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Ab initio calculation of the C1s photoelectron spectrum of C2H2
Nuclear Instruments and Methods in Physics Research B 213 (2004) 65–70 www.elsevier.com/locate/nimb
Ab initio calculation of the C1s photoelectron spectrum of C2H2 R. Colle
a,b,* ,
D. Embriaco a, M. Massini b, S. Simonucci c, S. Taioli
d
a
d
Dipartimento di Chimica Applicata, INSTM, Universit a di Bologna, Via Saragozza 8, I-40136 Bologna, Italy b Scuola Normale Superiore, I-56100 Pisa, Italy c Dipartimento di Matematica e Fisica, INFM, Universit a di Camerino, Via Madonna delle Carceri 7, I-62032 Camerino, Italy Dipartimento di Energetica, Nucleare e Controllo Ambientale, INFM, Universit a di Bologna, Via Risorgimento 4, 40136 Bologna, Italy
Abstract The C1s photoelectron spectrum of C2 H2 , measured by Kempgens et al. [Phys. Rev. Lett. 79 (1997) 3617] and characterized by the presence of a core-level splitting, has been calculated and interpreted using an ab initio quantummechanical method that allows detailed analysis of photoionization processes and accurate reproduction of spectral profiles. The anisotropy of the electron emission, measured in a hypothetical experiment in which the orientation of the molecules can be established, has been also predicted and analyzed. Ó 2003 Elsevier B.V. All rights reserved. PACS: 31.15.Ar; 33.60.Fy; 33.70.)w; 33.80.Eh Keywords: Molecular photoionization; Core-level splitting; Ab initio scattering wavefunctions
1. Introduction The continuous improvements in synchrotron radiation sources, soft X-ray monochromators and electron analyzers [1–4] have produced in recent years considerable progress in K-shell ionization spectroscopy of small molecules. The photon- and electron-energy resolutions which are now achievable allow one to gather relevant information also on subtle quantum-mechanical effects, such as those due to the occurrence of near-
* Corresponding author. Address: Scuola Normale Superiore, I-56100 Pisa, Italy. Tel.: +39-050-509-248; fax: +39-050563-513. E-mail address: [email protected] (R. Colle).
degenerate core–hole configurations, and make this type of spectroscopy very interesting and challenging for the theory. Recently, Kempgens et al. [5] have reported the first experimental identification of the core-level splitting in the Cls photoelectron spectrum of C2 H2 , a feature ascribed to the presence of two near-degenerate core–hole states, classified as 2 1 j2 Rg : 1r1 g i and j Ru : 1ru i in the independentparticle scheme. In this paper, we present an ab initio calculation and theoretical analysis of the C1s photoelectron spectrum of C2 H2 measured by Kempgens et al. [5]. The results of this study have been obtained using a quantum-mechanical approach [6–8] already successfully applied to the interpretation of XPS, Auger and autoionization spectra of atoms and molecules. In this paper, we
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01535-0
66
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
analyze the vibrational structure of the spectrum and we also predict the anisotropy of the electron emission measured in a hypothetical experiment in which the orientation of the ionized molecule has been established. In what follows, we briefly sketch the theory that is behind the calculation of the theoretical spectrum, than we compare the calculated spectrum with the experimental one analyzing its components and, finally, we present angular distribution patterns of electrons produced by a photoionization process in which the photons are polarized along a direction perpendicular to the molecular axis.
