Exterior complex scaling and the computation of continuum-continuum transition matrix elements involving converging or diverging operators This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1999 J. Phys. B: At. Mol. Opt. Phys. 32 3117 (http://iopscience.iop.org/0953-4075/32/13/303) Download details: IP Address: 128.248.155.225 The article was downloaded on 23/08/2013 at 16:12 Please note that terms and conditions apply. View the table of contents for this issue, or go to the journal homepage for more Home Search Collections Journals About Contact us My IOPscience
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Exterior complex scaling and the computation of continuum-continuum transition matrix
elements involving converging or diverging operators
This article has been downloaded from IOPscience. Please scroll down to see the full text article.
1999 J. Phys. B: At. Mol. Opt. Phys. 32 3117
(http://iopscience.iop.org/0953-4075/32/13/303)
Download details:
IP Address: 128.248.155.225
The article was downloaded on 23/08/2013 at 16:12
Please note that terms and conditions apply.
View the table of contents for this issue, or go to the journal homepage for more
Home Search Collections Journals About Contact us My IOPscience
J. Phys. B: At. Mol. Opt. Phys.32 (1999) 3117–3134. Printed in the UK PII: S0953-4075(99)99747-X
Exterior complex scaling and the computation ofcontinuum–continuum transition matrix elements involvingconverging or diverging operators
Michael Chrysos and Mickael FumeronLaboratoire des Proprietes Optiques des Materiaux et Applications, EP CNRS 130, Universited’Angers, 2 Bd Lavoisier, F-49045 Angers Cedex, France
Received 30 November 1998, in final form 24 March 1999
Abstract. A general methodology and a numerical procedure are presented for the accuratecomputation of continuum–continuum transition matrix elements〈E′, l′|T |E, l〉, whereT couplesany two scattering states of a single-variable Hamiltonian. The procedure makes use of the exteriorcomplex scaling of ther-coordinate [r → r0 + (r − r0) exp(iθ)], widely known for its preciousproperties in resonance quantization. It is applicable whatever the potential interaction, providedits functional form is globally analytic asymptotically. As couplingT , any holomorphic functionof the coordinate can be considered for which the matrix element is meaningful. This methodologyenables one to handle systematically not only converging operators, but also diverging ones (suchas the electric dipole moment in length form) for which standard real-coordinate integration fails.Graphical representation of the imaginary part of theθ -dependent complex matrix element againstits real part shows that the exact value of the integral is clearly marked by a cusp, loop or inflectionon a well definedθ -trajectory, analogously with resonance quantization.
1. Introduction
Continuum–continuum (CC) transition matrix elements, and their reliable evaluation, isan endless subject in contemporary atomic, molecular and laser physics, spectroscopyand atmospheric photochemistry. Transition matrix elements of single-particle radialwavefunctions with initial and final states in the continuum are involved in electronbremsstrahlung (Pratt 1983), non-perturbative (Grochmalickiet al 1986) or perturbativeapproaches (Deloneet al 1989) for multiphoton absorption processes (Agostiniet al 1984),computations of continuous absorption spectra of stars (Bell and Berrington 1987) and rare-gasmixtures (Frommhold 1993), as well as in van der Waals interaction-induced scattering Ramanspectra (Chrysoset al 1996, Gayeet al 1997).
