J . Phys. Chem. 1992, 96, 8712-8719
8712
ARTICLES Parametrization of Complex Absorbing Potentlais for Time-Dependent Quantum Dynamics
A.Vib6kt and G. G. Balint-Kurti* School of Chemistry, University of Bristol, Bristol, BS8 ITS,U.K. (Received: January 28, 1992)
Five different forms of complex absorbing potentials are examined and compared. Such potentials are needed to absorb wavepackets near the edges of grids in time-dependent quantum dynamical calculations. The extent to which the different potentials transmit or reflect an incident wavepacket is quantified, and optimal potential parameters to minimize both the reflection and transmission for each type of potential are derived. A rigorously derived scaling procedure, which permits the derivation of optimal potential parameters for use with any chosen mass or kinetic energy from those optimized for different conditions, is described. Tables are also presented which permit the immediate selection of the parameters for an absorbing potential of a particular form so as to allow a preselected (very small) degree of transmitted plus reflected probability to be attained. It is always desirable to devote a minimal region to the absorbing potential, while at the same time effectively absorbing all of the wavepacket and neither transmitting nor reflecting any of it. The tables presented here enable the user to easily select the potential parameters he will require to attain these goals.
1. Introduction
Time-dependent quantum dynamics is based upon the solution of the time-dependent SchrGdinger equation. This equation is finding increasing favor in discussing problems related to molecular scattering and photodissociation theory.'" The attraction of using the time dependent as opposed to the time-independent Schrridinger equation stems from its conceptual simplicity and the straightforward pictorial way in which the wavepackets involved may be used to illustrate the results of the computations and their relationship to the physical quantities under investigationn2V6A major advantage of the time-dependent approach lies in the fact that a single computation of the dynamics provides information concerning the computed cross sections over a wide range of energies, often even over the whole energy range of interest.2,7-9 The numerical techniques used to solve the time-dependent Schrridinger equation are normally based on the use of a grid representation of the time-evolving wavepackets and were first introduced into this field through the work of McCullough and Wyatt.Io They were later used in conjunction with Fast Fourier Transform techniques by Feit et al." K o s l ~ f f ' ~has - l ~used these methods extensively and has introduced the Chebychev expansion method, which we use here, for solving the time-dependent Schrridinger equation. In all such grid-based methods, whether or not they use Fourier transforms to perform the necessary differentiations of the wavepacket, a problem arises near the edges of the grid. It is not normally possible to use sufficiently large grids so that the wavepackets remain entirely on the grids for the entire length of time over which the propagation is performed. It is therefore necessary to introduce a procedure through which the wavepacket is artificially attenuated as it approaches the edges of the grid. Such procedures may involve a physical damping of the wavepacket at each time step by multiplying it by a function which decreases from 1.0 to zero near the grid edge,*V6 or, alternatively, a complex absorbing potential of the form V ( x ) = -if(x) may be introduced near the edge of the grid. Such potentials have been widely used for many purposes in scattering Address correspondence to this author. 'Permanent address: Institute of Theoretical Physics, Kossuth Lajos University, 4010 Debrecen, Hungary.
0022-365419212096-8712.$03.00/0
theory and are generally called 'optical potentials".16J7 The advantage of this latter approach lies in the fact that the damping is effectively applied in a more gradual manner as the computation of a single time step normally requires many operations of the Hamiltonian (and therefore of the potential) upon the wavepacket. Clearly it is desirable that the minimum possible length of grid be devoted to the absorbing potential. It is also essential that the computed results should not be distorted by the creation of reflected waves or by allowing any part of the wavepacket to be transmitted through the absorbing region and thereby hitting the edge of the grid. If Fourier transform techniques are being used to evaluate the action of the Hamiltonian operator on the wavepacket then the result of a part of the wavepacket reaching the edge of the grid will be the creation of an amplitudeat the opposite edge (this is referred to as aliasing in Fourier transfonn theory).18 Neuhauser and BearI9have previously examined a 'linear" form of absorbing potential and have derived conditions which satisfactory complex absorbing potentials must obey. This form of the potential has also been recently discussed by Childz0and in a previous paper by ourselves.2' An alternative approach to the problem of eliminating wavepacket amplitude near a grid boundary has been proposed by McCurdy and Stroud.z2 In this approach the coordinate themselves become complex near the edge of the grid. Another method of handling this problem is that of Heather and Metiuz3in which an inner and outer region are defined and the wavepacket amplitude is transferred gradually from the inner region grid to the outer region. The purpose of the present paper is to examine various different forms of complex absorbing potential and to establish the optimal parameters which permit the use of minimal absorbing lengths to attain a preselected (minimal) degree of reflection or transmission. The theory of the calculations is briefly reviewed in section 2. In particular this section outlines how potential parameters which have been optimized for a particular value of the reduced mass of a colliding pair of particles and for a particular energy may be correctly scaled to give optimal parameters for any other values of these physical quantities. In section 3 the results of the computations for five different types of complex absorbing potential are presented, largely in graphical form, and are qualitatively compared and discussed. Section 4 presents and discusses tables which enable the reader to readily determine the parameters 0 1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8713
