Rotational-Vibrational Levels of Diatomic Molecules Represented by the Tietz-Hua Rotating Oscillator

Transcription

1 J. Pys. Cem. A 1997, 101, Rotational-Vibrational Levels of Diatomic Molecules Represented by te Tietz-Hua Rotating Oscillator Josep A. Kunc* Departments of Aerospace Engineering Pysics, UniVersity of Soutern California, Los Angeles, California Francisco J. Gordillo-Vázquez Departamento de Física Atómica, Molecular y Nuclear, UniVersidad de SeVilla, SeVilla, Apdo. Correos 1065, Spain ReceiVed: September 1, 1996; In Final Form: NoVember 11, 1996 X Analytical expressions for te rotational-vibrational energy levels of diatomic molecules represented by te Tietz-Hua rotating oscillator are derived using te Hamilton-Jacoby teory te Bor-Sommerfeld quantization rule. In molecules wit moderate large values of rotational vibrational quantum numbers, te levels are in muc better agreement wit te results of numerical calculations tan te energies obtained from te common model of te rotating Morse oscillator. 1. Introduction Atom-molecule molecule-molecule collisions wit transfer of molecular rotational vibrational energy are important in plasma material processing, supersonic ypersonic flows, ligt sources, etc. Probabilities of te transitions depend on te molecular rotational-vibrational levels, te transitions involving moderately igly excited levels are often as important in gas kinetics as tose involving weakly excited levels. 1 Terefore, a general analytical expression for te molecular energies, accurate in a broad range of rotational vibrational quantum numbers, is of a common interest in pysics of ig-temperature gas. Suc expression would allow a substantial simplification of derivations of te transition probabilities for large number of atommolecule molecule-molecule collision systems for quick transparent evaluation of te main features of te collisions before tey are incorporated into complex timeconsuming collisional-radiative models of ig-temperature gas. Generality of te expression can be of great advantage in comparative studies of te molecular transitions in gases many different collision systems simultaneously influence te gas properties. Te main goal of tis work is to derive analytical expressions for molecular (diatomic) rotational-vibrational energy levels tat ave accuracy acceptable in applications better tan te accuracy of te commonly used expressions for te levels. We pay special attention to te reliability of te obtained expressions for moderate ig rotational-vibrational levels because tese levels often play crucial role in te gas rotationalvibrational dissociative kinetics. General analytical model potentials for diatomic molecules are available in literature (see ref references terein), te potential most common in applications is te internuclear potential of te rotating Morse oscillator L V M ef ) µr + D[1 - e) e-β(r-r ] (1) R is te internuclear distance, R e is te molecular bond lengt, β is te Morse constant, D is te potential well dept, Present address: Department of Aerospace Engineering, University of Soutern California, Los Angeles, CA X Abstract publised in AdVance ACS Abstracts, February 1, S (96) CCC: $14.00 µ is te reduced mass of te oscillator, L is its orbital angular momentum. Te rotational-vibrational levels resulting from a simplified solution of te Scrödinger equation for te rotating Morse oscillator can be given as E M V,J ) ω e( V+1 ) -ω e x e( V+1 ) +B e J(1 + J) - D e J (1 + J) () ω e, ω e x e, B e, D e are te usual spectroscopic constants, J V are te molecular rotational vibrational quantum numbers, respectively. Expression is a truncated series expansion solution of te Scrödinger equation for te rotating Morse oscillator. 