Download A computer program to calculate the total energy absorption cross

Computer Physics
Communications
Computer Physics Communications 74 (1993) 289—296
North-Holland
A computer program to calculate the total energy absorption
cross-section for the photodissociation of a diatomic molecule
arising from a bound state
repulsive state transition
using time dependent quantum dynamical methods
-~
Gabriel G. Balint-Kurti, Steven P. Mort and C. Clay Marston
1
School of Chemistry, The University of Bristol, Cantock’s Close, Bristol BS8 ITS, UK
Received 11 July 1992
A program is presented for calculating the total energy absorption cross-section for the photodissociation of a diatomic
molecule. The mechanism is assumed to involve the absorption of a photon of ultraviolet radiation which causes an
electronic transition in the diatomic molecule from a bound to a repulsive electronic state. The two atoms then fly apart
under the influence of the forces on the repulsive electronic state causing the molecule to break up into its atomic
fragments. Time dependent quantum dynamical methods are used in the calculation. These methods yield the complete
absorption spectrum from a single solution of the dynamics of the system. The computer program permits the calculation of
cross-sections for molecules in different initial vibrational states. The program is self-contained.
PROGRAM SUMMARY
Title of program: PHOTO
Memory required to execute with typical data: 1 Mbyte
Catalogue number: ACLC
No. of bits in a word: 64
Program obtainable from: CPC Program Library, Queen’s
University of Belfast, N. Ireland (see application form in this
issue)
No. of lines in distributedprogram, including test data, etc: 1885
Keywords: time dependent quantum dynamics, photodissociatiofl
Licensing provisions: none
Computer: Meiko i860 and SUN Sparc workstation (but
should work on any computer); Installation: Department of
Theoretical Chemistry, University of Bristol
Operating system: UNIX
Programming language used: Fortran 77
_________
Correspondence to: G.G. Balint-Kurti, School of Chemistry,
The University of Bristol, Cantock’s Close, Bristol BS8 iTS,
UK.
1 Present address: Max-Planck-institut für Strdmungsforschung, W-3400 Gdttingen, Germany.
OO1O-4655/93/$06.OO © 1993
—
Nature of physical problem
The program calculates the total energy absorption cross-section for radiation incident on a diatomic molecule as a function of photon energy. The radiation is absorbed by the molecule causing an electronic transition from an initial bound
electronic state to a repulsive one. A transition dipole moment function is included in the calculation. It is through the
mediation of this dipole function that the radiation interacts
with and is absorbed by the molecule. The molecule is assumed initially to be in a bound vibrational state. The wavefunction of this vibrational state is computed numerically from
the potential provided by the user. The user must also provide
the repulsive state potential energy curve as well as the form
of the transition dipole moment function. The computer program can be used as a modelling tool to extract information
Elsevier Science Publishers B.V. All rights reserved
290
G.G. Balint-Kurti et al.
/
Total energy absorption cross-section for the photodissociarion of a diatomic molecule
concerning potential energy curves and transition dipole moment functions from experimental data. It may also be used as
a starting point for writing computer programs for other
applications of time dependent quantum dynamics.
Method of solution
The Fourier Grid Hamiltonian method [1,2] is used to generate a selected vibrational state wavefunction using the bound
electronic state potential energy curve supplied by the user,
This wavefunction is then multiplied by the transition dipole
moment function (also supplied by the user) to yield an initial
wavepacket. The wavepacket is evolved forwards in time using
time dependent quantum dynamics [3,4]. Its time development
is governed by the (user supplied) repulsive state potential
energy curve. Grid methods are used throughout, both in the
computation of the initial vibrationally bound wavefunction
and in the subsequent solution of the time dependent
Schrddinger equation. The solution of the time dependent
Schrödinger equation is accomplished through the use of the
Chebychev polynomial expansion method of Tal-Ezer and
Kosloff [5,6]. The method requires the repeated operation of
the Hamiltonian operator on the initial or current wavepacket.