Several theoretical and numerical approaches have been developed so far for the evaluationof CC matrix elements. In some of those the emphasis is put on the simplicity of thecalculation, often resulting in analytical or semi-analytical expressions (Alderet al 1956,Deloneet al 1989, Bachmannet al 1995). In electron-impact collision processes, exactanalytical formulae and recursion relations can be obtained when a multipolar† transitionoperator couples two Coulomb functions with the same (Burgess 1974) or different (Ancaraniand Hervieux 1998) charges. When the asymptotic form of the continuum wavefunctions is
analytic, direct numerical integration may be possible beyond a certain radius (Stilley andCallaway 1970, Bellet al 1982). Numerical integration can also be performed via expansionsbased on repeated integration by parts (Aymar and Crance 1980, 1981). Finally, complexcoordinates can serve as a useful tool for evaluating CC matrix elements (Edwardset al1986, 1987), especially for tail integrals for which the wavefunctions take a simple analyticalform asymptotically (Silet al 1984, Gao and Starace 1987, Głazet al 1994, Ancaraniet al1998). In particular, Gao and Starace (1987) have computed electric dipole Coulomb matrixelements by carrying out complex rotations of the coordinate beyond a distancer0; for r > r0,a straight-line complex contour was employed. For their computations, accurate Coulombphasesσl = arg0[1 + l− iZ/k] were required (Gao and Starace 1987); at finite distances, thescattering wavefunctions were obtained by a specific procedure accounting for the evaluationof the wavefunction’s phase and amplitude. Although the procedure of Gao and Starace (1987)has the advantage of being able to deal with a diverging transition operator†, its applicabilityis restricted solely to the calculation of matrix elements between Coulomb wavefunctions.An alternative complex coordinate approach, making use of a sophisticated steepest-descentprocedure able to generate a specific optimal contour for each matrix element, has focusedon the computation of the far off-diagonal CC matrix elements, which are involved in thewings of interaction-induced scattering and absorption spectra from inert gaseous mixtures(Basileet al 1989, Głazet al 1994). It should be pointed out, however, that, in their works,Basile and Głaz dealt only with converging operators; furthermore, their method is able to dealsolely with potentials with a threshold, since, in their approach, initialization and normalizationof the wavefunction become possible with the assumption of constant phase and amplitudeasymptotically.
In the present paper, we attempt to describe a quite general methodology (and its numericalimplementation) for the accurate and systematic computation of CC transition matrix elements〈E′, l′|T |E, l〉, where a local operatorT couples any two scattering states of a single-variableHamiltonian. The procedure employs the exterior complex scaling (ECS) of ther-coordinate[r → r0 + (r − r0) exp(iθ)] (Nicolaides and Beck 1978, Simon 1979), widely known forits precious properties in resonance quantization (for a recent, well documented review oncomplex scaling methods in resonance quantization, see Moiseyev (1998)). Our approach isapplicable whatever the potential interaction, provided its functional form is globally analyticasymptotically. This offers the possibility of handling persistent potentials (Chrysos andLefebvre 1993) and saturating potentials (either short- or long-range) on equal footing‡. Asthe couplingT , any holomorphic function of the coordinate can be considered for which thematrix element is meaningful. This means that convergent operators (electric dipole momentin velocity or acceleration form, induced polarizability, induced dipole moment) and divergentoperators (electric dipole in length form, transition moments of molecular ions) can also behandled on an equal footing. When the imaginary part of the complex matrix element isrepresented graphically against its real part for various values of the rotation angle, the exactvalue of the integral is shown to be defined by a cusp, loop or inflection on a well definedθ -trajectory, analogously with resonance quantization. A suitable criterion is proposed whichis able to automatically lock these characteristic positions. The remainder of this paper isorganized as follows. In section 2, the principles of our methodology are summarized. The
† Conditionally convergent matrix elements (Gyarmati and Vertse 1971), like those which involve divergent operators(electric dipole moment in length form), can often be converted to absolutely convergent ones via length-to-velocityor length-to-acceleration transformations. However, for certain cases, a length form is preferable since it weights theasymptotic part most heavily where numerical wavefunctions are most accurate (Conneely and Geltman 1981).‡ Persistent potentials are those which do not reach a threshold value asymptotically. A typical example of a persistentinteraction is that of an atom in an external static electric field (Stark effect). Saturating interactions appear morefrequently than persistent ones.
Scaling and computation of CC transition matrix elements 3119
main steps of its numerical implementation are described in section 3. In section 4, twoapplications are illustrated and the criterion for convergence of the matrix element is developed.A brief conclusion is given in section 5.
