UFDC Home  Search all Groups  UF Institutional Repository  UF Institutional Repository  UF Theses & Dissertations  Vendor Digitized Files   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
PHOTODISSOCIATION OF POLYATOMIC MOLECULES: STATETOSTATE CROSS SECTIONS FROM THE SELFCONSISTENT EIKONAL METHOD By CLIFFORD DAVID STODDEN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1987 I would like to dedicate this work to my father from whom I learned to ask of nature why and how. ACKNOWLEDGMENTS I would like to thank some of the many people who in one way of another have helped or supported me in this work. First, I would like to thank Professor David A. Micha who supplied the basic formalism for this work as well as most of the funding. I spent many hours in discussions learning from his expertise in this field and developing the invaluable skill of how to patiently work through problems which arise in the research. I would like to thank those graduate students, postdoctoral associates and faculty who worked together to foster a cooperative atmosphere of research and learning at QTP. I would especially like to thank Professor PerOlov Lfwdin for providing the opportunity to attend the summer institute on Quantum Theory in Sweden. It was a unique and valuable learning experience as were the Sanibel Symposia. 9t Also, I would like to thank Professor Yngve Ohrn for his encouragement and advice, as well as the secretaries and staff. I would like to thank my family for their patience, many prayers, and support. I would like to especially thank mom and dad for always being there for me with support and love when I needed it. Other special thanks go to my fiancee Beth for her loving patience, prayers, and her constant encouragement. Finally, I would like to thank God for answering the prayers and for helping me keep everything in perspective. TABLE OF CONTENTS ACKNOWLEDGMENTS............................... ABSTRACT ............................................................ CHAPTERS I INTRODUCTION.............................. II CALCULATION OF THE PHOTODISSOCIATION CROSS SECTION USING THE EIKONAL WAVE FUNCTION........................ RadiationMolecule Interaction in the ElectronField Representation...................... The Eikonal Wave Function........................... III INTRODUCTION OF A TIME VARIABLE: TRAJECTORY EQUATIONS................................... A New Set of Variables ............................ The Eikonal Wave Function Along a Trajectory....... IV THE TRANSITION INTEGRAL IN ASPACE..................... Transformation to an Integral Over the Time Variable.................................. Generating the Jacobian Along a Trajectory......... Asymptotic Conditions.............................. V COLLINEAR MODELS OF POLYATOMIC PHOTODISSOCIATION........ Two Electronic States.............................. Three Electronic States............................ Symmetry Aspects and Cross Sections................ Page iii vi 1 4 4 14 18 18 22 29 29 31 39 47 48 50 52 VI SATISFYING ASYMPTOTIC CONDITIONS....................... 61 Statement of the Problem........................... 61 Method 1: Averaging Over a Period................. 62 Method 2: Construction of a WKB Internal Vavefunction................... 66 VII RESULTS: APPLICATION TO METHYL IODIDE................. 75 Coordinates........................................ 75 One Excited Surface................................ 83 Two Excited Surfaces............................... 115 VIII DISCUSSION AND CONCLUSIONS......................... 146 The 1ex Case....................................... 147 The 2ex Case...................................... 150 Considerations on Angular Distributions............ 155 Conclusions........................................ 158 APPENDICES A SHORT WAVELENGTH APPROXIMATION......................... 164 B COMPUTER PROGRAM FOR THE SCE METHOD.................... 166 BIBLIOGRAPHY................................................. 169 BIOGRAPHICAL SKETCH.......................................... 173 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PHOTODISSOCIATION OF'POLYATOMIC MOLECULES: STATETOSTATE CROSS SECTIONS FROM THE SELFCONSISTENT EIKONAL METHOD By CLIFFORD DAVID STODDEN December 1987 Chairman: Dr. David A. Micha Major Department: Chemistry A general method is presented for calculating statetostate cross sections for photodissociation of polyatomic molecules by visible and UV radiation. The method also provides branching ratios by selfconsistently coupling nuclear motion to transitions between several electronic surfaces. Cross sections are calculated using a transition integral between the initial ground state wavefunction of the molecule and a final wavefunction for the relevant states of the product fragments. The final state nuclear motion wavefunction with incoming wave boundary conditions is generated in an eikonal form along classical trajectories which follow an effective potential and contains the exponential of a common action integral and a pre exponential given by an amplitude matrix and a Jacobian describing the divergence of the trajectories. In a new development, the Jacobian of the variable transformation is generated exactly along a classical vi trajectory without requiring the simultaneous integration of adjacent trajectories. Two methods are presented for satisfying the asymptotic conditions of the eikonal wavefunction. In model calculations on CH3I the dynamics are described by two degrees of freedomthe relative position of I to CH3 and the umbrella mode of CH3. The first calculation involves a transition to a single dissociative excited 3 0 potential energy surface. The total and partial cross sections for an initial zero vibrational level over a range of photon energies are found to be in excellent agreement with results in the literature from an exact wavepacket calculation using the same empirical potential energy surfaces. Results are also presented for excitation from the first three excited vibrational levels of the ground electronic state. The second calculation involves an excitation to two coupled potential surfaces leading to CH3 and I(2 P3/2) or I (2 P/2). Quantum yields of .84 and .69 are calculated for I*( P1/2) production at photon wavelengths of 266nm and 248nm respectively. The crosssection curves for I( P3/2) production peak at higher energies than the results from a coupled channels calculation. Possible reasons for the discrepancy are presented. The method is also applicable to large polyatomic molecules. CHAPTER I INTRODUCTION This study addresses the problem of formulating and testing a computationally efficient and accurate theory for treating the photodissociation of polyatomic molecules with many internal degrees of freedom and many electronic states. The photodissociation event of interest will be limited to cases where incident radiation in the ultraviolet to visible range is absorbed by a molecule causing a transition to a dissociative state which gives a single atom and a molecular fragment as products. There are already many theories of photodissociation. They include quantum mechanical approaches [Heather and Light, 1983; BalintKurti and Shapiro,1981; Clary, in press] as well as classical and semiclassical trajectory approaches [Atabek et al., 1976; Mukamel and Jortner, 1976; Billing and Jolicard, 1984; Gray and Child, 1984; Henriksen, 1985]. Lee and Heller have also developed a theory for an exact time dependent approach using wave packets [Lee and Heller, 1982]. An exact approach in principle gives results which are more reliable than a semiclassical method. However, for large molecules with many internal degrees of freedom and many coupled excited states such calculations may be prohibitively complicated. The computational effort required for a coupled channels calculation for example increases by the power of the number of channels involved. A semiclassical method which employs classical trajectories requires much less computational effort than an exact approach for large systems. For this reason it is important to develop an efficient, accurate semiclassical method which is able to deal with many degrees of freedom and many coupled electronic surfaces. Of the semiclassical methods mentioned above none are at present able to effectively deal with more than one excited state potential energy surface. The photodissociation events of interest here have a bound initial electronic state with a certain set of vibrational quantum numbers and a final electronic state with a set of final vibrational quantum numbers for the molecular fragment. With this in mind another desirable feature of any photodissociation theory is to be able to set initial conditions as well as final conditions for the event, i.e. to calculate cross sections for state to state transitions. The formalism in the following treatment is designed, then, to include certain desirable features. These features can be summarized as the ability to efficiently deal with (i) many degrees of freedom, (ii) many coupled electronic states, and (iii) statetostate transitions. The particular method to be discussed uses the selfconsistent eikonal approximation. It is selfconsistent in that the electronic transitions are selfconsistently coupled to nuclear motions. The eikonal approximation simply means that the nuclear wave function is expressed as a modulus matrix times an exponential with a common eikonal phase and that the short wavelength approximation is applied. The general process of interest is the following photodissociation reaction, M(VM) + +(k,ao) A(vA) + X(X) (I.1) where M is a polyatomic molecule in a given electronic and nuclear state uv, A is an atom in electronic state vA, + is a photon with wavevector k and polarization o, and X is a molecular fragment in electronic and nuclear state vX. With the method presented in the following chapters one can treat photodissociation events as in Eq. (I.1) involving many atoms and several excited potential energy surfaces in a way which is computationally practical. CHAPTER II EVALUATION OF THE PHOTODISSOCIATION CROSS SECTION USING THE EIKONAL WAVEFUNCTION RadiationMolecule Interaction in the ElectronField Representation The interaction of a photon with a molecule can be modeled as a collision problem [Micha and Swaminathan, 1985]. In such a treatment we consider the photon to be occupying a finite volume V in space and the photon field to be contained within a cube of side length L. The molecule M(vM) enters this space and interacts with the field. The field is quantized so that the moleculefield system is described by a Hamiltonian Hl= HM + HF + HMF (II.1) The wavefunction for the molecular system is represented by ITM(Q)>, where the "ket" indicates electronic coordinates and 0 is a vector whose components are mass weighted Cartesian coordinates for nuclear motion. The Schrodinger equation for the molecular Hamiltonian HM is HM I',M(q)> = E ITM (2)> (11.2) where E is the total of the potential and kinetic energy of the isolated molecular system. In all subsequent notation the use of the symbol to underline a character will represent either a matrix or a collection of variables whose members represent components of a vector. The meaning in either case will be specified or obvious from the context. The second term in the Hamiltonian HF refers to the radiation field. It is given by [Loudon, 1973] HF H a + a ak (11.3) where a+ and a are the creation and annihilation operators of a photon with wavevector k, polarization a and energy HwkM.. The product of the annihilation and creation operators is the number operator which operates on an eigenstate I(ko)N) of the radiation field to give the number of photons in state (ka), where N is the number of photons. The photodissociation process treated here will involve one photon present in the initial state and none present in the final state. The Hamiltonian term H MF refers to the moleculefield interaction. In the dipole approximation this interaction term is (r = (r'r) E(r') d3 (11.4) MFp c f ~Le M c  where r is the position of the center of mass of the molecule M. The ~c factor DM is the dipole operator, which, in the center of mass frame at position r is M( = c ra (r r) (11.5) a where {c ) is a collection of charges at positions ({r}. The electric field operator E(r) in second quantized form is E(r) = E g() eak X(r) a X(1(r)1.6) k,o ~' k k g() = i [Ma/(2Co)1 1/2, (11.7) where es is the vacuum dielectric constant and e is the polarization unit vector with a=x,y for linearly polarized light and o.