- Email: [email protected]
A distorted-wave study for core-excitation processes in CO2 by electron impact
Journal of Molecular Structure (Theochem) 464 (1999) 49–57
A distorted-wave study for core-excitation processes in CO2 by electron impact T. Kroin a,*, S.E. Michelin a, K.T. Mazon a, D.P. Almeida a, M.-T. Lee b a
Departamento de Fı´sica, Universidade Federal de Santa Catarina, 88049, Floriano´polis, SC, Brazil Departamento de Quı´mica, Universidade Federal de Sa˜o Carlos, 13565-905, Sa˜o Carlos, SP, Brazil
b
Abstract In the present work, the distorted-wave approximation (DWA) is applied for the first time to study core-level excitations in 1;3 molecules by electron impact. More specifically, we have calculated differential and integral cross sections for the X 1 S⫹ g ! Pu (carbon 1s 2s g ! 2p u) transitions in CO2 in the 320–800 eV incident energy range. The frozen-core generalized oscillator strengths (GOS) for the transition leading to singlet excited states calculated using both the DWA and the first Born approximation (FBA) at 1290 eV are also reported. Our calculated DWA GOS are in general 20% lower than the corresponding FBA, results which seem to indicate that the FBA is not valid at that incident energy. Moreover, we have also calculated the ratios between the distorted-wave integral cross sections for the transitions leading to the triplet and the singlet excited states as a function of incident energy. The comparison of these data with the experimental ratio R(3A:1A) obtained by dividing the intensities of the autoionizing electrons from the same excited states in the 320–800 eV range is encouraging. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Core-excitation; CO2; Distorted-wave approximation PACS Number: 3480G
1. Introduction Inner-shell processes, involving the promotion or the removal of a 1s electron by both electron and photon impact have received considerable attention. As a result of the high degree of localization of these orbitals, the shapes and energies associated with such core-level excitation and ionization bands do not show significant changes as one moves from the gaseous state to the solid state [1]. Therefore, the excitation and ionization of the core-level electrons have played a significant role in the development of
* Corresponding author. Fax: ⫹ 55-48-331-9946. E-mail address: [email protected] (T. Kroin)
equipments for characterizing both solid and gaseous samples [2]. Despite the increasing experimental interest in this field of investigation [3–6], on the theoretical side very few studies involving core-excitation processes in molecules have been carried out. To date, a limited number of investigations have almost exclusively been at the first Born approximation (FBA) level [7]. Although the FBA can provide quite accurate generalized oscillator strengths (GOSs) for innershell transitions at very high incident energies, it is not expected to work out when incident energies approach the excitation threshold. In contrast, despite the recent development of several solid-based abinitio multichannel theories for treating both elastic and inelastic electron–molecule collisions [8–10] in
0166-1280/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(98)00534-X
50
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
the low-energy range, there are no reported applications for studying core-level excitation processes using these theories. Among various theoretical methods, the distortedwave approximation (DWA) have received considerable attention during the past few years. The DWA [11] and the first-order many-body theory [12], which is essentially equivalent to DWA, have been successfully and widely applied to calculate electronic excitation cross sections and coherence parameters for atomic targets in the intermediate and high energy range. Recently, the DWA has also been applied to investigation of electronic excitation of low-lying valence-shell orbitals in molecules by low energy electron impact. It was found [13–15] that the DWA calculations can in general lead to similar excitation cross sections to those obtained using the Schwinger multichannel method (SMC) at a few-state coupling level of approximation [16,17] for incident energies a few eV above the threshold. Also, comparison with available experimental data has shown that the DWA can in general reproduce the angular distribution of measured differential cross sections (DCS), although it usually overestimates the magnitude of the cross sections by a factor of 2–3. The reliability of the calculated cross sections increases with increasing incident energy. In view of the simplicity of the DWA, the extension of this method to studies involving core-excitation processes in molecules is clearly of interest. In this work, we report a DWA application of such processes in the CO2 molecule. More specifically, we have calculated differential and integral cross sections, as 1;3 Pu (carbon 1s well as the GOS, for the X 1 S⫹ g ! 2s g ! 2p u) transitions in CO2 molecule by electron impact in the 320– 800 eV energy range. We hope the present study would help with qualitative behavior and quantitative estimates that can be expected for these physical quantities. For the transitions studied herein, some experimental investigations were reported recently. For example, 1 the relative GOS for the X 1 S⫹ g ! Pu transition has been measured by Roberty et al. [7] using 1290 eV incident electrons. They have normalized the GOS for this transition to their calculated data using the full relaxed-core FBA. More recently, Almeida et al. [18] 1;3 Pu (carbon 1s 2sg ! have investigated the X 1 S⫹ g ! 2p u) transitions in CO2 molecule by observing the
autoionizing electrons via decay from the 1,3P u inner-shell excited states. These authors have reported ratios between the intensities of the autoionizing electrons from the triplet and singlet excited states, denoted by R (3-A : 1-A), as a function of incident energy at the scattering angle of 90⬚. As the final ionic state is formed via an inner-shell excited state, the autoionization yield depends on two transition probabilities, that for formation of the excited states by electron impact and that for their subsequent decay. However, it is expected that the contribution arising from the decay is independent of the impact energy. Hence, the measured R(3-A : 1-A) are in fact proportional to the ratios between the excitation cross sections for the transitions leading to the 3P u and 1P u states by electron impact. Thus, the comparison of the calculated ratios with the measured data may provide useful insight into the nature of these core-excitation processes. The organization of this article is the following: in Section 2 we briefly outline the used theory. The results are shown in Section 3, where we also present our concluding remarks.
