J. Phys. Chem. 1985, 89, 4201-4206
4201
Box Quantization and Resonance Determlnatlon: The Multichannel Case R. Lefebvre Laboratoire de Photophysique Mol&culaire.t Campus d’Orsay, 91 405 Orsay. France (Received: March 19, 1985; In Final Form: May 23, 1985)
The procedure based on box quantization to determine the resonance energies of a quantum system is extended to the multichannel case. This method exploits the behavior of real energies as the size of the box is changed. Several variants (either analytical or purely numerical) are examined. Various tests are made, which include the case of a rotationally predissociating van der Waals complex with a realistic potential and a model problem involving one closed channel and two interacting open channels for which there exists an exact analytical solution. The avoided crossings observed by previous investigators for the one-channel case are of quite general Occurrence and allow for an accurate extraction of the resonance parameters even in cases where no analytical discussion based on the box quantization conditions appears to be possible.
I. Introduction There is an ever increasing variety of methods to calculate the energies of the resonant states which are met in problems of molecular dynamics.’ Although great accuracy is guaranteed by several of these procedures a new technical development can give additional insight into the aspects of this fascinating concept. Such a development may also prove to be of particular efficiency in some specific situations. We consider in this paper the procedure consisting in exploiting the quantized energies obtained by enclosing a system in a box. The presence of a resonant state in some energy range manifests itself in a characteristic way in the dependence of these energies with respect to the size of the box. The analysis of the one-channel case served as a guide to Hazi and Taylor2 for extracting the information contained in the stabilization graphs obtained by adding more and more diffuse functions in a square integrable basis. The expressions they obtained were used subsequently with success by Truhlar3 and Maier et al.4 in some typical one-channel situations. Since it is technically possible to obtain in a multichannel case the eigenenergies corresponding to the boundary conditions imposed by the presence of a box,+’ it is interesting to extend this type of analysis to systems of a more complicated structure than those envisaged previously. An aspect of the method which may be of value for problems involving a large number of channels is that only real energies are involved. It has been shown recently in a somewhat different context” (analytical continuation of stabilization graphs obtained when the energies are plotted as a function of a scaling parameter) that a real energy approach may be superior to a direct complex energy approach in some situations with many channels. The same comment applies to the present developments. In the first part of the paper, problems involving one open channel, either alone or coupled to closed channels, are examined. The presence of the closed channels is only felt through the phase shift in the asymptotic form of the component wave function for the open channel so that the same analysis is valid for all these cases. Two kinds of approaches are possible. The information can be extracted from the behavior of the energies in regions of minimum slope (this is the original approach of Hazi and Taylor2). It is also possible to analyze the energies in regions of maximum slope, that is, close to the avoided crossings (or “anticrossings”) which are characteristic of the plots of energies vs. the size of the box. A new expression for the width is derived from this analysis. The anticrossings can also be exploited numerically by viewing them as reflecting the interaction between a constrained bound state and the complementary continuum, as explained by Maier et al.4 Tests of these various procedures are made and compared for a previously considered model potential and for the case of the ArH2 van der Waals complex described with a realistic potential. When the description of a system requires more than one open channel we are facing a much more complicated situation. Laboratoire du CNRS associt H I’UniversitE Paris-Sud.