Complex Absorbing Potentials needed for the specification of an absorbing potential to meet his individual requirements, while the final section summarizes the calculations. 2. Theory In this section, a scaling theory, presented in a previous paper,17 is briefly reviewed. This theory enables us to utilize optimal potential parameters, derived using a particular reduced collision mass and energy, to determine optimal parameters for any other desired combinations of masses and energies. The Schrixlinger equation may be written as
Introducing the following parametrization mI = tm
E, = aE and defining the new functions VI(X) = aV(fix)
(2)
*I(X) = B*(fixBx)
(3)
where
8 2 = ae
(4) it has been shown2' that an equation of identical form to the original one is recovered:
This shows that if an optimal complex potential has been obtained for a particular mass and energy combination, and for a particular range over which it acts, then this result may be transformed so as to apply to any other combination of mass and energy. Suppose for instance that the optimal potential was Vl(x) and that it operated over a length L I ,and that it has been found to be optimal for a mass ml and an energy E,. Then the optimal potential for mass m2 and energy E2 would be V2(x) = V,(@x)E2/EI,where fi = (E2m2/Elm1)1/2 = AI/A2 (where A is the wavelength associated with the relative motion of the scattering partners), and the new length over which it is taken to act would be L2 = L l / & We may summarize these results in a symmetric form: relationship of damping length for two different situations:
-LI= - L2 AI
(6)
A2
relationship of potential for two different situations:
In the case of the exponential function the normalization constant N was found numerically to be 13.22. The premultipliers, A,, are found by minimizing the sum of the reflection and transmission (see below). Note that if we take the range of the potential to be over the assigned damping length, L,then eq 13 takes the form:
In order to test the different potentials they were defined as beiig nonzero over some finite "damping" length, L,in the central portion of a one-dimensional grid, 0 Ix IL. A wavepacket was then "fired" at the potential from the left hand or negative x side. The motion of the wavepacket with time was determined by solving the time-dependent Schrixlinger equation using the Chebychev expansion technique as proposed by Kosloff and c ~ - w o r k e r s . l ~ - ~ ~ The wavepackets of necessity contain a range of energies, and numerical techniques were therefore used to propagate the wavepackets. In a previous work2I we have discussed the nth-order semiclassical Jeffreys-Wentzel-Kramers-Brillouin (JWKB) solution to the equivalent time-independent problem. The results obtained using the semiclassical JWKB approach are very similar to those obtained from the numerical solution of the time-dependent wavepacket quantum dynamics. Similar work has also been published by Child.20 The Chebeychev expansion technique has been developed for real potentials, and its validity for complex potentials such as those used in the present work has not been theoretically proved. In the calculations reported here the complex potentials (or the A, parameters of eqs 8-12) were scaled, according to the criteria required by the Chebychev expansion method,I2using the parameters determined from the maximum kinetic energy which can be accommodated by the spatial grid used. By monitoring the wavepacket as a function of time it was possible to clearly determine the portion which is reflected and that which is transmitted. The wavepackets used are of the form (at t = 0) +(x) = exp[-a(x - x ~ ) exp[ikx] ~]
The wave vector of the motion, k, is related to the kinetic energy by E = k2h2/(2m)
3. Qualitative Results In this section the reflecting and absorbing characteristics of five different forms of complex absorbing potentials are examined. The forms are I: 11:
V(n) = -~A,[R] V(R) = -iA2[Y2R2]
111:
V(X) = -iA3[2R3]
IV: V:
V($ = -iA5[N exp(-2/n)]
V ( X )= -iA4[5/2X"]
linear
(8)
quadratic
(9)
cubic
(10)
quartic exponential
(11) (12)
where R = x/L. The portions of the potential functions in square brackets have been normalized in the sense that their integral over the range 0 IR I1 yields unity, i.e.
(16)
It was found to be necessary to use narrower (spatial) wavepackets for higher energies because when this was not done the reflected wavepacket became spatially enmeshed with the incident one making it impossible to distinguish between them. The "a" or width parameter (eq 15) was computed according to the relationship a ( E ) = 1.0368E
(7)
(15)
(17)
Figure 1 shows the wavepacket and its time development for a kinetic energy of 0.1 au using a reduced collision mass of m = 1836.18 au. The initial wavepacket is shown in the first panel of the figure. The vertical scale in the figure is logarithmic, and the Gaussian wavepacket looks somewhat unfamiliar on this account. The "linear" complex potential is nonzero only in the central region of the grid between the limits 0 Ix I2.0 au. The second panel of the figure (Figure 1B) shows the wavepacket a t a later time when its leading edge has already passed through the complex absorbing potential, while the third panel shows the reflected and transmitted wavepackets at an even later time. In order to form a quantitative measure of the total reflected and transmitted probabilities we define the following quantities: reflection:
R(E,t) = joI+(x,t)12 -m dx
(18)
transmission:
T(E,t) = iml+(x,r)l2dx
(19)
As defined in the above equations, both the reflection and the transmission are time-dependent quantities. After a sufficient
ViMk and Balint-Kurti
8714 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
' 7 TABLE I: Values of Refkction and Trrasmbsion for Different Fo- of Conapkx A k r b i a g Potmtil (E+ 8-12)'
0
potential form linear quadratic cubic quartic exponential
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4
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v)
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0 -2 -I
W
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-3
