Photodynamic control using field optimized initial state: A mechanistic
investigation of selective control with application to IBr and HI
photodissociation
K. Vandana and Manoj K. Mishra
Citation: J. Chem. Phys. 110, 5140 (1999); doi: 10.1063/1.478409
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 110, NUMBER 11
15 MARCH 1999
Photodynamic control using field optimized initial state:
A mechanistic investigation of selective control with application
to IBr and HI photodissociation
K. Vandanaa) and Manoj K. Mishrab)
Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
~Received 13 July 1998; accepted 2 December 1998!
The probability density profiles from the optimal superpositions of the field free vibrational
eigenstates which maximize flux out of the desired photodissociation channels are examined for IBr
and HI molecules. Analysis of the structure in these optimal superposition states obtained by
applying the Rayleigh–Ritz variational procedure to the time integrated flux operator shows that the
transfer of probability density to appropriate areas of the Franck–Condon region on the excited
surfaces is responsible for selective flux maximization out of different channels. Localizing the
wave packet on the more repulsive part of the higher curve facilitates fast diabatic exit out of the
upper channel and transition to the less repulsive part promotes slow adiabatic exit out of the lower
channel. This mechanism is further probed by utilizing time dependent wave packet dynamics to
obtain absorption spectra and branching ratios using full Fourier transform of the autocorrelation
functions for these field optimized initial states. The results corroborate the central role of altered
spatial profile of the initial state in selective control of photodissociation. © 1999 American
Institute of Physics. @S0021-9606~99!00610-8#
I. INTRODUCTION
cessfully identified the optimal linear combinations for multicolor continuous wave ~cw! and Gaussian fields of different
intensities, frequencies, and phase differences and considerable enhancement in selectivity and product yield using these
field optimized initial states has been demonstrated.20,21 The
optimal combinations depend on the choice of the photodissociation objective, i.e., the linear combination which will
maximize flux out of the I1Br( 2 P 3/2) in the IBr differs
markedly from that which will maximize flux out of the excited I1Br* ( 2 P 1/2) channel.21 Similar considerations hold in
the photodissociation of HI as well.20
The FOIST scheme achieves selective flux maximization
by altering the spatial profile of the initial state to be subjected to the photolysis pulse and since changes in flux are
due to the flow of probability density, it is our purpose here
to analyze the probability density profiles of the optimal superpositions to try and isolate the mechanistic features which
may be responsible for selective control. These optimal linear combinations being field specific, a comprehensive
analysis will require the optimizations to be carried out and
the structural features of the optimal initial state to be analyzed for a sufficiently large number of field parameters. Instead, the time dependent wave packet ~TDWP! method pioneered by Heller22–24 permits the evaluation of branching
ratios for a large number of frequencies from a single
calculation,25 where, in our case, the initial wave packet to
be promoted to the excited electronic states is the optimal
superposition of the field free eigenstates for the given photolysis pulse and chosen photodissociation objective.
Such an identification of structural components of these
superpositions responsible for selective flux maximization
can provide critical insight into the mechanistic basis for
photodynamic control of reactions and is the principal focus
The possibility of selective product formation by controlling the underlying quantum dynamics through the use of
an appropriately designed laser field is a subject of intense
current interest and recent theoretical and experimental work
has burgeoned in many different directions.1–12 The different
established theoretical approaches to control have been reviewed recently.1 While the focus of these theoretical approaches has been on field design, the photodissociation
yield has also been found to be extremely sensitive to the
initial vibrational state from which photolysis is induced. Ex13 1 14
15,16
amples include the photodissociation of H1
2 , D2 , HI,
6,17,18
19
HOD,
and HCl in each of which the product yield has
been found to be sensitive to the initial vibrational state of
the molecule. Since the initial vibrational state which is subjected to photolysis is critical in determining the photodissociation products, it is reasonable to expect an appropriately
optimized linear superposition of these vibrational eigenstates to serve as a better initial condition for selective maximization of the desired product.
Towards this end, a scheme to establish the optimal linear mix of the field free vibrational eigenstates for the given
photolysis pulse and chosen photodissociation objective has
been pursued in our group20,21 whereby, the emphasis is
shifted from control through design of an appropriate field,
to control through the design of an optimal linear combination of the field free vibrational eigenstates for the chosen
photolysis pulse. Applications of field optimized initial state
~FOIST! based selective control to HI20 and IBr21 have suca!
CSIR Senior Research Fellow.
Electronic mail: mmishra@chem.iitb.ernet.in
b!
0021-9606/99/110(11)/5140/9/$15.00
5140
© 1999 American Institute of Physics
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J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
K. Vandana and M. K. Mishra
of this article. The results to be discussed are an attempt to
search for simplifying features so that the onus for selective
control of chemical reactions does not lie entirely on the
laser fields which thereby may come out to be complicated26
but the task of selective control is split into different
parts20,21 which could be comparatively easier to realize in
ordinary laboratory conditions.
