O /(m C + m O ). θ is the ngle between the moleculr xis nd R. The equilibrium distnce between the two toms H nd C cn be written s follows: r e ~ R e γ d cosθ e + γ 2 d 2 cosθ 2e, where subscripts nd 2 refer to HF nd CO molecules, respectively. Using Eq. (2) nd the eqution for equilibrium distnce r e on Eq. () nd conducting series expnsion on the exponents with x nd x 2, the overll interction potentil cn be pproximted s follows: VRθ (,, θ 2, x, x 2 ) V ( R, θ, θ 2 ) + V ( R, θ, θ 2, x ) +V( R, θ, θ 2, x 2 ) The first term in Eq. (3) cn be rewritten for potentil term tht includes only the reltive trnsltionl motion between the two molecules in θ nd θ 2 orienttion sttes s follows: V ( R, θ, θ 2 )D [ Q ( cosθ cosθ e ) Q 2 ( cosθ 2 cosθ 2e )] exp exp 2 --Q ( cosθ cosθ e ) --Q 2 2 ( cosθ 2 cosθ 2e ) exp (2) (3) (4) where, Q γ d / nd Q 2 γ 2 d 2 /. The second term of Eq. (3) cn be rewritten for potentil term for V R trnsition tht ssumes the vibrtionl-to-rottionl energy trnsfer within HF molecule. V ( R, θ, θ 2, x ) exp[q ( cosθ cosθ e ) Q 2 ( cosθ 2 cosθ 2e )] exp exp --Q 2 ( cosθ cosθ e ) --Q 2 2 ( cosθ 2 cosθ 2e ) exp (5) 2 x cosθ The lst term of Eq. (3), V 2 (R, θ, θ 2, x 2 ) is the potentil term for the V R process of CO; this ws not considered in this study. To determine the trnsltionl motion trjectory for vrious moleculr orienttions for θ nd θ 2, expressed s R(t), Eq. (4) cn be rewritten for the men orienttion for θ nd θ 2 s follows: 2π 2π V ( R) ( 2π) 2 U ( R, θ, θ 2 )dθ dθ 2 D 2 I ( Q )exp[ Q ( cosθ cosθ e ) Q 2 ( cosθ 2 cosθ 2e )] exp I 2 ( Q /2)exp 2 --Q ( cosθ cosθ e ) --Q 2 2 ( cosθ 2 cosθ 2e ) exp (6) where, I is the -order modified Bessel function. If R e * is the new equilibrium distnce for the verge orienttion for θ nd θ 2, t R R * dv(r)/dr nd then Eq. (7) cn be derived s follows: * I exp[ ( e )/2] ( Q /2)I ( Q 2 /2) I ( Q )I ( Q 2 ) exp --Q 2 e exp -- Q (7) 2 2 2e Further, using Eq. (7), Eq. (6) cn be rewritten s follows: V ( R) D * * * { exp[ ( R e R)/] 2exp[ ( R e R)/2]} where, D * DI [ 2 ( Q /2)I 2 ( Q 2 /2)/I ( Q )I ( Q 2 )]. When there is strong interction between two collision molecules due to hydrogen bonding by HF nd CO, the molecules will mostly form wek complex for the durtion of the collision rection ner R R *. This is cler contrst from the rections where strong repulsive potentil cuses short-distnce collision. Therefore, by tking the verge for θ 2 orienttion s ( 2π) 2π V ( R, θ, θ 2, x )dθ 2 using (8) 26, Vol. 6, No. 6