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c W
Y
-5
%
5
-6
-40
-30
-20
-10
0
:0
20
30
40
I
DISTANCE / A U
-1
I
i
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$ 0 $
W
0
W
3 a
W
::
2
55
B
A
-I -3
1 3
i
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-61
-30 -30
-20 -20
-10 -10
0 0
10 10
20 20
30 30
40 40
C
W -I
-1-
v)
W
0,
reflection and transmission. It should be noted, however, that even the very best choice of parameters for such a potential, acting only over a finite range, cannot altogether eliminate the incident wavepacket. From the figure we see that the amplitude of the wavepacket has been reduced by about 2 orders of magnitude (note again the logarithmic scale) from its original value. Table I provides a preliminary survey of the comparative efficacies of the five different forms of complex absorbing potential listed above (eqs 8-12). The same initial wavepacket, energy (0.1 au), mass, and damping length (2.0 au) was used in each case, and the quoted results all correspond to those obtained with the corresponding A, parameters optimized to yield a minimum value of the sum of the reflection plus transmission. The interesting point to note is that there is a substantial difference between the capabilities of the different potentials. In particular, the linear form of the complex potential is not capable of destroying the initial wavepacket nearly as thoroughly as any of the other forms. In our previous paperz1we have given a semiclassical JWKB analysis of reflection and transmission of waves by a complex potential. It was shown there that for a potential of form V(x) = +Ax" there is no reflection at all, to (n - 1)th order in JWKB theory, from the leading edge of the potential. More specifically, this means that a linear potential (V(x) = -ih)c a w a reflection in first-order JWKB theory while the other potentials considered do not. Figure 2 shows the reflection and transmission as a function of wavepacket kinetic energy for the linear form of the complex absorbing potential (eq 8). The AI parameter was optimized so as to minimize the sum of the reflection and transmission at a kinetic energy of 0.1 au. The details of the wavepacket parametem, mass,and damping length are given in the caption to the figure. The energy dependence of the reflection (Figure 2A) is the inverse of that of the transmission (Figure 2B). Thus, the reflection decreases with increasing energy while the transmission increases. Thus at low energies the complex potential is liable to create spurious reflected flux while at high energies the danger is that flux will be transmitted through it. The bottom panel of the figure (Figure 2C) shows the sum of the reflection and transmission. This quantity is lowest at E = 0.1 au, which is the energy at which the A I parameters of the potential were optimized. At the two extremes of the energy range examined (0.01 and 1.0 au), the smallest damping length (1.3 au) does not lead to a sufficiently complete annihilation of the wavepacket. The situation is improved by increasing the damping length, but this is seen to be much more effective at the low energy end than at high energies. Figure 3 presents the reflection and transmission as a function of wavepacket kinetic energy for the cubic (eq 10) form of the complex absorbing potential. The A3 parameter was optimized so as to minimize the sum of the reflection and transmission at a kinetic energy of 0.1 au. The details of the wavepacket parameters, mass, and damping length are the same as those used for the calculations shown in Figure 2. The qualitative form of the energy dependence is the same for the cubic potential (Figure 3) as for the linear one (Figure 2); however, the absolute values of the numbers are in general very much smaller for the cubic case. Thus, the cubic complex absorbing potential gives a value for the reflection plus transmission at E = 0.1 au which is between a factor of lo4 and lo" (depending on damping length) smaller than that given by the linear potential. It is also very noticeable
DISTANCE / A . U
"
0.179E-11 0.5848-12
-+I -40 -40
u V
transmission 0.505E-05 0.378E-10 0.831E-10
"The parameters of the potentials have been optimized to minimize the reflection plus transmission (eqs 18 and 19) at an energy of 0.1 au and using a damping length of 2.0 au. *In all tables the notation E-n is equivalent to XlO-"
4: - 4
d
reflection 0.280E-04b 0.361E-09 0.398E-09 0.348E-13 0.320E-11
J
-2-
-40
-30
-20
-10
0
10
20
30
40
DISTANCE / A . U .
' ' F
1. One-dimensional wavepacket propagation under the influence of a complex potential. The a h l u t e value of the (complex) wavepacket is plotted against distance. The complex potential is nonzero from x = 0 to x = 2.0 au. Note that the wavepacket amplitude (vertical scale) is logarithmic. (A) The initial wavepacket; (e) wavepacket as it starts to penetrate the complex potential region ( t = 1400 au); (C) reflected and transmitted wavepackets at a later time (r 3600 au). Potential parameters are found by minimizing the total reflected plus transmitted probabilities.
time however (see Figure lC), when the transmitted wavepacket has left the absorbing region, and the reflected packet has also 'bounced back" out of this region, they become constant as a function of time. These constant values are used when referring to the reflection and transmission in the tables and figures below. The AI parameter of the "linear" complex absorbing potential used in Figure 1 was optimized so as to minimize the sum of the
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8715
Complex Absorbing Potentials
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linear damping funODon
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Energy/a u
Energyla u
Figure 2. Reflection and transmission (eqs 18 and 19) as a function of kinetic energy of wavepacket for linear form of complex absorbing potential (eq 8). Curves for three different damping lengths are shown. The A, parameter in the potential was optimized for a kinetic energy of 0.1 au. A mass of m = 1836.18 au was used, and the wavepacket a parameter is given by eq 17. (A) Reflection as a function of energy; (B) transmission as a function of energy; (C) reflection plus transmission. Symbols: *, damping length 1.3 au; 0,damping length 2.7 au; 0 , damping length 4.0 au.
