A simplification of selective control using field optimized initial state with
application to HI and IBr photodissociation
K. Vandana and Manoj K. Mishra
Citation: J. Chem. Phys. 113, 2336 (2000); doi: 10.1063/1.482047
View online: http://dx.doi.org/10.1063/1.482047
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 113, NUMBER 6
8 AUGUST 2000
A simplification of selective control using field optimized initial state
with application to HI and IBr photodissociation
K. Vandanaa) and Manoj K. Mishra
Department of Chemistry, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
共Received 13 December 1999; accepted 20 April 2000兲
An examination of the dependence of total flux from competing photodissociation channels on the
photolysis field parameters and initial vibrational states for IBr and HI molecules reveals that, for a
range of field attributes, considerable selectivity and yield may be obtained by using only the ground
or the ground and the first excited vibrational states in the optimal linear combination constituting
the field optimized initial state 共FOIST兲. The new simplifications obviate the need for overtone
excitations or multicolor photolysis fields making it easier to implement FOIST experimentally.
Concrete specifications of field attributes for achieving selective control of IBr photodissociation
products is provided. © 2000 American Institute of Physics. 关S0021-9606共00兲01427-6兴
I. INTRODUCTION
channel specific and analysis of the FOIST scheme for IBr
and HI molecules has been presented earlier.22–27
Initial applications of the FOIST scheme have however
utilized a linear combination of the ground and first two
excited vibrational states ( v ⫽0, 1, and 2, respectively兲 requiring a complex fundamental plus overtone excitation,
making the experimental realization of these optimal initial
states somewhat difficult. Also, the photolysis field ⑀ (t)
⫽A 兺 2p⫽0 cos(␻⫺␻p,0)t with ប ␻ p,0⫽E p ⫺E 0 utilized in these
initial applications consists of a principal frequency ␻ and
another two colors separated by vibrational frequency difference between v ⫽0 (E 0 ) and v ⫽1 (E 1 ) and v ⫽0 and v
⫽2 (E 2 ). The additional frequencies ␻ ⫺ ␻ 10 and ␻ ⫺ ␻ 20
provide for molecules in the vibrational levels v ⫽1 and v
⫽2 to be transferred to the same final state as those in v
⫽0 under the influence of radiation with frequency ␻ . This
permits a coherent interference between these different paths
to the same final state enhancing selectivity and yield,22–27
but the need for a multicolor photolysis field also adds to the
complication of the FOIST scheme. The range of field parameters sampled in these initial applications22–27 has also
been somewhat limited.
It is therefore desirable to investigate, if over some range
of frequencies, for a single color photolysis field of moderate
intensity, an optimal superposition utilizing only the ground
vibrational state ( v ⫽0) requiring no vibrational excitation or
v ⫽0 and v ⫽1 vibrational states, requiring only the fundamental vibrational excitation for its experimental actuation,
which can provide adequate selectivity and yield. Towards
this end, it is our purpose in this article to map the dependence of flux in competing channels on v ⫽0, v ⫽1 and v
⫽2 initial vibrational states for the three color photolysis
fields employed in earlier FOIST applications22–27 as also for
a two color 关 ⑀ (t)⫽A 兺 1p⫽0 cos(␻⫺␻p,0t)兴 and single color
关 ⑀ (t)⫽A cos ␻t兴 simplifications thereof to try and identify
the field parameters which will make possible a simpler experimental realization of selective photodynamic control. A
probe of the underlying simplifying principles which can be
The dream of using lasers as molecular scissors for
cleaving bonds selectively is being pursued with much
vigor.1 Theoretical and experimental approaches based on
exploitation of laser attributes to achieve selective control of
photodissociation
products
have
been
reviewed
extensively.2–7 Most theoretical schemes focus on designing
an appropriate laser pulse for the desired objective.8–14 The
product yield and selectivity have however, also been shown
to be critically dependent on the initial vibrational state of
15
the molecule. Examples include photodissociation of H⫹
2 ,
⫹ 16
17,18
4,19,20
21
D2 , HI,
HOD,
and HCl, in which the dependence of the photodissociation yield on the initial vibrational
state of the molecule, has been demonstrated in some detail.