2. Theory The construction of the theoretical photoemission spectrum requires the calculation of the scattering wavefunctions fh1; . . . ; N jHj ; ~ k ig for the continuum states of the N -electron molecule involved in the process. These wavefunctions are defined asymptotically by the momentum ~ k of the ejected electron and the state jHj i of the residual ion. In core-ionization processes, however, the core–hole states produced by the primary photoionization are metastable, being coupled with the continuum states produced by the Auger decay which follows the primary electron emission. As a consequence, these states have finite lifetime and the corresponding spectral lines in the photoemission spectrum have non-negligible linewidths. From general considerations based on the scattering theory (see [6–9]), one can prove that the differential cross section of a photoemission process, in which a neutral molecule in its ground state j0i is ionized by a monochromatic (x), linearly polarized (k) radiation, can be approximated as follows: or0!j ðx; ~ kÞ / ~ ok
2C ^ k jHj ; ~ jh0jO k ij 2pj 2
ðx þ E0 Ejþ k Þ þ
C2j
;
ð1Þ
4
where jHj ; ~ k i is the final state of the process, E0 , Ejþ 2 and k ¼ k =2 are, respectively, the energy of the ground state of the neutral molecule, the energy of the state jHj i of the ion and the kinetic energy of
the ejected electron. Furthermore, Cj is the width b k ¼ ~ of the resonance and O ek RNi¼1~ ri the component of the electronic dipole operator along the polarization direction ~ ek . In deriving Eq. (1), we have used the so-called ‘‘two-step’’ approximation [9], that consists in neglecting the coupling between primary photoionization and subsequent Auger decay process, furthermore, here and in the following we have used the atomic system of units jej ¼ h ¼ me ¼ 1. If we consider now that in the Born–Oppenheimer approximation the wavefunction representing the state of a non-rotating molecule is the product of an electronic and a vibrational function, we see that the transition j0i ! jHj ; ~ k i, which the cross section (1) refers to, really corresponds to a set of transitions j0el ij01 ; 02 ; . . .i ! fjHelj ; ~ k ijvj1 ; vj2 ; . . .i; vj1;2;... ¼ 0; 1; . . .g, from the ground state of the neutral molecule to a bunch of vibrational states, each one classified (in harmonic approximation) by the numbers fvja g that give the degree of excitation in the vibrational modes a ¼ 1; 2; . . . supported by the electronic state jHelj i of the ion. It follows that the appropriate expression for the cross section of the process is a sum of contributions or0!j ðx; ~ kÞ 2 Cj b k jHel ; ~ / jh0el j O j k ij ~ 2p ok
X fvja g
jh0jfvja gij2 2
ðx Dfvja g k Þ þ
C2j
;
ð2Þ
4
h01 ; 02 ; . . . jvj1 ; vj2 ; . . .i
where h0jfvja gi and Dfvja g ¼ þ j j ½Ej ðelÞ þ v1 þ v2 þ ½E0 ðelÞ þ 01 þ 02 þ is the difference between the electronic plus vibrational energy of the ion in the state jHelj ijvj1 ; vj2 ; . . .i and the corresponding quantity of the neutral molecule in its ground state j0el ij01 ; 02 ; . . .i. Finally, Cj is the natural width of the metastable electronic state jHelj i. In deriving Eq. (2), we have assumed that, at room temperature, only the lowest vibrational state of the molecule is populated, furthermore, we have disregarded the contributions of the rotational states that are not resolved in these spectra. Finally, in the calculation of the matrix elements in Eq. (2), we have neglected the dependence of the electronic wavefunctions on the
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
