Collisional Energy Transfer between Electronic and Vibrational Degrees of Freedom BY P. G. DICKENS, J. W. LINNETT AND 0. SOVERS Inorganic Chemistry Laboratory, Oxford Received 18th January, 1962 A formal theory €or the collisional transfer of energy between electronic and vibrational degrees of freedom is outlined. Calculations are made with an ultra-simplified form of the interaction potential for the systems : (a) Wls) + H(1 s) -932s) + H(I $1, (b) Hg(3P1)+ BC +Htd3po) + BCt, (c) Na(2P) + BC -+Na(2S) + BCt. It is demonstrated that on the basis of the simplified interaction potential assumed, the exchange transfer between electronic and vibrational degrees of freedom is generally a very inefficient process, except for a near coincidence of vibrational and electronic energy levels.Conditions under which such resonance effects can be important are discussed. The object of this paper is to investigate the part played by the uptake of vibra- tional energy in the quecching of electronic excitation energy and in related processes. In particular, it is proposed to examine the conditions under which resonance effects such as those reported by Zemansky 1 might be expected. The quenching reactions and Hg(3P1) + BC+Hg(3Po) + BCt Na(2P) + BC-+Na(2S) + BCt where t represents a vibrationally excited molecule, are taken as examples. 1. GENERAL FORMULATION The co-ordinates employed in the description of a collision of an atom A with a diatomic molecule BC are as follows. r is the distance between the centres of mass of the colliding systems, r1 repre- sents an electronic co-ordinate of the atom, q is the normal co-ordinate for the vibra- tion of BC, do is the equilibrium distance between B and the centre of mass of BC.The axis of BC is assumed to lie along r with B nearest to A. For simplicity, the electronic states of the atom are assumed to be spherically symmetric and are described by the wave functions &(rl). The wave equation describing the collision process is where 4(rl) and ~ ( q ) are the corresponding wave functions for the internal motions concerned, Eo is the total internal energy of the isolated systems initially and H,, and H, are the corresponding Hamiltonians. Hereafter, $(rl,q) is written for 52P. G. DICKENS, J . W. LINNETT A N D 0. SOVERS 53 $(rl)~(q). p is the reduced mass of the colliding systems = mAmBC/(mA+mBC), and uo is the relative velocity of approach. V(r,rl,q) is the total interaction potential for the two systems averaged over the electron co-ordinates of BC.Rotation of BC is ignored. For a change in internal states of 0-n, a solution of eqn. (1) is sought of the form such that Ro+ exp (ikor cos O)+fo(6,$) exp(ikor)/r, r+ca and (4) Rn+f n(e9 4) ~ X P ( iknr)ir9 r+m in order that Ro should represent an incoming plane and outgoing elastically scattered spherical wave, and R, should represent an inelastically scattered spherical wave.2 ki is the wave number 2npvi/h, where vo and un are the relative velocities of approach and departure respectively (subsequently k is written for k, to simplify notation). The differential inelastic cross-section for the transition O-+n, on (6,00), is given by an(07u0)do = (k/ko) Ifn(e,6> l2 do, ( 5 ) where dw is an element of solid angle.The total inelastic cross-section, a,(u,), is given by integration of (5) over the surface of the unit sphere. Substitution of (3) into (1) leads to the set of equations where n Vin = V(r,rl,q)$i$nd?, J (7) and J. . . dz’ represents integration over the internal co-ordinates rl and q. Two common approximate methods available for the solution of the eqn. (6) are (a) the Born approximation, and (b) the method of distorted waves? (a) is suitable for very high energies of approach and (b) is valid when the off-diagonal elements Vff are < the diagonal elements Vti. A great simplification is found if V(rl,q,r) can be written in the form V(r)V(rl)V(q), in which case Yon can be written as uonV(r), where U0n is JV(r1)V(q)$o$,dz’. For actual molecules, the true V(rl,q,r) is an impossibly complicated function to work with, save in the very simplest cases,3 and progress can only be made by finding a simple but fairly accurate approximate form for V(rl,q,r). 