:h
cubic dampinp fundon
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cubic damping function
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2
001
001
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01
02
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02
01
Energy/a u
04
07
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01
02
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Energy/a u
Figure 3. Reflection and transmission (eqs 18 and 19) as a function of kinetic energy of wavepacket for cubic form of complex absorbing potential (eq 10). Curves for three different damping lengths are shown. The A , parameter in the potential was optimized for a kinetic energy of 0.1 au. A mass of m = 1836.18 au was used, and the wavepacket a parameter is given by eq 17. (A) Reflection as a function of energy; (B) transmission as a function of energy; (C) reflection plus transmission. Symbols: *, damping length 1.3 au; 0,damping length 2.7 au; 0 , damping length 4.0 au.
that increasing the damping length leads to a much more marked improvement of the degree of reflection in the case of the cubic potential (Figure 3, A and C) than in that of the linear potential (Figure 2, A and C). Figure 4 shows the best attainable values of the reflection plus transmission for different reduced values of the damping length using four different types of complex absorbing potentials (eqs 8-1 1). Figure 5 gives the same information for the exponential form of the complex potential. The calculations are performed at an energy of 0.1 au using a wavepacket whose details are given in the figure captions. From Figure 4 it can be seen that there is a lower limit of about 10” to the reflection plus transmission which is obtainable using a linear potential, no matter how large the damping length. The other forms of potential all permit much smaller values of this quantity to be attained, provided one is able or willing to use a sufficiently large damping length. As the damping length is reduced all the potentials perform more poorly
(a good performance equates to a small value of the reflection plus transmission) and at very small damping lengths the linear potential’s performance deteriorates at the slowcat pace. Thus, for very small damping lengths (not shown in the Figures 4 and 5 ) the linear potential is in fact the best; its optimal performance, however, is not very good in this situation. In our previous paper?’ we introduced a new exponential form of complex damping potential (q12). This form has the theoretical advantage that it is predicted not to lead to any reflection at all from its leading edge to all orders of JWKB theory?’ Figure 5 indeed shows that extremely low values of the reflection plus transmission are attainable using this potential, as long as sufficiently long damping lengths are used. The factor 2 in the exponent of the exponential potential (eq 12) was chosen 50 that the end of the damping region (X = 1) mrresponds to the inflection point of the potential function. A possible disadvantage of the exponential form of the complex potential lies in the fact that it
8716 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
TABLE 11: Optimized Parameters for Linear Complex Absorbing Potential (L8)" L/A A , / E R+T L/A AI/E R+T 1.00 2.200 2.87E-03 8.50 0.513 1.02E-05 1.50 1.810 1.27E-03 9.00 0.490 8.13E-06 2.00 1.500 6.46E-04 9.50 0.466 6.55E-06 2.50 1.260 3.70E-04 10.00 0.444 5.34E-06 3.00 1.080 2.31E-04 10.50 0.423 4.42846 3.50 0.946 1.55E-04 11.00 0.405 3.72E-06 4.00 0.845 1.09E-04 11.50 0.389 3.19E-06 4.50 0.770 7.96E-05 12.00 0.378 2.78E-06 5.00 0.714 5.96E-05 12.50 0.369 2.46E-06 5.50 0.672 4.54E-05 13.00 0.363 2.20E-06 6.00 0.638 3.50E-05 13.50 0.359 1.98E-06 6.50 0.610 2.71E-05 14.00 0.354 1.77E-06 7.00 0.585 2.11E-05 14.50 0.345 1.57E-06 7.50 0.561 1.65E-05 15.00 0.329 1.35E-06 8.00 0.537 1.29E-05
quartic damping function
Y
cubic damping function quadratic damping function
0
ViWk and Balint-Kurti
"The first column, L/A, lists the values of the reduced damping length. The second column, A I / E ,presents the optimized values of the potential premultiplier (eq 8) in reduced form. The last column, R T, gives the value of the reflection plus transmission obtained with these potential parameters. The parameters are determined by minimizing the values of the reflection plus transmission given in the final column.
+
\
0
2
4
6
6
10
12
14
16
Damping IengthNVave length
Figure 4. Minimized values of reflection plus transmission (eqs 18 and 19) obtained with four different types of complex absorbing potentials as a function of reduced damping length (Llh). The A, parameters of the potentials (see eqs 8-1 1) are optimized for every calculation so as to minimize the value of the reflection plus transmission. The figure shows the best values attainable for the reflection plus transmission for a particular form of complex potential and a particular damping length. The wavepacket kinetic energy was E = 0.1 au, the reduced collision mass was m = 1836.18 au, and the wavepacket a parameter is given by eq 17.