This wavepacket is represented by its values on a grid of
evenly spaced points and discrete fast Fourier transforms are
used in operating with the kinetic energy part of the Hamiltonian operator on the wavepacket [6].At each time step, as the
time development of the wavepacket progresses, the autocorrelation function is computed by taking the overlap of the
initial wavepacket with the wavepacket at the current time [4].
The cross-section is finally evaluated as the real part of the
Fourier transform of the autocorrelation function [3,4]. The
methods yield the absorption cross-section at all photon
energies of interest from a single solution of the time
dependent quantum dynamics.
Restrictions on the complexity of the problem
The program is written for a 1-dimensional problem and is
therefore limited to diatomics or systems being modelled
using one mathematical dimension. The number of grid points
must be a power of 2 (e.g. 128, 256) due to the nature of the
Fourier transform algorithm used [7].
Typical running time: 54 s on Meiko, 8 mm 9 s on SUN
References
[1] C.C. Marston and G.G. Balint-Kurti, J. Chem. Phys. 91
(1990) 3571.
[2] G.G. Balint-Kurti, C.L. Ward and C.C. Marston, Comput.
Phys. Commun. 67 (1991) 285.
[3] G.G. Balint-Kurti, R.N. Dixon and C.C. Marston, J. Chem.
Soc. Faraday Trans. 86 (1990) 1741.
[4] E.J. Heller, J. Chem. Phys. 68 (1978) 2066.
[5] H. Tal-Ezer and R. Kosloff, J. Chem. Phys. 81(1984) 3967.
[6] R. Kosioff, I. Phys. Chem. 92 (1988) 2087.
[7] W.H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetterling, Numerical Recipes (Cambridge Univ. Press, Cambridge, 1987).
LONG WRITE-UP
1. Theory
The one-dimensional time-dependent Schrödinger equation is given by
ö~I~(xt)
HcP(x t)
=
ih
=
~
(1)
t0).
(2)
The exponent is an evolution operator and can be
expanded as a polynomial series
-
—15
.
.
If the Hamiltonian is independent of time, this
can be solved formally to yield
CP(x, t)
dependent Schrödinger equation thus lies in the
ability to operate repeatedly with the Hamiltonian operator on the function P(x, t). The coefficients a~are Bessel functions [5,61. The upper
limit of the summation is determined by the
N
eHt~~~
=
~ a~P~(—iHt/h),
(3)
condition
10
The initial wavepacket is created according to
[1,2,81
~j~( ~,
0)
=
(
j.~ x) iP~(x,
E1),
(4)
where ~(x) is the transition dipole moment associated with the spectroscopic transition. The
wavepacket is then propagated using the above
scheme. The autocorrelation function, F~t),defined by
.
F(t)
where P~(x) are modified complex Chebychev
polynomials [5,6]. The key to solving the time
t
=
f
~*(x
t0) ~(x,
t) dx,
0
is computed at each time step.
(5)
G. G. Balint-Kurti et al.
/
Total energy absorption cross-section for the photodissociation of a diatomic molecule
The total absorption cross-section is then given
by the real part of the Fourier transform of this
autocorrelation function [3,41
2irv
+~
(6)
ffTOt(v)
Re~
eEt~F(t) dt
3ce~h 0
}
f
=
291
DATA RMIN/0.21D0/
PH000095
DATA RDAMP/4.ODO/
PH000096
DATA TTIME/320.ODO/
PH000098
NX is the number of grid points (see later).
NRECT is the number of time steps in the propa-
~,
gation.
where E
=
E1
+ hv. Using this approach the absorption lineshape is obtained over all photon
energies from a single computation of the molecular dynamics.
As the wavepacket progresses with time, from
small internuclear separations near the equilibrium position of the ground state electronic state
to larger separations, it inevitably approaches the
edge of the grid. For technical reasons, in order
to avoid “aliasing” [71,the wavepacket must be
annihilated or absorbed before it reaches the
edge of the grid. This task is accomplished by
using a complex absorbing potential of the form
1”complex
=
—
iAIIX\
kL/
(7)
—
where L is the length of the absorbing region
situated at the edge of the grid [9,10].
