Diagrammatic Representation of Electronic Correlations in Photoionization Process: Application to Scandium

Transcription

1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 2, August 15, 2011 Diagrammatic Representation of Electronic Correlations in Photoionization Process: Application to Scandium LIU Meng-Meng ( ) and MA Xiao-Guang ( ½) Department of Physics, Ludong University, Yantai , China (Received November 15, 2010; revised manuscript received December 20, 2010) Abstract The conversion rules under which an algebraic expression can be obtained from a corresponding photoionization Goldstone diagram have been given systematically in the present work. The electronic correlations in the photoionization processes then could be studied diagrammatically. The application to atomic scandium shows that the present theoretical scheme can give reasonable photoionization cross sections, which agree well with the experimental results. PACS numbers: Fb, t Key words: photoionization, electronic correlation, scandium 1 Introduction The electronic correlation has been studied by means of diagrammatic many-body perturbation theory techniques for fifty years. [1 4] This subject was very important because the energy arising from the instaneous correlations of the individual electronic motions has the same order of magnitude as most energies of chemical effect. Goldstone has already pointed out that the electronic correlation effects can be simply described by the linked diagram perturbation expansion in terms of the particle-hole diagrammatic formalism. [5] In order to translate a given correlation diagram to the corresponding algebraic expression, some diagrammatic rules and conventions should be introduced. [4 8] However, for studying the correlation effects in dynamic processes, especially, for the photoionization process, no systematic conversion rule has been given at all till now. Although Brueckner Goldstone diagrams provide a simple pictorial description of electronic correlation effects in terms of the particle-hole formalism and give some commonly used conventions, in the photoionization process, some different and explicit diagrammatic conventions are needed. In the present work, a set of more systematic conversion rules, which can deal with the interactions of photon with atoms and molecules has been given for the first time. With these diagrammatic rules and conventions, the photoionization and photoexcitation processes can be handled more physically and easily than the algebraic expressions. Open-shell atoms are more attractive than close-shell cases because they have non-spherical charge distribution. This means that more electronic correlation effects should be considered in photoionization process. With or without electronic correlation, the photoionization cross sections will be very different for open-shell atoms. The experimental studies of open-shell atoms are still relatively few due to the difficulty in producing a usable atomic beam of open-shell atoms. Recently, Whitfield et al. have performed high resolution experimental measurements on photoionization cross sections of open-shell atom scandium in the region of the 3p 3d giant resonance. [9] Some interesting electronic correlation effects which cannot arise in the closed-shell case have been observed. However, the existing theoretical results could not agree well with their experimental measurements. The discrepancies between theoretical and experimental results need to be explained by a more detailed theoretical study. In the present study, more electronic correlation effects, i.e., some important satellite channels in this region have been considered. For heavy atoms, the relativistic effect was very important. But, in the present work, this effect was not to be considered due to the lack of the relativistic single-electron orbitals. So in some double-peak regions, only one peak appears in the present theoretical results. As a whole, our theoretical results agree with the experiments. In the following sections, the diagrammatic conventions and rules for photoionization processes have been given systematically. The application to the scandium atom has proven that this pictorial description of electronic correlation was reasonable and correct. The correlation effects between 3p and 3d or 4s electrons in these photoionization processes have been studied and the results agree with experiments in principle. The present study was part of The University Science & Technology Planning Program of Shandong Province under Grant No. J10LB60 (self-financing) and partly supported by the Natural Science Foundation of Shandong Province under Grant No. ZR2011AM010 and 2009 Technology Innovation Fund (09L026) of Ludong University Author to whom correspondence should be addressed c 2011 Chinese Physical Society and IOP Publishing Ltd

2 No. 2 Communications in Theoretical Physics Electronic Correlation Diagrams and Conventions The interaction between photon and atom can be represented by the dipole transition matrix element. While the Coulomb interaction between two electrons, i.e., the electronic correlation (including exchange part) can be represented by the Coulomb matrix element. The electrons occupied in a single particle state can be excited to another state under these interactions. Considering onephoton ionization process, in diagrammatic representation, the ionization and excitation processes can be modelled by one dipole matrix element coupled with unlimited Coulomb matrix elements. Two types of the basic elements are included in the diagrams as shown in Fig. 1, one is the interaction line and the other is the single-electron orbital line. The horizontal solid line denotes the interaction between two electrons and the horizontal dash line with a dot at the end represents the interaction between an electron and a photon. The line with an arrow represents the single-electron orbital. Combining these two types of elements can give the ionization or excitation transition matrix element. Performing the translation from a given diagram to the corresponding algebraic expression should obey the following rules. Fig. 1 Two basic elements: the interaction line and the single-electron orbital line. Fig. 2 Some first- and second-order diagrams for photoionization processes. (i) The line with an arrow towards to the joint of interaction line denotes the initial ket orbital before interaction. (ii) The line with an arrow away from the joint of interaction line denotes the final bra orbital after the interaction. (iii) The solid horizontal lines without any ends represent the Coulomb interaction between two electrons, which then give out the Coulomb transition matrix elements v arbs = rs R 1 ab as shown in Figs. 1(a) and 1(b). It represents that in Fig. 1(a) the electron in the initial state a interacts with the other electron in the initial state b and finally one of them has been ionized into the final state r while the other has been ionized to the final state s. The excitation or de-excitation process can also be included into the diagrams. In Fig. 1(b), the Coulomb interaction between electrons in state a and b respectively makes one electron ionized into r state and the other excited to s state. Then the s electron interacts with the electron in the state c, the result was that the s electron has been de-excited into the state b and the c electron has been ionized into the t state.

