Dimensional erturbation theory for Regge oles Timothy C. Germann Deartment of Chemistry, University of California, Berkeley, California 94720 Sabre Kais Deartment of Chemistry, Purdue University, West Lafayette, Indiana 47907 Received 29 July 1996; acceted 2 October 1996 We aly dimensional erturbation theory to the calculation of Regge ole ositions, roviding a systematic imrovement to earlier analytic first-order results. We consider the orbital angular momentum l as a function of satial dimension D for a given energy E, and exand l in inverse owers of (D1)/2. It is demonstrated for both bound and resonance states that the resulting erturbation series often converges quite raidly, so that accurate quantum results can be obtained via simle analytic exressions given here through third order. For the quartic oscillator otential, the raid convergence of the resent l(d;e) series is in marked contrast with the divergence of the more traditional E(D;l) dimensional erturbation series, thus offering an attractive alternative for bound state roblems. 1997 American Institute of Physics. S0021-96069701102-1 I. INTRODUCTION In scattering theory, Regge oles corresond to singularities of the scattering matrix in the comlex angular momentum lane at real energies, roviding a descrition of resonances comlementary to the more familiar comlexenergy and real angular momentum viewoint. 1 3 One ersective or the other may be natural in a given situation. The usual artial wave exansion is most useful when only a few artial waves contribute significantly, as in electron-atom scattering, articularly at low energy. In such cases, a descrition of resonances as comlex energy states for each artial wave l may be the most aroriate view. On the other hand, as the system becomes more classical short wavelengths comared to the range of the otential, for instance in molecular collisions, a large number of artial waves hundreds or even thousands may become significant, diminishing the utility of the artial wave series. It is in this situation that the comlex angular momentum viewoint becomes most attractive, offering the ossibility of describing resonance scattering in terms of a few Regge ole ositions and residues for a recent alication to reactive molecular scattering, see Ref. 4. Comared to eigenvalue roblems involving bound states, or even comlex-energy resonances, relatively few methods exist for comuting Regge ole ositions and/or residues. This is due in art to numerical difficulties with such calculations, but also because of the great and early success of semiclassical methods. 5,6 Sukumar and Bardsley 7 alied a variational comlex rotation aroach to the calculation of Regge ole ositions; they also used direct numerical integration in the comlex coordinate lane to comute both ositions and via a fitting of the S-matrix in the vicinity of the ole the associated residues. Both of these techniques and also the semiclassical ones locate the oles by iteration in the comlex-l lane. Two other quantum mechanical methods, each combining numerical integration with a comlex-l root search, have been given by Bosanac 8 and Pajunen. 9 In contrast, the method described here requires no iteration, and thus no rior estimate of the ole location. Furthermore, the comutational effort is quite modest; the third-order erturbation theory exressions given in the Aendix ermit calculations by hand, which for the examles studied are as accurate as any of the available quantum methods. In Section II we describe the method, first reviewing the first-order treatment due to Kais and Beltrame, 10 then discussing the iterative method used to comute higher order terms. We note that the resent aroach is essentially equivalent to the semiclassical -exansion of Kobylinsky et al., 11 although the comutation of higher order correction terms is more transarent in the resent work. Section III resents numerical results for secific systems. We first consider bound