However, we can still expect anticrossings to be present for the reasons which explain them in situations with a single open channel. A quantization condition depends on both wavenumbers and phases. Fulfilling the condition for two successive levels in a region of slow variation of the phases is accounted for essentially by a change in wavenumbers. In a region of rapid variation of the phases (Le., close to a resonance) wavenumbers and phases are both contributing and we can therefore expect the levels to be closer because less change is now required in wavenumbers. We will attempt to make this analysis explicit for the case of one closed channel and two open channels. We will also examine the possibility of deriving the resonance energies (position and width) directly from the anticrossings. It will be shown on an example with two interacting open channels which has an exact analytical solution that this procedure is a very promising one. 11. The Single Open Channel Case
We recall briefly the analysis given by Hazi and Taylor: which applies for the case of a single open channel coupled to an arbitrary (including zero) number of closed channels. Assuming that the potential goes to zero faster than r-I, the component wave function of the open channel has the asymptotic form fi0(r)
- sin (kr +
7
+ 6)
(1)
where 7 is a slowly varying background phase shift and 6 is given by (we adopt here the notation E,, = E , - ir for a resonance energy)
r
tan 6 = -E - E,
(2)
where E, is the resonance position, which, for a Feshbach resonance, includes a shift with respect to the bound-state energy. The quantization condition is kR
+ 7 + 6 = mr,
m integer
(3)
where R is the size of the box. Ignoring the dependence of 7 vs. energy one derives easily2 from (2) and (3) the relation (1) For a recent account see: “Resonances in Electron-Molecule Scattering, van der Waals Complexes and Reactive Chemical Dynamics”; Truhlar, D. G., Ed.; American Chemical Society: Washington, DC, 1984; ACS Symp. Ser. No. 263. (2) Hazi, A. U.; Taylor, H. S.Phys. Reu. A 1970, 1 , 1109. (3) Truhlar, D. G. Chem. Phys. Leff. 1974, 26, 377. (4) Maier, C. H.; Cederbaum, L. S.; Domcke, W. J . Phys. E 1980, 13, L119. ( 5 ) Gordon, R. G. J . Chem. Phys. 1969, 51, 14. (6) Johnson, B. R. J . Chem. Phys. 1976, 64,4984. (7) Atabek, 0.; Lefebvre, R. Chem. Phys. 1980, 52, 199. (8) McCurdy, C. W.; McNutt, J. F. Chem. Phys. Lett. 1983,94, 306. For related work see: Simons, J. Chem. Phys. Leff.1981, 75,2465. Thompson, T. C.; Truhlar, D. G. Chem. Phys. Leff. 1982, 92, 71. Thirumalai, D.; Thompson, T. C.; Truhlar, D. G. J . Chem. Phys. 1984,80, 5864. Isaacson, A. D.; Truhlar, D. G. Chem. Phys. Leff. 1984, 110, 130.
0022-3654185 12089-4201%01.50/0 0 1985 American Chemical Societv
4202
Lefebvre
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 dE - _ dR
r
-2E1i2 +R
( E -E,)'
(4)
+ r2
The formula shows that ldE/dRI is minimum for E close to E,, this being a condition reinforced as r goes to zero. For E = E, expression 4 can be inverted to give
build a bridge between expressions 13 and 14 as follows. If the resonance energy is far enough from the threshold, for a radial problem the energies of the quasi-continuum are approximately E,,
The density of states at E
- n2n2/R?
-
p =
2E,Ii2
(15)
E, is
R:/2r2n,
(16)
with n, = ( R , / ~ ) E , ~ / ~ Thus E, can be obtained as an energy of minimum slope and r can be evaluated from E,, the size of the box R and the slope. As discussed by Hazi and Taylor2 and T r ~ h l a r for , ~ a narrow resonance the second term in the denominator may dominate the first so that there is the even simpler expression for the width
El r = --dR E = E ,
(6)
E,lJ2
We turn now to a different type of procedure to extract the resonance parameters from the curves giving the energies as a function of R . Consider two successive levels corresponding to the fulfillment of the conditions k R + TJ- + 6.. = m-ir (7a) k+R We assume 7,
+
+ 6+ = m+r
TJ,
-
with m, = m-
TJ- so that ( k + - k - ) R + 6, - 6- =
x
+1
(7b)
(8)
We can understand the occurrence of the characteristic anticrossings (see ref 4 for examples) from this equation. Our discussion will be similar to that given by Friedman et al.9 with an extension leading to a new expression for the width. Assume that we are in the energy range where there is a resonance. We know that there exists the possibility to find two phase shifts which differ by x for a certain size of the box R,. For a narrow resonance we are therefore able to fulfill relation 8 with almost no change in wavenumbers. This is equivalent to stating that there are two close energies. More generally, in the region of closest approach of two energies (the anticrossing), we write
k-2 = E, - V; k+' = E, + V
(9)
so that (from eq 2)
6, - 6- = R - 2 tan-' r / V
(10)
where tan-' is the principal value of the inverse tangent. Combining eq 8 and 9 we can extract r, given by
r = V tan
[l/z(k+ - k-)R,]
(11)
where R, is the size of the box a t the anticrossing. All the parameters in this formula are available from the box energies. As a simple variant of this, for a narrow resonance we may write
k+ - k- = V/2E,'J2
(12)
so that we have for this case
r
= pR,/2E,1/2
(13)
Finally there is also the possibility to view, as done by Maier et al.? the anticrossings as being due to the coupling between a discrete state and a continuum state. This golden rule interpretation gives for the width the expression
r
where p
(14) is the density of the quasi-continuum. It is possible to =x
p
~
(9) Friedman, M.; Rabinovich, A.; Thieberger, R. J . Phys. B 1983, 16, L381.