2. Principles
2.1. Continuum–continuum matrix elements in terms of standing and outgoing waves
Accurate computation of CC matrix elements〈E′, l′|T |E, l〉 is possible within a given modelof the interaction potentialV (r) and the coupling operatorT (r), providedV (r) andT (r) haveglobally analytic functional forms asymptotically. As couplingT , any holomorphic functionof the coordinate can be considered for which the matrix element is meaningful. When onefocuses solely on the one-electron problem the radial integrals read
fi→f =∫ ∞
0ψ(E′, l′; r) T ψ(E, l; r) dr. (1)
Quantitiesl and l′ denote orbital momentum quantum numbers of the electron or rotationalquantum numbers of a diatomic molecule, or a quasimolecular complex, in the initial (i) andfinal (f ) states, respectively; as wavefunctions, single-particle pure scattering functions areused, which are regular solutions of the radial Schrodinger equation†(
− d2
dr2+l(l + 1)
r2+ V (r)− E
)ψ(E, l; r) = 0. (2)
The interaction potential (in ru) is denoted byV (r).For the sake of simplicity, let us first consider the caseV (r) = 0. The scattering
wavefunctions of this potential are the spherical Bessel functions of the first kindjl(kr)
(Abramowitz and Stegun 1965). The energy-normalized,δ(E − E′), wavefunctions read
ψ(E, l; r) =√
1
πkkrjl(kr) =
√k
πrjl(kr) (3)
wherek = √E denotes the asymptotic wavenumber. Note in passing thatψ(E, l; 0) = 0(regularity at the origin). The matrix element takes the form
fi→f = 1
π√kk′
∫ ∞0k′rjl′(k′r) T (r) krjl(kr) dr. (4)
The computation of this integral can now be performed by using an idea developed originally byLandau and Lifshitz (1958) in the framework of Wantzel–Kramers–Brillouin (WKB) theory,and later by Yanget al (1990) who used exact wavefunctions. According to this approach theintegral reads
fi→f = 1
π√kk′
Re∫ ∞
0k′rhl′(k′r) T (r) krjl(kr) dr (5)
wherek′rhl′(k′r) denotes the spherical Bessel function of the third kind in P-form (or aspherical Hankel function of the first kind), which is defined as the linear combination
† This form arises from the transformationr → λr, whereλ = hc√
2m/h2. This is particularly suitable for molecularproblems since no constants or mass dependences occur. The new units are called reduced ones (ru); reduced lengthunits are given in cm1/2; reduced energy units are expressed in cm−1; m andh denote the reduced mass of the atom(or the ion, or the diatom) and the Planck constant, respectively. Numerically,r(cm1/2) ' 0.13 m1/2 (amu)r (Bohr).For atomic problems, ordinary atomic units (au) are equally convenient (¯h = m = 1) (see, for instance, Chrysos andLefebvre 1993).
3120 M Chrysos and M Fumeron
k′r[jl′(k′r) + iyl′(k′r)]; the quantityk′ryl′(k′r) represents the Bessel function of the secondkind in P-form (or a spherical Neumann function) and defines the irregular solution of the finalstate. In contrast to the Bessel and Neumann functions, which behave like standing waves
krjl(kr) −→kr→∞
sin(kr − 1
2 lπ)
(6a)
kryl(kr) −→kr→∞
− cos(kr − 1
2 lπ)
(6b)
the Hankel function of the first kind behaves asymptotically like a purely outgoing wave(Abramowitz and Stegun 1965)
krhl(kr) −→kr→∞
−i exp[i(kr − 1
2 lπ)]. (6c)
This property is the backbone of the approach to be developed when complex rotation of thecoordinate is applied.