nuclear coordinates. It is interesting to notice that, in this approximation, no interference effects appear among vibrational states associated with the intermediate electronic state. In order to construct a theoretical spectrum that can be quantitatively compared with the experimental photoemission spectrum, one has to include also the following contributions: 1. A sum over the electronic resonances that are embedded in the energy range of the incident radiation: in our problem, we have considered the two near-degenerate core–hole states 2 1 j2 Rg : 1r1 g i and j Ru : 1ru i; 2. A sum over the vibrational states belonging to the different vibrational modes supported by each electronic resonance; in our problem we have considered only the two totally symmetric C–C and C–H stretching modes without including bending modes that are not expected to be appreciably excited – see [5,10]; 3. An integration over all the directions k^ of emission of the photoelectron, for each kinetic energy of interest; 4. A convolution of the resulting cross section with ‘‘apparatus’’ functions that take into account the characteristics of the incident radiation and the finite resolving power of the electron analyzer. If we assume now that the incident radiation is constituted by a flux of photons almost perfectly polarized along e^k , with a small frequency spread around a given x, we can describe this flux by means of a normalized Gaussian function " 2 # 1 x0 x 0 pffiffiffi Gc ðjx xjÞ ¼ pffiffiffiffiffiffi exp ð3Þ c 2p c 2 whose full pffiffiffiffiffiffiffiffiffiffi width at half maximum ffi (FWHM ¼ 2c 2 ln 2) is proportional to the resolving power of the monochromator. Similarly, the finite resolving power of the electron analyzers can be taken into account by means of another Gaussian that gives the spread in energy of the spectrometers. A global convolution of the cross section with the two ‘‘apparatus’’ functions give the transition rate of the process to be compared
67
with the measured photoemission spectrum. An equivalent and simpler way of obtaining the same result is to calculate the transition rate of the process through the following expression: Z oW ðx;Þ ¼ M Gc ðjx0 xjÞdx0 o X Z or0!j ðx0 ;~ kÞ k dð k Þd~ ~ ok j¼g;u X ¼M jDk0;j ðÞj2 j¼g;u
X
2
jh0jfvja gij V
fvja g
Dfvja g þ x Cj pffiffiffi ; pffiffiffi ; c 8 c 2 ð4Þ
where M is a scaling factor adjusted on the intensity of the photon flux, g=u Rindicate the electronic 2 b k jHel ; ~ core–hole states, jDk0;j ðÞj2 ¼ d~ k jh0el j O j k ij dð k Þ is the electronic dipole matrix element averaged over the directions of emission, and V ðx; aÞ is a Voigt function that results from the convolution of a single Gaussian Gc , that takes into account the finite resolution of both monochromator and spectrometers, with the Lorentzian of Eq. (2). Such a Voigt function is defined as follows: Z 2 a 1 et V ðx; aÞ ¼ dt: ð5Þ p 1 ðt xÞ2 þ a2 We point out that the electronic wavefunctions for the continuum states of the process have been calculated using the method described in [6–8], while the ground state wavefunction comes out from an extended CI calculation and the vibrational functions have been obtained in harmonic approximation.
3. Results and discussion In Fig. 1, we compare the C1s photoelectron spectrum of C2 H2 , measured at 313 eV, with the theoretical spectrum calculated using Eq. (4) and a Gaussian function of FWHM ¼ 100 meV, a value that takes into account the finite resolving power of monochromator and electron spectrometers
68
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
10 -1
Intensity (arb. units)
8
C2H2 C1s h ν = 313 eV
experimental calculated g-state u-state
6
4
2
0 20.5
21 21.5 Kinetic energy (eV)
22
22.5
Fig. 1. Experimental (dots) and theoretical (red line) C1s photoelectron spectrum of C2 H2 . The blue and green lines are spectral profiles obtained from the convolution of the vibrational resonances associated, respectively, with the g- and u-core–hole states. The vertical lines give intensity and position of these resonances.
