An IPA procedure for bound-continuum diatomic transition intensities

Chemical Physics ELSEVIER

Chemical Physics 213 (1996) 295-301

An IPA procedure for bound-continuum diatomic transition intensities V.S. Ivanov 1, V.B. Sovkov Institute of Physics and Russian Centre of Laser Physics, St. Petersburg State University, 1 Ulyanovskaya Street, Petrodvorets, St. Petersburg 198904, Russian Federation Received I1 June 1996

Abstract

An inversion procedure for determining a repulsive diatomic molecule potential energy curve and a function of the electronic transition moment operator from structured bound-continuum transition intensity data is proposed. The method is based on the first-order perturbation inversion of node positions differences in an experimental spectrum and its zeroth order simulation. This approach is free from the restrictions of the RKR-like procedure of Child, Essen and Le Roy (J. Chem. Phys. 78 (1983) 6732). The method is tested for the 7Li2 (33[Ig(v = 17,N = 0) ~ a~Eu+) and 7Li2 + (3]~g(t;= 10, N = 10) ~ a 322,+ ) model bound-free transitions spectra.

1. Introduction

The determination o f molecular adiabatic potentials from spectroscopic experimental data is one of the leading problems in the theoretical molecular spectroscopy. The mostly used method to solve this problem in a case of diatomic molecules bound electronic states is the well-known method of Rydberg, Klein and Rees [ 1 ] (see also [2] ). The results of the RKR analysis can be refined by means of the IPA (inverse perturbation approximation) procedure [ 3 ], which connects the potential energy curve corrections with shifts of computed eigenenergies from their experimental positions, in a framework of the first-order perturbation approximation. Repulsive potentials are frequently determined from the analysis of the bound-free continuum structure.

I E-mail: [email protected].

The nature of structures in continuums was reviewed in detail in [4], for example. The simplest analysis of structured continuums is based on the classical reflection approximation [5], which is one of the Frank-Condon principle formulations (both coordinates and momenta of nuclei conserve during an electronic transition, see also [ 6 ] ) . In this approximation a transition intensity distribution can be estimated as a mere reflection of the interatomic distance density function (e.g., a squared nuclear wavefunction) from the difference potential of the transition (the final state potential energy function minus the initial state one). However, an accuracy of this approximation is only satisfactory for a very steep difference potential. The much more accurate method to determine a repulsive state potential function from an experimental structured continuum is the RKR-like procedure of Child, Essen and Le Roy [7]. This procedure is based on the uniform harmonic approximation [8], and al-

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lows to connect explicitly extrema positions in an observed spectrum with turning points of the repulsive potential. However, this method is also applicable to those systems only, where the difference potential is a monotone function of the internuclear coordinate. It is usually recommended [9,11] to use a fitting procedure for determining a repulsive potential energy curve in a case where the RKR-like procedure [7] is not accurate enough or applicable at all. The obvious inelegance of this approach has stimulated us to find another method, suitable for the analysis of structured continuums in those cases where the procedure [7] is not good. A new method for determining turning points of a repulsive potential energy function and a transition moment function is described below. This method uses the inverse perturbation approximation and is free from the restrictions of the RKR-like procedure [ 7]. The method is tested for the 7Li2 (23IIg(v = 17, N = 0) ~ a 3£u+ ) and 7Li2 ( 3 3 ~ + ( u = 10, N = 1,0) , a 3 £ S ) bound-free transitions spectra, experimentally measured by the PFOODR (perturbation facilitated optical-optical double resonance) technique [ 12,13].

alE> =JlE'> (°) (°)(~E' - - IflUsl ~ E>(°~ dE'. f

(4)

The consequent correction for the matrix element Cbf(P ) is

~°~(°~

flcb'( =J

de',

(5)

or flCbf (P) = ('B'li) C (0) ( p )

+P;C(b°)J

(°)(°>

(v') (°)(E'EI flui I _u ' E)(°) de'. (6)

Let us consider the frequency point v0(°), corresponding to the zeroth order spectrum node, i.e.

