PHOTODISSOCIATION OF POLYATOMIC MOLECULES:
STATETOSTATE CROSS SECTIONS FROM
THE SELFCONSISTENT EIKONAL METHOD
By
CLIFFORD DAVID STODDEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
I would like
to dedicate this work
to my father
from whom I learned
to ask of nature
why and how.
ACKNOWLEDGMENTS
I would like to thank some of the many people who in one way of
another have helped or supported me in this work. First, I would like
to thank Professor David A. Micha who supplied the basic formalism for
this work as well as most of the funding. I spent many hours in
discussions learning from his expertise in this field and developing
the invaluable skill of how to patiently work through problems which
arise in the research.
I would like to thank those graduate students, postdoctoral
associates and faculty who worked together to foster a cooperative
atmosphere of research and learning at QTP. I would especially like
to thank Professor PerOlov Lttwdin 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.
?!
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
t
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.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTERS
I INTRODUCTION 1
IICALCULATION OF THE PHOTODISSOCIATION CROSS SECTION
USING THE EIKONAL WAVE FUNCTION 4
RadiationMolecule Interaction in the
ElectronField Representation 4
The Eikonal Wave Function 14
III INTRODUCTION OF A TIME VARIABLE:
TRAJECTORY EQUATIONS 18
A New Set of Variables 18
The Eikonal Wave Function Along a Trajectory 22
IV THE TRANSITION INTEGRAL IN ASPACE 29
Transformation to an Integral Over
the Time Variable 29
Generating the Jacobian Along a Trajectory 31
Asymptotic Conditions 39
V COLLINEAR MODELS OF POLYATOMIC PHOTODISSOCIATION 47
Two Electronic States 48
Three Electronic States 50
Symmetry Aspects and Cross Sections 52
IV
VI SATISFYING ASYMPTOTIC CONDITIONS 61
Statement of the Problem 61
Method 1: Averaging Over a Period 62
Method 2: Construction of a WKB
Internal Vavefunction 66
VII RESULTS: APPLICATION TO METHYL IODIDE 75
Coordinates 75
One Excited Surface 83
Two Excited Surfaces 115
VIII DISCUSSION AND CONCLUSIONS 146
The 1ex Case 147
The 2ex Case 150
Considerations on Angular Distributions 155
Conclusions 158
APPENDICES
A SHORT WAVELENGTH APPROXIMATION 164
B COMPUTER PROGRAM FOR THE SCE METHOD 166
BIBLIOGRAPHY 169
BIOGRAPHICAL SKETCH 173
v
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
i
PHOTODISSOCIATION OF'POLYATOMIC MOLECULES:
STATETOSTATE CROSS SECTIONS FROM
THE SELFCONSISTENT EIKONAL METHOD
By
CLIFFORD DAVID STODDEN
December 1987
Chairman: Dr. David A. Micha
Major Department: Chemistry
A general method is presented for calculating statetostate
cross sections for photodissociation of polyatomic molecules by
visible and UV radiation. The method also provides branching ratios
by selfconsistently coupling nuclear motion to transitions between
several electronic surfaces. Cross sections are calculated using a
transition integral between the initial ground state wavefunction of
the molecule and a final wavefunction for the relevant states of the
product fragments. The final state nuclear motion wavefunction with
incoming wave boundary conditions is generated in an eikonal form
along classical trajectories which follow an effective potential and
contains the exponential of a common action integral and a pre
exponential given by an amplitude matrix and a Jacobian describing the
divergence of the trajectories. In a new development, the Jacobian of
the variable transformation is generated exactly along a classical
vi
trajectory without requiring the simultaneous integration of adjacent
trajectories. Two methods are presented for satisfying the asymptotic
conditions of the eikonal wavefunction.
In model calculations on CH^I the dynamics are described by two
degrees of freedomthe relative position of I to CH^ and the umbrella
mode of CH^. The first calculation involves a transition to a single
3
dissociative excited Qq 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
2 2
CH^ and I( or I ( ^1/2^' Quantum yields of .84 and .69 are
2
calculated for I*( P^/2^ production at photon wavelengths of 266nm and
2
248nm respectively. The crosssection curves for I( P^j) 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.
vii
CHAPTER I
INTRODUCTION
This study addresses the problem of formulating and testing a
computationally efficient and accurate theory for treating the
photodissociation of polyatomic molecules with many internal degrees
of freedom and many electronic states. The photodissociation event of
interest will be limited to cases where incident radiation in the
ultraviolet to visible range is absorbed by a molecule causing a
transition to a dissociative state which gives a single atom and a
molecular fragment as products.
There are already many theories of photodissociation. They
include quantum mechanical approaches [Heather and Light, 1983;
BalintKurti and Shapiro,1981; Clary, in press] as well as classical
and semiclassical trajectory approaches [Atabek et al., 1976; Mukamel
and Jortner, 1976; Billing and Jolicard, 1984; Gray and Child, 1984;
Henriksen, 1985]. Lee and Heller have also developed a theory for an
exact time dependent approach using wave packets [Lee and Heller,
1982]. An exact approach in principle gives results which are more
reliable than a semiclassical method. However, for large molecules
with many internal degrees of freedom and many coupled excited states
such calculations may be prohibitively complicated. The computational
effort required for a coupled channels calculation for example
increases by the power of the number of channels involved.
1
2
A semiclassical method which employs classical trajectories
requires much less computational effort than an exact approach for
large systems. For this reason it is important to develop an
efficient, accurate semiclassical method which is able to deal with
many degrees of freedom and many coupled electronic surfaces. Of the
semiclassical methods mentioned above none are at present able to
effectively deal with more than one excited state potential energy
surface. The photodissociation events of interest here have a bound
initial electronic state with a certain set of vibrational quantum
numbers and a final electronic state with a set of final vibrational
quantum numbers for the molecular fragment. With this in mind another
desirable feature of any photodissociation theory is to be able to set
initial conditions as well as final conditions for the event, i.e. to
calculate cross sections for state to state transitions.
The formalism in the following treatment is designed, then, to
include certain desirable features. These features can be summarized
as the ability to efficiently deal with (i) many degrees of freedom,
(ii) many coupled electronic states, and (iii) statetostate
transitions.
The particular method to be discussed uses the selfconsistent
eikonal approximation. It is selfconsistent in that the electronic
transitions are selfconsistently coupled to nuclear motions. The
eikonal approximation simply means that the nuclear wave function is
expressed as a modulus matrix times an exponential with a common
eikonal phase and that the short wavelength approximation is applied.
3
The general process of interest is the following photodissociation
reaction,
M(vm) + (k,a)> A(va) + X(vx) (1.1)
where M is a polyatomic molecule in a given electronic and nuclear
state \>^, A is an atom in electronic state \>A, is a photon with
wavevector k and polarization a, and X is a molecular fragment in
electronic and nuclear state With the method presented in the
following chapters one can treat photodissociation events as in
Eq. (1.1) 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 VAVEFUNCTION
RadiationMolecule Interaction in the ElectronField Representation
The interaction of a photon with a molecule can be modeled as a
collision problem [Micha and Swaminathan, 1985]. In such a treatment
we consider the photon to be occupying a finite volume V in space and
the photon field to be contained within a cube of side length L. The
molecule M(\>^) enters this space and interacts with the field. The
field is quantized so that the moleculefield system is described by a
Hamiltonian
H = Hm + Hp + Hmf (II.1)
The wavefunction for the molecular system is represented by 'fM(Q)>
where the "ket" indicates electronic coordinates and Q is a vector
whose components are mass weighted Cartesian coordinates for nuclear
motion. The Schrodinger equation for the molecular Hamiltonian H is
HM IVQ~ = E lYM(5>> (H2)
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.
4
5
The second term in the Hamiltonian Hp refers to the radiation
field. It is given by [Loudon, 1973]
+
a. a,
k,a k,a
(II.3)
where a,+ and a, are the creation and annihilation operators of a
k, a k, a
photon with wavevector k, polarization a and energy The
product of the annihilation and creation operators is the number
operator which operates on an eigenstate (k
field to give the number of photons in state (k
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 Hu refers to the moleculefield
interaction. In the dipole approximation this interaction term is
hmf(^c) = J $<Â£'> dV (II4)
where Â£c is the position of the center of mass of the molecule M. The
factor is the dipole operator, which, in the center of mass frame
at position r is
E
c
a
r
~a
Â£)
(II.5)
where {c ) is a collection of charges at positions {r }. The electric
cL
field operator E(r) in second quantized form is
6
E(r) Y, e() Xk(r) a* X^r)*]
k, a ~ ~
g(w) = i [Mw/(2e)]
1/2
(11.6)
(11.7)
where e0 is the vacuum dielectric constant and e^ is the polarization
unit vector with
circularly polarized light. The factor X in Eq. (II.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) (II.8)
The wavevector k has components 2rcjjVL with jÂ£=0,1,2... and E=x,y,z
corresponding to cyclic boundary conditions for a cube centered at the
origin of coordinates. Replacing Eqs. (II.5) to (II.8) in Eq. (II.4)
we obtain
HMF<^>  Â£ 8 V tD^(r ,k)
k, a
1/o ik.r
l/2 ~ ~c
Vea
k, a
1 ik.r
%(ra,k) V1/2e ~~Ca^a]
(II.9)
5^(ra,k) = E ca Â£a exp(ik.Â£a) (II.10)
a
The Hamiltonian in Eq. (II.1) acts on a state j) which describes
the state of both the molecule and the photon field. The initial
state i) and final state f) for a single photon event are given by
7
i> = lec y (k ff)> and lf> = IE; Efi
di.id
where p and p' are the initial and final momenta of the center of
*c *c
mass, u indicates the initial molecular state, and a indicates the
final state of the fragments. In Eq. (II.11) the state i>
corresponds to a molecule initially in the presence of a photon field
and the state f> to a dissociative excited state with no photon
present. The fragments in state f> have a relative final momentum
vector of The corresponding total energies are
(11.12)
E.
i
Ef (e4)2/(2mc) ei/<2ra) + E^
(11.13)
where m is the total molecular mass and m the reduced mass for the
c
relative motion of A and X of Eq. (1.1).
The dissociation cross section is the quantity of interest in a
photodissociation experiment. The initial and final states are
characterized by both electronic and vibrorotational energies of the
fragments. Thus the cross section to be calculated is a stateto
state (electronic, vibrational, rotational) resolved quantity at a
particular photon energy. The expression for this cross section is
[Rodberg & Thaler, 1967]
(11.14)
8
where is the transition rate and is the incoming photon flux in
the laboratory frame given by
= c/(2itM)
3
(11.15)
where c is the speed of light. To obtain Eq. (11.15) we have used
momentum normalization for which
(11.16)
The transition rate is given in terms of the transition amplitude T^
and Dirac's delta function with respect to energy,
(11.17)
The transition amplitude T^ is an integral over electronic and
nuclear coordinates and contains an operator T which, for low
intensity incident radiation, is simply the moleculefield interaction
of Eq. (II.9). For the single photon process to be considered here
only the first term in H^p is appropriate. A single photon event
corresponds to a weak electric field (i.e. low intensity). The form
of the transition amplitude, then, is
9
The last factor containes the Integral over which gives
lie IT
l I2 = (2rch)3 8
Therefore the integral in Eq. (11.14) becomes independent of gc with
the restriction that g = Ec + Kk. The cross section now contains an s
integral over g^ only,
= J4 g() (2it/M) (2rtM)~J
x J dpf. p2. 2 S(Ef E.), (11.20)
where p^ is the magnitude of g^. If we note that Mk << gc it
follows that g ' g With this very reasonable approximation and
using E^ = E^ we get from Eqs. (11.12) and (11.13) for p^
2m
= E E
U a
Mw
(11.21)
Transforming the integral in Eq. (11.20) by noting that
dEf = p^ dp^/m and using the definition of J^, the differential
cross section becomes
da
d S2
fi ,2 2nn,Pfi 1
= g()
Me
(11.22)
10
The factor in brackets will be denoted as indicating that the
transition is from a molecule initially in state i>=u> to fragments
in a final state f>=g^ a> in the asymptotic region. The final
state is designated by the states of the atom A and of the fragment X
asymptotically as anc* vxs^X^ w^ere ^ 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 y> for the molecule in the center of mass frame is
given. It remains, therefore, to obtain the final state wavefunction
The vector Q 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 rc> In the following notation,
however, we will be working strictly in the center of mass frame and
the symbol Q 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
(11.23)
This molecular eigenstate can subsequently be expanded in a basis of
the electronic states of M involved in the process, so that
1^(0 = tM(Q)> ^m(Q)
(11.24)
fM > = (  i>2> *^> ) is the matrix of electronic
wavefunctions, and
11
(11.25)
V
is the matrix of nuclear wavefunctions where h is the number of the
highest state energetically accessible in the reaction.
We will express as a sum of the nuclear kinetic energy
operator Knu and a term Hq which contains the nuclear repulsion and
all the energies of the electrons, including spinorbit coupling,
Hu = K +
M nu
(11.26)
Knu = (2M) 1
(11.27)
where M is an arbitrary mass which comes from mass weighting, and the
symbol V is a multidimensional gradient whose components are 3/3Q^,
i=0 to N. The gradient V to the second power is defined to be
2 2 2
(V) = 23 /3Q
(11.28)
The coefficients of the expansion contained in ^ are the nuclear
motion wavefunctions on the electronic potential energy surfaces.
These surfaces are defined by the diagonal elements of
(11.29)
12
When Eq. (11.23) is substituted into Eq. (II.2) and left
multiplied by <$M we have the following matrix differential equation
Sq + 5(QJ
+ H~o 
4M
(11.30)
We will Work in the diabatic picture [Smith, 1969] in which
G(Q)= = 0
(11.31)
This leaves us with the general matrix equation:
f 1_
f \
H_ o
1 2M
V
1
l /
* 5q eJ*h(Q) 0
(11.32)
When the coupling between the electronic ground state (T=l) and
the other electronic states is negligible, H can be separated by
blocks,
Hi: 0 0 ....
0 ^22 ^23** *
II
O'
0 H32 H33...
.0 v,
^ J
(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:
13
IV2 Iv *1 2 lv *t
(11.34)
We will define
Ygr(Q)> = *1(Q)> ^(Q) (11.35)
to be the ground state wavefunction, and
h
^ex(Q)> = *(Q)> (Q) = I j*r(Q)> *r(Q) (11.36)
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
Tex(Q)>. The matrix equation for the nuclear wavefunctions on the
excited electronic surfaces is
= 0
(II.37)
This is the equation we solve next using the common eikonal
approximation.
14
The Eikonal Vavefunction
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)(_) = *(Q) exp[i S(Q)/H] (11.38)
where the minus sign on 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
ih 2 ih H2 2 (VS)2
 2M S>* M 2mH X + + (V m = 0 (11*39)
We can also write this as
(11.40)
where 0 is the operator
0 = 1/2M (M/i V + VS) + V E
(11.41)
Note that 0 is in the form of a Hamiltonian minus energy
15
O = H E
(11.42)
H = T + V (11.43)
T = T(p,VS) (11.44)
where the normal momentum operator p=M/i V has been replaced by the
sum Â£ + VS thus defining a new momentum operator p' in the kinetic
energy operator T,
T =(p')2/2M
(11.45)
If we define the multidimensional classical momentum to be
P = 3S/3Q s VS (11.46)
we get the following for Eq. (11.39):
M 2 9 H (?P) u (P?) P
= ~2M X + J 2M ^ + r " + 2M % + ))Â£. (H.47)
In order to express Eq. (11.47) in the form of a HamiltonJacobi
equation we can carry out the following operations:
*+ X +
X+ 0 X + X+ 0+ X = 0 (11.49)
With the proper manipulation Eq. (11.49) becomes
16
X x 2V
M ~
m lpl
^ + IM
After dividing by 2)(+)( this can be put in the form of the Hamilton
Jacobi equation,
(P) /2M + ? = E
v ~ qu
(11.51)
where
V (js*x)'1{xt<r Vi m p
f i x *x
. (11.52)
It remains to solve for X(Q). To do this we first substitute Eq.
(11.51) into Eq. (11.47) to give
iH
 it Hp X ^ P.& ^ ^X + (V lVqu)X = 0 (11.53)
We can simplify Eq. (11.53) by introducing the following notation:
U = H/2M 7.P ,
(11.54)
Kp r = 5r,r(h2/2M)72Xr/Xr
(11.55)
Equation (11.53) becomes
/
M
T
1
P.7
V ) + K l(iU)
qu7 ~ ~ '
X = o
(11.56)
i
17
Equation (11.52) becomes the expression for an average potential
7 when we assume the second term to be negligible and make the short
wavelength approximation, for which 7^=0 (see appendix A), leaving us
with
7 = (X+*)1 X+v*
(11.57)
We will now apply the short wavelength approximation in which K is
negligible and 7^u becomes 7 to give
' p. 7
J jjj + (v IV) l(iU)
X = 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 Q at the
initial time. Initial values of the Q are denoted as q and are given
by the set
q = {q^, ^2 ^n^ (III.l)
where q^ sets the initial value of for a given trajectory, qÂ£ the
initial value of QÂ£, etc. One of the degrees of freedom in Q, namely
Qq, 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+l degrees of freedom in the problem.
Introducing this collection of variables is equivalent to
transforming from the set of spatial variables Q 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,
18
19
A a {t, qv q2, ... qN} a Â£t; q} .
(III.2)
Each trajectory evolves with the time variable t separately. The
other members of A will insure that each trajectory follows a unique
path in Qspace (even though a particular trajectory will cross others
many times).
It is convenient to define sets Q' and q' to be the subsets of Q
and q respectively which refer to internal variables. They will each
contain N' components
Q' a {Q1' Q2' .Q^j, }
(III.3)
(III.4)
where N' is the number of internal degrees of freedom in the system.
We can write the initial value of any the Q^' in terms of an angle q.'
defined by
Q.'(t=0) = (y.)'^ a sin(q.')
i ' v i7 vi 7
(III.5)
where A is the maximum value of Q.' for quantum number v. and y. is
vi i M i 1
the reduced mass for the i1"*1 internal mode. Note that for the Qi
corresponding to normal vibrational modes, if the potential energy
along this Q^' is harmonic in the asymptotic region (Qq*) then we can
write
(III.6)
20
where is the frequency along and t is the initial time. The
asymptotic region is that part of Qspace where the potential is
separable in Q', i.e.
9V(Q)/9Q' = f(Q') (III.7)
Thus Eq. (III.6) gives us the functional relationship for a
vibrational mode between Q' 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 momenta can be formulated using
Hamilton's equations. Ve define Eq. (11.51) to be the Hamiltonian H
and impose the restriction of energy conservation to obtain
(III.8)
This leads to Hamilton's classical equations of motion
dQ
dt
9H_
9P
(III.9)
and
dP
dt
9H_
9Q
(III.10)
Both Q and P can be written as a collection of variables,
Q 5 (Q0, Qr VV (III.11)
p = {PQ, px, p2...pn)
(III.12)
21
For each component of Q and P, then,
dt
(III.13)
dP.
i
dt
(III.14)
It is helpful at this point to discuss mass weighted coordinates.
We define the following to be mass weighted coordinates,
Qi
(HI.15)
P.=
i
(III.16)
where and P^ are the original Cartesian coordinates for the i*"*1
degree of freedom. The mass M can be arbitrarily set to some value
without affecting the results of the derivation. From Eqs. (III.15)
and (III.16) it follows that
dQ.(A)
Pi(A) = M (III.17)
Equation (III.17) shows the advantage of using mass weighted
coordinates in that the reduced mass vk 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
22
P.7
IT
N dQ.
& ^
d_
dt
(III.18)
The Eikonal Vavefunction Along a Trajectory
Having introduced the set A we can rewrite ^ using this new set
as
 \ J y(t') dt'
XI0(A)] = Xo C(A) e t (III.19)
where
Y = J 7.P (III.20)
and X is the value of X at t which is set by initial conditions. If
we replace Eqs. (III.18) and (III.19) into Eq. (11.58) we get
(H/i) dC/dt + (V 1V)C = 0 (III.21)
All that remains is to solve Eq. (III.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
(III.22)
23
Ue will also construct the nuclear wavefunction along the trajectory.
It now has the form
, n S(Q) 4 I l dt' n a(t;a)
r vQ) = X(Q) e = x C(t;a> e t en (III.23)
where S(Q)a( t ;<^). It will later be shown that the quantity involving
Y simplifies to a function of the momenta and other terms which can be
generated numerically along the trajectory.
The final part of left to construct is the factor
r
e
To do this we refer to the definition of P in Eq. (11.46) and
integrate to get
(III.24)
S(Q) S(Q)
(III.25)
1 N t ?
S[Q(t;g.)I = S[Q(t;a)] + Â£ J Pi dt s a(t;^ (HI26)
i=0 tov
where Q indicates all initial values of Q and t the initial value of
t. Thus, according to Eq. (III.26), to calculate
24
integrate the square of the momenta along each trajectory over the
time variable.
Ve will now develop a prescription for generating y. The
function y = Â¡7.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 Q to {t;q}.
Let us first recall that the term involving y appeared in as
a result of having transformed the differential equation from spatial
coordinates Q 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.
(III.27)
Let us look at one term in the sum on the right side of Eq. (III.27),
i.e. 3PV3Q^. Ve know from calculus that we can always relate a
partial derivative in one coordinate system with the partials 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:
3P.
i
(III.28)
25
Note that the i*"*1 column of the numerator in Eq. (III.28) contains
rather than Q^. The explicit form of the Jacobian in Eq (III.28) is
9Qo
9Qi
9Qn
9t
at
at
9Q0
9Qi
9qx
3qx
3qx
o
CM
rt>
3Qn
9qN
A
9qN
3qN
(III.29)
If we use Eq. (III.28) in Eq. (III.27) the product appears as the
following sum over i of the determinant in Eq. (III.28) divided by the
Jacobian,
9Qo
ap.
i
9qn
at
at *
at
9Qo
ap.
i
3Qn
aqx
3qx ""
3qx
9Qo
ap.
i
WN
9qN
9qN **
3qN
(III.30)
where J is the Jacobian of Eq. (III.29). The summation of
determinants in Eq. (III.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
26
3Q0
3\
9QN
3t '
3q j 3t
3t
N
3J V"1
SQi = L
2 i=0
3Q0
32i
9QN
3qx *
30
3qj3qx
32Q .
A
3qx
9qn
(III.31)
9qN *
9qj9qN
*" 9qN
The determinant in Eq. (III.31) looks
very similar to that in Eq.
(III.30) except that the i1*1
column contains double
derivatives
rather than the derivatives of
P. If
i
, however,
we
recall that
P^=M(dQ^/dt), then we can make
these determinants identical by
choosing qj = t to obtain
3Q0
32Qi
9QN
3t *
3t2
3t
N
3J V1
= L
3Q0
32Q.
