3‘, use RA R y ‘ Mmbgm State 3 ”@WW {,4 ovsnoua FINES: 25¢ per new per item ‘1]:I.\\\ A n - (”3‘ RETURNING LIBRARY MATERIALS. Phce in book return to remove charge from circulation records ) \ -« 3‘ _ :- 1‘ W. EOE, 1 2- 2!” N © 1981 KEITH L . PETERSON All Rights Reserved THEORETICAL CALCULATIONS OF POWER—BROADENED MICROWAVE LINESHAPES By Keith L. Peterson A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1981 V.‘ I 3. ‘/ ' (‘0) 'C ‘— (Iy' ‘ ' ‘S .> ‘ /,'/~.J ‘2 ABSTRACT THEORETICAL CALCULATIONS OF POWER—BROADENED MICROWAVE LINESHAPES By Keith L. Peterson The traditional method of analyzing high-power micro- wave transitions is to assume the lineshape is a sum of Lorentzians - one for each m-component pair connected by the microwave radiation. This assumption is tested by using extended Anderson Theory and infinite order energy sudden approximations to calculate relaxation cross sec- tions. These cross sections appear in the power-broadening term of a recently derived expression for high-power line- shapes which correctly takes into account the degeneracy of the rotational levels. The theoretical lineshape obtained with the relaxation coefficients is fit to a sum of Lorentz- ians using a computer program developed for fitting experi- mental lineshapes. The goodness of fit is a measure of the validity of the sum of Lorentzians approximation. The results for the J = 2 + 1 transition in DOS and the (J,K)= (3,3) inversion transition of NH3 show that in both modified Keith L. Peterson Anderson Theory and energy sudden approximations the value of Tl/T2 obtained by fitting to a sum of Lorentzians is a good approximation to the calculated values of Tl/T2' The values of T1 calculated above were based on the assumption that the population of the two levels connected by radiation remains constant. If this assumption is removed (inclusion of n-level effects) in the calculation of Tl’ the same conclusions as above may be drawn. In addition, however, the Tl/T2 ratios are brought into better agreement with experimental results. As expected, the n-level effects are more pronounced for OCS than for NH3. The sudden approximation has been applied to four-level double resonance experiments in NH3. The analysis Justifies several of the assumptions used previously in the analysis of these experiments. However, numerical calculations agree only qualitatively with experimental results. The sudden approximation gives values for cross sections that are too large. This is due to neglect of internal state energy differences. Two energy corrections to the sudden approximation are discussed. Their practical applica- tion to the calculation of relaxation parameters requires a complex numerical integration, a reversion to a hard sphere cutoff procedure, or an approximation that is difficult to justify rigorously, but which enables analytical evaluation of a required integral. A model of relaxation commonly used in NMR is transformed Keith L. Peterson to a spherical tensor basis. After introducing a correla- tion function for the intermolecular potential, the form of the relaxation parameters replicates that of the modified Anderson Theory. If a reasonable correlation function can be obtained the model offers an extremely simple method of calculating relaxation coefficients. Phase conventions are established for matrix elements in a previous work on microwave lineshapes. A previous derivation of steady-state absorption by a linear rotor in a static electric field is extended to symmetric tops with inversion. ACKNOWLEDGMENTS The author thanks Dr. R. H. Schwendeman for some valuable discussions concerning the computational aspects of this work and for allowing the author virtually complete freedom to work independently on the theoretical portions of this dis— sertation. Partial support of the National Science Founda- tion is gratefully acknowledged. Among other Chemistry Department personnel, S. Sand— holm and J. Leckey were notable for their willingness to provide enlightening discussions. The friendship of H. Larsen has been invaluable as has that of C. Chappelle whose patience, kindness and ability to create diversions provided my most pleasant moments. Finally, the members of my immediate family deserve a special thanks for providing their support and encourage- ment. ii TABLE OF CONTENTS Chapter Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . . . . . . . . . . . . . . X INTRODUCTION. . . . L . . . . . . . . . . . . . . . 1 CHAPTER I. CURRENT STATUS OF POWER- BROADENED MICROWAVE LINESHAPES AND RELAXATION PARAPflETERSo o o o o o o o o o o o o o 5 A. History and Summary of Steady- State Microwave Lineshapes. . . . . . . . . . . 5 B. History and Summary of the Multipole Relaxation Coefficients . . . . . . . . . 2A C. Summary of Available Methods for Calculating the Scattering Matrix D. Choice of Methods for Calculating AK. . . . 30 CHAPTER II. EXTENSION OF ANDERSON THEORY To THE CALCULATION OF AK . . . . . . . 32 A. Relation of the Liu-Marcus AK to Anderson Theory . . . . . . . . . . . . . . 32 B. AK in Terms of Anderson's P Matrix. . . . . 38 C. Evaluation of the P Matrix for Multipole-Multipole Intermolecular Potentials. . . . . . . . . . . . A6 D. Tensor Order Dependence of the AK . . . . . 52 E. Some Additional Properties of the Anderson AK . . . . . . . . . . . . . . . . 58 iii Chapter CHAPTER CHAPTER E.l. Expansion of the P Operator in Irreducible Tensors. E.2. Sudden Approximations in Anderson Theory Numerical Results for OCS and NH3 Systems . . . . . Numerical Results for OCS and NH3 Systems Within the Anderson Sudden Approximation . . . . . . . . III. A SIMPLE MODEL FOR THE RELAXATION COEFFICIENTS. . . . . . . . . . IV. APPLICATION OF AN ENERGY SUDDEN APPROXIMATION TO THE CALCULATION OFAK.............. Derivation of Equations and Numerical Results . . . . . . . . . . . Application of the Sudden Approxima- tion to Four-Level Double Resonance Experiments . . . . . . . . . . . Energy Corrections to the Sudden Approximation Scattering Matrix CHAPTER V. ADDITIONAL RESULTS. . . . A. Comparison of T2 for Transitions in 3 Static Electric Field in Linear and Symmetric Top Molecules Phase Conventions for Reduced Matrix Elements. . . . . . . . . . . APPENDIX A. EQUIVALENCE OF EQUATIONS (AZ) and (A3) . . . . . . . . APPENDIX B. CONVENTIONS FOR REDUCED MATRIX APPENDIX ELEMENTS . . . . C. REDUCTION OF EQUATION (“9) TO THE ANDERSON RESULT. . . iv Page 59 61 7O 92 96 113 113 136 1A3 156 156 163 169 173 17A Chapter APPENDIX D. APPENDIX E. APPENDIX F. APPENDIX C. APPENDIX H. APPENDIX I. REFERENCES. Page DERIVATION OF ANDERSON-LIKE EXPRESSIONS FOR AK . . . . . . . . . . 176 PROOF OF RESTRICTIONS ON MULTIPOLE ORDER OF POTENTIALS IMPOSED IN APPENDIX D . . . . . . . . . . . . . . 187 MATRIX ELEMENTS OF MULTIPOLE MOMENT OPERATORS FOR ONE- ENDED AND PARITY ADAPTED SYMMETRIC TOP EIGENFUNC— TIONS. . . . . . . . . . . . . . 19o SUMMARY OF RESONANCE FUNCTIONS AND HARD-SPHERE CUTOFF CALCULATION . . . . . 193 DETERMINATION OF THE SUDDEN AP- PROXIMATION BO . . . . . . . . . . 197 ANALYTICAL EVALUATION OF IMPACT PARAMETER INTEGRATION IN THE ENERGY CORRECTION TO THE SUDDEN APPROXIMATION. . . . . . . . . . . . . 200 202 Table II III IV VI \[II VIII LIST OF TABLES Conditions on AK for the Continued Fraction Lineshape to Reduce to a Sum of Lorentzians. K-Dependence of AK for Multipole- Multipole Potentials. ’ Assumed Parameters for OCS Calcula- tions . . . . . . . . . . . . . Relaxation Parameters for OCS Relaxation Parameters for the OCS J = 2 + 1 Transition: 2—Leve1 Ap— proximation . . . . . . . Matrices of b Coefficients for Calculation of A-Level Corrections for the OCS J = 2 + 1 Transition. (J i A; K = O). . . . . . . . . Relaxation Parameters for the OCS J = 2 + 1 Transition: A-Level Effects, J i A. . . . . . . . . Summary of Tl/T2 Calculations for the OCS J = 2 + 1 Transition. . vi Page 21 55 75 76 78 81 82 83 Table IX XI XII XIII XIV XV XVUI Page State to State Relaxation Parameters for the OCS J = 2 + 1 Transition. . . . . 85 Assumed Parameters for NH3 Ca1- culations . . . . . . . . . . . . . . . . 86 Relaxation Parameters for the (J,K) = (3,3) Inversion Doublet of NH 87 3. Matrices of b Coefficients for the NH3 (J,K) = (3,3) Transition for Calculation of A—Level Corrections: J i 5, K = O. . . . . . . . . . . . . . . 89 Summary of Tl/T2 Calculations for the NH3 (J,K) = (3,3) Transition. . . . . 90 State to State Relaxation Parameters for the NH3 (J,K) = (3,3) Transition: Dipole-Dipole Potential . . . . . . . . . 91 Values of T1 and T2 for the OCS J = 2 + 1 Transition Calculated by Anderson Sudden Approximation and Dipole-Dipole Potential . . . . . . . 93 Values of T1 and T2 for the NH3 (J,K) = (3,3) Transition Calculated by Anderson Sudden Approximation and Dipole-Dipole Potential . . . . . . . . . 9A vii Table XVII XVIII XIX XX IXXI XXII Page Relaxation Coefficients for the J = 2 + 1 Transition of OCS and the (J,K) = (3,3) Inversion Transition of NH3 Calculated in the Sudden Approximation. . . . . . . . . 128 Matrices of b Coefficients in the Sudden Approximation for Calculation of A—Level Corrections. OCS J = 2 + 1 Transition (J S A; K = O) and NH3 (J,K) = (3,3) Inversion Doublet (J i 5; K = O). . . . . . . . . . . . . . 130 Relaxation Parameters in Sudden Approximation for the OCS J = 2 + 1 Transition: A-Level Effects, J i A . . . . . . . . . . . . . . . . . . 131 Summary of Tl/T2 Calculations in Sudden Approximation for the OCS J = 2 + 1 Transition. . . . . . . . . . . 132 State to State Relaxation Parameters in Sudden Approximation for the OCS J = 2 + 1 Transition. . . . . . . . . . . 133 Summary of Tl/T2 Calculations in Sudden Approximation for the NH 3 (J,K) = (3,3) Inversion Doublet . . . . . 13A viii Table XXIII XXIV Page State to State Relaxation Parameters in Sudden Approximation for the NH3 (J,K) = (3,3) Inversion Doublet . . . . . 135 Sudden Approximation Calculations of Rate Constants for Four-Level Double Resonance Experiments in NH3. . . . . . . 1A2 ix LIST OF FIGURES Figure Page 1 Energy level scheme for a four-level double resonance experiment in NH3. . . . 137 2 Qualitative plot of f(k) vs. k for a dipole—dipole potential. k = 9?. . . . 153 INTRODUCTION Spectral lines in steady-state, gas-phase microwave absorption experiments are Characterized by a shape and width. Contributions to the width may come from uncer- tainty,(l) DOppler,(2) saturation,(3) and collision(u) broadening, as well as various experimental effects such as modulation broadening,(5) collisions of molecules with sample- cell walls,(6) and beam transit—times.(7) While it is pos- sible that these various broadening mechanisms may act in- dependently, they often act in concert. One example of this is the correlation of Doppler and collisional effects such that for certain conditions the spectral width is not merely the sum of the Doppler and collisional widths but instead a more complicated function of these quantities.(8'1u) Experimental conditions in microwave spectroscopy may easily be realized where only collisional and saturation effects are important in determining the lineshape. This dissertation will be concerned exclusively with these two effects. As will be seen later, the term in the lineshape expression that describes the saturation broadening is a fUnction of the incident microwave power and various col- Llisional relaxation cross sections. If the incident micro- Inave power is known, measurement of the linewidth is capable of giving information about the cross sections. Thus, line- width and/or lineshape measurements are probes of the dynamics of molecular collisions. This information can be used to gain information about intermolecular energy transfer and intermolecular potential energy surfaces. Steady—state absorption experiments are complementary to a host of other experiments which also yield information concerning col- lision dynamics. These include microwave-microwave(15-20) (21—26)double resonance, fluores— (31-38) (A0) and infrared-microwave cence,(27-3O) beam maser, molecular beam,(39) microwave transient effects, and transport proper- ties.(ul’u2) On a more practical level, lineshape measurements are useful as temperature probes in gases and plasmas,(u3) and in the study of planetary atmospheres. Carbon monoxide is pressure-broadened by carbon dioxide in the atmosphere of (AA) while in the Jovian atmosphere methane is (us—A8) Mars, broadened by several gases. Lineshapes have seen application in pollution analysis and are also useful in determining Optimal conditions for gas-laser opera— tion.(50-52) In view of the potential application of saturation- broadened and collision-broadened microwave transitions, it is desirable to have at hand a correct method of analyzing such lineshapes. It is well known that low-power lineshapes, i.e., those lineshapes where saturation effects are not important, can be analyzed in terms of a single Lorentzian whose width is proportional to a polarization relaxation cross sec- tion.(53) The inverse of this cross section is denoted T2 in analogy with NMR relaxation. The traditional assumption for analyzing high-power lineshapes where saturation ef- fects are important is that the lineshape can be expressed as a sum of Lorentzians — one for each m-component pair con- nected by the microwave radiation - each of which has its own T1 and T2.(5u) (T1 is a population relaxation time, again in analogy to NMR relaxation.) These assumptions are difficult to justify theoretically. Recently, an expression for power-broadened transitions has been derived which cor- rectly accounts for the degeneracy of the rotational levels.(55) The term which describes the power-broaden- ing contains various relaxation cross sections. The main thrust of this dissertation is the following: Procedures are develOped for the theoretical calculation of the cross sections in the exact lineshape expression. The procedures involve either an extended Anderson theory or an infinite order sudden approximation. Lineshapes are computed from the theoretical cross sections and are fit to a sum of Lorentzians(5u) by using a computer program developed to analyze experimental lineshapes. The goodness of fit is 6i measure of the validity of the sum of Lorentzians ap- IIPoximation. In addition to the lineshape calculations a model of relaxation commonly used in NMR is adapted to the microwave absorption case and is shown to be in quali- tative agreement with the extended Anderson theory results. Also, the effect of two adiabatic corrections to the infinite order sudden approximation is developed. Finally, because of the interest in microwave-microwave double resonance experiments in collisions causing the rotational angular momentum to change by two or more units, the infinite order sudden approximation is developed for use in calculating cross sections for such collisions. CHAPTER I CURRENT STATUS OF POWER-BROADENED MICROWAVE LINESHAPES AND RELAXATION PARAMETERS A. History and Summarygof Steady-State Microwave Line- shapes In any discussion of steady-state microwave line- shapes it is necessary first to distinguish a two—state approach from a two-level approach. In the absence of fields, rotation or rotation—vibration energy levels have at least a (2J + l)-fold degeneracy, where J is the total angular momentum. This degeneracy is usually termed m- degeneracy (m = -J, -J + l, . . . J) and is a result of the 2J + 1 possible projections of the angular momentum on a space-fixed z axis. In the sum of Lorentzians approxima- tion, each m-component pair connected by microwave radia- tion is treated as a two-state system. Equations of motion for a two-state density matrix are solved in the steady- state to obtain an expression for the absorption coefficient. Degeneracy of rotational levels is accounted for by summing over the m-component pairs. In the two-level approach, ‘the degeneracy is considered from the beginning by forming appropriate linear combinations of density matrix elements. Equations of motion for these linear combinations are solved in a manner analogous to the two—state case to obtain the absorption coefficient. The following summary of the two-state approach follows a review by Flygare and his coworkers.(5u) An incident radiation field e = E(z,t)cos[wt—kz+¢] (l) induces a macroscopic polarization P = PC cos[wt-kz+¢] + PS sin[wt-kz+¢] (2) In Equations Cl)and (2), z is a Spatial coordinate, t is time, w is the angular frequency of the field, k is the wave vector, ¢ is a phase, and PC and P8 are components of the polarization. By starting with the wave equation,(56) 828 _ 1 326 A11 3213 2 — j ——2' + H 2 3 (3) BZ C 3t C at it is possible to deduce that the absorption coefficient 0 as a function of frequency can be written as A P 1T0) a(w) = C .5. (A) (C is the speed of light.) It is assumed that the radiation interacts only with the dipole moments of the molecules. Then, for a sample of dipoles the polarization is a macroscopic dipole moment and can be written as P = Ntr(uo) (5) where N is the number of dipoles, u is the dipole moment, and p is the density matrix.