2. Program description
The program is self-contained and requires no
data input. The constants which define a particular problem are set in “DATA” and “PARAMETER” statements, or are specified in user supplied subroutines. These are discussed below. The
photodissociation of HC1 via the ~11
+ transition is used as an example. The variable parameters are chosen to represent this molecule.
—
~,
2.1. Data statements to be altered for new problems
ZMA and ZMB are the masses of the A and B
fragments, respectively.
RO is the equilibrium distance between the two
atoms in atomic units. This quantity is used only
as a typical measure of distance. All distances are
given in atomic units.
IVS is the quantum number of the initial vibrational state.
RLENGTH is the grid length expressed in units
of RO.
RMIN is the smallest internuclear distance or the
starting point of the grid expressed in units of
RO.
RDAMP is the starting point of the damping
region on the grid expressed in units of RO. This
region extends from RDAMP to the end of the
grid.
TTIME
is the total time of the propagation in
atomic units.
The program is set up to calculate the total
absorption cross-section as a function of energy
and write this data to file. Additional quantities
may also be evaluated after each step in the time
evolution. These are controlled by relevant parameters:
DATA ICHEX/1/
PH00011O
DATA INORM/ 1 /
PH0001 15
DATA IAUTOC/1/
PH00012O
If ICHEX 1 the energy or expectation value of
the Hamiltonian is computed at each time step
and written to a program specified file.
If ICHEX 0 the energy is not computed.
If INORM 1 the “norm” of the wavepacket,
N(t), defined by
=
=
PARAMETER
(NX 256, NRECT 40)
DATA ZMA, ZMB
/1836.9822D0, 64621.91975D0/
DATA RO/2.408558D0/
DATA IVS/0/
DATA RLENGTH/5.ODO/
=
=
=
PH000049
PH000089
PH000092
PH000093
PH000094
N(t)
=
f
+00
1*(x, t) P(x, t) dx,
(8)
0
is computed at each time step and written to a
program specified file.
G. G. Balint-Kurti et a!.
292
/
Total energy absorption cross-section for the photodissociation of a diatomic molecule
If INORM 0 the “norm” of the wavepacket is
not computed.
If IAUTOC 1 the absolute value of the autocorrelation function
(9)
SCRV1, SCRV2, SCRV3 and SCRV4 are analogous one-dimensional work arrays of length NX.
DZ is the calculated spatial increment on the
grid.
CZERO is the complex representation of zero as
defined in the program.
is computed at each time step and written to a
program specified file.
If IAUTOC 0 the absolute value of the auto-
C0O(I) will contain the values of the desired vibrational wavefunction evaluated at each grid
point X1.
=
=
P(t)
=
I
F(t)
=
correlation function is not computed.
Output to the standard output stream (file 6)
may be controlled with the IWRIT parameter.
DATA IWRIT/1/
PH0001O5
With IWRIT set to 0, the output will echo the
variables set by the user. Increasing the value to 1
will produce additional intermediate output which
may be useful for debugging purposes.
At large interatomic separations on the grid a
complex damping potential is used to absorb the
wavepacket. The magnitude of this potential is
specified by
DATA ZMIN/
—
0.3346D0/
PH000667
where —ZMIN is the A parameter in eq. (7).
The user is referred to refs. [9,10] for optimum
choice of damping potential to suit a specific
problem. If N(t) is evaluated (set INORM equal
to 1), then the user can observe the effect of the
complex damping potential on the wavepacket as
it is propagated along the grid. N(t) should fall to
zero as the wavepacket enters and is then consumed within the damping region.
2.2. Ground state wavefunction subroutine
The ground state vibrational wavefunction is
calculated in the subroutine “GROUND”.
CALL GROUND (NX,ZMA,ZMB,
ZMU,ZLZ,IVS,SCRV1,SCRM1,
SCRV2,SCRV3,SCRV4,SCRM2,
VMAX,RO,X0,DZ,CZERO,C00,
IWRIT)
2.3. Potential energy curve subroutines
“GROUND” requires the ground state potential energy curve evaluated at each grid point.