2.A SIMPLE APPROXIMATE FORM FOR THE INTERACTION POTENTIAL For close distances of approach Voo(r) may be expected to have the form Vo exp (- ar),4 where a is a constant. It is proposed to use for an approximate interaction potential (between atom A and atom B) a function of the form V = Vi (1 + prl)exp( - ad), (8) where d is the (instantaneous) distance between atoms A and B and p is a parameter with dimensions of reciprocal length and magnitude of ~ 1 0 8 cm-1.The physical model underlying this assumption is that around atom A is pictured a spherical shell of electron density, of radius r1, the position of which produces a modulation of the interaction potential for a fixed value of d. d can be written in the form,54 COLLISIONAL ENERGY TRANSFER d = r - d o - x ~ , where XB is the vibrational amplitude of B and is equal to Aq where A is constant and q is the normal co-ordinate. Hence, eqn. (8) can be rewritten V = Vo (1 + prl)exp( - ar) exp(aAiq) = V(r)V(q) V(rl). (9) Voo x Kn w V, exp( - ar), and Vonm V, uonexp( - ar). (10) As a consequence of the form of (9) it follows that Hence in this model there can be no crossing of potential-energy curves and thus it should only be applicable to systems in which the two electronic levels concerned are well separated in energy from one another and also from any other neighbouring states.The formal analogy with vibration-vibration energy transfer is immediate and the relevant formula of Witteman 5 and Herzfeld and Litovitz 6 can be applied. To test the validity of the form of the interaction potential (8), the collisional excitation of the 2s level of a hydrogen atom by another ground-state hydrogen atom was considered. Bates and Griffing3 have applied the Born approximation using the exact interaction potential, in the calculation of the total inelastic cross-section a,(vo) for the process exchange effects being neglected. Use of the Born approximation with the inter- action potential (8) leads to an expression for the differential inelastic cross-section of H( 1 s) + H( 1s) + H(2s) + H( 1 s), an(8,vo) = (k/ko)(2p/h2)2u&, where K = I K I, and K = k-ko is the vector difference between the final and initial wave vectors k and ko.That is, K2 = ki+k2-2kko cos 8. One finds 5 for the total cross-section Conservation of energy requires that where AE = En- Eo, and En and EO are the total internal energies of the n and 0 states respectively. To compare the two solutions, the parameters VO, a, and uon must be fixed, where kg - k2 = 2pAE/ii2 Bates and Griffing's calculated results are plotted in fig. 1, together with those derived from function (12). The value of a was chosen so that the maxima of the two curves coincide (the position of the maximum of (12) depends only on a).The product ( Vouon)2 was chosen to give the same values of a, at the maximum. The correspond- ing values were a = 3 x 108 cm-1, (VO~O,)~ = 1.22 x 10-18 erg2. If p-1 be taken as 1 A and 40 and $, are assumed to be hydrogen 1s and 2s functions respectively, then u&wO.O9 ; hence for internal consistency V; must be taken as 1.4 x 10-17 erg2. The absolute values of a and VO so chosen appear reasonable in comparison with other intermolecular force constant data.4 Of greater importance, however, is that the form of the interaction potential (9), which allows a separation of variables, leads to a reproduction of the salient features of electronic excitation in atomic col- lisions, namely, the sharp initial rise in the inelastic cross-section with increasingP .G. DICKENS, J . W. LINNETT AND 0. SOVERS 55 energy of approach at low incident energies followed by a much slower fall-off in cross-section at high energies.2 In the remaining sections we shall not be concerned with the absolute value of the electronic matrix element in U0n and this term can be looked upon as a scaling factor when considering the relative changes in cross-section for a given electronic energy change. loglo [incident energy (keV)] FIG. 1 .