2.2. Exterior complex scaling in the service of CC interactions
Exterior complex scaling (ECS) (Nicolaides and Beck 1978, Simon 1979) is often referred to asan efficient tool for the accurate localization of resonance positions and the computation of theirspectral widths and corresponding matrix elements (Moiseyev 1998). This is true for eithershape resonances (Lefebvre 1988, Kukulinet al 1989) or Feshbach resonances (Feshbach1962, Friedrich 1991), in both single-channel (Nicolaideset al 1990, Chrysos 1998) andmulti-channel situations (Atabek and Lefebvre 1980, Chrysos and Lefebvre 1993, Chrysoset al 1993). ECS is equally efficient when applied to static representations of the Hamiltonian(construction of the resonance wavefunction in terms of fixed or variationally optimized basissets) or to dynamical ones (where the wavefunction is built step by step through propagationon a grid). Whatever the case, it is often suitable to turn the real coordinate into the complexplane beyond a given distance and make the wavefunction become complex (Nicolaides andBeck 1978, Simon 1979). Diagonalization of a complex (not Hermitian) matrix (when basis-set methods are concerned), often comprising a real block that represents the bound part ofthe resonance wavefunction (see, for instance, Chrysoset al 1990 and references therein),or matching of the propagated quantities (when grid methods are concerned) (Atabek andLefebvre 1980 and references therein) give rise to resonance energies and widths. ECS canalso account for numerical or piecewise analytic potentials and couplings, provided the dataare fitted to a globally analytic form asymptotically. The mathematical grounds of the exteriorcomplex scaling should be searched in the context of the analycity properties of the integratedfunctions and analytic continuation of the physical quantities into the complex plane (Gyarmatiand Vertse 1971, Simon 1979).
According to the ECS, the variabler is scaled beyond a distancer0
r → ρ ={r for r 6 r0r0 + (r − r0) exp(iθ) for r > r0.
(7)
The asymptotic Hankel function (r > r0) then becomes
whereFk,l;k′,l′(r) andGk,l;k′,l′(ρ) designate the two primitives of equation (10); the integrationis carried out with ordinary Simpson’s quadrature for real functions.
3. Numerical implementation
Numerov’s three-point method (Numerov 1933) is applied to buildregular wavefunctions ofboth low- and high-lying states, as well as theoutgoingcombination of regular and irregularsolutions corresponding to the high-lying state. For this purpose, step-by-step propagation isperformed on anN -point spatial grid. In order to take account of the ECS of the coordinate avariable grid stephi is considered. It is given the form
hi ={h for i = 1, . . . , N0
h exp(iθ) for i = N0 + 1, . . . , N − 1(12)
whereh = |hi | = (rmax−rmin)/(N−1)andN0 is the grid point corresponding tor0 (r0 = hN0).Wavefunctions propagated inwards and outwards are built via the recursive relations
ψi−1 = βi
αiψi − γi
αiψi+1 (13a)
ψi+1 = βi
γiψi − αi
γiψi−1 (13b)
whereαi , βi andγi are given as (Romet al 1991)
αi ≡ α(ρi) = hi{1 + 1
12
(h2i−1 + hi−1hi − h2
i
)(E − Vi−1)
}(14a)
βi ≡ β(ρi) = (hi + hi−1){1− 1
12
(h2i−1 + 3hi−1hi + h2
i
)(E − Vi)
}(14b)
γi ≡ γ (ρi) = hi−1{1 + 1
12
(−h2i−1 + hi−1hi + h2
i
)(E − Vi+1)
}(14c)
andi = 2, 3, . . . , N − 1.
3.1. Direction of propagation
Regular solutions always increase with increasingr in the inner classically forbidden regionof the potential (for instance,ψreg ∼ rl+1 for a centrifugal potential). Outward propagationis able to build numerically a regular wavefunction, because a divergent quantity is generatedspontaneously in this direction and predominates by far in the linear combination of the two
3122 M Chrysos and M Fumeron
independent solutions of the Schrodinger equation (Chrysoset al1994). This quantity, havingthe property to decay asr decreases in the inner classically forbidden region of the potential,corresponds exactly to the regular wavefunction. Forr > r0, analytic continuation intothe complex plane shows that the rotated regular wavefunction is formally composed of onedecaying and one divergent term. Numerically, asr increases, the propagation generates thedivergent quantity spontaneously (Chrysoset al 1994), which dominates asymptotically overthe decaying term.
In contrast, only inward propagation can build a pure outgoing wave. This is because,formally, the outgoing wave has to decay asr increases beyondr0. Since numerically thepropagated quantity diverges in the direction of the propagation (which is now inward), thecorrect behaviour of a decaying solution is again generated spontaneously (Chrysoset al1994).