2. Theory The details of the basic theory used in the present work have already been presented elsewhere [13– 15,19–21] and will be only briefly described here. The differential excitation cross sections for electron–molecule scattering averaged over molecular orientations are given by: ds 1 Z ^ 0 2 a sin b d b d g f k d ; f dV 8p 2
1
where k^ 0f is the direction of scattered electron linear momentum in the Laboratory-Frame (LF), whereas the direction of the incident electron linear momentum is taken as zero. The (a , b , g ) are the Euler angles which define the direction of the molecular principal axis. The Body-Frame (BF) amplitude f k^i ; k^f is related to the T-matrix elements by the formula: f k^i ; k^f ⫺2p2 Tif :
2
Within the DWA framework, the transition T-matrix
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
is given by: ⫹ Tif 具w1 C ⫺ kf jUse jw0 Cki 典;
3
where w 0 and w 1 are the initial and final target wavefunctions, respectively. These wavefunctions are Slater determinants with appropriate symmetries. ⫺ C ⫹ ki and Ckf are the initial and final distorted continuum wavefunctions with the outgoing (⫹) and incoming-wave (⫺) boundary conditions, respectively. Use is the static-exchange potential operator. The distorted wavefunctions are solutions of the Lippmann–Schwinger equation: ^ C ^ Fk~ ⫹ G ^ 0 U Ck~ k~
being the free-particle Green’s operator with with appropriate boundary condition, and Fk~ is the ~ These plane wavefunction, with linear momentum k. wavefunctions Ck~ were calculated using the Schwinger variational iterative method (SVIM) [22] in the static-exchange field of ground-state target. The electronic excitation T-matrix can be partialwave expanded as: 0
Tif 2=p
X X il⫺l * ^ Tlml 0 m 0 Ylm ki Yl 0 m 0 k^f ; k k lm l 0 m 0 i f
5
where Tlml 0 m 0 is the partial T-matrix element given by ⫹ Tlml 0 m 0 具w1 C ⫺ k 0l 0 m 0 jUse jw0 Cki;lm 典: f
LF-scattering amplitudes and is given by X Bjmt t m 0 V 0 ⫺1m all 0 mm 0 ll 0 0mt jjt mt t
l 0 lm 0 m
ll 0 mm 0 j jt m 0t Ylml V 0
6
0
all 0 mm 0 n ← 0; k0 ⫺ 1=2p4p 2l 0 ⫹ 1 1=2 il ⫺l 具kf lm; njT jk0 l 0 m 0 ; 0典:
2 kf X 1 jt SMn Bmt m 0 n ← 0; k0 ; k 0 ; r^ 0 ; t k0 j m m 0 2jt⫹ 1 t
t
t
7 where ~jt ~l 0 ⫺ ~l is the transferred angular momentum during the collision, m 0 t and mt are the projections of jt along the laboratory and molecular axis, respectively. The S factor results from summing up over final and averaging over initial spin sublevels, and Mn is the orbital angular momentum projection degeneracy factor of the final target state. In Eq. (7), Bjmt t m 0 is t the coefficient of the jt-basis expansion of the
9
For the optically forbidden transitions which lead to the triplet excited states, only the exchange part of the T matrix is needed. The cross sections are calculated via Eq. (7) where a direct summation over (lm) up to some truncation parameters (lc, mc) is performed. In contrast, for the transitions leading to the dipoleallowed singlet excited states, the partial-wave expansion of the transition T matrices was also truncated at the same parameters (lc, mc). Contributions from higher partial waves were accounted for via the Born-closure procedure. In this procedure, the Bjmt t m 0 t is given by: X 0 t ^0 ⫺1m il⫺l 2l ⫹ 1⫺1 Bjtmt m 0 k^ 0 BBorn;j mt m 0 k ⫹ t
t
l 0 lmm 0
0 0 0 0 mm 0 l ⫺ m; l m jt m t TllS0 mm 0 ⫺ TllBorn
Finally, the LF–DCS is represented in a jt basis [23] as: ds n ← 0 dV
8
where the dynamical coefficients all 0 mm 0 for the transition from a initial target state 兩0典 to a final target state 兩n典 can be written in terms of fixed-nuclei partial-wave components of the electronic portion of the transition matrix elements as
4
G ^ 0
51
l0; l 0 mt jjt mt Yl 0 mt k^ 0 ;
10
t ^0 where BBorn;j mt m 0t k is the jt-basis representation of the Born scattering amplitude, defined as: t ^0 BBorn;j mt m 0 k t
2jt ⫹ 1 k Z ^ 0 Born ^ 0 ^ 0 jt* ^ 0 R ; k Dmt mt R ; dR f 8p2 ip 1=2 11
where Djmt*t m 0 R^ 0 are the usual rotation matrices. The t 0 mm 0 is the partial-wave Born T-matrix element TllBorn given by: 0 mm 0 具Sklm jUst jSkl 0 m 0 典; TllBorn
12
52
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
Fig. 1. GOS for the 2s g ! 2p u singlet transition in CO2 at 1029 eV. Solid line, present DWA results; long-dashed line, present FBA results; short-dashed line, the FBA results of Roberty et al. [7]; full circles, experimental results of Roberty et al. Normalized to their FBA results at the scattering angle of 3⬚.