(57) The symbol tr(x) denotes the trace of the matrix x. From Equations (2), (A) and (5) it is seen that if p can be determined, P, a(w) and there— fore, the lineshape can be obtained. The equation of motion for the density matrix is ih %% = [H,o] = Hp - DH (6) where H = HO-UE cos[wt—kz+¢]. For rigid rotors HO sup- ports the rotational levels of an unperturbed molecule. In writing H, the effect of collisions has been ignored. As in all previous two-state approaches, collisional effects will be added phenomenologically. When the two— level approach is considered, collisions will be treated Imore rigorously. Diagonal elements of the density matrix, 011, are proportional to the population of state i. To :see the meaning of the off-diagonal elements consider a two-state system with states 1 and f. The polarization of this system is O u o. p. P m tr(up) = tr{( if)( 11 lf)} 0 ofi Off “ti = “ifpfi + Ufipif (7) (The assumption has been made that the diagonal matrix elements of u are zero.) That is, the off-diagonal density matrix elements are related to the polarization. If Equa— tion (6) is written out in detail the following equations for density matrix elements in the interaction representa- tion(58) result. 1P ‘3?‘ = Eufi(pff-pii) " “Awpfi . 391i 3h ‘St' = ’Eufipif + Epfiuif (9) (The interaction representation is simply a device which allows the Operators to carry the time-dependence of Ho°) The rotating wave approximation, which assumes that the experimental apparatus has a limited ability to follow rapid time variation, has also been made. In Equation (8) Am is the difference between the frequency of the applied radiation and the resonant frequency of the spectral transition 1 + f. Equations (8) and (9) and the correspond- ing equations for Off and pif are equivalent to the follow- ing set of equations dPC EH7 + APPS = 0 where AN is the population difference between states i and f. (10) (11) (12) It is traditional to introduce the effects of collisions by phenomenologically adding relaxation times so that Equa- tions (10) - (l2) become dP __2 dt (13) (1A) (15) 10 A single relaxation time, T2, has been attributed to both P and PS, while a different relaxation time, T1, has been c attributed to the population difference. The presence of -ANO in Equation (15) is a statement that the perturbed population difference decays to an equilibrium population difference, ANO. Equations (13) - (15) can be solved in the steady- state by setting the time derivatives equal to zero. This results in an expression for PS: UifE(l/T2) ( 6) P m l s 2 1 2 2 2 T1 Ps is related to the absorption coefficient C(w) by Equa- tion (A) so that Equation (16) is essentially the lineshape in the two-state model. The sum of Lorentzian's approxima- tion consists of using Equation (16) for each m-component pair connected by radiation. If the saturation term in the denominator of Equation T (16) “IfB2(Tl) goes to zero (low-power conditions), the resulting exgression is essentially that derived by lkanVleck and Weisskopf.(59) If the assumption is made tfliat both the polarization and the population difference Imalax to equilibrium values at the same rate, i.e., T1=T2, Eqiuation (16) becomes the expression of Karplus and ll (61) (62) Schwinger,(6o) and Snyder and Richards. Townes was the first to suggest that two different relaxation times be used and this approach has been used extensively by Flygare and his coworkers in their analysis of microwave (5A) transient effect experiments. It is possible to be more rigorous in defining the ef— fects of collisions. Within the impact approximation the equation of motion for the density matrix including col- (63) lisions may be written as apfi Ifi —§E—_E“fi(pff'pii)'fiAwpfi'ifif33, Afif'i'pf'i' (1?) 391i *9 at ' ”Euripif+E“ifpfi‘if1 i Aiikkpkk (18) For the present it is sufficient to describe the A as thermally averaged products of scattering matrices. The impact approximation states that a collision is an instan— taneous event compared to the time between collisions. If the impact approximation is not valid the A are thermally averaged products of both on-shell and off-shell t matrix (elements. More precise discussions of the impact approxima- tixan are available.(6u) It will always be assumed in this dixssertation that the impact approximation is valid. It is ‘very difficult to establish rigorous limits of validity 12 for this approximation and it is almost impossible with present day methods to calculate the A when this approxima— tion is not valid. Experimentally, it is known that the low- power lineshape is Lorentzian when the impact approximation is expected to be valid. All high-power lineshapes con- sidered in this dissertation are obtained under conditions where the corresponding low-power lineshape is essentially Lorentzian. The A will be considered in great detail later as they form the principal tOpic of this work. For the moment it is sufficient to recognize that a two-state approximation applied to the A in Equations (17) and (18) (that is, restricting the summation indices f',1' and k to i and f) allows the relaxation times T1 and T2 to be expressed as 1_1 T‘ ' 2(Aiiii + Affff ' Affii - Aiiff) (19) III2 ( f'j j ) ( ) where Re(x) denotes the real part of x. Im(Afifi) gives “the line shift. The methods that will be used to calculate 1‘ are such that Im(A) E O. The above paragraphs summarize the two-state approach tn) analyzing high-power microwave lineshapes. If there 13 is no collisional coupling of the m states it is clear that each of the m-component pairs connected by radiation will evolve independently of the others and a sum of Lorent- zian's approximation is valid. This can be seen from the equations of motion for the density matrix when written to (63) include the degenerate m—states 1 3L (' m j m) = Aw (j m j m) at0 Jr ’ i rip f ’ i -E[o(Jim,jim)-O(meijfm)J . . . .,,.,, ,,.,, -1j§j§m'<<3fmaimlAlme Jim >>p(me Jim ) (21) f i i §gp = -E[o(3fm,jim) -O(jim3fm) ] -—i z <>p(j'm'j'm') (22) j'm' 1 1 (6A) ijmfjimi>> is a vector in Liouville space and is (defined by ljfmfjimi>>5|jfmf> (hereafter denoted by LM) made the first serious attempt to deal with the m degeneracy of the rota- tional levels. By forming the linear combinations j -m j j. K prim) >3 (—1) f f<2K+1>V2=(-l) ( O f i jf l)—- , (26) m.-m l which is merely an application of the Wigner—Eckart theorem. The quantum number Q is always equal to 0 if the microwave radiation is plane-polarized. The AK are defined by ‘1 Jf‘m%+Jf’mf <> =23: {-1) (2K+l) KQ ., ., . . Jr Ji K JfJi K K x A ,., . (27) m'-m'-Q IH'ml-Q f 1 fl ‘f ‘i f i \ ITKIS is a direct consequence of forming linear combinations (pf 'mefjimi>> in analogy with Equation (23), i.e ° 3 l3 J ,KQ>>= z: jr'mr 1/2 if JiK f i mfmi(-l) (2K+1)H_m - IJf mfjimi >> (28) 16 This can be inverted to give j —m J j K ijmfjimi>>=ZI(-l) f f(2K+l)l/2(mf i )|jfji;KQ>> (29) KQ f-mi—Q The Ag'i'fi are independent of the quantum number Q. This is a result of rotational invariance and has been discussed in detail by Ben-Reuven.(66) The problem with Equations (2A) and (25) is that p(K,Q) are coupled to p(jmj'm'). To get around this, LM noted that (30) Udll—J J J 1 2 f i m m -m o and assumed that in Equation (2A) the quantity J j 1 2( f i )[o(jimjim)-O(mejfm)] m m —m 0 could be replaced by l %[(2ji+l)- pii‘1pff(oo)3. Thijs assumption has been shown to be inadequate for analyzing 17 experimental data for several transitions in lSNH3.(67) This finding spawned the introduction of an alternative to the quantity Tl/T2’ namely qu/T2’ where q is related to the distributions of populations among the m states.(67) This parameter has since been shown to be an impractical method of analyzing line shapes and will not be discussed further. The next progress came when Bottcher gave a set of equations involving only p(KQ) and solved them exactly in the steady-state for the j = 1 + o transition.(68) The lineshape for this case is proportional to l (31) 2T T 2 1 2 A 2 2 l A + _ _ .— (Aw) (T2) + 9 ifE (T2 + T3) Comparison with the denominator of Equation (16) shows that the power broadening terms (the last term in the denominator are qualitatively similar. There is an additional relaxa- ‘tion time in (31), TA’ which describes the relaxation of off (20). Coombe and Snider<69> also considered the j = 1 + O traruxition and arrive at an expression for the lineshape whick1;is in agreement with that given by Bottcher.(68) Coombe and Snider also considered the general transition 3 + l +- J,(7O) The set of equations of motion for the 0(KQ) bexzomes very large as j increases. In the interest 18 of keeping the size of this set tractable Coombe and Snider assumed that all 0(KQ) with K greater than 2 could be ig— nored. This gave a set of eight equations which under two conditions reduce to a set of four equations. These give a lineshape identical to Equation (31) with the exception of numerical factors (which are a function of j) in the power broadening term. The two conditions are that plane polarized radiation be used and that the collision dynamics are the same in the j and j+l levels. This latter condi- tion is often referred to as a high-j approximation. Finally, Schwendeman<55> has derived an expression for power-broadened lineshapes which is valid for plane-polarized radiation and for any j. The expression may be used for either R branch (j = j + l + j) or Q branch (j = j + 3) transitions and does not make any high—j approximations. Thus, an expression is now available that exactly accounts for the m—degeneracy of the rotational levels involved in ‘the spectral transition. The power-broadening term is a rhinction of many AK. By calculating the AK and fitting the Iéesulting lineshape to a sum of Lorentzians, the validity cxf this approximation may be assessed. The equations of motion for the p(KQ) can be written 19 . ii 1 at pfi _ J -J. K K' 1 K' K 1 -Ep. )3{(-l)f l[<2K+1)(2K'+131/2 p (K') f1 K' O 0 j j j 11 f' i i {K' K 1 (K )} ji jfiHAff (32) j _'. K K' 1 -(-1) f Jl[(2X+1)(2K'+1)]1/2 ( ) ' o o o _ . AK 1 fif‘ip fi(K) , - iL (K) = -E— z:[ (Kw-(4)31-jf (K')J 1 at pii “fiK. pfi 0if j -j. K K' 1 K K' 1 x (-l) f l[(2K+l)(2K’+l)]l/2 < ) { } 00 O 'jfji'ji K o Iii the above p§k(K) is the equilibrium value of pkk(K), { :::} is a 6-j symbol,(71) and the label Q in pfi(KQ) has beaen.deleted because it is always zero as a consequence of pléine—polarized radiation. The following points are worth noting. The relaxation Of ea density matrix element labelled by K is governed by 20 1 AK and not by any AK for K' # K. This is a result of the assumption of rotational invariance. The only coupling to different values of K arises in the field dependent ' terms. The 3-j symbol (g g g) is zero if K + K' + l is odd. Therefore, the only values of K' in the summation over K' are K i 1. For the case of plane-polarized radia- tion this implies that diagonal matrix elements may have only even K while off-diagonal density matrix elements may have only odd K. The set of coupled equations (32) and (33) may be solved in steady-state to give an expression for the line shape. piiE(%:)AN Ps(l) m P ( ) (3A) 2 l 2 2 2 s 3 (Am) + (T5) + ufiE [PS(1)1 PS(K+2) PS(K+A) K K where *ggrij— is a function of PETXT77’ Afifi’ A1111, K K K . .A .ffff’ Aiiff and Affii' PS(K) Is the analogue ofPth§)PS s cuzcurring in Equations (2) and (A). The factor [ngITJ in Ekauation (3A) is in the form of a continued fraction and Ekluation (3A) will henceforth be referred to as the con- tiiuued fraction lineshape. Schwendeman was able to show truit the continued fraction lineshape reduces to a sum of Lorwentzians for the conditions on AK given in Table I.(55) 21 Table 1. Conditions on AK for the Continued Fraction Line- shape to Reduce to a Sum of Lorentzians. R Branch K All Afifi are equal All AK 1 iiii are equa K All Aiiff = O Q Branch All AK are equal fifi * All Tl-like relaxation times are equal 22 This set of conditions may not be the only set of conditions for which the continued fraction lineshape reduces to a sum of Lorentzians. It will be useful later to consider now the p(KQ) K and AK in slightly more detail. The details of A will be given in later chapters. For the present it is satisfactory to give a brief qualitative discussion. The p(KQ) are variously known as state multipoles or statistical tensors.(7l) They were first introduced by Fano.(72) While their applica- tion in chemistry has been rather limited, their applica- tion in physics includes discussions of the production of polarized particles in nuclear reactions,(73’7u) the re- distribution of resonance radiation,(75) angular distribu- tions of photoelectrons,(76-79) optical pumping(80‘82) and transport properties.(83’8u) Besides the previously mentioned work concerning microwave absorption, the work most closely related to this topic is that of Case et al.(85) who applied the state multipoles to the problem of determin- ing rotational state distributions in fluorescence experi- ments. The p(KQ) formed by Equation (23) are said to be the matrix elements of an irreducible tensorial basis. All Of the 2K + l componentsfku'a given K form an invariant set:,(66) i.e., under rotations they transform only among thennselves. It is easy to show from Equation (23) that 011(00) = ;o(Jimjim)//2ji+1 . (35) 23 This follows from the relation J J O _ _ ( ) = (-1)3 m<2i+1) 1/2 . (36) m-m 0 Therefore, (2ji+1)—l/2 pii(OO) is an average level popula- tion. At equilibrium the 0(Jimjim) are equal to each other for all m in which case the pii(KQ) are independent of K. A level that has nonzero pii(KQ) for K = 0 only is said to be unpolarized. A rotational level that has nonzero pii(KQ) for K > O is said to be polarized with a multipole moment, or simply moment of order K. The quantum number K may also be referred to as the tensor order. The off- diagonal p(KQ) are related to the macroscopic polarization induced by the applied radiation field. For a system of dipoles the dipole polarization may be written as P = Ntr(uo(10)) % uifofi(10) + “fioif(10) (37) in analogy with Equations (5) and (7). The state multi- 13018 has K = 1 here because u is a tensor operator of ‘tensor order 1. The A0 may be interpreted as being proportional iiii tC) the total collisional rate of transfer of molecules out Of‘ level i. The AIikk are proportional to the negative of true rate of collisional transfer from level k to level i. 2A This is a consequence of the unitarity of the scattering matrix, which may be expressed as E Aiikk E O . (38) Equation (38) simply states that the total population of molecules remains fixed. The AIikk for K > O are called mUltipole relaxation coefficients as thev describe how the diagonal elements of a state multipole of order K relaxes due to collisions, i.e., . 8 . K 1 FE P:: ~ -1 i Aiikkokk(KO) . <39) SiJnilarly, the Agifi describes the relaxation of the off— diiagonal elements of state multipoles of order K. For K ru>t equal to l the A§ifi are generalizations to arbitrary l fifi' Re3(A%ifi) is just the traditional low power linewidth, i.e. tendsor order of A This is of interest because l/J? This quantity has been the subject of considerable 2. attention. B. History and Summary of the Multipole Relaxation Qgefficients TThe first complete theory for AIifi for rotation and (53) Vibrwition—rotation levels was given by Anderson in 25 l9A9. The theory was amplified in 1962 by Tsao and Cur- nutte<86) (hereafter referred to as TC) who explicitly considered dipole—dipole, dipole-quadrupole, quadrupole- dipole, quadrupole-quadrupole, and dispersion intermolecular potentials. (The first four of these potentials will be abbreviated as U-u, u-Q, Q-u, and Q-Q.) Anderson made several key assumptions that are worth enumerating here. The most important assumption is that the impact approxima- tion is valid. As mentioned above this requires that the time of collision be negligible compared to the time be- tween collisions. This implies that only complete col- lisions need to be considered and in turn that only scatter- iJig matrices (or equivalently on—shell t matrices), and not oITf-shell t matrices are needed. Present day methods do TKDt allow calculation of off-shell t matrices for systems of‘ interest in microwave spectroscopy. The impact ap- prwoximation has been discussed in detail by Baranger(6u) wTuo also gave several expressions for estimating the validity lijnits of the approximation. Obtaining numerical esti- maixes from these expressions is almost as difficult as cal- culéiting AIifi itself so that, as explained earlier, the aSSLunption will be made here that the impact approximation is Ilalid for the conditions considered in this work. A germaral expression for the AK that does not depend on the impaxyt approximation was given by Fano<87) in 1963. An equilnalent expression, derived by different methods, was 26 given by Ben-Reuven in 1975.(88) Anderson also assumed that all molecules move along classical straight line paths. This implies that col- lisions resulting in changes in molecular internal states have a negligible effect on the trajectory. For rotational levels separated by energies : kT (Boltzmann constant times temperature), this is true. A discussion of this point may be found in Reference 89. The assumption of straight line paths has practical implications for numerical calcula- tions that are both good and bad. In the Anderson formula- tion the relaxation coefficient goes to infinity as the .intermolecular distance goes to zero. Circumventing this (iifficulty requires an artificial means of imposing uni— tnarity. The traditional method of doing this is to use a 'Wiard sphere cutoff." Details of the cutoff procedure will bee discussed later in this work and may also be found in tflie papers of Anderson<53> and TC. Baranger was apparently the first to remove the restric- ticni of classical straight line paths by treating all Peliitive molecular motion quantum mechanically.(6u) Baranger was also the first to formally exploit the cornsequences of rotational invariance. These ideas were caruried to completion by Ben-Reuven.