has a very shallow slope near its onset. It then grows very rapidly so that its integral (analogous to eq 13) gives a value of A 5 / 2 . This means that care should be taken when this potential is used to use a sufficiently small step size in the spatial grid so as not to introduce numerical inaccuracies, because of the relatively large curvature of the potential as it increases sharply toward the end of the damping region. All calculations must in any case be checked for convergence with respect to the grid spacing, and in our present calculations the curvature of the absorbing potential did not cause any noticeable problem. In every case we chose a grid spacing such that there were at least 15 grid points within the damping region, L. Several other analytic forms of exponential potential meet the criteria for zero reflection2' (e.g., v(f) = -i[N exp(-BIX)]), where n can take any value. If n > 1 then the sharp curvature of the potential is accentuated even further, while if n C 1 then the performance of the potential deterioratesto approach that of the Rm type functions, the smaller the value of n the worse the optimal value of the reflection plus transmission becomes. To some extent the other power types of potential also share this disadvantage of the exponential potential in that the higher the power o f f the lower the slope of the potential at it9 commencement and the more steeply it rises towards the end of the damping region. Figures 6 and 7 present the optimised potential premultipliers, A,, (see eqs 8-12) as a function of the reduced damping length. The premultipliers are presented in reduced form, A,/E. The results reported in Figures 6 and 7 correspond to the same calculations as reported in Figures 4 and 5. The captions of these latter figures list the details of the wavepacket used. These two figures, when taken in combination with the previous two (Figures 4 and 5 ) and the scaling relationships (eqs 6 and 7), in fact enable one to immediately determine the damping length and the potential energy parameters which will enable one to attain a preselected
TABLE III: Opti+zed Parameters for Quadratic Complex Absorbii Potential (Eq 9)" LIA A,/E R+T LlA A,IE R+T 1 .oo 2.750 1.29E-03 8.50 1.040 5.31E-11 1.50 2.640 7.14E-05 9.00 0.986 3.91E-11 2.510 6.14E-06 9.50 0.949 2.91E-11 2.00 2.50 2.380 7.74E-07 10.00 0.903 2.18E-11 2.230 1.368-07 10.50 0.870 1.64E-11 3.00 11.00 0.840 1.25E-11 3.50 2.090 3.16E-08 4.00 1.940 9.38E-09 11.50 0.813 9.49E-12 1.800 3.40E-09 12.00 0.788 7.29E-12 4.50 5.00 1.670 1.46E-09 12.50 0.764 5.66E-12 5.50 1.550 7.20E-10 13.00 0.741 4.488-12 6.00 1.440 3.97E-10 13.50 0.718 3.62E-12 6.50 1.340 2.39E-10 14.00 0.697 3.OOE-12 7.00 1.250 1.54E-10 14.50 0.678 2.58E-12 7.50 1.170 1.04E-10 15.00 0.663 2.32E-12 8.00 1.100 7.35E-11 "The first column, L/A, lists the values of the reduced damping length. The second column, A2/E,presents the optimized values of the potential premultiplier (eq 9) in reduced form. The last column, R + T, gives the value of the reflection plus transmission obtained with these potential parameters. The parameters are determined by minimizing the values of the reflection plus transmission given in the final column.
degree of wavepacket attenuation. To illustrate the necessary procedure, suppose that we have decided that we wish to use a cubic type of complex potential and that we need to attain a level of reflection plus transmission of or smaller. From Figure 4 we scan across at the "-10" line to find where it intersects the "cubic" curve and see that a reduce damping length of 7.5 is required. We now go over to Figure 6 and scan vertically upwards along the line corresponding to a reduced damping length of 7.5 until we again hit the cubic curve. The intersect occurs at about A 3 / E = 1.35. Thus, if we know the mass and energy involved in our proposed calculation, we may compute from this the corresponding wavelength (A = h/(2mE)Ij2). The damping length required is then given by L = 7.5A and the potential premultiplier A3 = 1.35E. 4. Procedure for Determination of Optimal Parameters for a Complex Absorbing Potential for a Specific Application The information given in Figures 4-7 would in fact permit the reader to determine the optimal potential parameters the reader requires for any desired application. They are, however, not given in the most useful form, as visual interpolation from a pair of graphs is not in general the most convenient way to obtain the
The Journal of Physical Chemistry, Vol. 96, No. 22, 1992 8717
Complex Absorbing Potentials
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Damping length/wave length Figure 5. Minimized values of reflection plus transmission (eqs 18 and 19) obtained using exponential complex absorbing potential (eq 12) as a function of r e d u d damping length (LIX). The AS potential parameter (see eq 12) is optimized for every calculation so as to minimize the value of the reflection plus transmission. The figure shows the best values attainable for the reflection plus transmission for a particular value of the damping length. The wavepacket kinetic energy was E = 0.1 au, the reduced collision mass was m = 1836.18 au, and the wavepacket a parameter is given by eq 17. TABLE IV Optimized Parameters for Cubic Complex Absorbing Potential (Eq 10)" L/X AJE R+T L/h AJE R+T 1.00 2.950 1.09E-02 8.50 1.070 3.95E-11 1.50 2.810 1.69E-04 9.00 1.020 2.39E-11 2.00 2.650 6.62E-06 9.50 0.983 1.46E-11 2.50 2.480 5.44847 10.00 0.948 9.05E-12 3.00 2.310 8.07E-08 10.50 0.916 5.79E-12 3.50 2.140 1.89E-08 11.00 0.885 3.86E-12 4.00 1.980 6.23E-09 11.50 0.856 2.70E-12 4.50 1.830 2.63E-09 12.00 0.827 1.99E-12 5.00 1.690 1.32E-09 12.50 0.798 1.54E-32 5.50 1.560 7.44E-10 13.00 0.770 1.24E-12 6.00 1.450 4.46E-10 13.50 0.743 1.02E-12 6.50 1.350 2.75E-10 14.00 0.720 8.32E-13 7.00 1.260 1.71E-10 14.50 0.702 6.47E-13 7.50 1.190 1.06E-10 15.00 0.694 4.54E-13 8.00 1.120 6.51E-11
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1
16