where Ust is twice the static potential (in atomic units) and Sklm are the partial-wave components of the freeparticle wavefunction. The ground-state configuration of CO2 is represented by a near-Hartree–Fock wavefunction. This wavefunction is generated by an SCF calculation using a standard [11s6p]/[5s3p] contracted Gaussian basis [24] for both carbon and oxygen atoms augmented by three s (a 0.0853, 0.0287 and 0.00473), three p (a 0.0551, 0.0183 and 0.003111) and two d (a 1.471 and 0.671), uncontracted functions on the oxygen centers and three s (a 0.0453, 0.0157 and 0.00537), two p (a 0.0323 and 0.00734) and two d (a 1.373 and 0.523) on the carbon center. At the experimental equilibrium geometry (RC – O 2.1944a0), this basis set gives an SCF energy of ⫺87.707 a.u. and a quadrupole moment of ⫺4.014 a.u. These values can be compared with the SCF results ⫺187.674 a.u. and ⫺4.013 a.u. of Lucchese and McKoy [25], respectively and also an SCF energy of –184.687 a.u. obtained by Roberty et al. [7]. The same basis set is also used to calculate the wavefunctions of the 1P u and 3P u excited states using the improved virtual orbital (IVO) approximation.
These orbitals were obtained by diagonalising the 2s⫺1 potential of the frozen-core basis. The calcug lated vertical excitation energies for the (2s g ! 2p u) (singlet and triplet) transitions at the equilibrium geometry of the ground state are 298.57 and 296.36 eV, respectively. For the singlet transition, our value can be compared with the experimental result (290.7 eV) of Wight and Brion [26] and Tronc et al. [27], and also with the calculated frozen-core result (301.7 eV) by Roberty et al. [7]. In addition, our calculated singlet–triplet energy splitting (2.21 eV) agrees reasonably well with the experimental value (1.51 eV) cited by Almeida et al. [18] and in references therein.
3. Results and discussion In Fig. 1 we compare our calculated GOS using 1 both the DWA and FBA for the X 1 S⫹ g ! Pu transition in CO2 at 1290 eV incident energy with the corresponding FBA results, as well as with the measured data of Roberty et al., normalized to their FBA results at scattering angle of 3⬚. Our FBA GOS lies about
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
53
Fig. 2. Present DWA integral cross sections for (2s g ! 2p u) (a) singlet transition and (b) triplet transition for e—CO2 scattering in the 320– 800 eV range.
20% higher than their FBA data. The difference is probably because of the different approaches used to represent the excited 1P u state. In their investigation, the excited 2p u orbital is obtained by a fully relaxed SCF calculation, whereas in our study the referred orbital is calculated via frozen-core IVO approximation. Although relaxation effects may play important roles in the excitation and ionization of core-level electrons, the use of a relaxed excited-state wavefunction in a DWA study will largely increase the computational efforts and thus is disregarded. Further, some feeling of the importance of the relaxation effects can be provided from the work of Schirmer et al. [28] on the K-shell photoionization study of CO molecule. In this study, the authors have reported calculated cross sections as well as the asymmetry parameters for the photoionizaton of one electron from the 2s (C1s) orbital using both the frozen- and
relaxed-core Hartree–Fock approximations. However, the comparison with experiments have shown that the measured data lie between the frozen- and relaxed-core results and seems to indicate that the target is not fully relaxed during the photoionization process. In addition, our calculated DWA GOS also lie about 20% below our FBA data and agree quite well with the FBA GOS of Roberty et al. The discrepancies seen between our DWA and FBA results seem to suggest that the high energy limit for the FBA being valid had not yet been attained at 1290 eV incident energy. In Fig. 2(a) and (b), we show the calculated ICS in the 320–800 eV energy range for the (2s g ! 2p u) (singlet and triplet) transitions in CO2, respectively. In the energy range covered in the present work, our calculated ICS for the transition leading to the 1P u state in CO2 continuously increase with increasing
54
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
Fig. 3. Comparison of the ratios calculated between the DWA ICS for triplet and singlet transitions with the measured R(3-A : 1-A), as a function of the incident energy in the 320–800 eV range. Dashed line, present DWA ratio; solid line, the calculated ratio multiplied by two; full circles, measured results of Almeida et al. [18].
incident energy. In contrast, the calculated ICS for the 1 X 1 S⫹ g ! Pu transition decreases rapidly with increasing energy which is expected for such singlet-to-triplet dipole-forbidden transitions. In Fig. 3, we compare our calculated ratios, obtained by dividing the ICS for the transition to
triplet excited state by the corresponding values for the singlet transition, with those experimental R(3A : 1-A) of Almeida et al. [18]. The calculated ratios agree qualitatively with the measured data. However, if we multiply the calculated ratios by a factor of two, the resulting values are also in quantitative agreement
Fig. 4. Present DWA differential cross sections for (2s g ! 2p u) singlet transition for e ⫺ CO2 scattering at 350 eV (short-dashed line), 400 eV (dashed line) and 500 eV (solid line).
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
55
Fig. 5. Same as Fig. 4, at 600 eV (dashed line) and 800 eV (solid line).
Fig. 6. Present DWA differential cross sections for (2s g ! 2p u) triplet transition for e ⫺ – CO2 scattering at 350 eV (shortest-dashed line), 400 eV (short-dashed line), 500 eV (dashed line), 600 eV (long-dashed line) and 800 eV (solid line).