(66) The key result or tflais work is that it is possible to write . . . K ((JIJIK'Q'lAIijiKQ>> = Af'i'fiaK'KaQ'Q (”0) 27 where Ag'i'f is defined by Equation (27). The point is i that the SK'K precludes the possibility of a sum over K' in the relaxation terms of Equations (32) and (33). K Almost all previous calculations of A have been for I Afifi’ l/T2. By far the most common calculations are those that (90-98) i.e., the cross section for low—power linewidth, employ Anderson's formulation as amplified by TC. (99) Goldflam et al. have used the close coupled (CC) and coupled states (CS) methods to calculate Raman cross sec- 2 tions Afifi for H2 perturbed by He. (The CC and CS methods (89) will be discussed shortly.) Nielsen and Gordon solved the time—dependent Schrodinger equation for a classical trajectory determined by a spherically symmetric potential. 1 l 2 and A"f'if for HCl per- They calculated Afifi’ 1,1,11, 1 turbed by Ar. The AK with K greater than one were calculated to rationalize the results of NMR relaxation and Raman (lOO) lineshape experiments. Shafer and Gordon calculated the same cross sections for H2 perturbed by He by using a CC method. Marcus and coworkers<101’lo2) have calculated 1 -l _ 1 O 0 0 o O Afifi and T1 ‘ 2(Aiiii + Affff ‘ Aiiff ' Arfii) “ iiii O - Aiiff for OCS and HCN perturbed by noble gas atoms. They used a semiclassical technique that required the calcula- A tion of complex valued trajectories. This technique was (lOA,lO5) The infinite (106) to developed by Marcus(103) and Miller. order sudden (IOS) approximation was used by Green calculate AIifi and various AIikk for OCS perturbed by 28 K iikk for a system of interest to microwave spectrOSCOpy. The noble gas atoms. This is the only calculation of A only other calculations of Agikk are those mentioned pre- viously for NMR relaxation. Finally, a few distorted wave (107) Born approximation calculations have been performed to rationalize experimental results for various transport ex— (108) periment properties. C. Summary of Available Methods for Calculating the Scatter- ing Matrix The AK are proportional to thermally averaged products of scattering matrices. The central problem in calculat- ing AK is to find a feasible method of obtaining the scattering matrix. There are a multitude of techniques for doing this. The close coupling method (CC) is the essentially exact, completely quantum mechanical method of (109’110) As it is calculating S, the scattering matrix. usually formulated the method consists of solving a set of N coupled second order differential equations, where N is the number of states included in the calculation. For rotational scattering the number of states increases rapidly with increasing j because of the m-degeneracy. The computer time required for solution of the differential equations rises approximately as N3. For this reason the CC method is practical at present only for light diatoms and symmetric tops perturbed by noble gas atoms. The coupled states (CS) 29 method attempts to reduce the number of equations by making an approximation on the orbital angular momentum (111) operator. This has the effect of reducing N by a factor of two. The next level of simplification is the infinite order sudden (IOS) approximation.(ll2'll6) By neglecting the energy difference between rotational levels and freezing the orbital angular momentum quantum number at an arbitrary value (there are several choices possible for this quantum number, some of which appear to be better than others) the set of N coupled equations becomes completely uncoupled. This results in considerable saving in computa- tional time so that diatomic, symmetric top, and asymmetric top molecules perturbed by noble gas atoms can be dealt with reasonably. A preliminary calculation involving H - H (M117) 2 2 has recently been reporte At the other end of the spectrum from fully quantal (118) In the typical methods are fully classical techniques. case Hamilton's equations of motion are integrated for a given set of initial conditions. The rotational quantum numbers are treated classically (i.e., continuously) with the result that the trajectories are "binned" to obtain transition probabilities. That is, for a given set of initial conditions all trajectories with a final rotational angular momentum between, for example, 2.5 and 3.5 are lumped together and are considered to have j equal to 3. The main disadvantage of this method is that a large number 30 of trajectories must be calculated-sometimes as many as one or two thousand. Also, any quantum effects will not be accurately considered. This is not expected to be a large problem, however, for the AK, as they are relatively highly averaged quantities. A calculation of transition prob— abilities has been performed for OCS perturbed by H2 treated (118) as a structureless perturber. The results were only qualitatively accurate. In an effort to include quantum effects in a classical (lOA,lO5) and Marcus have trajectory framework Miller independently developed a semiclassical technique that in— volves calculating complex-valued trajectories. Several calculations Of Agifi have been performed.(101’102) The theoretical values are smaller than the experimental low- power linewidths. In addition to the three broad categories outlined above there are many methods which treat certain degrees of freedom classically and others quantum mechanically, each with appropriate approximations. A very brief, representative sampling of these methods is in References 119-135. D. Choice of Methods for Calculating AK There are obviously many choices for the calculation of the scattering matrices and thereby, the AK. The method of choice should be relatively simple, inexpensive and capable 31 of giving reliable results. Two methods have been chosen with these considerations in mind. The first method is the Anderson theory, which will be extended to enable cal- culation not only of APifi but of all other AK as well. The theory meets the above criteria and has the additional advantage of being relatively familiar to microwave spectroscopists. The Anderson theory is capable of giving 8009 values for AIifi for many molecules although for some symmetric tops, most notably NH3, it is necessary to norma— lize computed values Of AIifi to one experimental value. The major drawbacks of the theory are that it requires an artificial method of imposing unitarity (this was discussed earlier, and will be discussed again later) and that because it is only a first order theory in the scattering matrix, a dipole-dipole potential will allow only collisional transi- tions where j changes by zero or one. To estimate the effect of these drawbacks the AK will also be calculated using a sudden approximation.(136’l37) As employed here dipole-dipole potentials and straight line paths are used so that a direct comparison with the Anderson theory may be made. The sudden approximation allows estimation of transitions where j changes by more than one. Other calculations to be presented have been outlined in the introduction. CHAPTER II EXTENSION OF ANDERSON THEORY TO THE CALCULATION OF AK A. Relation of the Liu-Marcus AK to Anderson Theory The A appearing in the continued fraction lineshape of Schwendeman<55> are the AK as defined by LM. Specific- ally, K K Af'i'fi = (VOf'i'fi> (”1) where Ji-j'+K—£' 0?,i,fi (1%) z 2 (—l) i (2Ji+l)(2Jf+1) k 22' J.J 1 f Jf Ji K Jf Ji K Ji Jf* X [éi'iéf'f ‘ Si'isf'f] ' ' t ‘ I ! Ji Jr A 31 Jr 2 (A2) In tile above expressions denotes a thermal average, v is the relative velocity, k = Pi),- is the rmflflritude of the relative collision wave vector with U (financeduced mass of the collision pair (i.e., system 32 33 molecule plus perturber molecule), K is the tensor order of the relaxation process, lower case subscripted j's are rotational quantum numbers of the system molecule, A and 2' are relative orbital angular momentum quantum numbers before and after the collision, respectively, Jk is the total angular momentum formed by coupling jk and A, and SJk is the scattering matrix which is diagonal in Jk and M. (M is the projection of Jk on a space-fixed axis.) Equation (A2) is valid for the case of a molecule perturbed by a structureless (143;, no internal states) perturber, such as a noble gas atom. Equation (A2) is exact in the sense that all degrees of freedom have been treated quan- tum mechanically. The Ag'i'fi must be related to an expression from An- derson theory. The simplest way to do this is to recognize 1 that Afifi and that the familiar expression from Anderson theory also is l/T2 for the Spectroscopic transition f + i, ggives 1/T2. One could then assume that all other Ag'i'fi (norrespond to certain modifications of the Anderson expres- sixmns. The Anderson theory expression for the AIifi was cnmiginally given in an uncoupled basis; Eifi’ a basis wherwe jk and A are not coupled to form Jk' It is shown in Apperndix A that the cross section in an uncoupled basis, 3A KK' n 1/2 Ji'Ji'mi-mi Oi'f'if = ('_2) Z Z [(2K+l)(2K'+l)] ('1) YA'mm' v V J1 J' K' J J K 1 f i f * X (m'—m' ) (m Q)[éi'iéf'f " Sivisfvf] 3 ([43) i f‘Q i‘mf‘ is equivalent to Equation (A2). In Equation (A3) the pos- sibility that K # K' is allowed. The derivation in Appen- dix A shows that K = K'. After performing a thermal average (Equation (Al)), Equation (A3) can be identified with (A27) of LM: 2 2 Af'i'fi ‘ 5; ggfidEapaE5f'f5i'iaa'a‘sf'a'fa 1 (AA) * Si'a'ia where ZdEapa constitutes a thermal average. (Although Equatign (A3) is written in the spherical tensor basis and iEquation (AA) is not, it is the thermal average which is of iJiterest at the moment and which is the same for either Equation (A3) or Equation (AA).) After taking a classical ILhnit; 243;, after replacing the quantum mechanical treat- ment (of relative translational motion and the use of the Quarnrum numbers A and 2' by a classical treatment of the relaiuive motion and the use of an impact parameter b, EQuaixions (113a) - (115) of Ben-Reuven(88) may be used to Convert the thermal average to 35 2nnfvf(v)dvfbdb . (A5) In Equation (A5) v is the relative collision velocity, f(v) is the distribution function for v, and b is the impact parameter. The conversion to a classical limit is important because the Anderson theory treats the relative motion classically. To summarize, Equations (Al) and (A2) may be replaced with the equivalent expressions, K K Aftitfi = 2NDfo3 f 1 1 (A9) The (conventions for reduced matrix elements are discussed irlAppendix B. Appendix C shows that Equation (A9) is identical to the expression given by Anderson<53) for the 1rIteraction of a molecule with a structureless perturber. Equation (A9) must be modified to include perturber 37 states when the perturber has internal structure. This (I 8) 3 AC. has been done by Ben-Reuven (139) and Fiutak and Van- Kranendonk. Modification of the notation slightly to agree with TC gives the final working expression for OK(i'f'if) K , , , , Jf’Jf‘+mf'm' f 1231+1 1/2 oj2(333fjijf)= mzmzmzm'(- -l) (2j2+l)123 +1 i f r 1 m2 m2QJ2 " ji jf K ji Jr K ix ( )(: [5. ..G. a 5m ,5 , V _ _ :_ _ J - J J 6J J m. lm' m m m mi mf Q mi mf Q iJ l f f 2 2 i mf f 2 2 ._1 (50) The 'T matrices in Equation (50) are analogues of the S matrrices in Equation (A9). The letter T has been used merefily to agree with the notation of TC. In Equation (50) it Srlould be noted that even when ji = ji and jf = jf, the Timajzrix elements can still be off-diagonal in perturber states. The subscript j2 has been added to OK in EQuation (50) beCHause the cross section is for a given j2 perturber 1eVel. The total cross section is 38 K K .,.,. . _ . .,., . 0 (JiJfJiJf) — 230(32)0j2(JiJfJiJf) (51) J2 where p(J2) is a Boltzmann factor for the perturber level j2. To obtain Ag'i'fi Equation (A6) is then used. B. AK in Terms of Anderson's P Matrix Iflow that the equivalence of the quantum mechanical AI'f'if (of LM and the semiclassical (iiii’ classical translational rnotion and quantum mechanical internal motion) OI'f'if (if Anderson<53> theory has been established, the Anderson- lLike expansions of the cross sections can be carried out. FRDllowing TC, let T = TO + T1 + T2 + . . . (52) O wherne T = (UO)'1Um with U and Um the evolution operators coruresponding to the Hamiltonians HO and Hm, respectively. Herea, H0 is the unperturbed internal state Hamiltonian for the Esystem molecule and Hm = H0 + HC where Hc is the Hamil- toniiin for the intermolecular potential. The equations of motixan for T and T-1 are in 3T = [(UO)'1HCUO]T 3E and 39 —1 . 3(T )_ -1 0—1 0 in T" -T ((1)) Rev 1 . (53) An iterative solution gives T0 = 1 , (5A) T1 = fig f[(UO)'1HC(t)UO]dt, (55) T2 = ;%J[(UO)‘1HC(t')Uoldt'ft(UQ)'1HC(t")UOJdt" , (56) etc. An operator P is defined by P = %-,C;[(UO)-1Hc(t)U01dt , (57) in which case T1 = -iP, Til = -T1 = iP, (58) T2 = a} P2, and T21 = T2 = I; P2 . (59) TheSe relations enable the Anderson-like expansions to be expressed in terms of P. The expansions are analogous to the expansion in TC for OIfif' Details of the expansions are given in Appendix D. The results are given here. A0 [ NIH o? (ifif) = . . -1 2 . . 2 (231+1)(2J2+l] (Jimi32m2'P IJim132m2> Z mim2 . , — . 2 . . +> 2 [(2Jf+1)(232+1)l 1<31m132m2IPIJimij2m2>' (60) j.-j!+n.—n! —l a? (i'i'ii) = _ 2 § 2 (-1) 1 1 1 l(2jé+1) I 2 miminini v. m2m2QJé (213+1)1/2 (3i ji K)(ji X 2ji+l ' i x - (61) U1 K o o o o o o —l 2 1111 = .+ ° ' ° ~ 0. ( ) z [(231 1)(232+1)] 2 mim2 n -n' i .i i oi - z z z (-1)j. 3T(232+1)1( ) < ) mimininiQ mi—ni—Q m -ni Q m2m 2J2 x (ji n. ij2m 2|Pljin ij2m 2> (62) Some general discussion of these results is warranted. First, there are in general two types of terms in each cross section. One is independent of K and corresponds to the "outer" terms of Anderson, the other depends on K and corresponds to the "middle" terms of Anderson. The outer terms are identically zero for UK (i'i'ii). The general J2 form of the cross sections given here is in agreement with the equations given by Ben-Reuven(l38) and Coombe, Snider and Sanctuary.(luo) Their expressions are written in ‘terms of transition (t) matrices, whereas the cross sec- txions given here were initially in terms of scattering (T) "matrices and later in terms of a P matrix. It is useful tc: show that the P matrix is equivalent to a lSt order .Deerturbation approximation to the t matrix. Let a wave function ¢(t) = U(t,t0)¢(t0) obey the SC hrodinger equat ion , 142 h a 1 7%111+ H'w(t) = 0 . (63) Then, the equation of motion for U(t,tO) is f] 3U(t,t0) I"“§E‘_‘ + H'U(t,t0) = o (64) with initial condition U(tO,tO) = 1. This is equivalent to - i t 1 V I I U(t,t0) — l-§§jEOH (t NJU;,tO)dt , (65) Which has the iterative solution, U(O)(t,to) 1, t U(l)(t,t0) = 1 - ,1; ftOH'(t')dt' , (66> or, _ °° (n) U(t,t0) - E U (t,t0) , (67) WhEre u(h) i n t 1;: 1501-1) (E) fl; fl; °°°€ H'(t')H'(t")...H'(t(n))dt(n)...dt'. o o o (68) “3 Equations (67) and (68) are equivalent to t U(t,t0) = P exp[-(%)f H'(t')dt'] (69) t o where here, P is the time ordering operator; i.e., P orders the upper limits of integration in Equation (68). The approximations in Equation (66) may be written (0) _ _ U - U0 - l (l) _ U - U0 + U1 (2) _ Sintilarly, with the definition 8 = U(+w, -m), s = 2 8n = P exp[-(%) I” H'(t')dt'] (71) n=0 —oo with 3(0) = 80 = 1, (1) s — 30 + 31, 8(2) = 80 + 81 + 82, etc. (72) 1114 (n) The matrix elements of S are (n) = 5fi + (sl)fi + ... + (Sn)f1 . (73) Then, the t matrix is defined by - 2nio(Ei-Ef)tf1 . (7U) Alternatively, Hfi = - 21ri and is the reason that Equation (38) is true. Several other points can be made concerning Equations ( 60)-(62). l) The factors [(2ji+l)(2j2+l)]-l in the outer terms giirise from an average over initial collisional states of t>c>th molecule and perturber. 2) In Equation (60) the only K-dependence comes from / ccaillisions that are simultaneously elastic in levels ji aiqci jf. This middle term gives rise to what have been 0513;1ed interference or correlation effects. If this term ifs zero, there is no K-dependence in the a? (ifif) cross 2 seecisions. More importantly, the Rydberg—Ritz principle 155 \falid, as discussed by Fano.(87) This means that the 16?Veels ji and jf relax independently of each other, and tkuat; the Liouville or "line-space" (i;e;, the need for four irmij.ces on AK) formalism is not needed. 3) It is useful to look at o? (1111). For this case 2 Q == 0. Use of Equation (36) in Equation (62) leads to a mid d 1e term of mi'mi -1 - z z (-1) [(231+1)(212+1)l m m! i 1 m2m2 2 (77) . o . 'c" 9 '°" ° ' ix (JimiJ2m2lPlJimiJ2m2>’ M6 which shows by reasoning similar to that in Reference (67) (iiii) is proportional to the total collisional 2 rate out of level i. that o? M) It is also of interest to set K = O in o? (i'i'ii). 2 (Then, use of Equation (36) in Equation (61) gives 0 '0'00 0 -1 o l/2(2Ji+l 1/2 , = — 2 + . ._____ 0J2(i 1 11) nzng ( J2 1) [(2Ji+l)(23 +l)l 2j1+1 i 13, 1 m2m2 2 >(31111321112lpljinij2m2> m. 3 ma X| (78) $71143 has the same form as the negative of the usual cross SENci:ion for a level-to—level collisional transition. The renormalization of the Lin—Marcus cross section 5) alfll<>ws a summation over Q. This in turn is the reason why tr“? outer terms are K-independent. 0' Evaluation of the P Matrix for Multipole-Multipole Intermolecular Potentials It remains to evaluate the matrix elements of P and 2? P Etnd.substitute the results into Equations (60)-(62). “7 Following TC the intermolecular potential is assumed to be expanded as k k k k l 2 1 HC =. 