Correlation effects are studied in electron scattering off the fluorine molecule. ... inner region contains all the electronic cloud of the target molecular states included in the ..... target ground state is represented by single closed-shell Slater determinant. .... Nestmann, B., Characterization of metastable anionic states within the ...
WDS'08 Proceedings of Contributed Papers, Part III, 186–190, 2008.
ISBN 978-80-7378-067-8 © MATFYZPRESS
Correlation Effects in R -matrix Calculations of Electron-F2 Elastic Scattering Cross Sections M. Tarana and J. Hor´acˇ ek Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.
Abstract. Correlation effects are studied in electron scattering off the fluorine molecule. Fixed nuclei approximation R -matrix calculations of the elastic collision cross sections are presented for a set of internuclear distances at three levels of correlation. The obtained results are used to study the role of electronic correlation on the properties of the 2 Σu resonance. The Feshbach-Fano R -matrix method of resonance-background separation is applied and the effect of inclusion of various levels of correlation on the energy and width of the 2 Σu resonance discussed. Data required for construction of the non-local resonance model (construction of a discrete state and its coupling to the continuum) which allows the calculation of inelastic processes such as dissociative electron attachment, associative detachment and vibrational excitation [Domcke, 1991] including the correlation are presented.
Introduction Understanding of the electron-molecule collisions is important for determining the energy balance and transport properties of electrons in low-temperature gases and plasmas under variety of conditions. Significant progress has been made in development of quantum scattering theory methods for resonant electron-molecule collisions based on ab initio methods of quantum chemistry. In particular, method for application of the Feshbach-Fano formalism to the R -matrix method (called Feshbach-Fano R -matrix - FFR) has been developed and successfully applied to the separation of resonances in the potential scattering [Kolorenˇc et al., 2005] as well as in the electronmolecule collision [Nestmann, 1998]. This method enables the extraction of quantities using which it is possible to construct the non-local resonance model (NRM) used to study the processes connected with the nuclear dynamics like dissociative electron attachment (DEA) and vibrational excitation (VE) [Kolorenˇc and Hor´acˇ ek, 2005]. Recently, Brems et al. [2002] studied the DEA and VE of the F2 molecule using the R -matrix method with FFR separation of the discrete state and successive construction of the NRM. The R -matrix calculation has been provided using the code of Nestmann et al. [1991] at the SEP (static exchange with correlation) level of correlation. This calculation indicates that the position of the crossing point of the ionic and neutral potential curve strongly influenced by the correlation included in the fixed nuclei calculation, is of particular importance in the nuclear dynamics calculation. The main aim of this work is to test the SD-MRCI (single-double multireference configuration interaction) approach to the correlation in the electron scattering off the fluorine molecule calculated previously on the SEP level and to study the correlation effects in the FFR separation.
Different models of correlation The R -matrix method is based on dividing the coordinate space into two regions using a spherical boundary, Ω, centred on the centre-of-mass of the target molecule. The radius of the boundary, rΩ , is chosen so that the inner region contains all the electronic cloud of the target molecular states included in the calculation. In standard, low-energy calculations, the wave function for N +1 electrons system in the inner region is given by the expansion: X X ψkN +1 = A aijk Φi (x1 ...xN )uij (xN +1 ) + bik χi (x1 ...xN +1 ) , (1) ij
i
where k represents the k th solution of the inner region Hamiltonian, A is the antisymmetrisation operator, xi are the spatial and spin coordinates of electron i, uij are continuum orbitals (COs) which represent the scattering electron [Tennyson and Morgan, 1999], aijk and bik are variational coefficients, Φi is the wave function of the ith target state and χi are L2 functions constructed from the target occupied and virtual molecular orbitals. These functions represent electron correlation and polarisation effects. In the first sum, the configuration state functions are constrained to give the correct (target) space and spin symmetry for the first N -electrons as well as the correct total, N + 1 electron space-spin symmetry. The target wave functions are expanded as linear combinations of the
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configurations φk : Φi (x1 ...xN ) =
X
cik φk (x1 ...xN ) ,
(2)
k
where the cik coefficients are determined by diagonalising the Hamiltonian of the molecular target. In the SE (static exchange) approximation no correlation is introduced in the scattering system. The R matrix basis set is written in the form (1), where Φ0 is the Hartree-Fock determinant |Φ0 i consisting of lowest N spinorbitals (a, b, . . . ), higher eigenstates Φi , i > 0 are not taken into account. The second sum in (1) is absent. The SEP approximation introduces the correlation between the molecule as a whole and the projectile, but not inside the molecular target. The SE space of configurations is augmented by states with singly excited target and scattered electron in virtual MO (k, l, . . . ). In this approximation the R -matrix basis states can be written in the following form: X X X |Ψa i = cka |Φ0 ki + cua |Φ0 ui + cbkla Φkb l , (3) k
u
bkl
where (u, v, . . . ) denote the COs. The matrix elements1 hΨa |HN +1 | Ψb iint of the R -matrix Hamiltonian in this basis contain terms corresponding to the (N + 1)-electron HF Hamiltonian, terms corresponding to the correlation
between the projectile and the target electrons as well as terms of form Φka m |HN +1 | Φlb m int = hkb||aliint introducing the additional correlation into the target. These elements are present also in the calculation of the N electron k neutral target ground state wave function in the basis set containing the |Φ0 i and all the mono-excitations Φa , but according to the Brillouin theorem they do not contribute to the |Φ0 i energy due to the fact that offdiagonal elements of the Fock operator are zero. On the other hand, in the (N + 1)-electron calculation these elements contribute to the R -matrix poles Ek . This difference in the correlation treatment of the neutral target and the scattering system leads to incorrect relative positions of the R -matrix poles with respect to the HF ground state energy of the neutral target molecule. The CI level of the correlation treatment introduces the correlation into the target in addition to the correlation between the projectile and the target. The target wave functions Φi in expansion (1) take the form of the MRCI expansion consisting of the single and double excitations from selected references into the virtual MOs. Inclusion of the double excitations improves the behaviour of the wave functions for larger internuclear distances in the target as well as in the scattering calculations.