(17)
Combining (16), (17), and (14) we recover (13). For all the estimates where use is made of an anticrossing, the real part of the resonance energy is calculated as the average of the two energies at the point of closest approach. The density is evaluated from this energy and the two adjacent ones, one above and one below the anticrossing. We turn now to tests of these various expressions on two types of examples: (i) single-channel model problems with a potential of the form Ug2e-'; (ii) a van der Waals complex requiring two channels, one closed, one open. For the one-channel cases, the real energies of the box quantization and the reference complex energies were obtained from the finite difference procedure proposed by Truhlar'O and extended later to complex integration paths." Multichannel results (either for real or complex energies) were all obtained from the Fox-Goodwin propagation technique with iterative matching.I2 (i) Single-Channel Model Problems. The lowest resonance state of the potential U,,r2e-' with U, = 15 (with the kinetic energy written - d 2 / d s ) is one of the favorite test cases for the resonance p r ~ b l e m . ~ JWe ~ consider also the cases with Uo = 20 (the resonance is narrower) and U, = 10 (the resonance is broader). Table I collects the results obtained from expressions 5 , 11, 13, and 14 for the resonance positions and the widths. The last column gives the resonance energies of the direct complex energy approach. There is an overall excellent agreement between all procedures, with a slight advantage to the stabilization procedure which deals efficiently with both narrow and broad resonances. Of particular interest for the developments given in the next section is the performance of the very simple expression 14 which uses only the splitting and the density of the quasi-continuum. ( i i ) A Two-Channel Example. Our two-channel example is the ArH2 van der Waals complex with the potential of Le Roy and Carley.I4 With the coupled equations written in the space-fixed frame of coordinates, the channels are labeled (uJ,l,J) where u is the vibrational quantum number of H2,j its rotational quantum number, 1 the relative angular momentum quantum number, and J the total angular momentum quantum number. Rotational predissociation (conversion of rotational energy into translational energy) in the first excited vibrational state of the diatomic corresponds to a coupling between the closed channel (1,2,2,0) and the open channel (1 ,O,O,O).Accurate determinations of the resonance energy are available.l5J6 Table I1 is organized like Table I. Three calculations are presented, the interchannel coupling being either that derived from the original potential or this coupling divided or multiplied by 2. This allows for a wide variation of the width. We note again the good achievements of the stabilization procedure and of the golden rule expression, particularly for the narrowest resonance. Both shifts and widths are well accounted for. It is interesting to note that the shifts and widths do not increase as fast as the square of the interchannel coupling and this non-Fermi behavior is well accounted for by the Fermi-like expression 14. This is not contradictory because (10) Truhlar, D. G. J . Compur. Phys. 1972, 10, 123. (11) Atabek, 0.; Lefebvre, R. Chem. Phys. Letr. 1981, 84, 233. (12) Atabek, 0.;Lefebvre, R. Phys. Reu. A 1980, 22, 1817. (13) b i n , R. A.; Bardsley, J. N.; Sukumar, C. V. J. Phys. B 1974, 7,2189. (14) Le Roy, R. .I. Carley, ; J. S.Adu. Chem. Phys. 1980, 42, 353. (1 5) Le Roy, R. J.; Corey, G. C.; Hutson, J. M. Faraday Discuss. Chem. SOC.1984, 73, 339. (16) Lefebvre, R. J . Phys. Chem. 1984, 88, 4839.