The methodology of the Rayleigh–Ritz variational
maximization of flux20 and the TDWP analysis22,25 is already
available in detail and only a skeletal outline is offered in
Sec. II. Section III discusses our results and some concluding
remarks are offered in Sec. IV.
and it is seen from this equation that the product yield may
be controlled by both altering the field dependent part F̂ or
the field free initial state c~0!. Earlier control schemes26–32
have attempted control over photodissociation entirely
through field manipulation for a fixed c~0!. In the FOIST
scheme, control over product yield is sought through preparation of the initial wave function c~0! as a coherent superposition of vibrational eigenstates of the ground electronic
state for the chosen photolysis pulse20,21 ~for the short femtosecond pulses to be considered here, rotational motion is
ignored!. Of course, the field itself may also be altered which
will change the nature of the optimal c~0!.
By expanding c~0! in a basis of (M 11) field free vibrational eigenfunctions,
II. METHOD
M
W , the effect
For molecules possessing a dipole moment m
W
of the radiation field e (t) may be obtained by solving the
time dependent Schrödinger equation ~TDSE!,
i\
]
c 5Ĥ ~ t ! c ,
]t
~1!
~7!
m50
flux maximization is reduced to the familiar Rayleigh–Ritz
variational optimization20 of $ c m % through diagonalization of
a (M 113M 11) matrix F whose elements are
F kl 5 ^ f k u F̂ u f l & 'Dt
W • eW ~ t !
Ĥ ~ t ! 5Ĥ 0 2 m
with H 0 being the field free Hamiltonian. The solution c at
some time T can be expressed as
c ~ T ! 5Û ~ T,0! c ~ 0 ! ,
~2!
where Û(T,0) is the ~not necessarily unitary! propagator and
c~0! is the wave function for the initial state of the molecule
to which the photodissociation pulse eW (t) is applied. Defining the time integrated flux operator F̂ as
F̂5
E
T
0
dtÛ ~ t,0! ĵÛ ~ t,0! ,
†
~3!
where
ĵ5
1
@ p̂ d ~ r2r d ! 1 d ~ r2r d ! p̂ #
2m
~4!
with m as the reduced mass, p̂ the momentum operator along
the reaction coordinate, and r d the grid point in the
asymptotic region where the flux is evaluated. In the case of
more than one possible dissociation channels, the operator ĵ
is channel specific and in a discrete representation of the
electronic curves on a spatial grid, r d denotes the grid point
appropriate to the desired channel. The time-integrated flux,
which is directly related to the product yield, is
T
0
dt ^ ĵ & t 5
5
E
E
T
0
T
0
dt ^ c ~ t ! u ĵ u c ~ t ! &
dt ^ c ~ 0 ! u Û † ~ t,0! ĵÛ ~ t,0! u c ~ 0 ! &
~5!
or
E
c~ 0 !5 ( c mf m ,
Nt
where
E
5141
T
0
dt ^ ĵ & t 5 ^ c ~ 0 ! u F̂ u c ~ 0 ! & ,
~6!
(
n50
^ c k ~ nDt ! u ĵ u c l ~ nDt ! & ,
~8!
with Dt as the step size for the numerical time propagation
and N t Dt5T.
The largest eigenvalue f max of F is the maximum product yield ~flux! and the corresponding eigenvector $ c max
m % is
the set of coefficients which define the optimal initial wave
function
M
c
~ 0 !5
max
(
m50
c max
m fm
~9!
constituting the superposition that will provide the maximum
achievable product yield f max out of the particular channel
specified by F̂ for the chosen field eW (t). As in other variational calculations, the larger the basis set size, the ‘‘better’’
the results. However, due to the difficulties in the simultaneous overtone excitation of many vibrational levels, the basis set expansion in Eq. ~7! is restricted to only the first few
vibrational eigenstates ~only the ground plus the first two
excited vibrational levels in our case! and so the size of the F
matrix is small ~333! and therefore computationally trivial
to diagonalize. The solution proceeds by propagating the full
TDSE including all electronic states M 11 times from t50
to t5T for each initial condition c (0)5 f m , m50,1,..., M
in the basis set expansion @Eq. ~7!#. During the propagation,
the matrix elements F kl are accumulated according to Eq. ~8!
to construct the time integrated flux matrix F which is diagonalized to obtain f max and c max(0). The total time T is chosen
to ensure near total dissociation of the molecule.
In the time dependent wave packet analysis ~TDWP!22
the Schrödinger equation
H exx l ~ 0 ! 5i\
]x l ~ 0 !
.
]t
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~10!