The critical role played by the initial vibrational state of
the molecule in the product selectivity and yield has motivated our development of a field optimized initial state
共FOIST兲-based approach to the control of photodissociation
reactions.22–27 In this FOIST-based procedure, the control of
photodissociation reactions is through the design of an optimal linear combination of a few selected vibrational states
prior to the application of the photolysis pulse whose attributes may be chosen for ease of experimental realization.
The optimal linear combination from the FOIST procedure
achieves selective flux maximization by mixing excited vibrational states to suitably alter the spatial profile of the initial state to facilitate Franck–Condon transition to appropriate region of the excited electronic state共s兲. The linear
combination leading to Franck–Condon transition to the
more repulsive part of the excited electronic state favors a
fast diabatic exit into the excited product channel, e.g., I ⫹
Br* in IBr photodissociation. The FOIST profile which leads
to a Franck–Condon transition to the less repulsive part of
the excited electronic state favors slow adiabatic exit into the
ground state product channel, e.g., I⫹Br in IBr photodissociation. These optimal linear combinations are field and
a兲
CSIR senior research fellow.
0021-9606/2000/113(6)/2336/7/$17.00
2336
© 2000 American Institute of Physics
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J. Chem. Phys., Vol. 113, No. 6, 8 August 2000
Selective control
utilized for an easily feasible approach to selective control is
another leitmotif.
The remainder of this article is organized as follows. The
following section provides an outline of the formal and numerical considerations underlying FOIST. In Sec. III, we
present the results from our attempts to simplify FOIST, and
a brief outline of the main findings in Sec. IV concludes this
article.
II. METHOD
ជ , the effect
For molecules possessing a dipole moment ␮
ជ
of the radiation field ⑀ (t) may be obtained by solving the
time-dependent Schrödinger equation,
iប
⳵
␺ ⫽Ĥ 共 t 兲 ␺ ,
⳵t
共1兲
where
ជ • ⑀ជ 共 t 兲 ,
Ĥ 共 t 兲 ⫽Ĥ 0 ⫺ ␮
with H 0 being the field-free Hamiltonian. The solution ␺ at
some time T can be expressed as
␺ 共 T 兲 ⫽Û 共 T,0兲 ␺ 共 0 兲 ,
共2兲
where Û(T,0) is the 共not necessarily unitary兲 propagator and
␺ (0) is the wave function for the initial state of the molecule
to which the photodissociation pulse ⑀ជ (t) is applied. Defining the time integrated flux operator F̂ as
F̂⫽
冕
T
0
dt Û † 共 t,0兲 ĵÛ 共 t,0兲 ,
共3兲
where
1
ĵ⫽
关 p̂ ␦ 共 r⫺r d 兲 ⫹ ␦ 共 r⫺r d 兲 p̂ 兴 ,
2m
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
冕
T
0
dt 具 ĵ 典 t ⫽
⫽
冕
冕
T
0
T
0
dt 具 ␺ 共 t 兲 兩 ĵ 兩 ␺ 共 t 兲 典
dt 具 ␺ 共 0 兲 兩 Û † 共 t,0兲 ĵÛ 共 t,0兲 兩 ␺ 共 0 兲 典 ,
f⫽
具 ␺ 共 0 兲 兩 F̂ 兩 ␺ 共 0 兲 典
,
具 ␺共 0 兲兩 ␺共 0 兲典
共7兲
and by expanding ␺ (0) in a basis of (M ⫹1) field-free orthonormal vibrational eigenfunctions, 兵 ␾ m 其 ,
M
␺共 0 兲⫽ 兺 cm ␾m ,
共8兲
m⫽0
maximizing f using the Rayleigh–Ritz linear variational optimization of 兵 c m 其 by substituting Eq. 共8兲 in Eq. 共7兲, and
differentiating the flux functional 关Eq. 共7兲兴 with respect to
兵 c m 其 leads to the eigenvalue problem28
共9兲
requiring diagonalization of a (M ⫹1 ⫻ M ⫹1) matrix F
whose elements are given by
F kl ⫽ 具 ␾ k 兩 F̂ 兩 ␾ l 典
Nt
⬇⌬t
⫽
共5兲
共6兲
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 ␺ (0). Also, the product yield in the
desired channel is the expectation value of a timeindependent Hermitian operator F̂. The time-integrated flux
兺
n⫽0
具 ␺ k 共 n⌬t 兲 兩 ĵ 兩 ␺ l 共 n⌬t 兲 典
冋 再
⳵
1
␺*
␺ 共r兲
k 共r兲
2␮
⳵r l
⫻
or using F̂ as defined in Eq. 共3兲,
product yield⫽ 具 ␺ 共 0 兲 兩 F̂ 兩 ␺ 共 0 兲 典 .