3 Even toug more accurate (iger-order) expansions of te solution ave been discussed in literature, eq is te most common in applications (tat is wy we compare below tis expression (instead of a iger-order expansion) wit te results of te present work). Te popularity of eq results from te fact tat te expression is simple it predicts quite accurate values of te weakly-excited rotational-vibrational levels of diatomic molecules tat reliable values of te molecular spectroscopic constants ω e ω e x e are available in literature (most of te oter constants in te iger-order expansion representing te rotational-vibrational energies of te Morse oscillator are not available in literature, calculation of teir accurate values is not trivial). In molecules wit moderate ig values of te rotational vibrational quantum numbers, eq is inaccurate. Terefore, we derive below a general analytical expression for rotational-vibrational levels of diatomic molecules, wic as accuracy acceptable in applications better tan te accuracy of eq. In order to do so, we use internuclear potential in wic te rotational energy is te centrifugal energy of te molecular rotation (as it is done in relationsip 1) te vibrational energy is given by te function suggested by Tietz 4,5 discussed in great detail by Hua; 6 tis function (called ereafter te Tietz-Hua potential) reduces to, respectively, te Rosen- Morse, 7 Morse, 8 Manning-Rosen 9 potentials for negative, zero, positive values of te potential parameter c (tis parameter is denoted by c in ref 6).. Te Tietz-Hua Potential Function If a diatomic molecule is represented by a rotating Tietz- Hua oscillator, ten te internuclear potential of te molecule 1997 American Cemical Society

2 1596 J. Pys. Cem. A, Vol. 101, No. 8, 1997 Kunc Gordillo-Vázquez TABLE 1: Spectroscopic Constants Oter Parameters of te Diatomic Molecules Studied in te Present Work a molecule HF(X 1 Σ + ) Cl (X 1 Σ + g ) I (X (0 + g )) H (X 1 Σ + g ) O (X 3 Σ ḡ ) V max J max M V max D µ/ b y e w ex e B e D e ω e R e β U TH U M E max/d a V max J max are te maximum vibrational quantum number te maximum rotational quantum number, respectively, for te rotating Tietz- Hua oscillator; V M max is te maximum vibrational quantum number for te Morse oscillator; c is te parameter in te potential 3; D is te well dept of te intramolecular potential (in cm -1 ); µ is te reduced mass of te molecule (in g); b is te constant in te Tietz-Hua function (in Å -1 ); y e ) b R e, ω ex e is te anarmonicity constant (in cm -1 ); B e D e are te rotational constants (in cm -1 ); ω e is te vibrational constant (in cm -1 ); R e is te molecular bond lengt (in Å); β is te Morse constant (in Å -1 ); U i ) U i - U /D is te average deviation of te potential U i from te corresponding exact potential U, te U i is te Tietz-Hua potential (te subscript TH) or te Morse potential (te subscript M), U is te potential obtained eiter from te RKR (ref 6 references terein) or from te ab initio calculations (ref 10 references terein); E max is te igest molecular energy assumed in te calculations of te parameter c. can be given as te first term on te rigt- side is te rotational (centrifugal) energy