Box Quantization and Resonance Determination
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 4203
TABLE I: Resonance Energies (in au) for tbe Potential Ug2e-'"
UO
A
10
5.0798 -0.108 2 6.8528 -0.2543 (-1) 8.3707 -0.4536 (-2)
15 20
B
D
C
E
5.0756
-0.1050 6.8515 -0.2497 (-1) 8.3712 -0.4593 (-2)
-0.09488
-0.09537
-0.2454 (-1)
-0.2515 (-1)
-0.4576 (-2)
-0.4577 (-2)
V
R C
5.0809 -0.1 095 6.8528 -0.2555 (-1) 8.3706 -0.4550 (-2)
P
13.395
0.1787
0.9507
11.06
0.1078
0.6893
13.84
0.4374 (-1)
0.7615
"Kinetic energy operator written as -d2/dr2. The various procedures are A, stabilization method;* B and C, analysis of the anticrossing according to either eq 11 or eq 13; D, golden rule expression (eq 14); E, direct complex energy quantization. The upper number in each doublet is the real part of the resonance energy, the lower number the imaginary part. Methods B, C, and D yield the same real part which is the arithmetical average of the two energies at the anticrossing. Numbers in parentheses are powers of ten. The case with U, = 15 has already been analyzed in ref 4 with procedures A and D. The three columns on the right give the values of the parameters (in au) needed for the derivation of the widths given in columns B, C, and D, namely R, (a size of the box producing an anticrossing), V (half the energy interval at the anticrossing), and p (the density of states). TABLE 11: Resonance Energies (in c d ) for the Ar-H2 van der Waals Complex with the Potential of Le Roy and Carley'4" C c/2 2c
A
B
3 15.9773 -0.5500 (-1) 315.9185 -0.1396 (-1) 316.2302 -0.203 6
3 16.0063 -0.5812 (-1) 315.9205 -0.1462 (-1) 316.1852 -0.2127
C
D
E
-0.5748 (-1)
-0.5471 (-1)
-0.1458 (-1)
-0.1390 (-1)
-0.204 3
-0.1932
315.9739 -0.5487 (-1) 315.9181 -0.1394 (-1) 316.2142 -0.2022
"The Column headings have the same meaning as in Table I. Calculations labeled C / 2 and 2C are made with the interchannel coupling either divided by 2 or multiplied by 2.
the bound state implicitly present in the golden rule (14) is not the state of the closed channel, but a state "dressed" by the coupling to the continuum. Before leaving the single open channel class of models, it is useful to indicate the limitations of the box quantization procedure. In ref 4 it was asserted that the potential 15r2e-' has only one resonance. This is not correct since there is in fact a "string of resonances" associated to this potential," the next two members having the energies 9.66962 - i2.23575 (au) and 10.55456 i6.77810 (au). The widths are larger than the intervals in the quasi-continuum for a size of the box in the asymptotic region. One could expect some localization to occur for a bunch of states within an energy range of the order of the width. However, the localization will be comparatively much less pronouncedZthan in the case of a narrow.resonance, so that the resonance effect will be diluted and unexploitable. This is very clear on the graph given for this potential by Maier et aL4 This is to be contrasted with the result of a complex rotation which produces a rigorous localization of resonance wave functions, irrespective of their widths. As a spectacular example of the contrast, Figure 1 gives the quantized energies for the potential 3rZe-', for a box size in the range 60-60.5 au. In this range, for the quasi-continuum states 0.14 au to close in energy to the resonance state one has p-' be compared with a width r 0.43 au so that we have r > p-'. No sign of the presence of a resonance is visible on the real energy diagram (left panel). The stable complex energy with the stable real part visible on the energy diagram with complex rotation (right panel) is 1.947646 - i0.430805.