operator F is field-dependent since it depends on Û and Û
ជ ជ
⬇e ⫺iHt/ប ⫽e ⫺i(H 0 ⫺ ␮ . ⑀ (t))t/ប , i.e., Û⫽Û( ⑀ជ (t)), and channel
specific since ĵ as seen in Eq. 共4兲 depends on r d which corresponds to the asymptotic grid point on the potential energy
curve of the electronic state corresponding to the chosen
channel.
Earlier control schemes8–14 have attempted control over
photodissociation entirely through field design for a fixed
␺ (0). In the FOIST scheme, control over product yield is
sought through preparation of the initial wave function ␺ (0)
as a coherent superposition of vibrational eigenstates of the
ground electronic state for the chosen photolysis pulse22–27
共for the short femtosecond pulses to be considered here, rotational motion is ignored兲. Of course, the field itself may
also be altered which will change the nature of the optimal
␺ (0).
Selective maximization of the product yield in the desired channel may therefore be recast as the maximization of
a Flux functional
FCÄf C,
共4兲
2337
再
⳵
␺ *共 r 兲
⳵r k
冎 册
冎
⫺ ␺ l共 r 兲
r⫽r d
,
共10兲
r⫽r d
with ⌬t as the step size for the numerical time propagation
and N t ⌬t⫽T is the total propagation time chosen to ensure
almost full dissociation. The vibrational eigenfunctions 兵 ␾ m 其
of the ground electronic state are obtained using the Fourier
grid Hamiltonian approach,29 and the derivatives in Eq. 共10兲
are evaluated using a nine-point Lagrangian interpolation
formula.30 As in other variational calculations, the larger the
basis set size, ‘‘better’’ the results. However, due to the difficulties in the simultaneous overtone excitation of many vibrational levels, the basis set expansion in Eq. 共8兲 is restricted to only the first three vibrational eigenstates (M ⫽ 0,
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2338
J. Chem. Phys., Vol. 113, No. 6, 8 August 2000
K. Vandana and M. K. Mishra
1, and 2兲 requiring the construction and diagonalization of a
3⫻3 flux matrix F. The largest of the three eigenvalues f max
of F is the maximum product yield 共flux兲, and the corremax
sponding eigenvector 兵 c m
其 is the set of coefficients which
define the optimal initial wave function
M
␺ max共 0 兲 ⫽ 兺 c mmax ␾ m ,
m⫽0
共11兲
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 ⑀ជ (t).
The split-operator fast Fourier transform31,32 with Pauli
matrix propagation33,34 algorithm was utilized to solve the
time-dependent Schrödinger equation. The formal and numerical aspects of the propagation scheme have been discussed in detail elsewhere33,34 and we only mention that using the propagation algorithm of Ref. 33, we propagate our
solution with ␾ 0 , ␾ 1 , and ␾ 2 as the initial states simultaneously to accumulate the F kl elements as prescribed by Eq.
共10兲. Diagonalization of the 3⫻3 complex Hermitian matrix
F provides the ␺ max(0) for the desired channel and chosen
field.
The results presented in the following section are from
modeling of 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 photofragmentation of HCl 共Ref. 35兲 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-magnitude and we have therefore found it judicious
to neglect rotational motion in this application.
III. RESULTS AND DISCUSSIONS
Photodynamics of IBr 共Refs. 14,23–27,36–43兲 and HI
共Refs. 17,18,22,24,44,45兲 molecules has been studied extensively. These molecules have served as systems with which
we have tested the effectiveness of the FOIST scheme
earlier.22–27 The IBr and the HI molecules have therefore
been chosen again in this attempt to simplify FOIST. The
details of potential and coupling parameters and the vibrational eigenenergies (E i ) and eigenfunctions ( ␾ i ) of these
molecules have been described in a previous article24 and we
shall cite only systemic details pertinent to this work. The
IBr photodynamics involves three potential energy curves,
one nonadiabatic coupling, and two optical coupling parameters, and is considerably simpler compared to that of HI
mediated by five potential energy curves, two nonadiabatic
couplings, and four optical coupling parameters, and is discussed first.