of te oscillator, te second term is te vibrational energy (te Tietz-Hua potential) of te Tietz- Hua oscillator wit V TH ef ) L µr + U TH (3) U TH ) D[ 1 - e-b (R-R e ) 1 - c e -b (R-R e )] R is te internuclear distance, R e is te molecular bond lengt, β is te Morse constant, D is te potential well dept, c is te potential constant discussed below. Te reason for coosing te Tietz-Hua potential in tis work is its remarkably good agreement (witin a broad range of te internuclear distance) wit te RKR (Rydberg-Klein-Rees) (ref 6 references terein) ab initio calculations (ref 10 references terein); see also discussions in refs According to tese works, te Tietz-Hua potential is muc more realistic tan te Morse potential in description of molecular dynamics at moderate ig rotational vibrational quantum numbers. Also, it was sown in refs 5 1 tat all practically significant analytical potentials allowing solution of te one-dimensional Scrödinger equation in terms of ypergeometric functions (te Tietz-Hua potential belongs to tis category) ave been already reported in te literature tat it is practically impossible to discover a new potential of tis kind. Tus, te Tietz-Hua potential seems to be one of te very best analytical model potentials for te vibrational energy of diatomic molecules. Terefore, we use it in te relationsip 3, wic is our coice of analytical approximation to te exact potentials (obtained from te RKR ab initio calculations) in diatomic molecules. Consequently, one of te (4) b ) β(1 - c ) (5) main tasks of tis work is to obtain accurate values of te parameters c from te exact potentials. Te parameter c of te Tietz-Hua potential is obtained ere, like in ref 6, from minimalization of te average value of te ratio U i ) U i - U /D, te internuclear potential U i is obtained from te Tietz-Hua function, U is te exact potential obtained from eiter RKR or ab initio calculations (te average is taken over all available data points of te exact potential). Tus, te degree of te overall agreement between te exact potentials te potentials 1 3 is measured ere by te ratios U i (see Tables 1 ; te corresponding values of c are given in Table 3). Te ratios are muc smaller tan 1 (typically, tey are smaller tan 0.05). However, one sould empasize tat a small value of suc a ratio does not automatically mean tat te sapes of te compared potential curves are equally close to one anoter in te entire considered range of te internuclear distance R. Tus, for example, potential 1 potential 3 for some molecules may ave very similar values of U i, but te potential curves does not ave to overlap in most of te range of te distance R. Tis lack of te overlapping can produce two (one for eac of te compared potentials) substantially different sets of te rotational-vibrational levels even toug te values of U i for te bot curves are very similar. Coice of te Morse constant β is not a simple issue. Solution of te Scrödinger equation for te Morse potential gives te following vibrational constants D ω e ) β[ π c µ] 1/ ω e x e ) β 8π cµ ω e ω e x e are in cm -1, D is in ergs, te rest of te quantities are in units of te cgs system. Tus, te constant β obtained from te expression 6 will differ from tat resulting from te expression 7 if te spectroscopic constants ω e, ω e x e, D are taken as te best available in literature; te latter (6) (7)