states, since by exressing l as a function of satial dimension D, for fixed energy E, the resulting l(d;e) function turns out to be much better behaved than E(D;l), which is the usual focus of dimensional erturbation theory. As a result, the l(d;e) series obtained here rovides an alternative method to traditional dimensional erturbation theory for the comutation of bound state energies. We demonstrate this ossibility for the quartic oscillator. Several rototyical calculations of comlex angular momentum Regge oles are then resented, demonstrating the raid convergence of the resent series to accurate quantum mechanical ole ositions. Concluding remarks are given in Section IV. II. DIMENSIONAL PERTURBATION THEORY Dimensional erturbation theory has been alied with great success to the calculation of ground and excited state eigenvalues for several systems, such as the helium atom 12,13 and the hydrogen atom in a magnetic field. 14,15 For the latter system, other roerties such as exectation values 15 and diole transition elements 16 have been accurately comuted by such methods. Resonance states have also been exlicitly treated, with several calculations of comlex energy eigenvalues for both central otentials and for the hydrogen atom in electric or arallel electric and magnetic fields. 17 21 J. Chem. Phys. 106 (2), 8 January 1997 0021-9606/97/106(2)/599/6/$10.00 1997 American Institute of Physics 599
600 T. C. Germann and S. Kais: Dimensional erturbation theory for Regge oles For sherically or cylindrically symmetric systems, an imortant feature of dimensional erturbation theory is the exact relationshi between satial dimension D and orbital angular momentum l or m). Such interdimensional degeneracies have been exloited to rovide an extremely accurate descrition of circular Rydberg states (mn11) of the hydrogen atom in a magnetic field. 15,16 In the resent work we will restrict ourselves to sherically symmetric otentials V(r), for which the D-dimensional Schrödinger equation, in reduced atomic units (1), is 1 2 d 2 dr 2 l1l 2r 2 VrrEr, 1 where (D1)/2 Refs. 22 and 23. The intimate link between angular momentum and dimensionality is evident in this form, since both l and only aear in the centrifugal term, and then only in the combination l. In the resent work we will rescale Eq. 1 such that this exact link is broken, in order to ermit an exansion of l in inverse owers of, but the raid convergence of such an exansion imlies that a simle connection remains. A. First-order erturbation theory We next review the alication of first-order dimensional erturbation theory to this roblem by Kais and Beltrame. 10 The Schrödinger equation Eq. 1 is not directly suitable for erturbation about, since in this limit the centrifugal otential diverges. This may be remedied by scaling the energy units by a factor of 2, namely ẼE/ 2, Ṽ(r)V(r)/ 2, giving 1 d 2 2 2 dr 2 2 2r 2 ṼrẼr0, 2 where (1(l1))(1l) and 1. Note that is defined such that it is unity in the hysical three-dimensional world, so that the scaling is basically a formality; with this in mind, we will dro the tildes from Ṽ(r) and Ẽ. This scaling does have the consequence of breaking u the interdimensional degeneracies resent in Eq. 1, which is necessary if l is to be exanded in a nontrivial series in. This exansion is most easily accomlished by first exanding as i0 i i. Subsequently, this series may be recast into a series in for l i i, i0 namely via the identities 0 1/2 0 1, 1 1 2 1 1 0 1/2, 3 4 5a 5b k2 1 k1 2 0 1/2 k k1 i i1 ki. 5c We see that in the infinite-dimensional ( 0) limit of Eq. 2, the derivative term vanishes, leaving the article at the minimum or more generally, a comlex stationary oint of an effective otential V eff r 0 2r 2 Vr. 6 Stationary oints of Eq. 6 are obtained by the condition 0 r m 3 Vr m, where the rime denotes differentiation with resect to r. Combining this with the requirement that V eff (r m )E0 gives the equation to be solved for r m : Vr m r m 2 Vr me0. 