J. Phys. Chem. 1993,97, 2453-2456
2453
Dimensional Scaling for Regge Trajectories S. Kais’ and G. Beltrame Department of Chemistry, Harvard University, Cambridge, Massachusetts 021 38 Received: October 6, 1992; I n Final Form: December 9, 1992
Using dimensional scaling, we were able to obtain a systematic expansion for Regge trajectories in 1 / ~ where , = (D- 1)/2 and D is the number of spatial dimensions. Bound states for the power-law potential were obtained from Regge trajectories by requiring the angular momentum quantum number to take on positive integer values. For scattering states, we calculated the positions of Regge poles for the Lennard-Jones (6,4) and (1 2,6) potentials. The results to first order in 1 / K were in good agreement with both semiclassical and quantum calculations. The same expansion was used to obtain the positions of Regge poles for complex optical potentials. The results for the Lennard-Jones (12,6) potential perturbed by an imaginary term were in excellent agreement with the semiclassical calculations. K
1. Introduction
In Regge pole theory, the energy E is treated as a real parameter and the angular momentum quantum number 1 is allowed to take on complex values.’ The physical interpretation of Zml is that the system decays exponentially with the scattering angle. The inverse of Zml is proportional to the ‘angular life” of the system, and Re1 determines the radius around which the angular decay In this theory, the differential cross section can be represented in terms of the poles of the scattering matrix in the complex 1 plane, together with a background integral. This theory has a wide range of applications, starting from high-energy elementaryparticles’ and nuclear physics4to atomic and molecular physia.5-9 The key quantities to be calculated are the Regge pole positions and their associated residues. At present, there are two main approaches: direct numerical integration of the Schrlklinger equationlo and semiclassical WKB approximations.l’J2 This paper examines a dimensional scaling treatment13 to calculate Regge trajectories. This approach is simple to implement and yields accurate results for the positions of Regge poles. In the same way as it has been applied for bound and resonance states,I4J5the method involves generalizing the Hamiltonian to D-dimensions, introducing appropriate D-scaled coordinates, and performing a 1/D expansion about the D m limit. The main difference is that here the energy is treated as a parameter, while the angular momentum is expanded in powers of 1 / (~K = (D - 1)/2) in order to obtain Regge trajectories directly. In section 2 we formulate the dimensional scaling procedure for Regge trajectories. In section 3 we show how to obtain bound state energies by requiring the angular momentum quantum number to have positive integer values. As an example, we give the energy formula to first order in powers of 1 / K for the powerlaw potential. In section 4 we obtain Regge trajectories in the complex angular momentum plane for scattering states. The Lennard-Jones (1 2,6) and (6,4) potentials are treated to illustrate the method. The results are in complete agreement with both quantum and semiclassical results. In section 5 we consider a Lennard-Jones potential perturbed by an imaginary term and show that our procedure, to first order in l / x , yields results that are in excellent agreement with the semiclassical results for a wide range of the strength of the imaginary term. In section 6 prospects for treating the scattering matrix and the residues of Regge poles are discussed.
-
2. Regge Trajectories
For spherically symmetric potentials, the radial Schriidinger equation in D-dimensions has the form
h=l where p is the reduced mass of the system, I is the orbital angular momentumquantum number, and K = (D- 1)/2. In Regge pole theory, I is allowed to take on continuous complex values but E is treated as a real parameter. The path traced out by I, (n is an index to distinguish between different poles) in the complex 1 plane as E varies is called a Regge trajectory. Scaling the energy and the potential (E = E / K ~p(r) , = V(r)/ K ~ ) ,eq 1 becomes
where
(
A(E) = 1
+
y )( + I(E) i) 1
K
-K
(3)
Taking E as a real parameter, eq 2 defines the implicit dependence of A ( E ) on E and K . Assuming that A ( E ) can be represented by an expansion in powers of 1 / ~ we , have
A(E) =
-
2
&(E)(
’)“
(4)
K
m=O
As D m, quantum fluctuations become unimportant and the particle remains localized at the bottom (r,) of the effective potential (5)
Minimizing V&),
we get the following expression for &,(E) d V( r)
Ao(E) = prm3p(rm),
p(r)= dr
(6)
where rm satisfies the equation 1 V(rm)+ Zrm V’(rm)- E = 0
(7)
-
This is the same equationobtained by performing a semiclassical approximationi6 for Regge trajectories in the limit h 0 for bound states, but now we also consider the possible complex solutions in order to describe Regge trajectories in the complex angular momentum plane.