-
-
111. The Multiple Open Channel Case
The analysis of the single open channel case is based on the simple relation between a single phase shift and the resonance parameters. With several open channels, it does not appear to be possible to perform a similar analysis, except in particular cases. The first part of this section is devoted to stressing this (rather negative) aspect. Nonetheless, as shown later, it is still possible to rely upon purely numerical methods exploiting the anticrossings of the stabilization graphs. This will be demonstrated on a model problem with one closed channel and two interacting open channels for which an exact analytical solution exists. (a) Box Quantization Condition for the Multiple Open Channel Case. Let us assume that we set up our problem in a basis of (17) Korsch, H. J.; Laurent, H.; Mohlenkarnp, R. J . Phys. E 1982, I S , 1.
Atabek, 0.;Lefebvre, R., unpublished results.
-2.51
I
> 0
60 0
60 2
604
600
60 2
60 4
0.U
Figure 1. Box size effect on the spectrum of the potential 3r%-' for R in the range 60-60.5 au: (a) real eigenenergies; (b) real parts of complex energies obtained with a complex rotation of 0.3 rad.
distorted waves, that is, with no open channels of kets laE) behaving asymptotically as lim (r,qlaE) = -03
[ "1
rh2ka
112
+a(q) sin (kar + 11,)
(18)
+,(q) contains the information on the internal states in channel a of two entities separated by distance r a n d of reduced mass k. The channel functions are normalized to a delta function of energy.ls The Hamiltonian has been separated into a term Ho defining the channels and a perturbation H1responsible for interchannel coupling. A standard treatment of scattering theoryig leads for the asymptotic form of the multichannel wave function to
(18) Landau, L. D.; Lifshitz, F. M. "Quantum Mechanics"; Pergamon: London, 1965; Section 21. (19) Fano, U.; Prats, F. J . Natl. Acad. Sci. India 1963,A33, 553.
4204
Lefebvre
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
K being the reaction operator of definition D
The coefficients M,(E) are to be chosen in a way depending on the type of asymptotic behavior required for the wave function since advantage can be taken of the essential degeneracy of order no. The standing wave representation of the wave function is convenient for a discussion of box quantization. We look for the eigenvectors of the “on the energy shell” K matrix obeying the relation
of the root of an energy, while the matrix elements giving the coupling between continuums are dimensionless. Thus all the matrix elements of K given in this set of equations are dimensionless. The eigenvalues of the matrix associated to K cannot be expressed in terms of the resonance shift and width present in the resonance energy E,,, = E,
+A
-
(27)
iT
which are given respectively byZ2
r=
r l ( W f i I 1E)I2+ TI(uIHi12E)I2
(21)
(28) + T*)(~EIH,)~E)I~ A = -r2[((UIHi11E)(lEIHI12E)(2EIHllu)+ ( ~ l H l ( 2 . EX)
with Mb,(E) being the bth component of the eigenvector corresponding to the eigenvalue A., The asymptotic wave functions now take the form
Consider the case with ( lEIH112E)= 0 (no interaction between open channels). The eigenvalues of the K matrix are
~ E I K I ~M,,(E) E ) = A, M A E ) b
1
(2EIHiI 1 E )( 1 ElHi Iu)) /( 1 + r21(1EIHi12E)12)1 ( 2 9 )
Xi =
I(ulH1I 1E)I2 + I(4H,12E)12
; X2 = 0
E - E,
(30)
The eigenphases are given by d,(q) M,,(E) ( 1
+ r2Am2)lI2sin (kir + 7, + 6,)
tan 6 i = (22)
with tan 6, = -rA,. The 6,’s are the eigenphases. The M,,(E)’s can be scaled to produce the desired normalization. There are no such wave functions since we can choose any of the no eigenvalues A,. Box quantization implies that the wave function is zero for r = R, if R is the size of the box. This is possible by making appropriate combinations of the independent solutions. The condition to be fulfilled is c
[l
+ r2X2]’/2 sin (k,R + 7, + A),
= 0 (23)
This can only be satisfied for arbitrary values of q if no
I C , M,,(E) [ 1
+ r2A,2]
sin (k,R
+ 7, + X),
=0
(24)
01
This is a set of homogeneous linear equations which have a nontrivial solution if det IM,,(E) sin (k,R + va + X,)l = 0 (25) The next step in the analysis would be to write this condition for two successive energy levels (for instance in the vicinity of an observed anticrossing) and to attempt to derive the resonance parameters from this pair of relations. One may convince oneself that this is a hopeless task by looking at the case of a closed channel coupled to two interacting open channels. In configuration interaction language,20in the limit of energy-independent couplings the relevant elements of the K matrix in the neighborhood of a bound state of energy E, belonging to the closed channel areZi
=
rl(ulHiJ1E)12 + * ( ~ l H i l 2 E ) l ~ E - E,
--.