+, for circularly polarized light. The factor X in Eq. (11.6) above is the plane wave of the photon inside the radiation volume and has the form Xk(r) = V1/2 exp(ik.r) (11.8) The wavevector k has components 2nj /L with j =0,1,2... and .=x,y,z corresponding to cyclic boundary conditions for a cube centered at the origin of coordinates. Replacing Eqs. (11.5) to (11.8) in Eq. (11.4) we obtain 1/2 ik.r HMF(Ec) = sg e A. DsEak) V1/2 e ak,a 1/2 ~ c +r D(rak) V e ak ] (11.9) D(rk) = ca a exp(ik.ra) (II.10) a The Hamiltonian in Eq. (II.1) acts on a state Ij) which describes the state of both the molecule and the photon field. The initial state ii) and final state If) for a single photon event are given by i> = p u (k a)> and If> = Ie Pf a> (II.11) where p and 2c are the initial and final moment of the center of mass, u indicates the initial molecular state, and a indicates the final state of the fragments. In Eq. (II.11) the state Ii> corresponds to a molecule initially in the presence of a photon field and the state If> to a dissociative excited state with no photon present. The fragments in state If> have a relative final momentum vector of efi. The corresponding total energies are Ei = e /(2mc) + Eu + MH (11.12) Ef = (e)2/(2mc) + fi/(2m) + Ez (11.13) where m is the total molecular mass and m the reduced mass for the relative motion of A and X of Eq. (I.1). The dissociation cross section is the quantity of interest in a photodissociation experiment. The initial and final states are characterized by both electronic and vibrorotational energies of the fragments. Thus the cross section to be calculated is a stateto state (electronic, vibrational, rotational) resolved quantity at a particular photon energy. The expression for this cross section is [Rodberg & Thaler, 1967] d fi d i1 Jd3 c J pfi Rfi (11.14) dQi d 'd ~f where Rfi is the transition rate and J. is the incoming photon flux in the laboratory frame given by J= c/(2nM)3 (11.15) where c is the speed of light. To obtain Eq. (11.15) we have used momentum normalization for which < clc> = (2nH)312 exp (iec rc/M) (11.16) The transition rate is given in terms of the transition amplitude Tfi and Dirac's delta function with respect to energy, Rfi = ITfi(2 (Ef E) (11.17) The transition amplitude Tfi is an integral over electronic and nuclear coordinates and contains an operator T which, for low intensity incident radiation, is simply the moleculefield interaction of Eq. (11.9). For the single photon process to be considered here only the first term in HMF is appropriate. A single photon event corresponds to a weak electric field (i.e. low intensity). The form of the transition amplitude, then, is Tfi = _1 eik.r = g() et The last factor contains the integral over r which gives c I Therefore the integral in Eq. (11.14) becomes independent of D with the restriction that g = C + Hk. The cross section now contains an integral over 2fi only, do d f 1 g()2 (2n/H) (2nh)3 x fdpfi p2i D afi A(r ',k).e I>'12 &(E Ei), (11.20) where pfi is the magnitude of gfi. If we note that Hk << it follows that ec'f c. With this very reasonable approximation and using Ef = E1 we get from Eqs. (11.12) and (11.13) for pfi 2 Pfi T = E E + (11.21) Transforming the integral in Eq. (11.20) by noting that dEf = pfi dpfi/m and using the definition of Ji, the differential cross section becomes defi 2 2nmpfi 2 dafi = g() 2 He j efiaDI(raoe;0).SO> 2 (11.22) The factor in brackets will be denoted as Dfi indicating that the transition is from a molecule initially in state i>=ju> to fragments in a final state If>=Ifi a> in the asymptotic region. The final state is designated by the states of the atom A and of the fragment X asymptotically as as(rA,u X) and vx=(ry,v) where r refers to an electronic state and v to a collection of vibrational quantum numbers. In this treatment we will assume that the initial bound state wavefunction Iu> for the molecule in the center of mass frame is given. It remains, therefore, to obtain the final state wavefunction The vector 0 in general contains components for the relative position R of A with respect to X, the internal degrees of freedom Q' of X, and the center of mass position r In the following notation, c however, we will be working strictly in the center of mass frame and the symbol 0 will refer only to relative and internal components. The final state wavefunction in the center of mass frame will be expressed in the coordinate representation, so that This molecular eigenstate can subsequently be expanded in a basis of the electronic states Pr of M involved in the process, so that IYVMq> = Mt )>> Wq> (11.24) IM > = ( > 2It> .. hIt> ) is the matrix of electronic wavefunctions, and  > = 1 (11.25) *2 is the matrix of nuclear wavefunctions where h is the number of the highest state energetically accessible in the reaction. We will express HM as a sum of the nuclear kinetic energy operator Knu and a term H0 which contains the nuclear repulsion and all the energies of the electrons, including spinorbit coupling, HM = Ku + BH (11.26) Knu = (2M)1 (im)2 (11.27) where M is an arbitrary mass which comes from mass weighting, and the symbol 7 is a multidimensional gradient whose components are 3/a1,, i=0 to N. The gradient V to the second power is defined to be (y)2 = Ea2/82 (11.28) The coefficients of the expansion contained in $4 are the nuclear motion wavefunctions on the electronic potential energy surfaces. These surfaces are defined by the diagonal elements of HQ = (11.29) When Eq. (11.23) is substituted into Eq. (11.2) and left multiplied by <&)I we have the following matrix differential equation (11.30) e 1 H a i [m 1961i hc ( i ( + GQ) + Q E J () = 0 We vill Vork in the diabetic picture [Smith, 1969] in which G(Q)=<$!(Q)I(H/i) I M(Q)> = 0 ~ M ~ ~n ~ ~ ~ This leaves us with the general matrix equation: 2f1 ]2 + E M(q) = 0 { 2M 7 J o^ 03 When the other blocks, the coupling between the electronic ground state (r=1) and electronic states is negligible, Hl can be separated by 1 BB! 0 H 0 0 H22 :32 0 .... 23" 1 0 H33** V (11.33) This gives us one equation for the ground state wavefunction and a set of coupled equations for the excited state wavefunction. Following this argument we will also separate the wavefunction expansion as follows: (11.31) (11.32) h IYm(Q)> = i> + E > r (11.34) r=2 We will define I'gr ()> = Il(q> (11.35) to be the ground state wavefunction, and h 1 <> = IS<()> (Q) = I lrQ> tr^ (11.36) r=2 to be the excited state wavefunction. By dividing the space of the electronic basis in this way we can isolate that part of the wavefunction which involves the excited states. Thus we need only solve that part of the matrix equation which deals with the function IT ex(Q0)>. The matrix equation for the nuclear wavefunctions on the excited electronic surfaces is + V + 1E) = 0 (11.37) This is the equation we solve next using the common eikonal approximation. The Eikonal Wavefunction The nuclear wavefunction with incoming wave boundary conditions can always be written in the form of a modulus matrix X(Q) times an exponential with a common phase S(Q). q)(0) = X(Q) exp[i S(Q)/M] (11.38) where the minus sign on t indicates that the boundary conditions chosen are those of an incoming wave. We now substitute Eq. (11.38) into Eq. (11.37), carry out the v2 operation. We then multiply by exp[iS(Q)/M] to obtain iM 2 im  Y2 (Q S)) 1(YS V) H2 2 (VS)2  2MY + M + (V E) = 0. (11.39) We can also write this as 0 = 0 , where 0 is the operator 2 0 = 1/2M (H/i V + VS) + V E Note that 0 is in the form of a Hamiltonian minus energy (11.40) (11.41) 0 = H E (11.42) H = T + V (11.43) T = T( ,VS) (11.44) where the normal momentum operator i=H/i 7 has been replaced by the .sum p + VS thus defining a new momentum operator p' in the kinetic energy operator T, T =(O')2/2M (11.45) If we define the multidimensional classical momentum to be P = as/a0 a vs (11.46) we get the following for Eq. (11.39): 2 (V.P) (P.) I S= vX + + i M 2M 0 + (V E)y. (11.47) In order to express Eq. (11.47) in the form of a HamiltonJacobi equation we can carry out the following operations: X+ 0 X + (IX+ 0 X)+ = 0 (11.48) X 0 + 0 X = 0 (11.49) With the proper manipulation Eq. (11.49) becomes M (_2 + 2V J + iH P.