0022-3654/93/2091-2453~04.00/0 0 1993 American Chemical Society
Kais and Beltrame
2454 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
TABLE I: Bound States for the fl Potential1 I
Eo/ 2.4625 4.5308 6.8140
0 1 2
exact
I
2.3936 4.4180 6.8303
3 5 7
Eo/ 9.4388 15.1 1 1 21.383
exact 9.4011 15.081 21.358
All values are in atomic units.
Higher order terms in the 1 / expansion ~ can be obtained by using the familiar tran~formation’~ x = d12(r - rm)/r,,, and expanding theeffective potential aroundx = 0. Now, substituting the expansions of Ve&) and A(E) into eq 2 and equating to zero the coefficients of the same powers of 1 / ~ one , can obtain the coefficients A,,,(E) in closed form. Regge trajectories can be obtained by expanding the scaled angular momentum a ( E ) = / ( E ) / Nin powers of 1 / ~
a ( E ) = a@)
+ a , ( E )1; + a 2 ( E )17+ ...
(8)
K
On substituting eq 4 and eq 8 into eq 3, we get a simple relation between the coefficients &(E) and a,,,(E). At last, the explicit expansion for Regge trajectories is
a ( E ) = (A;I2(E) - 1)
+
(2t;TE, + i) + ...
(9)
where
TABLE U Positions of Regge Poles for the Lennrrd-Joats (I2,6)Potentiala dimensional scaling
Rel, 180.018 119.243 118.519 111.841 111.227 116.658 116.140 115.614 115.260 114.891 114.585
n 0 1 2 3 4 5 6 1
8 9 10 a
c
= 4.0 X J.
Iml, 21.219 24.031 26.902 29.812 32.169 35.112 38.821 41.916 45.058 48.246 51.480
quantum
Rel, 180.012 179.239 118.522 111.866 177.272 116.142 116.271 115.811 115.544 115.216 115.074
lo-*’J; re = 4 X 1W cm; p
semiclassical
lml,
21.219 24.035 26.890 29.180 32.700 35.645 38.609 41.588 44.516 41.568 50.561
Rel, 180.015 179.242 118.525 177.869 117.275 176.745 116.279 175.880 115.541 115.219 115.016
= 4.311 X
Iml,, 21.218 24.034 26.889 29.119 32.699 35.644 38.608 41.587 44.515 41.561 50.560
g; E = 2 X
4) with the numerical integration results.’6 The agreement is good, especially for large 1. Equation 13 could also be applied to find Regge trajectories by considering the possible complex solutions of eq 7. In this context, it is identical (up to a constant term - I / z ) to the formula obtained by Connor and c o - ~ o r k e r s ~ ~ by means of a complex harmonic oscillator treatment. However, in their approach the potential is expanded around a point in the complex r plane whose position depends not only on E (as in the present treatment) but alsoon the particular poleone is interested in.
4. Scattering States
n = 0, 1, 2, ... Logarithmic perturbation theory provides an alternative way to evaluate higher order correction terms. Our expressions for the coefficients Am(,!?) are identical to those obtained by Kobylinsky and co-workers16by using an expansion of powers of
n.
3. Boundstates Bound state energy eigenvalues can be obtained by inverting eq 9 and letting the angular momentum 1 take on only positive integer values. In order to illustrate the method, we consider the power-law potential
V(r) = XI“, v # -2 (1 1) In the limit D .-c m, the particle remains fixed in the radial coordinate, thus executing only circular motion at a given value of the energy E. The radius of the orbit is the solution of eq 7. Its explicit expression for this particular case is
Scattering states with a fixed energy can be obtained by allowing the angular momentum quantum number to have complexvalues. The same procedure is used, with the radial coordinatenow allowed to be complex. As an example, we consider the familiar LennardJones (12,6) potential
V(r) = t[ 2I):(
- 2(
91
where t denotes the well depth and rethe radius at the minimum. Values of the parameters were chosen to approximate the elastic scatteringofK byHBr.18 For this particularpotential, thesolution of eq 7 can be written in closed form rm = [ a 4 f (16
- 2%)E
112
(15)
Using the value for r,,,calculated from this equation in conjunction with eqs 9 and 10, we obtained the results shown in Table 11. The second potential to be considered is the Lennard-Jones (694)
with parameters that approximate the elastic scattering of H+by Ar.I2 In this case, r,,, satisfies the equation
After evaluating &,(E) and h , ( E ) by means of eq 10, we arrive at the following expression for E at D = 3 (to first order in l / n )
For the Coulomb (v = -1) and harmonic oscillator (v = 2) potentials, eq 13 gives the exact energy formulas. These results areencouraging, sinceweareable toobtain thecompletespectrum from first-order calculations. In Table I, we compare the results from this simple formula (q13) for the quartic oscillator (v =
t
r:
- 3r: + 4 = 0
which could be solved in closed form, if desired. Following the same procedure outlined above, the expansion coefficientscan be readily calculated. Table I1 and Table I11 show that our results to secondorder are in very close agreement with both semiclassical and quantum calculations.12 For the leading pole the relative error is 0.00396 for the (12,6) potential and 0.03% for the (6,4) potential. This high accuracy shows the fast convergence of the expansion coefficients, especially for low values of n. As n increases, the agreement becomes less satisfactory, higher order terms being needed to achievethe same accuracy. Figure 1 shows the comparison with the semiclassical results for the positions of