Damping lengthWave length F i 6. Optimal potential premultipliers,A, obtained for four different types of complex absorbing potentials (see eqs 8-1 1) as a function of reduced damping length ( L l h ) . The wavepacket kinetic energy was E = 0.1 au, the reduced collision mass was m = 1836.18 au, and the wavepacket (Y parameter is given by eq 17. TABLE V Optimized Parameters for Quartic Potential (Eq 1 l ) O Llh AAIE R+T LIX 1.00 2.580 1.26E-02 8.50 1.50 2.500 5.21E-04 9.00 2.00 2.440 2.55E-05 9.50 2.50 2.390 1.49E-06 10.00 3.00 2.350 1.03E-07 10.50 3.50 2.310 8.64E49 11.00 4.00 2.260 8.66E-10 11.50 4.50 2.200 1.04E-10 12.00 5.00 2.170 1.5OE-11 12.50 5.50 2.120 2.57E-12 13.00 6.00 2.060 5.2OE-13 13.50 6.50 2.000 1.24E-13 14.00 7.00 1.990 3.44E-14 14.50 7.50 1.860 l.lOE-14 15.00 8.00 1.790 4.02E-3-15
Complex Absorbing AJE 1.720 1.650 1.580 1.510 1.450 1.390 1.330 1.280 1.240 1.200 1.160 1.130 1.100 1.063
R+T 1.66E-3-15 7.64E-16 3.88E-16 2.15E-16 1.28E-16 8.08E-17 5.36E-17 3.69E-17 2.61E-17 1.87E-17 1.34E-17 9.62E-18 6.81E-18 4.73E-18
"The first column, L/A, lists the values of the reduced damping length. The second column, A,/E, presents the optimized valuw of the potential premultiplier (eq 11) in reduced form. The last column, R T, gives the value of the reflection plus transmission obtained with these potential parameters. The parameters are determined by minimizing the values of the reflection plus transmission given in the final column.
+
"The first column, L/A, lists the values of the reduced damping length. The second column, AJE, presents the optimized values of the potential premultiplier (eq 10) in reduced form. The last column, R T, gives the value of the reflection plus transmission obtained with these potential parameters. The parameters are determined by minimizing the values of the reflection plus transmission given in the final column.
+
values of required quantities. In this section we therefore present essentially the same information in the form of five tables (Tables 11-VI), one table for each of the types of potential considered here. The tables are reasonable self-explanatory and their use should be straightforward. As an example, if one wishes to use an exponential complex absorbing potential (eq 12), one should consult Table VI. Suppose we require that the initial wavepacket be annihilate so that its
residual norm (reflection plus transmission) decreases to lO-'O. We scan down the final column, (R + T), of the table until the number in this column is less than our required value. We find the line
This line gives us the optimal parameters for a complex absorbing potential of exponential form which will meet the specified requirements. As indicated in the previous section, the reader should then determine the kinetic energy corresponding to his proposed
8718 The Journal of Physical Chemistry, Vol. 96, No. 22, 1992
a range of energies as possible. In such cases it is first necessary to establish the range of energies contained within the wavepacket to be used in the computation. This can be done by Fourier transforming the wavepacket and finding the values of the wavenumber (k) at which the square of the Fourier transform of the wavepacket corresponds to 1% (say) of its maximum value. This establishes a lower (E,) and an upper (E2)limit to the energy range within the wavepacket. Figures 2C and 3C show clearly that a complex potential which has been optimized for a particular energy is not optimal for use with either a lower or a higher energy. The figures show clearly that in order to attain the best overall performance we need to choose the complex absorbing potential parameters to be optimal for some intermediate energy within the range of energies contained within the wavepacket. Figures 2c and 3c are both log-log plots. In both cases the potential parameters have been optimized for an energy of 0.1 au. As the energy is changed away from the value for which the parameters were optimized the transmission reflection vs energy graph shows a linear behavior on the log-log plot in both directions. Both graphs show a similar behavior though with different slopes. Based on these two graphs it is possible to obtain a formula for the best energy value (E*) to use for choosing the optimal absorbing potential parameters. Thus, if the wavepacket spans a range of energies from E, to E2 then the optimal potential parameters should be chosen to correspond to an energy
exponential damping function 2 0-
‘t,
+
\
1.3-
1.24
1.14
E* = log-’ [0.6242 log (E,) + 0.3759 log (E&]
1.0-
0
ViMk and Baht-Kurti
9 ~ 0
,
I
2
,
,
,
4
, 6
1
8
’
10
8
8
’
12
16
14
Damping length/wave length Figure 7. Optimal potential premultipliers, A,, obtained using exponential complex absorbing potential (eq 12) as a function of reduced damping length (LJX). The wavepacket kinetic energy was E = 0.1 au, the reduced collision mass was m = 1836.18 au, and the wavepacket a parameter is given by eq 17. TABLE VI: Optimized Parameters for Exponential Complex Absorbing Potential (Eq 12)’
LIX
A5/E
R+T
L/X
AS/E
R+T
1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00
1.920 1.900 1.880 1.870 1.850 1.830 1.820 1.810 1.790 1.780 1.770 1.750 1.740 1.720 1.700
1.03E-02 1.12E-03 l.llE-04 1.05E-05 9.95E-07 9.78E-08 1.03E-08 1.17E-09 1.48E-10 2.08E-11 3.27E-12 5.78E-13 1.14E-13 2.49E-14 5.98E-15
8.50 9.00 9.50 10.00 10.50 11.00 11-50 12.00 12.50 13.00 13.50 14.00 14.50 15.00
1.680 1.650 1.630 1.590 1.550 1.510 1.470 1.420 1.370 1.310 1.260 1.210
1.56E-15 4.40E-16 1.32E-16 4.15E-17 1.378-17 4.64E-18 1.62E-18 5.83E-19 2.15E-19 8.19E-20 3.26E-20 1.38E-20 6.35E-21 3.27E-21
1.160
1.120
“The first column, L/X,lists the values of the reduced damping length. The second column, A 5 / E ,presents the optimized values of the potential premultiplier (q12) in reduced form. The last column, R + T, gives the value of the reflection plus transmission obtained with these potential parameters. The parameters are determined by minimizing the values of the reflection plus transmission given in the final column.