used in the experiment – see [5]. We see that the agreement between experimental and theoretical spectra is very close and the observed spectrum results from a delicate balancing of two quite different spectral profiles due to the vibrational progressions associated respectively with the g- and u-core–hole states of the ionized molecule. Moreover, our calculations make evident the vibrational structure that is behind the observed spectrum: the vertical lines drawn in Fig. 1 give, indeed, position and relative intensity of the transitions induced by the photon absorption, while the blue and green lines give the spectral profiles resulting from separate convolutions of the levels associated respectively with the g- and u-core–hole states. The linewidths of the two core–hole states resulting from our calculations are Cg ¼ 80 10 meV and Cu ¼ 68 8 meV. Fit procedures of the experimental data give a lifetime broadening referred to an ‘‘average’’ C1s hole state that is between 90 ± 10 meV – see [5] – and 106 ± 2 meV – see [10]. The vibrational frequencies used in our calculations are 260 and 420 meV, respectively for the C–C and the C–H stretching progressions associated with the g- and u-core–hole state. Finally, our calculations give an intensity ratio jDk0;u =
2
Dk0;g j ¼ 0:67 and a value of 96 ± 10 meV for the 2 Rg 2 Ru separation, that is consistent with the value of 105 ± 10 meV estimated by Kempgens et al. [5] and with that of 101.6 ± 0.8 calculated by Borve et al. [10]. Looking at the details of the spectrum, we observe that the highest peak in the spectrum of Fig. 1 is due to the purely electronic transitions fj0el ij0; 0i ! jHelj ; ~ k ij0; 0i; j ¼ g; ug, the well resolved shoulder, slightly above 21.5 eV, is due to the transitions fj0el ij0; 0i ! jHelj ; ~ k ij1; 0i; j ¼ g; ug that populate the first excited state of the C–C stretching mode associated with the two electronic core–hole states, plus a contribution from the transition j0el ij0; 0i ! jHelu ; ~ k ij0; 1i to the first excited state of the C–H stretching mode associated with the u-core–hole state. Finally, the small hump around 21.3 eV is due to the transitions fj0el ij0; 0i ! jHelj ; ~ k ij2; 0i; j ¼ g; ug that populate the second excited state of the C–C stretching mode associated with each electronic state and to the transition j0el ij0; 0i ! jHelg ; ~ k ij0; 1i to the first excited state of the C–H stretching mode associated with the g-core–hole state. Transitions to higher vibrational states of the C–C and C–H vibrational progressions have appreciably lower
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
69
Fig. 2. Polar plot of the angle resolved C1s photoelectron spectrum of C2 H2 in a plane defined by the molecular axis and the polarization direction of the light orthogonal to the molecular axis. The upper and lower part of the plot refer respectively to the g- and u-state of the residual ion.
intensities and contribute to the tail of the spectral profile in the low kinetic energy side of the spectrum. In Fig. 2, we show two angular distribution patterns of a photoelectron ejected in the continuum after absorption of one photon linearly polarized in a direction orthogonal to the molecular axis: one plot gives the angular distribution when P the residual ion is left in the j2 g : 1r1 g i state, while P the other when the ion is left in the j2 u : 1r1 u i state. The intensities of two angular distributions are shown as polar plots in the plane defined by the molecular axis and the polarization direction of the incident radiation. As expected from symmetry considerations, both the angular distributions have a node along the molecular axis, but the angular distribution associated with the ustate has also a node in the direction perpendicular to the molecular axis.
4. Conclusion The use of quantum-mechanical methods based on first principles allows an accurate reproduction
of K-shell photoionization spectra of molecules and the detailed analysis of the contributions that produce the measured spectral profiles. These theoretical methods give also crucial information on the dynamics of the photoionization processes and allow one to predict the results of sophisticated experiments, such as those in which angular distribution patterns of electrons ejected in photoionization or decay processes are recorded. Acknowledgements We acknowledge the support by MURST of Italy (PRIN-2000 and -2001). References [1] H.A. Padmore, T. Warwick, J. Electron Spectrosc. Relat. Phenom. 75 (1995) 9. [2] A. Kikas, S.J. Osborne, A. Ausmees, S. Svensson, O.P. Sairanen, S. Aksela, J. Electron Spectrosc. Relat. Phenom. 77 (1996) 241. [3] S.J. Schaphorst, A. Jean, O. Schwarzkopft, P. Lablanque, L. Andric, A. Huetz, J. Mazeau, V.J. Schmidt, Phys. B: At. Mol. Opt. Phys. 29 (1996) 1901.
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[4] S. Rioul, B. Rouvellou, L. Avaldi, G. Battera, R. Camilloni, G. Stefani, G. Turri, Phys. Rev. Lett. 86 (2001) 1470. [5] B. Kempgens, H. K€ oppel, A. Kivim€aki, M. Neeh, L.S. Cederbaum, A.M. Bradshaw, Phys. Rev. Lett. 79 (1997) 3617. [6] R. Colle, S. Simonucci, Phys. Rev. A 48 (1993) 392. [7] R. Colle, S. Simonucci, Recent Res. Devel. Quant. Chem. 2 (2001) 135.