8 In some cases, the solution of Eq. 8 may be obtained in a closed form; 10 otherwise a simle root search such as Newton s method may be emloyed. The leading order term 0 is then obtained from Eq. 7. Anticiating the extension of the erturbative treatment to higher orders, we exand the wavefunction in owers of 1/2 : x j0 j/2 j x. Fluctuations of r about r m are accounted for by introducing a dimension-scaled dislacement coordinate x, defined by rr m (1 1/2 x). If we also exand the otential in a Taylor 2 series about r m, multily Eq. 2 by r m and collect terms by owers /2 of 1/2, then one obtains H j j 0 j0 x 0, 9 where H 0 1 2 7 d dx 2 1 2 2 x 2 1 2, 10a H j1 r m j4 V j2 r m x j2 j2! 1 j i0 and 3 0 r 4 m Vr m 1/2. j2 2 j32i i 2 x j22i, 10b 11 The arenthesized terms in Eq. 9 may individually be set to zero, roviding an infinite set of differential equations to be solved. The leading (0) harmonic Schrödinger equation, H 0 0 (x)0, determines 1 : 1 2n1, 12
T. C. Germann and S. Kais: Dimensional erturbation theory for Regge oles 601 where n is the harmonic quantum number. Kais and Beltrame 10 have evaluated the first-order result 0 1 analytically for ower law otentials, and obtained numerical results for other otentials which are in excellent agreement with available quantum and semiclassical calculations. B. Higher order terms The 0 terms of Eq. 9 rovide the means for comuting the higher order erturbation coefficients i. This set of equations may be solved recursively by various techniques; a articularly elegant aroach which has recently been described for dimensional erturbation theory is the matrix method of Dunn et al. 24 The central idea behind this aroach is that the wavefunction exansion terms j (x) may be exressed in the harmonic oscillator basis set given by the leading order Hamiltonian H 0. In this basis, H 0 is obviously diagonal, and because the coordinate oerator x has a simle bidiagonal reresentation in a harmonic basis, the higher order Hamiltonians H j1 are also straightforward, since they are merely olynomials in x. There are thus no matrix elements to evaluate numerically, roviding a great savings in comutational effort comared to the variational method of Sukumar and Bardsley. 7 Denoting the matrix reresentation of H j as H j and the vector reresentation of j (x) asa j, Eq. 9 yields an infinite set of linear algebraic equations j0 H j a j 0, 0,1,2,..., 13 where 0 is the null vector. These equations must be solved iteratively for the a j and i. We note that an exression for a could be obtained immediately if H 0 had an inverse; however, the requirement H 0 a 0 0 means that the diagonal element of H 0 corresonding to the harmonic quantum number n is zero. Fortunately, the orthogonality condition 0 j 0 j 14 imlies that this element of the wavefunction vector a j is zero for j0. Therefore we may define K as a diagonal matrix with elements K jj jn1 jn 0 jn which has the roerty K H 0 a j a j ( j0). Now multilication of Eq. 13 by K gives a as a K H j a j. j1 15 The orthonormality condition Eq. 14 may also be used to obtain an exression for i. For even, H includes the undetermined constant term 1 2 (2)/2. By multilying Eq. 13 on the left by a 0 T and invoking Eq. 14, one obtains 0a 0T j1 H j a j. 16 This equation is automatically satisfied for odd due to the alternating arity of the H j, and consequently of the a j. For even, Eq. 16 fixes the value of the reviously undetermined term 1 2 (2)/2 in H. In the Aendix, we give analytic exressions for the first three terms i in the erturbation series for the two leading Regge oles. These exressions and analagous ones for higher oles may be obtained from the recursion relations given above; however, these relations are more easily imlemented directly on a comuter, ermitting comutation to any arbitrary order. In ractice, the achievable order is limited by the accumulation of roundoff error, in some cases to no more than tenth order. However, as we will now demonstrate, only a few terms are needed to give highly accurate results in most cases articularly for the leading Regge oles. III. RESULTS A. Bound states Bound states are identified by locating integral values of l(e). Kais and Beltrame 10 have noted that for ower law otentials, the first-order exression for l(e) may be analytically inverted, exactly reroducing the entire sectrum in the case of the Coulomb otential V(r)1/r: 1 E 2nl1 2. 