-Plot of cross-section against incident energy for collisional excitation of the H(2s) level : (I) present work, (11) Bates and Griffing (ref. (3)) 3. APPLICATION TO THE QUENCHING OF MERCURY RADIATION BY DIATOMIC MOLECULES For the low energies of approach encountered under gas kinetic conditions the Born approximation is not applicable.The method of distorted waves is more suitable since Von = Vo u,,exp(-ar)< Voo,T/nn. Making use of the analysis given by Witteman,s an expression for the total inelastic cross-section is found to be m o,(v,) = ( 4 n / k k ~ ) ( 2 p / ~ 2 ) 2 ~ & C (21 + 1)A&, l = O where 00 A,, = J v(r)FlkoFZkdr* Flko and Flk are auxiliary functions which satisfy the equation with the associated boundary conditions, [d2/dr2 + k2 - 1(1+ l ) / r 2 - (2p/fi2)Vnn(r)]Flk = 0, Flk--+ sin (kr-&En+tjlk) and Fzk = 0 at r = 0. r-+ 03 I is the usual angular momentum quantum number of the central field problem, and V(r) = Gn(r) = Vo exp( - ar).56 COLLISIONAL ENERGY TRANSFER An exact analytical expression can be found for the first term in the summation in (13).A convenient approximate expression for the complete sum as given by Witteman is where O0 = 27rko/a, 8 = 2nk/a and u$, and u$k are the matrix elements of Y(r1) and V(q) between the initial and final electronic and vibrational states respectively.* Conservation of energy requires ii2(k; - k2)/2p = AEvib-AEel = AE, where A&ib is the vibrational quantum gained or lost by the diatomic molecule and AEel is the electronic quantum lost or gained by the mercury atom. AE is the net energy transferred to translation. Negative values of AE correspond to an increase in relative translational energy. The total inelastic cross-section appropriate to a particular temperature is ob - tained by averaging a,(uo) over a Maxwellian distribution of velocities of approach, that is, for net deactivation Q* = +(p/kT)2 Jmcn(vo)v: exp ( - pvg/2kT)dvo0 (17) 0 The related cross-section for activation NUMERICAL CALCULATIONS Expression (17) was used in the calculation of the inelastic cross-section for the process, where * refers to electronic excitation and t to vibrational excitation ; AE is the net energy transferred to translation. Qd was calculated for a hypothetical series of diatomic molecules BC of the same mass but with different vibrational quanta.The electronic quantum removed from the excited mercury atom was taken to be 0.219 eV (23P1+23Po), and the vibrational quantum taken up by BC was allowed to vary in the range 0.219fO-2 eV. There appeared to be no sound a priori method for the estimation of the range of the interaction potential, a-1, and calculations were therefore made for several values of a employing the sort of magnitude used in the calculation of vibrational relaxation times in gases.6 A reduced mass of 25 a.m.u.was used since this value approximates those appropriate to the Hg + CO, N2, NO systems. Q d was calculated for 300°K and the integral in eqn. (17) was evaluated numerically. Hg** +BC*Hg* +BCt +AE, (18) Fig. 2 shows a plot of loglo Qi against AE, where and AE is the resonance defect previously defined. Qi is measured in units of 10-16 cm2. The quantity (u$,)2 will remain effectively constant for a series of molecules with the same mass but different vibrational frequencies ; (u$:)2 will * Current work on the numerical solution of eqn.(15) suggests that the approximation (16) considerably underestimates the value of a,(v,).P. G. DICKENS, J . W. LINNETT AND 0. SOVERS 57 vary with vibrational frequency - l / h which, at least in the range near AE = 0, is a much slower vaiiation than that of Q&. Hence, the general features of fig. 2 may be used in a discussion of the variation of Q d with the resonance defect for a one-quantum vibrational change in BC. The form of the curve of log Q d against AE is typical of a resonance effect, a consequence of which is that electronic-vibration exchange, involving the excitation of a single vibrational quantum, is an extremely improbable process except for those cases in which the vibrational quantum involved lies very close in energy to the AE (eV) FIG.2.