3 314 Communications in Theoretical Physics Vol. 56 (iv) The dashed line with a dot at the end represents the interaction between electron and photon, which gives the dipole matrix element z ar = r z a as shown in Fig. 1(c). It represents the excitation or ionization process of an electron in initial orbital a to the final orbital r by the dipole interaction. All the Coulomb interactions can be coupled with this dipole matrix. As a result, in the photoionization process, the dipole matrix element z should be replaced by the effective matrix z = z v. (v) Till now, all the interactions in photoionization and photoexcitation process can be described in these diagrams. Each interaction line gives a matrix element-a Coulomb interaction matrix or a dipole transition matrix. These interactions occur in order from bottom to top. At the same horizontal level, they occur in order from right to left. Then one writes these matrix-elements down on numerator from right to left according to the interactions sequence. (vi) Every interaction contributes an energy factor to the denominator except the last interaction. There are n-1 factors in the denominator corresponding to an n-th order diagram. The energy factors in the denominator can be determined by the following rules: draw a virtual line (the dash line as shown in Fig. 2) over an interaction line, all the crossed orbital line with this virtual horizontal line will contribute their energy to the denominator factors D = (Σǫ Σǫ ), where the ǫ are the one-electron orbital energies. If there is a dipole interaction under this virtual line, the energy terms should be written as D = (Σǫ Σǫ + ω). Each diagram will represent an electronic correlation process. The corresponding algebraic expression of a given diagram then can be obtained with these conversion rules. As an example of the application to these rules, let us consider some simple photoionization diagrams. In the simplest photoionization process, a single electron is ejected from an orbital of an atom or a molecule following the absorption of a single photon. This process can be described diagrammatically by Fig. 1(c) and the corresponding algebraic expression is r z a without any energy denominator according to the conversion rules (vi). This process is also called zero-order diagram because there is no any Coulomb interaction between electrons to be considered. Some important and first-order and second-order diagrams have been shown in Fig. 2. All these diagrams are the indirect photoionization processes and represent the electronic correlation effects. The diagrams of Figs. 2(a) and 2(b) represent the first-order diagrams and will give a coupling of the photoionization process (a r) with some single excitation transition processes (b s). The diagrams in Figs. 2(c) and 2(d) are the second-order diagrams because that there are two Coulomb interaction processes included. In Figs. 2(a) and 2(c), all the Coulomb interactions (i.e., the electronic correlations) occur before the dipole interaction. These are two of the auto-ionization processes. According to the conversion rules, the corresponding algebraic expressions for Figs. 2(a) and 2(c) respectively are z ar = s z ar = s,t z bs v arbs ǫ a ǫ r + ǫ b ǫ s, (1) z ct v arbs v bsct (ǫ a ǫ r + ǫ c ǫ t )(ǫ b ǫ s + ǫ c ǫ t ). (2) For the photoionization processes in Figs. 2(b) and 2(d), some resonant structures will be given when the photon energy is equivalent to the excitation energy. This can be seen clearly in algebraic expressions of Figs. 2(b) and 2(d), which can be converted respectively into z ar = s v arbs z bs ǫ b ǫ s + ω, (3) z ar = s,t v arbs z ct v bsct (ǫ a ǫ r + ǫ c ǫ t )(ǫ a ǫ r + ǫ b ǫ s + ǫ c ǫ t + ω). (4) Figure 2(b) will give a sharp resonance when the photon energy is equivalent to the absolute value of the energy difference between the states b and s. But in actual photoionization processes, a lot of electrons will interact with each other such as in Fig. 2(d), so the resonance will be broaden and shifted by these electronic correlations. As shown in the above algebraic expression Eq. (4), the resonance has been shifted by an energy ǫ a ǫ r + ǫ b ǫ s. In this diagrammatic representation, one can easily study and interpret the complex electronic correlations in photoionization processes. 3 Application and Discussion The single-electron orbitals 3p, 3d, and 4s in the ground configuration state 3p 6 3d4s 2 of scandium atom have been calculated respectively by using MCHF package. [10] The excited single-electron orbitals nl(l = s, p, d, f) in the final-excited configuration states 3p 5 3d4s 2 ns (n 5), 3p 5 3d4s 2 nd (n 3), and 3p 6 4s 2 np (n 4), 3p 6 4s 2 nf (n 4), and 3p 6 3d4snp (n 4) respectively have been obtained from the package MYBNDV [7] with their private final-ionic cores. In order to compare the calculated photoionization cross sections with the experimental results, in the present calculation, the energies of these ground, and excited singleelectron orbitals have been scaled with the experimental binding energies. [9] The continuum photoelectrons with different angular momentum leaving the same final-ionic