17 Note that here n reresents the radial quantum number, rather than the usual rincial quantum number. On the other hand, the usual dimensional erturbation theory treatment is limited to harmonic level sacings at first order, giving in the Coulomb case E 1 2 nl. 18 This erturbation series has a finite radius of convergence due to a second order ole at (nl) 1, excet for the nl0 ground state, whose series terminates at the zeroth order term E (1/2). This simle examle suggests that the resent alication of dimensional erturbation theory to Regge oles may actually have some ractical value as a method for bound states. Since the l(d;e) function seems to be much more well behaved i.e., more raidly convergent than the usual E(D;l) function, dimensional erturbation theory for l(d;e) may be used to locate the bound states at energies E where l(d;e) is integer-valued. This rocedure has been carried out analytically 10 for the case of first-order erturbation theory alied to ower-law otentials. Imroving the accuracy of this simle first-order treatment requires a rootfinding iteration using higher order erturbation results; Fig. 1 demonstrates this rocedure for the quartic oscillator otential, V(r)r 4. In order to demonstrate our claim that the exansion for l(d;e) is more raidly convergent than that
602 T. C. Germann and S. Kais: Dimensional erturbation theory for Regge oles TABLE II. Exansion coefficients j and artial sums given as l instead of l(l1)) for the leading (n0) Regge ole of the real LJ 12,6 otential with arameters k141.425 and E/5. j Re j Im j Re l Im l 0 32339.689922 7174.200104 180.422740 19.826695 1 204.511098 485.104426 180.014579 21.215196 2 1.134921 1.236568 180.011876 21.218939 3 0.027657 0.005082 180.011950 21.218917 4 0.000532 0.000637 180.011948 21.218915 5 0.000010 0.000031 180.011948 21.218915 quantum a 180.01195 21.21892 semiclassical b 180.015 21.219 a Reference 9. b Reference 6. FIG. 1. Real-valued Regge trajectories for the quartic oscillator, V(r)r 4. Bound states occur when the l n (E) curves ass through integervalued l. for E(D;l), Table I contains the first few artial sums for each aroach, at the n0, l3 bound state. We observe that direct summation of the E(D;l) erturbation series is divergent, although Padé summation roduces reasonable results at higher orders. On the other hand, the l(d;e) series is raidly convergent, whether the original series of Eq. 3 is summed to yield l(l1), which is then solved for l, or whether the series is converted to a series for l itself, Eq. 4. Of course, the roosed aroach suffers the disadvantage of requiring iteration of the energy until convergence to the desired accuracy is obtained. This is not uncommon; such a root-finding rocedure is also necessary in direct numerical integration of the Schrödinger equation. B. Lennard-Jones (12,6) otential As a secific examle of Regge ole calculations for comlex angular momentum resonances, we consider the Lennard-Jones 12,6 otential with an otional imaginary otical r s term Vr4 12 r r 6 iw r s, 19 TABLE I. Dimensional erturbation theory artial sums and, for E(D;l), diagonal N/N,N/N1 Padé sums for the quartic oscillator n0,l3 bound state at E9.4011602. E(D;l) l(d;e) order E series Padé sums series series 0 1.190551 1.190551 4.237050 3.710588 1 7.103215-0.300165 2.801682 2.985843 2 10.663506 16.051969 3.005709 3.000586 3 7.917819 5.359864 2.999838 3.000053 4 11.672436 9.598676 2.999959 2.999982 5 5.629352 9.259571 3.000010 