3 464 Chng Soon Lee Eq. (5) nd conducting series expnsion on the exponents with R e * R, eqution (9) cn be derived s follows: V ( R, θ, x ) Aexp( Q cosθ ) Bexp 2 --Q + 2Aexp( Q cosθ ) Bexp 2 --Q * [( R e R)/2] + 2Aexp( Q cosθ ) --Bexp Q * [( )/2] 2 + x cosθ (9) 2 where, A I ( Q /2)I 2 ( Q 2 /2)/I 2 ( Q )I ( Q 2 ) nd B I ( Q /2) I 2 ( Q 2 /2)/I ( Q )I ( Q 2 ). The first term of Eq. (9) is the potentil for pure V R energy trnsfer. The other terms of Eq. (9) refer to V R,T process, where the vibrtionl energy of HF is trnsferred to rottionl nd trnsltionl motions. The current collision model considers V R energy trnsfer, in R R * e due to strong hydrogen bonding, to be the min process in collision rection. Therefore, the potentil for V R energy trnsfer cn be expressed s follows: V ( θ, x ) Aexp( Q cosθ ) Bexp 2 --Q x cosθ () The introduction of Boson opertors ( +, ) provides the coordintes of rection system, xˆ ( h/2m ω ) /2 ( + + ), nd thus Eq. () cn be rewritten s follows: V ( θ, x ) Aexp( Q cosθ ) Bexp 2 --Q h /2 ( + + )F(t)( + + ) () 2M ω where, M nd ω re the reduced mss nd ngulr frequency of HF, respectively. PROBABILITY EQUATION FOR VIBRATIONAL RELAXATION TRANSFER The Hmiltonin for collision rection system expressed by Boson opertors cn be written s follows: H --hω (2 + +)+F(t)( + + )H + H'(t) (2) 2 where, H'(t)F(t)( + + ) refers to the perturbtion due to collision. The time-dependent Schrödinger eqution for collision rection system using Eq. (2) cn be written s follows: ih ψ v() t [ H + H () t ]ψ (3) t v () t The solution for this eqution cn be rewritten s follows: ψ v () t exp[ Gt () ( + + ) ]ψ v () t (4) where, G(t) is the imginry function determined by the perturbtion term H'(t) in Eq. (3) for time t s shown in Eq. (5). 2 i + Gt () -- (5) h Ft () e iδωt where, Δω ΔE/h. The probbility eqution for the vibrtionl relxtion of HF from the initil vibrtionl stte, v, to the finl vibrtionl stte, f, between the initil collision time, t, nd the finl collision time, t +, cn be written s follows: P v f lim ψ f t t () ψ v () t 2 ψ f () exp t [ Gt () ( + + ) ] ψ v ( t ) 2 (6) For Eq. (6), rottionl motion trjectory to obtin the trnsfer probbility, θ (t)ωt + θ, ws used. Here, Ω is the ngulr frequency of rottionl motion, (2E r /I) /2, nd I is the inerti moment. Using Eq. (6), the trnsfer probbility eqution for temperture (T) considering Boltzmnn distribution cn be expressed s follows: P v f ( T) ( kt) PE ( r ) exp( E r /kt)de r where, k is Boltzmnn constnt. CALCULATIONS AND RESULTS (7) The potentil constnts 6 used to clculte the vibrtionl relxtion probbility using Eq. (7) re D 3. kcl/mole nd 2. Å. The spectroscopic dt of HF nd CO were obtined from the stndrd tble. 3 The vibrtionl energy trnsfer rection of HF from the collision with CO is n exothermic rection, where the vibrtionl energy is relxed from high to low vibrtionl energy. Therefore, considering the imblnce from vibrtionl relxtion energy, ΔE, to trnsfer to rottionl energy level before nd fter the collision, E r * [(E r + ΔE) /2 + E r /2 ] 2 /4 ws used s the rottionl energy. The rte constnt to compre the theoreticl vlue obtined using Eq. (7) nd the experimentl vlues Journl of the Koren Chemicl Society

4 Vibrtionl Relxtion of HF (v3) + CO 465 of vibrtionl relxtion trnsfer probbility of HF (v 3) + CO rection system re s follows: If wve functions ψ 2 nd ψ 3 re hrmonic oscilltor wve functions with vibrtionl levels 2 nd 3, respectively, the first nd second terms of Eq (2) re ψ 2 () ψ t 3 () t nd ψ 2 () t ( + + ) ψ 3 () t 3. Here, the clcultions + for Boson opertors used ψ n ( n + ) /2 ψ n+ nd ψ n n /2 ψ n. However, if wve functions ψ 2 nd ψ 3 re nhrmonic oscilltor functions, the terms re ψ 2 () ψ t 3 () t nd ψ 2 () t ( + + ) ψ 3 () t The nhrmonic oscilltor wve function ψ n () t used in the clcultion is shown below. K ψ n () t ϕ n {[( n+ ) ( n+ 2) ( n+ 3) ] /2 ϕ n ( n+ ) 3/2 ϕ n + 9n 3/2 ϕ n [ nn ( ) ( n 2) ] /2 ϕ n 3 } (2) where, K ( α/hω) ( h/mω) 3/2, nd ϕ is the hrmonic oscilltor wve function in vibrtionl level, i. Moreover, α b 3 D e nd b ω e ( 2π 2 cμ/d e h) /2. D e is the equilibk d ( T) ZP v f ( T) 4.74 r * ( Tμ * ) /2 P v f ( T) tm - sec - (8) Here, subscript d on rte constnt, k d, shows tht this is nonrigid collision model, nd μ * is the reduced mss written s tomic mss unit (mu). For collision rdius, 6 r * 3.69 Å ws selected. 6 By substituting the vlues of μ * nd r * in Eq. (8) nd multiplying RT the rte constnt with cm 3 /molecule-s s the unit cn be expressed s follows: k cm 3 d ( T).8 T /2 P v f ( T) /molecule-s (9) Using Eq. (9), t 3 K the rte constnt for vibrtionl energy trnsfer, k d in HF (v 3)+CO HF (v 2)+ CO + ΔE rection systems ws clculted. Tble shows the result long with other experimentl nd theoreticl clcultion results. The clculted vlue using Eq. (9), ws k d cm 3 /molecule-s, which corresponds to the other experimentl nd theoreticl vlues. As shown in Tble, the rte constnt, 3. cm 3 /molecule-s, clculted using the qusiclssicl trjectory method hs lrger k d vlue thn our clcultions, even though the difference is smll. The experimentl vlues re 2.5, 2.8, nd cm 3 /molecule-s. These vlues re not significntly different, but re lso slightly higher thn our clcultions. This is probbly becuse only V R ws considered rther thn both the competitive pthwys, V R, T nd V R, in the clcultions of HF vibrtionl relxtion rection. The current clcultion, which only considered the V R mechnism, corresponds to the other experimentl nd theoreticl clcultion results, indicting tht HF (v 3) + CO vibrtionl relxtion rection minly occurs vi the V R rection pthwy. Indeed, ll the interction potentil terms relted to V R, T, including the term (R e * R)/2, in Eq. (9) were excluded, Tble. Rte constnts for the vibrtionl relxtion of HF (v 3) + CO nd HF (v 2) + CO t 3 K This work Willis-Thompson b Experiment (3.±.2) 2 (2.9±3.) 2c (2.8±.2) 2d (2.5±.3) 2e Rte constnts hve units of cm 3 /molecule-s. b Reference. c References 5 nd 6. d Reference 8. e Reference 7. nd only the -order term for (R e * R)/2 obtined through the series expnsion of exp[(r e * R)/2] ws used for the vibrtionl relxtion trnsfer rte constnt, k d, clcultion for HF. In other words, the V R mechnism tht does not include the trnsfer of vibrtionl relxtion energy to the internl trnsltionl motion energy is the process, when HF nd CO re in R R e * where molecules exist in wekly bonded dimer vi hydrogen bonding, in which the vibrtionl relxtion energy ΔE is trnsferred to the internl rottionl energy of HF when the vibrtionl level of HF is trnsferred from v 3 to 2. Further, the vibrtionl relxtion rection of HF owing to CO is n exothermic rection with high ΔE (3624 cm - ). In generl, s ΔE increses in vibrtionl energy trnsfer rection, it is more efficient to trnsfer the ΔE to rottionl motion level rther thn n internl trnsltionl motion. 4 Therefore, the clcultion results show tht the relxtion trnsfer rection is V R process, where the relxed vibrtionl energy of HF molecule is trnsferred to the internl rottionl level rther thn V R, T process. The current k d obtined from the V R rection pthwy is clcultion result tht ssumes HF s hrmonic oscilltor. In other words, for the wve functions of HF in the mtrix term of Eq. (6), the wve functions of hrmonic oscilltors were used. For exmple, when HF undergoes vibrtionl relxtion from v 3 to v 2, the series expnsion of the exponent in Eq. (6) results in probbility eqution for the vibrtionl relxtion of HF s shown below. P 3 2 E r ( ) ψ 2 () t + Gt () ( + + )]ψ 3 () t 2 (2) 26, Vol. 6, No. 6