[8] R. Colle, S. Simonucci, J. Phys. B 35 (2002) 4033. [9] T. Aberg, G. Howat, in: W. Melhorn (Ed.), Corpuscles and Radiation in Matter, Encyclopedia of Physics, Vol. 31, Springer, Berlin, 1982, p. 469. [10] K.J. Borve, L.J. Saethre, T.D. Thomas, T.X. Carroll, N. Berrah, J.D. Bozek, E. Kukk, Phys. Rev. A 63 (2000) 012506.
Ab initio calculation of the C1s photoelectron spectrum of C2H2 R. Colle
a,b,* ,
D. Embriaco a, M. Massini b, S. Simonucci c, S. Taioli
d
a
d
Dipartimento di Chimica Applicata, INSTM, Universit a di Bologna, Via Saragozza 8, I-40136 Bologna, Italy b Scuola Normale Superiore, I-56100 Pisa, Italy c Dipartimento di Matematica e Fisica, INFM, Universit a di Camerino, Via Madonna delle Carceri 7, I-62032 Camerino, Italy Dipartimento di Energetica, Nucleare e Controllo Ambientale, INFM, Universit a di Bologna, Via Risorgimento 4, 40136 Bologna, Italy
Abstract The C1s photoelectron spectrum of C2 H2 , measured by Kempgens et al. [Phys. Rev. Lett. 79 (1997) 3617] and characterized by the presence of a core-level splitting, has been calculated and interpreted using an ab initio quantummechanical method that allows detailed analysis of photoionization processes and accurate reproduction of spectral profiles. The anisotropy of the electron emission, measured in a hypothetical experiment in which the orientation of the molecules can be established, has been also predicted and analyzed. Ó 2003 Elsevier B.V. All rights reserved. PACS: 31.15.Ar; 33.60.Fy; 33.70.)w; 33.80.Eh Keywords: Molecular photoionization; Core-level splitting; Ab initio scattering wavefunctions
1. Introduction The continuous improvements in synchrotron radiation sources, soft X-ray monochromators and electron analyzers [1–4] have produced in recent years considerable progress in K-shell ionization spectroscopy of small molecules. The photon- and electron-energy resolutions which are now achievable allow one to gather relevant information also on subtle quantum-mechanical effects, such as those due to the occurrence of near-
* Corresponding author. Address: Scuola Normale Superiore, I-56100 Pisa, Italy. Tel.: +39-050-509-248; fax: +39-050563-513. E-mail address: [email protected] (R. Colle).
degenerate core–hole configurations, and make this type of spectroscopy very interesting and challenging for the theory. Recently, Kempgens et al. [5] have reported the first experimental identification of the core-level splitting in the Cls photoelectron spectrum of C2 H2 , a feature ascribed to the presence of two near-degenerate core–hole states, classified as 2 1 j2 Rg : 1r1 g i and j Ru : 1ru i in the independentparticle scheme. In this paper, we present an ab initio calculation and theoretical analysis of the C1s photoelectron spectrum of C2 H2 measured by Kempgens et al. [5]. The results of this study have been obtained using a quantum-mechanical approach [6–8] already successfully applied to the interpretation of XPS, Auger and autoionization spectra of atoms and molecules. In this paper, we
0168-583X/$ - see front matter Ó 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-583X(03)01535-0
66
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
analyze the vibrational structure of the spectrum and we also predict the anisotropy of the electron emission measured in a hypothetical experiment in which the orientation of the ionized molecule has been established. In what follows, we briefly sketch the theory that is behind the calculation of the theoretical spectrum, than we compare the calculated spectrum with the experimental one analyzing its components and, finally, we present angular distribution patterns of electrons produced by a photoionization process in which the photons are polarized along a direction perpendicular to the molecular axis.