calculation (or the range of kinetic energies which will be involved). From this and a knowledge of the reduced collision mass the reader may compute the effective wavelength (A = h / ( 2 n 1 E ) ~ /The ~). damping length needed is then given by L = 5SA, and the potential premultiplicative parameter (see eq 12) is given by 4 = 1.780E. In many, but not all, practical cases of interest, the problem of choosing an optimal complex absorbing potential is accentuated by the fact that the wavepacket used in the calculation contains a broad range of energies. This situation may ariae either through the nature of the physical problem or by design because we wish to use a single computation to gain information about as broad
(20) Surprisingly,this formula is valid for both the linear and the cubic complex absorbing potentials. Until such time as we have completed a similar study for the other potentials studied in this work we recommend the use of this formula for all of the potentials. 5. Summary The paper has presented a survey of five different forms of complex absorbing potentials intended for use with time-dependent quantum dynamical calculations. An important scaling procedure, first derived in a previous paper,2’ has been summarized (eqs 6 and 7). This scaling procedure is extremely important as it permits optimized complex potential parameters derived using a particular collision energy and collision mass to be ‘scaled” so as to provide the optimized potential parameters applicable to a different mass and energy combination. As a consequenceof this, if the potential parameters are reported in a “reduced” dimensionless form they are applicable to all energy and mass combinations. Section 3 discusses the capabilities of the different forms of complex absorbing potential (eqs 8-12). Quantities referred to as reflection and transmission are defined (eqs 18 and 19) which at large times measure the residual norm of the wavepacket, i.e., that portion of the wavepacket which has not been absorbed by the complex potential. The purpose of using a complex absorbing potential is to “gobble up” the incident wavepacket. The better the absorbing potential therefore the smaller the reflection plus transmission, R T, will be. The optimal potential parameters are therefore determined by minimizing this quantity. The most important results of the paper are presented in Tables 11-VI. These tables list the optimized potential parameters, in reduced form, for all five different types of complex absorbing potential considered. Section 4 discusses the usc of the tables, and a reading of this section, together with the tables, will enable a reader to choose the optimal absorbing potential parameters for any proposed calculation. Prior to the present work Neuhauser and Bear19 have made recommendations as to the optimal parameters to be used in conjunction with liiear complex absorbing potentials (eq 8). They have derived a condition which the complex potential parameters must obey. In the notation of the present paper their condition is
+
An examination of our Table I shows that all our recommended
parameters comply with the Neuhauser-Bear conditions.
J. Phys. Chem. 1992,96, 8719-8728
8719
(8) Kulander, K. C.; Heller, E. J. J . Chem. Phys. 1978, 69, 2439. (9) Balint-Kurti, G. G.; Dixon, R. N.; Marston, C. C. J . Chem. Soc., Faraday Trans. 1990,86, 174 1. (10) McCullough, A. E.; Wyatt, R. E. J . Chem. Phys. 1969, 51, 1253; 1971, 54, 3592. (1 1) Feit, M. D.; Fleck, J. A., Jr.; Steiger, A. J . Compur. Phys. 1982,47, 412. (12) Kosloff, R. J . Phys. Chem. 1988, 92, 2087. (13) Kosloff, D.; Kosloff, R.J . Compur. Phys. 1983, 52, 35. (14) Kosloff, R.; Kosloff, D. J . Compur. Phys. 1986, 63, 363. (15) Tal-Ezer, H.; Kosloff, R. J . Chem. Phys. 1984, 81, 3967. (16) Mott, N. F.; Massey, H. S.W. The Theory of Atomic Collisions; Oxford University Press: London, 1965. (17) Leforestier, C.; Wyatt, R. E. J . Chem. Phys. 1983, 78, 2334. (18) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: Cambridge. U.K., 1986. (19) Neuhauser, D.; Baer, M. J . Chem. Phys. 1989, 90, 4351. (20) Child, M. S. Mol. Phys. 1991, 72, 89. (21) ViMk, A.; Balint-Kurti, G. G. J . Chem. Phys. 1992, 96, 7615. (22) McCurdy, C. W.; Stroud, C. K. Compur. Phys. Commun. 1991,63, 323. (23) Heather, R.;Metiu, H. Chem. Phys. h i t . 1985, 118, 558.