5 466 Chng Soon Lee rium dissocition energy of the oscilltor. The vibrtionl relxtion rte constnt, k d, for the V R process ws clculted using the nhrmonic oscilltor wve function given in Eq. (2), which is cm 3 /molecule-s. This is pproximtely twice the vlue obtined from the clcultion ssuming HF s hrmonic oscilltor. As shown bove, this is becuse of the vlue of ψ 2 () t ( + + ) ψ 3 () t with 3 nd 2.65 differences when hrmonic nd nhrmonic oscilltor wve functions, respectively, re used. In generl, HF molecules re nhrmonic, nd thus k d probbly hs nhrmonic fctor. However, in relity, the result s the hrmonic oscilltor provided closer vlues to the experimentl dt. Acknowledgments. This reserch is finncilly supported by Chngwon Ntionl University in 25~26. REFERENCES. Yrdley, J. T. Introduction to Moleculr Energy Trnsfer; Acdemic Press: New York, U.S.A., Shin, H. K. J. Chem. Phys. 975, 63, Kim, Y. H.; Shin, H. K. J. Chem. Phys. 976, 64, Shin, H. K; Kim, Y. H. J. Chem, Phys. 98, 73, Lee, C. S.; Kim, Y. H. Bull. Kor. Chem. Soc. 992, 3,. 6. Benzel, M. A.; Dykstr, C. E. J. Chem. Phys. 983, 78, Sirkin, E. R.; Pimentel, G. C. J. Chem. Phys. 982, 77, Rybone, D.; Wtegonkr, S. J.; Setser, D. W. J. Chem. Phys. 988, 89, Dzelzklns, L. S.; Kufmn, F. J. Chem. Phys. 984, 8, 64.. Wllis, E. P.; Thompson, D. L. J. Chem. Phys. 992, 97, Kelley, J. D. J. Chem. Phys. 972, 56, Shin, H. K. Chem. Phys. Lett. 984, 8, Huber, K. P.; Herzberg, G. Moleculr Spectr nd Moleculr Structure IV. Constnts of Ditomic Molecules; Vn Nostrnd: New York, Wilkins, R. L. J. Chem. Phys. 977, 67, Smith, W. M.; Wrigley, D. J. Chem. Phys. 98, 63, Smith, W. M.; Wrigley, D. Chem. Phys. Lett. 98, 7, Arumen, E.; Rybone, D.; Setser, D. W. J. Chem. Phys. 992, 97, Journl of the Koren Chemicl Society

221B Lecture Notes WKB Method

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