56
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
with the measured data. Therefore, the present study not only confirms that the decay of autoionizing states is nearly independent of impact energy but also suggests that the efficiency of the decay process from the triplet excited state is about twice that of the singlet state. In Figs. 4–6 we present some selected DWA DCS for the (2s g ! 2p u) (singlet and triplet) transitions in CO2, respectively, in the 350–800 eV incident energy range. Although it is expected that the DCS for dipole allowed excitations are usually sharply peaked in the forward direction, reflecting the long-range dipole type of interaction, the present calculated DCS for the singlet transition at 350 and 400 eV are quite isotropic and also show some oscillations. Only at energies above 500 eV the calculated DCS present the characteristic behavior of dipole transitions. Unfortunately, there are no experimental or other theoretical results of both ICS and DCS for these core-excitation transitions available for comparison, which seriously limits the discussion. Considering the interesting behavior of the cross sections of these core-excitation processes, we are sure that additional theoretical and experimental studies of such processes should appear in the near future. Therefore, our present results would serve as a basis for comparison. In summary, the present work reports the first application of the DWA for studying the core-level 1;3 Pu (carbon 1s 2s g ! 2p u) transitions in X 1 S⫹ g ! CO2 in the 320–800 eV incident energy range. Also, the GOS for the transition leading to singlet excited state calculated within the frozen-core framework using both the DWA as well as the FBA at the 1290 eV are also reported. Our calculated FBA GOS are in general 20% larger than the corresponding FBA results calculated using the full relaxed wavefunction for the excited singlet state which clearly shows the important influence of the target relaxation effects on the core-level excitation processes. Nevertheless, previous studies on the K-shell photoionization in the CO2 molecule seem to indicate that the target may not fully relax during the excitation. Thus it is expected that the correct FBA GOS would lie between the values calculated using the frozen-core and the relaxed-core approximations. In contrast, our DWA GOS are also about 20% lower than our FBA data, which may indicate that the high-energy limit for the
validity of FBA is not yet achieved with 1290 eV incident electrons. Further, the comparison of the ratios between the DWA ICS calculated for the transitions leading to the triplet and the singlet excited states with those measured R(3-A : 1-A) of the intensity of the autoionizing electrons from the corresponding triplet and singlet excited states in the 320–800 eV range shows a very good qualitative agreement. Also, a quantitative agreement is seen between twice the calculated ratios and the measured data. As the measured R(3-A : 1-A) depends on two transition probabilities, that for formation of the excited states by electron impact and that for their subsequent decay, our study not only confirms the expectation that the decay of autoionizing states is nearly independent of impact energy but also suggests that the efficiency of the decay process from the triplet excited state is about twice that of the singlet state. Acknowledgements This research was partially supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq), FAPESP, FINEP-PADCT and CAPESPADCT. References [1] K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heden, K. Hamrin, U. Gelins, T. Bergmark, L.O. Werme, R. Manne, Y. Baer, ESCA Applied to Free Molecules, NorthHolland, Amsterdan, 1969. [2] J. Sto¨hr, in: R. Prins, (Ed.), X-Ray Absorption, Principal Application, Techniques of EXAFS, SEXAFS and XANES, Wiley, New York, 1986. [3] G.C. King, F.H. Read, in: B. Crasemann (Ed.), Atomic Inner Shell Physics, Plenum, New York, 1985. [4] M. Tronc, G.C. King, F.H. Read, J. Phys. B 12 (1979) 137. [5] D.A. Shaw, G.C. King, F.H. Read, D. Dvejanovic, J. Phys. B 15 (1982) 1785. [6] C.E. Blout, D.M. Dickinson, J. Electron Spectr. 61 (1993) 367. [7] H.M.B. Roberty, C.E. Bielschowsky, G.G.B. Souza, Phys. Rev. A 44 (1991) 1694. [8] S.D. Parker, C.W. McCurdy, T.N. Rescigno, B.H. Lengsfield III, Phys. Rev. A 43 (1991) 3514. [9] Q. Sun, C. Winstead, V. Mckoy, M.A.P. Lima, J. Chem. Phys. 96 (1992) 3531. [10] S.E. Branchett, J. Tennyson, L.A. Morgan, J. Phys. B 24 (1991) 3479.
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57 [11] K. Bartschat, D.H. Madison, J. Phys. B 20 (1987) 5839. [12] G.D. Meneses, C.B. Pagan, L.E. Machado, Phys. Rev. A 41 (1990) 4740. [13] M.-T. Lee, L.M. Brescansin, M.A.P. Lima, J. Phys. B. 23 (1990) 3859. [14] M.-T. Lee, S.E. Michelin, S.E. Kroin, T. Machado, L.M. Brescansin, J. Phys. B 28 (1995) 1859. [15] M.-T. Lee, A.M. Machado, M.M. Fujimoto, L.E. Machado, L.M. Brescansin, J. Phys. B 29 (1996) 4285. [16] M.A.P. Lima, T.L. Gibson, V. McKoy, W.M. Huo, Phys. Rev. A 38 (1986) 4527. [17] Q. Sun, C. Winstead, V. Mckoy, Phys. Rev. A 46 (1992) 6987. [18] D.P. Almeida, G. Dawber, G.C. King, Chem. Phys. Letters 233 (1995) 1.