[ Y+ (V)+X1l + XX 2Ey X = 0. (11.50) After dividing by 2X 4 this can be put in the form of the Hamilton Jacobi equation, (P)2/2M + V = E qu (11.51) where Vqu (x)1+ 2M + ) + i2M It remains to solve for )(Q). To do this we first substitute Eq. (11.51) into Eq. (11.47) to give iM i  V. M V2 + (y 1V )X = 0 qu (11.53) We can simplify Eq. (11.53) by introducing the following notation: U = H/2M V.P (11.54) Kpp = p(H(2/2.yX/Xm (11.55) Equation (11.53) becomes V ) + K 1(iU) = . qu ~ P. [ 7 ( + . }. (11.52) 2 V 2M (11.56) P.V S + (V 1 1M Equation (11.52) becomes the expression for an average potential V when we assume the second term to be negligible and make the short wavelength approximation, for which V2=0O (see appendix A), leaving us with = (Z )1 z+B (11.57) We will now apply the short wavelength approximation in which K is negligible and V becomes v to give P.V 7 _ S V + (< !V) 1(iU) = 0 (11.58) In chapter III we will solve Eq. (11.58) using trajectory equations after introducing a time variable within the new set of variables A. CHAPTER III INTRODUCTION OF A TIME VARIABLE: TRAJECTORY EQUATIONS A New Set of Variables The matrix equation which we just derived, Eq. (11.58), will be solved numerically using classical trajectories. In order to uniquely define any point along a trajectory, however, we need to introduce a time variable and indicate the values of each component of 0 at the initial time. Initial values of the 0 are denoted as c and are given by the set S= (q11 q' "' qN) (III.1) where q1 sets the initial value of Q,01 for a given trajectory, q2 the initial value of 02, etc. One of the degrees of freedom in Q, namely Q0, will be set by the relative distance between A and X which will be chosen to always have the same initial value for each trajectory. The subscript N is equal to one less than the number of degrees of freedom in the system for a center of mass reference frame, thus there will be a total of N+1 degrees of freedom in the problem. Introducing this collection of variables is equivalent to transforming from the set of spatial variables 0 to a new set of variables (t;q] such that Q=Q(t;q). The set q together with the time variable will be denoted A, A a {t, q q' 2, ... qN) a (t; C0 (III.2) Each trajectory evolves with the time variable t separately. The other members of A will insure that each trajectory follows a unique path in Qspace (even though a particular trajectory will cross others many times). It is convenient to define sets 0' and g' to be the subsets of 0 and c respectively which refer to internal variables. They will each contain N' components Q' a {(1 Q2...Q0N, (111.3) q {ql' q2' 2 q^N,} (111.4) where N' is the number of internal degrees of freedom in the system. We can write the initial value of any the QO' in terms of an angle q.' defined by 1/2 Q'(t=0) = () /2 Avi sin(q'i) (111.5) where A vi is the maximum value of QOi' for quantum number v. and ii. is the reduced mass for the i internal mode. Note that for the Qi corresponding to normal vibrational modes, if the potential energy along this Qi' is harmonic in the asymptotic region (Q0+) then we can write Qi'(t;q) = 1/2 Avi sin[(tto) + q.'] (111.6) where ai is the frequency along Q! and to is the initial time. The asymptotic region is that part of Qspace where the potential is separable in 0', i.e. ()/' = f(Q') (111.7) Thus Eq. (III.6) gives us the functional relationship for a vibrational mode between 0' and q' for a harmonic potential in the asymptotic region. Now that we have introduced the time variable we would like a prescription for determining the trajectories. The classical equations of motion for position and moment can be formulated using Hamilton's equations. We define Eq. (11.51) to be the Hamiltonian B and impose the restriction of energy conservation to obtain dH dt = 0 This leads to Hamilton's classical equations of motion dO d (111.9) dP and dt3 .H (III.10) Both Q and P can be written as a collection of variables, Q a (Q0' Q1' Q02" N (III.11) P a (P,' 1p P2" PN)} (III.12) For each component of 0 and P, then, d  (II.13) dt aP i a d  (III.14) It is helpful at this point to discuss mass weighted coordinates. We define the following to be mass weighted coordinates, Q. = X. / M (II.15) PXI P.=  nF (III.16) 1 nth where X. and PXi are the original Cartesian coordinates for the ith degree of freedom. The mass M can be arbitrarily set to some value without affecting the results of the derivation. From Eqs. (111.15) and (111.16) it follows that dt P (A) = M dt (III.17) Equation (111.17) shows the advantage of using mass weighted coordinates in that the reduced mass pi does not appear explicitly in the relationship between P and Q. Turning now to the differential equation derived in chapter II we can simplify Eq. (11.58) using the definition of 7 in chapter II and noting that P.V S N dQ i=odt ai dt (III.18) The Eikonal Wavefunction Along a Trajectory Having introduced the set A we can rewrite using this new set [Q(AQ)] = Xo C(A) e  ty(t') dt' to 1 y = V.PR and X is the value of X at to which is set by initial conditions. If we replace Eqs. (111.18) and (III.19) into Eq. (11.58) we get (111.21) All that remains is to solve Eq. (111.21) for C noting that the time dependence is through trajectories. As we travel along a particular trajectory we will be solving for C and V simultaneously because the form of V is now V = (C C)1C VC . where (111.19) (111.20) (H/i) dC/dt + (V 1V)C = 0 . (III.22) We will also construct the nuclear wavefunction along the trajectory. It now has the form / Y 5 S(Q) S Q = ) e S y()a,)= X e R 1 f ly dt' i = Xo C(t;i) e 7 t e where S(Q)*4(t;%). It will later be shown that the quantity involving y simplifies to a function of the moment and other terms which can be generated numerically along the trajectory. The final part of left to construct is the factor i e To do this we refer to the definition of P in Eq. (11.46) and integrate to get 0rdS = P dQ N S(Q) S(Qo) = i=0 fP.i dQ 2 1 N S[Q(t;q)] = S[Q(t0;q)] + i=0 J P2 dt a o(t;q) to where Q0 indicates all initial values of 0 and to the initial value of t. Thus, according to Eq. (11.26), to calculate a(t;g) we need only (111.23) (11.24) (111.25) (111.26) integrate the square of the moment along each trajectory over the time variable. We will now develop a prescription for generating y. The function y = V.P/M would seem to pose quite a problem due to the fact that it is in the exponent and contains partial differentials. It turns out, however, that the entire exponent can be reduced to a very simple expression containing the Jacobian for the variable transformation from 0 to (t;}). Let us first recall that the term involving y appeared in () as a result of having transformed the differential equation from spatial coordinates 0 to the set of coordinates A which contains a time variable. Using the definitions of P and V the product in y becomes N 3P. v. P( ) = E (111.27) i=0 i Let us look at one term in the sum on the right side of Eq. (111.27), i.e. 3aP/aQ We know from calculus that we can always relate a partial derivative in one coordinate system with the partial in another coordinate system providing we can construct the Jacobian of the transformation between the two systems. In this case the particular relation between partial derivatives is the following: a = o01" Q Ni" N S...q...111.28) aai J 0 01" .Q i ...i" QN] rt ,.. q"i .. qN Note that the ith column of the numerator in Eq. (111.28) contains P1 rather than Qi. The explicit form of the Jacobian in Eq (111.28) is ,j (o .. :i... N = ... qi...q I N 800 ai t .... at* ao0 ai 800 __i aqN 8q N If we use Eq. (111.28) in Eq. (111.27) the following sum over i of the determinant in Jacobian, N V P i=0 aQo 0 0 3t aaQO aq, aQO SN aP. 1 **. at .". ap. ap. *.* * ^N !oN at aQN N Oql aQN !qN N product appears as the Eq. (111.28) divided by the (111.30) where J is the Jacobian of Eq. (III.29). The summation of determinants in Eq. (111.30) calls to mind the expression one obtains when taking the derivative of a single determinant. The derivative of the Jacobian determinant with respect to q. is (111.29) 3QN at aq, aQN aqN at 3q" aq.t at N 80o a 3 N = Sql 8q 3qjq 3q1 (1I.31) i=0 800 2 i N ai0 N___ q N"" Saq qN N The determinant in Eq. (111.31) looks very similar to that in Eq. (III.30) except that the ith column contains double derivatives rather than the derivatives of P.. If, however, we recall that P =M(dQi/dt), then we can make these determinants identical by choosing q.=t to obtain 800 2 i 0N t a* 2 at N ao0 a2 i 3N at 3q1 "" at 3ql q (111.32) i=0 800 a0 i %N dqaN at ar N w aN If we now switch the order of differentiation on the double derivatives and replace Q. /at by P /M we get that N 8 J 1 . Ft M/ i=0 P Pq .... a . Nao aaoo aqN api at . ap. 1 api aN P aQN aqN (111.33) After comparing Eq. (111.33) with Eq. (111.30) we can express y in terms of the time derivative of the Jacobian which gives dJ (g. ?)J 1 1 dJ d ln(J) SJ dt dt (111.34) (111.35) Recall that the term involving y appears in the nuclear wavefunction () as an integral in the exponent. Replacing Eq. (111.35) in the integral gives  ty(t') dt' oe t e t td ln(J) = e t = e [ Iln(J) ln(Jo) I = exp{ [ In e arg(J) + ln I In i arg(Jo)) lnJo i( e (e I/w =exp [i arg(J) i arg(J)l 1/2 JI (III.36) where Jo refers to the value of the Jacobian at time to. Thus we have 7 Y(t') dt' Jo 1/2 e to = exp{2[ arg(J) arg(Jo) 1]. (111.37) Our nuclear wavefunction can now be constructed along a trajectory defined by A a (t;'}). The final form of this wavefunction is (t) Xo C(t;) Jo(O) 1/2 z(t;%) e a(t;C) (111.38) SJ(t;q) e e where J is the Jacobian at time t, Jo is the Jacobian at the initial time, and u contains the argument of the Jacobian as follows: u(t) = arg(J) arg(Jo) (111.39) The amplitude matrix C is selfconsistently solved for through Eq. (I1.21). The factor Xo is a constant whose value will be determined in chapter IV by initial conditions. The other terms are constructed from quantities generated along the trajectories. The nuclear wavefunction calculated in this manner will be used in the next chapter to calculate the transition integral which appears in the cross section of Eq. (11.22). CHAPTER IV THE TRANSITION INTEGRAL IN ASPACE Transformation to an Integral Over the Time Variable The quantity of interest to the experimentalist is the stateto state cross section. It is calculated using a transition integral between the ground state wave function *i and the function T This is calculated from the quantity Dfi which appears in Eq. (11.22). The general form of the transition integral is Dfi = J[* ]D.e *i dT (IV.1) where T refers to integration over all space. The subscript f has been introduced on () to indicate the asymptotic form of the wavefunction. The symbol D is a function of nuclear variables only. It is derived from the integral of initial and final electronic wave functions and the dipole operator D? over electronic coordinates. It has components I r'i)]= ,() 1()>. (IV.2) The integration over space dt in Eq. (IV.1) can be written as a multiple integral over the set of coordinates 0 z (Q0, Q1' "' ON)' Thus the transition amplitude is a multiple integral in (N + 1) variables which can be written as Dfi = f[) ]+ D.e i dQ (IV.3) In order to calculate this integral using trajectories we must again make a variable transformation from 0 to A as in the last section. The integral in Eq. (IV.3) becomes D f [ ]+ D.e I O 1 4". N )] 1A (IV.4) fi J f t q, ...q^N where 00 01 QN is the absolute value of the Jacobian of transformation and the collection of variables A is defined by A ={t,Q} where q={ql 2, .q N} contains the initial coordinates for a trajectory. To show explicitly that Dfi now contains an integral over time we write Dfi = [ tL)]De *i JI dt d% (IV.5) We can approximate the integral over dq by a sum over 6%. This leaves us with a sum of integrals over dt: D f = (qn) [t l D n.e .in 1iJ dt a E Df. (IV.6) n = n 1n n in n fi ,n n=1 0 n=1 Dfin = <( ) [ ]D nE in IJI dt (IV.7) fi,n nn For simplicity we have dropped the subscript f from t The label (n) which appears as a subscript in the summation refers to individual trajectories. Therefore, g n is a function