3.2. Initialization of propagation
Two initial conditions are required for each of the propagated quantities. Initialization ofoutward propagation is made at the first two points of the grid (r < r0). This is done by setting
ψreg(r1) = 0 and ψreg(r2) = ε (15)
whereε is an arbitrary real constant.Appropriate initialization of the inward propagation is trickier. The behaviour of both
regular and irregular components should be examined because it is the outgoing wave that isnow concerned. For this purpose, one can write asymptotically
ψreg(ρ) = A(ρ) sin(φ(ρ)) = 1
2iA(ρ)
{eiφ(ρ) − e−iφ(ρ)
}(16a)
ψirr(ρ) = −A(ρ) cos(φ(ρ)) = − 12A(ρ)
{eiφ(ρ) + e−iφ(ρ)
}(16b)
whereA andφ are two well behaved functions ofρ, depending parametrically onV (ρ) andE. The outgoing and incoming wave combinations read
ψout = ψreg(ρ) + iψirr(ρ) = −iA(ρ) eiφ(ρ) (17a)
ψin = ψreg(ρ)− iψirr(ρ) = iA(ρ) e−iφ(ρ). (17b)
Given that, forr →∞±iρ = ∓r sinθ ± ir cosθ (18)
and Re[±iφ(ρ)] retains the sign of Re[±iρ] in part of the upper half complex plane 0< θ <
θmax, the outgoing wave decays while the incoming one diverges asymptotically. The divergentexponential exp(−iφ(ρ)) dominates by far over the decaying exp(iφ(ρ)) asymptotically; onehas
ψreg(ρ) −→kr→∞
12ψin(ρ). (19)
By forming the productψoutψin and making use of equations (17a) and (17b), one obtains
ψout = A2
ψin= 1
2
A2
ψreg. (20)
The latter expression reads
ψout(ρ) ' 1
2πkloc(ρ)
1
ψreg(ρ)(21)
Scaling and computation of CC transition matrix elements 3123
where first-order WKB was applied (it was found to be sufficient) for the amplitude
A(ρ) ' 1√πkloc(ρ)
(22)
andkloc(ρ) =√E − V (ρ) (in reduced units ¯h2/2m = 1) denotes the local wavenumber atρ.
Equation (21) enables one to generate the outgoing solution of the high-lying state, via inwardpropagation. The initialization of the inward propagation is made at the last two points onthe grid,N andN − 1, according to equation (21). Having avoided the explicit computationof the outgoing wave as a linear combination of regular and irregular solutions (that wouldboth diverge asymptotically) accurate matrix elements can be evaluated whatever the potentialinteraction.
3.3. Normalization
For the normalization of the propagated quantities we proceed as follows: we localize the lastmaximum of the, as yet unnormalized, regular solution in the intervalr < r0. Wavefunctioncorrections beyond WKB can be neglected when the maximum is chosen far enough inr.An improved estimation of the maximum is possible by quadratic interpolation between gridpointsrm−1 andrm+1 that are adjacent to the maximum found atrm. The improved maximumis situated atrmax= −b/2a and is given byψmax
reg ≡ ψreg(rmax) = −(b2 − 4ac)/4a, wherea,
b andc are the coefficients of the parabolic equation that fits the propagated quantity atm−1,m andm + 1. Once the maximum is found, a normalized wavefunction is defined over theentire integration range
ψreg(ρ)→ Cψreg(ρ) and C = 1
ψreg(rmax)√πkloc(rmax)
. (23)
In this way, the value of the energy-normalized wavefunction atrmax becomes 1/√πkloc(rmax)
as it should, given that the sinusoidal dependence of equation (16a) is by construction unitythere (maximum of the wavefunction). Note that normalization of the outgoing wavefunctionhas already been accounted for, too, by construction (see equation (21)).
4. Illustrative numerical applications: results
Computations are made on a grid withN = 24 001. Results are reported for two effectivepotentials and two coupling operators for which exact results are available. These are: (a)V eff = l(l + 1)/r2 with T = 1/r2 (in ru) and1l = 0. As an example, we takel = l′ = 2,E = 0.2 cm−1 andE′ = 400 cm−1. This means that the final state wavefunction oscillatesabout 50 times more quickly than the wavefunction of the initial state; this is a common situationfor far-wing interaction-induced light scattering spectra (Chrysoset al 1996). (b) CoulombinteractionV eff(r) = −1/r + l(l + 1)/2r2 with electric dipole moment couplingT = r (in au)and selection rules1l = ±1 (Gao and Starace 1987). Several numerical combinations arechecked.