A. Selective control of IBr photodissociation
The potential energy curves for IBr are plotted in Fig. 1.
The total length of the spatial grid used is 10 a.u. 共2–12 a.u.兲
with grid spacing 0.0098 a.u. An optical ramp potential of
height 0.01 a.u. is applied at 10.1 a.u and extends up to 12
a.u. The propagation of the wave packet is carried out using
a pulse length of 480 fs 共20 000 a.u.兲 followed by further
FIG. 1. Potential energy curves for IBr.
propagation without the field for another 387 fs 共16 000 a.u.兲
using a step size of 0.0967 fs 共4 a.u.兲. The equilibrium internuclear distance in the ground electronic state X 1 兺 ⫹
0 共labeled 0兲 is 4.66 a.u. and the curves for the excited state
Y (O⫹ ) 共labeled 1兲 and B( 3 ⌸ 0 ⫹ ) 共labeled 2兲 cross at 6.08
a.u. The B( 3 ⌸ 0 ⫹ ) state 2 is coupled four times more strongly
with the ground state ( ␮ 01⫽0.25␮ 02) as compared to the
Y (O⫹) state 1. Also, the Y (O⫹) state is far off resonance
within the frequency band considered in this work. Almost
all the amplitude transferred from the ground electronic state
is therefore deposited on the B( 3 ⌸ 0 ⫹ ) level and it is the
initial dynamics on the B( 3 ⌸ 0 ⫹ ) potential energy curve that
dominates the photodissociation outcome. The limited
mechanistic details to be presented here are therefore inferred from arguments employing the dynamics ensuing
from the evolution of the wave packet on B( 3 ⌸ 0 ⫹ ) curve
alone. The amplitude transferred to the Y (O⫹ ) curve and
ending in the I ⫹ Br channel more or less completely stems
from intricate dynamics in the region of crossing between the
Y (O⫹) and the B( 3 ⌸ 0 ⫹ ) curves.
The flux in the I ⫹ Br and I ⫹ Br* channels using the
single 关 ⑀ (t)⫽A cos ␻t兴, two- 关 ⑀ (t)⫽A 兺 1p⫽0 cos(␻⫺␻p,0)t 兴 ,
and three-color 关 ⑀ (t)⫽A 兺 2p⫽0 cos(␻⫺␻p,0)t 兴 fields with A
⫽0.03 a.u. and ប ␻ p,0⫽ ⑀ p ⫺ ⑀ 0 with ⑀ i being the energy of
the ith vibrational state are plotted as a function of the field
frequency ␻ in Fig. 2. It can be seen that the flux in the
competing channels peak at different frequencies for all field
forms and initial vibrational states ( v ⫽0, 1, and 2兲 considered here. Also, as seen in Figs. 2共a兲 and 2共b兲, considerable
yield and selectivity may be obtained using just the v ⫽0 as
the initial state and a single-color photolysis field. The mixing of v ⫽1 and v ⫽2 in the optimal FOIST does enhance
yield, but both selectivity and yield may be ensured with
single color field and the v ⫽0 as the initial state with maximal yield at ␻ ⫽0.08 a.u. 共17 557 cm⫺1 ) for the I⫹Br channel 关Fig. 2共a兲兴 and ␻ ⫽0.09 a.u. 共22 386 cm⫺1 ) for the I ⫹
Br* channel 关Fig. 2共b兲兴. Moreover, the flux from this singlecolor photolysis field using the usual v ⫽0 initial state is just
as large as those using the three-state FOIST and two- and
three-color photolysis fields seen in Figs. 2共c兲–2共f兲.
The low frequency peak in the I ⫹ Br flux profile at ␻
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J. Chem. Phys., Vol. 113, No. 6, 8 August 2000
Selective control
2339
FIG. 3. Time evolution of wave function on the ground ( ␺ 0 ), the first
excited ( ␺ 1 ), and second excited ( ␺ 2 ) electronic states of IBr, under the
influence of single-color photolysis field ⑀ (t)⫽0.03 cos 0.08 t.