which is generated along the nth trajectory. Since each trajectory is defined by its values of % it can be said that (n) labels each unique set Sn' This can be seen by defining M(t) a t (tn) (IV.8) According to Eq. (IV.6) each trajectory n will contribute a certain amount Dfin to the cross section. This integral will be calculated along the nth trajectory by simultaneously generating each term in the integrand and integrating step by step over t. In order to do this the integrand must be composed of terms which can themselves be generated step by step over t. Generating the Jacobian Along a Trajectory In chapter III we developed the detailed form of ~ It was also shown how the factors C(t) and a(t) could be generated along the trajectory. The other factors appearing in ~ contain the Jacobian as an argument. This is the same Jacobian as that which appears in the transition integral of Eq. (IV.4) after transforming from dO to dA. It has not yet been shown how J can be calculated along each trajectory. This will be done next as we discuss a way of simultaneously generating all of the terms in J. The Jacobian of interest is the same as that in Eq. (111.29). The object is to numerically solve for each term and then to carry out the operations of the determinant. This is to be done at each time step along a given trajectory in order to obtain the Jacobian for that trajectory as a function of time. To generate numerically the terms in Eq. (111.29) we need to somehow express the time derivative of each term as a function of the other terms or of known functions. First let us define the quantities i i Q a 30./3q. and P. i a P./aq. (IV.9) where j = 0, 1, 2, ...N; i= 1, 2, ...N. Using the first definition in Eq. (IV.9) the Jacobian now appears as P 1 P"" N S01 ...1 01 .... JN = .N1 0N (IV.10) N N N 00 1.. ON To obtain the time derivatives of P. and Q let us start by recalling j j Eqs. (III.13) and (III.14). If we take the derivative of the left side of Eq. (111.13) with respect to q. and switch the order of differentiation we get 2 2 i 8Q0 ao. 3Q. S. (IV.11) 3aq3t 3t3qi at Taking the derivative of the right side of Eq. (IV.11) with respect to qi gives the relation aQ /at = 8P./83q = P (IV.12) Thus Eq. (IV.12) gives us one of the time derivatives we need. If we similarly take the derivative of the left side of Eq. (III.14) with respect to qi we get a2P. 32p pi 3___ ____ (IV.13) aqjat 3t3q. at Following the same procedure as before, we set Eq. (IV.13) equal to the derivative of the right side of Eq. (111.14) as follows: aP 2 3 a aq (IV.14) However, since we have no expression for the derivative of V with respect to %, an additional step must be taken. We know the quantity V only as an explicit function of Q. We also know that the derivative dependence of 0k on qi is one of the terms being generated (Qi=803Q3qi). Therefore we can expand 32 /aqiaQ. in terms of the 0Q by using the chain rule. It is important to note that the average potential V is a function of the components Cy(t;Q) of the amplitude matrix C(t;q) from Eq. (111.22). It is convenient to express Cr in terms of its real and imaginary parts Cp a K (Xp + iYp) (IV.15) where K is a constant to be specified later. With the definition of Eq. (IV.15) we note that V V(Q; X,Y) (IV.16) where X and Y are sets containing, respectively, the collection of Xr and Yr components of all Cp. Since V is a function of the Xr and Yr these must be included in the chain rule expansion. Define, F. a 8V/0Q. (the force along Q.) and substitute into Eq. (IV.14). Using the chain rule we get VF. N (8F. 0 (( F.) X Xr aF. Y S E1= \ + + r (IV.17) aqi k=O 8Qk C aqi r ax y3qi (ro,Yi i In Eq. (IV.17) we have introduced two new partial differentials which will also be numerically generated along each trajectory. We will give them the following notation: i axr i r X aq Y raqi (IV.18) Using Eqs. (IV.17) and (IV.18) and the definition of F. we get the following for the time derivative of P.: a j N 2 j 1 av Y V (IV.19) at k=0 k ao Jk r ar ri.j Tr Sy We have now expanded the partial derivative of Eq. (IV.14) in terms of quantities which we know how to calculate; however, we had to introduce the factors Xp and Y Of course, we have no analytical expression for these factors so we must again develop expressions for the time derivative of each and generate them numerically as we did with Q0 and P . 3 3' The first step is again to look at the time derivative of the function whose partial derivative we are considering. In this case we need the time derivatives of Xr and Y. To get an expression for these time derivatives we use Eq. (I1.21) and separate the real and imaginary parts given in Eq. (IV.15) to obtain dX . d X = Vr, Y (IV.20) dt r' aY dY . dt r = ? E rr, = Y  T' r where the matrix W is defined as W=V1V and we assume the matrix V defined in Eq. (11.33) to be real. Next we take the partial derivative of Eqs. (IV.20) and (IV.21) with respect to qi and switch the order of differentiation axr a dX rY r X (IV.22) aq q i (r Wrr, Yr' dt r 4 = r Y r + Vr, Y r (IV. 23) Once again we run into a situation where we must expand a partial derivative with respect to q.. By using the chain rule expansion of wrrT' q we obtain 3qi ^r E Li I r'1" k + E (ax rX ay rf r r' {i FT a'wr I r" ta"r, i "m ir rk(8kjX Y + Vrr, (IV.24) The expression in the equation above can be simplified by noting that the only terms in W that are dependent on X and Y are those containing 9, namely, the diagonal terms. We can indicate this by using a delta function rr, to get the final expression i {(rr, i a! ( x i }V i X r=E E 6 E ( r X ?+ IY rr k Qk k 1T r', rX, r" r T" J +Y (IV.25) r, At last we have an expression for which all terms can be calculated (either analytically or numerically). Using similar steps for Y we arrive at the time derivative rYr W av r Li a V r=w E (a0k 8rr Cax 41 r 18k J XY r r r" rYr" VJxrr X (IV.26) + E vrr, xr (IV. 26) Returning to Eq. (IV.19) we see that having derived Eqs. (IV.25) and (IV.26) we have an expression for the time derivative of P and thus for Q. in terms of quantities we can calculate. We are already generating all P. in time and so we have all the terms to calculate J along the trajectory. The only terms in D fin of Eq. (IV.7) which we have not yet talked about are those which we need to be given, namely, D .e and 4 . n 1,n The ground state wave function w. will be given as a function of Q; therefore, in. (t) is simply the value of J. at that point Q which iyn i1 corresponds to the nth trajectory at time t; i.e. i,n = i[q(t;n)] (IV.27) The factor D .e will similarly be given as a function of Q. n  We now have all the elements required for the transition integral. The entire expression in detailed form is D f = D fi n fi n fi,n (IV.28) where Jo 1/2 o Dn i[arg(J) arg(J )]/2 io (t)/M fn n e e x g()Dn(t).e *i,n(t) Jn dt (IV.29) The symbol Jo is defined as the value of the Jacobian at time to (the initial time) for the nth trajectory. Notice in Eq. (IV.29) that there will be a singularity in caused by the term 1J0j 1/2 each time J passes through zero. Remember that t" is constructed using the short wavelength approximation. This approximation breaks down whenever P goes to zero. This happens as a trajectory approaches a caustic where J goes to zero and t() passes through a singularity. Fortunately these singularities do not appear inside the transition integral of Eq. (IV.29) because of the Jacobian which comes from tranforming from dQ to dA. The fact that the singularities in t are integrated out in this way makes it numerically possible to use the semiclassical wave function () without having to integrate through poles. Asymptotic Conditions There is still a problem in the way t() behaves asymptotically. In order to satisfy asymptotic conditions the nuclear wave function must have the following form as R) and t4to: 1 0 *0(Q) = (2n))d/2 u (Q') e R" (IV.30) where d is the the dimensionality of the space that AX is constrained to (d=l, 2, or 3), u (Q') is the internal nuclear motion wave function (including internal rotation) of the fragment X in a vibrational/rotational state with quantum numbers v, Q' is the subset of 0 which contains all the internal modes of the fragment X, and R is the vector corresponding to the relative displacement of atom A from the center of mass of the fragment X. To get t( asymptotically we will look at how each term in Eq.(IV.29) behaves as Q9Q where Q refers to values of 0 which are in the asymptotic region. Asymptotically we can express the eikonal wave function as J 1/2 i i o e o 1 2 e a eg a (IV.31) a where the subscript "a" indicates that the various functions are in the asymptotic region. To solve for XoC let us recall Eq. (111.21) and look at the term (V 1V). Asymptotically, the off diagonal elements of V will be zero. If we choose the wave function initially to be entirely on excited surface r, then the term V becomes V 9 = Vr a E (IV.32) Q a r r ~ a The elements in V 1V = V then become [V ]ij = 8j(Ei Er) (IV.33) thus having off diagonal coupling elements equal to zero and a zero for the rth diagonal element. Because there are no off diagonal elements in the matrix equation asymptotically and because C will have the following form a = [r ,r (IV.34) 0 , Eq. (III.21) will reduce to dCr v= 0 ; (IV.35) 1 dt thus C a is constant in time. a,r The Jacobian has a very interesting and simple form asymptotically. In the asymptotic region any internal motion along Q. will not affect the relative motion along Q. and vice versa. Using this fact we can set the terms 80./aq! and OQ'./q. equal to zero where Sand q refer to relative variables. The Jaobian becomes 0 and q. refer to relative variables. The Jacobian becomes 41 P0 1 2 3 ""* N 800 301 a2 0 ...... 0 aql aq, a1 1 0ao 3, 1 Q2 0 ....... O aq2 aq2 aq2 J 1 0 Q3 3N (IV.36) a 0 0 0 3...  803 8aN 3q N' 3qN with the Jacobian determinant having a block for relative variables (i=0,1,2) and a block for internal variables. Thus the determinant reduces to a product of the Jacobians I = J 2 J' (IV.37) a a,R rt q1 2j a [1q q^ (IV.37) Outside the interaction region. the potential along 00, 01, and 02 does not change and so Po, Pi, and P2 are constant. In all of the following equations a superscript o indicates a value at the initial time. If we restrict the definition of q, and q2 such that q, = Qo and q2 Q0, then the determinant symbolized by JaR becomes simply 2 aR s2 smply the constant P. An analysis of the Jacobian J' for the asymptotic 0) a internal degrees of freedom shows that it will be dependent on the combined frequencies of the normal modes of vibration for the fragment. A specific example of the asymptotic Jacobian will be presented in chapter V for the simple case of a system with two degrees of freedom. For the present discussion it is sufficient to note that Ja is a periodic function dependent on an effective frequency & which is itself dependent on the frequencies of the normal modes. If r is the period of the internal modes then the value of J at to+r is equal to its value at to Based on the discussion above, the asymptotic Jacobian has the form (IV.38) Ja(t;q) = PO J(t;Q) where the (t;q) dependence of the Jacobian has been indicated. With this Jacobian the asymptotic nuclear wave function has the form 1 x')1 iv( S = a = X C e =a Tr,a X T J(t;q') The next term to look at is the action S. To asymptotically we can rewrite Eq. (111.24) as S(Q) = S(qo) + P dQ 9o i e .