5142
J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
K. Vandana and M. K. Mishra
is solved with x l 5 m 0l f (0) as the initial wave function on
the lth excited state and m 0l is the transition dipole moment
between the ground ~0! and the lth excited state. The field
free vibrational ground state f~0! in our case is the optimal
linear combination c max(0). The time evolution of the promoted wave function x l (0) is governed by
x l ~ t ! 5exp~ 2iH ext/\ ! x l ~ 0 ! ,
~11!
and the overlap ^ x l (0) u x l (t) & of the time evolving wave
function x l (t) with the initial x l (0) is called the autocorrelation function.
To ensure correct branching ratio,25 the autocorrelation
function is evaluated after a sufficiently large time interval t
such that the norm of the wave function on different curves
has stabilized and the system population is completely out of
the curve crossing region. The Fourier transform of the autocorrelation function yields frequency dependent partial absorption cross-section
s ~ v ! 5C v
E
1`
2`
e 2i ~ v 1E 0 ! t ^ x l ~ t ! u x l ~ t 1t ! & dt,
~12!
where C is a constant,22 v is the frequency of the incident
radiation, and E 0 is the energy corresponding to the initial
state. The branching ratio is given by the ratio of the sum of
partial photoabsorption cross-sections for the two channels.
We should however mention that the fields employed to
and c max
for both IBr and HI are quite intense
obtain c max
1
2
and since the TDWP formulation is perturbative, the branching ratios obtained using TDWP, in general, may not match
those using full solutions from the TDSE. We have however
shown20 that for HI, the branching ratios obtained as a function of field frequency using TDSE, with v 50, 1, and 2 as
initial states and the amplitude ~0.01 a.u.! employed in this
article are very similar to that obtained by Kalyanaraman and
Sathyamurthy16 using Heller’s TDWP technique.
These branching ratios peak at different frequencies20 for
v 50, 1, and 2 and in keeping with the reflection principle,
v 51 and 2 branching ratios have multiple maxima. An effective map of branching ratios as a function of field amplitude and frequency using TDSE can therefore be very demanding. Also, since the optimal linear combinations c max
1
and c max
combining v 50, 1, and 2 are field specific, such an
2
exercise is not meaningful in the present context specifically
since branching ratio is only one of the ingredients in the
unfolding of the mechanistic picture to be probed here. The
TDWP can provide the branching ratios for a broad frequency range from a single calculation and is our method of
choice for investigating the role of FOIST in selective control of photodissociation.
The split-operator fast Fourier transform with Pauli matrix propagation33–35 algorithm was utilized to integrate the
time dependent Schrödinger equation in both the FOIST and
the TDWP schemes. The FOIST scheme requires a full solution of the time dependent Schrödinger equation including
all the electronic states. In the TDWP calculation, the optimal wave packet c max(0) resulting from the FOIST scheme
is propagated only on the excited electronic states to obtain
the absorption spectrum and branching ratio resulting from
the use of c max(0) as the initial condition.
The results to be presented here are from modeling of
the molecules as rotationless oscillators. For the ultrashort
pulses employed here, the neglect of rotational effects may
be taken to be permissible, since a recent ab initio investigation of the photofragmentation of HCl36 has shown negligible influence of molecular rotation on the branching ratio.
In any case, treatment of rotational effects will increase the
computational complexities by at least an order of
magnitude37 and we have therefore found it judicious to neglect rotational motion in this initial application. Photodissociation of IBr21,26,38–43 and HI9–16,20 have been studied extensively and are our representative systems of choice. In the
following section we analyze some representative c max(0)
for IBr and HI to investigate the mechanistic underpinning of
photodynamic control of product selectivity and yield.
III. RESULTS AND DISCUSSION
The Rayleigh–Ritz variational procedure outlined in the
previous section provides the optimal linear combination of
initial vibrational eigenstates ~within the chosen manifold of
M 11 vibrational eigenstates! that selectively maximizes
flux out of the desired channel for the photolysis pulse of
choice. Such a linear combination leads to an altered spatial
profile for the initial wavefunction vis-a-vis those offered by
the eigenfunctions for the individual vibrational eigenstates.
This alteration in the spatial profile has to be at the core of
the mechanism for FOIST based flux maximization and since
changes in flux are due to the flow of probability density, we
attempt to understand the structural features responsible for
selective flux maximization by comparing the probability
density plots of the optimal superpositions with those from
individual pure vibrational eigenstates and by using these
field optimized linear combinations as initial states in the
TDWP calculation of the absorption cross-sections and
branching ratios.