22 C) 1 Y1 (1)Y12(2) , (79) k k l 2 l 2 l 2 A A k1 vvhere Y) (1) is a spherical harmonic of order kl, a func- l t:ion of the internal coordinates of molecule 1, and k kg CZAlAc is a factor that depends on the intermolecular dist- l 2 Eirice. tzrie charge distributions of the two molecules do not over— induction, The expansion (Equation 79) is valid as long as ].ezp. Therefore, it can express electrostatic, zar1d.dispersion forces but not exchange forces. From TC tliee matrix element of P is . . . (Akj) 11": k); a 2 12 kl k2 oo ""' x _ (ij) -. . .. ., , . " 1 a A(J11113111(1)21‘2~12)<31k1mi11”1111) ., , . x (80) #8 where a is a radial factor that depends on kl and k2 as k, Al and )2 as A and jl, ji, j2, and J5 as J; it must be evaluated for each potential. A(J'kj) is given by 3 ’3' 2k1'1 1/2 A(j'k j ) = (—1) ( MT: ) <3 k K OIJ'K > . 1/2 J j' k J —K 2k+ l = (-1) (L1,n ) (2j'+l)l/2 (K ) , (81) —K \ vqriere the relation a b c = (-1)a'b"Y(2o+1)1/2 (o B y) (82) has been used. Tsao and Curnutte have evaluated a; (ifif). The other 2 01°CHSS sections follow by analogy and are worked out in d63tiail in Appendix D. The results are given here. U9 K . . ' oj (1f1r) = 1 2 (21i+1)<215+1)|a1k1|2 . ., 2 . . 2 k. 2JH‘Jf k1 J2 32 k2 <231+1><2Jé+1>|ax ‘7' ( ) ( ) Kf—Kf O K2-K2 O (2ji+l)(2jf+l) A +Ag+kl+K + 2 z z (-1) (2Jé+1)a(klk2x A j) l6Tr kkj' 12 122 1112 2 _ 3' 3' k 5 J k 5 3' k 3 J K x a3 (21'5+:L>(2j§+1>laml‘2 J2 16n j j k k 1 2 1 2 A1A2 Jijikl2jgjék2 2 Ki-Ki o K2-K2 o (2ji+l)2 K+k1+1 1k 2 + __———§~— z z (-1) (2j' 2+1)la Jl l6n k1k2AlA2 V J2 51511‘1232521‘223131K x< ) ( > { } . (85) Ki—Ki o K2-K2 0 ji ji kl It is of interest to evaluate o? (iiii) and a? (i'i'ii). 2 EXY analogy with Equation (77) and Equation (78), O 2 03 (1111) 1-2 2 (231+1)(232 +1)|a*k3| ' Y 2 low Jij2klk2 A1‘2 31.33 k1 2 J2 32 k2 2 X Ki-Ki o K2-K2 0 ~ J 1 k 2 J " k 2 (2J1+1) Akj 2 1'1 1 232 2 \—2——k 2 X(21'+1)|a I <86) 161T11‘252 K K K K o 1' 1 0 2' 2 l 2 51 . ., 2. ., J1 Ji 1 J2 J2 k2 X< )( > . (87) K.—K. O K2-K2 0 Equations (86) and (87) can be obtained from Equations C 85) and (8“), respectively, by using the relation<7l> 31 32 J3 . J +3 +3 _ g . } = (-l) 1 2 3[(2j2+1)(233+1)] 1/2 (88) O J332 For the K = 0 case the structure of the middle term 00 (i'i'ii) collapses to the structure of an outer term. This is seen by comparing Equation (87) with the first term of (86). It should be noted that the matrix elements in Equa- tion (80) are valid for linear molecule eigenfunctions (Spherical harmonics) and for "one-ended" symmetric top eigenfunctions (proportional to rotation matrices). Proper eigenfunctions of symmetric tOp molecules should also be eigenfunctions of the inversion operator, which demands that they be linear combinations of the rotation matrices. 52 It can be easily shown that for dipole potentials, either proper or one-ended eigenfunctions give the same matrix elements. Quadrupole potentials work the same way. A discussion of these effects is relegated to Appendix F. ID. Tensor Order Dependence of the AK The motivation for studying the K-dependence of the /\}< may be seen by examining Table I. This table gives a sseat of conditions — hereafter called the Karplus—Schwinger— 'Ik3wnes conditions - on the K—dependence of AK under WhiCh ‘tldea continued fraction lineshape reduces to a sum of LADI°entzians. It was implied earlier that K—independence 817j;ses if there is no collisional interaction in one level. 191i.s is illustrated for oK(ifif) for a structureless per- tuPher. In this case Equation (M9) is appropriate and can be rewritten as K.. _ mf-mf'. 3'1fo JiJfK 0 (1f1f)— l -X '23 (-—l) m m! mimiQ mi mf-Q ml-m%—Q f f x (89) 53 The Kronecker deltas have been eliminated by using a sum rule over the 3-j symbols.(7l) The technique is illustrated in Appendix D. If there is no interaction in the jf level, = Gmfm% and the second term in Equation (89) becomes K Ji 3f K om! ' ) l l mi-mf—Q mi—mf_Q . -l . = -2; (231+1) . (90) m. l 'Ifldis is equivalent to one of the outer terms obtained pre- \rixously for oK(ifif). The other outer term would have tDeuen obtained if it were assumed that there was no inter- 51C13ion in the 3i level. The assumption = 5 , is equivalent to setting TO = 1, (See Equation (5“) Infqnf 831Ci Appendix D) and it may be said that oK(ifif) consists C”? three terms, the two outer terms, which describe how 143\7els 3i and jf evolve independently of each other, and t1163 middle term, which describes the aforementioned inter- fer‘ence effects . It is possible to discuss the K—dependence of the c17<>sss sections in terms of various intermolecular poten- ties-Il_s. Consider one of the products of 3-J symbols in the K K"dependent middle term of OJ (ifif) in Equation (83). 2 5U If the system molecule is linear so that Ki = Kf = O, the product may be written as 31 31 k1 jf'jf k1 ( )( ), O O O O O O which is zero if either ji+ji+kl or jf+jf+k1 is odd.(7l) That is, the K-dependent middle term is zero if kl is odd. If molecule 2 (the perturber) is linear, no definite con- clusions can be reached because the middle term may be inelastic in j2. Completely analogous arguments hold for the middle terms of Equations (8”) and (85). The result for 0K(iiii) is the same as for oK(ifif), while 0K(i'i'ii) is K-independent for kl = even. For symmetric tops with inversion (example, NH3) the Ioarity of the involved levels must be taken into account. (This is not explicitly indicated in the notation. The (zonsiderations are quite obvious and the results are the saime as for linear molecules. For symmetric tops without Lirlversion the situation is nebulous and no definite con- <3ZL11sions may be drawn. These results are summarized in TableIL For simple potentials a more quantitative description C’I‘ ‘the K-dependence may be given. The cross section I{ <7 (zifif) can be written as oK(ifif) = OK(ifif)Oi + 0K(ifif)of + CK(1fif)m (92) 55 Table II. K-Dependence of AK for Multipole-Multipole Potentials. R Branch (Aj = il) Linear molecules K Afifi no K-dependence for kl = odd K iiii K Aiikk no K-dependence for k 1 even Symmetric TOps without inversion no definite conclusions possible except for K = 0 (reduces to linear molecule) Q Branch (A3 = O) (inversion levels) results are as for linear molecules. 56 where the subscripts oi, of, and m denote the outer term for the i level, the outer term for the f level and the middle term, respectively. The only K—dependence is in the middle term and may be written as j ' K ( 1)K 1 Jr - . (93) iji k1 The major interest is in AEfif° Since AK is obtained by K . adding all 0 for each J2 and weighting them by a Boltz- mann factor it is possible to write K K—l K , , K-l . . Aifif ' Aifif = F: 0 (lflf)m " F3 O (ifif)m (9”) J2 J2 where the sum over 32 is meant also to imply weighting with a Boltzmann factor. Extraction of the K-dependent part of gK(ifif) enables Equation (9“) to be written as J1 JOf K AK AK”l = 2: (-1)K F('fk ) ifif ' ifif k 1 1 l Jf'Ji k1 ji jf K-l k . . l 3f Ji k1 'where F(ifkl) is everything in the middle term except 57 K J. ’ K Jf (-1) l 1 For a } . (F(ifkl) includes .2.) j j k f i l 32 sition, F (ifkl) has the same numerical value in the two terms on the right side of Equation (95) J. ij J K K—l _ K 1 i Aifif ’Aifif If ('1) { } -<-1)K'l{ .1 o .. k . Jle l Jf Let k1 = 1 only. As a result of the symmetry of symbols, given tran- each of so that } F(ifkl) Ji k1 (96) the 6—j { H w H }. 1 . . . l . . (97) 3f ji 1 ji Jf Jf Ji 1 J1 Jf From Table 5 of Edmonds<2Jf+2><2Ji><2gi+1><2ai+2>1 / x F(ifl) (99) Th d' lt f AK ' e correspon ing resu or iiii 15 K K-l MK .. A....-A.... = . , , F(111) . (100) 1111 1111 2Ji(2Ji+l)(2Jl+2) The result analogous to Equation (99) for Agikk is too cumbersome to be of use. A more useful result for Agikk may be obtained by realizing that it has no outer terms. For a single given potential the following ratio may be formed. K - 31 3f k1} Aiiff Jr 31 K _____ = . . (101) K-l J j k A. i f l iiff . , K 1 E. Some Additional Properties of the Anderson AK In anticipation of some later results concerning the application of a sudden approximation for calculating K .A the sudden approximation within the Anderson theory 59 will be discussed. Also, the P operator will be expanded in terms of irreducible tensors. This will be useful later in comparing the K-dependence of the Anderson theory with that of the sudden approximation. El. Expansion of the ngperator in Irreducible Tensors The matrix elements of P can be rewritten from Equation (80) as -Kl-K2-ml-m2 a(1kj) . . ., ... , = _ k2; ( 1) —_EF__ 1 2 A1A2 XE(2jl+l)(2ji+l)(2j2+l)(2jé+l)(2kl+l)(2k2+l)]l/2 31 3i K1 32 32 k2 J'1 3i K1 32 32 k2 x< >< >< >< )- - — — '_ - - K1 K1 0 K2 K2 0 m1 m1 Al m2 mé A2 Consider firsttfluasimple matrix element, _ Mm (-1)’K1 ”1 é;;jj7§[(231+1><21i+1)(2k1+1)11/2 X ( 1 :1 (51 Ji R1) (103) Kl-Kl 0‘) ml-mi-Al ' 60 If the left side of Equation (103) is expanded in irreducible (1U3) tensors by using equation 18.1 of Fano and Racah, i.e., if, J'1 3i K J -m +Q (jlm llPIji m l>=z:(—1) 1 1 (2K+l)l/2 ( KQ ml-mi—Q ) leji(KQ) (104) Then comparison of the right hand sides of Equation (103) and Equation (10“) shows that they are equal if K is iden- tified with k Q with Al and if 1, "51+K1'Q—(Akj) (231+1)(2Ji+1)(2k1 +1) 1/2 P- ,(KQ)=(-l) [ Hn (2K+l) 1131 J J' k X ( 1 1 l) . (105) Kl—Kl O lj,(KQ) is independent of m and m' as it should be. Here, Pj l 1 Upon returning to Equation (102) and applying Equation (104) twice, it is found that 31+32-m1—m2+Q+R =£§ 5%(-l) 3111 K) (j2jé L 1/2 x [(2K+l)(2L+l)] ( -mi_Q ) leji(KQ)PJ2J2(LR), (106) 61 where P- °v(KQ) P. .,(LR) = (-1)-J1+J2+K1+K2-Q_R a J131 J232 __EF—‘ l/2 ( (2jl+l)(2ji)(2j2+1)(2jé+l)(2kl+l)(2k2+l) X (2K+l)(2L+l) ) J1 3i K J2 35 L 1 >< > 0 Kl-Kl K2—K2 0 The conclusion to be drawn from the above is that expanding the matrix elements of P into a multipole potential is equivalent to expanding the matrix elements of P into ir- reducible tensors. E.2. Sudden Approximations in Anderson Theory Some interesting sum rules may be derived from Anderson theory if the sudden approximation<23f+1)1 1/2(2jé+l) . , 2 . v J2 32 k2 -Ki-Kf J2 J2 k2 x ( )==(—1) (235+1)( ) (11h) 6A Exactly analogous analyses hold for 0K(i'i'ii) and OK(1111). For oK(i'i'ii) an interesting sum rule may be derived. Writing OK(1'1'11) = ’1 2:2: G(i'i22'klk2K)Q(Akj) (115) 16n k k j' 1 2 2 A122 (compare to Equation (8U)) allows G to be written for K = 0 8.8 . 2 . . 2 j§_31 k1 J2 32 k2 G(i'i22'klk20) = (2ji+l)(2jé+l)( ) . Ki-Ki o K2-K2 o (116) Also, -j!-k +K G(0i'22'k1k2K) = <-1) 1 l (215+1)3/2(2J§+1> 2 2 ()31 k1 J2 32 k2 35.0 k1 .( )( ){ } o o o K2-K2 o o 31 K . 2 J2321‘2 = (2J§+1)(2J'i+1)l/2( >K053121 ° (“7) 65 Equation (117) allows oK(i'i'00) to be written as l6n2 k 5' 2 1 2 2 A1A2 . ., 2 (J2 J2 k2) X Q6 6 . (118) K2-K2 0 KO R131 Therefore, K . . .. J -J'-k +K o (1'1'11) = £Z(-l) l i 1 (2ji+l)(2ji+1)l/2 1 ' J‘! k 2 .' . k 31 1 1 31.31 1 X( ) < ) oK(i'i'00) (119) _ ’ " K1 K1 0 J1 J1 K . 2 , -1 Ji+K 1/2 J1 k1 k1 R1 31 k1 = 6 2 (-l) (2kl+l)(2ji+1) < : 1 7r K1 K1 0 ji kl K 0 . X 0 (1'1'00) . (120) This is an interesting result because it gives a oK(i'i'ii) cross section in terms of a oK(i'i'00) cross section. Setting K = 0 gives 66 . ' k 2 0 3/2 1/2 1 l 1 o (1'1'11) = -Z(2ji+1) (231+1) k 1 Ki-Ki o x 00(1'1'00) (121) Use of the above relations is valid only when Q(1kj) is independent of 3. This is the case in the sudden ap- proximation. In the Anderson theory Equation (121) is of limited utility because 00(i'i'00) cannot be calculated for ji > 2 (assuming potentials up to quadrupole). An analysis similar to the above can be carried out for oK(iiii)m. Defining the G function as kl+K 2 31 J1 kl . , _ _ . ., G(122 klkZK) - ( l) (2Ji+l) (232+l) { } . . 2 , , 2 31.31 k1 J2 32 k2 x ( > ( ) (122) Ki-Ki O K2-K2 0 gives 2 J2321‘2 t = -v G(022 klk2K) (232+l) ( ) leOSKO . (123) K -K 2 2 O Tlierefore, 67 , 2 K _1 kl+K 2 31 J1 k1 o (1111) = 2:(—1) (2J +1) m 16w2 k 2 1 K.-K. 0 l l 31 J1 k1 K x i } o (0000) m J1 J1 K 2 -1 2 J1 J1 0 J1 J1 0 0 - 2(2ji+l) o (0000) 160 K -K O . O m 1 J1 J1 = '12 00(0000)m (12M) l6fl and 0K(iiii) becomes K-independent in a sudden approximation. The analogous result for oK(ifif)m is K . . j ’K' o (ifif) = 1'2 GK(ofof) (—l) f l m 160 m (125) and does not appear to be useful. The sudden approximation allows simplification of the expressions given earlier for the AK. If the perturber molecule is treated with the sudden approximation, the sums over jé, mé and m2 may be carried out analytically. Writing oK(ifif) as 68 K mi—m! -1 J1.31 K 31‘ J1 K o (ifif) = 1 —22 (-l) l(2j2+1) ( )( ) _ v- v- mf mf Q mf mf Q o . -l o .' g . yo . . X (126) and using the relations<7l> Z Y. r' Y = - ' j'm' 32m2(’b ) Jéméhg) Mr g) (127) 2 2 and . -l l (2 +1) 2 Y. r Y. r = —— 32 m J2m2(m) J2m2(m) “fl (128) 2 allows (126) to be written as m.—m: 5131 K 5131 K oK(ifif) = 1 - 2: 2: (-l) l 1 l ) m.m!Q u" mimi mf-mi—Q m%-mi-Q f f , -l . y . x . (129) Analogous results clearly hold for the other cross sec- tions. The above results are valid only in the sudden ap- proximation, i.e., when the T operators do not depend on 32. The results for the cross sections in the sudden ap- proximation are given below. They are the analogs of 69 Equations (83) — (85). 2 J. j! k K . . _ 1 ., Akj 2 l l 1 o (ifif) — Z (2J.+1)Ia I 3211 "k A 1 J1 1 1 K —K 0 k A l l 2 2 (2j.+1)(2j +1) k +1 +1 +K + l 2 f X (-l) l l 2 a(k1k21112j) 16w klk 1 A j131 k1 31'51 k1 31.31 K K.-K.<1 K —K 0 j j. k1 l 1 f f f 1 (130) K (211+1)1/2<215+1)3/2 0 (1'1'11) = 2 x 16w klk2 K1K2 j.—j'+k +K+1 Ak' 2 x (-l) l i 1 la JI '1 . ., J1 J1 k1 3: J1 k1 x (131) - 0' 0 K1 K1 0 J1 J1 K 7O k +K+l . 2 K ... 0 (1111) = 1 z z <-1) 1 [.11le 160 k k 1 23. 1112 1 . , 2 . . Ji 311%. 1111.K1 x < ) { } (132) Ki—Ki 0 31 31 K The a factors are the same as before except that now they are evaluated by neglecting the energy spacing between the rotational levels of molecule 2. Calculation of the a factors will be discussed in section F of this chapter. The derivations in Equations (126)-(128) and the results in Equations (130)—(132) are also valid if one-ended sym- metric top eigenfunctions are used. This will be shown in Chapter IV where the sudden approximation is discussed in much greater detail. F. Numerical Results for OCS and NH3 Systems The theory developed in the previous sections of this chapter is applied here to the J = 2 + 1 transition in OCS and the (J,K) = (3,3) inversion transition in NH3. Before presenting the results the basic problem will be restated. Most previous analyses of power—broadened microwave lineshapes have assumed that the lineshape is a sum of 71 Lorentzians, one for each m-component pair. This is equivalent to assuming that the m-component pairs are not collisionally coupled. Fitting an experimental lineshape to this model allows extraction of a parameter denoted (Tl/T If the model is valid, (Tl/T should be very 2)O' 2)O close to the true Tl/T2° To test this model, the Anderson- like expansions derived here are used to calculate Tl/T2 and all of the other relaxation parameters that occur in the more exact continued fraction lineshape expression. These parameters are used to compute a lineshape which is then (5”) to obtain the parameter fit to a sum of Lorentzians (Tl/T2)O' If this (Tl/T2)O and the calculated Tl/T2 are equal, the model is presumed valid. Alternatively, the better the fit of the lineshape the more valid the model. There are two secondary purposes for calculating the K allows the relaxation parameters. Knowledge of all the A calculation by means of Equation (27), of the <>, which have an intuitive physical inter- pretation. In addition, knowledge of the AK allows the calculation of M-level effects, which have been discussed in Reference 55. A summary of these effects will be given shortly. Before discussing the results of the calculations it is worthwhile to discuss the method of calculation. The calculation may be split into four parts, the 72 calculation of the aAkJ factors (the resonance function in the parlance of TC), the calculation of the angular momentum coupling coefficients, the determination of the hard sphere cutoff and the calculation of the thermal average. The resonance functions are given explicitly in TC in terms of modified Bessel functions of the 2nd kind.(luu) TC also provides a table of these functions. Rather than interpolate from this table, explicit calculation of the functions is included in the program. To do this, the Bessel functions are expanded in terms of Tchebycheff poly- (1145) nomials. In practice, a Tchebycheff expansion is used to calculate the modified Bessel functions of the first kind, which are in turn used in another Tchebycheff expansion to calculate the desired modified Bessel functions of the second kind. The resonance functions are given in detail in Appendix H. Some of the angular momentum coefficients were cal- culated by using the general formula for 3—j coefficients givenim1Reference 71. Other coefficients were calculated by means of special case formulas.(7l) Determination of the hard sphere cutoff has been dis- cussed in many places.