9QN
3qx
3t 3q^
3qx
(III.32)
i=0
3Q0
2
3 Q .
9QN
A
3
9t 9qN
aqN
If we now switch the
order of
differentiation on the
double
derivatives and replace 3Q^/3t
by P./M
we get that
27
N
3J_ it"
at n_,
i=0
3P.
i
p
*0
3t
o
o
3P.
i
3qx
dq1
30
3P.
l
aqN '
3%
N
3qx
3Q,
9Qi
(III.33)
After comparing Eq. (III.33) with Eq. (III.30) we can express y in
terms of the time derivative of the Jacobian which gives
dJ
dt
?) J
(III.34)
Y
1 dJ d ln(J)
* ~ J dt dt
(III.35)
Recall that the term involving y appears in the nuclear
wavefunction as an integral in the exponent. Replacing Eq.
(III.35) in the integral gives
exp j [ ln( gi arg(J)j + In J  In arg(J)j lnj]
= expj j [i arg(J) i arg(J )]
1/2
(III.36)
28
where J refers to the value of the Jacobian at time t
Thus we have
1
~2
e
1/2
exp{j[ arg(J) arg(J) ]). (III.37)
Our nuclear wavefunction can now be constructed along a
trajectory defined by A a {t;^'}. The final form of this wavefunction
is
<{/ *(t;^) = X C(t;c[)
J(0;a)
J(t;^)
1/2 1 , N i , .
2u(t;^) R a
e e
(III.38)
where J is the Jacobian at time t, J is the Jacobian at the initial
time, and y contains the argument of the Jacobian as follows:
U(t) = arg(J) arg(J)
(III.39)
The amplitude matrix C is selfconsistently solved for through
Eq. (III.21). The factor X is a constant whose value will be
determined in chapter IV by initial conditions. The other terms are
constructed from quantities generated along the trajectories. The
nuclear wavefunction calculated in this manner will be used in the
next chapter to calculate the transition integral which appears in the
cross section of Eq. (11.22).
CHAPTER IV
THE TRANSITION INTEGRAL IN ASPACE
Transformation to an Integral Over the Time Variable
The quantity of interest to the experimentalist is the stateto
state cross section. It is calculated using a transition integral
between the ground state wave function and the function yt \ This
is calculated from the quantity which appears in Eq. (11.22).
The general form of the transition integral is
(IV.1)
where r 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 over electronic coordinates. It
has components
[D(Q)]ra = <*p(Q) D*1(Q)>.
(IV.2)
The integration over space dt in Eq. (IV.1) can be written as a
29
30
multiple integral over the set of coordinates Q {Qq, Q^, ... Q^}.
Thus the transition amplitude is a multiple integral in (N + 1)
variables which can be written as
Dfi = J[*f )]+ 2S *i d5 (IV.3)
In order to calculate this integral using trajectories we must
again make a variable transformation from Q to A as in the last
section. The integral in Eq. (IV.3) becomes
Dfi I 8S *i
,Q0 Q1
*
V
,qNj
dA
(IV.4)
where
Q0 Q1
V
qi qNj
is the absolute value of the Jacobian of
transformation and the collection of variables A is defined by A
={t,q} where q={q^,q2>q^} contains the initial coordinates for a
trajectory. To show explicitly that now contains an integral over
time we write
Dfi =  t]+D.e J dt dq (IV.5)
We can approximate the integral over dq by a sum over Aq. This leaves
us with a sum of integrals over dt:
Dfi n V *i, lJl dt
31
Dfi,n V Â£
_)]+D .e ip.
~n ~ i, n
J dt
1 n 1
(IV.7)
For simplicity we have dropped the subscript f from The
label (n) which appears as a subscript in the summation refers to
individual trajectories. Therefore, is a function which is
generated along the n1"*1 trajectory. Since each trajectory is defined
by its values of q it can be said that (n) labels each unique set q .
This can be seen by defining
)(t> a 4( )(t5an)
(IV.8)
According to Eq. (IV.6) each trajectory n will contribute a
certain amount Dc. to the cross section. This integral will be
ti ,n
calculated along the n1"*1 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 K 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 dQ to
dA. It has not yet been shown how J can be calculated along each
32
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. (III.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. (III.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
Q* s 9Q./3q. and P* s 3P./3q.
J J i J J1
(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
J
% Qr
t q. .
1
M
(IV.10)
To obtain the time derivatives of P^ and let us start by recalling
Eqs. (III.13) and (III.14). If we take the derivative of the left
side of Eq. (III. 13) with respect to q^ and switch the order of
differentiation we get
33
2 2 i
9 Qj 9 Qj 9Qj
3qi 3t 3t 3q 3t
(IV.11)
Taking the derivative of the right side of Eq. (IV.11) with respect to
q^ gives the relation
3Q*/3t = 3P./3q. = P*
3 J i ]
(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 q^ we get
2 2 i
3 P. 3 P. 3PT
J J m J
3q^ 3t 3t3q. ~3t
(IV.13)
Following the same procedure as before, we set Eq. (IV.13) equal to
the derivative of the right side of Eq. (III.14) as follows:
3P*
J
~3t
32 V
3q. 3Q.
3
(IV.14)
However, since we have no expression for the derivative of V with
respect to an additional step must be taken. Ve know the quantity
? only as an explicit function of Q. We also know that the derivative
dependence of on q^ is one of the terms being generated
(Q^=3Q^/3q^). Therefore we can expand 32V/3q^3Qj in terms of the
by using the chain rule.
34
It is important to note that the average potential 7 is a
function of the components Cp(t;q) of the amplitude matrix C(t;q) from
Eq. (III.22). It is convenient to express Cj. in terms of its real and
imaginary parts
Cr a K (Xr + iYr) (IV.15)
where K is a constant to be specified later. With the definition of
Eq. (IV.15) we note that
7 a 7(Q; X,Y) (IV.16)
where X and Y are sets containing, respectively, the collection of Xj,
and Yj, components of all Cj,. Since V is a function of the X^ and Yj.
these must be included in the chain rule expansion.
Define, a 37/30^ (the force along Q^) and substitute into
Eq. (IV.14). Using the chain rule we get
N
E
3F.)
1
E (
(3F.)
J
3Xr
+
(3F. j
J
k=0
r 9q. r l
UxJ
n v3(li
Uy_J
(IV.17)
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:
1
(IV.18)
35
Using Eqs. (IV.17) and (IV.18) and the definition of we get the
following for the time derivative of P^:
3P*
3t
N
E
k=0
32V
3Qk.3Qj
'X,Y
3
f \
3V
l3xr
9Qj
^ J /
(IV.19)
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 X^. and Yj,1. 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 Q* and P^.
J J
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 Xj, and Y^,. To get an expression for
these time derivatives we use Eq. (III.21) and separate the real and
imaginary parts given in Eq. (IV.15) to obtain
dxr ai7
= X V W Y
dt ~ T j* IT' 1T' 3Yf
111 = Y _V y x W_
dt r L* WIT' Ar 3Yr
(IV.20)
(IV.21)
where the matrix W is defined as W=V1V and we assume the matrix V
defined in Eq. (11.33) to be real. Next we take the partial
derivative of Eqs. (IV.20) and (IV.21) with respect to q^ and switch
the order of differentiation
36
!r_ a_
3q. 3q.
(IV.22)
3V
rr'
(IV.23)
Once again ve run into a situation where ve must expand a partial
derivative with respect to q^. By using the chain rule expansion of
we obtain
y
f ay )
d rr'
ni y
(ay ay
rr' i rr'
it
30.
Uk 41
*rtt
y y
[3xr rw f 3Yr
9
+ S wrr' Yp (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
V, namely, the diagonal terms. Ve can indicate this by using a delta
function to get the final expression
(y
Purr'l
n1 s V
3V
It
K J
Qk 5rr' p,
Y V 1
l9Xr"
9V_ i
3Ypi r"
t
+
(IV.25)
37
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
Y t =
fy
3wrp 1
lEk
l9Qk J
Q,. 
E
TP
bl
rn
3V_ i 3V_ Yi
3xp, Ar" + 3Yp, r*
+ 2 Wrr, Xp,
(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 Pj and
thus for in terms of quantities we can calculate. We are already
generating all in time and so we have all the terms to calculate J
along the trajectory. The only terms in of Eq. (IV.7) which we
have not yet talked about are those which we need to be given, namely,
D .e and
n ~ T,n
The ground state wave function will be given as a function of
Q; therefore, (t) is simply the value of *p. at that point Q which
*** i, n i "
corresponds to the n1"*1 trajectory at time t; i.e.
= *.[Q(t5a )] (IV.27)
1 y 11 1 II
The factor 5n.e will similarly be given as a function of Q.
We now have all the elements required for the transition
integral. The entire expression in detailed form is
E D
n
f i, n
(IV.28)
38
where
Â£i,n <*In> Â£<
i[arg(Jn) arg(J)]/2
ian(t)/H
e
x g()Dn(t).e ^.n(t) Jn dt (IV.29)
The symbol J is defined as the value of the Jacobian at time t (the
initial time) for the n^ trajectory.
Notice in Eq. (IV.29) that there will be a singularity in
o 1/2
caused by the term j /J each time J passes through zero.
Remember that 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 ~^ 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 are integrated out in
this way makes it numerically possible to use the semiclassical wave
function t/without having to integrate through poles.
Asymptotic Conditions
There is still a problem in the way behaves asymptotically.
In order to satisfy asymptotic conditions the nuclear wave function
must have the following form as R* and t>t:
39
*(Q) = (2nH) d/2 uv(Q') e
(IV.30)
where d is the the dimensionality of the space that AX is constrained
to (d=l, 2, or 3), uv(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 Q 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 asymptotically we
will look at how each term in Eq.(IV.29) behaves as QK) where Q
*** ~a
refers to values of Q which are in the asymptotic region.
Asymptotically we can express the eikonal wave function as
(IV.31)
where the subscript "a" indicates that the various functions are in
the asymptotic region. To solve for XC let us recall Eq. (III.21)
and look at the term (V IV). Asymptotically, the off diagonal
elements of V will be zero. If we choose the wave function initially
to be entirely on excited surface T, then the term V becomes
V
(IV.32)
40
The elements in V 17 = V then become
ny
= 5..(E. 
ij i
V
(IV.33)
thus having off diagonal coupling elements equal to zero and a zero
for the Tth diagonal element.
Because there are no off diagonal elements in the matrix equation
asymptotically and because C will have the following form
Q '
9r
lo )
a,r
(IV.34)
Eq. (III.21) will reduce to
dCr
d  *
(IV.35)
thus C is constant in time.
~a, r
The Jacobian has a very interesting and simple form
asymptotically. In the asymptotic region any internal motion along
will not affect the relative motion along and vice versa. Using
this fact we can set the terms 3Q^/3q^ and 3Q^/3q^ equal to zero where
and q^ refer to relative variables. The Jacobian becomes
41
po
P1
P2
P3
PN
8Q0
3Q1
3Q2
0
. 0
3qx
3q1
3qx
CD
o
o
3Q1
3Q2
n
. 0
3q2
3q2
3q2
u
0
0
0
3Q3

3QN
3q
3Q3
3QN
3%
3
(IV.36)
with the Jacobian determinant having a block, for relative variables
(i=0,l,2) and a block for internal variables. Thus the determinant
reduces to a product of the Jacobians
J = J _
iQ0 1 Q2'
J'
Q Q2 **
a a,R
[t q2J
a
5 q2 qN',
(IV.37)
Outside the interaction region the potential along Qq, Q^, and Q2 does
not change and so Pq, P^, 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^ = Q
and q0 = Q?, then the determinant symbolized by J D becomes simply
z z a,K
the constant Pq. An analysis of the Jacobian J^ for the asymptotic
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
42
frequency which is itself dependent on the frequencies of the normal
modes. If T is the period of the internal modes then the value of Ja
at t+x is equal to its value at t.
Based on the discussion above, the asymptotic Jacobian has the
form
Ja(t;q) = P J(t;a) (IV.38)
where the (t;q) dependence of the Jacobian has been indicated. With
this Jacobian the asymptotic nuclear wave function has the form
\
"2 u(t^}
e
i
R
e
ff(t;q)
.(IV.39)
The next term to look at is the action S. To see how S behaves
asymptotically we can rewrite Eq. (III.24) as
S(Q) = S(Q) +
(IV.40)
The momentum P is a multidimensional vector which can be separated
into the sum of two vectors,
P = PR + P' (IV.41)
where, P^ refers to the relative momentum between the atom A and
fragment X, and P' to the internal momentum of the fragment X. With
43
these definitions, then, the action integral becomes
S(Q) = S(Q) +
dR
This can be written as follows to isolate the term P
S(Q)
The term S(Q) can also be separated into relative and
contributions,
S(Q)
pr5 *
S'(Q')
With this separation Eq. (IV.43) becomes
S(Q) = PrR + S'(Q') + J~(PR PR)dR + J~ P'.dQ'
Since asymptotically PR=P^, we can write the action in
region as
rQ
s(q) pr*r + s'(q,) + r p,dQ'
Q
(IV.42)
dQ' .(IV.43)
internal
(IV.44)
(IV.45)
the asymptotic
(IV.46)
The only other term to look, at is y. Since it contains the argument
of J it will change each time J changes sign. Asymptotically this
44
means it will change by n each time the internal Jacobian J' changes
cl
sign. The frequency of this occurrence depends on the value of w.
To summarize the asymptotic wave function, then, we have the form
i R
R ?~r$
dQ'+S'(Q')j
(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 and \Â¡P at t=t.
 P.R a S'(Q')
 X Cr .*
1 P R
0/r.O. /'7ttW\<^/2 fnr\ R ~ *~
+ (Q ) = (2itM) uv(Q' ) e
Therefore, setting vj/ l(Q) = ^(Q) gives
a, i ~ ~
X Cr S (Q } = (2nH) d/2 uy(Q')
(IV.48)
(IV.49)
(IV.50)
 Sf(Q' )
Xo Cr = (2itM)_d/2 uv(Q') e R (IV.51)
2
The factor is a constant asymptotically. In order for Cj. to be
the amplitude of the wave function on electronic surface T it must be
initially set equal to one. If we put the resulting expression for X
back, into Eq. (IV.47) we get
45
*Â¡>r(A) = (2nM) d/2 uv(Q')
r/
1/2
x e
^ P R
H ~R~
P'
i,a
(t' i.q) ]
2
dt'
V
I
(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 ^ It
satisfies the asymptotic conditions at t (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 does not satisfy initial conditions away from the value
of R(t) due to the terms involving J' and P'^. This is a problem
because the transition amplitude D^, and hence the cross section,
should be independent of the value of R(t). 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
46
taking suitable averages over the initial relative distance. We will
return to this discussion in chapter VI where we will show how can
be made to be independent of R 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. (1.1)
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 T to be included and
consequently the size of the and $ matrices in Eq. (11.24).
In all of the models presented in this chapter the system will
have a fixed orientation in space throughout the dissociation. That
is, we will be invoking the infiniteorder sudden (IOS) approximation
in dealing with the rotational degrees of freedom of the molecule
[Pack, 1974], In addition we will be assuming only collinear
dissociation in which all motion of the molecule and the product
fragments occur along a line defined by the body fixed axis of
symmetry.
47
48
Two Electronic States
A simple model would be one in which there are only two
electronic states r=l,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 Cl 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 RsQq and no internal coordinates
Q'. The nuclear wave function on excited surface 2 simplifies to
i/)(t) = (2nM) 1/2
P(t)
1/2
>(t)
e
a(t)
(V.l)
A slightly more complicated model is one in which there are again
two electronic surfaces T= 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 rsQ^ and one relative coordinate RsQq. This model
corresponds to the photodissociation of a linear triatomic ABC
where A is the atom and BC the fragment X. An example of such a
system is the molecule CH^I (methyl iodide). This molecule can be
treated as a linear triatomic if one considers the to move as one
unit whose center of mass lies along the CI axis (see Fig.VII1).
This is equivalent to considering only the umbrella mode of CH^. It
49
turns out that this is a reasonable approach to CH^I dissociation
because experimentally it has been shown that the umbrella mode of CH^
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(t) = Av sin(q') a r (V.2)
and
Pr(t) = Av (o cos(q') a P (V. 3)
where A is the maximum value of r for vibrational level v, and w is
the asymptotic frequency of oscillation for uv(r). The range of q' is
0 to 2ji, which spans both positive and negative values of P (t) for
every r(t). The nuclear wave function along a trajectory will be
^ (t; q')
= (2jiK)
J'(q')
J'(t;q')
2 2M(t;q,) J[^(t;q')S'(Q'0)]
e e uv(q')
(V.4)
where uv(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'(tjq') = Pr(t;q')/o)
(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
50
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 momenta, and four to solve for the terms
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 T=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 T=1 to the
coupled surfaces labeled by T=2, 3. Recall from the transition
integral in Eq. (IV.1) that mathematically we will have a column
matrix (2x1) for 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 (Q)> with incoming wave boundary
6 X ***
conditions.
The wave function for this model will be
*<>(t;q') = (2nM)1/2 eiS'/M
C(t;q')
J(q')
J(t;q')
1/2 ip(t;q') J ff(t;q')
e e uy(q'). (V.6)
The transition integral for the collinear model with two degrees of
freedom and two excited electronic states can be written as
51
Dfi = Jlf )]+ SS* +ilJl dR dr
= J^2f ?21 + ^3f D311 lJl dR dr (V.7)
where and are the vector matrix elements of D and as before
isy and fs(gfi,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. (III.21) and (III.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 7. This
potential in turn depends on the mixing of surfaces 2 and 3 by the
matrix C and its adjoint. Thus the matrix C is selfconsistently
coupled to the potential 7. If C+C = 1, then we can say that the
quantity Cj.[Q(t)] represents the probability of the system being in
electronic state T. Therefore, since the potential 7 governs the
motion of a trajectory, we can say that the electronic transitions
between states 2 and 3 are selfconsistently coupled to the nuclear
motion and that ^ is being propagated on the average surface 7.
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
52
for the q in the Jacobian, and four equations for position and
momenta. 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
3 1
states referred to are, in the notation of Mulliken, the Qq and Q
states which are of symmetry species 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
CH^I is of species A^; therefore, the symmetry selection rule requires
that for matrix elements and respectively of Eq. (V.7) to be
nonzero
A1 x r(D) x A1 = A: (V.8)
E x r(D) x A1 = Ax (V.9)
where r(D^) is the symmetry species of the electric dipole moment
of the molecule. Equation (V.8) corresponding to a transition to the
53
3
Qq state is nonzero only for T(D^) a A^. According to the character
table for the point group C^v the species A^ 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 state requires that r(D^) = 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
(V.10)
(V.ll)
(V.12)
where e (cr'=x', y', z') are the body fixed unit vectors.
54
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
(V.13)
where we have arbitrarily set the perpendicular component of the
transition dipole along the body fixed x'axis for simplicity. We are
allowed to do this only because we are dealing with a pseudolinear
molecule for which the transition dipole is isotropic with respect to
rotation about the body fixed z'axis.
The direction of propagation of the electric field in the
laboratory reference frame will define the zaxis of this frame.
Experimentally this corresponds to the direction of a laser beam which
is crossed at right angles with a molecular beam of target molecules.
To obtain the experimental results that we will be comparing with in
chapter VIII [Shobotake et al.,1980; VanVeen et al., 1984] the
researchers have used laser light linearly polarized in the plane of
the crossed beams. We will define this to be the space fixed yz
plane as in Fig. (Vl). With these definitions we can use the Euler
angles to express the dot product as
2^ cosy cos0 cosa d2^ siny sina'
d^i sing sina
(V.14)
55
CH^I beam
VVV
Fig. (Vl) Diagram of collision angles in the spaced fixed frame.
56
where e^ is a unit vector along the spaced fixed yaxis. The angles
0 and a are the polar and azimuthal Euler angles respectively of the
body fixed system relative to the space fixed system. The angle y is
the Euler angle for rotation of the body fixed frame about the body
fixed z'axis.
The transition amplitude for the collinear model with two degrees
of freedom has been given by Eq (V.7). Since the dot product involves
a sum over two states of different symmetry the relative contributions
from P = 2 and P=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 will, however, be
parametrically dependent on the Euler angles a, g, and y of the
moleculefixed coordinate system with respect to the spacefixed
system. We will denote this angledependent integral in the IOS
(s)
approximation as D 7 (g, a, y) where the initial and final states
aÂ£f.,U
have been specified in detail as in Eq. (II. 11). Note that the
initial and final quantum states are specified by ys(l,v ,v ) and
x y
a5(T,v). By substituting Eq. (V.14) into Eq. (V.7) we obtain
Dip\,y ($>a,Y) = J[ ^2f (R r ^ d2l^R,r^ (cosycosgsina + sinycosa)
1
+ (R>r> 30 d^(R>r) sing sina] ^(R,r, g, a) J dRdr (V.15)
57
At this point it is necessary to define angles 9 and which
define the direction of the detector in the spacefixed frame. The
angles 9 and are also the polar and azimuthal angles respectively of
the vector These angles are distinct from g and a which define
the orientation of the molecule at the time of absorption of the
photon while 9 and define the direction of detection for the emitted
fragment. It is important to be very clear with these distinctions in
order to properly interpret the quantities being calculated. Recall
that in a photodissociation experiment it is the direction of the
incident radiation which defines the space fixed zaxis while the
direction of detection is the angle which the differential cross
section is dependent on.
In order to account for all possible orientations of the molecule
upon absorption of a photon, the square of the transition amplitude in
Eq. (V.15) above must be averaged in some way over all initial angles.
The simplest way to do this is to integrate over all angles a, g and y
and all rotational momenta p p0, and p using the classical thermal
a rg *y
partition function q^ for rotation and a Boltzman distribution
function of the classical rotational energy function H^. These
functions have the form [McQuarrie, 1976]
q = 8n2Ik_T (V.16)
r d
Hr = (p3 + P^/sin2g)/(2I) (V.17)
where I is the moment of inertia and k.D the Boltzman constant. The
D
momentum p^ does not appear because we are dealing with a linear
58
molecule. The appropriate integral is
2
f i th
1
a q
r
e
2
(V.18)
where g^ indicates that there is a dependence on 9 and . Since
is) 1
Dfi (0,a,y) is independent of pa and p^ in the IOS approximation the
integration over the angular momentum is straight forward and results
in
2
^ th
7T dotdgdy sing D
8n J
(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 g^ in
the subscript rather than the one dimensional scalar p^. 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 g and a.
We can predict therefore, that the only contribution to the cross
section amplitude in the direction of g^ will come from an
orientation of (g,a) which is coincident with the g^ 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 D^(g,a,y) from the collinear model. This
transition amplitude has two terms arising from the perpendicular and
parallel transitions,
59
(2) (3)
Df^($>a,Y) = '(cosy cosg sina + sinY cosa) + Dj^'sing sina
(V.20)
where we will define
Dfi,} = K'f(Rr) dr'i(Rr) VRr) lJl dR dr (v.2i)
Using this transition amplitude we will approximate Eq.(V.19) by
Dfi(0 4>) l^h ^JdedodY sing Df ( 0, a, Y) 2 S(cos9cos0)
8 it
$(a) .