The analysis to be presented here is for the photolysis of
both IBr and HI molecules employing a multicolor cw field
of the form e (t)5A ( 2p50 cos(v2vp,0)t where A is the amplitude, v the photodissociation frequency, and v p,05(E p
2E 0 )/\ is the Bohr frequency for transition between the p th
and the ground (0 th) vibrational energy levels. The vibrational eigenvalues E p and corresponding eigenfunctions f p
of the electronic ground state for both these systems are computed using the Fourier grid Hamiltonian ~FGH! method.44 A
total of three @M 52 in Eqs. ~7! and ~9!# initial vibrational
eigenstates f 0 ( v 50), f 1 ( v 51) and f 2 ( v 52) are considered for the superposition to derive the optimal initial state
for the desired photodissociation objective. The optimal linear combination maximizing flux out of the lower channel
~I1Br for IBr and H1I for HI to be labeled as 1! is denoted
by c max
and c max
represents the combination which maxi1
2
mizes flux out of the upper channel ~labeled 2! with
I1Br*/H1I* as the dissociation products.
The mechanistic underpinnings of the FOIST based selective control of IBr and HI photodissociation are further
probed by the use of c max
and c max
in the TDWP calculation
1
2
of branching ratios G~Br*/Br! and G~I*/I!. For the sake of
brevity, in further discussions, instead of writing x l (0)
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J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
K. Vandana and M. K. Mishra
5143
5m0lcmax
and x l (0)5 m 0l c max
repeatedly to signify the
1
2
and c max
to the
transport of the optimal wave packets c max
1
2
excited curves in the TDWP analysis, we will assume this to
and c max
are mentioned as the initial
be the case when c max
1
2
and c max
will
conditions. The figures labeled with c max
1
2
therefore have this same connotation that TDWP calculations
or x l (0)5 m 0l c max
as
have been done with x l (0)5 m 0l c max
1
2
the initial condition.
In the following subsections, we first analyze the simpler
IBr photodissociation modeled by only three potential energy
curves, two optical, and one nonadiabatic coupling. Results
from the controlled photodissociation of the HI molecule
modeled by five potential energy curves with three nonadiabatic and four optical couplings is taken up in the next subsection.
A. IBr
The potential energy curves @Fig. 1~a!#, the nonadiabatic
coupling, transition dipole moments, and other systemic parameters are same as those used in our previous work.21,26,39
1
The excited states B( 3 P 1
0 ) and B(O ) are nonadiabatically
coupled and their potential energy curves cross at R
56.08 a.u. The ground X( 1 S 1
0 ) state ~0! is optically coupled
to both the B(O 1 ) ~1! and the B( 3 P 1
0 ) ~2! states with the
transition dipole moment m 0150.25m 02 . Our FOIST scheme
for selective flux maximization out of the I1Br/I1Br* channels has been studied for a range of field amplitudes ~0.01–
0.1 a.u.! and frequency ~v! values ~0.081–0.093 a.u. or
17 770–20 410 cm21/563–490 nm!.21 All these frequencies
lie well above the crossing point and permit dissociation out
of both the channels.
The probability density plots for the first three vibrational states f 0 , f 1 , and f 2 of the IBr molecule are plotted
in Fig. 1~b! and the probability density profile from the optimal superpositions c max
and c max
which maximize flux out
1
2
of I1Br and I1Br* channels, respectively, for a field frequency v50.087 a.u. ~19 094 cm21/524 nm! and amplitude
A50.03 a.u. are displayed in Fig. 1~c!. This amplitude is the
lowest for which almost 100% dissociation21 occurs using a
pulse length of 480 fs ~20 000 a.u.!, followed by further
propagation without the field for another 387 fs ~16 000 a.u.!.
The probability density for the ground vibrational level f 0
peaks at 4.66 a.u., that for c max
at 4.8 a.u. and for c max
at
1
2
4.58 a.u. The optimal wave function c max
seems
to
maximize
2
flux out of channel 2 ~I1Br*! by localizing the probability
density to the left of that given by the f 0 wave function and
c max
maximizes flux out of channel 1 ~I1Br! by localizing
1
probability density to the right of the f 0 probability density
peak. Though the results presented here are for a single representative frequency, this trend of c max
peaking to the left
2
of f 0 and c max
to
its
right
persists
throughout
the frequency
1
range specified earlier. The probability density profiles for
c max
and c max
are mutually exclusive and much more com1
2
pact as compared to those from f 0 , f 1 , or f 2 . Furthermore, the probability density profiles from pure eigenstates
f 0 , f 1 , and f 2 subsume the spatial attributes of both c max
1
and c max
which explains why selective photodissociation
2
cannot ensue from the use of only one of these molecular
eigenstates as the initial state.
FIG. 1. ~a! Potential energy curves for the electronic states 0:X( 3 S 1
0 ),
1:B(O 1 ) and 2:B( 3 P 1
0 ) of IBr, ~b! probability density plots for the vibrational eigenstates f 0 , f 1 , and f 2 ~v 50, 1, 2! of the IBr electronic ground
state, and ~c! probability density plots for the optimized superpositions c max
1
and c max
for flux maximization through I1Br and I1Br* channels for the
2
three color field e (t)5AS 2p50 cos(v2vp,0)t where A50.03 a.u., v50.087
a.u., and v p,05(E p 2E 0 )/\ is the Bohr frequency for transition between the
p th and the ground (0 th) vibrational energy levels.