(86’93’9u’98) In the present cal- culations separate cutoffs for each tensor order are determined. That is, the cutoffs for Aifif and Aifif are calculated separately. The plural "cutoffs" is used be- cause a new cutoff is calculated for every perturber 73 rotational level. A problem arises in the calculation K of Ai'i'ii' For K = O, the relation 0 O _ A1111 11.7.1 A1'1'11 = O (133) must hold. The only way to insure that this condition holds is to use for given perturber rotational levels, the same cutoffs for Ag'i'ii as for Agiii' The cutoffs are calculated by a simple bisection iteration procedure. Therefore, the relaxation parameters obtained in this process depend on the upper and lower limits declared for the bisection. The limits used in the calculations reported hereanwaO K and 20 K. The uncertainty in the relaxation coefficients due to the choice of limits is estimated to be about 12 K2. Further details of the hard sphere cut- off calculation are presented in Appendix G. The thermal average consists of an average over the relative velocity distribution and a weighting by a Boltz- mann factor for each perturber rotational level. The cal- culations presented here ignore the velocity average. The assumption is made that at a given temperature all col- lisions occur at the mean relative velocity. This assump- tion should be quite good. Calculations of Aifif by (92) Cattani show a 2% difference for OCS between including the average and ignoring it. The difference for the (J,K) = (3,3) inversion line of NH3 is about 8%. The 7U Boltzmann factors used are those given for high tempera- ture limits by Townes and Schawlow.(1U6) Table III gives the assumed parameters used in the OCS calculation. The minimum value of “.13 K for 0 (hard sphere) is the gas kinetic diameter. If the calculated hard sphere cutoff is less than A.l3 3, the program de- faults to the value “.13 K and uses this as the hard sphere cutoff. Table IV is a tabulation of many of the relaxation parameters for OCS. These parameters incorporate all multipole—multipole potentials through quadrupole. The only parameters showing marked K-dependence are the K Ai'i'ii’ which also show a large dependence on rotational level. Condition (133) is satisfied very closely, as can be seen by considering, for example, A3222. 0 O O O O 02 A2222 + AM422 + A3322 + A1122 + A0022 = 2-8” A 1 a relatively small difference from zero. This is a result of the cutoff procedure described earlier. It is also worth noting that Equation (133) requires Ag'i'ii to be negative for i' # i. Finally, 0 where 5/3 = (2jf+l)/(2ji+l). 75 Table III. Assumed Parameters for OCS Calculations. “D = 0.71519 D Q = 1.0 x 10—26 esu cm2 BO = 6081.“9 MHZ T = 300K 312: “.6 x 10“ cm/s j (perturber) i 90 0 (hard sphere) 3 “.13 K Table IV. Relaxation Parameters for OCS.8 1111 K=0 K=2 K=“ K=6 K=8 0000 263.7 1111 263.2 263.6 2222 26“.9 265.2 265.1 3333 266.7 267.1 267.5 267.0 ““““ 269.8 269.9 270.1 270.3 269.9 5555 273.0 1100 -259.6 2211 -176.0 —103.6 3322 —155.6 —131.5 — 66.5 ““33 -150.0 —138.8 —10“.“ - “9.5 55““ -152.0 0011 - 86.0 1122 -103.3 - 60.8 2233 -110.5 - 91.5 — “6.1 33““ -115.8 —10“.5 - 78.6 - 37.2 ““55 -119.2 2200 “.“ 3311 — 2.7 — 1.3 ““22 - 2.3 - 1.7 - 0.6 5533 - 2.1 - 1.8 - 1.1 - 0.“ 66““ — 2.0 aThe entries in this table are AK n 32. 77 Table V gives the relaxation parameters necessary to calculate the power-broadened lineshape of the J = 2 + 1 transition of OCS. Comparison of Tables I and V shows that the first three Karplus-Schwinger-Townes conditions as well satisfied, while the fourth, i.e., all Agiff and K Affii equal zero, is not satisfied at all. The value of T31 calculated from the equationKSS) T (2j+3)bgf+(2j+1)bgi+[(2j+1)(2j+3)]l/2(bgf+bgi) 1 1(3+1>> where j and j' are the numbers in parentheses. All values are in 86 Table X. Assumed Parameters for NH3 Calculations. uD = 1.“68 D Q = -l.0 x 10"26 esu cm2 B0 = 9.933 cm—1 00 = 6.3 cm”1 T = 300 K 312 = 8.6“ x 10” cm/s j (perturber) : 15 0 (hard sphere) : “.“3 3 87 Table XI. Relaxation Parameters for the (J,K) = (3,3) Inversion Doublet of NH .a 3 K(+-+-) 70“.5 = 131, all K K(++++) 70“.“, all K O(++--) -657.3 1(++--) -602.5 2(++._) —u93.0 3(++——) -328.6 “(++__) -109.5 5(++—-) 16“.3 6(++——) “93.0 -1- T1 — 1361.7 Tl/T2 = 0.517 2-1eve1 approximation aThe values tabulated are A§f1,f, where ifi'f' are given in parentheses as the parity of the level. The left super- script is K (the tensor order). All values are in 32. 88 Schwinger-Townes conditions are only partially satisfied. The inversion doublets of NH3 are usually considered to be a very good approximation to a 2-leve1 system. To test this approximation, “-level effects have been included in a calculation of T11. The B11 and B11 matrices are given in Table XII. The new value of Tl/T2 of 0.582 is consistent with the view of the inversion doublet as a 2- level system. T was calculated from the formula given by 1 (55) Schwendeman iibff if fi -1 _ 2(bK K 'bK bK 1 T1 ‘ ff 11 if fi ' (1A0) (bK +bK +bK +bK ) A lineshape has been calculated by power-averaging the relaxation parameters. (Tl/T2)O obtained from fitting the theoretical lineshape to a sum of Lorentzians is given in Table XIII for three different pressures. As for OCS, (Tl/T2)0 is a fair approximation to (Tl/T2)O(exp). Relaxation parameters between two m states have been obtained from Equation (27). The results are given in Table XIV. As expected, the elastic reorienting collisions do not couple different m—component pairs. This is a direct result of using only dipole potentials. 89 Table XII. Matrices of b Coefficients for the NH3 (J,K) = (3,3) Transition for Calculation of “-Level Corrections: J i 5, K = 0. (33+)3 (33-) (“3+) (“3-) (53+) (53-) 70“.“ -657.3 0.0 — “7.0 0.0 0.0 -657.3 70“.“ - “7.0 0.0 0.0 0.0 B= 0.0 — “8.8 596.7 -“89.0 0.0 - 59.0 — “8.8 0.0 -“89.0 596.7 - 59.0 0.0 0.0 0.0 0.0 77.5 522.5 -388.3 0.0 0.0 - 77.5 0.0 ~388.3 522.5 (33+) (33-) ' 687.8 -671.8 B11= -67l.8 687.8 11/12 = .518 8The values in parentheses are J, K, parity for the cor- responding columns. The order of the rows is the same. 90 Table XIII. Summary Of Tl/T2 Calculations for the NH3 (J,K) = (3,3) Transition. w/o “-level Effects With “-level Effects D/mtorr (Tl/T2) (Tl/T2): (Tl/T2) (Tl/T2)O 20 .517 .567 .518 .568 30 .517 .550 .518 .552 “O .517 .5“5 .518 .5“6 (Tl/T2)O(exp) = 0.71 1 0.07 aExperimental parameters assumed for the determination of (Tl/T2)0 from the theoretical lineshapes are power = 15 MW, attenuation = 0.8. 91 Table XIV. State to State Relaxation Parameters for the NH3 (J,K) = (3,3) Transition: Dipole—Dipole Potential.a (+-+-) m/m' 0 3 0 70“.5 0 0 0 1 0 70“.5 0 O 2 0 0 70“.5 0 3 0 0 0 70“.5 (++++) m/m' 0 l 2 3 0 70“.“ 0 0 0 l 0 70“.“ 0 0 2 0 0 70“.“ 0 3 0 0 0 70“.“ (++——)b m/m' 0 1 2 3 0 .02 -328.66 - .01 01 1 -328.66 - 5“.77 -273.86 - 0 0 2 - .01 -273.86 -219.08 -16“ 33 3 - 01 - 0.0 -16“.33 -“92 96 (33““) m/m' 0 l 2 3 “ 0 - 2“.63 - 15.38 .02 0.0 .01 1 - 9.22 - 23.07 - 23.07 0.01 . 0.0 2 .01 - “.61 - 18.“5 - 32.30 0.0 3 .01 0.0 - 1.5“ - 10.77 - “3.07 aThe values tabulated are>. For the first three tables J=J'. All values are in bThe (m,m') = (0,-1) value which is not given here is -328.7. All other values not given are 0.0. 92 0. Numerical Results for OCS and NH3 Systems Within the Anderson Sudden Approximation The sudden approximation consists of neglecting internal state energy differences. When molecule 2 is treated in the sudden approximation the cross sections of interest are given by Equations (130) - (132). These equations are independent of molecule 2 quantum numbers. Therefore, weighting by a Boltzmann factor is not necessary. In addition only one hard sphere cutoff needs to be calculated for each cross section. Tables XV and XVI summarize the results of T1 and T2 calculations for OCS and NH3 in the sudden approximation for dipole-dipole potentials. The K-dependence of the relaxation parameters A§f1,f, is the same as the previous results, and values for these parameters are not given separately. 1 Calculated values of T.1 and T_ are larger than those 1 2 calculated by the normal Anderson theory. This is ex- pected from the properties of the resonance function for large values of internal state frequencies. It is in- teresting to note that the values of Tl/TZ are all very close to the Tl/T2 values calculated from the Anderson theory. The only exception to this behavior occurs in NH3 for the case that the internal energy differences in the system molecule are accounted for but those in the perturber molecule are not. That is, the sudden 93 Table XV. Values of T1 and T2 for the OCS J = 2 + 1 Transition Calculated by Anderson Sudden Ap- proximation and Dipole-Dipole Potential.a SUDDEN SYSTEM PERTURBER 1/T2 = 63“.0 l/Tl 961.9 'NO YES 11/12 = 0.659 YES YES l/T2 = 61“.0 1/Tl 93“.6 Tl/T2 = 0.659 2 aValues of 1/T2 and 1/11 are in K , 9“ Table XVI. Values of T1 and T2 for the NH3 (J,K) = (3,3) Transition Calculated by Anderson Sudden Ap— proximation and Dipole—Dipole Potential.a SUDDEN SYSTEM PERTURBER NO YES 1/T2 = 1216.5 l/T1 2“27.2 2““8.9 YES YES 1/12 = l“01.5 1/1l Tl/T2 = 0.572. a Values Of l/T2 and l/Tl are in 32. 95 approximation is invoked for the perturber molecule but not for the system molecule. In this case the calculated Tl/T2 is smaller than that calculated from the full Ander- son theory. A plausible explanation of this fact is as follows. The small value of Tl/T2 implies that T is too small or 1 equivalently that l/Tl is too big. The latter quantity can be made too large if the cross sections for J + 1 + J collisional transitions are too big. In NH3 the J + 1 + J energy gap is very large. All of this implies that the Anderson theory becomes poor at large energy gaps. The only quantity in the cross sections that depends on internal state energy differences is the resonance function which is calculated assuming a linear intermolecular trajectory. When the internal energy gap is large a large deviation from a straight line path is expected. Therefore, the above behavior Of Tl/T2 values may be taken as evidence that linear trajectories are not valid for collisional transitions exhibiting large internal energy changes. CHAPTER III A SIMPLE MODEL FOR THE RELAXATION COEFFICIENTS In the previous chapter a formalism based on Anderson theory was developed to calculate multipole relaxation co- efficients using multipole-multipole potentials. In this chapter a simple model will be developed which replicates the form of the Anderson theory results. The model utilizes the iterative solution to the equation of motion of the density matrix in the interaction representation. This is a common starting point for treating relaxation in nuclear magnetic resonance. The density matrix and inter- action potential are expanded in irreducible tensors. In addition, the potential is assumed to have an exponential correlation function, a root mean square strength or ampli- tude, and a characteristic decay time. These quantities will be defined later. The resulting equations exhibit the same form as the Anderson theory results. The dependence on tensor order is identical, while the resonance function and its associated numerical factors are represented in the form of a product of a root mean square amplitude and a decay time. This latter characteristic arises from the fact that the details of the collision are in effect 96 97 averaged out by the introduction of a correlation function for the potential. The following summary of the iterative solution to the equation of motion of the density matrix and its applica— tion to relaxation in magnetic resonance is taken from Abragam,(1u7) Weissbluth,(lu8) and Redfield.(1u9) Since only collisions are considered here (and not the inter- action of the system with a radiation field), the Hamil- tonian can be written as H = H0 + V (1“1) where V is the intermolecular potential and H0 supports the internal rotational states of the molecule. The Hamiltonian H0 may also describe a static external field. In the interaction representation the density matrix may be written iHOt/h -1HOt/fi 01(t) = e 0(t)e , (1“2) where the subscript I denotes the interaction representa— tion. The equation of motion for pI(t) is 3pI(t) 31-5 T = [VI(t)’ 01(13):] 3 (1113) 98 which has the formal solution ° t 01(t) = oI(tO) - % 4%)dt[v1(tl),pl(tl)l . (1““) Equation (1““) can be iterated to give DI(t) = oI(tO)— % ftto dt[V1(tl)0I(tO)] _- 2 t t 1 dtlf l 0 t dt2[VI(t1),[VI(t2),oI(tO)]] + ... (1“5) This can in turn be differentiated, resulting in do (t) _. ° __%6__ = 3%[V1(t),01(t0)] + (3%)2ii)dt'EVI(t)’[VI(t')’ oI(tO)11 + . . . (1“6) Following the usual argument of nuclear magnetic resonance relaxation, it will be assumed that the ensemble average of VI(t) is zero, so that the first term in Equation (1“6) is zero. If this assumption is not valid, the first term merely produces a frequency shift, which may be either ignored or incorporated into a redefinition of H0 and H1. Hereafter, all quantities are assumed to be in the interaction representation and the subscript I will be 99 dropped. By assuming that the first term in Equation (1“6) is zero and by setting h = 1, Equation (1“6) can be rewritten as 0.. SE = -[A§[VEVeindT . (1A9) (19' 0 q q' In going from Equation (1“7) to Equations (1“8) and (1“9) it has been assumed that the intermolecular potential has multipole character k and can be written 100 k k V(t) = 2(-1)qv (t)v . (150) q ’q q 0 k 0 O O In Equation (150), V_q(t) is an expans1on coeffic1ent and k vq is a unit tensor of order k and component q. Any effort to obtain an absolute numerical rate of change of the density matrix requires evaluation of Equation (1“9) with subsequent substitution into Equation (1“8). The goal of the present work is to obtain a simple model capable of giving easily calculated numerical re- sults and tensor order dependence of the state multipoles. To accomplish this the correlation function<150> -T/T k k Sq,_q(-l)qv2e C = (2k+l)§ (151) is introduced for the potential expansion coefficients. In Equation (151) Tc is a correlation time and v is an average intermolecular interaction strength. With this correlation function koq'(w) may be evaluated as ‘ 8 _ .(-l)qV2 “ (iw-l/T )T q 2 0q_q.(—l) v To = ( (2k+l 1 1‘1ch 101 Obviously, (SO_q.(-1)qv2 qu,(0) =( ’ 2k+l )TC . (152) The next step is to transform Equation (1“8) into a spherical tensor basis. The potential has already been expanded in such a basis. The analogue of Equation (150) for the density matrix is k q . (153) k p = 23 (-l)qp_ v kq q k . . where, as before, V0 is a unit tensor of order k. There— fore, double commutators containing unit tensors of the k k k form [v ,[v 1, v 1] must be evaluated. This can be q“ 91 92 accomplished by using the following relation, given by Judd:(151) k k :17 n_ v_ [v 1(22'),v 2(2"1"')1 = z: (-1)21 +1 1 q3 q1 q2 R q 3 3 - 1/2 k1 k2 k3 x [(2kl+l)(2k2+1)(2k3+1)] ( ql Q2-Q3 kl+k2+k3+2+g'+2"+2"' k1 k2 k3 X [6£'£!1(-1) 2,"!2’ 2’" k3 k1 k2 k3 k3 X V (88"')-5 1!! } V (KHQ')1 . (15“) q AK q 3 2" 2' 8 3 102 To understand the meaning of the 2' it is useful to con— sider the v2 in greater detail. The unit tensors v: can be written . j j. k J -m f l vk = 1 (-1) f (2k+l)l/2 ( mm' )ijm>11 = ZZA(—1) 1 5 X 1 { } V W (1.7) i f f i f f kn . k1 . k [un(fl)[vq (1f)vq . k k k 2(fi)]l = + zzA(-1)f‘l { 1 2 3} 1 2 k k k l 3 5 k k +k +k +k +i+f X { }v05(fi) + ZZA(-l) l 2 u 5 kl k2 k3 ku k3 k5 k - . Q5 1 1 f 1 f 1 k k k k +k +k [v “<11)[v 1(ff)v 2(fi)]] = _ZZA(_1) 1 2 3 q“ q1 q2 k1 k2 k3 k4 k3 k5 k 5 . x { } { }V (fl) (169) qs 1 f f f 1 1 k k k k +k +k [v ”(ff)[v 1(ii)v 2(fi)]] = —zzA(—1) 3 u 5 q“ q1 q2 k k2 R3 1 k“ k3 k5 k5 { } { } V (fi) (170) q5 f 1 1 1 f f 108 k u k k k +k-+k +i+f [un(if)[v 1(fi)v 2<1i>11 = ZZA(-1) 1 2 3 q1 q2 k1 k2 k3 k3+ku+k5 k3 ku k5 k5 X }[(-1) v (ii) - . q5 1 f 1 i i f -: } vq (ff)] (171) . 5 f f 1 k k k k k k 1 2 3 [vou(fi)[v l(if)v 2(ii)]] = “mm-Di"f { } 1 f i k k k k k k 3 l4 5 k k +k +k 3 u 5 k x [{ ] vq5<11> - (-1) 3 u 5‘ } v05(ff)] 1 1 f ' 5 f f 1 ‘5 (172) where the summations are over k3, k5, q3 and q5 and A is given by A = (2k3+l)[(2kl+l)(2k2+l)(2ku+1)(2k5+1)]l/2 5 -q -q x ) (~1) 3 5 . (173) Q1 q2-q3 qu q3—05 109 By using the procedure shown in detail for the case of all 2's equal, Equations (166) — (172), respectively, may be shown to give rise to the following relaxation terms. no on f. 2 1 k2 (f0) 11 11 1 + v Tc 2i+l p Q2 1 ff ff f' + 2 1' k (f') l V Tc 2f+l p q2 1 1+1 i+f k . . . 2 (-1) {-1) 2 . f1 1f fl * “V Tc [“18337“ + "ififiif‘JO-Q2(fl) k f f i+f+k+k2 { } pk2 11 ff £1 + -v2Tc(-l) -0 (fi) . *2 R2 1 1 2 i+f+k+k2 k f f k2 ff 1- fi + -v TC(-l) . o_q (f1) ‘ . . 2 R2 1 1 k+k k f 1 . . .. 2 2 ( k 1f f1 11 + -v TC[(-l) fi } p_: (ff) — 2 1 k2 f ( 1)“f —¥L—- k2 (11)] ' ’ 21+1 p-q2 2 k+k2 k f 1 k2 11 if 11 + -v T [(-1) p (ff) — k i+f l 2 <-1) 21+1 o_q2(ii)] (17M) (175) (176) (177) (178) (179) (180) 110 2 k+k2 k l i 1 k2 i1 11 11 + —2v Tc[(-l) - 21+lJp-q2(li) k i i 2 (181) The left hand sides of expressions (17h) — (181) are a shorthand for denoting the double commutator; i.e., k 88' 8"8"' 8iV8V = [v§(£8'), [v:(8"8"'), vq2(81V8V)]]. Also, 2 in Expressions (17M) — (181) i and f are shorthand for 8 = 3i and jf, respectively. Through Equations (148) and (153), Equations (17“) - k (178) describe the time dependence of p g - 2 with the notation of Chapter II shows that (-l)qu(fi) = (fi). Comparison pfi(KQ). Equations (17“) - (176) have the form of an outer term of oK(ifif); i.e., in each case the coefficient k 2 of p_q (fi) is independent of tensor order and has been 2 averaged over the degenerate initial rotational states of the system molecule. The factor V210 corresponds to the quantity . 2 . (232+1> XX [UH-J of Chapter II. Equations (179) - (181) describe the time k dependence of the diagonal coefficients pq2(ii). With 2 these comments, each of the equations can be given an 111 interpretation. Equations (17“) and (175) correspond to the the elastic contributions to the outer terms of oK(ifif), while Equation (176) is the inelastic contribution to this quantity. Equations (177) and (178) correspond to the middle terms of oK(ifif) and contain elastic contributions from levels 1 and f simultaneously. Because k is the multi- pole order of the potential (R1 of the previous chapter) and k2 is the tensor order of the relaxation (K of Chapter II), it is seen that Equations (177) and (178) replicate the tensor order dependence of oK(ifif). The first term of both Equations (179) and (180) corresponds to 0K(i'i'ii), while the second term of these equations cor- responds to the inelastic contributions to the outer terms of oK(iiii). Finally, the first term of Equation (181) corresponds to the middle term of oK(iiii), while the second term is the elastic contribution to the outer term of 0K(iiii). Again, the tensor order dependence is exactly the same as the Anderson theory results. 