(V.22)
Note that the (0,$) angular dependence has been approximated by using
delta functions of cosG and . The dependence of 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
Df .(0,) ^h = Jdgda sing [ 2 (cos20 sin2a + cos2a)
+ 2 D^2) 2 sin20 sin2a ] S(cos9cos0) S(<>a) (V.23)
It is the expression in Eq. (V.23) that will be integrated over
solid angle S 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
60
"ti ii5Â¡/()2 Jd0 lDfi<5>l?h
All that remains is to carry out the integral over 9 and * 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
2 Tt M
'fi ~ 3Mc p
fi
g() { D
(2) .2
fi I
4Pi2
(V 25)
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 vj/~^(t;Â£') seemed to satisfy asymptotic conditions
only at the beginning of a trajectory (t=t). We also hinted that
this problem would be dealt with for a specific model and a general
solution presented. In this section we will show how asymptotic
conditions can be satisfied for a two coordinate linear model with one
relative coordinate R and one internal coordinate r. This is the same
model proposed in chapter V. Two different methods will be proposed
to accomplish this. Each method can in principle be applied to any
twocoordinate linear model regardless of the number of coupled
electronic surfaces.
Recall from Eq. (IV.52) that the asymptotic form of ^ does not
seem to be equivalent to the form of Eq. (IV.30) except at t. In
other words 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:
(VI.1)
61
62
The root of the problem is the fact that u in Eq. (VI.1) does not
3.
equal uv, where uy 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
/ 0\
uy(r )
P^tV0)
p'(t;r)
1/2 1 Sk I'o
3 ua[r(t;ru)] .
(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 uy at time t=t and t=t+x, 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
vv '
nuclear vibrational motion. There is a frequency w and a period x
associated with this vibrational motion where
63
x =
2 n
w
(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 uv(r) at t = t;
therefore, if we can show that Eq. (VI.2) is also periodic over the
same x then we know that the wavefunction ^ ^ will meet asymptotic
conditions at every t = t + nx, where n is an integer. We will do
this by showing that each factor in u is periodic over x.
cl
The first factor to consider is the exponential of iy/2. The
value of y will increase by 2it after each period x, therefore, the
exponent decreases by in. The other exponential factor contains the
2
integral of P' This integral increases by (2vji + n) after a period
x, where v is the vibrational quantum number. Combining the integral
2
of P' with y/2 after every period x gives for the nth period,
t+nx
ip(t +nx) + J Q [P ]^dt = nni + n(2vit + n)i
= ni(n + 2vrt + it) = nv2ni
(VI.4)
From Eqs. (VI.4) and (VI.2) we can conclude that the exponents in u
cl
add up to an integral multiple of 2ni after every period x and, thus,
do not affect the value of u& at times t=t+nx in the asymptotic
region. The only other factor involves the square of the momenta and
will have the same periodicity as the internal wavefunction uv(r)
because the vibrational momentum P'(r) has the same period x as the
vibrational motion.
64
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 r. Since the function uv[r(t;r0)] is also periodic
over a time t, then, the following equality holds:
, _o o.
uy(t ;r )
P (t
P (x ; r )
v n '
1/2  ',
= uv[r(Tn;r0)] (VI.5)
where tn = t+nx. The equality in Eq. (VI.5) shows that asymptotic
conditions are met at every tn (n = 0,1,2...).
Since satisfies asymptotic conditions periodically, then the
transition integral will reflect this periodicity and it is
expected that will be a periodic function of the initial relative
coordinate R(t) for the bundle of trajectories. Calculations of
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(t). Remember that R(t)
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(t) s
R, the first method will consist of simply averaging the square of
the function D^(R) over a distance in R corresponding to one period
2
of oscillation in r. This average will be denoted D^ave and
defined by
65
fi'ave
r
R(T)
Dfi(R):
(VI.6)
where is the number of equally spaced values of R in the average.
Note that time t = 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(0) > R(t). The equation for the
relationship between R(0) and R(t) of Eq. (VI.6) is
R(t) = R(0) + gJE
nR
(VI.7)
where P is negative.
2
From Eq. (VI.7) we can see that the quantity lDf^ lave is
independent of R. Thus, if we use it in the calculation then the
cross section will also be independent of R. For the special case of
an internal potential which is harmonic (or any symmetric potential)
asymptotically, the transition integral D^(R) is cyclic over only
half a period; i.e.
Df.(0) Dfi(J)
(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 ua having the form of the WKB bound state
wavefunction.
The WKB wavefunction for a bound internal state has the form
[Messiah, vol.I],
uWKB(r)
(VI.9)
t?
where S =
JP (r) dr
a
SWKB
T
4
k = constant = 2
and M 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
67
} = *+ + jL
(VI.10)
s
H
, and Â£ = +, 
where e'
Asymptotically we will separate the action into relative and internal
contributions,
SCa = WR> + SaC(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 Uy^g(r). Asymptotically
~l
1 1
i T,0
i i '
T0
2 2^a
K P R
r rr S rr S .
n a+ n a
Xo
xIa
J
J
a
e
e
1
O 1
+
0)
o +
1
(VI.11)
where we have used
SRa =PR
(VI.12)
Equation (VI. 11) must be compared to v of Eq. (IV.30) in order to
isolate ua(r) which in this case will be equivalent to Uy^g(r).
68
Setting Xja = (2jiH) ^2 we obtain
ua(r>
' n,
1 L
2 2 ya
r VI Sa+ H sa]
P (t)
e
K e +Ke
Pa(t)
(IV.13)
By comparing Eq. (VI.5) with the VKB function in Eq. (VI.1) we have
the requirement that
ip'>F K
S u
3+ 3
i
K ^a
.TT 2 .
,,0
+ K e
. h 2 .
ii it
is is
J Vi M
K
(VI.14)
or
E exP
c=+,
i(SaC/H V2 + pz)
= E exP
c=+,
Ci [ p'dr ViJ
/Vi
a
(VI.15)
where it is implied that
KelpC =
Kp'(t) 1/2 = K
(VI.16)
69
and pj. is a constant to be determined. To prove the validity
of Eq. (VI. 15) and determine p^. we will first determine the form of
r
S aÂ£. To do this we will assume that Sa^.(r) is the asymptotic
f o
internal action which is calculated from a trajectory where P (r ) =
Cp'(r).
If we define S^r) as
S^.(r) = J Pj.dr (VI.17)
a
then we obtain
r
a
where P^. is the internal momentum whose value at r is C P^(r) . By
examining a trajectory whose initial internal coordinate and momentum
are r and P (r)  we can arrive at an expression for S (r) in terms
t
of Sy^g(r) and A where
r
SWKB(r) = J lP (r) ldr (VI.19)
a
/
and A is defined as the number of times that the momentum P (r) has
changed sign during the trajectory from r to r. As we follow this
70
trajectory we will rewrite the internal momentum in the action
integral as
P'(r) =
(VI.20)
where Â£ is the sign of P We will also use the relation
a
J P^(r) dr = Mrt(v + 1/2) (VI.21)
a
where v is the vibrational quantum number. The expression for S (r)
cl't
we arrive at is
Sa+(r) = 5
5' (r\ H
bVKBU; 4
lf
_u rth ajiM
Tin + + A
(VI.22)
We can also follow the trajectory with initial conditions r and
IP (r ) I to obtain S (r). Using the same relations we obtain
SL
r
s
a
(r)
SWKB(r)
hit
4
/
(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 Â£. For Eq.
(VI.22) Â£ = 1 for even values of A and 1 for odd values of A. This
71
makes the sum (A + 1/2 IJ2)it an integer multiple of 2rt. Thus, this
t
part of Sa+ can be dropped because the exponential of 2Jti is one. The
same is true of the sum (A + 3/2 Â£y2) in Eq.(VI.23) because for S
cl
we find E = 1 for even values of A and E = 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 y is
u = arg[P (r)] arg[P (r)] = An (VI.24)
d
Substituting Eq. (VI.24) into the left side of Eq. (VI.15) and using
S from Eqs. (VI.22) and (VI.23) gives
sc,
WKB 4
/H
i
f \
it
r + p
4 +
i^_
t
S WKB
itM
" 4
/H
i
f \
3 it
"4 + p
e
^ t
+
e

e
* /
i
SVKB
= e
x
e
wkb
(VI.25)
where E^ 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+ = Ji/4 (VI.26)
p = 3JI/4 (VI.27)
The final step is to note the relationship between E+ and E_. Having
/
assumed that Sa+ is calculated from the trajectory whose initial
72
momentum is P+(r) = P(r)  we have prescribed that j^(t,r) will
be constructed from two trajectories whose position and momenta are
always opposite in sign asymptotically; i.e.
P_(r) = P+(r)
(VI.28)
As a consequence of this judicious choice we have that
= K+ (VI.29)
Noting that cos(E) = cos(E) where E is any phase and using Eq.(VI.29)
we arrive at the equality
' JtM
VKB 4
/H
nM
bWKB ~ 4
e
* ^
+
e
 
1
= 2C0S
It
4
i
it
4
l
svkb/m
n
4
= 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)
73
*c(t;r0)
IP;
i
h
x K
(VI.30)
where S^(Q) =
wavefunction asymptotically it is required that
K
o
(VI.31)
The wavefunctions and have been defined as those generated from
the trajectories beginning at r and r with initial momenta P (r) 
and P (r)  respectively. Therefore, the function ^^(t;r) is
constructed from two trajectories by adding the wavefunctions
generated along each using the proper value of from Eqs. (VI.26)
and (VI.27).
The wavefunctions
differential equation as does \t;r), because they differ only by
the constant exp(ip^). In this method the transition integral of Eq.
(IV.5) is independent of R so there is no need to average. This is
due to the way meets asymptotic conditions at all (R,r) in the
asymptotic region. The form of in this region being
74
4^(t;r0)
(2jiH) 2 eM P R
tP (t;r)
X cos
*jV
' a
il
4
(VI.32)
The majo'r advantage in this method over the first one presented is
that there is no average over R 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 VKB approximation is poor for small quantum
numbers and the symmetric potential approximation poor at large
quantum numbers there will probably be a certain range of quantum
numbers for which this method gives optimal results.
CHAPTER VII
RESULTS: APPLICATION TO METHYL IODIDE
In this chapter the Self Consistent Eikonal Method is applied to
the collinear photodissociation of methyl iodide. Statetostate
cross sections are calculated for two different models. The first
corresponding to excitation from the ground electronic state to a
single dissociative excited electronic state. This shall be
designated as the 1ex case. The second model calculation involves
excitation from the ground state to two dissociative excited
electronic states which are coupled (the 2ex case). Computational
details such as the selection of initial conditions are also
presented.
Coordinates
Methyl iodide is treated as a linear triatomic as in the treatment
by Shapiro and Bersohn where the umbrella mode of the CH^ group is
modelled by an "equivalent effective bond" resulting in a linear
triatomic problem with two vibrational coordinates. Essentially, the
model replaces the three hydrogen atoms by an effective mass (of three
hydrogens) located in the plane of the real hydrogens and collinear
with the CI bond axis. The coordinates used are the Jacobi
coordinates. In the 2coordinate linear model the Jacobi coordinates
correspond to the distance R from the iodine atom to the center of
75
76
mass of CH^ 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, PD and r, P for relative and internal variables
respectively. The distance R is given in terms of r and the
CI distance R^ as follows:
R = R + 0.2011r (VII.1)
These coordinates are illustrated in Fig. (VII1). Corresponding to
each coordinate there is a reduced mass defined by
Mr (mch3) 7 (Mch3i)
(VII.2)
Mr = (MH3) (McP 7 (MCH3)
(VII.3)
Recall that in chapter VI the problem of requiring to
satisfy asymptotic conditions at all Q was addressed and solved by two
different methods. The method of averaging over a period was used in
all the calculations for which results will be presented in this
chapter. The integral cross section expression for the collinear
model is given by Eq. (V.25). As discussed in chapter V the effect of
the symmetry of the excited states on the cross section is most
important when considering two or more excited states of different
symmetry. For the 1ex case the effect of integrating over angles is
simply to multiply the cross section by a constant 1/3. For the 2ex
case, however, the additional effect is of have a sum over the squares
of the perpendicular and parallel contributions to the transition
amplitude.
Fig. (VII1) Jacobi coordinates for CH
78
When satisfying the asymptotic conditions using method 1 the
squares of the contributions are given by the following sum
Rh i
(T')
D^T
f i
2 _
1
Nr
E 1 I2
L i'
n 0
(VII.4)
where the interval from Ra to is divided into equally spaced
distances and R > Ru. The value of R, is chosen arbitrarily outside
a b b
the interaction region. The length of the interval (R& R^) is such
that the CH^ vibrational motion goes through half a period as R goes
from R to R, i.e.
a b
R (t = O = R,
R (t + T /2) = R,
PR T
Rb = Ra + \
(VII.5)
where x is the period of the umbrella motion of CH^ asymptotically and
is the incoming asymptotic relative momentum. Recall the variable
transformation from the set Q to a set A defined in Eq. (III.2). In
the case of the two coordinate linear model this corresponds to the
transformation
(R, r} > {t, w1)
(VII.6)
79
where w^ is an angle defined by
r(t) = Ay sin wr (VII.7)
v is the vibrational quantum number for the internal "umbrella" mode
of CH^, and 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) = Ay sin[w(t t) + w^] (VII.8)
Pr(t) = Mr o) An cos [w(t t) + vj (VII.9)
Pr(t) = Mr ai Ay cos (w1) (VII.10)
where
(A) =
(VII.11)
Each trajectory is distinguished by the angle w^. A particular
value of w^ gives the initial internal position r(t) and momentum
Pr(t). The range of w^ is from 0 to 2n. Thus, a grid of w^ values
will span all classically allowed values of r along with both the
positive and negative corresponding momenta at each r. For example,
the value of r has associated with it both Pr and Pr0 .
80
This bundle of trajectories will all begin at the same asymptotic
value of R with the same initial time t. As an example of this a
bundle of 10 trajectories is shown in Fig. (VII2). There will be
such bundles used in each calculation of D^ \ For example if the
range of w^ is divided up into N equally spaced intervals then for
t = 0 there will be N trajectories propagated from R = R&. After
a value for D^ '(R ) I has been calculated another bundle of
trajectories is propagated from
R R(^Â¡ ) = Rc and another D^f ^Rc) calculated. This
R
continues until NR such bundles have been propagated and = R(t/2)
is the next R. Note that F* ^(R^)^ 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
CH^ fragment. Remember that 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.
i
(VII.12)
E.
i
+ E + hv
V
E
f
E
a
(VII.13)
+
(VII.14)
81
Fig. (VII2) Trajectories on the excited electronic potential ene gy
surface.
82
where is the energy of the CH^I molecule plus photon system
initially (in the ground state), is the energy of the system after
having absorbed the photon, E^ is the energy of the CH^I molecule
initially for quantum state y=(l,vx,vy), E^ is the combined energy of
the fragments in the final quantum state a=(r,v), Mc is the mass of
CHI and p^aP?. The numbers T and v label the I electronic state and
i i k
the CH^ 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
P = P + MkD. P .
c c Ph c
(VII.15)
Equation (VII.12) then reduces to
E + hv
y
po2
R
2M
E = 0
a
(VII.16)
which gives the following equation for P:
PR I2Hr(Eu hv Eet)l1/2
(VII.17)
For a given initial and final quantum state and Ea are fixed and
the photon energy hv can be varied from calculation to calculation,
giving a curve of cross section vs. photon energy.
83
One Excited Surface
The only states considered in the first calculation (1ex) are the
3 3
ground state and Qq excited state. The Qq state leads to the
* 2
fragments CHq + I ( P^/2^ whe t*ie ground state leads to
2
CH3+ I( P3/2^ as shown in Figure (VII3). The potential surfaces
given in analytical form were determined'empirically by Shapiro and
Bersohn [Shapiro and Bersohn, 1980]. The ground state surface is
V11 (RCI r) = De {exp[0.899(RCI 4.043)] l}2
+  {k + (.1100) exp[0.4914(RCI 4.043)]}
x {r 0.6197 exp[0.4914(RCI 4.043)]}2 D E* (VII.18)
2 3
k 2 the force constant = 0.0363 e /a^
D = 0.0874 e2/an
e 0
E* = 0.0346 e2/a
and the variable R^,^=(R .2011 r) is the carbon to iodine distance.
All energies are in Hartrees and lengths in Bohr units.
3
The Qq 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
( n e) a
Fig. (VII3)
Diabatic potential energy curves for CHjI at r=0
Fig. (VII4) Diabatic potential energy surfaces for the ground and QQ states
of CH3I.
Fig. (VII5) Contours of the surfaces in Fig. (VII4).
88
89
following transformation [Lee and Heller, 1982]
r >
X
7.830
0.1762'
y.
,0.6183
4.939 .
\
( \
R
/
'Rrv'
where
R = 4.043 a0
Rcx = r 0.6197 a0
and R = R + .2011 R^ .
(VII.20)
Lee and Heller have calculated the four lowest vibrational
eigenvalues for the ground electronic state. They are
v )
y
E (a.u.)
(0,
0)
.00399346
(1,
0)
.00643926
(2,
0)
.00885801
(0,
i)
.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,
T00(x, y) = 0.9966110 gQ0(x, y)
0.0816282 g1Q(x, y) 0.0101739 gQ1 (x, y)
(VII.21)
90
where g (x, y) is the product of harmonic oscillator wave functions
vxvy
for quantum numbers vx 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 S and r using the determinant of
the rotation matrix in Eq. (VII.20). Results will be shown for
excitation from this (0, 0) level and compared to those of Lee and
Heller. Results for excitation from the next three vibrational levels
(1,0), (2,0), and (0,1) of the ground state will also be presented in
this chapter. For these higher vibrational levels a simple product of
harmonic oscillator functions was used for the ground state
vibrational wave function.
The equations of motion for the 1ex case are
r =
R =
3H
9PR
P
r
3H
3V
ex
3r 3r
(VII.22)
(VII.23)
(VII.24)
(VII.25)
where Vgx is given by Eq. (VII.19). The Jacobian is numerically
calculated from
91
R r
U w
= P,
y
t _
3r
3w1
\ 1/
P
r
3R
ar
3w^
3w^
 P
r
f
as
3wn
fE Q1 fr Q0
1 P1
U1 *1
Q1 P1
u0 *0
1 a2y l
P1 = Q1 + _2
2
51
asa?
3r
ex qJ
2 2
.. a v azv ,
1 ex ex
pn = ~~3 Qo + Q1
0 as2 u 3R3? 1
(VII.26)
(VII.27)
(VII.28)
(VII.29)
where the following notation of chapter IV has been used:
Q
1
0
P
1
0
i 3P
P1 a L
1 3w1
(VII.30)
(VII.31)
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 Pr/w (VII.32)
In Fig. (VII6) the function J (dotted line) is compares to the exact
cl
calculation of J for a single trajectory. From this figure we can see
that J coincides with J until the interaction region, then begins to
3l
deviate slightly. It is not until the trajectory begins to exit the
Fig. (VII6) The Jacobian for one trajectory; exact (solid line), and J (dotted line).
3.

PHOTODISSOCIATION OF POLYATOMIC MOLECULES:
STATETOSTATE CROSS SECTIONS FROM
THE SELFCONSISTENT EIKONAL METHOD
By
CLIFFORD DAVID STODDEN
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1987
I would like
to dedicate this work
to my fatherâ€”
from whom I learned
to ask of natureâ€”
why and how.
ACKNOWLEDGMENTS
I would like to thank some of the many people who in one way of
another have helped or supported me in this work. First, I would like
to thank Professor David A. Micha who supplied the basic formalism for
this work as well as most of the funding. I spent many hours in
discussions learning from his expertise in this field and developing
the invaluable skill of how to patiently work through problems which
arise in the research.
I would like to thank those graduate students, postdoctoral
associates and faculty who worked together to foster a cooperative
atmosphere of research and learning at QTP. I would especially like
to thank Professor PerOlov Lttwdin 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.
?!
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
t
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.
in
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTERS
I INTRODUCTION 1
IICALCULATION OF THE PHOTODISSOCIATION CROSS SECTION
USING THE EIKONAL WAVE FUNCTION 4
RadiationMolecule Interaction in the
ElectronField Representation 4
The Eikonal Wave Function 14
III INTRODUCTION OF A TIME VARIABLE:
TRAJECTORY EQUATIONS 18
A New Set of Variables 18
The Eikonal Wave Function Along a Trajectory 22
IV THE TRANSITION INTEGRAL IN ASPACE 29
Transformation to an Integral Over
the Time Variable 29
Generating the Jacobian Along a Trajectory 31
Asymptotic Conditions 39
V COLLINEAR MODELS OF POLYATOMIC PHOTODISSOCIATION 47
Two Electronic States 48
Three Electronic States 50
Symmetry Aspects and Cross Sections 52
IV
VI SATISFYING ASYMPTOTIC CONDITIONS 61
Statement of the Problem 61
Method 1: Averaging Over a Period 62
Method 2: Construction of a WKB
Internal Wavefunction 66
VII RESULTS: APPLICATION TO METHYL IODIDE 75
Coordinates 75
One Excited Surface 83
Two Excited Surfaces 115
VIII DISCUSSION AND CONCLUSIONS 146
The 1ex Case 147
The 2ex Case 150
Considerations on Angular Distributions 155
Conclusions 158
APPENDICES
A SHORT WAVELENGTH APPROXIMATION 164
B COMPUTER PROGRAM FOR THE SCE METHOD 166
BIBLIOGRAPHY 169
BIOGRAPHICAL SKETCH 173
v
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
i
PHOTODISSOCIATION OF'POLYATOMIC MOLECULES:
STATETOSTATE CROSS SECTIONS FROM
THE SELFCONSISTENT EIKONAL METHOD
By
CLIFFORD DAVID STODDEN
December 1987
Chairman: Dr. David A. Micha
Major Department: Chemistry
A general method is presented for calculating statetostate
cross sections for photodissociation of polyatomic molecules by
visible and UV radiation. The method also provides branching ratios
by selfconsistently coupling nuclear motion to transitions between
several electronic surfaces. Cross sections are calculated using a
transition integral between the initial ground state wavefunction of
the molecule and a final wavefunction for the relevant states of the
product fragments. The final state nuclear motion wavefunction with
incoming wave boundary conditions is generated in an eikonal form
along classical trajectories which follow an effective potential and
contains the exponential of a common action integral and a preÂ¬
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 CH^I the dynamics are described by two
degrees of freedomâ€”the relative position of I to CH^ and the umbrella
mode of CH^. The first calculation involves a transition to a single
3
dissociative excited Qq 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
2 * 2
CH^ and I( or ^ ( ^1/2^' Quantum yields of .84 and .69 are
2
calculated for I*( P.^) production at photon wavelengths of 266nm and
2
248nm respectively. The crosssection curves for I( P^j) 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.
vii
CHAPTER I
INTRODUCTION
This study addresses the problem of formulating and testing a
computationally efficient and accurate theory for treating the
photodissociation of polyatomic molecules with many internal degrees
of freedom and many electronic states. The photodissociation event of
interest will be limited to cases where incident radiation in the
ultraviolet to visible range is absorbed by a molecule causing a
transition to a dissociative state which gives a single atom and a
molecular fragment as products.