This need for a suitable mixing of vibrational states for
selective control of photodissociation is also seen in the optimal control theory based calculations on IBr26 where the
additional frequency components of the optimal field separated from each other by IBr ground state vibrational spacings @Fig. 3~b! of Ref. 26#, large expectation value for the
internuclear distance on the electronic ground state corresponding to vibrational stretch for highly excited vibrational
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5144
J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
levels of IBr @Fig. 3~d! of Ref. 26# and extremely intense
fields required to achieve this @Fig. 3~a! of Ref. 26# in the
very beginning of the control procedure point to the same
central role of initial mixing of vibrational states in achieving
and c max
repselective control. In our FOIST scheme, c max
1
2
resent the premixing of vibrational states required for selective control with an additional advantage that the photolysis
pulse may be chosen beforehand for practical convenience.
or the
In the FOIST based selective control, it is the c max
1
max
c 2 which are transported to the excited electronic states. In
the case of IBr, the B( 3 P 1
0 ) state ~2! is coupled four times
more strongly with the ground state ( m 0150.25m 02) as compared to the B(O 1 ) state ~1!. Also, the B(O 1 ) state is far off
resonance within the frequency band considered by us and
the amplitude transferred to the B( 3 P 1
0 ) level will therefore
be much larger and dominate the photodissociation outcome.
The broad mechanistic details may therefore be inferred from
arguments employing the dynamics ensuing from the evolution of the wave packet on the B( 3 P 1
0 ) curve alone. We may
2
therefore infer from the structural features of u c max
2 u portrayed in Fig. 1~c! that for any given frequency, transitions
will occur to the
from the initial state represented by c max
2
energetically higher or steeper, more repulsive region of the
excited B( 3 P 1
0 ) potential energy curve. The excited molwill therefore traverse the
ecule described by c max
2
1
)
–
B(O
)
crossing
with
greater velocity and exit out
B( 3 P 1
0
of the excited I1Br* channel compared to where the molwhich transfers it to a relatively
ecule is represented by c max
1
smoother region of the B( 3 P 1
0 ) state potential energy curve
thereby facilitating a slower adiabatic exit out of the lower
I1Br channel.
This interpretation of the control mechanism utilizing
c max
and c max
as the initial states is consistent with the
1
2
analysis of the frequency dependence of IBr photodissociation as a function of the molecular radial velocity using the
Landau–Zener theory presented by Devries et al. where increase in photodissociation yield out of the I1Br* channel
with increase in frequency is well correlated with an increase
in radial velocity at the crossing point.45 From Fig. 1~c!, it is
max 2
2
obvious that u c max
2 u and u c 1 u are localized in a mutually
exclusive manner and that the optimization leads to a significant altering of the spatial profiles to excite the molecule to
the region most suited for directing flux out of the desired
channel.
These optimal linear combinations being field specific, a
comprehensive analysis will require that the optimizations be
carried out and the structural features of the optimal initial
state be analyzed for sufficiently large number of field parameters. Such an attempt will be extremely demanding ~specially for IBr! and the proposed central role of the FOIST
generated altered spatial profile of the initial state to be subjected to the photolysis pulse is best checked by the application of the TDWP method22 which permits the calculation of
absorption cross-section and branching ratios at all frequencies from a single calculation. The TDWP calculations for
the curve crossing problems however require that the branching ratios be calculated from Fourier transform of autocorrelation function ^ x l ( t ) u x l ( t 1t) & after a time t such that the
norm of the wave functions have stabilized for all excited
K. Vandana and M. K. Mishra
FIG. 2. Norm evolution on different potential energy curves of IBr with ~a!
c max
and ~b! c max
as the initial condition; position expectation values ^ r & t on
1
2
different curves of IBr with ~c! c max
and ~d! c max
as the initial condition.
1
2
states and are completely out of the curve crossing region.25
To ensure that this indeed is the case, in Figs. 2~a! and
2~b! we have plotted the norm ^ x l (t) u x l (t) & from TDWP
as the initial conditions.
analysis using both c max
and c max
1
2
Figures 2~a! and 2~b! present the evolution of these norms on
both the excited states 1 @ B(O 1 ) # and 2 @ B( 3 P 1
0 ) # , respectively, and it is seen that the norms have indeed stabilized at
the end of 170 fs by which time the position expectation
values ^ r & t 5 ^ x l (r,t) u r u x l (r,t) & / ^ x l (t) u x l (t) & , with both
c max
@Fig. 2~c!# and c max
@Fig. 2~d!# as initial condition are
1
2
well past the crossing point. Further examination of Figs.