2 depends on the correlation function The quantity v , and therefore also on the indices 8'. Suppose that the above results are to be applied to the calculation of oK(ifif) for the case of a dipole-dipole potential. The relevant equations are Equations (17A) — (178). In Chapter II it was noted that matrix elements of the dipole moment operator between the same rotational levels of a linear molecule are zero; i.e., =o. 112 Therefore, only Equation (176) can be non-zero. The point 2 is that the v in Equations (17M), (175), (177) and (178) 2 is zero while the v in Equation (l76) is nonzero. Con- siderations such as these must be taken into account in the determination of v2. CHAPTER IV APPLICATION OF AN ENERGY SUDDEN APPROXIMATION TO THE CALCULATION OF AK A. Derivation of Equations and Numerical Results In Chapter II the Anderson theory was extended to enable the calculation of Ag'i'fi' The major weakness of this theory and in general any perturbation technique that uses linear trajectories is that the scattering matrix is not unitary. This necessitates the evaluation of a hard sphere cutoff. If the use of a hard sphere cutoff is satisfactory for Agifi and Agiii’ then its use for . Ag'i'ii is questionable. For Agifi and Agiii the cutoff essentially is that value of the impact parameter for which the probability for scattering(either elastic or inelastic) K is one. In calculating A. 1'i'ii - individual transition probabilities - the transition probability goes to zero as the impact parameter goes to zero. This is because for small b there is a large number of possible transitions, making the probability for any one transition small. (This is just the opposite case from moderately large impact parameters where only collisions with small Aj 113 11“ are likely.) This behavior is not reflected in the cutoff procedure described in Chapter II. It is difficult to incorporate such behavior because the probabilities usually do not go to zero in a simple fashion. These problems can be avoided by using an exponential (152,15u) approximation to the S matrix s = exp(2in) = 2(2in)n/n! , I (182) n where an element of the phase shift matrix n is(15u) nij(b,¢) = n0(b)6ij - EELZDAvij[v(t),e(t),¢]exp(1wt)dt. (183) The integral is taken over a classical trajectory determined by a spherically symmetric potential V0. In Equation (183) AVij is the matrix element of AV = V - V0, where V is the full potential and V0 is the part used to determine the trajectory, ”O is the phase shift corresponding to V0 and w = (Ei - Ej)/fi with E1 the internal energy of state i. The z-axis is assumed to be parallel to the initial velocity. This means that the polar angle, a, and the intermolecular distance, v, depend on time, but the azi- muthal angle, ¢, does not. This chapter will treat only linear trajectories so that comparison with the previous Anderson theory results may be made. In this case VO = 0, ”0 = 0, and AV = V, 115 so that the working equation for n is _ -1 m i nij(b3¢) - — f_ooe 2h wtvij[r(t),6(t),¢1dt. (181) Comparison with Chapter II shows that nij is just minus one-half the P matrix element used there. Writing out the first few terms of the expansion (182) as . 2 s = 1 + 21m + 13%92—.+ ... (185) clearly shows the relation of the current results to the 1 - iP + (iP)2/2 + Anderson theory, 8 It is very difficult to evaluate the infinite sum in Equation (182) analytically. (To the best of this author's knowledge it is not possible. The phase shift may be evaluated using the WKB approximation, but this requires use of numerical techniques.) If the "sudden approximation" is invoked, the problem is simplified considerably.(133’l36) The sudden limit is the limit where the molecular orienta- tion remains fixed during the collision. That is, the rotation time is slow compared to the collision time. For an atom and a rigid linear rotor, the scattering matrix in the sudden approximation is S (b,¢) = (186) J'Jm'm 116 where for linear trajectories the sudden phase shift is n(b,¢,6m,¢m) = 3%{:V[r(t),e(t),¢,em,¢m]dt . (187) The free linear rotor wavefunctions are ij(em’¢m)’ so that e and ¢m describe the (fixed) orientation of the molecule. m A comparison of Equations (18“) and (187) shows that in Equation (187) the exponential has been set equal to one. This implies that the sudden approximation will be valid for small internal energy spacings. It has already been mentioned that transitions with small Aj occur at large impact parameters, where the trajectories are linear to a very good approximation. At small impact parameters col- lisions with large Aj are possible and the corresponding trajectories will be non-linear. Therefore, collisions for large Aj will suffer in two respects when calculated in the sudden approximation with linear trajectories. First, the assumption of linear trajectories will break down, and second, becomes large and setting the ex- wij ponential to one may not be valid. It will be shown later that probabilities calculated in the sudden approximation are too large. Equation (186) is difficult to use directly because many integrals must be evaluated. However, because the factor exp(2in) is a function of em and ¢m it is possible to expand it as 117 exp(21n) = i; flu qu(em¢m) (188) with the coefficients fAU given by . 2W . fAu = &; sinemdem A) exp[21n(b,¢,6m,¢m)]Y§u(em¢m)d¢m (189) The S matrix results from taking matrix elements of exp(2in); i.e., matrix elements of Equation (188). If this is done, 8- = J'jm'm = Z fAU Au = z; (-1)m' [(23'+1)(23+1)(2 +1)]1/2 Au an A J J' A (190) x6 ex 1». O O m—m' u The last step is a standard result and was used earlier when the matrix elements of the P operator were evaluated for multipole-multipole potentials. (155) Cross has used the above formulation to evaluate transition probabilities P for dipole—dipole potentials. J'J After integrating Pj'j over the impact parameter, these 118 (It is important to note however, that P.. does not correspond to A9...; JJ 1111 . . O quantities correspond to Ai'i'ii' this will be discussed later.) A brief review of Cross' derivation will be given before developing the formulas for the general . . . K relaxat1on coeff1c1ents Ai'f'if' The transition probability from state j to state j' is given by 2 2 . . b = T . g = o o PJ,J( ) |S J.J] (191) J'J 6 3'3 If the Pj'j are averaged over the degenerate m states, the probability is (192) . . b = 2'+ ' . PJ.J( ) ( J 1) z T By using Equations (190), (191) and (192) it is easy to show that 35% 2 . . = " , 1 PJ.J(b) i (23 +1) ( > FA(b) ( 93) 0 0 o where _ -1/2 2 FA - zl(un) flu — dxol . (19M) 1.] Equation (193) is valid for a linear molecule - atom system. 119 It will be generalized to symmetric tops later. If Equa- tion (193) is integrated over the impact parameter, the result is A? J'J'J'J" J J' A 2 O ., co Aj'j'jj = 21T 2 (2.] +1) ( ) f0 FA(b)bdb (195) A 000 For a system of two linear rotors the transition prob— ability may be written 1 = ° - — °v v-v v P5131(b) [(2Jl+l)(2J2+l)1 ma'mzm'|<11m132m2| 1 1 2 2 -1 J2 . . . 2 x |exp(21n)-1|jlm132m2>| (196) where exp(2in)—l is just the T operator. Since only mole— cule l is observed, the internal states of molecule 2 may be summed over by using Equations (127) and (128). This gives 1 -1 2 ' = 74—— + -v 1 - _ 1 x dn2 (197) For linear perturbers the volume element dQ2 is sin62d62d¢2. 120 Cross<155> showed that for a dipole-dipole potential, the function Fx(b) may be evaluated as _ 1 n . . _ 2 Fx(b) _ (A+§) &)[Jx(y s1n8) 510] sin82d82 (198) where 2U U2 y = —_lE—— 3 (199) hb v and jA(x) is a spherical Bessel function. It is related to the normal Bessel function by W 1/2 JA(X> = (5;) Jx+1/2(X) (200) . O . Finally, Aj'j'jj is given as O 2 /h )Z(2" jl 3i A 2A (201) AJ'J'jJ ( "U1U2 V A Jl+l) (O O O) A, where Tr2 A0 ‘ EH n2 -1 AA = 1r[(2x+3)(2A-1)] (A # O) n2 AtOt = 2 AA = 7r . (202) 121 The derivation of Equation (201) followed that given by Cross. A corresponding equation for a general relaxa- tion coefficient may be evaluated in the following manner. The general cross section is J -J'+m -m' _ oK(i'f'if) = z 2 2m (- 1) f f f f(232+1) 1 $fQJ2 K K 231+1 1/2 ) [23;11] Q '1" )(:1 J1“ Y 1 x[6 aj 5 '8 5 . - X l (203) After the molecule 2 states have been summed over (by again using Equation (127) and (128)), (203) can be written as J J K j j +m m' 1 f 8(if'1f)=1%;nzz' (-1)ffff( ) mimiQ m —mf-Q mfmf Ji jf' K 2ji+l 1/2 * X [231+l [aJij'anJ' 5mim'5m m jfmfls [Jr f mi-m'-Q' i x 1dQ2 (20“) 122 Use of the relation J J' A J J' A = _ m . ., 1/2 éj'jém'm i ( 1) [(2J+l)(2J +1)] ( )( >510 O O C m-m' u (205) for 6 -.6m m’ and 6 ,6m m" use of Equation (190) for jiJim 1m 1 Jfme fm f the matrix elements of the scattering operator, and use of Equation (D8) enables the cross section to be written as x , , K+u+J -J' o (l'f'lf)= 3% ffE (- 1) f f(253+1)[(2jf+1)(2j£,+1)31/2 MA J1 J) A Jf.J} A J1 J5 A 1 2 X ( >( { ) [(E—ffx - 6A03d92 o o o o o 0‘ " K U Jf 3f (206) By using Equation (206) o K(i' i' ii) and AK can be i' i 'ii obtained as OK(i'i'ii) = _X(_1)K(2J1+l)3/2(2ji+1)1/2 A J1 J; A2 Ji 31 A x < ) ‘ } FA (207) o o o 31 31 K and 123 K _ K . /2 . 1/2 11.1.11 - —(2wulu2/hv) i <-1> (2Ji+1§ (231+1) 0" . 31316231391A x < > { AA . (208) o'. O O O 31 31 K The coefficients Agfif and AK cannot be written in terms iiii of AA' This is seen by comparing Equations (206) and (198). When A = 0, the terms in square brackets in these equations are not the same. It is shown in Appendix H that Agfif K and Aiiii can be written in a form similar to Equation (208) with AA replaced by BA where BA = AA for A # 0 d B ”‘2 a“ 0 "127° If the system molecule is a symmetric top with one- ended symmetric top eigenfunctions, the results are almost the same as those given above. In particular, the sudden 8 matrix element is now S = j'jk'km'm {fou<3'kvmvIYAu(em,¢m)|jkm> A: (-1)k'+m'[(211'+1)(12l;1r+1)(2A+1)]1/2 u 33' A j j' A x( >( )fxu. (209) k-k' 0 m-m' u 12m In the cross section, 6 ,6 ,6 , is replaced by 5131 kiki mimi k!+m! o 0 1/2 ..5 ,5 ,- 2 (—1) 1 l[(2J.+1)(2J'+1)] 131 kiki mimi Au 1 i . ., . ., 31 31 A 31 Ji A x( )( 6A0 (210) k.-k! o l l The angular momentum algebra is the same as that leading to Equation (206). The analogue of Equation (206) is K+A+j —j' K ., ,. = -1 f f . o (1 f if) H? ff ;I(-l) (235+l) u J- 5' A J J' A J- J! A . ., 1/2 1 i f f 1 1 x [(23f+1)(23f+1)] ki-ki O kf'kf O 3% 5f K x [3- f2 - a Jan (211) Er— Au A0 2 The same comments regarding the functions A and B for A A . K K the COfoiCIGNtS A1111 and Aifif for the linear rotor case apply to the present case. Proper consideration of symmetric top molecules in- volves the parity-adapted symmetric top eigenfunctions Ijkm€> = N€[ljkm>-+e| J-km>] (212) 125 where N6 = l for k = e = 0 NE = 1//2 for k > o and e = :1 By using these functions, the scattering matrix in the sudden approximation may be written as Sjtjkvkmvmm = . ., J' Au 3 5' ; J J' A J J' A J J' A x: + 5' +6 +ee' k-k' -k-k' k' 0 —k k' 0 In this discussion only k = k' will be considered. In Au m-m' u (218) this case _ k+m' (23+1)(2j'+1)(2A+1) 1/2 Sj'jk'km'me'e ’ NEN€v('1) f; E “n J J 5' A J J' A I x [l + ee'(—l)J+J +Alfxu (21H) k-k O m-m' u The cross section in Equation (211) becomes 126 K . . '1 K+A+j —j' o (1'f'1f)= ENE N€,N€ Ne, ff 2 (-1) f f f f 1 1 Au jf+jf+x 31”?A v _ v _ -v J, 33 A Jf~Jf A J, 53 A} ki-ki O k -kf O f 3% Jf K X [Hg f - 5AOJdQ2 . (215) All of the previous equations are valid for the case that molecule 2 is a symmetric top. Instead of Equations (127) and (128) the following two equations are used to sum over the internal states of molecule 2. * 2jé+l Dj2 j' Z Z 2 Dk m,(aBY)Dk 2mé(a' B' y' ) = 6(d- -d' ) jék'mé 8n 2 2 k2 X 5(8-8')5(Y-Y') (216) 2j +1 Dj j2 22 2 l . X (aBY)D (dB ) = -—- Z 1. 2J2.” k m 8W2 Dk2m2 k2m2 Y 8W2 k 2 2 2 . 2 = (232+l)/80 (217) In the application of these two equations two points 127 must be considered. First, the factor (un)-l in Equation (197) becomes (81r2)_l in the present case. However, the volume element dQ2 for a symmetric top is da sine dB dy so that after integration over a the (14K)-1 factor remains. Second, it appears from Equation (217) that an additional factor of (2j2+l) will be introduced. However, since the quantity Pj'j of Equation (196) is averaged over all degenerate states of the perturber an extra factor of (232+1)‘1 should be inserted in Equation (196) when the perturber is a symmetric top. This additional factor is necessary because of the average over k2. The preceding analysis has been applied to the cal- culation of all relaxation coefficients for the J = 2 + 1 transition of OCS and the (J,K) = (3,3) inversion transition of NH3. The coefficients are given in Table XVII. The assumed parameters for the calculations are the same as for the extended Anderson theory calculations and are given in Tables III and X, respectively. The values of T for OCS l and NH were calculated from Equations (135) and (1A0), 3 respectively. All of the relaxation coefficients here are larger than the corresponding quantities obtained from the modified Anderson theory of Chapter II. In addition the Aifif are larger than the eXperimental low-power linewidth. It is expected that cross sections calculated from a sudden approximation will be larger than those calculated from a theory where the energy differences between internal states 128 .Iu++ ++|n ++++ mmmm hoeumwwhecm wmmumm mean an» new mmmm nmehecm wee .mmmm hoe mmehpcm was on Hmoeecmee new mmmm use mmmm hoe nmfihecm mnen .mm CH .e.eew< mew mesmeoeeemoo when Hmmw.o u m9\He o.meH n awe ammooo. u He un++ o.mH: m.ome :.mm . m.mwmn :.oa:: m.HHmu H.3mm- mmmm m.mmHH m.mmme m.wmme :.mmma o.HomH o.meHH m.e:efi mmmm m a mz maee.o n me\He m.omm u Awe eefioo.o u He uuuuuuuuuuuuuuu :.Hmm H.Hmm m.omm :uunuu mama uuuuuuuuuuuuuuuuuuuu :.mmHu m.msan m.mmmn Hamm uuuuuuuuuuuuuuuuuuuu w.ss u H.moau m.smfiu mmaa .......... m.mmm e.wem w.wmm w.mmm o.mom mmmm uuuuuuuuuuuuuuuuuuuu e.mmm H.Hmm m.mom HHHH moo m m a m m H o “H.323 x m.c0Hpmeflxopoo< cooosm one CH Umpmazoamo mmz mo coHpHmchB coampm>zH Am.mv u Ax.ev me» new moo so caehencmhe H + m u a was com mesmeoeeemoo coeewxmfimm .HH>x wanes 129 are considered. This will be discussed later. The other major difference from the Anderson theory results is in the tensor order dependence of the cross sections. While K the K-dependence of A. 1,1,11 is similar in both theories, the Agfif and Agiii show a larger variation with K in the sudden approximation. (The fact that the 3_3_3_3_, 3+3+3+3+ and 3+3_3+3_ cross sections are the same may be easily demonstrated by using Equation (21M).) Table XVIII gives the B matrices obtained from four- level corrections to OCS and NH3. Tables XX and XXII describe and compare the fits of the theoretical lineshapes to a sum of Lorentzians. The procedure is the same as that used in Chapter II. As before, the four-level effects are larger for OCS than for NH3. However, the effects of four-level corrections for OCS are smaller than the same effects in the Anderson theory calculations. A plausible explanation of this is that because the sudden approxima- tion neglects all internal state energy differences it effectively already treats the system as a many level one. The sudden value of 0.68 for Tl/T2 gives rise to a (Tl/T2)O of 0.71 which is in excellent agreement with the experimental value. If one assumes that the sudden ap- proximation is a valid description of the collision dynamics, the value of Tl/T2 obtained from a lineshape experiment which has been analyzed by fitting to a sum of Lorentzians would be 0.68. 130 Table XVIII. Matrices of b Coefficients in Sudden Ap— proximation for Calculation of U-Level Cor— rections. K: Doublet (J i 5; K = 0. OCS J = 2 + 1 Transition (J i A; O) and NH3 (J,K) = (3,3) Inversion OCS ma (2) (0) (3) (1:) 505.17 -137.77 -321.U7 - 23.66 — 8.89 -229.62 507.95 - 76.5“ -1U7.55 - 26.58 B = -107.16 - 15.31 535.79 - 5.10 - 2.32 - 55.20 -206.57 - 35.72 509.31 -152.68 - 26.66 — “7.85 - 20.87 -196.30 510.03 (1) (2) B' “35.37 —163.07 11 —271.79 u2u.91 NH3 (33+)b (33-) (u3+) (u3—) (53+) (53-) 11u7.9 -56u.1 —170.6 — 69.8 -28.0 _ 22.6 -56H.1 11U7.9 - 69.8 -170.6 -22.6 - 28.0 B = -219.3 - 89.8 12ou.6 -382.u —226.3 — 61.8 - 89.8 -219.3 —3A2.u 12OU.6 - 61.8 -226.3 - “3.9 - 35.6 -276.6 - 75.5 1212.0 -238.0 - 35.6 - “3.9 - 75.5 -276.6 -238.0 1212.0 (33+) (33-) v = 1084.72 -618.56 B11 -618.56 108u.72 8The numbers in parentheses above the matrix are .for the corresponding columns. The rows are in order. {Phe symbols in parentheses above the matrix are Inarity for the corresponding columns. The rows Same order . the J values the same J, K) are in the 131 Table XIX. Relaxation Parameters in Sudden Approximation for the OCS J = 2 + 1 Transition: U-Level Effects, J i A. 2 K aK (3 ) 1 520.5 3 531.“ 11 22 12 21 2 K bK bK bK bK (fl ) O “35.37 U2u.91 -163.07 -271.79 2 532.01 U98.U1 - 81.71 -136.17 A 518.92 -1- T1 — 6AA.38 Tl/T2 = .8078 132 Table XX. Summary of Tl/T2 Calculations in Sudden Ap- 2 + 1 Transition. proximation for the OCS J w/o U—Level With A-Level Effects Effects a a p/mtorr Tl/T2 (Tl/T2)O Tl/T2 (Tl/T2)O 60 .761 .802 .808 .850 80 .761 .800 .808 .8u8 100 .761 .799 .808 .8A6 (Tl/T2)O (exp) = 1.0a : 0.10 aExperimental parameters assumed for the determination of (Tl/T2)0 from the theoretical lineshapes are power = 10.00 Mw, attenuation = 0.800. 