3 Te Tietz-Hua Rotating Oscillator J. Pys. Cem. A, Vol. 101, No. 8, TABLE : Spectroscopic Constants Oter Parameters of te Diatomic Molecules Studied in te Present Work a molecule N (X 1 Σ + g ) NO(X Π r) O + (X Π g ) N + (X Σ + g ) NO + (X 1 Σ + ) V max J max M V max D µ/ b y e w ex e B e D e ω e R e β U TH U M E max/d a Meaning of te symbols is te same as in Table 1. TABLE 3: Coefficients c, ω 0, ω 1 in Expressions molecule c ω 0 ω 1 HF(X 1 Σ + ) Cl (X 1 Σ + g ) I (X (O + g )) H (X 1 Σ + g ) O (X 3 Σ ḡ ) N (X 1 Σ + g ) NO(X Π r) O + (X Π g ) N + (X Σ + g ) NO + (X 1 Σ + ) constants are not consistent wit te model of te Morse oscillator because tey are usually obtained from measurements ab initio calculations. (Te expressions 6 7 give te same values of β only wen te Birge-Sponer relationsip, D ) ω e /4ω e x e, is applied. However, te Birge-Sponer approximation can lead to errors in values of D tat can be as large as 30%). In tis work, we calculate β from te relationsip 6 (Tables 1 ) because te constants ω e D available in literature are usually more accurate tan te anarmonicity constants ω e x e (also, expression 6 is identical wit te corresponding expression obtained from comparison of te vibrational force constants for armonic Morse oscillators at R ) R e ). Te anarmonicity constants ω e x e used in te present calculations are tose obtained from te best available data (typically, tese constants differ from tose predicted by expression 7 if β is obtained from expression 6). Te reason for tis coice of ω e x e is te fact tat all te exact (RKR ab initio) calculations of te internuclear potentials discussed in tis work used te best available values of te anarmonicity constants. 3. Te Rotational-Vibrational Energy Levels Using te Hamilton-Jacoby teory te Bor-Sommerfeld quantization rule, Porter et al. 13 studied, using te model of te rotating Morse oscillator, te rotational-vibrational dynamics of diatomic molecules. In tis paper, we apply teir semiclassical approac to diatomic molecules wit intramolecular potential given by relationsip 3. Subsequently, te Hamiltonian of te rotating Tietz-Hua oscillator wit angular momentum L can be written as H ) p r µ + L µr [ + D 1 - e-b(r-re) 1 - c e -b (R-R e )] p r ) µ(dr/dt) is te oscillator linear momentum. (8) Te equilibrium internuclear distance R * of a molecule in te Jt rotational level can be obtained from te derivative wic, after introducing a new variable becomes dv ef (R) dr dv ef (R) dr ) 0 (9) C J ) b L µd Wen J f 0, C J f 0, eq 11 can be written 1 - e [ -y* e ye (10) ) [ 1 - e -y* e ye (1 - c e -y * e y e ) 3] (1 - c ) - ey * e -y e C J y * 3 ) 0 (11) y * ) b R * y e ) b R e (1) (1 - c e -y * e y e ) 3] (1 - c ) ) 0 f 1 ) e-y *+y e y * ) y e (13) wic suggests te possibility of an asymptotic solution of te transcendental equation 11 wic depends on a small parameter (c ). Suc a solution can be obtained as an asymptotic expansion in terms of tis small parameter. 14 We propose te following expansion y * ) y e + B C J + D C J (14) te values of te coefficients B D can be obtained by substituting eq 14 into eq 11 using te fact tat C J (Jf0) f 0. (Note tat expression 14 is also valid for te Morse oscillator wen c ) 0.) As a result, one as e -B C J -D C J - e -B C J -D C J (1 - c (1 - c e -B C J -D C J ) 3 ) ) (y e + B C J + D C J ) 3 Since C J (Jf0) f 0, we can use te expansion of e x = 1 + x + x / +... limit te expression for te cubic function C J (15)