2. Theory The construction of the theoretical photoemission spectrum requires the calculation of the scattering wavefunctions fh1; . . . ; N jHj ; ~ k ig for the continuum states of the N -electron molecule involved in the process. These wavefunctions are defined asymptotically by the momentum ~ k of the ejected electron and the state jHj i of the residual ion. In core-ionization processes, however, the core–hole states produced by the primary photoionization are metastable, being coupled with the continuum states produced by the Auger decay which follows the primary electron emission. As a consequence, these states have finite lifetime and the corresponding spectral lines in the photoemission spectrum have non-negligible linewidths. From general considerations based on the scattering theory (see [6–9]), one can prove that the differential cross section of a photoemission process, in which a neutral molecule in its ground state j0i is ionized by a monochromatic (x), linearly polarized (k) radiation, can be approximated as follows: or0!j ðx; ~ kÞ / ~ ok
2C ^ k jHj ; ~ jh0jO k ij 2pj 2
ðx þ E0 Ejþ k Þ þ
C2j
;
ð1Þ
4
where jHj ; ~ k i is the final state of the process, E0 , Ejþ 2 and k ¼ k =2 are, respectively, the energy of the ground state of the neutral molecule, the energy of the state jHj i of the ion and the kinetic energy of
the ejected electron. Furthermore, Cj is the width b k ¼ ~ of the resonance and O ek RNi¼1~ ri the component of the electronic dipole operator along the polarization direction ~ ek . In deriving Eq. (1), we have used the so-called ‘‘two-step’’ approximation [9], that consists in neglecting the coupling between primary photoionization and subsequent Auger decay process, furthermore, here and in the following we have used the atomic system of units jej ¼ h ¼ me ¼ 1. If we consider now that in the Born–Oppenheimer approximation the wavefunction representing the state of a non-rotating molecule is the product of an electronic and a vibrational function, we see that the transition j0i ! jHj ; ~ k i, which the cross section (1) refers to, really corresponds to a set of transitions j0el ij01 ; 02 ; . . .i ! fjHelj ; ~ k ijvj1 ; vj2 ; . . .i; vj1;2;... ¼ 0; 1; . . .g, from the ground state of the neutral molecule to a bunch of vibrational states, each one classified (in harmonic approximation) by the numbers fvja g that give the degree of excitation in the vibrational modes a ¼ 1; 2; . . . supported by the electronic state jHelj i of the ion. It follows that the appropriate expression for the cross section of the process is a sum of contributions or0!j ðx; ~ kÞ 2 Cj b k jHel ; ~ / jh0el j O j k ij ~ 2p ok
X fvja g
jh0jfvja gij2 2
ðx Dfvja g k Þ þ
C2j
;
ð2Þ
4
h01 ; 02 ; . . . jvj1 ; vj2 ; . . .i
where h0jfvja gi and Dfvja g ¼ þ j j ½Ej ðelÞ þ v1 þ v2 þ ½E0 ðelÞ þ 01 þ 02 þ is the difference between the electronic plus vibrational energy of the ion in the state jHelj ijvj1 ; vj2 ; . . .i and the corresponding quantity of the neutral molecule in its ground state j0el ij01 ; 02 ; . . .i. Finally, Cj is the natural width of the metastable electronic state jHelj i. In deriving Eq. (2), we have assumed that, at room temperature, only the lowest vibrational state of the molecule is populated, furthermore, we have disregarded the contributions of the rotational states that are not resolved in these spectra. Finally, in the calculation of the matrix elements in Eq. (2), we have neglected the dependence of the electronic wavefunctions on the
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