4 No. 2 Communications in Theoretical Physics 315 cores as the above excited states have been prepared by using the package MARHUZ. [7] Finally the Coulomb matrix elements v and dipole matrix elements z can be calculated from the program GFCID. [7] The effective dipole matrix elements for different diagrams then can be obtained by using the above diagram conversion rules. The photoionization cross sections then could be calculated by the following expression σ(ω) = 8πω ck z 2, (5) where k(= 2(ǫ + ω)) represents the ejected photoelectron momentum. Fig. 3 The theoretical partial absolute photoionization cross sections of the 4s, 3p, and 3d main lines 4skp, 3pks, 3pkd, 3dkp, and 3dkf of atomic scandium. The total photoionization cross sections of these main lines are compared with experiments. The theoretical partial photoionization cross sections of the 4s, 3p, and 3d main lines 4skp, 3pks, 3pkd, 3dkp, and 3dkf of atomic scandium are shown in Fig. 3. The label such as 4skp in the present work means that a 4s electron has been ionized from the ground configuration state 3p 6 3d4s 2 by a photon and becomes a p photoelectron with continuum momentum k. all these five main lines can be represented by the pure photoionization diagram in Fig. 1(c) with the corresponding algebraic expression-only a pure dipole matrix element. So there is no any electronic correlation effect taken to be considered. The behavior of these photoionization cross sections then can be determined only by the bound and continuum electronic wavefunctions. [11] Cooper et al. have shown how the nature of these one-electron radial wavefunctions can cause dynamic variations in the photoionization cross sections. [11] Especially for the 3pkd process, there is a very broad maximum covering the whole energy region. This means that the resonance will be very wide when we consider the excitation of the 3p electron into 3d. While for other four processes, the main lines are very smooth and decrease slowly as the energy increases and thus the excitation resonance due to the 3p 3d transition will be very clear in these four photoionization processes. However, only considering the sum of these five main lines will not give the accurate results when we compare with the experimental results shown in Fig. 3(f). The electrons in actual scandium atom will interact with each other. Some satellite lines due to the 3p electron excitation have been observed in experiments. More electronic correlations should be considered theoretically in the photoionization processes. In Fig. 4, the 3p nd, ns excitations have been taken into account in the 4skl and 3dkl photoionization processes by using all the correlation diagrams in Fig. 2. It is immediately apparent by comparing Fig. 4 and Fig. 3(f)

5 316 Communications in Theoretical Physics Vol. 56 that the theoretical results with considering electronic correlations in Fig. 4 agree with experiments while the ones without in Fig. 3(f) agree very poor. Fig. 4 A comparison of the experimental and theoretical cross sections in the 3p excitations region. The electronic correlations in the diagrams of Fig. 2 have been taken into account. In Fig. 4, the double-peak structures shown in experiments in the low energy region are due to the spin-orbit interaction while the present theoretical calculations have not taken this effect into account. The experimental cross sections have been scaled because that the values obtained by experiments are relative ones. In the whole, the total cross sections of 4skp, 3dkp, and 3dkf with electron correlations in the 3p excitation region agree with the experimental results. The most structures due to the 3p electron excitation have been predicted by the present diagrammatic technique. And more electronic correlation diagrams considered, more accurate the total cross sections will be. 4 Conclusion The present work has given some rules and proven that the electronic correlation effects in photoionization processes could be studied explicitly by the diagrammatic scheme. These diagrams could be converted to the algebraic expressions, which can be calculated by computer under the conversion rules. The application to atomic scandium shows that the present theoretical scheme can give reasonable photoionization cross sections, which agree with the experimental results. The difference between the theoretical and experimental results might be due to inadequate correlations. In the future work, more electronic correlations by diagrammatic representation should be considered. References [1] K.T. Chung, Phys. Rev. Lett. 78 (1997) [2] H.P. Kelly and R.L. Simons, Phys. Rev. Lett. 30 (1973) 529. [3] H.P. Kelly and A. Ron, Phys. Rev. A 5 (1972) 169. [4] E.R. Davidson and D.W. Silver, Chem. Phys. Lett. 52 (1977) 403. [5] K.A. Brueckner, Phys. Rev. 97 (1955) [6] J. Goldstone, Proc. R. Soc. (Lond.) A 239 (1957) 267. [7] H.P. Kelly, Comput. Phys. Commun. 17 (1979) 99. [8] H.P. Kelly, Phys. Scr. T 14 (1987) 109. [9] S.B. Whitfield, K. Kehoe, R. Wehlitz, M.O. Krause, and C.D. Caldwell, Phys. Rev. A 64 (2002) [10] C. Froese-Fischer, Comput. Phys. Commun. 14 (1978) 145. [11] U. Fano and J.W. Cooper, Rev. Mod. Phys. 40 (1968) 441.