FIG. 2. Flux out of I⫹Br 共a兲, 共c兲, 共e兲/I⫹Br* 共b兲,共d兲,共f兲 channels from photodissociation of IBr induced by photolysis field ⑀ (t)⫽0.03兺 np⫽0 cos(␻
⫺␻p,0)t where n⫽0, 1, and 2 共single-, two-, and three-color fields, respectively兲 as a function of principal frequency ␻ and different initial vibrational
states. The filled symbols represent flux from FOIST-based optimized linear
combination of the vibrational states indicated next to them.
⫽0.08 a.u. with v ⫽0 as the initial state 关Fig. 2共a兲兴 could be
attributed to the low-energy adiabatic exit of the wave packet
initially transferred to the B( 3 ⌸ 0 ⫹ ) state out of the Y (O⫹)
共I⫹Br兲 channel. As the frequency is increased, the initial
transfer of the wave packet will be to the more and more
repulsive part of the B( 3 ⌸ 0 ⫹ ) electronic state with the wave
packet executing a faster diabatic exit out of the higher I ⫹
Br* channel 关Fig. 2共b兲兴. Between 0.085–0.105 a.u. frequency interval, the flux in the I⫹Br* channel is much
higher than that in the I⫹Br channel for all initial states and
field types 关Figs. 2共a兲– 2共f兲兴. However, as the frequency increases beyond 0.105 a.u., the flux in two channels become
increasingly equal. A recent study of IBr wave packet
dynamics39,40 has found the redistribution of amplitude in the
crossing region to be extremely sensitive to the field attributes and simple explanation of the type based on prevalence of adiabatic/diabatic behavior adduced earlier may not
be completely valid. In any case, this is not our purpose here
and instead we reiterate the considerable yield and selectivity
seen in Figs. 2共a兲 and 2共b兲, whereby selective control may be
achieved for IBr with just the normal v ⫽0 initial state and a
single-color photolysis field.
To understand the mechanism underlying the surprising
effectiveness of the v ⫽0 initial state and the single-color
photolysis field, we have charted the evolution of the wave
function on the ground and the excited electronic states in
Fig. 3 which reveals that under the influence of the singlecolor photolysis field, ␺ 1 (0)⫽ ␾ 0 is distorted into ␺ 1 (t)
which is a linear combination of other vibrational states. The
considerable mixing of the first ( ␾ 1 ) and second excited
( ␾ 2 ) vibrational states may be quantified by computing
c 1 (t)⫽ 具 ␾ 1 兩 ␺ 0 (t) 典 ,
and
c 2 (t)
c 0 (t)⫽ 具 ␾ 0 兩 ␺ 0 (t) 典 ,
⫽ 具 ␾ 2 兩 ␺ 0 (t) 典 . That 兩 c 0 (t) 兩 ⫽0.4286, 兩 c 1 (t) 兩 ⫽0.2071, and
兩 c 2 (t) 兩 ⫽0.1660 at t⫽100 fs makes obvious the considerable mixing of ␾ 1 and ␾ 2 in ␺ 0 . Furthermore, while the total
amplitude in the ground electronic state decreases with increase in time due to the amplitude transfer to the excited
electronic state being lost to dissociation, the extent of mixing of ␾ 1 and ␾ 2 in ␺ 0 (t) keeps increasing as seen from
兩 c 0 (t) 兩 ⫽0.1612, 兩 c 1 (t) 兩 ⫽0.07186, and 兩 c 2 (t) 兩 ⫽0.1019
at t⫽200 fs, and 兩 c 0 (t) 兩 ⫽0.03432, 兩 c 1 (t) 兩 ⫽0.05462, and
兩 c 2 (t) 兩 ⫽0.04407 at t⫽400 fs. The results of Fig. 3 and the
兩 c i (t) 兩 values at t⫽100, 200, and 400 fs thus re affirm our
contention24 that even with v ⫽0 as the initial state, mixing
of other vibrational eigenstates to form a modified initial
state lies at the core of the mechanism underlying selective
control.