There are already many theories of photodissociation. They
include quantum mechanical approaches [Heather and Light, 1983;
BalintKurti and Shapiro,1981; Clary, in press] as well as classical
and semiclassical trajectory approaches [Atabek et al., 1976; Mukamel
and Jortner, 1976; Billing and Jolicard, 1984; Gray and Child, 1984;
Henriksen, 1985]. Lee and Heller have also developed a theory for an
exact time dependent approach using wave packets [Lee and Heller,
1982]. An exact approach in principle gives results which are more
reliable than a semiclassical method. However, for large molecules
with many internal degrees of freedom and many coupled excited states
such calculations may be prohibitively complicated. The computational
effort required for a coupled channels calculation for example
increases by the power of the number of channels involved.
1
2
A semiclassical method which employs classical trajectories
requires much less computational effort than an exact approach for
large systems. For this reason it is important to develop an
efficient, accurate semiclassical method which is able to deal with
many degrees of freedom and many coupled electronic surfaces. Of the
semiclassical methods mentioned above none are at present able to
effectively deal with more than one excited state potential energy
surface. The photodissociation events of interest here have a bound
initial electronic state with a certain set of vibrational quantum
numbers and a final electronic state with a set of final vibrational
quantum numbers for the molecular fragment. With this in mind another
desirable feature of any photodissociation theory is to be able to set
initial conditions as well as final conditions for the event, i.e. to
calculate cross sections for state to state transitions.
The formalism in the following treatment is designed, then, to
include certain desirable features. These features can be summarized
as the ability to efficiently deal with (i) many degrees of freedom,
(ii) many coupled electronic states, and (iii) statetostate
transitions.
The particular method to be discussed uses the selfconsistent
eikonal approximation. It is selfconsistent in that the electronic
transitions are selfconsistently coupled to nuclear motions. The
eikonal approximation simply means that the nuclear wave function is
expressed as a modulus matrix times an exponential with a common
eikonal phase and that the short wavelength approximation is applied.
3
The general process of interest is the following photodissociation
reaction,
M(vm) + â™¦(kf A(va) + X(vx) (1.1)
where M is a polyatomic molecule in a given electronic and nuclear
state \>^, A is an atom in electronic state \>A, is a photon with
wavevector k. and polarization a, and X is a molecular fragment in
electronic and nuclear state With the method presented in the
following chapters one can treat photodissociation events as in
Eq. (1.1) 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 VAVEFUNCTION
RadiationMolecule Interaction in the ElectronField Representation
The interaction of a photon with a molecule can be modeled as a
collision problem [Micha and Swaminathan, 1985]. In such a treatment
we consider the photon to be occupying a finite volume V in space and
the photon field to be contained within a cube of side length L. The
molecule M(\>^) enters this space and interacts with the field. The
field is quantized so that the moleculefield system is described by a
Hamiltonian
H = HM + Hf + HMF â€¢ (II<1)
The wavefunction for the molecular system is represented by 'f'M(Q)>f
where the "ket" indicates electronic coordinates and Q is a vector
whose components are mass weighted Cartesian coordinates for nuclear
motion. The Schrodinger equation for the molecular Hamiltonian Hâ€ž is
HM IVQ~Â» = E lYM(5>> â€™ (H.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.
4
5
The second term in the Hamiltonian Hp refers to the radiation
field. It is given by [Loudon, 1973]
+
a. a,
k,a k,a
(II.3)
where a,+ and a, are the creation and annihilation operators of a
k, a k, a
photon with wavevector k, polarization a and energy The
product of the annihilation and creation operators is the number
operator which operates on an eigenstate (k
field to give the number of photons in state (k
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 Huâ€ž refers to the moleculefield
interaction. In the dipole approximation this interaction term is
HMF(^c) =  J * 5 dV (II4)
where ris the position of the center of mass of the molecule M. The
factor is the dipole operator, which, in the center of mass frame
at position r is
5â€ž  E ca r S(ra  r) (II.5)
a
where {c ) is a collection of charges at positions {r }. The electric
cL
field operator E(r) in second quantized form is
6
E(r) . E e(â€œ) Xk(r)  a* X^r)*]
k, a ~ ~
g(w) = i [Mw/(2eâ€ž)]
1/2
(11.6)
(11.7)
where e0 is the vacuum dielectric constant and e^ is the polarization
unit vector with
circularly polarized light. The factor X in Eq. (II.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) . (II.8)
The wavevector k has components 2rcjjVL with jÂ£=0,1,2... and E=x,y,z
corresponding to cyclic boundary conditions for a cube centered at the
origin of coordinates. Replacing Eqs. (II.5) to (II.8) in Eq. (II.4)
we obtain
Hmf(Â£c) Eg V [D'(ra,k) V"
k, a
1/o ik.r
1/2 ~ ~c
k, a
. ik.r^
D~M
(II.9)
5^(ra,k) = E ca Â£a exp(ik.ra) (II.10)
a
The Hamiltonian in Eq. (II.1) acts on a state j) which describes
the state of both the molecule and the photon field. The initial
state i) and final state f) for a single photon event are given by
7
i> = Ibc u (k ff)> and lf> = IEc Efi â€œ>
(II.11)
where p and p' are the initial and final momenta of the center of
*c *c
mass, u indicates the initial molecular state, and a indicates the
final state of the fragments. In Eq. (II.11) the state i>
corresponds to a molecule initially in the presence of a photon field
and the state f> to a dissociative excited state with no photon
present. The fragments in state f> have a relative final momentum
vector of The corresponding total energies are
(11.12)
E.
i
Ef . (ei)2/(2m(.) . ei/<2m) + E^
(11.13)
where m is the total molecular mass and m the reduced mass for the
c
relative motion of A and X of Eq. (1.1).
The dissociation cross section is the quantity of interest in a
photodissociation experiment. The initial and final states are
characterized by both electronic and vibrorotational energies of the
fragments. Thus the cross section to be calculated is a stateto
state (electronic, vibrational, rotational) resolved quantity at a
particular photon energy. The expression for this cross section is
[Rodberg & Thaler, 1967]
(11.14)
8
where is the transition rate and is the incoming photon flux in
the laboratory frame given by
= c/(2itM)
3
(11.15)
where c is the speed of light. To obtain Eq. (11.15) we have used
momentum normalization for which
(11.16)
The transition rate is given in terms of the transition amplitude T^
and Dirac's delta function with respect to energy,
(11.17)
The transition amplitude T^ is an integral over electronic and
nuclear coordinates and contains an operator T which, for low
intensity incident radiation, is simply the moleculefield interaction
of Eq. (II.9). For the single photon process to be considered here
only the first term in H^p is appropriate. A single photon event
corresponds to a weak electric field (i.e. low intensity). The form
of the transition amplitude, then, is
9
The last factor containes the Integral over which gives
lie â€¢ IT
l I2 = (2rch)3 8
Therefore the integral in Eq. (11.14) becomes independent of gc with
the restriction that g ' = Ec + Mk. The cross section now contains an %
integral over g^ only,
= V g(Â») (2it/M) (2rtM)~J
x J dpf. p2. 2 S(Ef  E.), (11.20)
where p^ is the magnitude of g^. If we note that Mk << gc it
follows that g ' gc With this very reasonable approximation and
using E^ = E^ we get from Eqs. (11.12) and (11.13) for p^
1Ãœ
2m
= E  E
U a
Mw
(11.21)
Transforming the integral in Eq. (11.20) by noting that
dEf = p^ dp^/m and using the definition of J^, the differential
cross section becomes
da
d S2
fi , ,2 Ã 2nn,Pfi 1
= g(Â«)
Me
(11.22)
10
The factor in brackets will be denoted as indicating that the
transition is from a molecule initially in state i>=u> to fragments
in a final state f>=g^ a> in the asymptotic region. The final
state is designated by the states of the atom A and of the fragment X
asymptotically as anc* vxs^Xâ€™^ w^ere ^ 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 y> for the molecule in the center of mass frame is
given. It remains, therefore, to obtain the final state wavefunction
The vector Q 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 rc> In the following notation,
however, we will be working strictly in the center of mass frame and
the symbol Q 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
(11.23)
This molecular eigenstate can subsequently be expanded in a basis of
the electronic states fj, of M involved in the process, so that
IVQ)> = ltM(QÂ» *m(9>
(11.24)
fM > = (  i>2> ...$ji> ) is the matrix of electronic
wavefunctions, and
11
(11.25)
V
is the matrix of nuclear wavefunctions where h is the number of the
highest state energetically accessible in the reaction.
We will express as a sum of the nuclear kinetic energy
operator Knu and a term Hq which contains the nuclear repulsion and
all the energies of the electrons, including spinorbit coupling,
Hu = K +
M nu
(11.26)
Knu = (2M) 1
(11.27)
where M is an arbitrary mass which comes from mass weighting, and the
symbol V is a multidimensional gradient whose components are 3/3Q^,
i=0 to N. The gradient V to the second power is defined to be
2 2 2
(V) = 23 /3Qf â€¢
(11.28)
The coefficients of the expansion contained in ^ are the nuclear
motion wavefunctions on the electronic potential energy surfaces.
These surfaces are defined by the diagonal elements of
(11.29)
12
When Eq. (11.23) is substituted into Eq. (II.2) and left
multiplied by <$M  we have the following matrix differential equation
Sq + Â«<2>
+ H~o 
iM  0 â€¢
(11.30)
We will Work in the diabatic picture [Smith, 1969] in which
G(Q)= = 0
(11.31)
This leaves us with the general matrix equation:
f 1_
f \
H_ o
1 2M
â– â€” V
1
l /
â™¦ 5q  eJ*h(Q)  0
(11.32)
When the coupling between the electronic ground state (T=l) and
the other electronic states is negligible, H_ can be separated by
blocks,
Hi: 0 0 ....
0 ^22 ^23** *
II
O'
0 H32 H33...
.0 v,
^ â€¢ â€¢ J
(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:
13
IV2Â»  Iv *1 * 2 lv *r â€¢
(II.34)
We will define
Ygr(Q)> = *1(Q)> ^(Q) (11.35)
to be the ground state wavefunction, and
h
^ex(Q)> = *(Q)> 4(Q) = I j*r(Q)> *r(Q) (11.36)
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
Tex(Q)>. The matrix equation for the nuclear wavefunctions on the
excited electronic surfaces is
= 0
(II.37)
This is the equation we solve next using the common eikonal
approximation.
14
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)(_) = *(Q) exp[i S(Q)/M] (11.38)
where the minus sign on Â¿ 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
ih 2 iH H2 2 (VS)2
 2M S>* " M " 2MH X + JmX +  m = 0 (H39)
We can also write this as
(11.40)
where 0 is the operator
0 = 1/2M (M/i V + VS) + V  E
(11.41)
Note that 0 is in the form of a Hamiltonian minus energy
15
O = H  E
(11.42)
H = T + V (11.43)
T = T(p,VS) , (11.44)
where the normal momentum operator p=M/i V has been replaced by the
sum Â£ + VS thus defining a new momentum operator p' in the kinetic
energy operator T,
T =(p')2/2M
(11.45)
If we define the multidimensional classical momentum to be
P = 3S/3Q s VS (11.46)
we get the following for Eq. (11.39):
M 2 9 H (?P) K (P2) P
^ = ~2M  X + l 2M 'X + i ~M + 2M % + (~ Â®)X* (H.47)
In order to express Eq. (11.47) in the form of a HamiltonJacobi
equation we can carry out the following operations:
*+ Ã“ X + (X+ Ã“ X>+ = 0 (11.48)
X+ 0 * + X+ 0+ X = 0 . (11.49)
With the proper manipulation Eq. (11.49) becomes
16
X v2. 2V
M ~
m lpl
^ + IM 
After dividing by 2^+)( this can be put in the form of the Hamilton
Jacobi equation,
(P) /2M + ? = E
v ~ qu
(11.51)
where
V â– Â»* â™¦ P* â™¦ ng i
X+ ? *  (?X)+ X
. (11.52)
It remains to solve for X(Q). To do this we first substitute Eq.
(11.51) into Eq. (11.47) to give
iH
ft X  H P.&  1m + (V  lVqu)X = o (11.53)
We can simplify Eq. (11.53) by introducing the following notation:
U = H/2M 7.P ,
(11.54)
Kp r = 5r,r(h2/2M)72Xr/Xr
(11.55)
Equation (11.53) becomes
/
M
T
1
P.7
V ) + K  l(iU)
qu7 ~ ~ '
X = o
(11.56)
i
17
Equation (11.52) becomes the expression for an average potential
7 when we assume the second term to be negligible and make the short
wavelength approximation, for which 7^=0 (see appendix A), leaving us
with
7 = (X+*)1 X+YX
(11.57)
We will now apply the short wavelength approximation in which K is
negligible and 7^u becomes 7 to give
' p. 7
Â£ IT + (V  IV)  l(iU)
X = 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 Q at the
initial time. Initial values of the Q are denoted as q and are given
by the set
q = {q^, ^2â€™ ^n^ (III.l)
where q^ sets the initial value of for a given trajectory, q^ the
initial value of QÂ£, etc. One of the degrees of freedom in Q, namely
Qq, 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+l degrees of freedom in the problem.
Introducing this collection of variables is equivalent to
transforming from the set of spatial variables Q 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,
18
19
A a {t, qlt q2, ... qN} = {t; %) .
(III.2)
Each trajectory evolves with the time variable t separately. The
other members of A will insure that each trajectory follows a unique
path in Qspace (even though a particular trajectory will cross others
many times).
It is convenient to define sets Q' and q' to be the subsets of Q
and ^ respectively which refer to internal variables. They will each
contain N' components
Q' a {Q1', Q2'â€¢â€¢QÂ¿,}
(HI.3)
(III.4)
where N' is the number of internal degrees of freedom in the system.
We can write the initial value of any the Q^' in terms of an angle q,'
defined by
Q.'(t=0) = (y.)'^ a . sin(q.')
i ' ' v i7 vi '
(III.5)
where A . is the maximum value of Q.' for quantum number v. and y. is
vi i M i 1
the reduced mass for the i1"*1 internal mode. Note that for the Qi
corresponding to normal vibrational modes, if the potential energy
along this Q^' is harmonic in the asymptotic region (Qq*Â°) then we can
write
(III.6)
20
where is the frequency along and tÂ° is the initial time. The
asymptotic region is that part of Qspace where the potential is
separable in Q', i.e.
9V(Q)/9Q' = f(Q') (III.7)
Thus Eq. (III.6) gives us the functional relationship for a
vibrational mode between Q' 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 momenta can be formulated using
Hamilton's equations. Ve define Eq. (11.51) to be the Hamiltonian H
and impose the restriction of energy conservation to obtain
(III.8)
This leads to Hamilton's classical equations of motion
dQ
dt
9H_
9P
(III.9)
and
dP
dt
9H_
9Q
(III.10)
Both Q and P can be written as a collection of variables,
Q = (Q0> Qr Q2'*QN} (III.11)
p = {PQ, px, p2...pn) â€¢
(III.12)
21
For each component of Q and P, then,
dt
(III.13)
dP.
i
dt
(III.14)
It is helpful at this point to discuss mass weighted coordinates.
We define the following to be mass weighted coordinates,
Qi
(HI.15)
P.=
i
(III.16)
where and P^ are the original Cartesian coordinates for the i*"*1
degree of freedom. The mass M can be arbitrarily set to some value
without affecting the results of the derivation. From Eqs. (III.15)
and (III.16) it follows that
dQ.(A)
Pi(A) = M . (III.17)
Equation (III.17) shows the advantage of using mass weighted
coordinates in that the reduced mass vk 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
22
P.7
IT
N dQ.
Â£> ^
d_
dt
(III.18)
The Eikonal Vavefunction Along a Trajectory
Having introduced the set A we can rewrite % using this new set
as
 \ J y(t') dt'
*[Q(A)] = XÂ° C(A) e tÂ° (III.19)
where
Y = J V.P â€¢ (III.20)
and XÂ° is the value of X at tÂ° which is set by initial conditions. If
we replace Eqs. (III.18) and (III.19) into Eq. (11.58) we get
(H/i) dC/dt + (V  1V)C = 0 . (III.21)
All that remains is to solve Eq. (III.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
(III.22)
23
Ue will also construct the nuclear wavefunction along the trajectory.
It now has the form
, , n S(Q) 4 I l dt' n a(t;a)
r ;(Q) = X e en (III.23)
where S(Q)Â»a(t;^). It will later be shown that the quantity involving
Y simplifies to a function of the momenta and other terms which can be
generated numerically along the trajectory.
The final part of left to construct is the factor
r Â»
e
To do this we refer to the definition of P in Eq. (11.46) and
integrate to get
(III.24)
S(Q)  S(QÂ°)
(III.25)
1 N t ?
S[Q(t;g.)I = S[Q(t0;a)] + jj E J Pi dt s (ill.26)
i=0 tov
where QÂ° indicates all initial values of Q and tÂ° the initial value of
t. Thus, according to Eq. (III.26), to calculate
24
integrate the square of the momenta along each trajectory over the
time variable.
Ve will now develop a prescription for generating y. The
function y = Â¡7.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 Q to {t;q}.
Let us first recall that the term involving y appeared in
a result of having transformed the differential equation from spatial
coordinates Q to the set of coordinates A which contains a time
variable. Using the definitions of P and 7 the product in y becomes
N 3P.
(III.27)
Let us look at one term in the sum on the right side of Eq. (III.27),
i.e. 3PV3Q^. Ve know from calculus that we can always relate a
partial derivative in one coordinate system with the partials 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:
3P.
i
(III.28)
25
Note that the i1"*1 column of the numerator in Eq. (III.28) contains
rather than Q^. The explicit form of the Jacobian in Eq (III.28) is
9Qo
9Qi
9Qn
9t
at
at
9Q0
9Qi
9qx â€¢â€¢â€¢â€¢
3qx *â€¢â€¢â€¢
3qx
o
â– â€¢ CM
rt>
3Qn
9qN â€”
A â€¢ â€¢ â€¢ â€¢
9qN
3qN
(III.29)
If we use Eq. (III.28) in Eq. (III.27) the product appears as the
following sum over i of the determinant in Eq. (III.28) divided by the
Jacobian,
9Qo
ap.
i
9qn
at
at â€¢â€¢â€¢â€¢
at
9Qo
ap.
i
^N
aqx
3qx ""
3qx
9Qo
ap.
i
WN
9qN â€¢â€¢â€¢â€¢
9qN *'â– '
3qN
(III.30)
where J is the Jacobian of Eq. (III.29). The summation of
determinants in Eq. (III.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
26
SQ0
3qn
st 'â€¢â€¢â€¢
sqjst â€¢â€¢â€¢â€¢
St
SQ0
32Qi
3Qn
3qx
Sq.Sqi *â€¢â€¢â€¢
SqL
sÂ¿0
s2q .
3QN
3qN â€¢â€¢â€¢'
3qj3qN ""
(III.31)
The determinant in Eq. (III.31) looks very similar to that in Eq.
(III.30) except that the i11*1 column contains double derivatives
rather than the derivatives of P^. If, however, we recall that
P^=M(dQ^/dt), then we can make these determinants identical by
choosing qj = t to obtain
SJ
St
SQq
3QN
St 'â€¢â€¢â€¢
r% â€¢ â€¢ â€¢ â€¢
St
St
SQ0
3qn
Sqx â€¢â€¢â€¢â€¢
St Sqx *â€¢â€¢â€¢
Sqx
sq0
2
S Q .
3QN
A â€¢ â€¢ â€¢ â€¢
3
3t 3qN
(III.32)
If we now switch the order of differentiation on the double
derivatives and replace SQ^/St by P,/M we get that
27
N
3J_ _ it"
at ~ nÂ¿_,
i=0
3P.
i
p
ro
3t
o
o
3P.
i
aqx *â€¢â€¢â€¢
dq1
3Â¿0
3P.
l
aqN
3%
N
?%
3qx
3Q,
9Qi
(III.33)
After comparing Eq. (III.33) with Eq. (III.30) we can express y in
terms of the time derivative of the Jacobian which gives
dJ
dt
?) J
(III.34)
y
* _ 1 dJ _ d ln(J)
* ~ J dt ' dt
(III.35)
Recall that the term involving y appears in the nuclear
wavefunction as an integral in the exponent. Replacing Eq.
(III.35) in the integral gives
exp  j [ ln( gi arg(J)j + In J   In arg(JÂ°)j  lnjÂ°]
= expj j [i arg(J)  i arg(J )]
1/2
(III.36)
28
where JÂ° refers to the value of the Jacobian at time t
Thus we have
1
~2
e
1/2
exp{j[ arg(J)  arg(JÂ°) ]). (III.37)
Our nuclear wavefunction can now be constructed along a
trajectory defined by A a {t;^'}. The final form of this wavefunction
is
<{/ *(t;^) = XÂ° C(t;c[)
JÂ°(0;a)
J(t;^)
1/2 1 N i , â€ž .
2u(t;a) H
e e
(III.38)
where J is the Jacobian at time t, JÂ° is the Jacobian at the initial
time, and y contains the argument of the Jacobian as follows:
U(t) = arg(J)  arg(JÂ°)
(III.39)
The amplitude matrix C is selfconsistently solved for through
Eq. (III.21). The factor XÂ° is a constant whose value will be
determined in chapter IV by initial conditions. The other terms are
constructed from quantities generated along the trajectories. The
nuclear wavefunction calculated in this manner will be used in the
next chapter to calculate the transition integral which appears in the
cross section of Eq. (11.22).
CHAPTER IV
THE TRANSITION INTEGRAL IN ASPACE
Transformation to an Integral Over the Time Variable
The quantity of interest to the experimentalist is the stateto
state cross section. It is calculated using a transition integral
between the ground state wave function and the function \ This
is calculated from the quantity which appears in Eq. (11.22).
The general form of the transition integral is
(IV.1)
where r 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 over electronic coordinates. It
has components
[D(Q)]ra = <*p(Q) DÂ¿*1(Q)>.
(IV.2)
The integration over space dt in Eq. (IV.1) can be written as a
29
30
multiple integral over the set of coordinates Q Â» {Qq, Q^, ... Q^}.
Thus the transition amplitude is a multiple integral in (N + 1)
variables which can be written as
Dfi = J[*f )]+ 2S *i d5 * (iv.3)
In order to calculate this integral using trajectories we must
again make a variable transformation from Q to A as in the last
section. The integral in Eq. (IV.3) becomes
Dfi  I 8S *i
,Q0 Q1
t
â– V
,qNj
dA
(IV.4)
where
f Q0 Q1
â€¢V
qi â€¢â€¢â€¢%;
is the absolute value of the Jacobian of
transformation and the collection of variables A is defined by A
={t,q} where q={q^,q2>â€¢â€¢q^} contains the initial coordinates for a
trajectory. To show explicitly that now contains an integral over
time we write
Dfi =  t]+D.e J dt dq . (IV.5)
We can approximate the integral over dq by a sum over Aq. This leaves
us with a sum of integrals over dt:
Dfi â– Â¿/â€˜V J^'VSnS *i,â€ž Jâ€žI it *
31
Dfi,n â– Â«V [ l*Ã¼
_)]+D .e ip.
1 ~n ~ i, n
J dt
1 n 1
(IV.7)
For simplicity we have dropped the subscript f from The
label (n) which appears as a subscript in the summation refers to
individual trajectories. Therefore, is a function which is
generated along the n1"*1 trajectory. Since each trajectory is defined
by its values of q it can be said that (n) labels each unique set q .