2. Theory The expression for the intensity distribution in a spectrum of an electronic transition from a selectively populated rovibronic state is B (v) = Bo }Cbf (~')12

Let us assume that the final state potential energy curve Uf is known in some zeroth approximation (we shall characterize the zeroth approximation values by the superscript " ( 0 ) " ) . According to the quantum mechanical perturbation theory, a small correction flU: to U f(°) leads to the first-order correction for a wavefunction

,

(1)

(.o

=o

(7)

For this frequency the first term in (6) turns to zero, and the sign P of the principal value integration in the second term can be omitted. So, we have

where v is the light frequency, B0 is a coefficient (if B0 = 8¢r3/3h 2, B(v) is the Einstein coefficient for photoabsorption; if B0 = 4~rZv3/3hc 3, B(v) is the Einstein coefficient for spontaneous photoemission), Cbf ( v ) is the bound-free transition matrix element

.<., (.o
Cbf (P) = (U IMI E),

Let us assume additionally that flUf(r) changes slowly with internuclear distance r relative to the wavefunctions iE'). Then

(2)

where Iv) is the bound state wavefunction of nuclear motions, IE) is the repulsive state one, amd M is the transition moment function. The initial state energy E,,, the final state energy E and the light frequency v are connected by

E~, - E = ehu,

(3)

where h is the Planck constant, • = - 1 for photoabsorption and • = 1 for photoemission.

<°'

x

Eo(°) - E'

dE'.

(8)

(o) ( E' IflUf [ E(o°)) (o) ,~ flUf ( r(oO,) (3(E' - E(o°) ) , (9) where %(o) is the zeroth approximation turning point, corresponding to the energy E0(°). In a small vicinity of the point v~ °), the following equations are valid:

297

VS. lvanov, V.B. Sovkov/Chemical Physics 213 (1996) 295-301

C~°) (t,) ~ Z

dv

Eo( ° ) - E

,

(10)

where t'o is the node position in the experimental spectrum. Substituting ( 9 ) - ( 1 1 ) into (8), we get in a firstorder approximation

(2) Choice of the zeroth order approximate potential energy function U~°) ( r). (3) Determination of the zeroth order transition moment function M(°)(r) using (13) with the zeroth order wavefunctions IE)(0) (r). (4) Direct calculation of the zeroth order intensity distribution in the spectrum and estimation of the nodes positions %(0) (and E0(°)). (5) Determination of the coordinates r(0°), where =

~0

~Uf(r(0°>)

~Eh(),(o°)-uo).

(12)

Eq. (12) is the main result of the consideration. It allows us to correct the zeroth order potential function U}°) (r) according to position differences of the nodes in the calculated and experimental spectra. This approach as sumes that the i ni ti al state potential Ui(r) and the transition moment function M(r) are known or, at least, that M(r) does not influence the node positions appreciably. Notice that the latter assumption has been also adopted in [7]. But the whole inversion procedure should include the determination of M(r) as well. The approach to this problem that has been proposed in [7] uses essentially the approximation of the one transition point for every p. Hence, this approach has the same restrictions as the whole method [7]. Therefore, in a general case another procedure for determining M ( r ) should be derived. We suggest to use an exact quantum mechanical relation

M(r)

= f (v [gl E)
dE '

(13)

where the matrix element (vlMIe) can be determined as a square root of the experimentally observed ICbf(V)] 2 (Eq. ( 1 ) ) . The proof of (13) is absolutely evident and does not need any additional elucidations.

3. Computational algorithm According to (12), (13), the procedure for determining a repulsive potential energy function Uf(r) can be performed in a following scheme. (1) Estimation of the nodes positions u0 (and the corresponding energies E0) in an experimental spectrum.

°

(6) Construction of the new approximation for Uf(r) so that its values in the points r(o°) are equal to E0's. The steps ( 3 ) - ( 6 ) should be repeated iteratively until the calculated spectrum reproduces the experimental one well enough. With the aim to reduce the intermediate calculations errors, it is advisable to place the knots of the net where M(r) is calculated by (13) in points, close to the bound state wavefunction Iv)(r) extrema. It is also advisable to apply a smoothness procedure to a functional dependence of E0's on r(00~),s at every step of the computational loop - this will assure the requirement of 6Uf(r) to be smooth. The procedure described above can be often simplified by taking into account the next circumstances. (1) The influence of M(r) on the node positions in a spectrum is relatively small as usual [4]. Hence, there is no necessity to refine M(r) at every step of the loop. When the zeroth order spectrum is far enough from the experimental one, several first steps can be done in the Condon approximation M ( r ) = const. (2) The determination of the nodes positions v0(°) does not need the whole spectrum calculation at every step; moreover, it does not need an exact normalization of wavefunctions. There are many possibilities for the choice of a zeroth order approximation U~°) (r). It can be taken from nonempirical calculations, estimated by using the reflection or RKR-like analysis or by a simple fitting procedure with a low precision.