A justification for the considerable selectivity and yield
using ground vibrational state and single-color photolysis
field may be imputed to the fact that IBr has a large reduced
mass and the optically brightest excited electronic state
B( 3 ⌸ 0 ⫹ ) 共labeled 2 in Fig. 1兲, has a shallow well. This leads
to slow jostling of some parts of the wave packet in the
B( 3 ⌸ 0 ⫹ ) well providing ample time for amplitude dumping
from the excited electronic state onto different vibrational
levels of the ground electronic state. This field-induced
dumping from B( 3 ⌸ 0 ⫹ ) state facilitates a mixing of vibrational eigenstates in the X 1 ⌺ 0 ⫹ electronic ground state well.
Hence, in the case of molecules like the IBr involving slow
photodynamics, there is enough time for the single-color
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2340
J. Chem. Phys., Vol. 113, No. 6, 8 August 2000
K. Vandana and M. K. Mishra
FIG. 4. Potential energy curves for HI.
photolysis field to produce a different field-dependent optimal linear combination of field-free vibrational states in the
electronic ground state well, which, when charted either experimentally or computationally as has been done here, will
help identify the field attributes permitting selective control
of IBr photodissociation without any need for further
FOIST-based optimization.
B. Selective control of HI photodissociation
Potential energy curves for HI are plotted in Fig. 4. The
nonadiabatic/optical coupling elements used for selective
control of HI photodissociation here are same as those utilized earlier.22–25 The total length of the grid used is 9 a.u.
共1–10 a.u.兲 with the grid spacing being 0.035 a.u. An optical
ramp potential is applied at 8.5 a.u extending up to 10 a.u.
and height 0.05 a.u. The wave packet is propagated for a
pulse length of 58 fs 共2400 a.u.兲 and further propagation for
another 232 fs 共9600 a.u.兲 without the field and with a time
step of 0.02 fs 共0.8 a.u.兲.
Just as in the case of IBr photodissociation, we have
studied the dependence of flux in the H⫹I and H⫹I* channels using three-, two-, and single-color photolysis fields for
different initial conditions which are plotted in Fig. 5. It can
be seen that the flux in competing channels peak at different
frequencies thereby ensuring selectivity. A strong evidence
of sensitivity to initial vibrational quantum number in conformity with the reflection principle18 is also observed. However, flux in H⫹I or H⫹I* channel using just ␾ 0 as the
initial state and single-color photolysis field is much less
compared to that obtained using the FOIST-based optimal
linear combination of vibrational eigenstates as the initial
state and multicolor photolysis fields. Hence in the case of
HI, the need for multicolor photolysis field and multivibrational FOIST to achieve selectivity and yield is underscored
by Figs. 5共a兲–5共f兲.
For mechanistic insights, we have once again plotted
nuclear wave functions on different electronic states at selected times and for different field intensities in Fig. 6, and it
can be seen that there is only a simple diminution of the
␺ 0 (0)⫽ ␾ 0 amplitude with passage of time and increase in
FIG. 5. Flux out of H⫹I 共a兲,共c兲,共e兲/H⫹I*共b兲,共d兲,共f兲 channels from photodissocation of HI induced by photolysis field ⑀ (t)⫽0.01兺 np⫽0 cos(␻⫺␻p,0)t
where n⫽0, 1, and 2 共single-, two-, and three-color fields, respectively兲 as a
function of principal frequency ␻ and different initial vibrational states.
Conventions of Fig. 2 apply.
field intensity, which suggests that unlike for IBr, there is no
mixing of vibrational eigenstates in the ground electronic
well. This lack of mixing of other vibrational states is further
confirmed by the calculation of c 0 (t), c 1 (t), and c 2 (t) defined earlier for IBr. That for the field with A⫽0.01 a.u. 关Fig.