In order to push forward the characteristics of the method, the results are summarized inthe form of rich graphical information. Comparison is made with exact values and with thoseobtained via other specific methods. Figure 1 shows a typical ECS contour withr0 = 5 auandθ = 0.1 rad. The regular wavefunction of the centrifugal potentiall = 3 is representedin figure 2 for the low-lying stateE = 0.2 cm−1; in (a) no ECS has been used; in (b) ECSwith θ = 0.1 rad andr0 = 5 ru has been applied. Full and broken curves denote the real
3124 M Chrysos and M Fumeron
Figure 1. A typical ECS contour. The coordinate has been rotated byθ = 0.1 beyondr0 = 5 au.
Figure 2. Regular wavefunction of the low-lying state,E = 0.2 cm−1, pertaining to the matrixelement (C) of table 1; (a) with no ECS; (b) with ECS. Full curves, real parts; broken curves,imaginary parts. Once ECS is applied both real and imaginary parts diverge.
Scaling and computation of CC transition matrix elements 3125
and imaginary parts of the wavefunctions, respectively. One clearly sees the divergence ofboth real and imaginary parts when ECS is applied. The high-lying state regular wavefunctionwith l′ = 3 andE′ = 400 cm−1 is shown in figure 3(a) without ECS and (b) with ECS(θ = 0.1 rad andr0 = 5 ru). Note that both real and imaginary parts of the rotated regularwavefunction diverge far more quickly than those corresponding to the low-energy state. Therotated outgoing wavefunction is represented in (c). Its rapid decay, beyondr0, is clearly seenon the figure. It is this rapid convergence of the rotated outgoing wave that makes the integralconverge. Figure 4 shows the quantity to be integrated,ψout(E
′, l′; ρ)T (ρ)ψreg(E, l; ρ) (in ru),
Figure 3. Wavefunctions of the high-lying state,E′ = 400 cm−1, pertaining to the matrixelement (C) of table 1; (a) regular solution with no ECS; (b) regular solution with ECS and(c) outgoing solution with ECS. Full curves, real parts; broken curves, imaginary parts. OnceECS is applied, both real and imaginary parts of the regular solution diverge; this divergence is farmore rapid than for the state withE = 0.2 cm−1; instead, the outgoing solution quickly decreasesbeyondr0. It is this rapid convergence of the rotated outgoing solution that makes the productψout(E
′, l′; ρ)T (ρ)ψreg(E, l; ρ) converge.
3126 M Chrysos and M Fumeron
Figure 3. Continued.
Figure 4. Quantity to be integrated,ψout(E′, l′; ρ)T (ρ)ψreg(E, l; ρ) (in ru), for the matrix element
(C) of table 1; (a) with no ECS; (b) with ECS.
Scaling and computation of CC transition matrix elements 3127
Tabl
e1.
Com
pute
din
tegr
als
usin
g(i)
the
cent
rifug
alef
fect
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land
the
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2an
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)th
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mb
inte
ract
ion
with
,as
coup
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dive
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t(D
)el
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inle
ngth
formr
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rre
pres
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tive
valu
esof
para
met
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dl.
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paris
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mad
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obta
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with
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rsp
ecifi
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ithex
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reem
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lcom
pute
del
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tsfo
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timal
valu
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the
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tion
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pt;
with
noE
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(sim
ply
com
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obta
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sar
e(l
engt
h)×(e
nerg
y)−1
(C)
and(
leng
th)−
2×(e
nerg
y)−1
(D).
Spe
cific
atio
nsT
his
met
hod
Gao
and
Sta
race
(198
7)a
EE′
ll′
θ=
0in
tegr
alθ o
ptIn
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ompl
expl
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By
part
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xact
C0.
2cm−1
400
cm−1
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−1.2
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ru0.
020
1.194×
10−7
ru—
—1.