nuclear coordinates. It is interesting to notice that, in this approximation, no interference effects appear among vibrational states associated with the intermediate electronic state. In order to construct a theoretical spectrum that can be quantitatively compared with the experimental photoemission spectrum, one has to include also the following contributions: 1. A sum over the electronic resonances that are embedded in the energy range of the incident radiation: in our problem, we have considered the two near-degenerate core–hole states 2 1 j2 Rg : 1r1 g i and j Ru : 1ru i; 2. A sum over the vibrational states belonging to the different vibrational modes supported by each electronic resonance; in our problem we have considered only the two totally symmetric C–C and C–H stretching modes without including bending modes that are not expected to be appreciably excited – see [5,10]; 3. An integration over all the directions k^ of emission of the photoelectron, for each kinetic energy of interest; 4. A convolution of the resulting cross section with ‘‘apparatus’’ functions that take into account the characteristics of the incident radiation and the finite resolving power of the electron analyzer. If we assume now that the incident radiation is constituted by a flux of photons almost perfectly polarized along e^k , with a small frequency spread around a given x, we can describe this flux by means of a normalized Gaussian function " 2 # 1 x0 x 0 pffiffiffi Gc ðjx xjÞ ¼ pffiffiffiffiffiffi exp ð3Þ c 2p c 2 whose full pffiffiffiffiffiffiffiffiffiffi width at half maximum ffi (FWHM ¼ 2c 2 ln 2) is proportional to the resolving power of the monochromator. Similarly, the finite resolving power of the electron analyzers can be taken into account by means of another Gaussian that gives the spread in energy of the spectrometers. A global convolution of the cross section with the two ‘‘apparatus’’ functions give the transition rate of the process to be compared
67
with the measured photoemission spectrum. An equivalent and simpler way of obtaining the same result is to calculate the transition rate of the process through the following expression: Z oW ðx;Þ ¼ M Gc ðjx0 xjÞdx0 o X Z or0!j ðx0 ;~ kÞ k dð k Þd~ ~ ok j¼g;u X ¼M jDk0;j ðÞj2 j¼g;u
X
2
jh0jfvja gij V
fvja g
Dfvja g þ x Cj pffiffiffi ; pffiffiffi ; c 8 c 2 ð4Þ
where M is a scaling factor adjusted on the intensity of the photon flux, g=u Rindicate the electronic 2 b k jHel ; ~ core–hole states, jDk0;j ðÞj2 ¼ d~ k jh0el j O j k ij dð k Þ is the electronic dipole matrix element averaged over the directions of emission, and V ðx; aÞ is a Voigt function that results from the convolution of a single Gaussian Gc , that takes into account the finite resolution of both monochromator and spectrometers, with the Lorentzian of Eq. (2). Such a Voigt function is defined as follows: Z 2 a 1 et V ðx; aÞ ¼ dt: ð5Þ p 1 ðt xÞ2 þ a2 We point out that the electronic wavefunctions for the continuum states of the process have been calculated using the method described in [6–8], while the ground state wavefunction comes out from an extended CI calculation and the vibrational functions have been obtained in harmonic approximation.
3. Results and discussion In Fig. 1, we compare the C1s photoelectron spectrum of C2 H2 , measured at 313 eV, with the theoretical spectrum calculated using Eq. (4) and a Gaussian function of FWHM ¼ 100 meV, a value that takes into account the finite resolving power of monochromator and electron spectrometers
68
R. Colle et al. / Nucl. Instr. and Meth. in Phys. Res. B 213 (2004) 65–70
10 -1
Intensity (arb. units)
8
C2H2 C1s h ν = 313 eV
experimental calculated g-state u-state
6
4
2
0 20.5
21 21.5 Kinetic energy (eV)
22
22.5
Fig. 1. Experimental (dots) and theoretical (red line) C1s photoelectron spectrum of C2 H2 . The blue and green lines are spectral profiles obtained from the convolution of the vibrational resonances associated, respectively, with the g- and u-core–hole states. The vertical lines give intensity and position of these resonances.