This can be seen by defining
Â¿ )(t> a 4( )
(IV.8)
According to Eq. (IV.6) each trajectory n will contribute a
certain amount Dc. to the cross section. This integral will be
ti ,n
calculated along the n1"*1 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 ij/ K It was
also shown how the factors C(t) and
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 dQ to
dA. It has not yet been shown how J can be calculated along each
32
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. (III.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. (III.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
Q* s 9Q./3q. and P* s 3P./3q.
J j i J J1
(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
J
% Qr
t q .
1
M
(IV.10)
To obtain the time derivatives of P^ and let us start by recalling
Eqs. (III.13) and (III.14). If we take the derivative of the left
side of Eq. (III. 13) with respect to q^ and switch the order of
differentiation we get
33
2 2 i
9 Qj _ o Qj _ 3Qj
3qi 3t 3t 3q , 3t
(IV.11)
Taking the derivative of the right side of Eq. (IV.11) with respect to
q^ gives the relation
3Q*/3t = 3P./3q. = P*
3 J i ]
(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 q^ we get
2 2 i
3 P. 3 P. 3PT
J J m J
3q^ 3t 3t3q. ~3t
(IV.13)
Following the same procedure as before, we set Eq. (IV.13) equal to
the derivative of the right side of Eq. (III.14) as follows:
3P*
J
~3t
32 V
3q. 3Q.
3
(IV.14)
However, since we have no expression for the derivative of V with
respect to an additional step must be taken. Ve know the quantity
? only as an explicit function of Q. We also know that the derivative
dependence of on q^ is one of the terms being generated
(Q^=3Q^/3q^). Therefore we can expand 32V/3q^3Q^. in terms of the
by using the chain rule.
34
It is important to note that the average potential 7 is a
function of the components Cp(t;^) of the amplitude matrix C(t;^) from
Eq. (III.22). It is convenient to express Cj. in terms of its real and
imaginary parts
Cr a K (Xr + iYr) (IV.15)
where K is a constant to be specified later. With the definition of
Eq. (IV.15) we note that
7 a 7(Q; X,Y) (IV.16)
where X and Y are sets containing, respectively, the collection of Xj,
and Yj, components of all Cj,. Since V is a function of the X^ and Yj.
these must be included in the chain rule expansion.
Define, a 37/30^ (the force along Q^) and substitute into
Eq. (IV.14). Using the chain rule we get
N
E
Ã3F.)
1
+ E (
f3F.)
J
3Xp
f3F. j
J
k=0
r 9q. T l
UxJ
UyJ
(IV.17)
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:
1
(IV.18)
35
Using Eqs. (IV.17) and (IV.18) and the definition of we get the
following for the time derivative of P^:
3P*
3t
N
E
k=0
a2v
3Qk3Qj
'X,Y
Ã3
f _ \
3V
laxr
9Qj
^ J /
(IV.19)
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 X^. and Yj,1. 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 Q* and P*.
J J
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 Xj, and Y^,. To get an expression for
these time derivatives we use Eq. (III.21) and separate the real and
imaginary parts given in Eq. (IV.15) to obtain
dxr â€¢
â€”â€” = X  V W Y  â€”â€”
dt ~ T " Â£ IT' 1T' " 3Yf
II = Y  V V X  W_
dt T â€ $ WIT' aT' ~ 3Yf
(IV.20)
(IV.21)
where the matrix W is defined as W=V1V and we assume the matrix V
defined in Eq. (11.33) to be real. Next we take the partial
derivative of Eqs. (IV.20) and (IV.21) with respect to q^ and switch
the order of differentiation
36
!fr_ a_
3q. 3q.
(IV.22)
3V
rr'
Yr, + Vrr,
(IV.23)
Once again ve run into a situation where ve must expand a partial
derivative with respect to qp By using the chain rule expansion of
we obtain
Ã y
idV )
d rr'
ni , y
i3Wrr'
wi , 9WIT'
it
30.
Uk 41
TV!
y y
l3xr"
rM + 3Yrâ€ž
1 JM!
+ E wrr' Yp â€¢ (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
V, namely, the diagonal terms. Ve can indicate this by using a delta
function to get the final expression
(y
i3urr'l
n1 s V
3V
It
K J
Qk 5rr' Â¿Â¡
Y V 1
i9xr"
9V_ i
9Yp, r\
t
+
(IV.25)
37
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
Y t =
(y
Ã3wrp 1
lEk
l9Qk J
Q,. 
E
TP
bl
rn
3V_ i 3V_ Yi
3xp, Ar" + 3Yp, r*
+ 2 Wrr, Xp,
(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 Pj and
thus for in terms of quantities we can calculate. We are already
generating all in time and so we have all the terms to calculate J
along the trajectory. The only terms in of Eq. (IV.7) which we
have not yet talked about are those which we need to be given, namely,
D .e and
The ground state wave function will be given as a function of
Q; therefore, (t) is simply the value of *p. at that point Q which
*** i, n i
corresponds to the n1"*1 trajectory at time t; i.e.
= *.[Q(t;a )] . (IV.27)
1 y II 1 ~ II
The factor Dn.e will similarly be given as a function of Q.
We now have all the elements required for the transition
integral. The entire expression in detailed form is
E D
n
fi ,n
(IV.28)
38
where
Â°Â£i,n Â«V C>>
i[arg(Jn)  arg(JÂ°)]/2
ian(t)/H
e
x g(w)Dn(t).e ^i)n(t) Jn dt (IV.29)
The symbol JÂ° is defined as the value of the Jacobian at time tÂ° (the
initial time) for the n^ trajectory.
Notice in Eq. (IV.29) that there will be a singularity in
o 1/2
caused by the term j /J each time J passes through zero.
Remember that 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 ~^ 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 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 behaves asymptotically.
In order to satisfy asymptotic conditions the nuclear wave function
must have the following form as R*Â° and t>tÂ°:
39
*Â°(Q) = (2JiH) d/2 uv(Q') e
(IV.30)
where d is the the dimensionality of the space that AX is constrained
to (d=l, 2, or 3), uv(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 Q 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 asymptotically we
will look at how each term in Eq.(IV.29) behaves as QK) , where Q
~cl ~cl
refers to values of Q which are in the asymptotic region.
Asymptotically we can express the eikonal wave function as
(IV.31)
where the subscript "a" indicates that the various functions are in
the asymptotic region. To solve for XÂ°C let us recall Eq. (III.21)
and look at the term (V  IV). Asymptotically, the off diagonal
elements of V will be zero. If we choose the wave function initially
to be entirely on excited surface T, then the term V becomes
V
(IV.32)
40
The elements in V  17 = V then become
ny
= 5..(E. 
ij i
V
(IV.33)
thus having off diagonal coupling elements equal to zero and a zero
for the Tth diagonal element.
Because there are no off diagonal elements in the matrix equation
asymptotically and because C will have the following form
â– Q '
9r
lo )
a,r â€™
(IV.34)
Eq. (III.21) will reduce to
dCr
dâ€” Â° *
(IV.35)
thus C _ is constant in time.
~a, r
The Jacobian has a very interesting and simple form
asymptotically. In the asymptotic region any internal motion along
will not affect the relative motion along and vice versa. Using
this fact we can set the terms 3Q^/3q^ and 3Q^/3q^ equal to zero where
and q^ refer to relative variables. The Jacobian becomes
41
po
P1
P2
P3
* â€˜ PN
8Q0
3Q1
3Q2
0
. 0
3qx
3q1
3qx
a>
O
O
3Q1
3Q2
n
. 0
3q2
3q2
3q2
u
0
0
0
3Q3

3qn
3q
3Q3
3QN
3%
3<Â»n
(IV.36)
with the Jacobian determinant having a block for relative variables
(i=0,l,2) and a block for internal variables. Thus the determinant
reduces to a product of the Jacobians
J = J n
iQ0 Â°1 Q2'
J'
QÃ Q2 **â€¢
a a,R
[t q2J
a
5Ã q2 qN',
(IV.37)
Outside the interaction region the potential along Qq, Q^, and Q2 does
not change and so Pq, P^, 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^ = QÂ°
and q0 = Q?, then the determinant symbolized by J D becomes simply
the constant Pq. An analysis of the Jacobian J^ for the asymptotic
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
42
frequency w which is itself dependent on the frequencies of the normal
modes. If T is the period of the internal modes then the value of Ja
at tÂ°+x is equal to its value at tÂ°.
Based on the discussion above, the asymptotic Jacobian has the
form
Ja(t;q) = PÂ° JÂ¿(t;a) (IV.38)
where the (t;q) dependence of the Jacobian has been indicated. With
this Jacobian the asymptotic nuclear wave function has the form
\
"2 u(t^}
e
i
R
e
t;q)
.(IV.39)
The next term to look at is the action S. To see how S behaves
asymptotically we can rewrite Eq. (III.24) as
S(Q) = S(QÂ°) +
(IV.40)
The momentum P is a multidimensional vector which can be separated
into the sum of two vectors,
P = PR + P' (IV.41)
where, P^ refers to the relative momentum between the atom A and
fragment X, and P' to the internal momentum of the fragment X. With
43
these definitions, then, the action integral becomes
S(Q) = S(QÂ°) +
dR
This can be written as follows to isolate the term PÂ°
S(Q)
The term S(QÂ°) can also be separated into relative and
contributions,
S(QÂ°)
PR'?Â° *
S'(Q'Â°)
With this separation Eq. (IV.43) becomes
S(Q) = PrR + S'(Q'Â°) + J~(PR PR)dR + J~ P'dQ'
Since asymptotically PR=P^, we can write the action in
region as
rQ
S(Q) PRR + S'(Q'Â°) + \~ p,dQ'
QÂ°
(IV.42)
dQ' .(IV.43)
internal
(IV.44)
(IV.45)
the asymptotic
(IV.46)
The only other term to look at is y. Since it contains the argument
of J it will change each time J changes sign. Asymptotically this
44
means it will change by n each time the internal Jacobian J' changes
cl
sign. The frequency of this occurrence depends on the value of w.
To summarize the asymptotic wave function, then, we have the form
^ P^ R
R ?~r Â£
â€¢ dQ'+S'(Q'Â°)j
(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 and \Â¡P at t=tÂ°.
 PÂ°.RÂ° a S'(Q'Â°)
 XÂ° Cr .* â€˜ Â«
1 PÂ° RÂ°
â€¢ 0/r.O. ,nrÂ°\ R ~ *~
* (Q ) = (2nM) uv(Q' ) e
Therefore, setting vj/ 1(QÂ°) = ^(QÂ°) gives
a, i ~ ~
XÂ° Cr S (Q } = (2nH) d/2 uy(Q'Â°)
(IV.48)
(IV.49)
(IV.50)
 Sf(Q' )
Xo Cr = (2itM)_d/2 uv(Q'Â°) e R . (IV.51)
2
The factor is a constant asymptotically. In order for Cj. to be
the amplitude of the wave function on electronic surface T it must be
initially set equal to one. If we put the resulting expression for XÂ°
back, into Eq. (IV.47) we get
45
*Â¡>r(A) = (2nM) d/2 uv(Q'Â°)
r/Â°
1/2
x e
^ PÂ® R
H ~Râ€˜~
P'
i,a
2
dt'
V
I
(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 ^â€¢ It
satisfies the asymptotic conditions at tÂ° (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 does not satisfy initial conditions away from the value
of R(tÂ°) due to the terms involving J' and P'^. This is a problem
because the transition amplitude D^, and hence the cross section,
should be independent of the value of R(tÂ°). Remember that R(tÂ°)
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
46
taking suitable averages over the initial relative distance. We will
return to this discussion in chapter VI where we will show how can
be made to be independent of RÂ° 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. (1.1)
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 T to be included and
consequently the size of the Â¿ and $ matrices in Eq. (11.24).
In all of the models presented in this chapter the system will
have a fixed orientation in space throughout the dissociation. That
is, we will be invoking the infiniteorder sudden (IOS) approximation
in dealing with the rotational degrees of freedom of the molecule
[Pack, 1974], In addition we will be assuming only collinear
dissociation in which all motion of the molecule and the product
fragments occur along a line defined by the body fixed axis of
symmetry.
47
48
Two Electronic States
A simple model would be one in which there are only two
electronic states r=l,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 Cl 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 RsQq and no internal coordinates
Q'. The nuclear wave function on excited surface 2 simplifies to
i/)(t) = (2 jxM ) 1/2
P(t)
1/2
>(t)
e
*( t)
(V.l)
A slightly more complicated model is one in which there are again
two electronic surfaces T= 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 rsQ^ and one relative coordinate RsQq. This model
corresponds to the photodissociation of a linear triatomic Aâ€”BC
where A is the atom and BC the fragment X. An example of such a
system is the molecule CH^I (methyl iodide). This molecule can be
treated as a linear triatomic if one considers the to move as one
unit whose center of mass lies along the CI axis (see Fig.VII1).
This is equivalent to considering only the umbrella mode of CH^. It
49
turns out that this is a reasonable approach to CH^I dissociation
because experimentally it has been shown that the umbrella mode of CH^
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(tÂ°) = Av sin(q') a rÂ° (V.2)
and
Pr(tÂ°) = Av (o cos(q') a PÂ° (V. 3)
where A is the maximum value of r for vibrational level v, and w is
the asymptotic frequency of oscillation for uv(r). The range of q' is
0 to 2ji, which spans both positive and negative values of P (tÂ°) for
every r(tÂ°). The nuclear wave function along a trajectory will be
^ (t; q')
= (2jiK)
J'Â°(q')
J'(t;q')
2 2M(t;q,) )S'(Q'Â°)]
e e uv(q')
(V.4)
where uv(q') is the value of the harmonic oscillator function at r(tÂ°)
for vibrational quantum number v. The internal Jacobian for this
model is given by
J'(tjq') = Pr(t;q')/o>
(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
50
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 momenta, and four to solve for the terms
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 T=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 T=1 to the
coupled surfaces labeled by T=2, 3. Recall from the transition
integral in Eq. (IV.1) that mathematically we will have a column
matrix (2x1) for 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 (Q)> with incoming wave boundary
6 X ***
conditions.
The wave function for this model will be
*<>(t;q') = (2nM)1/2 eiS'/M
C(t;q')
JÂ°(q')
J(t;q')
1/2 iy(t;q') J cx(t;q')
e e uy(q'). (V.6)
The transition integral for the collinear model with two degrees of
freedom and two excited electronic states can be written as
51
Dfi = JlÃf )]+ Sâ€™S* +ilJl dR dr
= J^2f ?21 + ^3f D311 â€¢ lJl dR dr (V.7)
where and are the vector matrix elements of D and as before
isy and fs(gfi,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. (III.21) and (III.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 7. This
potential in turn depends on the mixing of surfaces 2 and 3 by the
matrix C and its adjoint. Thus the matrix C is selfconsistently
coupled to the potential 7. If C+C = 1, then we can say that the
quantity Cj.[Q(t)] represents the probability of the system being in
electronic state T. Therefore, since the potential 7 governs the
motion of a trajectory, we can say that the electronic transitions
between states 2 and 3 are selfconsistently coupled to the nuclear
motion and that ^ is being propagated on the average surface 7.
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
52
for the q in the Jacobian, and four equations for position and
momenta. 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
3 1
states referred to are, in the notation of Mulliken, the Qq and Q
states which are of symmetry species 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
CH^I is of species A^; therefore, the symmetry selection rule requires
that for matrix elements and respectively of Eq. (V.7) to be
nonzero
A1 x r(DÂ¿) x A1 = A: (V.8)
E x r(DÂ¿) x A1 = Ax (V.9)
where r(D^) is the symmetry species of the electric dipole moment
of the molecule. Equation (V.8) corresponding to a transition to the
53
3
Qq state is nonzero only for T(D^) a A^. According to the character
table for the point group C^v the species A^ 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 state requires that r(D^) = 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
(V.10)
(V.ll)
(V.12)
where e , (cr'=x', y', z') are the body fixed unit vectors.
54
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
(V.13)
where we have arbitrarily set the perpendicular component of the
transition dipole along the body fixed x'axis for simplicity. We are
allowed to do this only because we are dealing with a pseudolinear
molecule for which the transition dipole is isotropic with respect to
rotation about the body fixed z'axis.
The direction of propagation of the electric field in the
laboratory reference frame will define the zaxis of this frame.
Experimentally this corresponds to the direction of a laser beam which
is crossed at right angles with a molecular beam of target molecules.
To obtain the experimental results that we will be comparing with in
chapter VIII [Shobotake et al.,1980; VanVeen et al., 1984] the
researchers have used laser light linearly polarized in the plane of
the crossed beams. We will define this to be the space fixed yz
plane as in Fig. (Vl). With these definitions we can use the Euler
angles to express the dot product as
2^ cosy cosg cosa  d2^ siny sina'
d^i sing sina
A
/
(V.14)
55
CH^I beam
VVV
Fig. (Vl) Diagram of collision angles in the spaced fixed frame.
56
where e^ is a unit vector along the spaced fixed yaxis. The angles
0 and a are the polar and azimuthal Euler angles respectively of the
body fixed system relative to the space fixed system. The angle y is
the Euler angle for rotation of the body fixed frame about the body
fixed z'axis.
The transition amplitude for the collinear model with two degrees
of freedom has been given by Eq (V.7). Since the dot product involves
a sum over two states of different symmetry the relative contributions
from P=2 and P=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 will, however, be
parametrically dependent on the Euler angles a, 0, and y of the
moleculefixed coordinate system with respect to the spacefixed
system. We will denote this angledependent integral in the IOS
â€¢ . (s)
approximation as D 7 (0,a,y) where the initial and final states
oÂ£f.,U
have been specified in detail as in Eq. (II. 11). Note that the
initial and final quantum states are specified by ys(l,v ,v ) and
x y
a5(T,v). By substituting Eq. (V.14) into Eq. (V.7) we obtain
Dip\,y (^â€™a>Y) = J[ (Râ€™ r > 9Â» 40 d21(R,r) (cosycos0sina + sinycosa)
^'L 1
+ (R>r> 3Â»a) d^(R,r) sin0 sina]
57
At this point it is necessary to define angles 9 and which
define the direction of the detector in the spacefixed frame. The
angles 9 and are also the polar and azimuthal angles respectively of
the vector These angles are distinct from g and a which define
the orientation of the molecule at the time of absorption of the
photon while 9 and define the direction of detection for the emitted
fragment. It is important to be very clear with these distinctions in
order to properly interpret the quantities being calculated. Recall
that in a photodissociation experiment it is the direction of the
incident radiation which defines the space fixed zaxis while the
direction of detection is the angle which the differential cross
section is dependent on.
In order to account for all possible orientations of the molecule
upon absorption of a photon, the square of the transition amplitude in
Eq. (V.15) above must be averaged in some way over all initial angles.
The simplest way to do this is to integrate over all angles a, g and y
and all rotational momenta p , pot and p using the classical thermal
or rgâ€™
partition function q^ for rotation and a Boltzman distribution
function of the classical rotational energy function H^. These
functions have the form [McQuarrie, 1976]
q = 8n2Ik_T (V.16)
r d
Hr = (p3 + P^/sin2g)/(2]:) (V. 17)
where I is the moment of inertia and k.D the Boltzman constant. The
D
momentum p^ does not appear because we are dealing with a linear
58
molecule. The appropriate integral is
2
f i ' th
1
a q
r
e
2
(V.18)
where g^ indicates that there is a dependence on 9 and . Since
is) 1
DÂ£i (8Â»a,Y) is independent of pa and p^ in the IOS approximation the
integration over the angular momentum is straight forward and results
in
(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 g^ in
the subscript rather than the one dimensional scalar p^. 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 8 and a.
We can predict therefore, that the only contribution to the cross
section amplitude in the direction of g^ will come from an
orientation of (8,a) which is coincident with the g^ 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 D^(8,a,y) from the collinear model. This
transition amplitude has two terms arising from the perpendicular and
parallel transitions,
59
(2) (3)
Df^($>oc,Y) = '(cosy cosg sina + sinY cosa) + Dj^'sing sina
(V.20)
where we will define
Df['} = J**,f(R,r) dr,1(R,r) *.(R,r) j  dR dr . (V.21)
Using this transition amplitude we will approximate Eq.(V.19) by
Df i(0, 4Â») \Zth = ijJdedadY sing Df.(g,a,Y) I2 Â¿(cosGcosg)
8 it
S(a) .
(V.22)
Note that the (0,$) angular dependence has been approximated by using
delta functions of cosG and . The dependence of 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
Df .(9,) 2h = jdgda sing [ dÂ£2^ 2 (cos20 sin2a + cos2a)
+ 2 D^2) 2 sin20 sin2a ] S(cosGcosg) S(<>a) . (V.23)
It is the expression in Eq. (V.23) that will be integrated over
solid angle S 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
60
"ti  iiÃ5Â¡/(â€œ)2 Jd0 lDfi<5>l?h â€¢
All that remains is to carry out the integral over 9 and 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
2 Tt Mâ€ž
'fi ~ 3Mc p
fi
g(w) { D
(2) .2
fi 1
4fi2
(V. 25)
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
only at the beginning of a trajectory (t=tÂ°). We also hinted that
this problem would be dealt with for a specific model and a general
solution presented. In this section we will show how asymptotic
conditions can be satisfied for a two coordinate linear model with one
relative coordinate R and one internal coordinate r. This is the same
model proposed in chapter V. Two different methods will be proposed
to accomplish this. Each method can in principle be applied to any
twocoordinate linear model regardless of the number of coupled
electronic surfaces.
Recall from Eq. (IV.52) that the asymptotic form of ^ does not
seem to be equivalent to the form of Eq. (IV.30) except at tÂ°. In
other words 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:
(VI.1)
61
62
The root of the problem is the fact that u in Eq. (VI.1) does not
3.
equal uv, where uy 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
/ 0\
uy(r )
P^tV0)
P (t; r )
1/2
3 ua[r(t;r0)] .
(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 uy at time t=tÂ° and t=tÂ°+x, 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
vv '
nuclear vibrational motion. There is a frequency w and a period x
associated with this vibrational motion where
63
x =
2 n
w
(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 uv(r) at t = tÂ°;
therefore, if we can show that Eq. (VI.2) is also periodic over the
same x then we know that the wavefunction ^ ^ will meet asymptotic
conditions at every t = tÂ° + nx, where n is an integer. We will do
this by showing that each factor in u is periodic over x.
cl
The first factor to consider is the exponential of iy/2. The
value of y will increase by 2it after each period x, therefore, the
exponent decreases by in. The other exponential factor contains the
2
integral of P' . This integral increases by (2vji + n) after a period
x, where v is the vibrational quantum number. Combining the integral
2
of P' with y/2 after every period x gives for the nth period,
r,tÂ°+nx
ip(t +nx) + J Q [P ]^dt = nni + n(2vit + n)i
= ni(n + 2vrt + it) = nv2ni
(VI.4)
From Eqs. (VI.4) and (VI.2) we can conclude that the exponents in u
cl
add up to an integral multiple of 2ni after every period x and, thus,
do not affect the value of u& at times t=tÂ°+nx in the asymptotic
region. The only other factor involves the square of the momenta and
will have the same periodicity as the internal wavefunction uv(r)
because the vibrational momentum P'(r) has the same period x as the
vibrational motion.