(IV.39) see how S behaves (IV.40) The momentum P is a multidimensional vector which can be separated into the sum of two vectors, P = P + P' ~ ~R  (IV.41) where, PR refers to the relative momentum between the atom A and fragment X, and P' to the internal momentum of the fragment X. With these definitions, then, the action integral becomes S(Q) = S(Q) + PR dR 0 + P' dQ' Q This can be written as follows to isolate the term P R R S() = 0 0.R P S(Qo) + (P 0).dR R R ~ R R o + J P'.dQ' .(IV.43) o The term S(Qo) can also be separated into relative and internal contributions, SQo) = S.Ro0 + S'(Q'o) tiR S'Q) (IV.44) With this separation Eq. (IV.43) becomes 5(0) = R. + S'(Q'O) R+ L R PR).dR ro Since asymptotically PR=PR, we can write region as S(Q) 0R R + S'(Q') + R; KR  P'.dQ' 0 + P'.dQ' 0 (IV.45) the action in the asymptotic (IV.46) The only other term to look at is U. Since it contains the argument of J it will change each time J changes sign. Asymptotically this (IV.42) means it will change by n each time the internal Jacobian J' changes a sign. The frequency of this occurrence depends on the value of W. To summarize the asymptotic wave function, then, we have the form Jo i P Po.R P'.dQ'+S'Qo)) = x C e e e (IV.47) In order to satisfy asymptotic conditions the function in Eq. (IV.47) must be equivalent to the general asymptotic form in Eq. (IV.30). Let us first equate the two functions ar and at t=t a, i i Po Ro0 1 SI(gO) (Q) = X C e e (IV.48) i po.Ro o0) =o (2n)d/2 u (Q) e (IV.49) Therefore, setting J (Q) 0 ) gives X S'(Q'o) X Cr e R = (2nM)d/2 u (Qro) i s'(Q'O) X Cp = (2un) vd/2 U(Q') e (IV.51) The factor Cp is a constant asymptotically. In order for ICr12 to be the amplitude of the wave function on electronic surface r it must be initially set equal to one. If we put the resulting expression for Xo back into Eq. (IV.47) we get (IV.50) d/2 ,oo 1/2 e = (2aM) uv(Q'o) J1 e dt i R, Mn E [PI!a(t'IL)12 dt' x e e (IV.52) where the product P'.dQ' has been expanded in a sum over i and dQ' replaced by P'/M dt. Equation (IV.52) is the general asymptotic form of 4). It satisfies the asymptotic conditions at to (i.e. at the initial value of R). The conditions represented by Eq. (IV.52), however, apply in the entire asymptotic region and not just at the initial value.of R which is chosen somewhat arbitrarily in that the only prescription is for it to be outside the interaction region. Clearly the asymptotic form of .v) does not satisfy initial conditions away from the value of R(to) due to the terms involving J' and P'.. This is a problem because the transition amplitude Dfi, and hence the cross section, should be independent of the value of R(to). Remember that R(t ) refers to the relative distance between the fragment X and the atom A after photodissociation of the molecule. The problem of satisfying asymptotic conditions becomes even more complicated as the number of internal degrees of freedom in X increases (N' becomes larger). The problem is not unique to this method, however, as it is also present in purely classical calculations of transition probabilities. An approximate way of solving it in such classical treatments is by taking suitable averages over the initial relative distance. We will return to this discussion in chapter VI where we will show how Dfi can be made to be independent of Ro by two different methods. Up to this point the treatment of the eikonal wave function has been perfectly general in the number of degrees of freedom allowed and in the number of excited states involved in the molecule M. The only constraint in this regard has been that the bond broken in the photodissociation event be one between a molecular fragment X and an atom A. The feasibility of a more general treatment of the problem involving photodissociation of a molecule into two molecular fragments X and X' is certainly within the scope of this theory and should be pursued. However, such a treatment is outside the scope of this dissertation. Also outside the scope of this dissertation is the general solution to the complicated problem of satisfying asymptotic conditions for any number of degrees of freedom N. The present work has been developed in terms general enough so that such topics can be pursued as extensions of the basic formalism presented here. The first step in solving the problem of asymptotic conditions in general, as in any investigative work is to limit the discussion to special model cases which are more easily solved. In the next section we will present various simple models of photodissociation so that the problem of meeting asymptotic conditions can be solved in a special case and results can be generated to compare with previous results from other known theories. In this way the accuracy of the theory can be tested and possibilities for a more general solution can be proposed. CHAPTER V COLLINEAR MODELS OF POLYATOMIC PHOTODISSOCIATION The general photodissociation event as symbolized by Eq. (I.1) can be greatly simplified by imposing various constraints on the molecule M and the fragment X. The factors which can be constrained include the following: i) the number of internal degrees of freedom in X, ii) the spatial orientation of the system throughout the process (i.e. free vs. fixed orientation), iii) the number of coupled electronic states in M (and X) involved in the process. Factors i and ii will affect the number of coordinates in the set q. Factor iii will affect the number of states r to be included and consequently the size of the t and f matrices in Eq. (11.24). In all of the models presented in this chapter the system will have a fixed orientation in space throughout the dissociation. That is, we will be invoking the infiniteorder sudden (IOS) approximation in dealing with the rotational degrees of freedom of the molecule [Pack, 1974]. In addition we will be assuming only collinear dissociation in which all motion of the molecule and the product fragments occur along a line defined by the body fixed axis of symmetry. Two Electronic States A simple model would be one in which there are only two electronic states r=1,2 (ground and excited states) involved, no internal degrees of freedom, and no rotation in the fragment X or in the molecule M allowed (i.e. one dimensional motion). An example of such a model would be the photodissociation of a single bond in one dimension. The photodissociation of a single CI bond in a polyatomic is such an example and has already been studied using the Self Consistent Eikonal Method [Swaminathan and Micha, 1982]. In such a case there is one relative coordinate ROQ and no internal coordinates Q'. The nuclear wave function on excited surface 2 simplifies to S1/2 i i ( ) ( 1/2 Po 1/2 U(t) g o(t) P (t) = (2nH) 1 (t) e e (V.1) A slightly more complicated model is one in which there are again two electronic surfaces r= 1 and 2, but now with one internal degree of freedom in Q'. In this case there are a total of two coordinates, one internal raQ1 and one relative coordinate R=Q%. This model corresponds to the photodissociation of a linear triatomic ABC where A is the atom and BC the fragment X. An example of such a system is the molecule CH3I (methyl iodide). This molecule can be treated as a linear triatomic if one considers the H3 to move as one unit whose center of mass lies along the CI axis (see Fig.VII1). This is equivalent to considering only the umbrella mode of CH3. It turns out that this is a reasonable approach to CH3I dissociation because experimentally it has been shown that the umbrella mode of CH3 is virtually the only one excited during photodissociation [Shobotake, et al., 1980]. The set A in this model will consist of {t;q'} where q' is the angle defined by r(to) = Av sin(q') a ro (V.2) and Pr(to) = Av w cos(q') a Po (V.3) where A is the maximum value of r for vibrational level v, and w is the asymptotic frequency of oscillation for u (r). The range of q' is 0 to 2n, which spans both positive and negative values of P (to) for every r(to). The nuclear wave function along a trajectory will be J'o(q') u(t;q') l[(t;q')S'(Q'o) S (t;q) = (2mr) e e u (q') J'(t;q') (V.4) where u (q') is the value of the harmonic oscillator function at r(t ) for vibrational quantum number v. The internal Jacobian for this model is given by J'(t;q') = P r(t;q')/w (V.5) The equation for () above is exactly the same as the general form of Eq. (IV.32) except that the function C is not a matrix, and the set A is composed of (t;q'). This model will require the simultaneous solution of 8 differential equations. There will be four equations for position and moment, and four to solve for the terms Q. in the Jacobian. Three Electronic States A more complicated model for the photodissociation of a polyatomic is one which involves the same number of coordinates, but an additional electronic surface r=3. Because we now have two excited potential energy surfaces we have to consider the coupling between them. The transition will be from the ground state surface r=1 to the coupled surfaces labeled by r=2, 3. Recall from the transition integral in Eq. (IV.1) that mathematically we will have a column matrix (2x1) for t() multiplying a row matrix (1x2) for the dipole operator D in the integrand. Thus, the nuclear wave function on surfaces 2 and 3 will be the matrix of Eq. (11.25) whose elements are the expansion coefficients for Y ex(Q)> with incoming wave boundary conditions. The wave function for this model will be ()(tq') = (2nM)112 eiS'(O0o)/ Jo(q') 1/2 _i (t;q') R a(t;q') C) Jq') e e u (q'). (V.6) J(t;q') The transition integral for the collinear model with two degrees of freedom and two excited electronic states can be written as Dfi = J' (I+ D.e i IJ dR dr = f[2if D21 + 3f D311 e* i I dR dr (V.7) where D21 and D31 are the vector matrix elements of D and as before imp and fa(pfi,a). The fact that we have two surfaces brings up the question of which surface to propagate the trajectories on. Since the transition can be to either surface and we want to construct a wave function with amplitude on both, then both surfaces must be involved in the dynamics of the problem. If we refer to Eqs. (111.21) and (111.22) we can see how this is possible. The equation for generating the amplitude part of the wave function (C) depends on the average potential V. This potential in turn depends on the mixing of surfaces 2 and 3 by the matrix C and its adjoint. Thus the matrix C is selfconsistently coupled to the potential V. If C+C = 1, then we can say that the quantity ICy[Q(t)] 2 represents the probability of