4 1598 J. Pys. Cem. A, Vol. 101, No. 8, 1997 Kunc Gordillo-Vázquez (y e + B C J + D C J ) 3 to linear quadratic terms of C J. After some algebra, one obtains [B C J + D C J - (3/)B C J ](y e 3 + 3y e B C J + 3y e B C J + comparison of te coefficients in te terms wit te same powers of C J gives Te ratio y * /y *n (y *n ) b R *n, R *n is te numerical solution of eq 11) for diatomic molecules is increasing wit increase of te rotational quantum number J. Te values of te ratio are usually not greater tan about 1.03, except for te H (X 1 Σ g + ) HF(X 1 Σ + ) molecules te ratio at J J max is about 1.05 (J max is te maximum value of te molecular rotational quantum number). Using te above we calculated te values of te ratios U i E max /D (E max is te maximum value of te molecular energy taken into account in te ab initio RKR calculations used in tis work) for several diatomic molecules (see Tables 1 ). In order to apply te approac of ref 13, we introduce a new variable, ξ * (R) ) exp(-b (R - R * )), use te following series expansion of 1/(b R) about ξ * ) 1 R *, te internuclear distance corresponding to te minimum of te intramolecular potential of te molecule wit J > 0, is given by te relationsips Substituting te expansion 18 into eq 8 solving for p r (te radial momentum of te vibrating molecule) at fixed value of te rotational-vibrational energy H ) E V,J gives 3y e D C J )(1 - c ) ) C J [1 + 3c (B C J + D C J - (1/)B C J - 1) + 3c (B C J - B C J - D C J + 1) + B ) (1 - c ) y e 3 3c 3 (D C J - (3/)B C J + B C J - (1/3))] (16) D ) 3 B - 3 B + 3B c (1 - c ) y 3 e y e (17) 1 (b R) ) 1 (b R * ) + (b R * ) 3(ξ * - 1) - 1 (b R * ) 3(1-3 b R * ) (ξ * - 1) +... (18) p r )((µ) 1/ [ E 1 - ξ V,J - L (A * + B * ξ * - C * ξ * ) - D( ξ 1/ * e* 1 - c ξ * ) ] ) e -b (R * -R e ) (19) (0) L ) p [J(J + 1) - Λ ] (1) Λ is te quantum number for te axial component of te molecular electronic angular momentum, te coefficients A *, B *, C * are A * ) [ b R * (b R * - 3)] b R * C * () B * ) [ 1 + b R * (b R * - 3)] C * (3) Te energy of te rotating Tietz-Hua oscillator is calculated below using te Bor-Sommerfeld quantization rule V is te vibrational quantum number. As sown in te Appendix, te quantization rule leads to te following expression for te rotational-vibrational levels of te diatomic molecules µ is te reduced mass of te molecule, u ) c Te values of te energy levels obtained from expression 6 are in very good agreement (wit accuracy better tan 1%) for all molecules considered ere (except te H (X 1 Σ g + ) moleculessee discussion below) in te entire range of vibrational, 0 e V e V max (V max is te maximum vibrational quantum number of te rotationless molecule), rotational, 0 e J e J max, quantum numbers, wit te corresponding energy values obtained from numerical solution of te Scrödinger equation 15 for te potential 3. Numerical calculations sow tat te dependence of te functions F F 3 on V is weak for all values of V between 0 V max. Terefore, assuming in F F 3 tat V)V max /, one obtains anoter expression for te rotational-vibrational energies of diatomic molecules C * ) 1 µb R * 3( 1-3 b R * ) (4) N V ) 1 π I p r dr ) ( V+ 1 ) p (5) E V,J ) D + L (A * - 3u C * - u B * - uc * ) - [ 4] (V+1 / ) pb (µ) - F 1/ 1 - F L - F 3 L F 3 ) (6) F 1 ) k 1-4 Dµ -4µ D 1/ + k (7) -µ F ) -4µ D 1/ + k [ φ + F φ 1 1/] 1 (8) D -µφ 1 D 1/ (-4µ D 1/ + k )[ F + F φ 1 1 (9) ξ e* D] φ 1 ) 6u C * + 3u B * + 6uC * + ub * + C * (30) φ )-(8u C * + 3u B * + 6uC * + B * ) (31) k 1 ) up (V+ 1 / ) b (3) k ) up(v + 1 / )b (µ) 1/ (33) E V,J ) D + L B J B J - (F 0 x 1 - F 1 - F L - F 3 L4 ) (34) B J ) ω (b R * ) 4 (35) B J ) (b R * ) - b R * (7u + u + 3) + 15u + 6u + 3 (36)