4. Numerical examples and conclusions As a numerical test, the above method was applied to synthetic data for photoemission from the v =

298

ES. lvanov, V.B. Sovkov/Chemical Physics 213 (1996) 295-301

17, N = 0 rovibrational level of the 23IIu electronic state of 7Li2 into continuum levels of the a 3~ ,+ state. The synthetic "experimental" intensities were generated by using the programs [ 14] for the NumerovCooley-Blatt method; the RKR potential of the 23Hg state has been calculated using the molecular constants from [13], the nonempirical potential energy curve of the a 3~,+ state has been taken from [15] and the nonempirical transition moment function has been taken from [ 16]. Notice that in this test case the spectral intensity distribution has an interference structure (the difference potential is nonmonotone), i.e., the method [7] is not applicable. As a zeroth approximation U f ) (r) we have chosen the extrapolation into the short-range region by the Morse function of those points from [ 13] that lay below the dissociating limit D,. ~ 8517 cm -~ of the a 3Y + state (the shallow well of this state has the minimum ~ 8190 cm -j at re ..~ 4.17 ~ ) . The Pade approximation technique [ 17] along with the method of a singular value decomposition (SVD) has been applied to the intermediate approximate potentials as a smoothness procedure. The exact potential energy curves, the zeroth order approximation and the final IPA results are shown in Fig. 1. During the calculation we have determined the zeroth order transition moment function M (°~ (r) using (13) with the zeroth order wavefunctions [E)(0) and performed 16 steps of the IPA procedure without an intermediate correction of M(r). After the 16th step M(r) has been corrected to get the next approximation M(l)(r) and 9 additional steps of the IPA procedure have been performed without further correction of M(r). The "experimental" spectrum, the zeroth approximation spectrum and the 16th approximation spectra before and after correction of M(r) are shown in Fig. 2. The latter coincides in the figure scale with the 25th approximation spectrum and the "experimental" one. The approximate transition moment functions M (°) (r) and M (~) (r) are compared with the "exact" one in Fig. 3. Some deviations of the M (t) (r) points from M(r) can be explained by the fact that the integration step has not been short enough (it has been chosen 10 cm -1, what is more or less characteristic for an experiment) and by neglecting the discrete portion of the spectrum in (13). The distances between the final approximate turning points and the "exact" ones are shown in Fig. 4. The convergence of

~lb

=--"35000

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E

/'/" '~'\' . ' /

ii

la//

30000 -

12000-

-

25000 -< - -- ~3Q

2/3b

20000-

10000

\

15000-

\

10000-

5000

'

i

2

'

i

3

2.5

,

i

4

,

i

5

'

i

6

'

i

7

\

3.0

'

i

8

,

i

9

3.5

,

i

10

'

i

,

11 r

(AI

Fig. 1. Potential energy curves of the a3.E +, 23He and 333`+ states of 7Li2, which have been used in calculations, l a - 23IIg state; lb - 33~ + state; 2 - a3E + (the nonempirical points from I151 and their spline-interpolation); 3a - the difference potential for the 23[1g ~ a3E + transition; 3b - the difference potential for the 333`+ --* a33`+ transition; 4 - the zeroth approximation for the a33`u+ state potential; 5 - the final result of the first test calculation (see text for details).

the most short-range turning point towards the "exact" one as a function of the step number is shown in Fig. 5. The conclusion is that the IPA procedure with the M(r) determination converges to the exact data satisfactorily enough and gives very high accuracies the turning points deviations from the "exact" positions do not exceed 0.0025 A here, what is of the same characteristic value as in the test calculation of [7] and even smaller if measured in the scale of the Frank-Condon zone width (the distance between the bound state turning points is ,-- 2.85 ,~ in our case). The simulated spectrum of the 7Li2 (23Hg(v = 17, N = 0) , a 3Y,+ ) transition is compared with the experimental data from [ 13] in Fig. 6. An additional calculation we have made was the determination of the transition moment function M(r) -

V.S. lvanov, V..B. Sovkov/Chemical Physics 213 (1996) 295-301

INIENSI]Y

(photons/see/era ')

i

~<

q 'i

299

0.002

,,2 ~"° 0 001

i'

"! ,I {

-

1

-2

"!