the system being in electronic state r. Therefore, since the potential V governs the motion of a trajectory, we can say that the electronic transitions between states 2 and 3 are selfconsistently coupled to the nuclear motion and that () is being propagated on the average surface V. This model will require the simultaneous solution of 16 differential equations. Specifically, there will be four equations to solve for the real and imaginary parts of C; eight equations to solve for the Qj in the Jacobian, and four equations for position and moment. In the next section we will consider this same model but for the case when the two excited electronic states are of different symmetry. Symmetry Aspects and Cross Sections Having more than one excited electronic state introduces an additional complication if the states are of different symmetry. For example in methyl iodide it has been shown experimentally that the dominant transitions are from the ground electronic state to two excited electronic states with different symmetry types. The excited states referred to are, in the notation of Mulliken, the 3 0 and 10 states which are of symmetry species A1 and E respectively. In order for an electronic transition to be symmetry allowed the direct product of the species for initial and final states with the species for the transition operator must be totally symmetric. The ground state of CH3I is of species Al; therefore, the symmetry selection rule requires that for matrix elements D31 and D21 respectively of Eq. (V.7) to be nonzero Al x r(DA) x A = A (V.8) E x r(D) x A = A (V9) where r(DM) is the symmetry species of the electric dipole moment D' of the molecule. Equation (V.8) corresponding to a transition to the 3 0 state is nonzero only for r(DA) a Al. According to the character table for the point group C3v the species Al corresponds to D'. The term D' refers to the transition dipole operator along the body fixed z z'axis which coincides with the symmetry axis of the molecule. In this type of transition then, the electric field must have a component oriented parallel to the body fixed z'axis. We will refer to this as the parallel transition. Similarly Eq. (V.9) corresponding to a transition to the 1 state requires that r(DM) = E. According to the character table the x and y components of the transition dipole moment (D' and D') have E type symmetry. Thus for Eq. (V.8) to be nonzero ~x ~y the electric field vector must have a component perpendicular to the body fixed axis and this is designated as the perpendicular transition. Because of the product D.e of Eq. (V.7) the relative magnitudes of these transitions obviously depends on the orientation of the molecule with respect to the electric field vector. The factor D is a column matrix of vectors which, for the present model of methyl iodide, consists of D r<2P  (V.10) D1 M 1 (V.11) D, = e D' + e D + e ,D' (V.12) M x' x ~y' y z' where ea, (a'=x', y', z') are the body fixed unit vectors. In order to carry out the dot product in Eq. (V.7) the elements of Eq. (V.10) will be defined in terms of unit vectors as 221 d21 'Sx (V.13) u31 d31 e' where we have arbitrarily set the perpendicular component of the transition dipole along the body fixed x'axis for simplicity. We are allowed to do this only because we are dealing with a pseudolinear molecule for which the transition dipole is isotropic with respect to rotation about the body fixed z'axis. The direction of propagation of the electric field in the laboratory reference frame will define the zaxis of this frame. Experimentally this corresponds to the direction of a laser beam which is crossed at right angles with a molecular beam of target molecules. To obtain the experimental results that we will be comparing with in chapter VIII [Shobotake et al.,1980; VanVeen et al., 1984] the researchers have used laser light linearly polarized in the plane of the crossed beams. We will define this to be the space fixed yz plane as in Fig. (V1). With these definitions we can use the Euler angles to express the dot product as D 21'e0 21 ex,.e D.e = = rd e.e ~~ 31.e 3 e3z,.e 21 cosy cos3 cosa d21 siny sin (V.14) d31 sino sine 55 CH3I beam VVV hv  / T Detector Fig. (V1) Diagram of collision angles in the spaced fixed frame. where e is a unit vector along the spaced fixed yaxis. The angles 0 and a are the polar and azimuthal Euler angles respectively of the body fixed system relative to the space fixed system. The angle y is the Euler angle for rotation of the body fixed frame about the body fixed z'axis. The transition amplitude for the collinear model with two degrees of freedom has been given by Eq (V.7). Since the dot product involves a sum over two states of different symmetry the relative contributions from r'=2 and r'=3 are, as we have shown, dependent on the orientation of the molecule with respect to the direction of the electric field. If we invoke the IOS approximation we can allow the molecule to have different orientations in space and retain the same form for the transition amplitude. The value of Dfi will, however, be parametrically dependent on the Euler angles a, 0, and y of the moleculefixed coordinate system with respect to the spacefixed system. We will denote this angledependent integral in the IOS approximation as D(s) (0,a,y) where the initial and final states have been specified in detail as in Eq. (II.11). Note that the initial and final quantum states are specified by u=(l,vx vy) and os(r,v). By substituting Eq. (V.14) into Eq. (V.7) we obtain D) ,s 1 (XB,'Y) = f[1P*f(R,r,9,<) d21(R,r) (cosycososina + sinycosa) + *3f(R,r,0,a) d31(R,r) sino sina] *i(R,r,,a) IJ dRdr (V.15) At this point it is necessary to define angles 9 and + which define the direction of the detector in the spacefixed frame. The angles 9 and + are also the polar and azimuthal angles respectively of the vector pfi. These angles are distinct from 0 and a which define the orientation of the molecule at the time of absorption of the photon while 9 and f define the direction of detection for the emitted fragment. It is important to be very clear with these distinctions in order to properly interpret the quantities being calculated. Recall that in a photodissociation experiment it is the direction of the incident radiation which defines the space fixed zaxis while the direction of detection is the angle which the differential cross section is dependent on. In order to account for all possible orientations of the molecule upon absorption of a photon, the square of the transition amplitude in Eq. (V.15) above must be averaged in some way over all initial angles. The simplest way to do this is to integrate over all angles a, 0 and y and all rotational moment p po, and p using the classical thermal partition function qr for rotation and a Boltzman distribution function of the classical rotational energy function H These functions have the form [McQuarrie, 1976] r =8 2IkBT (V.16) H = (p + p 2sin2 )/(21) (V.17) where I is the moment of inertia and kB the Boltzman constant. The momentum p does not appear because we are dealing with a linear molecule. The appropriate integral is Dfi h d ddadyddp dpa 1 eHr/(kBT) JD(s ,a,y) 12 (V.18) where pfi indicates that there is a dependence on 9 and *. Since Dfi (O,a,) is independent of p and pg in the IOS approximation the integration over the angular momentum is straight forward and results in ID fi 2 = daddy sing ID ((3',ay)2 (V.19) The collinear models introduced in this chapter describe one dimensional motion (motion confined to a line) yet the perturbing influence (the electric field) is a vector in three dimensional space. Note that in the integral in Eq. (V.19) we have not yet invoked the collinear approximation. This is indicated by using the vector Dfi in the subscript rather than the one dimensional scalar pfi. At this point we will impose the restrictions for the collinear model. In a collinear model we know that the products fly off along a straight line whose orientation in the spaced fixed frame is given by a and a. We can predict therefore, that the only contribution to the cross section amplitude in the direction of gfi will come from an orientation of (0,a) which is coincident with the pfi orientation given by (9,+). With these considerations we will approximate the integral in Eq. (V.19) by an integral over the square of the transition amplitude Dfi(3,a,y) from the collinear model. This transition amplitude has two terms arising from the perpendicular and parallel transitions, Dfi(,pca,y) = Df 2)(cosy cos0 sinm + siny cosa) + D sinP sina (V.20) where we will define D r = (R,r) drl,(R,r) *i(R,r) IJI dR dr (V.21) Using this transition amplitude we will approximate Eq.(V.19) by IDf (9,#)h = 1 ddxdady sing IDfi(0,a,y) 2 6(cos9cos0) 8() . (V.22) Note that the (9,+) angular dependence has been approximated by using delta functions of cos9 and *. The dependence of Dfi on a and 0 has been explicitly determined in Eq. (V.20). Carrying out the integration first over y eliminates all cross terms giving the equation ID(9,) = dda sin [ ID(2)12 (cos20 sin2M + cos2) RE fi i9 9 9 + 2 ID 2 sin2 0 sin a ] I(cos9cos0) 6(4a) (V.23) It is the expression in Eq. (V.23) that will be integrated over solid angle Q to obtain the amplitude which will finally be used to calculate the integral cross section. The expression for the integral photodissociation cross section within the collinear model and in the IOS approximation, then, is 2nMR 2 dg Dfi( 1 = fi pg(o) d ))Df(Q)th (V.24) All that remains is to carry out the integral over 9 and f. After doing this we arrive at an expression which has the same weighting factors for the parallel and perpendicular contributions to the cross section. The final expression for the integral cross section is fi = 2RMMR g(()2 { ID 2) 2 + ID )12 } (V.25) "fi 3Mc l ffi This is the expression we will use in the next chapter to calculate the integral cross section for the photodissociation of methyl iodide using a model with two excited electronic states. CHAPTER VI SATISFYING ASYMPTOTIC CONDITIONS Statement of the Problem At the end of Chapter IV we began to address the need to impose conditions on the asymptotic form of ( In doing so we encountered the problem that / (t;g') seemed to satisfy asymptotic conditions only at the beginning of a trajectory (t=to). We also hinted that this problem would be dealt with for a specific model and a general solution presented. In this section we will show how asymptotic conditions can be satisfied for a two coordinate linear model with one relative coordinate R and one internal coordinate r. This is the same model proposed in chapter V. Two different methods will be proposed to accomplish this. Each method can in principle be applied to any twocoordinate linear model regardless of the number of coupled electronic surfaces. Recall from Eq. (IV.52) that the asymptotic form of t does not seem to be equivalent to the form of Eq. (IV.30) except at to. In other words a () except at