64
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 r. Since the function uv[r(t;r0)] is also periodic
over a time t, then, the following equality holds:
, _o o.
uy(t ;r )
P (t
P (x ; r )
' nâ€™ '
1/2   u Hh Jto MV'0Â»2111'
= uv[r(Tn;r0)] (VI.5)
where tn = tÂ°+nx. The equality in Eq. (VI.5) shows that asymptotic
conditions are met at every tn (n = 0,1,2...).
Since satisfies asymptotic conditions periodically, then the
transition integral will reflect this periodicity and it is
expected that will be a periodic function of the initial relative
coordinate R(tÂ°) for the bundle of trajectories. Calculations of
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(tÂ°). Remember that R(tÂ°)
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(tÂ°) s
RÂ°, the first method will consist of simply averaging the square of
the function D^(RÂ°) over a distance in R corresponding to one period
2
of oscillation in r. This average will be denoted D^ave and
defined by
65
fi'ave
r
R(T)
Dfi(RÂ°):
(VI.6)
where is the number of equally spaced values of RÂ° in the average.
Note that time tÂ° = 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(0) > R(t). The equation for the
relationship between R(0) and R(t) of Eq. (VI.6) is
R(t) = R(0) + gJE
nR
(VI.7)
where PÂ° is negative.
2
From Eq. (VI.7) we can see that the quantity D^ avg is
independent of RÂ°. Thus, if we use it in the calculation then the
cross section will also be independent of RÂ°. For the special case of
an internal potential which is harmonic (or any symmetric potential)
asymptotically, the transition integral D^(RÂ°) is cyclic over only
half a period; i.e.
Df.(0)  Df.()
(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 ua having the form of the WKB bound state
wavefunction.
The WKB wavefunction for a bound internal state has the form
[Messiah, vol.I],
uWKB(r)
(VI.9)
ft
where S =
JP (r) dr
a
SWKB
TÃœÃ
4
k = constant = 2
and M 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
67
} = *+ + 4_
(VI.10)
â€” s
H
, and C = +, 
where e'
Asymptotically we will separate the action into relative and internal
contributions,
SCa = SRa + SaC(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 Uy^g(r). Asymptotically
*Z~l
1 1
i T,0â€ž
i ' i '
T0
2 2^a
K P R
r rr S w S .
n a+ n a
Xo
xIa
J
J
a
e
e
1
O 1
+
0)
o +
1
(VI.11)
where we have used
SRa =PR â€˜
(VI.12)
Equation (VI.11) must be compared to y of Eq. (IV.30) in order to
isolate ua(r) which in this case will be equivalent to Uy^g(r).
68
Setting Xja = (2nH) we obtain
ua(r) =
P'(tÂ°)
Pa
1 1
e
2 â€œ 2 Ma
Ã s
H a+
KÂ° e + KÂ° e
+
I S
H a
(IV.13)
By comparing Eq. (VI.5) with the VKB function in Eq. (VI.1) we have
the requirement that
1
p'
S V
3+ 3
i
K ^a
.TT " 2 .
,,o
+ K e
. h â€œ 2 .
iS iS
J K H
K
(VI.14)
or
E exP
Cf+,
i(SaC/M ' V2 +
= E exP
c=+,
Ci [ p'dr 
/M
a
(VI.15)
where it is implied that
KÂ°elpC =
KÂ°p'(tÂ°) 1/Z = K
(VI.16)
69
and pj. is a constant to be determined. To prove the validity
of Eq. (VI. 15) and determine p^. we will first determine the form of
t
S aÂ£. To do this we will assume that Sa^.(r) is the asymptotic
f o
internal action which is calculated from a trajectory where P (r ) =
Cp'(rÂ°).
If we define S^r) as
sj.(rÂ°) = J P^dr (VI.17)
a
then we obtain
r
a
where P^. is the internal momentum whose value at rÂ° is C P^(rÂ°) . By
examining a trajectory whose initial internal coordinate and momentum
are rÂ° and P (rÂ°)  we can arrive at an expression for S (r) in terms
3 +
t
of Sy^g(r) and A where
r
SWKB(r) = J lP (r) ldr (VI.19)
a
r
and A is defined as the number of times that the momentum P (r) has
changed sign during the trajectory from rÂ° to r. As we follow this
70
trajectory we will rewrite the internal momentum in the action
integral as
P'(r) = Sp'(r)
(VI.20)
where Â£ is the sign of P . We will also use the relation
a
J P^(r) dr = Hrt(v + 1/2) (VI.21)
a
where v is the vibrational quantum number. The expression for S (r)
a +
we arrive at is
Sa+(D = 5
5' (r\ _
bWKBU' 4
lf
_u nM ajiM
Tin + â€” + Aâ€”
(VI.22)
We can also follow the trajectory with initial conditions rÂ° and
â€” IP (r ) I to obtain S (r). Using the same relations we obtain
SL
r
s
a
(r)
SWKB(r)
Mil
4
/
(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 Â£. For Eq.
(VI.22) Â£ = 1 for even values of A and 1 for odd values of A. This
71
makes the sum (A + 1/2  Â£/2)n an integer multiple of 2rt. Thus, this
t
part of Sa+ can be dropped because the exponential of 2Jti is one. The
same is true of the sum (A + 3/2  Â£y2) in Eq.(VI.23) because for S
we find Â£ = 1 for even values of A and Â£ = 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
u = arg[P (r)]  arg[P (rÂ°)] = An . (VI.24)
d
Substituting Eq. (VI.24) into the left side of Eq. (VI.15) and using
S . from Eqs. (VI.22) and (VI.23) gives
sc,
nM
WKB " 4
/H
i
f \
n
r + p
4 +
t
S WKB
itM
" 4
/H
i
f \
3 n
"4 + p
â–
e
^ t
+
e
â€¢
e
i
SWKB
= e
l
e
WKB
(VI.25)
where Â£^ 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+ = Ji/4 (VI.26)
p = 3JI/4 . (VI.27)
The final step is to note the relationship between Â£,+ and Having
/
assumed that Sa+ is calculated from the trajectory whose initial
72
momentum is P+(rÂ°) = P(rÂ°)  we have prescribed that Â¿j^(t,rÂ°) will
be constructed from two trajectories whose position and momenta are
always opposite in sign asymptotically; i.e.
P_(r) = P+(r)
(VI.28)
As a consequence of this judicious choice we have that
= l+ (VI. 29)
Noting that cos(E) = cos(E) where E is any phase and using Eq.(VI.29)
we arrive at the equality
itM
VKB 4
/H
ItM
bWKB ~ 4
e
*â– ^
+
e
 
1
= 2C0S
ll
4
i
it
4
l
svkb/m
It
4
= 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)
73
*c(t;rÂ°)
f
1
2
,o
_
2JÃœÃ
C
J_
>
J
1
2
IP;
i
h
<^(t;rÂ°)
x K
(VI.30)
where S^(Q) = crj.(t;rÂ°). To satisfy the form of the WKB internal
wavefunction asymptotically it is required that
K
o
(VI.31)
The wavefunctions and have been defined as those generated from
the trajectories beginning at rÂ° and rÂ° with initial momenta P (rÂ°) 
and P (rÂ°)  respectively. Therefore, the function ^^(t;rÂ°) is
constructed from two trajectories by adding the wavefunctions
generated along each using the proper value of from Eqs. (VI.26)
and (VI.27).
The wavefunctions
differential equation as does <{/ \t;rÂ°), because they differ only by
the constant exp(ip^). In this method the transition integral of Eq.
(IV.5) is independent of RÂ° so there is no need to average. This is
due to the way meets asymptotic conditions at all (R,r) in the
asymptotic region. The form of in this region being
74
4â€œi(t;r0)
(2 tiH ) 2 eM P R
tP (t;rÂ°) 
X cos
' a
il
4
(VI.32)
The majo'r advantage in this method over the first one presented is
that there is no average over RÂ° 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 VKB approximation is poor for small quantum
numbers and the symmetric potential approximation poor at large
quantum numbers there will probably be a certain range of quantum
numbers for which this method gives optimal results.
CHAPTER VII
RESULTS: APPLICATION TO METHYL IODIDE
In this chapter the Self Consistent Eikonal Method is applied to
the collinear photodissociation of methyl iodide. Statetostate
cross sections are calculated for two different models. The first
corresponding to excitation from the ground electronic state to a
single dissociative excited electronic state. This shall be
designated as the 1ex case. The second model calculation involves
excitation from the ground state to two dissociative excited
electronic states which are coupled (the 2ex case). Computational
details such as the selection of initial conditions are also
presented.
Coordinates
Methyl iodide is treated as a linear triatomic as in the treatment
by Shapiro and Bersohn where the umbrella mode of the CH^ group is
modelled by an "equivalent effective bond" resulting in a linear
triatomic problem with two vibrational coordinates. Essentially, the
model replaces the three hydrogen atoms by an effective mass (of three
hydrogens) located in the plane of the real hydrogens and collinear
with the CI bond axis. The coordinates used are the Jacobi
coordinates. In the 2coordinate linear model the Jacobi coordinates
correspond to the distance R from the iodine atom to the center of
75
76
mass of CH^ 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, PD and r, P for relative and internal variables
respectively. The distance R is given in terms of r and the
CI distance R^ as follows:
R = R + 0.2011r . (VII.1)
These coordinates are illustrated in Fig. (VII1). Corresponding to
each coordinate there is a reduced mass defined by
Mr â€œ (mch3) 7 (Mch3i)
(VII.2)
Mr = (MH3) 7 (MCH3)
(VII.3)
Recall that in chapter VI the problem of requiring to
satisfy asymptotic conditions at all Q was addressed and solved by two
different methods. The method of averaging over a period was used in
all the calculations for which results will be presented in this
chapter. The integral cross section expression for the collinear
model is given by Eq. (V.25). As discussed in chapter V the effect of
the symmetry of the excited states on the cross section is most
important when considering two or more excited states of different
symmetry. For the 1ex case the effect of integrating over angles is
simply to multiply the cross section by a constant 1/3. For the 2ex
case, however, the additional effect is of have a sum over the squares
of the perpendicular and parallel contributions to the transition
amplitude.
Fig. (VII1) Jacobi coordinates for CH
78
When satisfying the asymptotic conditions using method 1 the
squares of the contributions are given by the following sum
Rh i
D
f i
2 _
1
Nr
E 1 D<['V) I2
L i'
n o â€ž
(VII.4)
where the interval from Ra to is divided into equally spaced
distances and R > Ru. The value of R, is chosen arbitrarily outside
a b b
the interaction region. The length of the interval (Rg  R^) is such
that the CH^ vibrational motion goes through half a period as R goes
from R to R, , i.e.
a bâ€™
R (t = O = R,
R (tu + T /2) = R,
PR T
Rb = Ra + \
(VII.5)
where x is the period of the umbrella motion of CH^ asymptotically and
is the incoming asymptotic relative momentum. Recall the variable
transformation from the set Q to a set A defined in Eq. (III.2). In
the case of the two coordinate linear model this corresponds to the
transformation
(R, r} > {t, w1)
(VII.6)
79
where w^ is an angle defined by
r(tÂ°) = Ay sin wr (VII.7)
v is the vibrational quantum number for the internal "umbrella" mode
of CH^, and Ay 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  tÂ°) + w^] (VII.8)
Pr(t) = Mr o) An cos [w(t  tÂ°) + wj (VII.9)
Pr(tÂ°) = Mr (o Ay cos (v1) (VII.10)
where
(A) =
(VII.11)
Each trajectory is distinguished by the angle w^. A particular
value of w^ gives the initial internal position r(tÂ°) and momentum
Pr(tÂ°). The range of w^ is from 0 to 2n. Thus, a grid of w^ values
will span all classically allowed values of r along with both the
positive and negative corresponding momenta at each rÂ°. For example,
the value of rÂ° has associated with it both PrÂ° and â€” Pr0 .
80
This bundle of trajectories will all begin at the same asymptotic
value of RÂ° with the same initial time tÂ°. As an example of this a
bundle of 10 trajectories is shown in Fig. (VII2). There will be
such bundles used in each calculation of D^ â€™ \ . For example if the
range of w^ is divided up into N equally spaced intervals then for
tÂ° = 0 there will be N trajectories propagated from RÂ° = R&. After
a value for D^ '(R ) I has been calculated another bundle of
trajectories is propagated from
RÂ° = R(^j ) = R and another D^f ^Rc) calculated. This
R
continues until NR such bundles have been propagated and = R(t/2)
is the next RÂ°. Note that F* ^(R^) 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
CH^ fragment. Remember that 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.
i
(VII.12)
E.
i
+ E + hv
M
E
f
E
a
(VII.13)
+
(VII.14)
81
Fig. (VII2) Trajectories on the excited electronic potential ene gy
surface.
82
where is the energy of the CH^I molecule plus photon system
initially (in the ground state), is the energy of the system after
having absorbed the photon, E^ is the energy of the CH^I molecule
initially for quantum state y=(l,vx,vy), E^ is the combined energy of
the fragments in the final quantum state a=(r,v), Mc is the mass of
CHI and pr.3PÂ° The numbers T and v label the I electronic state and
3 f i R
the CH^ 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
P = P + MkD.  P .
c c Ph c
(VII.15)
Equation (VII.12) then reduces to
E + hv
y
po2
R
2Mâ€ž
E = 0
a
(VII.16)
which gives the following equation for PÃ:
PR I2Hr(Eu â™¦ hv E^l"2
(VII.17)
For a given initial and final quantum state and Ea are fixed and
the photon energy hv can be varied from calculation to calculation,
giving a curve of cross section vs. photon energy.
83
One Excited Surface
The only states considered in the first calculation (1ex) are the
3 3
ground state and Qq excited state. The Qq state leads to the
* 2
fragments CH^ + I ( P^/2^ whÃ¼e t*ie ground state leads to
2
CH3+ I( P3/2^ as shown in Figure (VII3). The potential surfaces
given in analytical form were determined'empirically by Shapiro and
Bersohn [Shapiro and Bersohn, 1980]. The ground state surface is
V11 (RCIâ€™ r) = De {exp[0.899(RCI  4.043)] l}2
+  {k + (.1100) exp[0.4914(RCI  4.043)]}
x {r  0.6197 exp[0.4914(RCI  4.043)]}2 D E* (VII.18)
2 3
k 2 the force constant = 0.0363 e /a^
D = 0.0874 e2/an
e 0
E* = 0.0346 e2/a
and the variable R^^=(R  .2011 r) is the carbon to iodine distance.
All energies are in Hartrees and lengths in Bohr units.
3
The Qq 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
( n e) a
Fig. (VII3)
Diabatic potential energy curves for CH3I at r=0
Fig. (VII4) Diabatic potential energy surfaces for the ground and QQ states
of CH3I.
vO
CO
Fig. (VII5) Contours of the surfaces in Fig. (VII4).
88
89
following transformation [Lee and Heller, 1982]
r >
X
7.830
0.1762'
y.
,0.6183
4.939 .
\
( \
R
/
vLV'
where
R = 4.043 a0
Rcx = r  0.6197 a0
and R = R + .2011 R^ .
(VII.20)
Lee and Heller have calculated the four lowest vibrational
eigenvalues for the ground electronic state. They are
v )
y
E (a.u.)
(0,
0)
.00399346
(1,
0)
.00643926
(2,
0)
.00885801
(0,
i)
.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,
T00(x, y) = 0.9966110 gQQ(x, y)
0.0816282 g1Q(x, y)  0.0101739 gQ1 (x, y)
(VII.21)
90
where g (x, y) is the product of harmonic oscillator wave functions
vxvy
for quantum numbers vx 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 S and r using the determinant of
the rotation matrix in Eq. (VII.20). Results will be shown for
excitation from this (0, 0) level and compared to those of Lee and
Heller. Results for excitation from the next three vibrational levels
(1,0), (2,0), and (0,1) of the ground state will also be presented in
this chapter. For these higher vibrational levels a simple product of
harmonic oscillator functions was used for the ground state
vibrational wave function.
The equations of motion for the 1ex case are
r =
R =
3H
9PR
P
r
3H
3V
ex
3r 3r
(VII.22)
(VII.23)
(VII.24)
(VII.25)
where Vgx is given by Eq. (VII.19). The Jacobian is numerically
calculated from
91
ÃR r
U w
= P,
y
r _
3r
3w1
\ 1/
P
r
3R
3r
3w^
3w^
 P
r
f
3R
3wn
â– fE Q1  fr Q0
Ã³1  P1
U1 " *1
Q1  P1
u0 â€œ *0
â€¢1 a2y l
P1 = Q1 + _2
2
31
3S3r
3r
ex qJ
2 2
.. 3 V  3ZV ,
r\ 1 ex â€ž1 ex â€ž1
pn = ~~3â€” Qn + Q1
0 3fT U 3R3r 1
(VII.26)
(VII.27)
(VII.28)
(VII.29)
where the following notation of chapter IV has been used:
Q
1
0
P
1
0
i 3P
P1 a â€”Â£
1 3wx
(VII.30)
(VII.31)
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 Pr/w . (VII.32)
In Fig. (VII6) the function J (dotted line) is compares to the exact
cl
calculation of J for a single trajectory. From this figure we can see
that J coincides with J until the interaction region, then begins to
3l
deviate slightly. It is not until the trajectory begins to exit the
Fig. (VII6) The Jacobian for one trajectory; exact (solid line), and J (dotted line).
3.
94
interaction region that the two differ significantly, and even then
the difference is not great. If this behavior of J is typical of all
trajectories then it can be assumed that for this potential surface
the approximate Jacobian is sufficient for the calculation of the
transition integral. This assumption is shown to be correct according
to trial calculations using Ja in place of J in which the difference
between the results was less than 5%. We would not expect this
behavior, however for an excited state potential energy surface for
which the variation in one of the position variables along a
trajectory is large from one trajectory to the next. For example it
can be predicted by observing the upper surface in Fig. (VII4) that
the factor 3R/3w^ in Eq. (VII.26) will be very close to zero
throughout a trajectory. For all of the results in this chapter,
however, we have used the exact form for J.
We will be solving Eqs. (VII.22)(VII.25) and (VII.27)(VII.29)
along each trajectory making a total of 8 simultaneous linear
differential equations for this model of 2 coordinates and one excited
surface. A modified version of the subroutine DE was used to
integrate these equations. It is based on a Adam's procedure and
follows a PredictorCorrector sequence with variable step sizes
[Shampine and Gordon, 1975]. Figure (VII2) shows some typical
trajectories beginning at R^. For all calculations R^ was chosen to
0
be 14.0 a.u. (7.4 A) which is well outside the interaction region.
For the cross section calculation of CH^I photodissociation in the
1ex case each statetostate cross section required the following to
95
obtain a value which was converged to less than 5%
Ntr ' 50
Nr  5
where Ntj_ is the number of trajectories per bundle.
Thus a total of only 250 trajectories were needed for each cross
section calculation to converge to within an error less than the
maximum statistical error
100 y
H250
~ 6.3%
Results for the one excited surface cross section are shown in
Figs. (VII7) to (VII14). Figure (VII7) shows partial cross section
results from excitation of the ground (0,0) state to dissociation with
internal vibrational state v, where v goes from 0 to 8. These results
can be compared to those obtained by Lee and Heller plotted in Fig.
(VII8). The scale on both figures is chosen so that the peak height
of the total cross section is equal to one as seen in Fig. (VII9)
where the total cross section curves are compared.
It is clear from Figs.(VII7) and (VII8) that the results from
the SelfConsistent Eikonal (SCE) method are in very good agreement
with the "exact" wave packet calculations of Lee and Heller. Our
results give the same trends for the relative peak heights, showing
v=4 to have the highest peak, followed in order by v=3,5,2, and 6.
The peak positions and shapes also appear to be very close to those of
Lee and Heller. A careful examination of the peak positions of the
partial cross section curves shows that all of the curves from the SCE
method peak at slightly higher energies than the curves from the
calculations of Lee and Heller. The average of these differences is
found to be approximately 200cm This is also the difference in the
Fig. (VII7) Partial cross section vs. photon energy from SCE method for the 1ex
Excitation from the ground (0,0) state to excited electronic state
and final DH~ vibrational levels: v=0 (.), v=l ( + ), v=2(*), v=3 (o)
v=4 (X), v=5 ([]), v=6 (A), v=7 (0), v=8 (II). All intensities have
been divided by the maximum total intensity.
case.
9
Fig. (VII8) Same as Fig. (VII7) for the results of Lee and Heller.
Fig. (VII9) Total cross section curves for the 1ex case from the SCE
method (solid line), and from Lee and Heller (dotted line).
, O I râ€”T i i i I I I I I
33 34 36 36 37 38 39 40 41 42 43 44 46
ENERGY (/1000 cml)
101
Fig. (VII10) Partial cross sections at X=266nm for: SCE (*), Lee and Heller (.),
experiment (+). The experimental results of Sparks et al. are
normalized to one. The theoretical results are all divided by the
maximum intensity.
103
xt3THS3/(a)so
Fig. (VII11) Partial cross sections at X=248nm labeled as in Fig. (VII10).
105
XBmgo/(A)so
106
peak positions for the total cross section curves of Fig. (VII9). An
energy difference of 200cm'*' is very small compared to the energy of
the photon (~.5% of h\>). However, it is significant in relation to
the vibrational energy levels which are separated by only about
630cm*". A comparison of the values for FWHM values shows that the
partial cross section curves for the SCE method are slightly narrower
(by about 100300cm'*') than for the curves of Lee and Heller. This is
also true of the total cross section curves for which the FWHM values
differ by 300cm'*'. Figure (VII10) and (VII11) show experimental
results for partial cross sections at two different photon energies
along with the results from the SCE method and from Lee and Heller.
These results show that the observed CH^ vibrational distribution
peaks at v=2 for photon wavelengths X=266nm and 248nm. The calculated
results differ at both wavelengths, peaking instead at v=3 and v=5
respectively. The present results, however, agree very well with
those of Lee and Heller at these wavelengths. Note that the relative
intensities of the two theoretical results are preserved for each
figure. The disagreement between the theoretical and experimental
results can be attributed primarily to the potential energy surface
[Shapiro, 1986].
Results for excitation from the ground (1,0), (2,0) and (0,1)
states to dissociation with internal vibrational states from v=0 to 8
are shown in Figs. (VII12) to (VII14). These results are also
divided by the peak height of the total (0,0) curve. The cross
section vs. photon energy curves are very interesting with respect to
the way they reflect the nodal structure of the ground state
wavefunction. Each of the partial cross section curves in Fig.(VII
12) from excitation of the first vibrationally excited (1,0) ground
state show two separate peaks. This reflects the fact that the (1,0)
Fig. (VII12) Partial and total cross sections from the SCE method for the 1ex case.
Excitation from the (1,0) vibrational level of the ground electronic
state to the JQq excited electronic state with "v" levels labeled as
in Fig. (VII7). All intensities have been divided by the maximum
total intensity from Fig. (VII7).