This information is provided by the subroutine
SUBROUTINE GSTATE(R, Vi)
PH0006O8
where R is the internuclear separation and Vi is
the value of the ground state potential calculated
by the subroutine at this separation. This self
contained subroutine must be supplied by the
user.
A similar subroutine to calculate the excited
potential energy at each grid point must also be
supplied.
SUBROUTINE ESTATE(R, V2)
PH000585
The arguments are the analogues of those in
“GSTATE”.
3.4. Transition dipole moment subroutine
The user must supply a self-contained subroutine to evaluate the transition dipole moment for
the spectroscopic transition at specified values of
the internuclear separation.
SUBROUTINE DIPMOM(R, DIP)
PH000772
This is called with a particular value of R
and returns the value of the transition dipole
moment in the variable DIP.
=
PH000164
SCRM1 and SCRM2 are NX by NX work arrays
required to calculate the bound vibrational wavefunction.
2.5 Testing convergence
In order to test the reliability of results it is
necessary to test convergence with respect to
/
G. G. Balint-Kurti et al.
Total energy absorption cross-section for the photodissociation of a diatomic molecule
several parameters. The first of these is the number of grid points.
PARAMETER(NX
=
256,...)
PH000049
A test run with the value of NX doubled should
be performed to check consistency. NX must be
a power of 2.
The “grid length” is defined by
293
2.6. Program output
The main part of the program output is the
computed absorption cross-section which is currently written to file 7 (IUNITA). The energy
resolution of the cross-section is defined in the
following way. If ~t is the length of each time
step given by
TITIME
ZLZ
=
RLENGTH*RO
PH000139
and the start of the grid by
X0
=
RMIN * RO
(10)
NRECT’
then the energy span of the spectrum (z~E)is
such that
~t
=
PH00015O
The two parameters are set in terms of multiples
of RO which is the equilibrium distance. A test
run should be made with a decreased value of
RMIN and also (separately) with an increased
value of RLENGTH.
The Chebychev expansion procedure used for
evaluating the time evolution operator [5,61 requires that a maximum value for the potential be
defined. This is done with the statement
VMAX 10.D0/HTEV
PH000198
This line sets the maximum value of the potential
to 10.0 eV. The user should check that this value
is sufficiently high by increasing it until there is
no change in the results.
The total time of the propagation is determined by the TTIME parameter. This value needs
to be large enough for the fragmentation process
to complete. It is advisable to examine the absolute values of the autocorrelation function (set
IAUTOC equal to 1) when choosing a suitable
size for TT’IME. The final value of the autocorrelation function should be very small compared to
the initial one, otherwise TTIME should be in=
creased. In particular, problems may arise through
the failure of the complex absorbing potential to
fully absorb the wavepacket incident on it. This
may lead to both reflected or transmitted waves.
The reader should consult the two papers by
Vibók and Balint-Kurti [9,10] for the optimal
choices of parameters for the complex absorbing
potential.
h
(ii)
NRECT < h
~E
(12)
2 X TTIME
Note that h i in atomic units.
The energy resolution, ~E, can be made as
small as desired by padding out the autocorrelation function with zero values at large times. In
the present program a nominal 4096 time steps
are used for this purpose, though only the first 40
values of the autocorrelation function are actually
computed.
=
=
~iE
=
~
(13)
Any change to the value of ITIME will change
the energy scale in the output unless NRECT is
adjusted in the same proportion. NRECT is not
constrained to a power of 2.
The file numbers to which the cross section,
energy, autocorrelation and wavepacket norm
data are written are determined by
DATA IUNITA,IUNITC,IUNITF,
IUNITN/7, 8, 9, 10/
PH000i25
Acknowledgements
We are grateful to the SERC for the provision
of funds to purchase the Meiko i860 computer on
which this program was developed. S.P. Mort
294
G.G. Balint-Kurti et al.
/
Total energy absorption cross-section for the photodissociation of a diatomic molecule
thanks the SERC for a studendship. We also
thank R.N. Dixon for useful discussions.
References
[1] G.G. Balint-Kurti, R.N. Dixon and C.C. Marston, J.