the initial time. By comparing the two functions and removing common terms we arrive at the following expression for the internal part of the asymptotic wavefunction: 1/2 1 h E [t ] dt' U (q) J' e e 2 to a(91) (VI.1) J' The root of the problem is the fact that ua in Eq. (VI.1) does not equal u where u is the harmonic oscillator vibrational wavefunction for quantum number v. To simplify things we will begin by limiting the discussion to the two dimensional linear model. Within this model Eq. (VI.1) reduces to t 2 1/2 to ['(t;ro)]dt' u (r0) P'(t0,ro) e e P (t;ro) ua[r(t;ro)] (VI.2) Now that we have identified the problem we will proceed to propose two methods to deal with it. The first method recognizes that ua is cyclically equal to uv at time t=to and t=t0o+, where x is the period of oscillation of u In the second method we alter the form of ) to involve a sum of two exponentials whose phases are equal in magnitude but opposite in sign. Using this form we can construct a WKB internal wavefunction to approximate u (r). Method 1: Averaging Over a Period The wavefunction u (r) is a vibrational function for the diatomic nuclear vibrational motion. There is a frequency w and a period T associated with this vibrational motion where T 2n (VI.3) Strictly speaking this period is only present asymptotically, but this is the only region we are concerned with when setting initial conditions. We know that Eq. (VI.2) is equal to u (r) at t = to; therefore, if we can show that Eq. (VI.2) is also periodic over the same T then we know that the wavefunction ( will meet asymptotic conditions at every t = to + nt, where n is an integer. We will do this by showing that each factor in ua is periodic over T. The first factor to consider is the exponential of iu/2. The value of u will increase by 2n after each period T, therefore, the exponent decreases by in. The other exponential factor contains the integral of P'2. This integral increases by (2vn + n) after a period T, where v is the vibrational quantum number. Combining the integral of P,2 with U/2 after every period T gives for the nth period, iz(t0+n)+ L[P 2dt = nni + n(2vn + n)i 2 MM to = ni(n + 2vn + n) = nv2ni (VI.4) From Eqs. (VI.4) and (VI.2) we can conclude that the exponents in ua add up to an integral multiple of 2ni after every period T and, thus, do not affect the value of ua at times t=t +n in the asymptotic region. The only other factor involves the square of the moment and will have the same periodicity as the internal wavefunction u (r) because the vibrational momentum P'(r) has the same period T as the vibrational motion. We have now looked at each time dependent factor on the left side of Eq. (VI.2) and have shown that each returns to its initial value after every period T. Since the function u [r(t;ro)] is also periodic over a time t, then, the following equality holds: r 1/2 ; 0n P;r) ( o )2dtn o P (to0 2 n Mh to n u (t ;r ;r e e = u [r(Tn;ro)] (VI.5) where Tn = to+nt. The equality in Eq. (VI.5) shows that asymptotic conditions are met at every T (n = 0,1,2...). Since t satisfies asymptotic conditions periodically, then the transition integral Dfi will reflect this periodicity and it is expected that Dfi will be a periodic function of the initial relative coordinate R(to) for the bundle of trajectories. Calculations of Dfi with various initial values of R show this to be true. This behavior does not corresponds to physical reality, however, because the cross section should be totally independent of R(to). Remember that R(to) in the two coordinate model is the distance between A and BC after the photodissociation event. In order to calculate a transition integral independent of R(to) N R the first method will consist of simply averaging the square of the function Dfi(Ro) over a distance in R corresponding to one period of oscillation in r. This average will be denoted D fi and defined bave defined by IDfive 1 R 0) IDf (R) 2 (VI.6) NR R(r) where NR is the number of equally spaced values of R in the average. Note that time to = 0 has been arbitrarily assigned to one end of the interval. To be consistent with a model having incoming wave boundary conditions we will require that R(O) > R(T). The equation for the relationship between R(0) and R(T) of Eq. (VI.6) is Po0 R(t) R(0) + M (VI.7) MR where Po is negative. From Eq. (VI.7) we can see that the quantity ID fiav is independent of Ro. Thus, if we use it in the calculation then the cross section will also be independent of Ro. For the special case of an internal potential which is harmonic (or any symmetric potential) asymptotically, the transition integral Dfi(Ro) is cyclic over only half a period; i.e. Dfi(O) = Dfi() (VI.8) For such a case, then, the average in Eq. (VI.6) will be over the interval from R(0) to R(T/2). 66 Method 2: Construction of a WKB Internal Wavefunction Let us look again at Eq. (VI.2). The idea behind the second method for meeting asymptotic conditions is to construct () in a way that would result,in pa having the form of the WKB bound state wavefunction. The WKB wavefunction for a bound internal state has the form [Messiah, vol.I], uWKBr) k Cos (r) dr  w =' =J a k r i i " 2 i e S + S (VI.9) 2F^p e [e KK r where S =  P (r)dr  a SVKB 4 k = constant = 2 r and Mr is the reduced mass for motion along r. Obviously we cannot reproduce the sum of two exponentials in the WKB wavefunction unless we use a sum of two functions in Our modified wavefunction, which we will call will have the form + (VI.10) S where = e and = +,  Asymptotically we will separate the action into relative and internal contributions, Sa= SR(R) + Sa(r) where the subscript "a" indicates the values of each variable or function in the asymptotic region. We can show that this modified wavefunction satisfies the same differential equation as Before exploring these assumptions, however, we will show how satisfies asymptotic conditions for all asymptotic values of R. Let us start with the detailed asymptotic form of so as to separate out that part we can equate with the WKB internal wavefunction uWKB(r). Asymptotically 1 1 i poR i S SoT2 &a R r a+ a Ia = aXa e e Koe +Koe (VI.11) ,a a J a where we have used SRa = PR (VI.12) Equation (VI.11) must be compared to *0 of Eq. (IV.30) in order to isolate ua(r) which in this case will be equivalent to uWKB(r). Setting X a = (2nH)1/2 we obtain 1 1 i p (t 2 a M a+ u (r) = t2 e oK eK a P (t) + i ' S Sa + Ko e j By comparing Eq. (VI.5) with the WKB function in Eq. (VI.1) we have the requirement that [ La a] i a] + _ +  iS "r iSt is^ (VI.14) exp i(S /H /2 + pa) a=+ , P a r = exp Ui JP' l dr MH /K C=+, a (VI.15) where it is implied that KoeC = Ko Ko 0p (to0) 12 = K (IV.13) (VI.16) and p. is a constant to be determined. To prove the validity of Eq. (VI.15) and determine pC we will first determine the form of S a. To do this we will assume that S aC(r) is the asymptotic internal action which is calculated from a trajectory where P (r ) = CIP'(ro). If we define S C(r) as 0 r S (r) = J Pdr (VI.17) a then we obtain r SaI(r) =J P dr (VI.18) a where P, is the internal momentum whose value at r is ZIP(ro). By examining a trajectory whose initial internal coordinate and momentum are r and IP (ro) we can arrive at an expression for S (r) in terms a+ of S KB(r) and A where r S (r) = IP'(r) dr (VI.19) a and & is defined as the number of times that the momentum P (r) has changed sign during the trajectory from ro to r. As we follow this trajectory we will rewrite the internal momentum in the action integral as P (r) = IP'(r) (VI.20) where & is the sign of P We will also use the relation a I P'(r)dr = Kn(v + 1/2) (VI.21) a where v is the vibrational quantum number. The expression for S a+(r) we arrive at is ',. R 1 nH RH a+(r) = B(r) + + + + + (VI.22) We can also follow the trajectory with initial conditions ro and I P (ro)I to obtain S (r). Using the same relations we obtain a S A + (VI.23) Sa(r) = [SKB(r) + ) + + (VI.23 The two Eqs. (VI.22) and (VI.23) are close to the form we need to construct the WKB internal wavefunction. In order to simplify these equations it is important to note the relationship of A to &. For Eq. (VI.22) & = 1 for even values of A and 1 for odd values of A. This 71 makes the sum (A + 1/2 V/2)r an integer multiple of 2n. Thus, this part of Sa+ can be dropped because the exponential of 2ni is one. The same is true of the sum (A + 3/2 V/2) in Eq.(VI.23) because for S a we find = 1 for even values of A and & = 1 for odd values of A. At this point it is necessary to look back at Eq. (VI.15) and note that the asymptotic form of u is Pa = arg[P'(r)] arg[P (ro)] = Ax (VI.24) Substituting Eq. (VI.24) into the left side of Eq. (VI.15) and using S a from Eqs. (VI.22) and (VI.23) gives + WKB / + P VKB 44 e I e + e I e i K ] / i [ ]/M (VI.25) = e + e where & is the sign of P We are now very close to showing how the two sides of Eq. (VI.25) are equal. One obvious step left is to determine the values of p and p In order to eliminate the exponentials involving p, we set p+ = n/4 (VI.26) p = 3n/4 (VI.27) The final step is to note the relationship between E& and Having assumed that Sa+ is calculated from the trajectory whose initial assumed that Sa+ is calculated from the trajectory whose initial I momentum is P (ro) = IP(ro) we have prescribed that t (t,ro) will be constructed from two trajectories whose position and moment are always opposite in sign asymptotically; i.e. P (r) = P (r) (VI.28) As a consequence of this judicious choice we have that = F (VI.29) Noting that cos(E) = cos(E) where Z is any phase and using Eq.(VI.29) we arrive at the equality & ] /K]' & I ^ n11 /H e + e  = 2os KB/ n] I"S ?K] i[S/  =e + e thus proving Eq. (VI.15). After so may steps it is helpful now to sum up the prescription for this second method of satisfying asymptotic conditions. We have derived a specific form for the of Eq. (VI.10) 1 1 1 ,(t;r) = (2nj 2 C 2 e ip a (t;ri x Ko e e (VI.30) where S (Q) = o (t;ro). To satisfy the form of the WKB internal wavefunction asymptotically it is required that 1 Mo 1 Ko = 2j P(to I (VI.31) The wavefunctions *. and p have been defined as those generated from the trajectories beginning at ro and ro with initial moment P (ro)I and IP (ro)I respectively. Therefore, the function T (t;r ) is constructed from two trajectories by adding the wavefunctions generated along each using the proper value of p from Eqs. (VI.26) and (VI.27). The wavefunctions +,.