108
Fig. (VII13) As in Fig. (VII12) for level (2,0).
110
Fig. (VII14) As in Fig. (VII12) for level (0,1).
33 34 36 36 37 38 39 40 41 42 43 44 46
E (/lOOO cm1)
112
113
bound state wave function has one node in the normal coordinate
composed primarily of the ICH^ stretch. Of even greater interest is
the structure of the total cross section curve also plotted in
Fig. (VII12). This curve has a symmetric shape with two peaks
separated by 3950cm'' and one mimima corresponding to an energy of
about 38700cm''. iocation 0f the (1,0) minima is the same (within
an error tolerance of +50cm'') as that of the (0,0) peak. It lies
exactly midway between the two peaks. The peak at the higher energy
is slightly less intense than the one at lower energy.
The cross section curves in Fig. (VII13) for excitation from the
(2,0) vibrational state also exhibit the same nodal structure as the
electronic ground state vibrational wavefunction. Each curve now
shows three peaks and two minima. The total cross section curve has
two peaks of similar height and a third, less intense, peak positioned
between them. The less intense peak is at 38150cm'". This is at
about 300cm''' lower energy than midway between the two more intense
peaks which are at 35300cm'' and 41600cm''. Of the two intense peaks
the one at higher energy is slightly more intense.
The cross section curves of Fig. (VII14) are for excitation from
the ground (0,1) state. In this state the part of the vibrational
wavefunction corresponding primarily to ICH^ stretch, the xmode, has
no nodes while that corresponding primarily to the CH^ umbrella
motion, the ymode, has one node. Each of the curves in Fig. (VII14)
has only one peak. The total cross section curve has one broad
asymmetric peak which appears to be a superposition of two peaks
positioned only 1300cm' apart. If this is the case then by
114
inspection we can place the more intense peak to be centered at about
38900cm^ +100cm^ and the other at 37600cm^ +100cm'*'. The midpoint
between the two peaks would then be 38250cm ^.
The only other theoretical results available to compare with for
excitation from the first three vibrationally excited levels of the
ground state are from a coupled channels calculation by Shapiro
[Shapiro, 1981]. Unfortunately in generating these results there was
an error in the calculation. This error was later corrected but only
the results for the (0,0) level of the electronic ground state were
published [Gray and Child, 1984]. The original published results do,
however, show the same trends as the present SCE results. The peak
positions for the partial absorption curves are close to those for the
present results and all of the cross section curves have the same
number of peaks as the present results. The major difference is in
the distribution of the final vibrational levels. The results of
Shapiro in general show the maximum intensity to occur one or two
vibrational quanta lower than the present results.
There are some distinctive features of these curves that we have
observed here. Each set of curves tends to exhibit the same nodal
structure as that part of the ground state vibrational wave function
corresponding to the ICH^ stretching motion. For example the (2,0)
level has a wave function with two nodes in the x direction and the
cross section curves for excitation from this level all have two
minima. Another feature is the trend that the total crosssection
curves are centered at successively lower energies as we go from the
(0,0) to (0,1) levels. All three total crosssection curves for
(1,0), (2,0), and (0,1) show slight asymmetry in that the peaks at
115
higher energies are more intense that those at lower energies. These
features will be discussed again in the last chapter.
Two Excited Surfaces
The second calculation involves excitation from the ground state
to two coupled dissociative states (2ex case). One of the excited
3
states is the Qq state as in the 1ex calculation. The other is the
1 2
Qj state which leads to the fragments CH^ + I ( P)â€¢ All three
surfaces (T = 1, 2, 3) are empirical potentials by Shapiro [Shapiro,
3
1986]. The ground and Qq potentials are slightly different from
those used in the previous calculation. The analytical expression for
the potentials are
vn(Rcr r> = De  E*
+ D {exp[0.87094(RrT  4.04326)]l}2
+ j {k + 0.119584 exp[0.4914(RCI  4.04326)]}
x{r  0.619702 exp[0.4914(RCI  4.04326)]} 2 (VII.33)
â€ž r, 0..71398 exp[0.38597(R)1
V22(Râ€™ r) â€œ {1 + exp[1.5(R  4.5)]}
+ 0.82978 exp[1.5 (R  .2011 r)] + 0.048149 exp[0.5 (R.2011r)]
+ 2 k r2  0.034642
(VII.34)
116
V33 (R, r) = 16.972 exp[1.361(R)]
+ 206.68 exp[2.217(R  0.2011r)]  37.286exp[1.7146(R  0.2011r)]
+  k r2 (VII.35)
V
where k = 0.036225, De = 0.0874, and E* = 0.034642.
Because we are working in the diabatic electronic representation of
Eq. (11.31) there are off diagonal coupling elements in the V block of
the electronic Hamiltonian matrix Eq. (11.33). This matrix is real
and symmetric, therefore, V23 = V32. The coupling as given by Shapiro
is only dependent on R,
V23 = A exp[0.5(R  4.2)]
A = 2.1 x 103 . (VII.36)
The relative strengths of the transition dipole matrix elements are
given also as a function of R. The equations for dp,^ are for the
parallel transition
d31 =1/(1+ exp[2(R  9.8)] ) (VII.37)
and for the perpendicular transition
d21 = 0.48304/( 1 + exp[2(R  9.8)] )
(VII.38)
117
In the 2ex case the final state of the fragments is denoted not
only by the internal vibrational quantum number v but also by the
electronic state T of the iodine atom. Since the final state of the
fragments determines initial conditions for the calculation there must
be some way to indicate which potential energy surface a trajectory is
on initially. The nuclear motion wavefunction provides the necessary
quantity which gives the amplitude on one or the other potential
surface. From Eq. (V.6) we have
Â£<_)(t; wx) = (2JlM) 1/2 Ar(t;Wl)
1/2
2P(t;w1)
Â¿S'(Q'Â°) ,
i/H cr(t;w^)
W e
where we have set C a Aj. indicating a particular choice for the
asymptotic value of C. The matrix form of A is
From normalization we set
(VII.39)
A+A = 1 = A2A2* + A3A3* (VII.40)
+ *
and note that d(A A)/dt = 0. Therefore the quantity ApAp
(T' = 2, 3) is the probability amplitude of the system being in
electronic state T'. One of the initial conditions, then, is the
o 2
value of A2 and A3 at t = t . For example, in the CH3 + I (
118
channel we have A2(tÂ°) = 1, (tÂ°) = 0. As an example of the time
'fc 'ft "fc
dependence of Ap,Ap, we have plotted curves of AjAj and A^A^ vs time
in Fig. (VII12) for final states corresponding to X=266nm, v=2,
(v ,v ) = (0,0), and w..=4ji/3. In Fig. (VII15a) the chosen final
x y i
electronic state is T=2 which requires that A2(tÂ°)=l and A^Ct0)^.
Similarly Fig. (VII15b) corresponding to T=3 requires that A2(tÂ°)=0
and A^it0)^. Notice that the time required to reach the interaction
region is slightly longer for the case T=3. This is to be expected
since the asymptotic potential energy for T=3 is larger than for T=2.
This figure shows that for a typical trajectory corresponding to final
state (T,v) the amplitude A^Ap stays at the same value as at tÂ° until
the interaction region is reached. In the interaction region the
â˜…
value ApAp goes through a sharp decrease, then begins to oscillate
with a steadily increasing frequency and decreasing amplitude
fluctuation. The period of oscillation ranges from about 600 to 200
a.u. where 1 a.u. = 2.42 x lO'^s. These oscillations rapidly tend
toward a constant value for the amplitude which varies from one
trajectory to the next. Based on an examination of numerous
trajectories it has been observed that the value for Ap*Ap usually
settles to some value between 0.6 and 0.8. According to the time
scale in Fig. (VII15) the photodissociation process takes place
within approximately 0.05 ps. This is in good agreement with a
calculated excitedstate lifetime of .07ps by Dzvonik et al. [Dzvonik
et al., 1974].
The relationship between the average potential and Ap, is
implicit in the definition of V of Eq. (11.22). For the present
calculation we obtain the following:
Fig. (VII15)
The value of ApAp for T'= 2 (dotted line) and T'=3 (solid line).
Initial conditions are (a) T=2, and (b) T=3.
(a)
120
(b) Figure (VII15) (continued)
121
122
?(Q; A2, A3) = {V22A2A2 + V33 A3A3
â€¢k * * *
+ ^23 ^2 ^3 + A2 A3 (A2^2 + A3A3^
(VII.41)
It is convenient to express the complex quantity Ar,as (Xr,+iYr,).
r m 1 r
Thus, the effective potential can be written as
V = A+ V A (A +A)
+ tv 1
[v22 (x2 + y2) + v33 (x3 + y3) + 2v23 (x2x3 + y2y3)]
2 2 2 2
(Xf + Yi + X2 + Y2 >
. (VII.42)
The differential equation for A is the same as that for C in
Eq. (III.21),
ha
M/i ^ + (V) A = 0
(VII.43)
where W = V  1 V =
/y y
22 23
y y
132 W33J
(VII.44)
giving the following matrix elements: U23=^32=^23 anc^
From Eq. (VII.43) we get the following for Xp and Yp:
*2 * Ã =
(VII.45)
i2  gi
(VII.46)
123
% â– Ã (Ã2 V23 * Y3 U33> â€™ %
(VII.47)
*3 *  ff * ' f
(VII.48)
From Eqs. (VII.45) to (VII.48) we can see that Xp and Yp are conjugate
variables in the Hamiltonian.
The differential equations for generating R, r, PD, P of nuclear
k r
motion and Qg, Q^, Pg, P^ of the Jacobian for the 2ex case are
ex
similar to those for the 1ex case. The only differences are that V
is replaced by V and that the time derivative of P^ contains the
additional sum over T due to the dependence of V on Xp, and Yp, as in
Eq. (IV.19). The explicit form of these time derivatives are
dIi
dt
/ 9 >
32V
nl
2 1
rv
n1 9
( \
_2
Ur J
Qo +
V V
,3r 3R,
ui + axâ€ž
V V ^
.35,
bl
3
f N
3V
Y1 3
f >
3V
yl
3
( \
3V
+
.35,
*2 + 3X~
X, Y J
.35.
X3 f
hi
3Y3
.3R>
X,Y
(VII.49)
and
dp;
dt
<1
nl
32V
nl 9
( \
3V
,3R3ry
Qg +
V V
Â¿i2.
Qi + axT
V V ^
,3r.
X,Y
3
f ^
3V
Y1 3
OJ
<1
yl 3
( \
3 V
* ^
,3r.
2 + 3X
V V
.3?,
3 + 3Y,.
V V
,3r.
Y V
(VII.50)
124
where the factors Xp and Yp defined in chapter IV are explicitly,
(VII.51)
The presence of these factors in Eqs. (VI.49) and (VII.50) leads to
four additional differential equations to generate for each Xp and
Yp along the trajectories. Equation (IV.25) and (IV.26) give the
time derivative equations for them. Thus, for the 2ex case a total
of 16 differential equations were solved simultaneously to generate
position, momenta, and the Jacobian along each trajectory.
To obtain results which were converged to within 5% the 2ex case
required about the same number of trajectories as the 1ex case, for
every statetostate cross section. This convergence was reached when
the number of trajectories in each bundle was set to 50 (Ntj,=50).
Convergence in the number of initial relative values needed in the
average over RÂ° was again reached at NR = 5. Thus, a total of only
250 trajectories were used for each statetostate cross section.
All results for the 2ex case are presented in Figs. (VII16) to
(VII21). In Fig. (VII16) all results for the SCE method are shown
as curves of cross section vs photon energy. The intensities for all
of these curves have been divided by the maximum intensity for the
total cross section. The curves in Figs. (VII16) and (VII17) are
Fig. (VII16) Partial cross sections for SCE from the 2ex case where T=2.
All intensities are divided by the maximum total intensity.
Curves for each "v" are labeled as in Fig. (VII7).
1.00 M â– ^rI rr* \ ,1 I t I I 1
34 35 36 37 3B 30 40. 41 42 43 44 45
E C/1000 cm1)
126
Fig. (VII17) As in Fig. (VII16) for T=3.
I /iiaâ€”â€” r t >â€”rÂ»i. . i zh
34 35 30 37 30 39 40 41 42 43 44 45
E C/1000 cm1)
128
Fig. (VII18) Total cross sections from SCE for T=2 (.), T=3(+), and for the
sum of these (*).
TCS/TCSmax
OCT
131
partial photodissociation cross sections for final states (r,v)=(2,v)
and (3,v) respectively. In these two figures the final vibrational
quantum number v for the CH^ fragment ranges from zero to five. The
trend for both T=2 and T=3 is for the maximum in the vibrational
distribution to shift from v=l to higher v as the photon energy
increases. The most intense partial absorption for T=2 is predicted
to occur at about E=42000cm'*' and v=3, while for T=3 the most intense
peak is at an energy E=39000cm^ and for quantum number v=2. A
general feature of the SCE results is for the partial cross sections
for r=2 to peak at energies which are about 20002500cm''' higher than
those for T=3. The curves for T=2 are also much less intense than for
r=3.
Looking at the total cross sections in Fig. (VII18) which are
the result of summing the partial cross sections we see the same
features. The total cross section curve for T=2 (.) has a peak at an
energy E2=41900cm^ which is 2500cm''' higher than for T=3 ( + ) which
peaks at E2=39400cm'''. The curve for the total cross section has a
maximum at an energy between E^ and E^ of 39800cm
The features exhibited by these results correspond well with
experimentally known facts about methyl iodide. For instance it is
well known experimentally that the transition to the ^0 potential
3
energy surface is weak compared to that for the Qq surface leading to
I [Mulliken, 1940]. The MCD absorption spectrum for CH^I has been
estimated to be composed of about 25% intensity from "'Q and 75% from
3
Qq [Gedanken and Rowe, 1975]. The results from Fig. (VII18)
indicate that for the total photodissociation cross section about 68%
* 2 2
shows up as I ( P^/2^ yield and 32% as I( ^3/2^ * '^'ie difference in
132
the intensity distribution between the observed absorption spectrum
and the calculated dissociation cross section is only about 9%. This
difference is actually to be expected because the absorption spectrum
and the dissociation cross section measure two related but different
quantities. The relationship between the absorption and
photodissociation cross sections will be discussed later.
In order to help determine the reliability of these results we
will compare them with those of Shapiro [Shapiro, 1986] from a coupled
channels type calculation using the artificial channel method
[Shapiro, 1973]. The three graphs in Fig. (VII19) are comparable to
those of Fig. (VII16) to (VII18). It is evident that many of the
features are the same. For example about 74% of the total intensity
comes from the I yield. This is more than we predict. Another
similar feature between the present results and those of Shapiro is
the trend for the relative population of CH^ vibrational states to
tend toward higher v as the energy increases.
The most interesting aspect of this comparison, however, are the
differences between the two sets of theoretical results. The most
striking difference is the peak position for the total intensity of
2
I( P3/2) which is at a much lower energy according to Shapiro's
results than for the SCE results. This corresponds to the T=3 peak
for the SCE results. According to Shapiro's data the peak for I
(37700cm is at an energy 550cm^ lower than the peak for I*
production (38250cm ^). This is the opposite of what we have seen
from the SCE results where the curve for I is at a higher energy than
that for I . The peak position of the T=2 curve, then, is not at all
verified by Shapiro's data and represents a major difference in the
Fig. (VII19) All figures (a), (b), and (c) correspond to those in Figs. (VII16) to
(VII18) for the results of Shapiro.
1.00 r flr rtfi I n.L..rr^~l â€”â€”Uffl  I a r*i1 li nâ€”
34 35 35 37 30 , 39 40 41 42 43 44 46
E (/1000 cm1)
(a)
134
(b) Figure (VII19) (continued)
135
(c) Figure (VII19) (continued)
136
137
two theoretical results which cannot be explained simply by numerical
inaccuracies. The position of the T=3 peak is somewhat verified by
Shapiro's results, however. The difference between Shapiro's peak and
the SCE results is not nearly as great as for T=2 but there is still a
variance of about llOOcm^. This is on the order of roughly two
vibrational quanta of energy.
The total photodissociation curve has a peak position which
differs by much more (~1700cm''") than the T=3 curves due to the effect
of the curves for T=2. The total photodissociation cross sections are
equivalent to total absorption cross sections because the intensity of
the radiation which is absorbed at a particular photon energy must
translate directly into an intensity of total fragments detected. For
this reason the curves for total a can be compared to the
experimentally observed absorption curves. An absorption curve from
MCD spectra [Gedanken and Rowe, 1975] is able to show the breakdown of
* 2 2
the spectrum in terms of the transition to the I ( aÂ°d I( ^3/2)
surfaces. This curve shows the total absorption to peak at 38500cm^
which is very close to the peak from Shapiro's data at 38100cm
This is to be expected because Shapiro fit his data to these
absorption curves in order to construct an empirical potential energy
surface. The experimental partial absorption curves follow the trend
2
of the present SCE results in that the peak positions for I*( P^^)
2 12
and for I( ) are separated by 3300cm with the I( P^^) peak at
higher energy. An interpretation of these different results will be
given in the next chapter.
138
We have compared experimental cross sections with the present SCE
results and with Shapiro's results for A=266nm in Fig. (VII20) and
for A=248nm in Fig. (VII21). Experimental photodissociation cross
sections for CH^I are only available for photon wavelengths of \=266nm
and 248nm. Although the curves of cross section vs energy for the SCE
method differ substantially from Shapiro's results the trends of cross
section vs final v are very similar for a given photon wavelength.
The intensity distribution for I* yield vs v is nearly identical with
Shapiro's at both wavelengths. The intensity distribution for I yield
vs v shows some discrepancy. The SCE results for 266nm and 248nm peak,
at v=2 and v=3 respectively for the I yield while Shapiro's both peak
at v=2. The largest discrepancy is in the quantum yield at each
^f
wavelength. The SCE data gives a yield for I of 84% at 266nm and 67%
at 248nm. Shapiro's data shows the opposite trend in that the I*
yield is less (70%) at 266nm than at 248nm (87%).
The comparison with experiment is much better in the 2ex
calculation than in the 1ex calculation. There are still some
important discrepancies however. At 266nm the peak for I* yield
occurs at v=2 as do both theoretical results. The peak for I yield is
at v=3 which is one quanta higher than the theoretical results. At
'fc
248nm the experimental data for I and for I yields both have a
maximum at v=2. The SCE results agree completely while Shapiro's
results show a maximum at v=3 for I yield. According to the
vibrational distribution, then, the SCE results are slightly closer to
experimental observation than Shapiro's at 248nm.
Fig. (VII20) Partial cross sections at X=266nm compared to experiment of
Sparks et al. (dotted line) for T=2 (upper curves) and T=3 (lower curver).
Theoretical results (solid lines) are from (a) SCE method, and (b) Shapiro.
Maximum intensities are normalized to one.
(a)
140
y
(b) Figure (VII20) (continued)
141
Fig. (VII21) As in Fig. (VII20) for X=248nm; (a) SCE, (b) Shapiro.
(a)
143
(b) Figure (VII21) (continued)
144
145
The quantum yields experimentally for both wavelengths are 71%.
As would be expected the quantum yields from the SCE method have the
opposite trend as those from Shapiro's data. The SCE method gives
quantum yields of 84% and 67% respectively at 266nm and 248nm while
Shapiro obtains 70% and 87% respectively.
CHAPTER VIII
DISCUSSION AND CONCLUSIONS
In chapter II through IV we have presented the general
semiclassical formalism to calculate statetostate cross sections for
the nonadiabatic photodissociation of a polyatomic molecule into an
atom and a molecular fragment. In Chapter V we adapted the discussion
from the more general case to the specific case of a linear triatomic
molecule. We discussed the wavefunction and the transition integral
assuming collinear motion and invoked the IOS approximation to carry
out an average over spatial orientations. In Chapter VI we presented
two possible methods for satisfying the asymptotic conditions of the
wavefunction and developed them for the model of a collinear
triatomic. In Chapter VII we discussed in detail the application of
the SCE method to the collinear photodissociation of methyl iodide as
a test of the method. Results were presented along with other
theoretical results and experimental data using first a model with one
excited surface (1ex). To further test the reliability of using an
average potential, results were presented for calculations based on
the same model but with two excited surfaces
(2ex). In the first part of this chapter we will attempt to explain
the various features of these results and the deviations from other
theoretical and experimental results for the photodissociation of
ch3i.
146
147
The 1ex Case
The theoretical results presented in Fig. (VII6) by Lee and
Heller are from an exact time dependent wave packet calculation [Lee
and Heller, 1982]. These results were later verified in a revised
quantum mechanical coupled channels calculation by Shapiro and in two
semiclassical calculations [Gray and Child, 1984; Henriksen, 1985].
We can conclude, therefore, that the exact results of Lee and Heller
are reliable. In light of this it is most encouraging to find the SCE
results in such close agreement. Considering the manner in which the
internal part of the nuclear motion wave function for the excited
electronic state compares to the expansion in harmonic oscillator
wavefunctions used in the exact treatment, it is very significant that
the SCE results correctly predicted the relative intensities for the
partial crosssections. Using harmonic oscillator wavefunctions in
the nuclear motion wave function ^ only to set initial conditions,
the SCE method correctly predicts the overlap of with the
vibrational wave function ^of the ground electronic state. This
supports the assumption that a bound state wavefunction can be
adequately described by a sum of exponentials of the action integral
along classical trajectories which span the classically allowed region
of coordinate space [Knudson, Delos, and Noid, 1986]. Indeed many of
the assumptions made are validated in this application.
Of particular importance is that assumption that the most
important contribution to the transition integral comes from the
classically allowed region of coordinate space and that a finite
number of classical trajectories will adequately span this space.
This assumption is verified to the extent that only 250 trajectories
were needed to give converged results which compare extremely well
with the exact results.
148
The small differences which show up in the comparison also give us
information about the effect of these assumptions. The small shift in
peak positions to higher energies as well as the smaller values for
FWHM can both be explained by the fact that the trajectories, and thus
the wavefunction, do not extend into the nonclassical region. The
contribution to the crosssection from the tail of the quantum
mechanical wavefunction is not accounted for. According to the
results this has a greater effect on the crosssections at lower
energies than it does at high energies. The partial crosssection
curves from the SCE method fall off more rapidly on the low energy
side than the exact results do. If we picture the overlap of ^ with
the wavefunction we can see that at lower energies the quantum
mechanical tail at small R would overlap the vibrational wavefunction
in the ground electronic state more than at high energies. This is
also the reason we see the peaks shifted to slightly higher energies
relative to the exact results. This shift of 200 cm'*' practically
speaking is almost insignificant if one considers that the error bars
on similar spectroscopic data are estimated to be 100 cm'*' for peak
positions and 10% for curve widths (~300cm*') [Gedanken and Rowe,
1975].
In comparing the theoretical results for the 1ex case with
experimental data at two wavelengths we find significant disagreement.