Chem. Soc. Faraday Trans. 86 (1990) 1741.
[2] E.J. Heller, J. Chem. Phys. 68 (1978) 2066.
[3] CC. Marston and G.G. Balint-Kurti, J. Chem. Phys. 91
(1990) 3571.
[4] G.G. Balint-Kurti, CL. Ward and CC. Marston, Comput. Phys. Commun 67 (1991) 285.
[5] H. Tal-Ezer and R. Kosioff, J. Chem. Phys. 81(1984)
3967.
[6] R. Kosloff, J. Phys. Chem. 92 (1988) 2087.
[7] W.H. Press, B.P. Fiannery, S.A. Teukolsky and W.T.
Vetterling, Numerical Recipes (Cambridge Univ. Press,
1987).
[8] R.M.V. Krishna and RD. Coalson, Chem. Phys. 120
(1988) 327.
[9] A. Vibók and G.G. Balint-Kurti, J. Chem. Phys. 96 (1992)
7615.
[10] A. Vibók and G.G. Balint-Kurti, J. Phys. Chem., to be
published.
G.G. Balint-Kurti et al.
/
Total energy absorption cross-section for thephotodissociation of a diatomic molecule
TEST RUN OUTPUT
All quantities used in the program are in au
(unless otherwise stated).
The energy absorption cross sections are output in
Angstroms squared and the photon energy is given in
cm-i
Number of grid points in lD grid (NX) = 256
Range of grid (ZLZ) = 12.04279000
Starting point for grid (X0) = 0.50579718
End point of grid = 12.54858718
Spatial increment (DZ) =
0.04704215
Reduced nass of AB (ZMU) = 1786.20640467
Vibrational state selected (IVS) = 0
Number of points for wavefunction evaluation (NEND) = 246
Corresponding grid length (ZFGH) = 11.57236852
Eigenvalue of ground vibrational state (EIG) = -0.16293500
Damping to start at grid point * (NDAI4P) 194
Damping region Starts at 9.63197398
Minimum momentum on grid (ZKMIN) =
-66.78250798
Increment in momentum space (1)1(Z) =
0.04704215
Maximum kinetic energy on grid (TMAX) =
1.24842889
Number of time steps for propagation (NREC~) = 40
Total time over which wavepacket is propagated (ITIME) = 320.00000
Length of each time step (ITrIME)= 8.00000000
Initial autocorrelation
0.14514183, 0.00000000
Argument for Bessel functions (ALPHA) = 6.46367432
Number of terms in Chebychev expansion (NTERNS) = 28
Midpoint of energy range (E0) =
0.80795929
Scaling factor (ESC) =
0.80795929
EOSC (E0/ESC) =
1 .00000000
Phase factor (CPF) = 0.98375603,-0.17951067
Energy before time propagation
3*
=
0.01944248, 0.00000000
Absorption cross section (Fl’ analysis)
3*
Total number of time steps computed = 40
Length of each time step = 8.0000000000000000
Total time represented on FT grid = 32768.000000000000
Maximum energy (cm-i) represented on FF grid = 172375.516066500600
Maximum absorption cross section (Ang.sqrd) = 0.03685085
Corresponding photon energy (cm-i) = 65092.6656
Total absorption cross section data
Photon energy (cm-i) vs cross section (Ang.sqrd.)
63367.22708546898
63409.31095169616
63451.39481792332
63493.47868415050
63535. 56255037768
63577.64641660485
63619.73028283202
63661.81414905918
63703.89801528636
3.5072053876433314E-002
3. 5156840695574065E-002
3.52396603483l4363E-002
3 .5320501474248589E-002
3. 5399353063334235E-002
3.5476204458183398E-002
3. 555 1045356247736E-002
3.5623865811896797E-002
3. 5694656238388821E-002
295
296
G.G. Balint-Kurti eta!.