3.000002 6 15.270904 9.350154 3.000000 3.000000 where is the well deth and the radius at the otential minimum. By taking as our unit of distance and the reduced mass as the unit of mass, we can write the otential in these reduced atomic units as Vr4r 12 r 6 iwr s, 20 where, W and E are measured in the energy units 2 / 2. The relevant dimensionless arameters may be taken as the wavenumber k2e, the energy-to-well deth ratio E/, and the strength of the otical term, W. First, we take the leading ole for the commonly studied case k141.425, E/5 and W0, which may be comared with earlier quantum 6,9 and semiclassical 6 results. Table II contains the first few exansion coefficients j,as well as their artial sums converted from l(l1) to l). The first order result comares well with the semiclassical calculation, as noted reviously by Kais and Beltrame. 10 The raid convergence of the succeeding terms is striking; thirdorder erturbation theory corresonding to the simle analytic exressions given in the Aendix already yields a ole location in comlete agreement with the most accurate revious quantum calculation that we are aware of. 9 Table III lists converged ole ositions for the comlex otical otential with k53.401, E/2.462, W5000 and s20. The full number of digits of convergence of Padé TABLE III. Regge ole ositions for the comlex LJ 12,6 otential with k53.401, E/2.462, W5000 and s20. Padé aroximants u to 10/10 converge to the number of digits shown. For comarison, artial sums for n0 converge to l 0 76.93675234.6776463i. n Re l n Im l n 0 76.93675231616172 4.677646258571630 1 76.691242656436 7.521212950414 2 76.56189829690 10.36977300855 3 76.539384466 13.214757685 4 76.61961547 16.047554217 5 76.7988748 18.8579303 6 77.07177 21.635323 7 77.43098 24.37015 8 77.868 27.0545 9 78.372 29.683 10 78.935 32.253
T. C. Germann and S. Kais: Dimensional erturbation theory for Regge oles 603 aroximants u to 20th order in, for either the or l series are resented, in order to illustrate the diminishing usefulness of dimensional erturbation theory for the higher oles. Again, these results are in comlete agreement with the results of Pajunen, 9 excet for some insignificant differences in the final digit for some of the higher oles. 26 IV. DISCUSSION The semiclassical methods of Connor, Delos and coworkers aear to yield sufficiently and often remarkably! accurate Regge ole ositions and residues for most ractical calculations, with the ossible excetion of the leading ole residue, which has been found to be in error by as much as 10%. Nevertheless, the ossibility of efficient quantum calculations is always desirable. Unfortunately, the revious quantum methods 7 9 all roceed via a root search in the comlex-l lane, which means that 1 an estimate of the ole osition is required, and 2 iteration is required. Furthermore, the methods involving numerical integration in the comlex-r lane 7,9 require great care in the choice of integration ath, with roer attention aid to the Stokes and anti-stokes lines, for instance. In contrast, the method resented here see also Refs. 10 and 11 offers the best of both worlds: a simle yet accurate analytical exression, with easily comuted higher order correction terms. Particularly for the leading Regge oles, which are often the most imortant contributions to the observable cross section, just a few terms in the 1/ exansion yield ole ositions more accurately than revious quantal calculations. And because no iteration of the ole osition or numerical quadrature is required, evaluation of these terms should be simler to carry out than even the semiclassical method. Whenever using Regge theory in ractice, the locations of the oles are only half of the story; the associated residues of the oles are vital in obtaining scattering cross sections. All of the available techniques both quantum and semiclassical roceed by first locating the oles and subsequently comuting their residues by numerical methods, either