(t;r ) of course satisfy the same differential equation as does ((t;ro), because they differ only by the constant exp(ip ). In this method the transition integral of Eq. (IV.5) is independent of Ro so there is no need to average. This is due to the way y meets asymptotic conditions at all (R,r) in the asymptotic region. The form of t( in this region being 1 i 44(t;ro) = (2nh)2 e1 poR M E,a r 1cIP(t;ro ) x cos (r)dr (VI.32) a The majdr advantage in this method over the first one presented is that there is no average over Ro needed and thus not so many trajectories are required. There are two main weaknesses in this approach however. The first one is that the internal wavefunction is being approximated by a WKB wavefunction. This can be a poor approximation especially at low vibrational quantum numbers. The second weakness is the restriction that the internal potential be symmetric asymptotically. In most cases the internal potential is only approximately symmetric asymptotically. This particular approximation would be especially poor at high vibrational quantum numbers. Since the WKB approximation is poor for small quantum numbers and the symmetric potential approximation poor at large quantum numbers there will probably be a certain range of quantum numbers for which this method gives optimal results. CHAPTER VII RESULTS: APPLICATION TO METHYL IODIDE In this chapter the Self Consistent Eikonal Method is applied to the collinear photodissociation of methyl iodide. Statetostate cross sections are calculated for two different models. The first corresponding to excitation from the ground electronic state to a single dissociative excited electronic state. This shall be designated as the 1ex case. The second model calculation involves excitation from the ground state to two dissociative excited electronic states which are coupled (the 2ex case). Computational details such as the selection of initial conditions are also presented. Coordinates Methyl iodide is treated as a linear triatomic as in the treatment by Shapiro and Bersohn where the umbrella mode of the CH3 group is modelled by an "equivalent effective bond" resulting in a linear triatomic problem with two vibrational coordinates. Essentially, the model replaces the three hydrogen atoms by an effective mass (of three hydrogens) located in the plane of the real hydrogens and collinear with the CI bond axis. The coordinates used are the Jacobi coordinates. In the 2coordinate linear model the Jacobi coordinates correspond to the distance R from the iodine atom to the center of mass of CH3 and the distance r from the carbon atom to the plane of the three hydrogens. In this chapter the mass weighted variables will be denoted R, PR and r, Pr for relative and internal variables respectively. The distance R is given in terms of r and the CI distance RCI as follows: R = RCI + 0.2011r (VII.1) These coordinates are illustrated in Fig. (VII1). Corresponding to each coordinate there is a reduced mass defined by R = (M) (MCH3) / (CH3 I ) (VII.2) Mr = (MH3) (MC)/ (MCH3) (VII.3) Recall that in chapter VI the problem of requiring () to satisfy asymptotic conditions at all Q was addressed and solved by two different methods. The method of averaging over a period was used in all the calculations for which results will be presented in this chapter. The integral cross section expression for the collinear model is given by Eq. (V.25). As discussed in chapter V the effect of the symmetry of the excited states on the cross section is most important when considering two or more excited states of different symmetry. For the 1ex case the effect of integrating over angles is simply to multiply the cross section by a constant 1/3. For the 2ex case, however, the additional effect is of have a sum over the squares of the perpendicular and parallel contributions to the transition amplitude. Fig. (VII1) Jacobi coordinates for CH3 . When satisfying the asymptotic conditions using method 1 the squares of the contributions are given by the following sum D ') 2 R b D fi (R) 12] (VII.4) R0R a where the interval from Ra to Rb is divided into NR equally spaced distances and Ra > Rb. The value of Rb is chosen arbitrarily outside the interaction region. The length of the interval (Ra Rb) is such that the CH3 vibrational motion goes through half a period as R goes from Ra to Rb, i.e. R (t = to) = R R (t + T /2) = Rb P 0 R = R + R (VII.5) b a 2MR where T is the period of the umbrella motion of CH3 asymptotically and PR is the incoming asymptotic relative momentum. Recall the variable transformation from the set 0 to a set A defined in Eq. (111.2). In the case of the two coordinate linear model this corresponds to the transformation (R, r) 4 {t, wl)} (VII.6) where v1 is an angle defined by r(to) = A sin wl, (VII.7) v is the vibrational quantum number for the internal "umbrella" mode of CH3, and A is the value of r at the classical turning point. Because the internal potential is asymptotically harmonic the internal position and momentum are given by r(t) = Av sin[w(t to) + wl] (VII.8) P, (t) = Mr w A cos [W(t to) + w1] (VII.9) P (to) = rM A cos (w1) (VII.10) where S= f (VII.11) r Each trajectory is distinguished by the angle w1. A particular value of wl gives the initial internal position r(t ) and momentum Pr(to). The range of w1 is from 0 to 2n. Thus, a grid of v1 values will span all classically allowed values of r along with both the positive and negative corresponding moment at each r For example, the value of ro has associated with it both IP r0 and Pro . This bundle of trajectories will all begin at the same asymptotic value of R with the same initial time t As an example of this a bundle of 10 trajectories is shown in Fig. (VII2). There will be NR such bundles used in each calculation of ID r) 2. For example if the range of w1 is divided up into Ntr equally spaced intervals then for to = 0 there will be Ntr trajectories propagated from Ro = Ra After a value for IDr) (R )12 has been calculated another bundle of trajectories is propagated from R = R( ) = R and another ID rfR )2)2 calculated. This continues until NR such bundles have been propagated and Rb = R(r/2) is the next R0. Note that ID r)(Rb) 12 is not included in the average. The initial relative momentum is given by conservation of energy and is negative indicating that the iodine atom is moving toward the CH3 fragment. Remember that t is being propagated in such a way that the initial conditions of the trajectories correspond to the system after the photodissociation event. Thus, the system is followed backward in time. The total energy of the molecule plus photon system is conserved as shown by the following E. Ef = 0 (VII.12) P 2 E = C + E + hv (VII.13) i 2M '2 2 E ac Pfi E c+ + E (VII.14) f 2M 2MR a 1.0 0.7 0.3 L 0.0 0.3 0.7 1.0 3 4 5 6 7 8 9 10 11 Fig. (VII2) Trajectories on the excited electronic potential energy surface. where E. is the energy of the CH3I molecule plus photon system initially (in the ground state), Ef is the energy of the system after having absorbed the photon, E is the energy of the CH3I molecule initially for quantum state p=(l,vxv y), E is the combined energy of the fragments in the final quantum state a=(r,v), Mc is the mass of CH31 and pfiP .P The numbers r and v label the I electronic state and the CH3 vibrational state respectively. As in chapter II we assume that the momentum due to the photon is much smaller than the center of mass momentum and so I P = P + HkPh P (VII.15) Equation (VII.12) then reduces to o2 E + hv  E =0 (VII.16) U 2MR which gives the following equation for P o 1/2 PR = [2MR(E + hv E )] (VII.17) For a given initial and final quantum state E and E are fixed and the photon energy hv can be varied from calculation to calculation, giving a curve of cross section vs. photon energy. One Excited Surface The only states considered in the first calculation (1ex) are the ground state and 3 0 excited state. The 3 state leads to the fragments CH3 + I* ( P1/2) while the ground state leads to CH3+ I( P3/2) as shown in Figure (VII3). The potential surfaces given in analytical form were determined'empirically by Shapiro and Bersohn [Shapiro and Bersohn, 1980]. The ground state surface is V11 (RCI, r) = De (exp[0.899(RCI 4.043)] 1}2 + 1 {k + (.1100) exp[0.4914(RC 4.043)]} x (r 0.6197 exp[0.4914(RCI 4.043)]}2 De E* (VII.18) 2 3 k a the force constant = 0.0363 e /a0 D = 0.0874 e2/a0 E* = 0.0346 e2/a and the variable RCI=(R .2011 r) is the carbon to iodine distance. All energies are in Hartrees and lengths in Bohr units. The 3 0 surface is given by V33 (R,r) = 9.618 exp (1.40R) + 2.604 exp (1.20R + 0.24r) + 1/2 (0.0362)r2 (VII.19) To have meaningful vibrational quantum numbers for the ground state the R, r motion must be transformed to normal modes. This was done by Lee and Heller who arrived at the x and y modes given by the 0.30 0.25 0.20 0.15 S0.10  0.05 0.00 0.05 0.10 0.15 3 4 5 6 7 9 10 R (a.u.) Fig. (VII3) Diabatic potential energy curves for Cl3I at r=0. .J 0 C b( 0 a, $4 0 o (U 0* F M O H H bd "4 I 1a 11 0 to Ci *i. 0 rz. 88 I1I _ II if I / \ I \ \ / II l ' I \ u I i /1  . following transformation [Lee and Heller, 1982] 7.830 0.1762 1 (VII.20) 01y.6183 4.939 ) Cx) where R = RCI 4.043 ao RCX = r 0.6197 a. and R = R + .2011 RCX Lee and Heller have calculated the four lowest vibrational eigenvalues for the ground electronic state. They are (v v E (a.u.) (0, 0) .00399346 (1, 0) .00643926 (2, 0) .00885801 (0, 1) .00949585 Also given by Lee and Heller is the vibrational wavefunction for the (0, 0) state in terms of harmonic oscillator wave functions in the normal coordinates, Y00(x, y) = 0.9966110 g00(x, y) 0.0816282 g10(x, y) 0.0101739 g01 (x, y) (VII.21) where g (x, y) is the product of harmonic oscillator wave functions xy for quantum numbers v and v The wavefunction in Eq. (VII.22) has been normalized with respect to the coordinates x and y. Before using it in the transition integral it must be renormalized with respect to the mass weighted Jacobi coordinates R and i using the determinant of the rotation matrix in Eq. (VII.20). Results will be shown for excitation from this (0, 0) level and compared to those of Lee and Heller. Results for excitation from the next three vibrational levels (1,0), (2,0), and (0,1) of the ground state will also be presented in this chapter. For these higher vibrational levels a simple product of harmonic oscillator functions was used for the ground state vibrational wave function. The equations of motion for the 1ex case are r P (VII.22) P r r r aH 1 R P (VII.23) aPR MR R 8V P = v ex (VII.24) r r r c a V where Vex is given by Eq. (VII.19). The Jacobian is numerically calculated from 91 r R r Jt w 1 A 8i 1 1w7 3w1 = (R M (aa '1 l = l = R Q1 (VII.26) 1 1 11 1 01 P 1 0 P 0 (VII.27) 2 2V 1 ex 1 ex 1 P1 = e 00 + 2 1 (VII.28) 2ai 8i 1 ex 1 ex 1 PO = 2 0 + 0 Q1 (VII.29) where the following notation of chapter IV has been used: Q = A 01  (VII.30) 0 a1 1 aw1 1 1 1 R 1 r 0 aw 1 av (VII.31) 1 1 It is possible to arrive at an approximate expression for the Jacobian by looking at the asymptotic form of the terms in Eq. (VII.26). The expression for the asymptotic Jacobian is Ja = PR P / (VII.32) In Fig. (VII6) the function Ja (dotted line) is compares to the exact calculation of J for a single trajectory. From this figure we can see that J coincides with Ja until the interaction region, then begins to deviate slightly. It is not until the trajectory begins to exit the C r*4 0 *( r4 0 r M 0 4, 0 O 0 (0 a) 10 *i 41 co 11 o 93 1 4b @ S OS? M8 I I (;)r 