r I
tan 6, = 0
E - E,’
Since the phases are now related in a simple way to the resonance parameters, it is possible to analyze an anticrossing in a way similar to that of section 11. The quantization condition derived from ( 2 5 ) for two channels with one of the eigenphases (6,) equal to zero is M i l ( E )M2,(E) sin (klR + v i ) sin (k,R+ u2 + 6,) M I 2 ( E )M 2 1 ( E sin ) ( k i R + q I + 6,) sin(k2R + v 2 ) = 0 (32) Assuming that the two energies of an anticrossing can be written E..=E,-K E+=E,+V (33) and ignoring the (weak) energy dependence of the M,,(E)’s, a combination of the quantization conditions for levels E- and E+ leads to 1 1
+ -rV cot (ki-R + V I - ) +
’ P
V
cot (kz-R
+ 72-)
1-
-
r
- cot (kl+R + T I + ) V
’ P
1
+ V cot (k2+R + 02+)
(34)
where kl* and q l t are the wavenumbers and phase shifts in channel 1 at energies Et, while kZ* and v2* are the wavenumbers and phase shifts in channel 2 at energies E+ All quantities entering this relation except r are available either from an inspection of an anticrossing ( V , kit, kZt, R ) or from a study of the uncoupled channels (qla, q2+). One may also note that if the two channels are governed by the same potential the relation is trivially satisfied and does not yield r. However, this is a very simple case which can be treated by going back to eq 32. Two kinds of bound levels are expected with either kR
+ 7 + 6,
= mr
m integer
(35a)
or
(2ElKllE) = (2EIH,IlE) (2ElK12E) =
D) + ( ~ E I HEI-I(uIHil1E) E,
(2E111:!u)( u p ,!ZE) E - E,
In an E-normalized scheme*O the matrix elements expressing coupling of a discrete state to a continuum state have the dimension (20) Fano, U. Phys. Reu. 1961, 124, 1866. (21) Atabek, 0.;Lefebvre, R. Chern. Phys. 1981, 55, 395
kR
+ 7 = m’r
m’integer
(35b)
since 6, = 0. No channel index is necessary for k or 7 in the assumption of the same potential for the two open channels. This corresponds to the possibility of combining the two channels in two different ways, to define a new channel decoupled from the bound state and another one producing a resonance of width P. An example belonging to this category will be obtained as a particular case of a model to be developed now and which will (22) Beswick, J. A,; Lefebvre, R. Moi. Phys. 1975, 29, 1611.