5 Te Tietz-Hua Rotating Oscillator J. Pys. Cem. A, Vol. 101, No. 8, F 0 ) pb µ (37) F ) A J (1 + ω 0 x + ω 0 x )( F 1 ) D(1 + ω 0 x 1 + ω 1 x 1 ) (38) -η 1J + η J D D(1 + ω 0 x + ω 1 x )) (39) F 3 ) A J η J D (1 + ω 0 x + ω 0x ) ( F + A J η J D D(1 + ω 0 x + ω 1 x )) (40) x 1 )V+1/ (41) x ) (V TH max + 1)/ (4) Figure 1. Rotational-vibrational energies E V,J (dased line, eq 34) E M V,J (dotted line, eq ) te energy obtained from numerical solution of te Scrödinger equation for te potential 3 (solid line) for te N (X 1 Σ + g ) molecule. Te zero rotational-vibrational energy corresponds to te dissociation continuum of te rotationless molecule. V J are vibrational rotational quantum numbers, respectively. ω 0 ) pb c µd (43) ω 1 ) ω 0 ( 1-1 c ) (44) ω A J ) (45) D (b R * ) 4 ω ) b µ (46) η 1J ) b R * (0u + 6u + 4) - (4u + 18u + 6) (47) η J ) b R * (18u + 14u + 1) - (36u + 30u + 3) (48) te maximum value of te vibrational quantum number in te Tietz-Hua oscillator is (see Appendix) V TH max ) 1 - c - (1-3c ) 1/ - 1 ω 0 (c - 1) (49) One sould mention tat te maximum vibrational quantum number V M max of te Morse oscillator, obtained from te function, can be taken as te integer closest to, smaller tan V M max ) ω e - 1 ω e x e (50) In order to evaluate te accuracy of te expression 34, we calculated te energies E V,J of 10 diatomic molecules molecular ions representing a broad range of spectroscopic constants (Tables 1 ). Te results for two of te molecules (N (X 1 Σ + g ) H (X 1 Σ + g )) are sown in Figures 1, togeter wit te rotational-vibrational eigenvalues obtained from numerical solution of te Scrödinger equation for te potential 3 wit te energies predicted by eq. (Te accuracy of te expression 34 for te N (X 1 Σ + g ) molecule is Figure. Rotational-vibrational energies E V,J (dased line) E M V,J (dotted line) te energy obtained from numerical solution of te Scrödinger equation for te potential 3 (solid line) for te H (X 1 Σ + g ) molecule. Meaning of te symbols is te same as in Figure 1. very close to te accuracies of te expression for all, except H (X 1 Σ g + ) HF(X 1 Σ + ), molecules listed in Tables 1 ). As can be seen in Figures 1, expression 34 is always more accurate in describing te rotational-vibrational levels of diatomic molecules tan relationsip. (Te accuracy of te Morse model increases wit decrease of te degree of te rotational-vibrational excitation of te molecules, wile te agreement of te relationsip 34 wit te numerical solution of te Scrödinger equation for te potential 3 is quite good in te entire range of te vibrational rotational quantum numbers.) Expression 34 is less accurate for low V in te case of te H (X 1 Σ g + ) HF(X 1 Σ + ) molecules (te accuracies of te expression for tese two molecules are similar). Terefore, it is more accurate to use in suc cases te expression 6 wit te sum F 1 - F L - F 3 L 4 being replaced by te exact expression 65 (see Appendix Figure 3). One sould notice te significant differences between te values of V max obtained in te present work tose resulting from te model of te Morse oscillator (Tables 1 ). Te difference can be as big as te one in te I (X(0 g + )) molecule V max ) 96 (te present model) 174 (te Morse model). Tis is a clear failure of te Morse model because te realistic value of V max in te I (X(0 g + )) molecule is about 107 (see ref 16).