Dimensional Scaling for Regge Trajectories
The Journal of Physical Chemistry, Vol. 97, No. 10, 1993 2455
TABLE IIk Positions of Regge Poles for tbe Lennard-Jones (64) Potential’
l ” ” ” ” ’ ” ’ ’ 1
~
dimensional scaling n
0 1 2 3 4
5 10 14
Iml,
Rel, 97.484 96.690 95.905 95.131 94.367 93.613 89.995 87.284
12.394 14.021 15.683 17.382 19.117 20.888 30.284 38.450
quantum
semiclassical
Iml, 12.398 14.025 15.687 17.384 19.114 20.877 30.112 37.860
Rel, 97.519 96.726 95.948 95.187 94.447 93.730 90.582 88.693
Iml, 12.398 14.025 15.687 17.384 19.114 20.877 30.112 37.859
Rei, 97.520 96.725 95.948 95.188 94.448 93.731 90.583 88.693
“ r = 6 . 9 7 2 3 5 X 1 0 - 1 9 J ; r e =1.31233X1Ck8cm;c= 1.613178X g; E = 8.010 51 X l@Ig J.
Imt
180
181
182
183
184 185
186
Rd
120
Figure 2. Location of Regge poles from n = 0 to 10 for the complex potential given in eq 18. The values of the parameters are C = 6000, E = 10000.515, e = 8000.4123, and Y = 12. Semiclassical results are indistinguishablefrom the ones obtained by the dimensional scaling method in this scale.
100
TABLE Iv: Positions of the Leading ( n = 0) Regge Poles for the Lennard-Jones(12,6) Potential plus a Perturbation
80
4CI-12
dimensional scaling
60
C
15 I
210
250
330
290
1
370
Ret Figure 1. Location of Regge poles for n = 0 to 20 for the Lennard-Jones (1 2,6) potential. Values of E / € range from 10 to 30. In the scale of the figure, semiclassicalresultsI2are indistinguishable from the ones obtained by the dimensional scaling method.
the leading poles for the (12,6) potential from n = 0 up to n = 20. Values of the ratio E / c are 10, 15, 20, 25, and 30.
5. Opticnl Potential As an exampleof a singular complex potential, we consider the Lennard-Jones (1 2,6) potential perturbed by an imaginary ry term
The &dimensional radial SchrMinger equation can be-written in terms of the dimensionless quantities R, c, C, and E in the following way:
iC-k) R’
q ( R ) = O (19)
where we chose 2 = 10 000.515, = 8000.4123, and Y = 12 to be able to comparewith the available semiclassicalresults.I9Table IV gives the first-order calculationsfor the positions of the leading (n = 0) Regge poles as C varies between 0 and 50 000. Again, the agreement with the values obtained from semiclassical calculations19is excellent: four to five digits for the leading pole. Figure 2 comparesthe positions of the first 11 poles in the complex 1 plane for C = 6000 with semiclassical results. 6. Discussion
Dimensional scaling offers a remarkably simple means to obtain Regge trajectories. Our results for bound states, scattering states, and complex optical potentials are in good agreement with both quantum and semiclassical results. To first order, dimensional scaling procedure requires merely finding the minimum of an effective potential function and the curvature at that point. The quantum results involved direct numerical integration of the
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10 OOO 20 OOO 30 000 40 000 50 000
Relo 180.006 180.791 181.567 182.337 183.091 183.824 184.536 185.231 185.902 186.550 187.175 192.352 196.157 199.147 201.620
semiclassical
Imlo
21.211 20.837 20.513 20.238 20.006 19.813 19.655 19.530 19.430 19.355 19.299 19.369 19.845 20.37 1 20.87 1
Relo 180.015 180.797 181.575 182.342 183.095 183.830 184.544 185.238 185.909 186.557 187.182 192.360 196.161 199.153 201.629
Imlo 21.218 20.843 20.519 20.243 20.01 1 19.818 19.661 19.535 19.436 19.360 19.304 19.373 19.848 20.373 20.873
Schrijdinger equation and matching the inward and outward solutions in thecomplex coordinatespacetoobtain theS-matrix.*O In the semiclassical calculations, the quantization formula for the Regge poles was obtainedby applying the parabolicconnection formula20 and imposing the outgoing wave only boundary condition.12 In this method an initial guess for the position was necessary and the turning points were calculated iteratively by Newton’s method in the complex r plane. Prospects for applying dimensional scaling to treat scattering problems appear promising. In Regge pole theory, the scattering amplitude can be represented as a background integral and a sum over Regge pole positions and their associated residues.2 Work is still in progress to calculate the S matrix within the framework of the dimensional scaling method. Knowledge of the S matrix will allow us to calculate the residues of the Regge poles, since in the neighborhood of these poles the S-matrix S(I) is dominated by a first-order pole with residue rn21
S(I) = ‘n
I - I,
The large D limit has already proven its efficiency in providing good estimates for the scattering length and phase shifts for a class of potentials which can support up to one bound state.22 Dimensional scaling has not yet been fully exploited in the field of quantum scattering, and we hope to present additional applications in future work.