57
[19] A.W. Fliflet, V. McKoy, Phys. Rev. A 21 (1980) 1863. [20] T.N. Rescigno, C.W. McCurdy, V. McKoy, J. Phys. B 7 (1974) 2396. [21] M.-T. Lee, V. McKoy, J. Phys. B 15 (1982) 3971. [22] R.R. Lucchese, G. Raseev, V. Mckoy, Phys. Rev. A 25 (1982) 2572. [23] U. Fano, D. Dill, Phys. Rev. A 6 (1972) 185. [24] T.H. Dunning, J. Chem. Phys. 55 (1971) 716. [25] R.R. Lucchese, V. Mckoy, Phys. Rev. A 25 (1982) 1963. [26] G.R. Wight, C.E. Brion, J. Electron Spec. 3 (1974) 191. [27] M. Tronc, G.C. King, F.H. Read, J. Phys. B 12 (1979) 137. [28] J. Schirmer, M. Braustein, V. McKoy, Phys. Rev. A 41 (1990) 283.
A distorted-wave study for core-excitation processes in CO2 by electron impact T. Kroin a,*, S.E. Michelin a, K.T. Mazon a, D.P. Almeida a, M.-T. Lee b a
Departamento de Fı´sica, Universidade Federal de Santa Catarina, 88049, Floriano´polis, SC, Brazil Departamento de Quı´mica, Universidade Federal de Sa˜o Carlos, 13565-905, Sa˜o Carlos, SP, Brazil
b
Abstract In the present work, the distorted-wave approximation (DWA) is applied for the first time to study core-level excitations in 1;3 molecules by electron impact. More specifically, we have calculated differential and integral cross sections for the X 1 S⫹ g ! Pu (carbon 1s 2s g ! 2p u) transitions in CO2 in the 320–800 eV incident energy range. The frozen-core generalized oscillator strengths (GOS) for the transition leading to singlet excited states calculated using both the DWA and the first Born approximation (FBA) at 1290 eV are also reported. Our calculated DWA GOS are in general 20% lower than the corresponding FBA, results which seem to indicate that the FBA is not valid at that incident energy. Moreover, we have also calculated the ratios between the distorted-wave integral cross sections for the transitions leading to the triplet and the singlet excited states as a function of incident energy. The comparison of these data with the experimental ratio R(3A:1A) obtained by dividing the intensities of the autoionizing electrons from the same excited states in the 320–800 eV range is encouraging. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Core-excitation; CO2; Distorted-wave approximation PACS Number: 3480G
1. Introduction Inner-shell processes, involving the promotion or the removal of a 1s electron by both electron and photon impact have received considerable attention. As a result of the high degree of localization of these orbitals, the shapes and energies associated with such core-level excitation and ionization bands do not show significant changes as one moves from the gaseous state to the solid state [1]. Therefore, the excitation and ionization of the core-level electrons have played a significant role in the development of
* Corresponding author. Fax: ⫹ 55-48-331-9946. E-mail address: [email protected] (T. Kroin)
equipments for characterizing both solid and gaseous samples [2]. Despite the increasing experimental interest in this field of investigation [3–6], on the theoretical side very few studies involving core-excitation processes in molecules have been carried out. To date, a limited number of investigations have almost exclusively been at the first Born approximation (FBA) level [7]. Although the FBA can provide quite accurate generalized oscillator strengths (GOSs) for innershell transitions at very high incident energies, it is not expected to work out when incident energies approach the excitation threshold. In contrast, despite the recent development of several solid-based abinitio multichannel theories for treating both elastic and inelastic electron–molecule collisions [8–10] in
0166-1280/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(98)00534-X
50
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
the low-energy range, there are no reported applications for studying core-level excitation processes using these theories. Among various theoretical methods, the distortedwave approximation (DWA) have received considerable attention during the past few years. The DWA [11] and the first-order many-body theory [12], which is essentially equivalent to DWA, have been successfully and widely applied to calculate electronic excitation cross sections and coherence parameters for atomic targets in the intermediate and high energy range. Recently, the DWA has also been applied to investigation of electronic excitation of low-lying valence-shell orbitals in molecules by low energy electron impact. It was found [13–15] that the DWA calculations can in general lead to similar excitation cross sections to those obtained using the Schwinger multichannel method (SMC) at a few-state coupling level of approximation [16,17] for incident energies a few eV above the threshold. Also, comparison with available experimental data has shown that the DWA can in general reproduce the angular distribution of measured differential cross sections (DCS), although it usually overestimates the magnitude of the cross sections by a factor of 2–3. The reliability of the calculated cross sections increases with increasing incident energy. In view of the simplicity of the DWA, the extension of this method to studies involving core-excitation processes in molecules is clearly of interest. In this work, we report a DWA application of such processes in the CO2 molecule. More specifically, we have calculated differential and integral cross sections, as 1;3 Pu (carbon 1s well as the GOS, for the X 1 S⫹ g ! 2s g ! 2p u) transitions in CO2 molecule by electron impact in the 320– 800 eV energy range. We hope the present study would help with qualitative behavior and quantitative estimates that can be expected for these physical quantities. For the transitions studied herein, some experimental investigations were reported recently. For example, 1 the relative GOS for the X 1 S⫹ g ! Pu transition has been measured by Roberty et al. [7] using 1290 eV incident electrons. They have normalized the GOS for this transition to their calculated data using the full relaxed-core FBA. More recently, Almeida et al. [18] 1;3 Pu (carbon 1s 2sg ! have investigated the X 1 S⫹ g ! 2p u) transitions in CO2 molecule by observing the
autoionizing electrons via decay from the 1,3P u inner-shell excited states. These authors have reported ratios between the intensities of the autoionizing electrons from the triplet and singlet excited states, denoted by R (3-A : 1-A), as a function of incident energy at the scattering angle of 90⬚. As the final ionic state is formed via an inner-shell excited state, the autoionization yield depends on two transition probabilities, that for formation of the excited states by electron impact and that for their subsequent decay. However, it is expected that the contribution arising from the decay is independent of the impact energy. Hence, the measured R(3-A : 1-A) are in fact proportional to the ratios between the excitation cross sections for the transitions leading to the 3P u and 1P u states by electron impact. Thus, the comparison of the calculated ratios with the measured data may provide useful insight into the nature of these core-excitation processes. The organization of this article is the following: in Section 2 we briefly outline the used theory. The results are shown in Section 3, where we also present our concluding remarks.