6 1600 J. Pys. Cem. A, Vol. 101, No. 8, 1997 Kunc Gordillo-Vázquez c f ) c ξ e* (E V,J - A * L ) + B * L c + C * L - ξ e* D = B * L c + C * L - ξ e* D (55) d f ) D - B * L - c (E V,J - A * L ) = D - B * L (56) e f ) E V,J - A * L - D (57) Figure 3. Rotational-vibrational energies E V,J (dased line, eq 6 wit te sum F 1 - F L - F 3 L 4 being replaced by te exact expression 65), E M V,J (dotted line) te energy obtained from numerical solution of te Scrödinger equation for te potential 3 (solid line) for te H (X 1 Σ + g ) molecule. Meaning of te symbols is te same as in Figure Summary Summarizing te above, one can say tat relationsip 34 is a good analytical approximation to te rotational-vibrational levels of diatomic molecules (including molecular ions) in te ground electronic states. Te relationsip sould also be able to predict reasonable values of te rotational-vibrational levels of te electronically excited diatomic molecules tat can be treated as te rotating Tietz-Hua oscillators. Acknowledgment. We tank Alex Dalgarno, Evgenii E. Nikitin, Ricard Zare for valuable comments. Tis work was supported by te Air Force Office for Scientific Researc, Grant F Contract 3019B, by te Pillips Laboratory, Air Force Material Comm, troug te use of te MHPCC under Grant F One of us, F.J.G.V., tanks te Spanis CICYT for financial support wile a postdoctoral researcer at USC. Appendix In tis Appendix we summarize te matematical transformations leading to te expressions 6 34 for te rotationalvibrational energies of diatomic molecules, according to te model of te rotating Tietz-Hua oscillator. Canging te variable r in eq 19 to ξ * one as N V )- (µ)1/ ξ *> {[EV,J ξ*< - L (A * + B * ξ * - C * ξ * )] (1 - c ξ * ) - D(1 - ξ * ) } 1/ /[ξ * (1 - c ξ * )] dξ * (51) ξ *< ξ *> are roots of equation p r ) 0. To obtain te roots, we first rearrange te terms in eq 19 so tat te equation can be written as te following fourt-order polynomial f 1 (ξ * ) ) a f ξ * 4 + b f ξ * 3 + c f ξ * + d f ξ * + e f ) 0 (5) te simplification of te coefficients c f d f resulted from te fact tat c is a small parameter. Let us now exp f 1 (ξ * ) in power series about ξ * ) 1 retaining only terms wic are linear or quadratic in ξ * Tis allows one to evaluate te integral 51 as follows (u ) c, ξ *< ξ *> are now te roots of equation f (ξ * ) ) 0), wic leads to te limits of integration are Te relationsip 51 can now be written as, after some algebra f 1 (ξ * ) = f (ξ * ) ) φ 1 ξ * + φ ξ * + φ 3 (58) φ 1 ) 6a f + 3b f + c f, φ )-8a f -3b f +d f, φ 3 ) 3a f +b f +e f (59) - (µ)1/ ξ *> f 1 ξ*< ξ * (1 - uξ * ) dξ * = - (µ)1/ ξ *> f ξ*< ξ * (1 - uξ * ) dξ * ) - (µ)1/ ξ *> f ξ*< ξ * (1 - uξ * ) dξ * (60) - (µ)1/ [(- -φ 1 + -φ 3 u ) + - φ 1 - φ u - φ 3 u ] ξ *> ) -φ - φ - 4φ 1 φ 3 φ 1 N V )-(- -φ 1 + -φ 3 u )- (61) ξ *< ) -φ + φ - 4φ 1 φ 3 φ 1 (6) -φ 1 -φ u-φ 3 u ) p(v+1 / )ub (µ) 1/ (63) wit a f ) c ξ e* C * L (53) b f ) B * L c ξ e* + c C * L (54) - -φ 3 ) p(v +1 / )b (µ) 1/ - up (V+ 1 / ) b -µφ (64) -4µ -φ 1 + up(v + 1 1/ / )b (µ)

