Photodissociation of polyatomic molecules

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Photodissociation of polyatomic molecules state-to-state cross sections from the self-consistent eikonal method
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Thesis (Ph. D.)--University of Florida, 1987.
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Includes bibliographical references.
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by Clifford David Stodden.
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Vita.

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Full Text







PHOTODISSOCIATION OF POLYATOMIC MOLECULES:
STATE-TO-STATE CROSS SECTIONS FROM
THE SELF-CONSISTENT 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 Per-Olov 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........................

Radiation-Molecule Interaction in the
Electron-Field 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 A-SPACE.....................

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 1-ex Case....................................... 147

The 2-ex 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:
STATE-TO-STATE CROSS SECTIONS FROM
THE SELF-CONSISTENT EIKONAL METHOD

By

CLIFFORD DAVID STODDEN

December 1987

Chairman: Dr. David A. Micha
Major Department: Chemistry



A general method is presented for calculating state-to-state

cross sections for photodissociation of polyatomic molecules by

visible and UV radiation. The method also provides branching ratios

by self-consistently 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 freedom--the 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 cross-section 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;

Balint-Kurti 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) state-to-state

transitions.

The particular method to be discussed uses the self-consistent

eikonal approximation. It is self-consistent in that the electronic

transitions are self-consistently 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


Radiation-Molecule Interaction in the Electron-Field 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 molecule-field 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 molecule-field

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) = V-1/2 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) V-1/2 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 vibro-rotational energies of the

fragments. Thus the cross section to be calculated is a state-to-

state (electronic, vibrational, rotational) resolved quantity at a

particular photon energy. The expression for this cross section is

[Rodberg & Thaler, 1967]


d fi

d i-1 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 molecule-field 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 (II.18)








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 efia|DI(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>=I|fi 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



= IV(Q;,fi)> (11.23)


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 spin-orbit 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 Hamilton-Jacobi

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 Q-space (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[(t-to) + q.'] (111.6)










where ai is the frequency along Q! and to is the initial time. The

asymptotic region is that part of Q-space 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
J-I


(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 self-consistently 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 A-SPACE


Transformation to an Integral Over the Time Variable


The quantity of interest to the experimentalist is the state-to-

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=V-1V 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 (a0-k 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 A-X 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 Q-9Q 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)
ti-R 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;- K-R


- 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 infinite-order 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 A--B-C

where A is the atom and B-C 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 C-I axis (see Fig.VII-1).

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 e-iS'(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 self-consistently

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 self-consistently 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 z-axis 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 y-z

plane as in Fig. (V-1). 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. (V-1) Diagram of collision angles in the spaced fixed frame.








where e is a unit vector along the spaced fixed y-axis. 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

molecule-fixed coordinate system with respect to the space-fixed

system. We will denote this angle-dependent 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 space-fixed 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 z-axis 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 e-Hr/(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(cos9-cos0) 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(cos9-cos0) 6(4-a) (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

two-coordinate 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


S-VKB 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 Z|IP(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
1-cIP(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. State-to-state

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 1-ex case. The second model calculation involves

excitation from the ground state to two dissociative excited

electronic states which are coupled (the 2-ex 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 C-I bond axis. The coordinates used are the Jacobi

coordinates. In the 2-coordinate 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

C-I distance RCI as follows:



R = RCI + 0.2011r (VII.1)



These coordinates are illustrated in Fig. (VII-1). 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 1-ex case the effect of integrating over angles is

simply to multiply the cross section by a constant 1/3. For the 2-ex

case, however, the additional effect is of have a sum over the squares

of the perpendicular and parallel contributions to the transition

amplitude.






































Fig. (VII-1) 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. (VII-2). 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. (VII-2) 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 (1-ex) 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 (VII-3). 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. (VII-3) 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





1-1


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 1-ex 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. (VII-6) 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
*-(














r-4

0













r-
M
-0
-4,
0
O-






0










(0
a)

















10



















*i-
41
co


















1-1








o-





93



















1

-4b


















@ S OS? M8
I I
(;)r