2. Theory The details of the basic theory used in the present work have already been presented elsewhere [13– 15,19–21] and will be only briefly described here. The differential excitation cross sections for electron–molecule scattering averaged over molecular orientations are given by: ds 1 Z ^ 0 2 a sin b d b d g f k d ; f dV 8p 2
1
where k^ 0f is the direction of scattered electron linear momentum in the Laboratory-Frame (LF), whereas the direction of the incident electron linear momentum is taken as zero. The (a , b , g ) are the Euler angles which define the direction of the molecular principal axis. The Body-Frame (BF) amplitude f k^i ; k^f is related to the T-matrix elements by the formula: f k^i ; k^f ⫺2p2 Tif :
2
Within the DWA framework, the transition T-matrix
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
is given by: ⫹ Tif 具w1 C ⫺ kf jUse jw0 Cki 典;
3
where w 0 and w 1 are the initial and final target wavefunctions, respectively. These wavefunctions are Slater determinants with appropriate symmetries. ⫺ C ⫹ ki and Ckf are the initial and final distorted continuum wavefunctions with the outgoing (⫹) and incoming-wave (⫺) boundary conditions, respectively. Use is the static-exchange potential operator. The distorted wavefunctions are solutions of the Lippmann–Schwinger equation: ^ C ^ Fk~ ⫹ G ^ 0 U Ck~ k~
being the free-particle Green’s operator with with appropriate boundary condition, and Fk~ is the ~ These plane wavefunction, with linear momentum k. wavefunctions Ck~ were calculated using the Schwinger variational iterative method (SVIM) [22] in the static-exchange field of ground-state target. The electronic excitation T-matrix can be partialwave expanded as: 0
Tif 2=p
X X il⫺l * ^ Tlml 0 m 0 Ylm ki Yl 0 m 0 k^f ; k k lm l 0 m 0 i f
5
where Tlml 0 m 0 is the partial T-matrix element given by ⫹ Tlml 0 m 0 具w1 C ⫺ k 0l 0 m 0 jUse jw0 Cki;lm 典: f
LF-scattering amplitudes and is given by X Bjmt t m 0 V 0 ⫺1m all 0 mm 0 ll 0 0mt jjt mt t
l 0 lm 0 m
ll 0 mm 0 j jt m 0t Ylml V 0
6
0
all 0 mm 0 n ← 0; k0 ⫺ 1=2p4p 2l 0 ⫹ 1 1=2 il ⫺l 具kf lm; njT jk0 l 0 m 0 ; 0典:
2 kf X 1 jt SMn Bmt m 0 n ← 0; k0 ; k 0 ; r^ 0 ; t k0 j m m 0 2jt⫹ 1 t
t
t
7 where ~jt ~l 0 ⫺ ~l is the transferred angular momentum during the collision, m 0 t and mt are the projections of jt along the laboratory and molecular axis, respectively. The S factor results from summing up over final and averaging over initial spin sublevels, and Mn is the orbital angular momentum projection degeneracy factor of the final target state. In Eq. (7), Bjmt t m 0 is t the coefficient of the jt-basis expansion of the
9
For the optically forbidden transitions which lead to the triplet excited states, only the exchange part of the T matrix is needed. The cross sections are calculated via Eq. (7) where a direct summation over (lm) up to some truncation parameters (lc, mc) is performed. In contrast, for the transitions leading to the dipoleallowed singlet excited states, the partial-wave expansion of the transition T matrices was also truncated at the same parameters (lc, mc). Contributions from higher partial waves were accounted for via the Born-closure procedure. In this procedure, the Bjmt t m 0 t is given by: X 0 t ^0 ⫺1m il⫺l 2l ⫹ 1⫺1 Bjtmt m 0 k^ 0 BBorn;j mt m 0 k ⫹ t
t
l 0 lmm 0
0 0 0 0 mm 0 l ⫺ m; l m jt m t TllS0 mm 0 ⫺ TllBorn
Finally, the LF–DCS is represented in a jt basis [23] as: ds n ← 0 dV
8
where the dynamical coefficients all 0 mm 0 for the transition from a initial target state 兩0典 to a final target state 兩n典 can be written in terms of fixed-nuclei partial-wave components of the electronic portion of the transition matrix elements as
4
G ^ 0
51
l0; l 0 mt jjt mt Yl 0 mt k^ 0 ;
10
t ^0 where BBorn;j mt m 0t k is the jt-basis representation of the Born scattering amplitude, defined as: t ^0 BBorn;j mt m 0 k t
2jt ⫹ 1 k Z ^ 0 Born ^ 0 ^ 0 jt* ^ 0 R ; k Dmt mt R ; dR f 8p2 ip 1=2 11
where Djmt*t m 0 R^ 0 are the usual rotation matrices. The t 0 mm 0 is the partial-wave Born T-matrix element TllBorn given by: 0 mm 0 具Sklm jUst jSkl 0 m 0 典; TllBorn
12
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T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
Fig. 1. GOS for the 2s g ! 2p u singlet transition in CO2 at 1029 eV. Solid line, present DWA results; long-dashed line, present FBA results; short-dashed line, the FBA results of Roberty et al. [7]; full circles, experimental results of Roberty et al. Normalized to their FBA results at the scattering angle of 3⬚.