Box Quantization and Resonance Determination
The Journal of Physical Chemistry, Vol. 89, No. 20, 1985 4205
serve as a test for a purely numerical approach (Le., embodied in expression 14 for the width). ( b ) A Soluble Model with a Closed Channel Coupled to Two Interacting Open Channels. This soluble model results from two choices: (i) choosing the same potential for the two interacting open channels; this circumstance leads for the width to a Fermi-like expression with the bound state coupled to a displaced continuum state; (ii) choosing the potentials so that there is an analytical form for this bound-free coupling. Consider condition (ii). It is well-known23 that the shift and width of a resonance associated with harmonic oscillator states coupled to the continuum states of a linear potential can be given, in the limit of small interchannel coupling, an analytical form. Of particular importance in the present context is the bound-free overlap which for the zero point level is, as shown by Stueckelberg,23related to an Airy function according to
For the zero-point level of a harmonic potential coupled to two interacting continuums associated with identical linear potentials, the width is therefore
ro= 27x21 ; ( I ~ : ) I ~ I ~ = ~ ~ + ~
where (OIE) is the overlap aril’ it.-de given i n eq 36. The shift is affecting this amplitude through The choice of the turning point r,. A different view of the probleili is obtained by looking at the poles of the resolvent in a configuration interaction approach. Let I1E) and ( 2 E ) represent the eigenkets of the first two equations in the set 37 when the off-diagonal elements of the potential matrix are ignored. The continuum-continuum coupling is (IE(HI12E’)= [6(E - E’)
(42)
The poles are obtained as solutions of the equation E = E,
(DIE) = I
(41)
+ (ulR(z)lu)
(43)
,
withz4 R ( z ) = PHIP (36) This expression is written in units such that energies are in cm-’ while the kinetic energy operator is -dz/dr2. 0 is the zero-point energy in cm-’ and y is lF11/3 with F the slope of the linear potential in ~ m - ~ / *r,.is the turning point of the continuum state of energy E and re the equilibrium position of the oscillator both in cm’/z. The width of the resonance associated with the zero-point level of the harmonic potential is, in the Condon approximation, proportional to the squared modulus of this overlap.23 Let us now look at the consequences of using identical potentials for the two open channels (which of course differ by some quantum number of the internal states). We assume the off-diagonal elements of the potential matrix to be real and independent of r. A first view of the problem is obtained from a manipulation of the three coupled equations
with the following choice for the potential matrix of elements Up&)
6
u(r)=
U,(r)
t (38)
U,(r) is a repulsive potential associated with channels 1 and 2, U,(r) is an attractive potential associated with channel 3 while l and x are real constants with the dimension of an energy. Subtracting the second equation from the first decouples the channel $z(r) - Gl(r). Consideration of the third equation and of the sum of the first two equations leads to the two coupled equations
+
with qs(r)= 2-’/2($1(r) tJ2(r)).This shows that the continuum state degenerate with a bound state of the attractive potential is a state of the original repulsive potential pushed down by -,$. In the Fermi limit the width of a resonance is therefore
r, = 2
7 ~( u~ I 1E ) I ~ I ~ = ~ , + ~
(40)
(23) Stueckelberg, E. C. G. Phys. Rev. 1932,42, 518. Sink, M. L.; Bandrauk, A. D.Chem. Phys. 1978, 33, 205.
+ PHIQ(z - QHQ)-’QHlP
(44)
with P = lu) (ul and Q associated to the complementary continuum space. Introducing
GQ = Q ( z - Q f f Q Y ’ Q
(45)
the four matrix elements of GQ necessary for the evaluation of (ulR(z)lu) are found to be, from standard techniques: ( Z - E’36(E’- E”) (1EqGQJlE”)= ( ~ E I G Q ~ ~=E ” ) ( Z - E’?’ - F2 ( IEIGQIZE”) = ( 2 E 1 G ~ l l E ” )=
@(E’- E”) (Z
- E!’)’ - l2
The equation yielding the poles (eq 37) takes the form
This is an equation very similar to the result of Fano” for a bound state coupled to a single continuum, except for the factor of 2 and the energy shift. In the limit of a smooth dependence of (ulE) with respect to energy in a range of the order of the width (this condition has to be verified a posteriori), one can calculate from (48) by letting z to go to E io+ both a resonance shift and a width, the latter quantity being in fact that given by eq 40. The energy shift occurring in this relation is linked to the singularity of the coupling leading to so-called persistent effeckZ5 The parameters used for testing the ability of box quantization to give the resonance energies are as follows: Attractive potential: that of a harmonic oscillator with a zero-point energy 0 of 1000 cm-I for a reduced mass of 7.5 au. Repulsive potentials: linear potentials of slope F = 65000 cm-I/A intersecting the harmonic potential at the equilibrium distance and becoming constant at p = 0.269 91 1 A (exactly 0.18 cm1i2in the units described after eq 36). Such plateaus make it possible to define asymptotic wavenumbers while being of no effect on the resonance (the right turning point of the zero-point level is -0.047 A). The coupling x (cf. eq 38) is taken equal to 50 cm-I. The coupling 4 has been given the values (in cm-I) 0, 100, 1000, and 2000. Three kinds of calculations of the width are performed (a) according to the analytical formula (eq 36 and 41), (p) using direct Siegert quantization with complex rotation (in which case the linear potentials can be used throughout the integration range), and (y) from the Fermi-like expression (14) which makes use of an anticrossing. The localization of the anticrossings is performed by first calculating the quasi-continuous spectrum as a function of box size. This spectrum is obtained by the finite difference methodlo
+
(24) Mower, L. Phys. Rev. 1966, 142, 199. (25) Van Hove, L. Physicn 1955, 22, 345.