directly or by fitting the S-matrix in the vicinity of the oles. The resent analytical aroach takes advantage of the seudo-bound state character of the oles, and thus aears ill-suited for the calculation of residues, which requires knowledge of the S-matrix or equivalently, the wavefunction for comlex angular momentum close to the Regge ole. One ossible aroach is to use the resent theory to locate the oles accurately, and then use an alternative method, such as direct numerical integration, to comute the residues; this is in the same sirit as the aroach taken by Sukumar and Bardsley. 7 ACKNOWLEDGMENT T.C.G. is suorted by a Research Fellowshi from the Miller Institute for Basic Research in Science. APPENDIX: ANALYTIC j EXPRESSIONS THROUGH THIRD ORDER Here we resent analytic exressions for the first two Regge oles, through third order in. The zeroth order term 0 is obtained as described in the main text, by minimizing the effective otential function in Eq. 6. For convenience in the resulting exressions, we define j4 j3 v j 2 2 r 01 j m j2! V j2 r m. For the leading (n0) Regge ole, 1, 2 5 2 3v 1 3 2 v 2 11 4 v 1 2, 3 1 16 82136v 1182v 1 2 396v 1 3 465v 1 4 82v 2 84v 2 2 120v 3 60v 4 424v 1 v 2 684v 1 2 v 2 260v 1 v 3, and for the next (n1) ole, 1 3, 2 45 2 27v 1 15 2 v 2 71 4 v 1 2, 3 3 16 498776v 11358v 1 2 2556v 1 3 1875v 1 4 804v 2 220v 2 2 600v 3 140v 4 2632v 1 v 2 2484v 2 1 v 2 820v 1 v 3. We note that these exressions are equivalent to those from the -exansion of Kobylinsky et al., 11 with the identification r 0 g 1/2 0 (E) and v j 1 2 (1) j a j. 1 J. R. Taylor, Scattering Theory Wiley, New York, 1972,. 302 314. 2 K.-E. Thylwe, in Resonances: The Unifying Route Towards the Formulation of Dynamical Processes, Foundations and Alications in Nuclear, Atomic and Molecular Physics Lecture Notes in Physics 325, edited by E. Brändas and N. Elander Sringer, Berlin. 1989,. 281 311. 3 J. N. L. Connor, J. Chem. Soc. Faraday Trans. 86, 1627 1990. 4 D. Sokolovski, J. N. L. Connor, and G. C. Schatz, J. Chem. Phys. 103, 5979 1995. 5 J. B. Delos and C. E. Carlson, Phys. Rev. A 11, 210 1975. 6 J. N. L. Connor, W. Jakubetz, and C. V. Sukumar, J. Phys. B 9, 1783 1976. 7 C. V. Sukumar and J. N. Bardsley, J. Phys. B 8, 568 1975. 8 S. Bosanac, J. Math. Phys. 19, 789 1978. 9 P. Pajunen, J. Chem. Phys. 88, 4268 1988. 10 S. Kais and G. Beltrame, J. Phys. Chem. 97, 2453 1993. 11 N. A. Kobylinsky, S. S. Steanov, and R. S. Tutik, Phys. Lett. 235B, 182 1990. 12 D. Z. Goodson, M. Lóez-Cabrera, D. R. Herschbach, and J. D. Morgan III, J. Chem. Phys. 97, 8481 1992. 13 D. Z. Goodson and D. K. Watson, Phys. Rev. A 48, 2668 1993. 14 T. C. Germann, D. R. Herschbach, and B. M. Boghosian, Comut. Phys. 8, 712 1994. 15 T. C. Germann, D. R. Herschbach, M. Dunn, and D. K. Watson, Phys. Rev. Lett. 74, 658 1995.
604 T. C. Germann and S. Kais: Dimensional erturbation theory for Regge oles 16 T. C. Germann, J. Phys. B 28, L531 1995. 17 V. S. Poov, V. D. Mur, A. V. Shcheblykin, and V. M. Weinberg, Phys. Lett. 124A, 771987. 18 V. M. Vanberg, V. D. Mur, V. S. Poov, A. V. Sergeev, and A. V. Shcheblykin, Theor. Math. Phys. 74, 269 1988. 19 V. M. Vanberg, V. S. Poov, and A. V. Sergeev, Sov. Phys. JETP 71, 470 1990. 20 S. Kais and D. R. Herschbach, J. Chem. Phys. 98, 3990 1993. 21 T. C. Germann and S. Kais, J. Chem. Phys. 99, 7739 1993. 22 D. R. Herrick and F. H. Stillinger, Phys. Rev. A 11, 421975. 23 D. R. Herrick, J. Math. Phys. 16, 281 1975. 24 M. Dunn, T. C. Germann, D. Z. Goodson, C. A. Traynor, J. D. Morgan III, D. K. Watson, and D. R. Herschbach, J. Chem. Phys. 101, 5987 1994. 25 J. N. L. Connor, D. C. Mackay, and C. V. Sukumar, J. Phys. B 12, L515 1979. 26 As ointed out to us by Professor J. N. L. Connor rivate communication, the systematic discreancies in the ole ositions noted by Pajunen Ref. 9 in comaring with the earlier quantum results of Connor et al. Ref. 25 are due to a slight difference in the conversion factors used.