63745.98188151354
63788.06574774071
63830.14961396788
63872.23348019506
63914.31734642223
63956.40121264940
63998.48507887658
64040.56894510375
64082.65281133093
64124.73667755809
64166.82054378527
64208 .90441001244
64250.98827623962
64293.07214246679
64335. 15600869396
64377.23987492113
64419.32374114831
64461 .40760737548
64503.49147360266
64545.57533982983
64587.65920605699
64629.74307228417
64671.82693851135
64713.91080473852
64755.99467096569
64798.07853719287
64840.16240342003
64882.24626964721
64924.33013587438
64966.41400210156
65008.49786832873
65050.58173455589
65092.66560078307
65134.74946701025
65176.83333323742
65218.91719946459
65261 .00106569177
65303 .08493191894
65345.16879814611
65387.25266437329
65429.33653060046
65471.42039682763
65513.50426305480
65555.58812928197
65597.67199550915
65639.75586173632
65681.83972796350
65723.92359419067
65766.00746041784
65808.09132664501
65850.17519287219
65892 .25905909936
65934.34292532653
65976 .42679 155371
66018 .5 1065778088
66060.59452400805
66102.67839023523
/
Total energy absorption cross-section for the photodissociation of a diatomic molecule
3. 5763407409733904E-002
3.58301 10462449147E-002
3. 5894756897205171E-002
3. 5957338580364179E-002
3 .60l7847745408754E-002
3.6076276994261809E-002
3. 6132619298497037E-002
3. 6i86868000440048E-002
3. 62390l68l4159974E-002
3.628905982635 1694E-002
3. 633699i497i08492E-002
3. 6382806660585396E-002
3.6426500525553 192E-002
3 .6468068675843346E-002
3. 6507507070684057E-002
3.65448i2044927537E-002
3.6579980309i68945E-002
3. 6613008949757227E-002
3 .6643895428698364E-002
3.66726375834511 15E-002
3. 66992336266l6i66E-002
3. 6723682145518764E-002
3.6745982i0l68551lE-002
3. 6766132830216042E-002
3.6784 134039049989E-002
3 .6799985808129904E-002
3. 68i3688588460895E-002
3. 682524320i06778iE-002
3. 6834650835850240E-002
3. 6841913050337086E-002
3.684703 1768340206E-002
3. 6850009278509364E-002
3. 6850848232788377E-002
3. 684955l644773895E-002
3. 6846122887977700E-002
3. 6840565693993424E-002
3. 6832884150568859E-002
3 .6823082699584786E-002
3.68ill66l3494l599E-002
3 .6797139600354613E-002
3.678 1008587059502E-002
3.6762778931428854E-002
3. 67424568i250i038E-002
3. 672004874942289’TE-002
3. 6695561598807075E-002
3. 6669002552005843E-002
3.6640379132302239E-002
3. 660969919202020iE-002
3. 6576970909554869E-002
3. 6542202786324654E-002
3.6505403643646177E-002
3.64665826 19533949E-002
3 .6425749165425771E-002
3 .6382913042835793E-002
3 .6338084319936397E-002
3.629 127336807053 1E-002
3.62424908581961 14E-002
66144.76225646240
66186.84612268957
66228.92998891675
66271 .01385514392
66313.09772137110
66355. 18158759827
66397.26545382544
66439.34932005260
66481 .43318627978
66523.51705250696
66565.60091873414
66607.68478496130
66649.76865118848
66691 .85251741564
66733.93638364282
66776.02024986999
66818.10411609718
66860.18798232434
66902.27184855151
66944.35571477868
66986.43958100586
67028.52344723303
67070.60731346020
67112.69117968738
67154.77504591455
67196.85891214172
67238.94277836890
3. 6i9l747757263922E-002
3. 6139055324530683E-002
3. 6084425107808772E-002
3. 6027868939654296E-002
3 .5969398933495038E-002
3. 5909027479699963E-002
3 .584676724i59i949E-002
3.5782631 151405279E-002
3. 5716632406189705E-002
3. 5648784463662507E-002
3. 5579101038010652E-002
3. 5507596095644238E-002
3.54342838509033 1OE-002
3. 5359178761719484E-002
3. 5282295525234292E-002
3. 5203649073375823E-002
3.5l2325456839542iE-002
3.5041 127398366088E-002
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