133 Table XXI. State to State Relaxation Parameters in Sudden Approximation for the OCS J = 2 + l Transition.a (2121)a m/m' -l 0 l 0 l8u.1 212.6 18u.1 1 159.u 18u.1 159.u (1111) m/m' -l 0 l 0 - 9.2 523.5 - 9.2 l -l8.u - 9.2 532.7 (2222) m/m' -2 -l 0 l 2 — 7.0 3.0 527.9 — 3.0 - 7.0 - 1.5 -1l.2 - 3 0 533.u — 9 - 3.0 - 1.5 - 7.0 - 9.8 529.2 (1122) m/m' —2 -1 0 l 2 0 - l 7 -52 5 -69.A -52 5 - 1 7 l - 5 l - 3 A -18.6 -50 8 -100.0 (2211) m/m' -l 0 l 0 -3l.0 —115.8 -3l.0 l - 5.6 -87.5 -8A.7 2 - 8.5 - 2.8 -l66.6 ‘ aThe values tabulated are <> where J agd J' are the numbers in parentheses. All values are in 2. l3“ Table XXII. Summary of Tl/T2 Calculations in Sudden Ap- proximation for the NH3 (J,K) = (3,3) In- version doublet. w/o U-Level With U-Level Effects Effects a p/mtorr (Tl/T2) (Tl/T2)O (Tl/T2) (Tl/T2)O 30 .682 .705 .686 .712 MO .682 .70A .686 .709 (Tl/T2)O(exp) = 0.71 i 0.07 aExperimental parameters for the determination of (Tl/T2)0 from the theoretical lineshapes are power = 15.0 Mw, attenuation = 0.800. Table XXIII. 135 State to State Relaxation Parameters in Sudden Approximation for the NH3 (J,K) = (3,3) Inversion Doublet.a’b b (+—+-) mfin' -3 -2 -l 0 l 2 3 0 -0.7 -2“.3 - 2.6 1203.1 - 2.6 - 2“.3 - 0.7 l -0.5 - 0.2 -29.6 - 2.6 1212.0 - 18.6 -l2.7 2 0.0 - 0.8 - 0.2 - 2“.3 - 18.6 1222.“ -30.6 3 0.0 0.0 - .5 - 0.7 - 12.7 - 30.6 1192.5 (++-—)C m/m' —3 —2 -l 0 l 2 3 0 -2.3 - 2.3 -277.“ 0.0 -277.“ — 2.3 - 2.3 l —0.1 - “.5 0.0 —277.“ - “8.2 -229.“ = “.5 2 -0.1 0.0 - “.5 - 2.3 —229.“ -185.7 -l“2.1 3 0.0 -00.1 - 0.1 - 2.3 - “.5 -1“2.l -“15.0 aThe values tabulated are <> where J = 3 and the parity is indicated in the parentheses. All values are in 32. bThe values for (++++) and (--—-) are the same as the values for (+-+-). cThe values for (—-++) are the same as the values for (++—-). 136 State to state relaxation parameters are given in Tables XXI and XXIII. It is interesting to note that the elements <> (where 3 may also denote parity for the case of NH3) for Am 3 l are all very small relative to those elements diagonal in m. This is reminiscent of the results obtained from the Anderson calculations. In the latter case the elements for Am 2 l are small because the Anderson theory is a first order theory. One might expect that in an infinite order theory such as the sudden approximation the off-diagonal elements would be large. That they are small implies that the m—component pairs are not significantly coupled by collisions. B. Application of the Sudden Approximation to Four-Level Double Resonance Experiments The most powerful and general method of observing rotationally inelastic scattering is microwave—microwave double resonance. A brief summary of a four-level double resonance experiment in NH3 will be presented here. Oka has published a complete review of these experiments.(15) A four-level double resonance experiment on NH3 is depicted schematically in Figure l. The double arrow represents a microwave pump beam which tends to equalize the populations of the pair of inversion levels that it connects. This change in populations is transferred to 137 + l J+I.K 81$ 3 ' - 2 J+I.K Figure 1. Energy level scheme for a four-level double resonance experiment in NH3. 138 other levels by collisions, and is monitored by observing the change in absorption intensity, A1, of a weak radiation field (the signal). To a good first approximation the observed change in intensity can be related to collisional rates among four levels, the two pump levels and the two signal levels. The quantity which is measured experimentally is ka(15) n = AI/I. 0 has shown that + n = \2 + + (218) s k +k +2k +k a Y 8 E if the following conditions hold. 2 + K13 k2, - kY + klu k23 k0 k3u E ku3 = k k+ z k+ B B — B In these equations knm is the rate constant for collision- + ally-induced transitions from state n to m. The symbol ka denotes the collisional rate from a higher to a lower level and is related to k; by 139 = exp(-AE/kT) . (220) W‘ W Q+Q+ The condition kg 5 kg is a statement that the energy dif- ference between the levels of an inversion doublet is very small. Oka has calculated n by using a simplified Anderson theory with dipole-dipole potentials.(156) For this case, k E 0 and k E 0, so that Y E —vp k+ ”=6 —,——°‘-—. (221) s k +2k a 8 Although this calculation gives the algebraic sign of n in agreement with experiment the calculated magnitude of n is about five times too large. With the sudden approxi- mation developed in this chapter it is possible to cal- culate n by using Equation (218) The collisional rates k occurring in Equation (218) are just the A9'i'ii with the appropriate 1 and i' indices 1 given earlier in this chapter. Specifically, 0 21ml“2 ji+ji+A “1'1'11 = ( hv . 2 2 ' )(2Ji+l)NE N X [l+€.€!(-l) ' °.+ '+A ' +"+A 3' 31 A 31.31 A Jl 31 l(-l)JiJi ( l ) ( )AA 0 X [l+€.€!(—l) 1 l k-1< 0 1“0 Four rates need to be considered. These are AO(jk- + j+lk-) and AO(jk+ + j+lk+), which are dipole allowed in first order perturbation theory, and AO(jk- + j+lk+) and Ao(jk+ + j+lk-), which are dipole forbidden. The minus and plus signs in the current notation denote e and not the parity of the level. Equation (222) can be used to give the following expressions for the rate constants. 0 3+1 j A 2 A (Jk- + J+lk-) = c z [1+(-1)“1l2 ,(223) A k - k 0 J+l J A AO(jk+ + j+1k+) = C X [l+(-l)x+1]2 A A’ (22“) A J+l J AO(jk- + j+lk+) C 2 [1-(-l)“l]2 j):A , (225) J+l J AO(jk+ + j+lk-) = c z [l-(-1)“1l2 AA. A (226) where 2"“1112 c = (——————)(2j+l)(% ) . (227) hv In Equations (223) and (22“) only terms with A odd contribute, while in Equations (225) and (226) only terms with A even contribute. Also Equations (223) and (22“) are equal, as are Equations (225) and (226). Therefore, in the sudden 1“1 approximation the first two conditions given in Equation (219) are satisfied. It is useful to compare these results with the Ander- son theory expression for Ao(i'i'ii) (dipole—dipole inter- action) . ., J2 J2 1 2" Z( Ji+l) 1 2 J' 2 k.—k. k2—k2 0 16K 00(1'1'11) m |aAk3|2 (228) The only part of this expression that depends on e is laxkjl2. The dependence on e is due to the energy spacing of the inversion doublets. Therefore, there should be differences between the two dipole allowed transitions. However, the computational results presented in Chapter II show that to a resolution of about 0.2 A2 there are no dif— ferences. The results of the sudden approximation calculation of the rate constants k are summarized in Table XXIV. Oka's calculation using a simplified Anderson theory shows the best agreement with the experimental value. The agree- ment is probably merely fortuitous, because only resonant collisions have been considered. Inclusion of non-resonant collisions would lead to a larger value as discussed by Oka.(156) 1“2 Table XXIV. Sudden Approximation Calculations of Rate Constants for Four—Level Double Resonance Experiments in NH3,a 1: ka 170.6 2'. = 6 .8 RY 9 k = 6“. 8 5 l k = 10 .0 g 3 kg-ki ( + + Y ) = 0.0685 k +k +2k +k SUDDEN a Y B E k; —:—*—— = 0.0567 R +2k OKA d B OBSERVED = 0.0112 aThe levels involved are J = 3 and K = 3 in Figure l. l“3 C. Energy Corrections to the Sudden Approximation Scatterinngatrix The sudden approximation to the scattering matrix neglects energy differences between internal states. Therefore, it is expected to give transition probabilities that are too large. Because the sudden approximation relaxa- tion coefficients are so easy to calculate it is of in— terest to try to correct the approximation by reintroduc- ing consideration of internal state energy differences. (157) have made an attempt at doing this. De Pristo et al. A brief description of their approach and applications to relaxation coefficients will be given here. Later, some other ways of incorporating energy corrections will be considered. It is difficult to evaluate the accuracy of energy corrections because there is little experimental data on transition probabilities and because there are very few accurate fully quantum mechanical calculations of transi- tion probabilities for systems of interest to microwave spectroscopists. The argument of DePristo et al. is as follows. The perturbation series for the exact S matrix in the inter- action representation is 1““ 5mm. = 6m. - in‘1 L: exp(inmm.t)vrsnm,(t)dt — h‘2 z f°° exp(iw t)VS (t) It exp(iw t') m" ...00 mm" mm" _| j m2mé 2 o o -l . 2 = Z 2 +1 2 + v v-v v m3' 3' [( J ) J2 1)] Il , m2m2 2 (2M2) with a similar equation involving SS and PS. The analogue of Equation (2ND), written in terms of cross sections, is 0(3) = —O—(—P§)—O(SS) (21:3) 0(P ) In Equation (2M3) C(88) is the cross section evaluated from a scattering matrix computed in the sudden approximation. 150 The quantity 0(8) is an exact value of the corresponding cross section. Now, 0(P) and C(PS) are cross sections cal— culated by using the approximation in Equation (238). Equation (2A3) will be useful if the ratio o(P)/O(PS) is more exact than either of its factors. From Chapter II 0(P) may be written (for a dipole-dipole potential) . . -l . . 0(2) = 2' z [(2J+l(2J2+l)] |l2 mm ., m2m'J2 . 2 . . 2 J J’ 1 J2 Jé l = C 2: B (2j'+1)(2j'+1) 2 l o o 0 0 0 A2 (2AM) where 2 c = (plu2/3th2) It was shown by TC that 2 2 2 2 z B = 2[K (k) + MK (k) + 3Ko(k)] . (2“5) l l A1*2 2 1 Defining f(k) as f(k) = g k“ 2: Bill, A1A2 enables 0(P) to be written as J.J" 1 232.331 0(P) = 8C .2 f(k)(2j'+l)(2jé+l) J' 2 O O O O 0 (2M6) In the sudden limit w and therefore k goes to zero and f(O) is one. Therefore . . 2 . 2 3'1 :32 .jé (P8) = 80 z (2j'+l)(2jé+l) " J2 o o o o o 2 J J' 1 =80(2j'+l) (2“7) 0 O O Substitution of Equations (2A6) and (2&7) into Equation (2U3) gives , 2 32 Jé 3- 0(8) = [ll f(k)(2jé+l) 10(85) . (2A8) J2 o o 0 In Equation (2MB), 0(8), f(k) and C(88) are understood to be evaluated at the same given impact parameter. The interesting features of Equation (2A8) are most clearly seen if molecule two is considered to be adequately described by the sudden approximation. In this case the sum over 35 in Equation (2MB) may be carried out giving 0(8) = f(k)o(SS) 152 A qualitative plot of f(k) vs. (k) is given in Figure 2 for a dipole-dipole potential. The most significant difference between Equation (2&8) and Equation (235) is that while in Equation (235) 0(3) is smaller than 0(55) for all values of m at a given b, this is no longer true in Equation (2MB). (wj'j 3 O in Equation (235).) For values of m such that k is less than k0 Equation (2M8) predicts that 0(8) is larger than C(83). If it is desired to extend Equation (2MB) to relaxation coefficients, the equation must be averaged over the impact parameter; i.e., J2 jé l 2 2n A ho(S)db = 2n;l(2jé+l) if bf(k)o(SS)db J2 o o o (2H9) The integral on the right side of Equation (2&9) involves integrals of the form °°72C 2wb L,b JA(;§)Kn(?7)db , (250) where C is a constant, n is an integer and A is a half- integer. All attempts to integrate Equation (250) analyti- cally have failed. It is possible to analytically evaluate an approximation to Equation (2H9). Because 0(SS) is proportional to FA’ the integral on the right side of Equation (2A9) is f(k) Figure 2. 153 _--—--—-e-- Ii 3? ‘O J? Qualitative plot of f(k) vs k for a dipole- dipole potential. k = g? . 15& 4? bf(k)FAdb , which may be approximated as gff(k)dbtf bFAdb . (251) The second integral in Equation (251) is just the integral that was evaluated earlier for the pure sudden approxima— tion. (If the integral was approximated as {fbf(k)db{fFAdb, the introduction of a hard sphere impact parameter would be necessary.) The first integral may be evaluated analytically. The details are given in Appendix I. The result is (X) h f(k)db (%)(%)(l.5)3[1‘(1.5)3u (252) v 2.77586(5) In this approximation Equation (2&9) becomes 52 i; 1 2 MS) = 2 (235m <§><§><1 5>3[1~<1.5>1“% 32 o 0 o 2 J2 x A(SS) (253) where 03 is a Boltzmann factor. 2 155 One of the advantages of the sudden approximation is that the sum 0V9? 32 and 35 may be neglected. (The per- turber states have been summed over by using the closure relations for spherical harmonics and rotation matrices given earlier.) This advantage is lost in Equation (253). The essential prediction of Equation (253) is that A(S) m (%)A(SS) (251) When a is small, A(S) is much greater than A(Ss) while when w is large, A(S) is much less than A(SS). Intuitively, only the latter limit seems reasonable. To attempt to establish the validity of approximation (Equation (251)) the exact integral should be evaluated numerically at least once. To compute a cross section, however, the exact integral will have to be evaluated for every allowed value of A. Because |j-j'|:kij+j', this could become very expensive. CHAPTER V ADDITIONAL RESULTS In this chapter two additional results will be pre— sented. They are concerned with the equations of motion for the density matrix. First, the equations of motion for a symmetric top Q-branch transition in a static external electric field will be derived. The second part of the chapter discusses some phase conventions for reduced matrix elements and their relation to previously derived equations of motion for state multipoles. A. Comparison of T2 for Transitions in a Static Electric Field in Linear and Symmetric Top Molecules The case of a linear rotor in a static external elec— tric field has been discussed in LM. Before discussing the analogous case for symmetric tops, a brief summary of the LM results will be presented. In a static electric field and plane polarized radiation the (jfm) + (jim) and (jf-m) + (ji-m) spectral transitions are coupled. By using the Wigner—Eckart theorem it is possible to show that 156 157 = (255) This result and the following linear combination of density matrix elements oi(JiJf) = p(jimjfm)to(Ji-mjf-m) (256) can be used to write the equations of motion as 0 a ’ ° ° ' o o i §€Qi(JfJi) = wfipi(JfJi)-€COSwt x [tr(jiji)-pi(jfjf)l : (o I ) o i — + _ O C iAfifiD- JfJi lkfiifo:(lilf) (257) a 0 o o . o o 0 i -a—t01(Jj-Ji) = ’ECoswt[pi(JfJi)'pi(Ji-Jf) o 0 I i o O . i o o X (meluIJim>]-1Aiiiipi(JiJi)—lxiiffpi(JfJf) (258) In the above equations the m-dependence of pi(jijf) has been suppressed. The A: are defined as i . . O O O Afifi=<>i<<3f_mJi—mlAlmeJim>> (259) 158 i + A d 1' i fiif’ iiff 3“ iiii' Equation (27) may be used in Equation (259) to give with analogous equations for A 2 jf 3i K+jf+ji K = Z (2K+l) [li(-l) JAfifi K i Afifi (260) m- m 0 'th 1 ‘1 t' f i i i w1 s ml ar equa ions or Afiif’ Aiiff and Aiiii' Equa- tions (257) and (258) decouple the pair of transitions (jfm) + (jim) and (jf—m) + (ji—m) from the other possible transitions among m states. They also show that 0+ and p_ are uncoupled. The polarization for this case can be written P = 0+(JiJf) + (JimIUijm>p+(jfji) (261) so that only the equations of motion for 0+ are needed. The system can be treated analogously to a two-state system and gives for relaxation times 1 + T ‘ Rexfifi l + + + + T1 ‘ 2‘Aiiii‘*ffii**ffff‘*iiff) (262) In the above it was assumed that the strength of the field was such that the spectral lines arising from different values of m are non overlapping. The dipole moment matrix element for a symmetric top 159 evaluated with a parity-adapted basis may be written J. = (-l) 1 if . (263) The following discussion is restricted to Q-branch transi- tions in a symmetric top with inversion (like NH3). For this case, J J = (-1)3’m Eif . (26&) m-m Also + J J l = (-1)1 m pif —m m 0 , + J j 1 = H)” 1 a (265) if m-m 0 where the symmetry properties of the 3-j symbols and the fact that m is an integer have been used. The conclusion from Equations (26&) and (265) is that = - . (266) 160 The polarization can be written "U I ‘ tr(uo) ; [o(meJim)+o(Jimem)l (267) where the sum over m is restricted to m and -m. In Equa— tion (267) the symbols ji and jf include, in addition to the j valueof the state, the quantum number k and the parity. Equation (267) can be evaluated to give P = 6_(ifii)+6_(iiif). (268) Therefore, the equations of motion for p_ are needed. Use of Equation (21) of Chapter I and Equation (266) of the present chapter gives the equation of motion for p_(jfji) as i 5%o_(JfJi)=wfip_(JfJi)-€COSwt x [0+(J1Ji)-p+(JfJf)] ' iAEifip-(iji) ‘ 1AEiifp—(jijf) (269) This result differs from Equation (257) in that here D_(JfJi) is coupled to p+(jiji) and 0+(ijf). In a manner 161 similar to that used to obtain Equation (269) the equation of motion for 0+(jj) is 3 . - . . . . i §Efl+(JiJi) = -ecoswt[o_(JfJi)-o_(JiJf) x 1 + . . . + . - 1A 1 1 p+(jiji) is coupled to p_(jfji) which is given in Equation (269). Therefore Equation (269) and Equation (270) are the equations of motion necessary to describe this system. The relaxation times are seen to be 1 _ T2 = 39(Afifi) .3; _ l + + + + T1 ‘ 2(Aiiii ‘ Affii + Affff ‘ Aiifi‘) (271) This is the same T1 as for the linear rotor. The l/T2 differs from the linear rotor case in that here the minus combination is needed. For comparison Afifi and Afifi are given as 3f Ji K jf+j.