Acknowledgment. We are grateful to the SERC for a grant of computer time on the Rutherford Laboratory computers and to the Computational Science Initiative of the SFRC for funds used to purchase of a Meiko MK860 computer. A.V. thanks the Royal Society and the Soros foundation for a fellowship which funded her study visit to the U.K. We thank Professor R. N. Dixon for helpful discussions.
References and Notes (1) Kulander, K. C., Ed. Time-DependentMethods for Quantum Dynamics. Compur. Phys. Commun. 1991,63. (2) Dixon. R. N.; Marston, C. C.; Balint-Kurti, G. C. J. Chem. Phys. 1990, 93, 6520. (3) Sun, Y.;Judson, R. S.;Kouri, D. J. J. Chem. Phys. 1989, 90, 241. (4) Neuhauser, D.; Baer, M. J. Chem. Phys. 1989, 91,4651. (5) Neuhauser, D.; Judson, R. S.; Jaffe, R. L.; Baer, M.; Kouri, D. J. Chem. Phys. Lcrr. 1991,176, 546. (6) Gray, S.K.; Wozny, C. E. J . Chem. Phys., in press. (7) Heller, E. J. J . Chem. Phys. 1978, 68, 2066.
Vibrational Activation in the Radlationless Decay of the S,, S,, T,, and T, States of Aromatic Thlones in Solution: Red Edge Effects Marian Szymanskit and Ronald P . Steer* Department of Chemistry, University of Saskatchewan, Saskatoon, Saskatchewan, Canada S7N 0 WO (Received: July 10, 1992)
The quantum yields of phosphorescence, +p, of six aromatic thiones (xanthione, xanthioned,, benzopyranthione, pyranthione, tetramethylindanethione, and 4-bromotetramethylindanethione)have been measured accurately as a function of excitation wavelength and temperature in two solvents (3-methylpentaneand perfluoro-l,3-dimethylcyclohexane).The values of +p decrease on the red edge of the S2 SOabsorption band, particularly for pyranthione which has the shortest S2lifetime. So and the finite rate of vibrational activation required This red edge effect is attributed to competition between fast Syo before S, can decay nonradiatively via SV, Sp. The quantum yield of phosphorescence gradually increases with increasing excitation wavelength over the SI S, absorption region as the fraction of triplet formed directly by excitation in the underlying TI So system increases. Excitation on the red edge of the TI So absorption results in a dramatic increase in +r This second red edge effect is interpreted in terms of the nonequilibrium dynamic behavior of the coupled TI-S,-T2 system. Finite triplet sublevel spin depolarization rates may play an important role in determining the difference in the dynamics of the system when excited initially to TI compared with SI.
-
-
-
--
+
1. Introduction Aromatic thiones are proving to be useful models for studying the photophysics and photochemistry of polyatomic molecules in solution.14 These compounds often exhibit a set of well-resolved UV-vis absorptions consisting of strong, symmetry-allowed transitions to S2and higher singlet states in the near-W, a weak electric-dipole forbidden transition to SIin the visible, and a still weaker but nevertheless readily observable transition to TI in the red or near-IR. Because these electronic transitions are relatively widely spaced, selective photoexcitation of the thiones to TI, SI, S2, and sometimes higher states is often possible5even in solution at room temperature when the individual absorption bands are relatively broad. Prompt fluomcence from Sz,bllphosphorescence from T1,12-20triplet-triplet transient a b s o r p t i ~ n , E-type ~.~~~~~ thermally activated delayed fluorescence (TDF) from S,,23and P-type delayed fluorescence from S24 have all been observed in these compounds and constitute a set of convenient, sensitive methods for following the rates of their excited-state relaxation. We have previously elucidated many of the photophysical and photochemical properties of the thiones in solution by measuring their emission quantum yields, excited-state lifetimes, and photolysis rata as a function of thione and solvent structure, quencher 'On leave from the Institute of Physics, A. Mickiewin University, Poznan,
Poland.
To whom correspondence should be addressed. 0022-3654/92/2096-87 19$03.00/0
concentration and structure, and temperature. In particular, we5J7-20and others'2-16,21,22 have shown that the T I state is produced subsequent to excitation to any of the higher singlet states accessible in the near-W. The quantum yield of triplet production is a weak function of the nature of the state initially populated and tends to decrease slightly as one proceeds to excite progressively higher states. Direct radiationless decay of S2 to s,, accounts for a small part of the inefficiency in TI production when S, (n L 2 ) is excited initially5-6.8*g in thiones possessing H atoms in their 'fl"-positions. However, a number of apparently contradictory observations have been reported concerning triplet decay times and phosphorescence quantum yields when these thiones are excited to S1and TI under a variety of conditions. First, there is some discrepancy in the values reported for the SI T, intersystem crossing efficiency, Values of & = 1 have been measured or inferred in several previous s t ~ d i e s . ~ ~ . ~ ~ , ~ ~ However, we have measured the absolute quantum yields of phosphorescence, at selected excitation wavelengths5and find that they are somewhat smaller when excitation occurs in the SI S, absorption band system than when excitation occurs directly to TI. The latter observation could imply that $ J < ~ 1 ~ and that S1and T I therefore decay at least in part by independent paths. However, the logarithm of the ratio of the intensity of TDF to that of phosphorescence varies linearly with T I and yields an activation energy which is approximately q u a l to the solventrelaxed SI-Tl electronic energy gap.23 Moreover, the lifetimes,
--
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+,,
0 1992 American Chemical Society