where Ust is twice the static potential (in atomic units) and Sklm are the partial-wave components of the freeparticle wavefunction. The ground-state configuration of CO2 is represented by a near-Hartree–Fock wavefunction. This wavefunction is generated by an SCF calculation using a standard [11s6p]/[5s3p] contracted Gaussian basis [24] for both carbon and oxygen atoms augmented by three s (a 0.0853, 0.0287 and 0.00473), three p (a 0.0551, 0.0183 and 0.003111) and two d (a 1.471 and 0.671), uncontracted functions on the oxygen centers and three s (a 0.0453, 0.0157 and 0.00537), two p (a 0.0323 and 0.00734) and two d (a 1.373 and 0.523) on the carbon center. At the experimental equilibrium geometry (RC – O 2.1944a0), this basis set gives an SCF energy of ⫺87.707 a.u. and a quadrupole moment of ⫺4.014 a.u. These values can be compared with the SCF results ⫺187.674 a.u. and ⫺4.013 a.u. of Lucchese and McKoy [25], respectively and also an SCF energy of –184.687 a.u. obtained by Roberty et al. [7]. The same basis set is also used to calculate the wavefunctions of the 1P u and 3P u excited states using the improved virtual orbital (IVO) approximation.
These orbitals were obtained by diagonalising the 2s⫺1 potential of the frozen-core basis. The calcug lated vertical excitation energies for the (2s g ! 2p u) (singlet and triplet) transitions at the equilibrium geometry of the ground state are 298.57 and 296.36 eV, respectively. For the singlet transition, our value can be compared with the experimental result (290.7 eV) of Wight and Brion [26] and Tronc et al. [27], and also with the calculated frozen-core result (301.7 eV) by Roberty et al. [7]. In addition, our calculated singlet–triplet energy splitting (2.21 eV) agrees reasonably well with the experimental value (1.51 eV) cited by Almeida et al. [18] and in references therein.
3. Results and discussion In Fig. 1 we compare our calculated GOS using 1 both the DWA and FBA for the X 1 S⫹ g ! Pu transition in CO2 at 1290 eV incident energy with the corresponding FBA results, as well as with the measured data of Roberty et al., normalized to their FBA results at scattering angle of 3⬚. Our FBA GOS lies about
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
53
Fig. 2. Present DWA integral cross sections for (2s g ! 2p u) (a) singlet transition and (b) triplet transition for e—CO2 scattering in the 320– 800 eV range.
20% higher than their FBA data. The difference is probably because of the different approaches used to represent the excited 1P u state. In their investigation, the excited 2p u orbital is obtained by a fully relaxed SCF calculation, whereas in our study the referred orbital is calculated via frozen-core IVO approximation. Although relaxation effects may play important roles in the excitation and ionization of core-level electrons, the use of a relaxed excited-state wavefunction in a DWA study will largely increase the computational efforts and thus is disregarded. Further, some feeling of the importance of the relaxation effects can be provided from the work of Schirmer et al. [28] on the K-shell photoionization study of CO molecule. In this study, the authors have reported calculated cross sections as well as the asymmetry parameters for the photoionizaton of one electron from the 2s (C1s) orbital using both the frozen- and
relaxed-core Hartree–Fock approximations. However, the comparison with experiments have shown that the measured data lie between the frozen- and relaxed-core results and seems to indicate that the target is not fully relaxed during the photoionization process. In addition, our calculated DWA GOS also lie about 20% below our FBA data and agree quite well with the FBA GOS of Roberty et al. The discrepancies seen between our DWA and FBA results seem to suggest that the high energy limit for the FBA being valid had not yet been attained at 1290 eV incident energy. In Fig. 2(a) and (b), we show the calculated ICS in the 320–800 eV energy range for the (2s g ! 2p u) (singlet and triplet) transitions in CO2, respectively. In the energy range covered in the present work, our calculated ICS for the transition leading to the 1P u state in CO2 continuously increase with increasing
54
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
Fig. 3. Comparison of the ratios calculated between the DWA ICS for triplet and singlet transitions with the measured R(3-A : 1-A), as a function of the incident energy in the 320–800 eV range. Dashed line, present DWA ratio; solid line, the calculated ratio multiplied by two; full circles, measured results of Almeida et al. [18].
incident energy. In contrast, the calculated ICS for the 1 X 1 S⫹ g ! Pu transition decreases rapidly with increasing energy which is expected for such singlet-to-triplet dipole-forbidden transitions. In Fig. 3, we compare our calculated ratios, obtained by dividing the ICS for the transition to
triplet excited state by the corresponding values for the singlet transition, with those experimental R(3A : 1-A) of Almeida et al. [18]. The calculated ratios agree qualitatively with the measured data. However, if we multiply the calculated ratios by a factor of two, the resulting values are also in quantitative agreement
Fig. 4. Present DWA differential cross sections for (2s g ! 2p u) singlet transition for e ⫺ CO2 scattering at 350 eV (short-dashed line), 400 eV (dashed line) and 500 eV (solid line).