194×
10−7
rub
D0.
016
au0.
059
au1
2−5
94.6
au0.
627
114.
193
au11
4.24
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91.9
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114.
240
auc
D0.
016
au0.
059
au2
1−1
453.
9au
0.61
833
.446
au33
.414
au34
.836
au33
.413
auc
D0.
1au
0.7
au0
1−1
0.56
au0.
556
1.29
928
8au
1.29
930
au1.
293
30au
1.29
930
auc
D0.
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0.7
au2
1−1
4.90
au0.
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783
au0.
087
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781
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087
82au
c
aM
etho
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ped
spec
ifica
llyfo
rth
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oulo
mb
prob
lem
(Gao
and
Sta
race
1987
).b
Hoc
hsta
dt19
86(p
274)
.c
Land
auan
dLi
fshi
tz(1
958)
(mat
hem
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ices
).
3128 M Chrysos and M Fumeron
Figure 5. (a) Evolution of the imaginary part of the complex matrix element against its realpart for different values ofθ , in the interval fromθ = 0 (no rotation) toθ = 1 rad in steps of1θ = 10−3 rad. The trajectory exhibits a clear cusp pattern whose real part marks precisely thevalue of the matrix element. The region around the characteristic point has been blown up in theinset (an arrow indicates precisely its position). At the cusp, the curvature of the trajectory becomesa maximum. This is equivalent to the minimization of the quantity1s(θ), where1s(θ) denotesthe distance between the trajectory points corresponding to adjacent rotation anglesθ andθ +1θ .(b) The quantity1s(θ) (plotted per unit1θ ) as a function ofθ . The centrifugal potential withl = l′ = 3 has been taken as the effective interaction; as coupling, the 1/r2 operator has beenconsidered; initial and final energies were chosen asE = 0.2 cm−1 andE′ = 400 cm−1.
(a) without ECS and (b) with ECS. Its substantial confinement inside a very restricted intervalof the scaled coordinate enables one to evaluate the CC matrix element by simple three-pointSimpson’s quadrature, provided the optimal rotation angle is found and a sufficiently densegrid is used.
How to evaluate the optimal angle of an ECS contour may not be an easy task. Whendifferentθ are checked, deviations of the computed results are obtained, especially for relativelyshort box sizes. However, once the imaginary part of the complex matrix elements is plotted
Scaling and computation of CC transition matrix elements 3129
Figure 6. Same as in figure 5 but for the Coulomb potential and a divergent coupling (the electricdipole operator in length form). Specifications:l = 2, l′ = 1, E = 0.1 au andE′ = 0.7 au. Acusp is found again, localized precisely at 0.087 83 au. Note that when no rotation is applied thevalue of the integral is erroneous, differing from the exact result by several orders of magnitude. In(b) the quantity1s(θ) (plotted per unit1θ , with1θ = 10−3 rad) is represented. The minimum ofthis function marks the optimal value of the rotation angle. The region of the minimum has beenblown up in the inset.
against the corresponding real part for different values of the rotation angleθ , well definedθ -trajectories are found which reveal striking pathological patterns marked by cusps, loops orinflections. Similar structures are also met in a different physical context, as fingerprints ofa manifestation of a resonance (Doolen 1975, Moiseyevet al 1978); within that context, atthe peculiar point of the trajectory, real and imaginary parts of the plotted quantity are seen tocorrespond to the resonance energy and resonance half-width (some representative resonancetrajectories are illustrated in figure 1 of Bylicki (1991)); it is noteworthy that cusp conditionshave also been found (and used) in calculations of optimal scattering transition probabilityamplitudes (Peskin and Moiseyev 1992). Here, to our pleasant surprise, the real part of the
3130 M Chrysos and M Fumeron
Figure 7. Same as in figure 6 but with specifications:l = 0, l′ = 1,E = 0.1 au andE′ = 0.7 au.