2~c! and 2~d! reveals that the position expectation value on
curve 1 increases gradually till the amplitude on curve 2
reaches the crossing point with a portion crossing over to the
lower r values of curve 1. This crossed over amplitude at
lower r value decreases the average position expectation
value on curve 1. A comparison of ^ r & t from Figs. 2~c! and
2~d! shows that as surmised earlier, approach to the crossing
point is indeed much faster when the molecule is represented
by c max
2 .
The norm on the excited curves are well past the crossing point and have completely stabilized by 170 fs. The wave
function on each curve at the end of 170 fs were therefore
stored and used to calculate the total autocorrelation functions ( i ^ x i ( t ) u x i ( t 1t) & which are plotted in Figs. 3~a! and
3~b! with c max
and c max
as the initial states, respectively. As
1
2
can be seen from Fig. 3~a!, with c max
as the initial condition,
1
the autocorrelation function decreases to zero at around 18 fs
and with c max
as the initial condition, the fall to zero is
2
approximately twice as rapid ~10 fs!. This is as we would
expect from the time evolution of ^ r & t in Figs. 2~c! and 2~d!
discussed earlier where the molecule when described by
c max
is placed on the steeper part of the excited curve and
2
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J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
K. Vandana and M. K. Mishra
5145
max
FIG. 5. Branching ratio G ~Br*/Br! with f 0 , c max
as the initial
1 , and c 2
conditions.
FIG. 3. Total autocorrelation functions for IBr with ~a! c max
and ~b! c max
as
1
2
the initial condition.
rolls down more rapidly and therefore should lose the correlation with x i (0) much faster in comparison with when it is
max
represented by c max
1 . This loss of correlation in case of c 2
max
is much more complete as compared to that for c 1 since
the longer time autocorrelation function with c max
as the
2
initial state is approximately two orders of magnitude
smaller compared to that with c max
as the initial condition.
1
The total absorption cross-sections obtained from full
Fourier transform of autocorrelation functions presented in
Fig. 3 are plotted in Fig. 4. The absorption cross-section
obtained using c max
as the initial condition is smooth with
2
only a negligible interference pattern around 600 nm. The
absorption spectrum from c max
as the initial condition con1
tains a series of sharp peaks characteristic of predissociation
dynamics40 in the higher wavelength region due to a slower
1
approach to the B( 3 P 1
0 ) – B(O ) crossing. The longer
wavelength region is dominated by predissociation in the
1
vicinity of the energy values around the B( 3 P 1
0 ) – B(O )
crossing leading to the complicated interference pattern seen
in the c max
absorption profile. In contrast, the smoothness of
1
the absorption spectrum from c max
as the initial condition
2
stems from the initial placement of the c max
wave packet on
2
the steeper part of the excited curve which facilitates faster
diabatic exit with little time for interference. The absorption
spectrum from f 0 as the initial condition has also been plotted and compares well with other calculated39,40 and
experimental42,43 absorption spectra for IBr. The absorption
spectra peak at the wavelengths corresponding to the vertical
max
Franck–Condon transition energies with c max
1 , c 2 , and f 0
as the initial states.
The branching ratios G~Br*/Br! with f 0 , c max
and c max
1
2
as the initial conditions are plotted in Fig. 5. At all energy
values, G~Br*/Br! is much larger in magnitude with c max
as
2
the initial condition compared to that with f 0 or c max
as
the
1
initial condition and G~Br*/Br! is uniformly smaller with
c max
as the initial condition as compared to c max
or f 0 as the
1
2
initial condition. This does seem to suggest that preparation
of the initial state for a suitable photolysis pulse using the
FOIST scheme can provide selective control through constructing appropriate linear combination of vibrational states.
The field optimized superposition modifies the probability
density profile to enable Franck–Condon transitions to appropriate region of the excited electronic state potential energy curves to enable a faster diabatic ( c max
2 ) or slower adiabatic ( c max
)
exit
to
maximize
flux
out
of
the I1Br*/I1Br
1
channel, respectively.
B. HI
max
FIG. 4. Total absorption spectrum for IBr with f 0 , c max
as the
1 , and c 2
initial conditions.
The potential energy curves @Fig. 6~a!# and nonadiabatic/
optical coupling elements used for our investigation of the
HI photodissociation here are the same as those utilized
earlier.15,16,20 Probability density profiles from f 0 , f 1 , and
f 2 eigenfunctions included in the optimization manifold are
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5146
J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
K. Vandana and M. K. Mishra
FIG. 7. Norm evolution on different potential energy curves of HI with ~a!
c max
and ~b! c max
as the initial condition; position expectation values ^ r & t
1
2
with ~c! c max
and ~d! c max
as the initial condition.
1
2
FIG. 6. ~a! Potential energy curves for the electronic states 0: 1 S 0 , 1: 3 P 1 ,
3
2: 1 P 1 , 3: 3 P 1
0 and 4: S 1 of HI. ~b! Probability density plots for f 0 , f 1 ,
and f 2 , ~v 50, 1, 2! vibrational eigenstates of the HI electronic ground
state, and ~c! probability density plots for the optimized superpositions c max
1
and c max
for flux maximization through H1I and H1I* channels for the
2
field e (t)5AS 2p50 cos(v2vp,0)t with A50.01 a.u. and v50.20 a.u. for HI.