Based on the reliability of the theoretical results it can be
concluded that the discrepancy arises from either a weakness in the
two coordinates collinear model or in the empirical potential of
Shapiro and Bersohn. As has been mentioned in chapter V the only
vibrational mode which is significantly excited upon dissociation of
CH^I is the umbrella mode, which is a symmetric vibration parallel to
the axis of symmetry of the molecule. As a result the collinear model
is expected to adequately describe the dynamics of dissociation for
methyl iodide. We have also assumed that the rotational energy in the
parent molecule is very small. Experimentally the CH^I molecules are
supersonically cooled leaving rotationally cold parent molecules. The
source of the discrepancy therefore must be the parametrized potential
energy surfaces. Shapiro even agreed with this conclusion [Shapiro,
3
1986] and subsequently reparametrized the Qq excited state surface.
This new surface is used in the 2ex calculations along with a surface
for the state.
For the 1ex case we also generated crosssections for excitation
from the first three vibrationally excited levels of the ground state.
Unfortunately the other theoretical data is in error although the
general trends do agree. The features of these curves are very
reasonable and can be explained. As we mentioned in Chapter VII each
of the curves of crosssection vs. photon energy seems to exhibit a
nodal pattern similar to that of the ground state nuclear motion
wavefunction for the normal mode x which corresponds primarily to the
ICH^ stretch. This behavior is due to the fact that different photon
energies correlate with different classical turning points for the I
CH^ motion. Each photon energy in effect causes the nuclear motion
wavefunction to extend over a different portion of the ground
electronic state potential in the direction.
The trend for the total crosssection curves to be centered at
successively lower energies is due to the fact that the energy gap
between the dissociative state and each vibrationally excited level of
the ground state is successively smaller. The shifts, however, are
generally about half of what would be expected from the energy
150
difference between levels. It is not clear why this is the case. The
reason for the asymmetry of the peak intensities toward higher
energies is also not clear but may be due again to the absence of a
tail in the nonclassical region. This asymmetry is so slight,
however, that it could just as well be due to numerical inaccuracy
The 2ex Case
The calculations for the 2ex case were different in many respects
3
from the 1ex case. As was mentioned before the Qq surface was
reparametrized by Shapiro to obtain better agreement with experiment.
This calculation also involved another excited state surface (^Q)
3
which, in the diabatic electronic representation, crosses the Qq
surface. As a result this calculation allowed for the possibility of
3 1
"leakageâ€ from the Qq surface to the Q surface and viceverse. The
additional surface called for more differential equations to be solved
(16 compared with 12 for the 1ex case) so there was more CPU time
required per trajectory. The same number of trajectories were
required for convergence as in the 1ex case yet about twice as much
CPU time was required ( 80s/job on the IBM3080). A thermal average
over initial orientations and an integration over detection angles
were also found to be necessary for the 2ex case because the excited
states are of different symmetry. For these reasons it is difficult
to directly compare results for the two cases. One similar feature
between the two cases however, is the trend for the maximum intensity
to occur at successively higher values of the quantum number v as the
photon energy is increased.
As we mentioned in chapter VII for the 2ex case there is a large
discrepancy between the results of Shapiro and the SCE results. The
best way to begin to discuss this discrepancy is to compare both sets
of results to the experimentally determined absorption curves from the
MCD experiment [Gedanken and Rowe, 1975]. The absorption curves
3 1
essentially show the relative amplitude on the Qq and Q electronic
surfaces immediately after absorption of the photon but prior to
dissociation of the excited molecule. As the molecule is dissociating
the relative amplitudes will be redistributed between the two excited
surfaced due to the curve crossing shown in Fig. (VII3a). This is
where the interesting dynamics take place. The redistribution as the
molecule dissociates is obtained in the SCE method from the behavior
of the amplitudes A^ and A^ as shown in Fig. (VII12).
According to the SCE results there is no radical redistribution of
amplitudes as the molecule dissociates. Evidence of this is shown by
the peak for the ^Q crosssection curve which is at about the same
3
energy as the corresponding absorption curve. The Qq crosssection
curve, however, has its peak at about 1000 cm''â€™ higher than the
absorption curve so there has been some redistribution of amplitudes.
Further evidence of this is given by the difference in the relative
1 3
intensity of the Q and Qq curves for a particular photon energy.
For example at a photon wavelength of 266nm the experimental
3 1
absorption curves give an intensity ratio for Qq to Q of about 20.0
while the SCE results give a corresponding crosssection ratio of 6.0.
At a wavelength of 248nm the absorption ratio is 4.0 while the SCE
crosssection ratio is 2.4. Comparisons such as this lead us to the
conclusion that there is a substantial amount of "leakage" from the
3 1
Qq surface to the Q surface during dissociation caused by the curve
152
crossing. This redistribution is apparently greater at lower energies
and decreases at higher energies. At energies higher than about 41500
11 3
cm , where the Q intensity becomes greater than the Qq intensity,
this trend is reversed. The results indicate that the "leakage" at
11 3
energies above 41500 cm is from the Q surface to the Qq surface.
We will discuss the deviations of the SCE results from those of
Shapiro in terms of the redistribution of amplitudes during
dissociation. Shapiro has included partial absorption curves in the
published results. Because he used the MCD results to parametrized
the potential energy surfaces his partial absorption results are
nearly identical to the experimentally obtained curves of Gedanken and
3
Rowe. The Qq crosssection curve also peaks at about the same energy
as the partial absorption curve. The crosssection curve for the *"Q
state, however, has a peak which is at a much lower energy than the
absorption peak. The difference is approximately 4000 cm'*'! The
ratios of crosssection intensities at 266nm and 248nm are
respectively 2.3 and 6.7. This shows that in Shapiro's results there
3 1
is much more "leakage" from the Qq surface to the Q surface at low
energies than in the SCE results and much less leakage at higher
energies than for the SCE results. In fact already at around an
energy of 40000 cm *" the leakage trend has reversed and the dominant
1 3
process in the redistribution is for leakage from the Q to the Qq
surface. Recall that this reversal also occurred in the SCE results
but at an energy around 41500 cm"*'.
Both sets of results, therefore indicate a redistribution of
3 1
amplitude from the Qq surface to the Q surface at low energies and
1 3
from the Q surface to the Qq surface at high energies but Shapiro's
results show a much more radical redistribution than the SCE results.
Both sets of results also show similar trends in the vibrational
distribution at each photon energy. At X = 266nm the distribution of
intensity vs. vibrational quantum numbers for both the I and I*
channels from the SCE results are identical to those from Shapiro's
results with respect to the order of intensities. It is evident from
Fig. VII15 that the SCE quantum yield is higher than that of Shapiro.
Although it does not appear so by inspection, it turns out that
Shapiro's quantum yield at 266nm is almost identical to the
experimental value of 71%. At 248nm Fig. (VII16) shows the situation
to be reversed in so far as the SCE quantum yield is close to the
experimental yield of 71% and Shapiro's yield is much higher. From
this point of view the experimental quantum yields at these
wavelengths do not greatly favor either set of results. Since the
experimental quantum yields are the same at both photon energies we
cannot even conclude that there is a trend experimentally for the
quantum yield to increase or decrease with energy. Other experimental
results are of little help in this regard as there is not good
agreement between them especially at 248nm [Riley and Wilson, 1972;
Ozronik et. al., 1974; Baughcum and Leon, 1980; Hunter et. al., 1978;
Zhu et. al., 1987].
The results from the 2ex case show better agreement with
experiment than the 1ex case. According to Shapiro this is due more
3
to the improved Qq potential energy surface than to the effect of the
additional excited electronic surface ^Q. The results indicate,
however that there is still room for improvement on the surfaces.
Judging from Shapiro's quantum mechanical calculations the SCE
method does not seem to correctly model the effect of the curve
crossing on the redistribution of amplitudes as the molecule
154
dissociates. Ve can narrow the discrepancy down to the curve crossing
effect because the SCE results for one excited surface are in such
good agreement with the exact calculation. This failure to properly
account for the curve crossing is very difficult to understand,
however. In calculations on a different curve crossing problem in
which the collinear collision of H+ + H0 was studied, Olson and Micha
found the SCE method to give good results compared with more exact
calculations [Olson and Micha, 1984]. In this particular calculation
the transition integral was not used and the wavefunction was not
explicitly calculated. It was, therefore, a direct test for the
validity of using an effective potential coupled to nuclear motion to
describe the dynamics of curve crossing.
Another possibility is that Shapiro's results are in error with
respect to the amount of redistribution caused by the curve crossing.
The absorption step in his calculations cannot be doubted as they
correctly reproduce the MCD results. However, for dissociation to CH^
+ I it is difficult to see how there could be such a large shift in
the cross section peak relative to the peak for an optical transition
to this surface. As we mentioned previously this represents a radical
redistribution of amplitudes. Lacking other theoretical or
experimental results to validate Shapiro's the possibility of error is
not out of the question. It should also be noted that Shapiro may
have incorporated some sort of average over angular momentum states.
The formalism presented in his publication [Shapiro, 1986] includes a
dependence on initial and final angular momentum quantum numbers. It
is not clear from the article how this was incorporated into the
results we have been comparing to. If indeed they are the result of
an average over initial total angular momentum and a sum over final
total angular momentum this may have had an affect on the
redistribution of amplitudes which we are not properly considering.
Considerations on Angular Distributions
In applying the formalism of the SCE method in chapter VII we
used the collinear model to calculate the cross section. In order to
deal with the angular dependence of the transition integral we
performed a simple thermal average over initial orientations of the
molecule. We also pointed out that the cross sections were calculated
for the photodissociation of rotationally cold molecules. For those
cases where the molecules initially have a nonnegligible rotational
energy it may be important to deal with the angular and rotational
states of the system before and after the photodissociation process.
One way of incorporating the angular momentum states in the
evaluation of the transition integral is to begin with an expansion of
the final state nuclear motion wave function in terms of spherical
harmonics. For example the wave function corresponding to an atom
diatom system could be expanded as
Ãf(R.r)
E
lm
Y* (Â» )
lm' Efi'
W5â€™i!PÂ£i)M ,.a
Ã± R (l)
PfiR
(VIII.1)
where R and r in the space fixed frame are the vectors corresponding
to the atom relative to the center of mass of the diatomic, and the
location of the diatomic atoms relative to each
156
other. The solid angle u> gives to the direction of
i
in the spacefixed frame. The quantum numbers 1 and m refer to the
orbital angular momentum and a refers to the electronic, vibrational,
and rotational state of the fragment. The next step is to transform
the function to the body fixed frame using rotation matrix
elements D^. The equation for this transformation is
J,M,2
(VIII.2)
where J and M are the quantum numbers of the total angular momentum J
and 2 is the projection of J on the body fixed axis. The angles a, 0
and y are the Euler angles of the body fixed coordinate system
relative to the space fixed system and X is the angle between R and r.
If this approach is taken then the transition integral will separate
into a sum of terms composed of an integral over the angles of (a,0,y)
and another integral over the variables (R,r,X)* The first integrand
will contain a sum of rotation functions from the final wavefunction,
the angular dependent part of D.e^, and a rotational function from the
initial state wave function. The initial wavefunction will be of the
form
2J.+1 J.
*.(R,r) = â€”iâ€” D 1 (
8n i i
where is the initial total angular momentum quantum number and
and 2^ are the quantum numbers for the projection of on the space
fixed and body fixed axis respectively. The first integral then will
157
serve as a weighting factor for the second integral which will contain
TM O
^lm â€™ VH.r.X) and the part of D.e dependent on (R,r,X)Â« The
JMQ
function will be calculated using the SCE method where the
effective hamiltonian will be dependent on J and 2.
An alternative to this approach is to calculate the entire
function <^(R,r) including the rotational part using the SCE method.
This will mean propagating the trajectories in terms of all degrees of
freedom (except the center of mass motion). The transition integral
will be
D.e^ ^(R,r) dR dr
(VIII.A)
where the vectors R and r in the final wave function appear in the
eikonal phase as
^f(R,r) = X(R,r) e
j[S(RÂ°,rÂ°) + JpR.dR + JVr.dr ]
(VIII.5)
The integral will then be expressed in terms of the set A as
'Q'
J'i'f(A) D.e^ ^(A)
J
A
dA
(VIII.6)
where As{t; 0R, <>R, cj'} and 0R and 4>R are orientation angles of R in
the spaced fixed frame.
There is a certain amount of freedom in defining aj . One
convenient set is c[' = {w^, 0Â°, 4>Â°} where 0^ and r are the angles
in and w^ is the same as in Eq. (VII.7). The final wavefunction
in terms of A is
n 5 ff(A)
+f(A) = Xj(a') Cf(A) en
(VIII.7)
where
Xf(a') = (2rm)
(VIII.8)
The values of 9Â°, v1 , 9Â°, and 4>Â° will be evenly sampled in
k i\ i r r
defining a bundle of trajectories. The magnitude and direction of the
initial incoming momenta will be set for each trajectory by
conservation of energy and total angular momentum. The initial and
final total angular momentum must be chosen to follow the selection
rule Jc = J., J.+1.
f i l
Conclusions
In this contribution we have presented a semiclassical formalism
to calculate statetostate cross sections for the photodissociation
of polyatomic molecules such as in Eq. (1.1). This formalism employs
the calculation of a transition integral from initial and final
wavefunctions. The electronic part of the wavefunction is expanded in
a diabatic electronic basis with coefficients corresponding to nuclear
motion wavefunctions. The nuclear motion wavefunctions for the
electronic excited states are constructed in the form of a preÂ¬
exponential amplitude matrix multiplied by the exponential of a common
phase. Electronic transitions between coupled excited states during
dissociation are described using an effective potential which is self
consistently coupled to nuclear motion along trajectories. This
formalism has been shown to have certain desirable features as stated
in the introduction. These features include computational efficiency,
the ability to deal with many degrees of freedom and many coupled
electronic states and the ability to specify initial and final quantum
states.
In further development of the basic theory we have established the
precise form of the preexponential X in terms of the Jacobian of a
transformation from the set of coordinates Q to the set of variables
A. We have developed the equations for numerically calculating this
Jacobian exactly as a function of time along a trajectory. We have
also further developed the formalism by proposing two possible ways of
satisfying asymptotic conditions. As applications, we have expressed
the precise form of the wavefunction and the transition integral for a
collinear model of dissociation. Within this collinear model we have
described the angular dependence of the transition dipole for both
parallel and perpendicular transitions and have derived an expression
for the cross section by performing a thermal average over initial
orientations of the molecule.
To test the various aspects of this method we performed
calculations for the dissociation of methyl iodide within the
collinear model and compared the results with those from previous
theoretical and experimental work. For the calculation involving only
one excited surface we obtained excellent agreement with other well
established theoretical work. This calculation therefore establishes
the validity of using an eikonal wavefunction for nuclear motion on an
electronic surface having several degrees of freedom. It also
established the validity of using this eikonal wavefunction to
calculate transition amplitudes. The small deviations from exact
results gave us an indication of the effect of ignoring that part of
the vavefunction in the nonclassical region, i.e. lower cross
sections on the low energy side. The magnitude of this effect,
however, was negligible. Also verified from the 1ex calculation is
the method of averaging over RÂ° to satisfy asymptotic conditions as
well as the method of calculating the exact Jacobian along a
trajectory.
It should be noted that in the 1ex case the photodissociation
crosssections for excitation from the excited vibrational levels
(v ,v ) of the ground electronic state always required the same
x y
bundles of trajectories on the upper surfaces as were used for the
(0,0) level, but with the excitation energies selected according to
(v ,v ). This convenient feature eliminates the need to perform a
x y
completely separate calculation for each ground state vibrational
level and increases the computational efficiency of the calculation.
The results of cross section vs. photon energy for excitation from
various excited vibrational levels of the ground state exhibit the
nodal structure of the ground state vibrational wavefunction along the
dissociation coordinate.
As we have discussed the results from the 2ex calculation are not
well verified by the exact quantum mechanical calculation of Shapiro.
This seems to indicate that there is a problem in describing the
redistribution of amplitudes using an average potential coupled to
nuclear motion. When compared to the experimental absorption curves,
however, the SCE results are intuitively reasonable; perhaps even more
so than Shapiro's results. Given that there have been no other
theoretical cross sections published to verify or refute Shapiro's
results there is a possibility that they are in error. There is a
need, then, for a calculation to be performed by another method using
these same empirical surfaces and couplings in order to verify or
refute Shapiro's results for the case of two excited surfaces.
Assuming, however, that they are correct it must be concluded that the
aspect of the formalism which describes electronic transitions using
an effective potential must be modified.
The SCE results for the 2ex case do show some similarities to
Shapiro's. The trend of the intensities for final vibrational levels
is verified. Also verified by Shapiro are the overall magnitudes of
the quantum yields for I* production which hover around 70% to 80%
depending on the photon energy.
The SelfConsistent Eikonal method has been shown to be a
computationally practical way of calculating cross sections for
photodissociation of polyatomic molecules. It seems quite reliable
for determining trends in the partial cross sections and in estimating
the magnitude of quantum yields. The SCE method has been shown to be
very accurate for photodissociation cases with only one excited
surface.
The method now should be applied to other systems to test other
features of the theory. One extremely simple application would be
CD^I using the same model potential energy surfaces as for CH^I. This
would test whether the theory correctly predicts the mass effects of
photodissociation. Another more involved application would be the
photodissociation of CF^I. The dynamics of this application must
involve another internal degree of freedom besides the umbrella
vibrational mode. The CF stretch vibration is very close in energy
to the umbrella mode, and is probably significantly excited during
dissociation [Van Veen et al., 1985]. It may also be necessary to
include a bending mode for the CF^I dissociation. In that case a full
threedimensional treatment such as we have outlined in the last
section would be necessary. A full threedimensional calculation on
CF^I using a wave function expansion has already been performed by
Clary [Clary, to be published]. Once the details of the formalism
have been worked out for nonlinear triatomics it can be extended to
other polyatomics with many degrees of freedom. There are a great
number of polyatomic systems for which experimental data is available.
More experimental results are needed, however, in order to establish
the reliability of theoretical methods. It would be particularly
useful for the present application to have more consistent data on
quantum yields for CH^I photodissociation as a function of photon
energy.
In closing it is appropriate to summarize some of the appealing
aspects of the SCE method as applied to photodissociation. It is a
semiclassical method which employs classical trajectories to calculate
nuclear motion wavefunctions. As such it combines the desirable
features of classical calculations with those of a quantal approach.
From the classical viewpoint it provides insight into the dynamics of
the reaction, i.e. motion of the nuclei during dissociation and the
time frame for such motion. The use of wavefunctions allow for
quantum mechanical features such as phase interference effects and
statetostate selection. Probably one of the most appealing aspects
of this method over other classical and semiclassical methods is the
way in which the transitions between coupled electronic surfaces are
incorporated. Unlike some other classical or semiclassical methods
which rely in part on physical arguments [Tully and Preston, 1971] the
expression for the effective potential as well as for the equation for
the amplitudes come directly from the matrix differential equation.
Now that this method has been tested against exact calculations it is
hoped that it will hold up well against further tests and become an
important tool in predicting photodissociation cross sections and
quantum yields particularly in cases where there are too many degrees
of freedom for more exact methods to handle and where several coupled
excited states are involved.
APPENDIX A
THE SHORT WAVELENGTH APPROXIMATION
In Eq. (11.58) we have assumed the short wavelength
approximation. In this approximation we require that
H2 Â¿2
<< 
VS
2M X
2M
or
(A. 1)
Â« 1
(A.2)
If we use the HamiltonJacobi form of Eq. (11.51) we can define a
reduced wavelength
X =
H
K
l
VS
2M(EV)
Thus we have the requirement that
(A.3)
Â« 1
(A.4)
164
165
This is the normal condition for the short wavelength or classical
approximation [Messiah, I]. The conditions of validity require ^ to
be short compared with the "transverse" dimensions of the wavefront.
In addition to this is the requirement that
V* Â«1 . (A.5)
These requirements are deduced from the optical analogy whereby curves
of constant action S form wavefronts and the rays drawn perpendicular
to these fronts are parallel to VS. It is required that the curvature
of the wavefront should be small compared to l/>t.
If Eq. (A.4) is valid then we are dealing with a case in which
the wavelength is short and the amplitude X is slowly varying. It
turns out that the amplitude will be roughly proportional to the
1/2
quantity (P) . With this in mind we can make the general statement
that the classical approximation calls for a large but slowly varying
momentum. Obviously this approximation will break down near classical
turning points of onedimensional motions where the momentum goes to
zero. It will also break down in regions where the potential energy
surface changes rapidly with distance thus causing the momentum to
change rapidly.
APPENDIX B
COMPUTER PROGRAM FOR THE SCE METHOD
Main Program:
166
167
Subroutine DYNSIM:
Call
XNLIN1
Â±__
Call
DEPHOT
168
The program PH0T02 is the main program vhich merely starts and
stops the program and calls the driver DYNSIM.
The subroutine DYNSIM is given in some detail in the preceeding
flowchart. It begins by reading from unit 2 the namelist STATES which
contains data defining the final state of the system. From this data
all initial conditions are generated and a DO loop over RÂ° is entered.
For .each calculation there are NRMAX bundles of trajectories; each
with a DO loop over NMANY individual trajectories. A trajectory is
propagated in the time variable with DEPHOT, and contributes an amount
CREAL and CIMAG to the real and imaginary parts respectively of the
integral. After NMANY such contributions have been added the cross
section SIGh
the R to give ASIGMAscx,;. .
o 6 f 1
The subroutine XNLIN1 merely puts all initial values in common
blocks to be used by XNLIN2.
The subroutine DEPHOT is a modified version of the well
documented subroutine DE. It is used to integrate the differential
equations. It uses subroutines INTRP, STEP, and FPRIME.
The subroutine FPRIME containes all of the time derivatives for
coordinates, momenta and amplitudes as well as for all Qj, p, and
Yp of the Jacobian equations. It is this subroutine which containes
all of the information about the potential surfaces.
The subroutine XNLIN2 performs the actual calculation of the real
and imaginary parts of the transition integral for a single
trajectory. It uses subroutines JCBIAN, TRAPZ, and PSINV.
The subroutine SIGGEN takes the real and imaginary parts of D^
and computes the cross section SIG. The variable ASIGMA is the final
statetostate integral cross section for a given photon energy
averaged over RÂ°.
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BIOGRAPHICAL SKETCH
Clifford David Stodden was born on June 9, 1959, in Kansas City,
Missouri. He attended high school in Kansas City and graduated in
1977. That same year he was accepted to Rockhurst College in Kansas
City and began his undergraduate education. In the Spring semester of
1980 he attended Fordham University in New York as an honors exchange
student. In the spring of 1981 he graduated from Rockhurst College
with a Bachelor of Science degree in chemistry.
In the fall of 1981 he began his graduate work at the University
of Florida. He joined the Quantum Theory Project in 1982. In the
fall of 1983 he attended the International Summer Institute in Quantum
Chemistry held in Uppsala, Sweden. Upon his return he successfully
completed the qualifying exam for admission to candidacy and began his
doctoral research.
173
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
David A. Micha, Chairman
Professor of Chemistry and
Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
Professor of Chemistry and
Physics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
This dissertation was submitted to the Graduate Faculty of
the Department of Chemistry in the College of Arts and
Sciences and to the Graduate School and was accepted as
partial fulfillment of the requirements for the degree of
Doctor of Philosophy.
December 1987
Dean, Graduate School
UNIVERSITY OF FLORIDA
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