Acknowledgment. We thank Dudley Herschbach for granting us the privilege to work with him. This paper is dedicated in honor of his 60th birthday. S.K.thanks M.S. Child for his hospitality and fruitful discussions about Regge pole theory and
2456 The Journal of Physical Chemistry, Vol. 97, No. 10, 1993
the future usage of dimensional scaling in scattering theory. In addition, S.K.thanks J. N. L. Connor for useful discussions. Support for this work was provided by the Office of Naval Research, Grant NOOO 14-90-5-4025.
References 8 d Notes (1) Newton, R. G. The Complex J-Plane; Benjamin: New York, 1964.
(2) Connor, J. N. L.J. Chem. Soc. Faraday Trans. 1990,86, 1627. (3) ,Squires, E. J. Complex Angular Momentum and Particle Physics; Benjamin: New York, 1963. (4) Rowley, N.; Marty, C. Nucl. Phys. 1976, 266A, 494. ( 5 ) Sukumar, C. V.;Lin, S.L.; Bardsley, J. N. J . Phys. 1975, B8, 577. (6) Connor, J. N. L.;Delos, J. B.; Carlson, C. E. Mol. Phys. 1976,31, 1181.
(7) Connor, J. N. L.;Farrelly, D.; Mackay, D. C. J . Chem. Phys. 1981, 74, 3218. (8) Thylwe, K.-E.; Connor, J. N. L.J. Chem. Phys. 1989. 91, 1668. (9) Thylwc, K.-E.; In Rcsonanw: The Unlni/vingRoute Towards the Formulationof Dynomical Processes;Brandas, E., Elander, N., Eds.;Lecture Notes Lecture in Physics 325; Springer: Berlin, 1989; p 281. (IO) Sukumar, C. V.;Bardsley, J. N. J . Phys. B At. Mol. Phys. 1975. 8, 568.
Kais and Beltrame (11) Delos, J. B.; Carlson, C. E. Phys. Reo. 1975, A l l , 210. (1 2) Connor. J. N. L.; Jakubetz, W.; Sukumar, C. V.J . Phys. B At. Mol. Phys. 1976, 9. 1976.
(13) Herschbach, D. R. In Dimensional Scaling in Chemical Physics; Herschbach, D. R., Avery, J., Goscinski, 0.. Eds.; Kluwcr: Dordrecht, in press. (14) Herschbach, D. R. J. Chem. Soc., Faraday Discuss. 1987,84,465. (15) Kais, S.;Morgan, J. D., 111; Herschbach, D. R. 1.Chem. Phys. 1991, 95.9028. Kais, S.;Herschbach, D. R. J . Chem. Phys. in press. (16) Kobylinsky, N. A.; Stcpanov. S.S.;Tutik, R. S.Phys. Lett. 1990, 8235, 182. (17) Goodson. D. 2.; Herschbach, D. R. Phys. Reo. Lett. 1987,58,1628. (18) Bernstein, R. B.; k i n e , R. D. J . Chem. Phys. 1968, 49, 3872. (19) Connor, J. N. L.;Mackay, D.C.;Thylwe, K.-E.J. Chem. Phys. 1986, 85, 6368. (20) Child, M. S.Molecular Spectroscopy Specialist Periodical Reports; Barrow,R. F.,Long, D. A.,Millcn.D. J.,Eds.;TheChemicalSociety: London, 1974. (21) Frautschi, S.C. Reggc Poles andS-Matrix Theory; Benjamin: New York, 1963. (22) Sukhatme, U.P.; Lauer, B. M.; Imbo, T. D. Phys. Rev. 1986,033, 1166. (23) Connor, J. N. L.; Jakubetz, W.;Mackay, D. C.; Sukumar, C. V.J . Phys. B At. Mol. Phys. 1980. 13, 1823.