Conventions of Fig. 1~c! apply.
plotted in Fig. 6~b!. The 1 S 0 ground state ~labeled 0! is optically coupled to all the four excited states 3 P 1 , 1 P 1 , 3 P 0 ,
and 3 S 1 ~labeled 1, 2, 3, and 4, respectively!. Furthermore,
the states 3 P 1 (1) – 1 P 1 (2), 3 P 1 (1) – 3 S 1 (4), and
1
P 1 (2) – 3 S 1 (4) are nonadiabatically coupled with each
other, while the 3 P 1
0 state ~3! has no nonadiabatic coupling
with any other state. The outermost crossing between states 2
and 4 which will control the final flux redistribution occurs at
R53.83 a.u.
and c max
from our
The altered spatial profiles for c max
1
2
investigations spanning the frequency range of 0.16–0.26
a.u. ~or 36 000–56 000 cm21/285–175 nm! are similar and
2
we offer a representative probability density plot of u c max
1 u
max 2
21
and u c 2 u for v50.20 a.u. ~44 000 cm /228 nm! and A
and c max
once again rep50.01 a.u. in Fig. 6~c! where c max
1
2
resent the field optimized initial states which maximize flux
out of the lower H1I and the higher H1I* channels, respectively. The probability density of the ground vibrational level
2
peaks at 3.08 a.u., that for the u c max
1 u maximizing flux out of
channel 1 H1I/states 0, 1, and 2! peaks at 3.29 a.u. and that
2
for u c max
2 u maximizing flux out of channel 2 ~H1I*/states 3
and 4! at 3.01 a.u., i.e., just like in the case of IBr discussed
earlier c max
is peaked once again to the left and c max
to the
2
1
right of the f 0 peak.
In Figs. 7~a! and 7~b! we present a plot of the variation
in the individual norm ^ x i (t) u x i (t) & on each excited state
and also the total norm in the H1I/H1I* channels resulting
from the use of c max
and c max
as initial states in the TDWP
1
2
calculation. As seen in Fig. 7~a!, with c max
as the initial
1
state, there is a net transfer of amplitude from H1I* channels ~states 1 and 2! to H1I channel ~state 4!. The 3 P 1
0 ~3!
has no nonadiabatic coupling to any other state and therefore
experiences no change in the norm transferred to it. There is
a steady depletion from both states 1 and 4 into 2 initially,
but within 5 fs the norm on all these repulsive curves for this
extremely light ( m 'm H ) system is well past the outermost
crossing point @Fig. 7~c!# and unlike in the case of the heavy
and slow moving IBr, the norms stabilize much more quickly
and the system covers an average distance of 12.5 a.u. within
25 fs. Results from the TDWP analysis with c max
as the
2
initial condition are plotted in Fig. 7~b! where a fast depletion from state 2 into both state 4 ~H1I*! and state 1 ~H1I!
is clearly seen. The build up in H1I* channel is therefore
entirely due to depletion from the more repulsive 1 P 1 state
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J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
FIG. 8. Total autocorrelation functions for HI with c max
and c max
as the
1
2
initial conditions.
and selectivity will be assisted by tuning the laser to the
frequency corresponding to Franck–Condon transitions to
1
P 1 state. The molecule when described by c max
has to
2
travel greater distance but still does traverse the outermost
crossing between the states 2 and 4 faster than when described by c max
1 , however, due to the low reduced mass of
HI and consequently very fast motion as also considerable
overlap between the spatial profiles of c max
and c max
for HI,
1
2
the net change in the ^ r & t at 25 fs is only marginally higher
with c max
as the initial state. At around 25 fs, the norms are
2
well stabilized and as seen from the position expectation
value plots of Figs. 7~c! and 7~d! the wave function is indeed
well past the outermost crossing. The total autocorrelation
and c max
as the
functions ( i ^ x i ( t ) u x i ( t 1t) & with both c max
1
2
initial condition have therefore been calculated with t525 fs
and are displayed in Fig. 8.