0.000

- -3

00Ol

-0002

21000

22000

23000

24000

214 v (cm

26

218~-

3,0

312

')

Fig. 2. The model 7Li2 (23IIx(v = 17, N = 0) ~ a-a~,,+ ) transition spectra used in the test of the IPA method (see text for details): 1 - the "'experimental" synthetic spectrum; 2 - the zeroth approximation spectrum; 3 - the 16th order spectrum before the M(r) correction; 4 - the 16th order spectrum after the M(r) correction.

34

r 0 (A) Fig. 4. D i s t a n c e s b e t w e e n the final a p p r o x i m a t e n o d e s p o s i t i o n s a n d the e x a c t o n e s f o u n d in the IPA test c a l c u l a t i o n ( s e e text f o r details). 005

I

0.04 30 L,

2.5

0.03 -

-10

20

o

1.5

002

-lb

--

'!

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.

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.

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.

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,

.

°

i

5

10

15

20

25

Step number

Fig. 5. Distance between the most short-range approximate turning point and its "exact" positions as dependent on the IPA step number (see text for details).

Fig. 3. The transition moment function M(r) of the Li2 (23Iie ~ a3E + ) transition: l a - nonempirical ("exact") points from 1161; lb - their Pade approximation; 2 - the function M d ) / ( r ) , which has been determined at the zeroth step of the IPA procedure; 3 - the function M (1) ( r ) , which has been determined at the 16th step of the IPA procedure (see text for details).

for the7Li2 3 -3~ , + (v 10, N 10) ~ 3 + transition from the experimental data [ 12] (Fig. 7). The RKR potential function of the 3 3E;g+ state has been taken from [ 12], the final state potential has been the same as in the previous calculation. Unfortunately, we have been unable to apply Eq. (13) satisfactorily because o f the following reasons: the relatively low precision of the graphical experimental data, the influence of the transition into the b31-[, state and a rather large portion of the intensity concentrated in the discrete portion of the spectrum. But due to the good

2

1

i

t-

4020

4460

4900

Wavelength [~)

Fig. 6. T h e 7Li2 ( 2 3 1 1 g ( t , = 17, N = 4 ) ----+(13~ + ) t r a n s i t i o n s p e c tra: 1 - the experimental spectrum from 1131; 2 - the simulated spectrum as described in text.

V.S. lvanov, V.B. Sovkov/Chemical Physics 213 (1996) 295-301

300

.7 0 0

(/)

4450 Wavelength

[~,]

Fig. 7. The spectra of the 33E+(v = 10, N = 10) --* a3y,,+ transition: 1 - the experimental spectrum from 12], 2 - the simulated spectrum using the M(r) function determined by the quasiclassical procedure (see text for details).

1.4

The main conclusion is that the proposed method gives very reliable results in determining the repulsive potential energy curves and the transition moment functions and can be efficiently combined with other techniques to obtain the best quality data. Another result of the consideration is the possibility to obtain Uf(r) and M(r) simultaneously from the one structured continuum both in the case of a monotone difference potential (see also [7] ) and of a nonmonotone one. This is valid when the properties of smoothness of Uf (r) and M ( r ) are fulfilled. Indeed, Eq. (13) shows that the unique function M(r) can be found for any U f ( r ) at most coordinate points except for the points where Jv)(r) = 0. This is the requirement for these poles to be compensated by the analogous features of the numerator of (13) along with the smoothness of the functions Uf(r) and M(r), which make the statement at the beginning of this paragraph valid.

3z~ 1.2

12 o

1.0

Acknowledgements

o 3

0.8 0,6 ~.

0.4

•

0.2 0.0

I

[

I

I

2.5

30

3.5

40

r (A)

Fig. 8. The transition moment function M(r) of the 33~,+ (v = 10, N = 10) ~ a3E + transition: l - the points determined by the quasiclassical method of [711 from the experimental graphical data of [12,13]; 2 - their Pade approximation; 3 - the result of the equation (13) application to the synthetic data (see text for details).

reflection nature of the spectrum we have been able to use the quasiclassical procedure of [7]. The determined points of the M(r) function and their Pade approximation are shown in Fig. 8, the spectrum generated using this M(r) is compared with the experimental one in Fig. 7. Afterwards we have applied Eq. (13) to these synthetic data to obtain the M(r) points shown in Fig. 8 as well. Their deviations from the Pade approximant is of the same value as the input points; these deviations originated mainly from neglecting the discrete part of the intensities.