Lefebvre
4206 The Journal of Physical Chemistry, Vol. 89, No. 20, 1985
open channels are identical and can be obtained from eq 49 at the two energies E, =F V,q- = -25.325 03 rad and q+ = -25.278 37 rad. The wavenumbers in the open channels are identical. The threshold energy being -17 544.232 cm-’, the wavenumbers at the two energies E, F V are k- = 136.061 87 cm-’l2 and k+ = 136.298 91 cm-’/2. One may then check that these quantities fulfill the two relations
TABLE III: Widths (in cm-I) of the Lowest Resonance for a Harmonic Oscillator Coupled to Two Interacting Continuums”
E
(CY)
(P)
(7)
0 100 1000 2000
2.728 2.720 2.416 1.714
2.731 2.723 2.419 1.714
2.717 2.733 2.436 1.732
a [ is the continuum-continuum coupling (in cm-I). (CY)refers to the analytical expression for the width (eq 41); (0) is obtained through direct complex energy quantization; (y) comes from an analysis of an anticrossing according to eq 14.
using the repulsive potential with an appropriate shift and can be controlled from a knowledge of the phase which, for this model, is analytically deducible from a matching of the wave function for the linear potential (an Airy function) to that of the plateau (a sine function): q = tan-’ [ - F 1 / 3 k A i ( - a ) / A i ’ ( - a ) ]
kP
(49)
with a = F’i2(f - rt)
(50)
This quasi-continuous spectrum is also used for estimating the densities. The widths obtained under procedures (a),(p), and ( 7 ) are given in Table 111. They amply demonstrate that the argumentation given for the one-channel case4 applies to more complicated situations. One may look further into the meaning of an anticrossing when there is no continuum-continuum coupling, since this case, as discussed above, can be reduced to a pseudosingle open channel situation. The width given in Table I11 for f = 0 was obtained from the two energies E- = 968.6014 cm-’ and E+ = 1033.1601 cm-’, with for the position of the anticrossing R = 0.508 75 cm’/*. The eigenphase 6, at the two energies E, F V (cf. eq 31) is 6,- = 0.08451 rad and = 3.05708 rad. The eigenphase is zero in this particular case. The phases in the
+ q- + 6’k+R, + q+ + 61, kR,
= 13.99957~ = 14.99901~
This shows that box quantization operates here very much like in the single open channel case of section 11.
IV. Conclusion We have shown that the resonances occurring in both the single channel and multichannel cases can be obtained in a very similar way from the technique of box quantization. There are clearly some limitations in this approach. We can expect it to be efficient if the resonances are well isolated. Broad and overlapping resonances will affect the energy patterns in a way which would require a much more complicated analysis. However, this is a fate which is common to many approaches to the resonance problem (for instance phase shift analysis or Argand plots). Another comment concerns the procedure used for box quantization. The quantized energies discussed in this paper for the multichannel case were obtained from an iterative propagation plus matching technique. There is another possibility which should be particularly efficient for the treatment of a large number of channels: this is the development of the wave function in the basis for the free particle eigenfunctions of the box. This was done in ref 4 for the one-channel situation. There should be no difficulty for extending this at least to the treatment of systems with two degrees of freedom. Acknowledgment. This work has been supported by a grant of computing time on the CCVR Cray.