In the case of HI, since all the four excited curves are
repulsive, the possibility for recurrences is negligibly small
and, the autocorrelation plots in Fig. 8 fall to zero much
faster due to a quick dephasing of the wave packet in the
coordinate space. Figure 9 contains the total absorption
cross-sections obtained from the full Fourier transform of the
K. Vandana and M. K. Mishra
5147
max
FIG. 10. Branching ratio G~I*/I! with f 0 , c max
as the initial
1 , and c 2
conditions.
autocorrelation function where the absorption spectrum with
f 0 as the initial states is once again in excellent agreement
with experimental absorption profile for HI,15 and Fig. 10
and c max
portrays the branching ratio G~I*/I! with f 0 , c max
1
2
as the initial conditions. The magnitude of G~I*/I! over a
large frequency range is indeed much less with c max
as the
1
initial condition compared to that from f 0 or c max
as
the
2
initial conditions and the G~I*/I! ratio with c max
as
the
initial
2
condition is much larger than that obtained with f 0 or c max
1
as the initial conditions. Hence the linear combination of
vibrational eigenfunctions leading to c max
favors the forma1
tion of H1I products whereas the linear combination c max
2
favors H1I* formation as expected from the analysis presented earlier for IBr.
The use of FOIST, even for this much more complicated
system leads to requisite alterations in the spatial profiles so
that they peak at the internuclear distances required to facilitate Franck–Condon transitions to appropriate portion of the
excited state potential curves enhancing photodissociation
out of the desired channel.
IV. CONCLUDING REMARKS
max
FIG. 9. Total absorption spectrum for HI with f 0 , c max
as the
1 , and c 2
initial conditions.
Our investigation of the selective control of IBr and HI
photodissociation using the optimal superpositions selected
by the Rayleigh–Ritz variational procedure for maximization
of flux out of the desired channel for the chosen field reveals
that the selective maximization is effected through localization of the probability density at internuclear distances which
enable Franck–Condon transitions to appropriate region of
the excited states. Transfer of the wave function to the more
steep, repulsive part of the excited potential energy curves
favors high velocity diabatic exit into the higher channel.
Localization away from the repulsive wall favors slow adiabatic exit into the lower channel. Uniformity in the results
obtained for the two fairly disparate systems investigated
here seems to confirm the utility of the FOIST approach
advocated here. The nascent mechanistic notions linking selectivity to appropriate modification of the initial state have
been further examined by an analysis of the evolution of
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5148
J. Chem. Phys., Vol. 110, No. 11, 15 March 1999
norm and position expectation values, autocorrelation functions, resulting absorption spectrum and branching ratios and
this central role for modified spatial profile in selective control provides a new possibility for experimental exploration.
The experimental realization of the optimal initial states
is however a completely uncharted area at this time. In an
earlier article20 we have presented the formulas to obtain
field parameters required to achieve these field optimized
initial states and the optimal control31 approach may also be
easily and profitably employed to attain this FOIST comprising of only three vibrational levels. We however believe that
while the theoretical tools are useful, the central result from
our investigation is that instead of putting the entire onus of
selective control on a theoretically designed laser pulse
which may not be easy to realize in practise, the approach
where different vibrational population mixes are experimentally obtained and subjected to readily attainable photolysis
pulses leading to an empirical experimental correlation between selectivity attained for diverse photolysis pulses and
initial vibrational population mix used, represents a more
promising and desirable alternative. Our results, we hope
will spur such experimental tests and a concerted partnership
between field and initial state shaping is required to better
realize the chemical dream5,6 of using lasers as molecular
scissors and tweezers to control chemical reactions. It is our
hope that the approach advocated here will merit experimental attention where instead of attempting selective control by
using an active field manipulating a passive molecule in the
ground vibrational state, experiments will be planned to use
a variety of population mixes as the initial state. The mechanistic notions rooted in our intense field optimal control and
FOIST results are corroborated by the perturbative TDWP
analysis and thereby become accessible to easy experimental
checks without requiring ultrashort high intensity lasers.
Should a pattern of the kind where altered spatial probability
density profiles of the type studied here are experimentally
confirmed to lead to selective control, generation of these
profiles can be reduced to finding a suitable linear combination of known vibrational eigenfunctions without requiring
any time dependent quantum mechanical calculations whatsoever. In an extremely optimistic scenario, since the profile
of the standard simple harmonic oscillator eigenfunctions is
well known, a lot can be done by mere inspection as well.
Also, we have utilized three vibrational states, but there
could be frequency ranges where only two vibrational states
play a dominant role and further simplification may be obtained by examining flux as a function of initial vibrational
states for an even easier shaping of the optimal linear combination.
In conclusion, the Rayleigh–Ritz variational maximization of flux by generating an optimal spatial profile for the
initial wave function offers a new and flexible alternative for
laser assisted selective control of chemical reactions. It is our
hope that the FOIST based approach analyzed here will attract requisite experimentation and will assist in keeping the
K. Vandana and M. K. Mishra
dream of controlling chemical reactions by modifying the
underlying quantal dynamics, alive, and attractive for further
pursuit.
ACKNOWLEDGMENTS
M.K.M. acknowledges financial support from the Board
of Research in Nuclear Sciences ~BRNS 37/19/97-R & D-II!
of the Department of Atomic Energy, India. K.V. acknowledges the support from CSIR, India ~SRF 2-14/94~II!/
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1
2
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