We would like to acknowledge deeply Professor Li Li and Professor A.M. Lyyra for providing us the experimental 7Li2 (335;+(v = 10, N = 10; v = 9, N = 10) --~ a 3~u+ ) and 7Li2 (23Hg(v = 17, N = 4; v = 4, N = 11; v = 2, N = 4) ---, a 3~ ,+ ) spectra, as well as Professor A. Dalgarno, whose kind letter has stimulated us for this research. This work was supported by the Russian Foundation for Basic Researches.

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V.S. Ivanov, V.B. Sovkov/Chemical Physics 213 (1996) 295-301 A.S. Dickinson, J. Mol. Spectrosc. 44 (1972) 183; W.G. Richards and R.E Barrow, Proc. Phys. Soc. (London) 83 (1964) 1045; J.A.C. Gallas, Phys. Lett. A 124 (1987) 290; S.M. Kirschner and J.K.G. Watson, J. Mol. Spectrosc. 51 (1974) 321; J.N. Huffaker, J. Mol. Spectrosc. 71 (1978) 160; C. Schwartz and R..I. Le Roy, J. Chem. Phys. 81 (1984) 3996; M.S. Child and D.J. Nesbitt, Chem. Phys. Lett. 149 (1988) 404; R.M. Rotb, M.A. Ratner and R.B. Gerber, Phys. Rev. Lett. 52 (1984) 1288. 13] M.M. Kosman and J. Hinze, J. Mol. Spectrosc. 56 (1973) 93; H. Helm, in: SASP'84: Symp. Atom. & Surface Phys., Maria Aim, Salzburg, 29 Jan.-4 Feb. 1984, Innsbruck, p. 16; C.R. Vidal and H. Scheingraber, J. Mol. Spectrosc. 65 (1977) 46; G. Gouedard and J. Vigue, Chem. Phys. Lett. 96 (1983) 293; I.P. Hamilton, J.C. Light and K.B. Whaley, J. Chem. Phys. 85 (1986) 5151; C.R. Vidal and W.C. Stwalley, J. Chem. Phys. 77 (1982) 883; J.M. Hutson, J. Phys. B. 14 (1981) 851. 14] J. Tellinghuisen, Photodissociation and Photoionization, (Ellis Horwood, Chichester, 1985) p. 299.

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151 N.S. Baylis, Proc. Roy. Soc. (London) A 158 (1937) 551. 161 M. Lax, J. Chem. Phys. 20 (1952) 1752. [71 M.S. Child, H. Esssen and R.J. Le Roy, J. Chem. Phys. 78 (1983) 6732. [8] M.S. Child, Mol. Phys. 35 (1978) 759; P.M. Hunt and M.S. Child, Chem. Phys. Lett. 58 (1978) 202. 191 N.E. Kuzmenko and V.V. Eremin, in: Collision and Radiation Processes with Excited Particles (Riga, 1987) p. 103 [in Russian ]. 110] R.J. Le Roy, R.G. Macdonald and G. Bums, J. Chem. Phys. 65 (1976) 6699. [ 11 ] V.S. Ivanov and V.B. Sovkov, Opt. & Spectrosc. 74 (1993) 52. [121 A. Yiannopoullou, K. Urbanski, A.M. Lyyra, Li Li, B. Ji, J.T. Bahns and W.C. Stwalley, J. Chem. Phys. 102 (1995) 3024. [131 Li Li, private communication (1995). [ 14] M.S. Aleksandrov, V.A. Elokhin and V.S. Ivanov, Dep. AllUnion Institute of Scientific and Technical Information, No. 4213-81 (1981) pp. 71-82. 115] I. Schmidt-Mink, W. Miiller and W. Meyer, Chem. Phys. 92 (1985) 263. [16] L.B. Ratcliff, J.L. Fish and D.D. Konowalow, J. Mol. Spectrosc. 122 (1987) 293. [171 J.A. Baker and P. Graves-Morris, Pade Approximation (Addison Wesley, New York, 1981)