6共a兲兴, 兩 c 0 (t) 兩 ⫽0.99, 兩 c 1 (t) 兩 ⫽0.00548, and 兩 c 2 (t) 兩
⫽0.00293 for t⫽5 fs, 兩 c 0 (t) 兩 ⫽0.90, 兩 c 1 (t) 兩 ⫽0.00280,
and 兩 c 2 (t) 兩 ⫽0.000431 for t⫽50 fs, and 兩 c 0 (t) 兩 ⫽0.88,
兩 c 1 (t) 兩 ⫽0.00493, and 兩 c 2 (t) 兩 ⫽0.00235 for t⫽250 fs confirms that for HI, there is negligible mixing of ␾ 1 and ␾ 2 in
␺ (0)⫽ ␾ 0 . We attribute this to the fact that in case of molecules like HI, with completely repulsive excited potential
energy curves and a very small reduced mass, photodynamics evolves at a very fast rate. The amplitude flows down the
repulsive excited states without being dumped back into
other ground state vibrational levels, and distortion of
␺ 1 (0)⫽ ␾ 0 through mixing with field-induced amplitude
dumping from higher excited states is not significant. FOIST
and multicolor photolysis field may therefore be necessary
for molecules like HI.
The requirement of FOIST-based control for molecules
like HI involves overtone excitation which is quite difficult
to realize experimentally. A simplification in such a case
may be sought by studying the variation of flux for a range
of frequencies using a linear combination of only the ground
( ␾ 0 ) and the first excited ( ␾ 1 ) vibrational state. As can be
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J. Chem. Phys., Vol. 113, No. 6, 8 August 2000
Selective control
2341
IV. CONCLUDING REMARKS
An investigation of the dependence of IBr and HI photodissociation on photolysis field attributes and initial vibrational states suggests that a computational investigation of
the type presented here can be an effective input for designing simpler experiments for selective photodynamic control.
The results make it possible to infer that for IBr and other
similar molecules with large reduced mass, slow photodynamics evolving on excited curves that permit sufficient time
to dump nuclear amplitude from the excited electronic state
onto the ground electronic state leading to a mixing of vibrational states in the ground state potential well, and even a
single-color photolysis field with appropriate frequency and
amplitude may provide considerable selectivity and yield
with v ⫽0 as the initial state. The field-induced distortion of
␺ 0 (t) mimics the FOIST result over some frequency range
quite well. For HI and other similar molecules with small
reduced mass and fast photodynamics evolving on repulsive
excited curves, FOIST and multicolor photolysis fields might
be necessary. In such cases, a simplification may be attempted by studying the flux over a range of frequencies. For
field attributes where the contribution from ␾ 2 is minimal, an
optimal linear combination of only the ground and the first
excited vibrational states can deliver the same selectivity and
yield as the FOIST using ␾ 0 , ␾ 1 , and ␾ 2 . This will require
only a fundamental excitation which can be implemented
more easily and our results will hopefully attract speedy experimental tests.
ACKNOWLEDGMENTS
K.V. acknowledges support from CSIR, India 共SRF
2-14/94共II兲/E.U.II兲. M.K.M. acknowledges financial support
from the Board of Research in Nuclear Sciences 共BRNS 37/
19/97-R & D-II兲 of the Department of Atomic Energy, India.
FIG. 6. Time evolution of wave function on the ground ( ␺ 0 ) and excited
electronic states ( ␺ 1 , ␺ 2 , ␺ 3 , and ␺ 4 ) of HI, for photolysis fields 共a兲 ⑀ (t)
⫽0.01 cos 0.23t, and 共b兲 ⑀ (t)⫽0.03 cos 0.23t.
seen from Fig. 5共e兲, in the frequency range 0.19–0.25 a.u,
the difference in the flux obtained from the H⫹I channel
using FOIST-based optimal linear combination of only ␾ 0
and ␾ 1 and the linear combination of ␾ 0 , ␾ 1 and ␾ 2 as
initial state is very small and hence the optimized linear
combination of only ␾ 0 and ␾ 1 can serve as an initial state
without significant loss of yield or selectivity. This diagnosis
is true even in the case of flux out of the H⫹I* channel,
where in the frequency range 0.185–0.22 a.u. 关Fig. 5共f兲兴, the
flux from FOIST using the two and three vibrational states
are almost the same. This suggests that in the case of molecules like HI, where FOIST is necessary for selective control and flux maximization, optimal linear combination of
just ␾ 0 and ␾ 1 involving fundamental excitation alone may
be sufficient over a range of frequencies to obtain desired
yield and selectivity, and a study like the one presented here
can play a useful role in this simplification.
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