7 Te Tietz-Hua Rotating Oscillator J. Pys. Cem. A, Vol. 101, No. 8, Let us denote te last term in eq 64 by F V,J (k 1 ) up (V+ 1 / ) b k ) up(v + 1 / )b (µ) 1/ ) exp it wit respect to L. Te resulting expression is F V,J ) k 1 - µφ -4µ -φ 1 + k (65) F V,J = F 1 + F L + F 3 L (66) ω 1 ) ω 0 ( 1-1 c ) (79) ω A J ) (80) D (b R * ) 4 ω ) b µ (81) η 1J ) b R * (0u + 6u + 4) - (4u + 18u + 6) (8) wit Solving eq 64 for φ 3 taking into account eqs 65-71, one obtains te expression 6 for te rotational-vibrational levels of te diatomic molecules Since in diatomic molecules te eqs can be approximated, respectively, by F 3 ) F 1 ) k 1-4 Dµ -4µ D 1/ + k (67) -µ F ) -4µ D 1/ + k [ φ + F φ 1 1/] 1 (68) D -µφ 1 D 1/ (-4µ D 1/ + k )[ F + F φ 1 1 (69) ξ e* D] φ 1 ) φ 1 / L ) 6u C * + 3u B * + 6uC * + ub * + C * (70) φ ) φ / L )-(8u C * + 3u B * + 6uC * + B * ) (71) E V,J ) D + L (A * - 3u C * - u B * - uc * ) - [ 4] (V+1 / ) pb (µ) - F 1/ 1 - F L - F 3 L (7) c 1 µd pb (V+1 / ),1 (73) F 1 ) D(1 + ω 0 x 1 + ω 1 x 1 ) (74) -η 1J + η J D 1) F (75) F ) A J (1 + ω 0 x 1 + ω 0x 1 )( F 3 ) A J η J D (1 + ω 0 x 1 + ω 0x 1 )( F + A J η J D 1) F (76) x 1 ) (V+ 1 / ) (77) ω 0 ) pb c µd (78) Numerical calculations sow tat te dependence of te functions F F 3 on V is weak for all values of V between 0 V max (V max is te maximum vibrational quantum number of te rotationless molecule). Terefore, one can assume in relationsips tat V)V TH max /. In te case of te Tietz-Hua oscillator, te maximum vibrational quantum number V TH max F 1 wit Numerical solution of eq 84 gives te maximum vibrational quantum number V TH max wic is very close to te corresponding value obtained from analytical solution of te following equation Tis analytical solution, te smallest positive root of eq 87, is Taking te above into account, te relationsip 7 leads to te main expression 34 of tis work for te rotationalvibrational energies of diatomic molecules η J ) b R * (18u + 14u + 1) - (36u + 30u + 3) (83) can be obtained from te function 7 by replacing F 1 by (because of relationsip 73), de V,J)0 ) 0 f -(F dx 0 x 1 - F 1 1 )( F 0 - df 1 dx 1) ) 0 (84) V TH max F 0 ) pb µ (85) df 1 dx 1 ) ω 0 D + ω 1 x 1 D (86) E V,J)0 - D ) 0 (87) ) 1 - c - (1-3c ) 1/ - 1 ω 0 (c - 1) (88) E V,J ) D + L B J B J - (F 0 x 1 - F 1 - F L - F 3 L4 ) (89) B J ) ω (b R * ) 4 (90) B J ) (b R * ) - b R * (7u + u + 3) + 15u + 6u + 3 (91) x ) (V TH max + 1)/ (9)

8 160 J. Pys. Cem. A, Vol. 101, No. 8, 1997 Kunc Gordillo-Vázquez References Notes F 0 ) pb µ (93) F ) A J (1 + ω 0 x + ω 0 x ) ( -η η J 1J + D D(1 + ω 0 x + ω 1 x )) (94) F 3 ) A J η J D (1 + ω 0 x + ω 0x ) ( F + A J η J D D(1 + ω 0 x + ω 1 x )) (95) (1) Nonequilibrium Vibrational Kinetics; Capitelli, M., Ed.; Springer- Verlag: New York, () Liu, Y.; Sakib, F.; Vinokur, M. Pys. Fluids A 1990,, (3) Pekeris, C. L. Pys. ReV. 1934, 4, 98. (4) Tietz, T. J. Cem. Pys. 1963, 38, (5) Natanson, G. A. Pys. ReV. A 1991, 44, (6) Hua, W. Pys. ReV. A 1990, 4, 54. (7) Rosen, N.; Morse, P. M. Pys. ReV. 193, 4, 10. (8) Morse, P. M. Pys. ReV. 199, 34, 57. (9) Manning, M. F.; Rosen, N. Pys. ReV. 1933, 44, 953. (10) Levin, E.; Partridge, H.; Stallcop, J. R. J. Termopys. Heat Transfer 1990, 4, 469. (11) Pack, R. T. J. Cem. Pys. 197, 57, 461. (1) Wu, J.; Alassid, Y.; Gursey, F. Ann. Pys. 1989, 196, 163. (13) Porter, R. N.; Raff, L. M.; Miller, W. H. J. Cem. Pys. 1975, 63, 14. (14) Nayfe, A. H. Introduction to Perturbation Tecniques; Jon Wiley Sons: New York, (15) Le Roy, R. J. LEVEL 5.1: A Computer Program for SolVing te Radial Scrödinger Equation for Bound Quasibound LeVels Calculating Expectation Values Franck-Condon Intensity Factors; University of Waterloo Cemical Pysics Researc Report CP-330R-199. (16) Appadoo, D. R. T.; Le Roy, R. J.; Bernat, P. F.; Gerstenkorn, S.; Luc, P.; Verges, J.; Sinzelle, J.; Cevillard, J.; D Aignaux, Y. J. Cem. Pys. 1996, 104, 903.