plotted quantity at the cusp, loop or inflection marks the exact value of the matrix element,analogously with resonance quantization. The imaginary part of the plotted quantity at thisposition also has a physical meaning: it corresponds precisely to the value of the integral∫ ∞
r0
ψirr (E′, l′; r)T ψ(E, l; r) dr
where ψirr (E′, l′; r) denotes theirregular solution of the Schrodinger equation (2) forthe final state. The origin of these characteristic patterns is not clear to us yet. Thejustification of a relation between the peculiar point of theθ -trajectory and the true value of thecorresponding CC integral should be searched for in terms of Puiseux expansions and Cauchy–Riemann conditions (Moiseyev 1998, p 263). In practice, an effective criterion for localizingautomatically the characteristic point on aθ -trajectory is based on the observation that, there,the density of points is maximized. This means that, at the characteristic point, the trajectory
Scaling and computation of CC transition matrix elements 3131
Figure 8. Same as in figure 6 but with specifications:l = 2, l′ = 1, E = 0.016 au andE′ = 0.059 au.
length1s = 1s(θ) corresponding to two neighbouring rotation anglesθ andθ +1θ (with1θ constant) has to reach its absolute minimum. Table 1 shows the computed integrals for thecentrifugal and Coulomb potentials and for two different coupling operators (one convergentand one divergent) and for representative values of parametersE andl. Comparison is madewith results obtained with other specific methods, as well as with exact values. As we can see,very satisfactory agreement is obtained for all computed elements for optimal values of therotation angle; values obtained with no ECS (by simply settingθ = 0) are of course erroneous.The inconvenience of our procedure lies in its relatively high numerical effort; this is a generalconclusion for all complex coordinate approaches and has already been stressed by Basileet al (1989) and Głazet al (1994) in relevant studies. Only a few values of integrals can becomputed per CPU second on a small HP 2094A workstation. This fact seems to restrict thepractical utility of our algorithm to computations of a reduced number of isolated integrals forspecific tasks, rather than to simulations of processes where a large number of matrix elementsare needed for converged calculations.
3132 M Chrysos and M Fumeron
Figure 9. Same as in figure 6 but with specifications:l = 1, l′ = 2, E = 0.016 au andE′ = 0.059 au.
Figure 5 refers to the centrifugal effective potential and 1/r2 operator; (a) shows theθ -trajectory of the matrix element, whereas (b) represents the spacing1s (normalized per unit1θ , with1θ = 10−3 rad) as a function of the rotation angle. Figures 6–9 illustrate differentCC matrix elements with the Coulomb interaction (Z = 1) and the electric dipole momentoperator in length form,r. For all figures, (a) representθ -trajectories and (b) the quantities1s(θ) as a function ofθ . In the insets, the regions around the peculiar points of the trajectoriesare blown up for clarity. Since1s(θ) serves as a sensitive probe for detecting the behaviourof the complex matrix elementY (θ), the sharp peaks observed in certain figures reflect theappearance of some jagged structure on the correspondingθ -trajectories for specific values ofθ , due to the highly nonlinear dependence ofY (θ) with the parameterθ .
Scaling and computation of CC transition matrix elements 3133
5. Conclusion
This work had as its aim the accurate computation of CC matrix elements〈E′, l′|T |E, l〉 as anattempt to handleconvergentanddivergentcouplings, andsaturatingandpersistentpotentials,on an equal footing. To this end, a constraint-free methodology and a black-box numericalprocedure were developed, provided the potential was globally analytic asymptotically, thefunction T (r) was holomorphic and the matrix element was meaningful. There were fourmain steps in our methodology: (a) we built the regular solutions of the Schrodinger equationfor initial and final states; (b) we formed the outgoing solution of the Schrodinger equation forthe high-energy state; (c) by means of exterior complex scaling of the coordinate, we forcedthe outgoing solution to die off asymptotically and (d) we determined the exact value of thematrix element via aθ -trajectory plot. The latter step became possible by making use of theobservation that, analogously with resonance quantization, a striking cusp, loop or inflectionstructure marked the optimal value of the integral. Apart from its direct practical utility inthe computation of isolated integrals for specific tasks, this work is expected to allow for afuller perception on aspects involving continuum–continuum interactions and, via the complexplane, the better understanding of analogies between resonance properties and concepts of thecontinuum. Work is in progress in our institute.
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