+K lgifi = i (2K+l) [l-(-l) 1 3A};ifi (272) 162 and O o 2 Jr Ji ' - + Jf+Ji+K (273) A . . = z (2K+1) 1+ -1 K ° fifi K (m m :>[ ( ) 1Afifi For Q-branch transitions in a symmetric top with inversion, Afifi is needed for l/T2 and only terms with odd K contrib— ute to Equation (272). For R-branch transitions in a linear POtOP, Afifi is needed and again only terms with K odd contribute to Equation (273). Equations (272) and (273) suggest a way of experi- mentally obtaining Agifi for K greater than 1. To be concrete, Equation (273) will be considered for R-branch transitions in linear rotors. The following comments apply equally well to Equation (272) for Q-branch symmetric top transitions. Re(A%ifi) is the low-power linewidth ob- tained in zero field. For a l + 0 transition K = 1 only K for Afifi. For a 2 t 1 transition and plane—polarized radiation K = 1, 3 for Agifi' Re Afifi may be obtained from a zero field lineshape experiment. This enables Agifi to be obtained when a non—zero field lineshape experiment is performed. For a 3 + 2 transition K = l, 3, 5. After Re A%1fi is obtained from a zero field experiment there are two remaining unknowns. However, there are three dif- ferent m-component: pairs which may be probed. This en- ables a set of linear equations to be set up from Equation (273), with one equation for each |m|. Obtaining Agifi 163 in this way would be a useful check on the theoretical calculations presented earlier. In the Anderson theory K the Agifi and Aiiii are similar in form so that indirect comparison with the AK iiii may also be made. B. Phase Conventions for Reduced Matrix Elements In this section a discussion of conventions for re- duced matrix elements and their relation to equations of motion for the density matrix is presented. Equations (&.1) and (&.2) of LM may be written for plane-polarized radiation and non-overlapping lines as 1 3L (’ m' m) = w (' mj m)-€coswt<' ml I' m> atp Jr Ji rip Jr i Jr uz Ji x [0(JimJim)—0(meme)1 -j_£}2f>p(j%m'jim') (27&) jfjim . 3 . . . . i 350(JimJim) = —ecoswt[p(mejim) -o(Jimem)1 .4_ Z 2 <>p(j'm'j'm') . (275) J'm' Equation (27&) is identical to Equation (1) of l6& Schwendeman.(55) Equation (2) of Schwendeman is 8 . . . . . . . . i §Eo(JimJim) = -€coswt<3fm|uzIJim>[o(meJim)-O(Jim3fm)1 -i. Z Z <>p(j'm'j'm') (276) j'm' 3- Comparing Equation (276) with Equation (275) shows that in Equation (276) the assumption = (277) has been made. The adjoint of a tensor operator is defined as . -o . + . (-1)p ‘ _ * (_l)p q (278) where q is the z—component of the tensor order k. As discussed in Brink and Satchler,(7l) there are two choices for the parameter p; p = O or p = k. These choices will be denoted convention I and II, respectively. If p = O, the following relations hold: _ , * (-1) q , (279) (-l)j'j'* : (280) 165 while if p = k, = (-l)k-q* (281) and = <-1>3‘J"k* . (282) In the previous four equations is a reduced matrix element and obeys the same equation as fifi of Chapter I(Equation (26)). It is noted that if Equation (279) is adopted then Equation (280) and not Equation (28) must be used. (Equations (280) and (282) follow from Equations (271) and (281), respectively, by use of the Wigner-Eckart theorem.) In the following the z-component (i.e., q = O) of the dipole moment will be considered. It will also be assumed that is real. Then, Equations (279) - (282) give the following results. Convention I (p = O): (Jimluloljfm> = (jfmluloljim> (283) J _ -J __ 6,, = (-1> 1 f ufi (281) 166 Convention II (p = l): (jimluloljfm> = “(jfmluloljim> (285) “if = fifi (286) Equation (3&) of Schwendeman<55> is _ - “J (K) = -euficoswt21C§;,[pfi(K')-(-l)ji f (K')l K Dir i.2_ atpii K -i i Aiikkokk(K) (287) In Equation (287) the Cii. are defined as J J k k' l K K' 1 fi _ r” .i ' CKK' (-1) [(2K+1)(2K +1)] 1/2 { O O (3 3f J1 J1 (288) It is possible to derive an equation of motion for pii(K) without choosing a convention. This is (neglecting collisions) a _ jf-ji fi 1 §tpii(K) - -€coswt £1 (-1) CKK' x [Eifpfi(x') - fifioif(x')l . (289) If convention I is used in Equation (289), the result is 167 Equation (287); i.e., the result is Equation (3&) of (55) Schwendeman. This establishes the phase conventions used in that work. Finally, it is interesting to note that the commuta- tion relation given in Equation (15&) of Chapter III may be used to obtain the equations of motion for the state multipoles. This is done by recognizing that these equa- tions are essentially just the commutators [pii(K)“fi(l)]’ [pff(K)ufi(l)] and [pif(K),ufi(1)]. These commutators may be easily evaluated to give, respectivelv, j _J K 1 K l< l K' ./§ 2 [(2K+1)(2K'+l)]l/2(-1) f { } K O O O Ji'Ji 31 x pfi(K') . - (290) ' -J /3 2 [(2K+l)(2K'+l)]l/2(-1)Jf i K 1 K K 1 K' K' } O O 0 J1 3f 3f X pfi(K') (291) K 1 K K 1 K'} E 2 [(2K+1)(2K'+1)Jl/2 [ K! O O 0 Ji 31 5f k 1 K' x 011(K') -{ }pff(x')l . (292) if J, J, 168 The factor /3 arises from a different normalization of reduced matrix elements. APPENDICES APPENDIX A EQUIVALENCE OF EQUATIONS (&2) AND (&3) In this appendix the equivalence between Equations (&2) and (&3) of Chapter II is established. By using the four equations, it Q" J! 2'-jf-M% ‘f = 23 (-l) f f(2J +l)1/2 J M f f f mf m—lVIf ijtJfo> , (A2) 2-3 -M. Ji 1 Ji |jimilm> 2: (-l) i 1(2J.+1)1/2 J M. l i 1 m m-M [jiniMi> , (A3) and 169 170 V I ' £"11'M1 1/2 J k J1 x . (5) * Also, S. .S ,f can be written as 171 0 -0' o -0 _ _ '- _ y 3., s - z z 2 (-l) l i f'f J J'J J' i i f f v M! MiMiMf”f x [(2J +1)(2J'+1)(2J +1)(2J'+l)11/2 i i f f t t t 0' . J 2 J1 Jl 2 J1 Jr 2' Jf 3f 2 J1. X 1 '_V _l v !_1 _ i m Mi ml m M1 f m Mf f m Mf SJi SJf X i'i f'f (A6) Use of Equations (A5) and (A6) and the fact that the scatter- ing matrix must be diagonal in Jk and M k allows Oi'f'if to be written as 1 j.-j!+j -jf—j.-j'-M.-M!-M —M'—m.-m x (-1) 1 l f 1 f 1 1 f f 1 1(2J1+1)(2Jf+1) 1/2 ¥H_3f K 31.3% K' ji 2 J X [(2K+1)(2K'+l)] i .. _ V- '_ _ mi mf Q mi mf Q i m M1 31.1' Ji 3f'1 Jr 51‘1' Jr J J * x [8 8 -S i S f 1 i'i f'f i'i f'f v v_ _ v 1_ i m Mi f m Mf mf m Mf (A7) 172 Finally, use of the relation ji+jf+2+mi+mf+m 3f 1 Jr 31 1 J1 J1 jf K 2? 2 <-1) m.m m 1 f mf m—M mi m-Mi mi-mf-O (A8) twice and summation over the resulting two 3-j symbols results in Equation (U2). APPENDIX B CONVENTIONS FOR REDUCED MATRIX ELEMENTS In this appendix the LM and TC conventions for reduced matrix elements are compared. According to Equation (U.lO) of LM, the LM convention for reduced matrix elements is . j'-m' = (—l) —m Q'm (B1) 1 This is Equation (5.“.1) of Edmonds.(142) The TC convention is obtained from Equation (81) of TC as (jimiluzljfmf> F' J l J- J +m.+l f 1 = (-1) f 1 (23144)“2 F' f 0’m1 3 l J j -m. i f = (-1) i 1(29-11+1)1/2 F' -mi 0 m (B2) Equation (B2) differs from Equation (Bl) by the factor (231+1)1/2. 173 APPENDIX C REDUCTION OF EQUATION (“9) TO THE ANDERSON RESULT This appendix shows that Equation (A9) of the text reduces to the expression given by Anderson (Equation (U7') of Reference (53)) and TC (Equation (88)) for 01(jijfjijf). If the appropriate substitutions (i.e., K = l, j; = ji’ etc. are made in Equation (A9), then 1 mf—mf «111(5r' 31 K 3r o (ifif) = Z Z (-1) 5 m'5m . m m' mr f imi i i m.-Q-m m!-Q-m' m 2Q 3- l ‘ f m 4n' Ji K Jf f f . ’ Z Z (-l) mm' f f f f V mfmf x . (Cl) The first term of Equation (Cl) is Just JiKJf JiKJf JiJfK .31ij Z Z = XX m m i f- 1‘Q'm mi‘Q‘mf mi-mf-Q mi’mf’Q Q = ’3 T2'KL+15 = 1 (C2) Q 17“ 175 Use in turn of the three relations, a b c = (-l)a"b'Y(2c+1)'l/2 , (03) a B Y b+ + 1/2 = (-l) BES:+%] , (CA) and = (-l)a+b-C , (C5) allows the second term of (Cl) to be expressed such that (Cl) can be written (jflmeljimi> 01(ifif) = 1 — z z (231+1) x (06) mef i which in a slightly different notation is exactly the expression given by Anderson and Tsao and Curnutte. APPENDIX D DERIVATION OF ANDERSON-LIKE EXPRESSIONS FOR AK This appendix presents the details of the Anderson— like expansions of the cross sections given above in Equa- tions (60)-(62) and Equations (83)-(85). The terminology follows that of TC. The expansion T = TO + T1 + T2 + is substituted into the product T'lT giving [(T’l)O + (T—l)l + (T-l)2 + ...](TO + T1 + T2 + ...). The order of the expansion of UK is determined by the sum of the sub- scripts on the various terms of T and will be denoted as On- oK(ifif) Zeroth order: let T = T0 = l and T.1 = T61 = l. mf-m% -1 3i 3f K o = 1 - z z z (-1) (23 +1) 0 , , 2 m.m,m m 1 3; f‘f" i—mf-Q Ji jf K 5 6 5 X dmrmr mimi m2mé 323é t_m!_Q i f ~ 176 177 1 - z 2(2j2+1)-1 mim f m.-m _ m.—m _q m2Q f 1 f J1 Jf K Jijf K = l - Z Z m.m f - _ h _ 1Q 1 mf Q mi mf Q +K 6 6 = 1 _ Z 1129 = 0 Q=-K (2K+l) . -l _ -l . First order: let T - T0 = l and T = T1 = -1P or -1 . T = 1? and T = l. The first combination (i.e., T61 = l and T1 = -ip) gives mf—mg, -1 31% 313:“ o =-222 {—1) (23 +1) 1 m m'm m' 2 i i f f .-m -Q m!-m'-Q t -' (l f ‘ l f X 5. ..5 ,6 , J2J2 m2m2 mfmf 1 1 2 2 i 1 2 2 If mf = m%, as required by the factor Smfm%, the two 3-j symbols require that m. = m. Therefore, 01 can be written 2 l 1' 178 -1 JiJf K JiJf K 01 = i E Z (2j2+l) mim f o mi-mf" i-mf- m2Q X (jfmf32m2'Pljimij2m2> ’ OI” _ . . . -1 . . . 01 - 1 £:£:[(2Ji+1)(232+1)] ‘i 2 (D1) where 5. 6 C O m. Ji Jf 51 ij 3131 mi 1 Z = (231+l) m Q f .—mf— H—m — has been used, By a,completely analogous procedure, the combination T0 = LT;l = 1P gives . . -l . . . - 1 m2; [(2Jf+l)(232+l)] . (D2) f 2 Second Order: Let T-1 = l, T = -P 2, T.1 = -P g, T = 1; or T" = 1?, T = -iP. 179 The first combination gives 02 = - z z 2(-1) f(232+1)1 ' 1 mimimfmf m. —mf— Q mi-m%-Q m m'Q j' 1 2 2 2 X 6. 6 6 3232 mfm% mimi i i 2 2 2 i i 2 2 1 _1 jiJf K j. = 5 2323(2J2+1) mimf m -m '1’m i f f m Q 2a X (jimij2m2lp 2|J1m1J2m 2 l 0 O -1 Q . 2 — .+ + 21§1§ [(2Jl l)(2J2 l] , (D3) 1 2 Similarly, the second combination gives 1: z z[(2j +l)(2j +1)]’1<3 m ° m IP2|j m j m > (Du) 2 f 2 f £32 2 f r 2 2 ° mfm2 For T"1 = 12 and T = —iP, mf—me‘ -1 ji Jf' Ji Jf K o = —. z z z (-1) (23 +1) 2 m m'm m' 2 i i f‘f m.—m —o mi—m%-Q mm'Q J! l f 2 2 2 X (jrmf32m2lPlme232m2> ’ (D5) 180 which cannot be simplified further. This completes the Anderson-like expansion of OK(ifif). The first order term is pure imaginary and therefore contributes only to the lineshift. It will not be considered further here. Com- bination of Equations (D3)—(D5) gives Equation (60) of Chapter II. oK(i'i'ii) In Equation (50) of Chapter II the symbol 3 in o K(J. ijfjiJ f) and in 5313', etc. should be considered to be i a set of quantum numbers. If j # j', the outer terms drop out giving the purely middle term, Equation (50). Both zeroth and first order contributions to oK(i'i'ii) are zero in the same manner as for 0K(i'f'if). Second Order: Let T-1 = l and T = - l 2; T = l and 2 T’1 = - %p2; and T"1 = 1P and T = -iP. Again, the first two combinations are zero, while the last combination gives j -j. '+n -n' 2j'+l _. z z z (-1) i i i 1(232+1) 1 [231+111/2 miminini 1 2m2Q J2 (:2:‘ji f>mj j! -ni -Q q-ni— ' " ' V' " '°' 1 ' X <31n132m2|Pl312132m2><31m132m2|P|31m132m2> ’ (D6) 181 which is Equation (61) of Chapter II. oK(iiii) The zeroth order term is zero by reasoning analogous to that used for the other cross sections. First Order: Let T‘1 = 1 and T = -iP; T = 1 and T'1 = 1P. The first combination gives n.-n! :1 J1 J1 X 21 51 K 01 = — z z z (—1) l 1(2j2+1)- m,m'n.n{ l i l 1 mi-ni-Q i-ni-O 7 " m2m2Q J2 x 6 ,6 ,6. . nini m2m2 J232 i i 2 2 1 1 2 2 o o o -1 o o = 2 + o lerf [( J1 l)(232+1)] 1 2 A similar calculation shows that the second combination gives the negative of this result, so that the first order term is identically zero. Second Order: Let T‘1 = 1 and T = _ %P2 T"1 = - %P2; and T”1 = i? and T = -iP. The first two combinations are completely analogous to the ; T = l and corresponding part of the second order expansion of OK(ifif) and give 1 . . -1 . 2 . i 2 182 and 1- z [(21 l+1)(2 +1)J1< n (p2 > (D8 2 n,“ J2 J11321112 IJ1111321112 ’ ) i 2 respectively. The third combination gives ni-ni -l Joi J1 J.i Joi K - z z z (-1) (232+1) m.m'n n! l i i 1 mi-ni— mi-ni- V '! m2m2Q J2 . . . V x <32k 2K2O|12K2 > i 2 1 2 1112 J- J' k 1 i l JiJ2k1k2 K.-K. 0 Al 1 1 , 2 2 J2 K2 X (D10) -K o 183 with a similar term for jf. The middle term is Ji+jf+K [(2ji+l)(2jf+1)]l/2 k1+11112 (-1) 2 z z (-1) ° v 16“ klk2J2 A1A2 x a(klk211121)a(klk2-Allgj') . . , 2 . . . . W4 . 0' . O A(J.’1k131)A(J2k232) X A <' !n A'l V '><' y A'I.' '>2ji+l 1/2 X J11‘1 1 131“: J2K2”2 2 J2m2 [5311?] After using the definition of A(abc) given in Equation (81), using Equation (82), and setting kl = ki, k2 = ké, X1 = Xi and A2 = A5 @he validity of which is discussed in Appendix E), the above expression may be written J.-ji+mi-ni la(}\kj)|2 oK(i'i'ii) = - z z z (-1) l 2 mimiklk2 l6n n1‘19112 15c; 231+1 1/2 . . ji ji kl 2 x EEK—:T] (2k1+1)(231+l)(2Ji+D(2Jé+l) 1 K -K. o 185 . . 2 . . . . . . J2 32 k J1 J1 K\ 31.31 K 31.31 k1 J1 J1 k1 K2 K2 0 m1 n1 Q 1 “1 Q 1 m1 11 “1“n1 11 The product of the last four 3—J symbols and the factor m.-n! (-l) l 1 may be written as " °' ' o o ( )Ji+ji+kl+mi-ni J1 J1 J1 J1 J1 51 k1 -l _ v t _ _ _ - m1 n1 Q +n1 m1 ~ m1 m1 11 J1 31 k1 x (D12) ’ni “1‘11 Now, the sum rule<71) A b C a B B A c X Z (_l)a+8+y+c+c V ' QBYG B a B! Y 0' B‘Y _B a Y! a b 01 sec GY'Y' a b c x — 1 1 (D13) (2c+l) a' 8' vi A B C is used with A = ji, B = 3%, C = K, a = ji, b = 31’ c = kl, a = mi, 8 = mi, y = Q, a' = -mi, 8' = mi, and y' = 11' With this rule and the observations that 186 a+8+Y = mi + m1 + Q and -mi + ni + Q = O, which in turn . 0 ' = ?_ _ = — ' implies that ni mi Q and therefore that mi ni m1 m1 + Q, it is possible to sum Equation (D12) over mi, mi, n n! and Q to give the right side of Equation (D13). i’ 1’ ThisgfiveS‘the final expression for oK(i'i'ii) as Equation (8U). The outer term of 0K(iiii) is just twice one outer term of oK(ifif). Therefore, it can be written , 2 1 . J1 31 k z z |a<1kJ)|2<21'+1)<21'+1) 16112 k k 5' 1 2 1 2 i K.—Ki O 1 1 1v 1 l 212 J'2 32 k2 x K2-K2 O The middle term of 0K(iiii) may be obtained by the same procedure used to obtain 0K(i'i'ii). The only change is that ji is replaced by 31' APPENDIX E PROOF OF RESTRICTIONS ON MULTIPOLE ORDER OF POTENTIALS IMPOSED IN APPENDIX D In appendix D the restrictions k1 = k1’ k2 = k5: 11 = 11 and 12 = 1% were imposed in the derivation of 0K(i'i'ii). In this appendix these restrictions are shown to be rigorous. If the assumption is made that the above conditions do not hold, 00(i'i'ii) may be written as j -j!+mo—m' _ 00(1'1'11) = _ Z Z Z X (_1) 1 l l 1(2j2+1) l ' v v m1m1k1k21‘11‘2 ' ' ' m2111211121112 X awkwam'k'j) X A(31k131)1(31k13111(32k23é)’1(32k23é) o . 1 1 Y 1 X <11k1m111131m1><31k1m111ljim1> x <32k2m212ljémé><32k2m21é135mé> 187 188 The symmetry properties of the demand that A =A' 1 1 rewriting the above expression and X2 = X' 2. Clebsch-Gordon coefficients With these restrictions, in terms of 3-j symbols gives '1 't O -(21 +1)(2j!+1) . . J1 J1 i 2 1 Z Z aAkJaxk'J(2jé+1) ' l6n mimik1k2 m'-m! O m m'k'k' i 1 2 2 1 2 ' j21112 J1 J1 O J'1’31‘Lm1'm1 x (-1) [(2k1+1)(2ki+l)(2k2+1) ml-mi O 1/2 '1 :11 11 3131 k :11 .12 k2 x (2ké+1)] , . . . . J2 J2 k2 31.31 k1 31.31 k1 J2 J2 k2 x - ' - K2-K2 O mi-mi A m1 mi 1 m2 mé 12 J2 32 k2 x _ 1 m2 m2 12 Summation over m2 and mé then demands that k2 = ké and gives 189 O _(2j.+1)(2j’+1) j -j!+m -m! o (i'i'ii) = l 6 2 i Z Z Z (-1) i l i l I 1 n mimik1k2 ' I j21111112 Akj xk'j . 1/2 J1 11 O x a a (23211)[(2k1+1)(2k1+1)1 mi-mi 0 Finally, Equation (D13) may be used to show that k1 = k1. These results may be easily generalized to arbitrary K and also applied to oK(iiii)m. APPENDIX E MATRIX ELEMENTS OF MULTIPOLE MOMENT OPERATORS FOR ONE-ENDED AND PARITY ADAPTED SYMMETRIC TOP EIGENFUNCTIONS This appendix considers matrix elements of dipole and quadrupole moments for both "one-ended" and proper sym- metric top eigenfunctions. Following the notation of Anderson<53> parity-adapted symmetric top eigenfunctions are written as (W _ W +K _K) (Fl) 11).}. = 3'— (W+K + 11)-K); 1p_ = .1. f2 /2 where w+K are the usual symmetric top eigenfunctions for J = J and k = :K. Therefore, _ 1 , 1 (M — 72 (11+ + w_), w_K 5 (1+ - w_> . (F2) By using these relations and the fact that the dipole moment n has odd parity, it is easy to show that = %[<+|Ul-> +<-|U|+>1 , <-K|u|—K> = - %[<+|u|-> + <-|p|+>] 190 191 That is, = -<-KluI—K>. (F3) Equation (F3) can be demonstrated directly using 1/2 [(2J+1)(2J'+1)(2j+1) t+Mt )4," J = (—1)K x (FM) with j = l and m = 0. Equations (F1) and (F2) can also be used to show that <+|u|-> %[- + <-K|u|K> —<-K|u|—K ] = . (F5) The last step follows from application of Equation (FA) to obtain = <-K|u|K> = O and 192 ==-<-KIuI—K> . (F6) Equation (F5) shows that the matrix elements of the dipole moment operator may be taken using either "one-ended" or proper eigenfunctions. By using Equations (F1) and (F2) and the fact that the quadrupole moment Q has even parity, it is easy to show that <—KIQI-K> (F7) and <+|Ql+> %[<+K‘QI+K> + <-KIQl-K> +<+KIQI-K> + <—K|Q|+K>]. Equation (FA) may be used with j = 2 to show that <+|Q|+> = <+K|Q|+K> (F8) APPENDIX C SUMMARY OF RESONANCE FUNCTIONS AND HARD—SPHERE CUTOFF CALCULATIONS This appendix will give explicit expressions for the alkj factors occurring in the expressions for oK(i'f'if) in Chapter II. Comparison in turn of Equation (108) of TC with Equations (133), (150) and (161) of TC leads to the following identifications for dipole—dipole, dipole- quadrupole and quadrupole-quadrupole interactions, respectively: 1 Ak' 2 ———:la JI %( 2)2 f k), - ; (G1) 32112 = “5 “V b6 5:3 5 (G2) and . Q Q 1 AkJ 2 1 2 2 1 _— a = (_ _) f (k) , (2-9, . (G3) 32-12' I 115 “V b8 3 wb In the above equations k-—77, where w is an internal state energy difference, and b is the impact parameter. The 193 19A functions fl(k), f2(k) and f3(k) are defined as 1 A 2 2 2 f1(k) E k [K2(k) + 1Kl(k) + 3K0(k)] 12(1) 61 k2[K§(k) + 6K§(k) + 15K§ + 10K§(k)1 1 8 2 2 2 2 f3(k) — 236E k [Ku(k) + 8K3(k) + 28K2(k) + 56Kl(k) + 2 35K01 g f(s) 2 2 u 2 x r[%1 . (II) The first integral in Equation (251) becomes 4? f(k)db = % if k“[x§ + ”Ki + 3Kg]db u) A l °° )4 2 u) 0° U 2 m = (V) [E f0 b K2(;,-b)db + [O b Kl(§l_—b)db A 2 w 3 00 + H L, b KO 513)de Repeated use of Equation (Il) leads to 200 201 = (a?) 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