used in the experiment – see [5]. We see that the agreement between experimental and theoretical spectra is very close and the observed spectrum results from a delicate balancing of two quite different spectral profiles due to the vibrational progressions associated respectively with the g- and u-core–hole states of the ionized molecule. Moreover, our calculations make evident the vibrational structure that is behind the observed spectrum: the vertical lines drawn in Fig. 1 give, indeed, position and relative intensity of the transitions induced by the photon absorption, while the blue and green lines give the spectral profiles resulting from separate convolutions of the levels associated respectively with the g- and u-core–hole states. The linewidths of the two core–hole states resulting from our calculations are Cg ¼ 80 10 meV and Cu ¼ 68 8 meV. Fit procedures of the experimental data give a lifetime broadening referred to an ‘‘average’’ C1s hole state that is between 90 ± 10 meV – see [5] – and 106 ± 2 meV – see [10]. The vibrational frequencies used in our calculations are 260 and 420 meV, respectively for the C–C and the C–H stretching progressions associated with the g- and u-core–hole state. Finally, our calculations give an intensity ratio jDk0;u =
2
Dk0;g j ¼ 0:67 and a value of 96 ± 10 meV for the 2 Rg 2 Ru separation, that is consistent with the value of 105 ± 10 meV estimated by Kempgens et al. [5] and with that of 101.6 ± 0.8 calculated by Borve et al. [10]. Looking at the details of the spectrum, we observe that the highest peak in the spectrum of Fig. 1 is due to the purely electronic transitions fj0el ij0; 0i ! jHelj ; ~ k ij0; 0i; j ¼ g; ug, the well resolved shoulder, slightly above 21.5 eV, is due to the transitions fj0el ij0; 0i ! jHelj ; ~ k ij1; 0i; j ¼ g; ug that populate the first excited state of the C–C stretching mode associated with the two electronic core–hole states, plus a contribution from the transition j0el ij0; 0i ! jHelu ; ~ k ij0; 1i to the first excited state of the C–H stretching mode associated with the u-core–hole state. Finally, the small hump around 21.3 eV is due to the transitions fj0el ij0; 0i ! jHelj ; ~ k ij2; 0i; j ¼ g; ug that populate the second excited state of the C–C stretching mode associated with each electronic state and to the transition j0el ij0; 0i ! jHelg ; ~ k ij0; 1i to the first excited state of the C–H stretching mode associated with the g-core–hole state. Transitions to higher vibrational states of the C–C and C–H vibrational progressions have appreciably lower
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Fig. 2. Polar plot of the angle resolved C1s photoelectron spectrum of C2 H2 in a plane defined by the molecular axis and the polarization direction of the light orthogonal to the molecular axis. The upper and lower part of the plot refer respectively to the g- and u-state of the residual ion.
intensities and contribute to the tail of the spectral profile in the low kinetic energy side of the spectrum. In Fig. 2, we show two angular distribution patterns of a photoelectron ejected in the continuum after absorption of one photon linearly polarized in a direction orthogonal to the molecular axis: one plot gives the angular distribution when P the residual ion is left in the j2 g : 1r1 g i state, while P the other when the ion is left in the j2 u : 1r1 u i state. The intensities of two angular distributions are shown as polar plots in the plane defined by the molecular axis and the polarization direction of the incident radiation. As expected from symmetry considerations, both the angular distributions have a node along the molecular axis, but the angular distribution associated with the ustate has also a node in the direction perpendicular to the molecular axis.
4. Conclusion The use of quantum-mechanical methods based on first principles allows an accurate reproduction
of K-shell photoionization spectra of molecules and the detailed analysis of the contributions that produce the measured spectral profiles. These theoretical methods give also crucial information on the dynamics of the photoionization processes and allow one to predict the results of sophisticated experiments, such as those in which angular distribution patterns of electrons ejected in photoionization or decay processes are recorded. Acknowledgements We acknowledge the support by MURST of Italy (PRIN-2000 and -2001). References [1] H.A. Padmore, T. Warwick, J. Electron Spectrosc. Relat. Phenom. 75 (1995) 9. [2] A. Kikas, S.J. Osborne, A. Ausmees, S. Svensson, O.P. Sairanen, S. Aksela, J. Electron Spectrosc. Relat. Phenom. 77 (1996) 241. [3] S.J. Schaphorst, A. Jean, O. Schwarzkopft, P. Lablanque, L. Andric, A. Huetz, J. Mazeau, V.J. Schmidt, Phys. B: At. Mol. Opt. Phys. 29 (1996) 1901.
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