T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
55
Fig. 5. Same as Fig. 4, at 600 eV (dashed line) and 800 eV (solid line).
Fig. 6. Present DWA differential cross sections for (2s g ! 2p u) triplet transition for e ⫺ – CO2 scattering at 350 eV (shortest-dashed line), 400 eV (short-dashed line), 500 eV (dashed line), 600 eV (long-dashed line) and 800 eV (solid line).
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T. Kroin et al. / Journal of Molecular Structure (Theochem) 464 (1999) 49–57
with the measured data. Therefore, the present study not only confirms that the decay of autoionizing states is nearly independent of impact energy but also suggests that the efficiency of the decay process from the triplet excited state is about twice that of the singlet state. In Figs. 4–6 we present some selected DWA DCS for the (2s g ! 2p u) (singlet and triplet) transitions in CO2, respectively, in the 350–800 eV incident energy range. Although it is expected that the DCS for dipole allowed excitations are usually sharply peaked in the forward direction, reflecting the long-range dipole type of interaction, the present calculated DCS for the singlet transition at 350 and 400 eV are quite isotropic and also show some oscillations. Only at energies above 500 eV the calculated DCS present the characteristic behavior of dipole transitions. Unfortunately, there are no experimental or other theoretical results of both ICS and DCS for these core-excitation transitions available for comparison, which seriously limits the discussion. Considering the interesting behavior of the cross sections of these core-excitation processes, we are sure that additional theoretical and experimental studies of such processes should appear in the near future. Therefore, our present results would serve as a basis for comparison. In summary, the present work reports the first application of the DWA for studying the core-level 1;3 Pu (carbon 1s 2s g ! 2p u) transitions in X 1 S⫹ g ! CO2 in the 320–800 eV incident energy range. Also, the GOS for the transition leading to singlet excited state calculated within the frozen-core framework using both the DWA as well as the FBA at the 1290 eV are also reported. Our calculated FBA GOS are in general 20% larger than the corresponding FBA results calculated using the full relaxed wavefunction for the excited singlet state which clearly shows the important influence of the target relaxation effects on the core-level excitation processes. Nevertheless, previous studies on the K-shell photoionization in the CO2 molecule seem to indicate that the target may not fully relax during the excitation. Thus it is expected that the correct FBA GOS would lie between the values calculated using the frozen-core and the relaxed-core approximations. In contrast, our DWA GOS are also about 20% lower than our FBA data, which may indicate that the high-energy limit for the
validity of FBA is not yet achieved with 1290 eV incident electrons. Further, the comparison of the ratios between the DWA ICS calculated for the transitions leading to the triplet and the singlet excited states with those measured R(3-A : 1-A) of the intensity of the autoionizing electrons from the corresponding triplet and singlet excited states in the 320–800 eV range shows a very good qualitative agreement. Also, a quantitative agreement is seen between twice the calculated ratios and the measured data. As the measured R(3-A : 1-A) depends on two transition probabilities, that for formation of the excited states by electron impact and that for their subsequent decay, our study not only confirms the expectation that the decay of autoionizing states is nearly independent of impact energy but also suggests that the efficiency of the decay process from the triplet excited state is about twice that of the singlet state. Acknowledgements This research was partially supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq), FAPESP, FINEP-PADCT and CAPESPADCT. References [1] K. Siegbahn, C. Nordling, G. Johansson, J. Hedman, P.F. Heden, K. Hamrin, U. Gelins, T. Bergmark, L.O. Werme, R. Manne, Y. Baer, ESCA Applied to Free Molecules, NorthHolland, Amsterdan, 1969. [2] J. Sto¨hr, in: R. Prins, (Ed.), X-Ray Absorption, Principal Application, Techniques of EXAFS, SEXAFS and XANES, Wiley, New York, 1986. [3] G.C. King, F.H. Read, in: B. Crasemann (Ed.), Atomic Inner Shell Physics, Plenum, New York, 1985. [4] M. Tronc, G.C. King, F.H. Read, J. Phys. B 12 (1979) 137. [5] D.A. Shaw, G.C. King, F.H. Read, D. Dvejanovic, J. Phys. B 15 (1982) 1785. [6] C.E. Blout, D.M. Dickinson, J. Electron Spectr. 61 (1993) 367. [7] H.M.B. Roberty, C.E. Bielschowsky, G.G.B. Souza, Phys. Rev. A 44 (1991) 1694. [8] S.D. Parker, C.W. McCurdy, T.N. Rescigno, B.H. Lengsfield III, Phys. Rev. A 43 (1991) 3514. [9] Q. Sun, C. Winstead, V. Mckoy, M.A.P. Lima, J. Chem. Phys. 96 (1992) 3531. [10] S.E. Branchett, J. Tennyson, L.A. Morgan, J. Phys. B 24 (1991) 3479.
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