-Plot of relative cross-section Qi against resonance defect (AE) for Hg+BC. electronic quantum removed (AE,l/hc = 1770 cm-1). Those exchanges are most probable for which a minimum of energy is transferred to translation. This con- clusion is in accord with experimental experience.8 The order of efficiencies of the diatomic molecules, N2, CO and NO found in the quenching of 3P1-+3Po mercury radiation is NO> CO> N2.8 This is understandable since this is the order of in- creasing resonance defect (?NO = 1904 cm-1, 9 ~ 0 = 2170 cm-1, C N ~ = 2360 cm-1). The relative efficiencies read off from fig. 2 (for a = 5 x 108 cm-1) are QNO = 26Qco = 250Q~,, those found experimentally are QNO = 6Qco = 129 QN,. Values for other a’s are shown in table 1. The calculated result is very sensitive to the value of a chosen.Quenching of the same radiation by inert gas atoms would be predicted by this calculation to be quite negligible since such atoms have no energy levels close in value to the electronic quantum removed and hence all the excitation energy must be transferred to translation (AE = -0.219 eV). This conclusion is again in agreement with experiment; inert gas atoms are invariably found to be inefficient quenchers .4 To illustrate the relative efficiencies of multi-quantum transfers of vibrational energy, calculations were made for a series of hypothetical diatomic molecules with fundamental vibration frequencies of 500, 1000, and 1500 cm-1 respectively. Relat- ive total inelastic cross-sections for reaction (1 8) were computed corresponding to the58 COLLISIONAL ENERGY TRANSFER uptake of 1, 2, 3, etc., quanta of vibrational energy.S.H.O. matrix elements (u&~)Z were calculated. The results are shown in table 2. It is clear that there are two opposing effects : (a) a multiple quantum jump, which leads to a small AE, provides a favourable translational factor Qi, but (b) a multiple quantum jump is associated with a much smaller vibrational matrix element than is a single quantum process. In general, factor (a) appears to be the more important; the maximum cross-section TABLE 1-RELATIVE CROSS-SECTIONS FOR THE PROCESS Hg(3P1)+ BC+Hg(3Po)+ BC? molecule loglo(u~b)z loglo Q~(AZ) loglo ~ i ( U ; ; i ; ) ) ' ( ~ z ) rel. cross-section 3 N2 - 2-64 - 3.50 -6.14 1 co - 2.60 - 1.85 - 4.45 49 NO - 2.55 + 1.20 - 1.35 6 . 2 ~ 104 5 N2 - 2.20 - 1.50 - 3.70 1 co - 2.16 - 0.55 - 2.71 9.8 NO - 2.10 + 0-80 - 1-30 250 7 N2 - 1 *90 - 0.70 - 2.60 1 co - 1.87 -0.15 - 2-02 3.8 NO - 1.81 + 0.60 - 1.21 25 TABLE 2.-RELATIVE CROSS-SECTIONS, Qi, FOR TRANSFER OF A H@P1 + 3 P ~ ) ELECTRONJC QUANTUM TO THE TZTH VIBRATIONAL LEVEL OF THREE HYPOTHETICAL MOLECULES AE, eV vibrational frequency of molecule, cm-1 500 0 -0.219 1 -0.157 2 - 0.095 3 - 0.033 4 + 0.029 5 + 0.09 1 lo00 0 - 0.21 9 1 - 0.095 2 + 0.029 3 +0.153 1500 0 -0.219 1 - 0.033 2 +0*153 p = 25 a.m.u., loglo Qi(A2) - 9-50 - 6-90 - 3.90 - 0.35 + 0.35 -2.15 - 9.50 - 3.90 -415 - 9.50 - 0.35 -4.15 + 0.35 a = 5 A-1, 0 - 9.50 - 1.47 - 8.37 - 3.24 - 7.14 - 5.19 - 5.54 - 7.26 - 6.91 - 9.43 - 11.58 0 - 9-50 - 1.77 - 5.67 - 3.84 - 3.49 - 6.09 - 10.24 0 - 9.50 - 1.95 - 2.30 - 4.20 - 8.35 T= 300°K.will derive from a process in which AE is minimized. However, for the three vibrators considered, the smallest AE in the particular case results from the transfer of respectively 3, 2, and I vibrational quanta. The relative cross-sections for these most favourable transfers fall in the ratios 1 : 1.1 x 102 : 1.7 x 103. That is a single quantum jump which can also make AE very small will produce the largest cross- section. Anharmonicity will tend to make the differences between the vibrational matrix elements for single and multi-quantum jumps smaller (see 5 4) and thus cause the requirement of near resonance to become more rather than less important.P . G . DICKENS, J .W. LINNETT AND 0. SOVERS 59 4. QUENCHING OF SODIUM RADIATION BY DIATOMIC MOLECULES The quenching reaction Na(2P) + BC+Na(2S) + BC', (19) differs from (18) principally in that a much larger electronic quantum (2 eV) is involved. From the discussion in 5 3, it is clear that use of the simple interaction potential (8) will lead to a very small cross-section for this process. This is because any vibrational transition which makes AE sufficiently small, and hence Q; large, will be offset by a very small vibrational matrix element for a multi-quantum jump (a 0-7 vibrational transition would be needed with N2 to bring AE-0). The results of a calculation displayed in table 3 demonstrate this. The calculation was made TABLE 3 RELATIVE CROSS-SECTIONS, FOR TRANSFER OF A Na(2P-tzS) ELECTRONIC QUANTUM TO THE IZTH VIBRATIONAL LEVEL OF A N2 MOLECULE n loglo Q;(u ::)' (A') Morse S.H.O.AE, eV log10 ~J(8.z) S.H.O. Morse 0 1 2 3 4 5 6 7 8 9 - 2.095 - 1.804 - 1.517 - 1.234 - 0.954 - 0.677 - 0.404 -0.135 +0*131 + 0.393 - 18.10 - 15.95 - 13.65 - 11.40 - 8.85 - 6-25 - 3.25 + 0-20 + 0.40 - 2.30 0 - 204 - 4.38 - 6-90 - 9.54 - 12.28 - 15.10 - 17.99 - 20.93 - 23.93 0 - 2.02 - 3.79 - 5.40 - 6.88 - 8.26 - 9.57 - 10.81 - 11.98 - 13.11 - 18.10 - 17.99 - 18.03 - 18.30 - 18.39 - 18.53 - 18.35 - 17.79 - 20.53 - 26.23 - 18.10 - 17.97 - 17.44 - 16.80 - 15.73 - 14.51 - 12.82 - 10.61 - 11.58 - 15-41 for a Na+N2 collision at 2500°K with a taken as 6 x 108 cm-1. Vibrational matrix elements were calculated both for a S.H.O. and for a Morse oscillator. The effect of anharmonicity is to increase uZb for the 0-7 transition by a factor of about 104, but even so the resulting cross-section would be extremely small ( m 10-10 A2).This result is not in agreement with experiment since N2 is found to be a fairly efficient quencher of Na(2P-G) radiation.ss9 A reason for this discrepancy is not hard to find. The form assumed for the interaction potential in (8) requires V&m Vnn. Hence for all values of r, where En and EO are the total internal energies in the final and initial states respec- tively. Hence, a large cross-section can only be found within the framework of this assumption for cases in which the vibrational energy taken up, preferably in 0-1 quantum change, is rn Suppose on the other hand VOO and Vnn can differ considerablyover some range of Y, and suppose further that En - EO in the absence of any vibrational change is large, that is AEel>hvvib.Then for some region of Y, (and hence for which En-EowO). may become very small, equal to 6, say. If 6=hvvib, an efficient transfer involving the uptake of a single vibrational quantum would then be possible, although En - Eo> hvvib. Such a situation is equivalent to the near crossing of the potential energy60 COLLISIONAL ENERGY TRANSFER curves corresponding to the initial and final electronic states. That is for cases in which VOO and Vnn are very different, no correlation is to be expected between the efficiency of electronic-vibration energy transfer and the near coincidence of a vibra- tional energy level with the energy difference between initial and final electronic states.This correlation might be expected only where VOO- Vnn for all r and hence where there is little tendency towards a crossing of the potential energy curves. Such a situation seems to arise in the quenching of H~(~PI-'~Po) radiation but not for the quenching of Na(ZP4S) radiation. A comprehensive theoretical treat - ment must be based, therefore, on a detailed knowledge of the relevant potential energy surfaces for each collision pair. These data are rarely available. The simple interaction potential used here, though incomplete, does enable one to relate the particular case of electronic-vibrational energy transfer to general scattering theory and to look at the factors affecting efficiency of transfer in these terms. We wish to thank the Royal Society and the Imperial Chemical Industries for providing calculating machines. P.G.D. and O.S. thank the Pressed Steel Company and the National Science Foundation (U.S.A.) respectively for the award of Post- Doctoral Fellowships. 1 Zemansky, Physic. Rev., 1930, 36, 919. 2 Mott and Massey, The Theory of Atomic Collisions (O.U.P., 1949). 3 Bates and Griffing, Proc. Physic. SOC. A , 1953, 66, 961. 4 Hirschfelder, Curtiss and Bird, The Molecular Theory of Gases and Liquids (Wiley, New York, 5 Witteman, J. Chem. Physics, 1961, 35, 1. 6 Herzfeld and Litovitz, The Absorption and Dispersion of Ultrasonic Waves (Academic Press, 7 Jackson and Mott, Proc. Roy. SOC. A, 1932, 137,703. 8 Laidler, The Chemical Kinetics of Excited States (O.U.P., 1955). 9 Clouston, Gaydon and Hurle, Proc. Roy. SOC. A, 1959, 252, 143. 1954). New York, 1959).