Application to the fluorine molecule In all the calculations the R -matrix sphere of radius rΩ = 10 Bohr has been used. Since the dipole moment of F2 as well as its polarisability are negligible, it is sufficient to consider the scattered electron free in the outer region. The target is represented by the cc-pVTZ basis set [Dunning, 1989], the continuum basis set used in all the calculations has been taken from the previously published R -matrix calculation [Brems et al., 2002]. In the expansion (1) only the ground electronic state of the target has been included. Due to the 2 Σu symmetry the lowest partial wave contributing to the scattering in our basis set is the p-wave, the d-wave does not contribute to the scattering. These two facts allow the single channel scattering calculation. The R -matrix poles Ek as well as transition density matrix used to calculate the R -matrix amplitudes [Nestmann et al., 1991] have been calculated using the MRCI program package DIESEL [Hanrath and Engels, 1997]. The R -matrix amplitudes as well as elastic scattering eigenphases and cross sections have been calculated using R -matrix program package by Nestmann et al. [1991]. Calculations at the different levels of correlation have been carried out. Due to technical complications with implementation of the MRCI method it is usually possible to obtain only 4 lowest R -matrix poles and corresponding amplitudes at the SEP and CI levels in case of the fluorine molecule. The higher R -matrix poles and amplitudes are approximated by corresponding 16 terms calculated on the SE level. The FFR method has been used to calculate the discrete state. The Σres region has been taken such that contains four lowest R -matrix states. The separation of the cross sections and phase shifts to the resonance and background parts has been calculated as well as the discrete state-continuum coupling and discrete state energies, necessary for construction of the NRM.
Results At the SE level of correlation 16 R -matrix poles have been calculated for every considered internuclear separation R near the equilibrium internuclear distance Req = 2.67 Bohr. The integral cross sections of the electron-F2 collision are presented in Fig. 1 for several geometries. The 1 The
subscript int denotes that the integration is carried over the inner region.
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8 R = 2.30 R = 2.40 R = 2.50 R = 2.60 R = 2.67 R = 2.70
(a) 25
Total Background (Σ’res) Resonance (Σ’res) Background (Σres) Resonance (Σres)
6 5 σ (Å2)
σ (Å2)
20
(b) 7
15
4 3
10
2 5
1
0
0 0
2
4
6
8
10
12
0
2
4
Energy (eV)
6
8
10
12
Energy (eV)
Figure 1. Cross section (a) at the SE level as a function of E at several internuclear separations R (Bohr) and corresponding FFR separation (b) for geometry R = 2.4 Bohr.
30 ×0.1
35
R = 2.30 R = 2.40 R = 2.50 R = 2.60 R = 2.67 R = 2.70
(a)
25
Total Background (Σres) Resonance (Σres)
(b) 30 25
σ (Å2)
σ (Å2)
20
15
20 15
10 10 5
5
0
0 0
2
4
6
8
10
12
E (ev)
0
2
4
6
8
10
12
E (ev)
Figure 2. Cross sections (a) calculated on the SEP level as a function of E for several internuclear separations R (Bohr) and corresponding FFR separation (b) for internuclear separation R = 2.4 Bohr
resonance structure becomes more narrow as R approaches to the crossing point of the potential curve of neutral molecule ground state with the negative ion. The FFR method has been used to separate the discrete state. Two calculations have been carried out. In one of them the Σres region includes lowest four R -matrix poles in all considered geometries. For small internuclear separations (R < Rcrit ) the contribution of the lowest R -matrix eigenstate |Ψ1 i to the discrete state should be negligible. However, estimation of the overlaps between the eigenstates of the unperturbed Hamiltonian and R matrix eigenstates causes unreasonably large value of coefficient c1 (for more detailed discussion see [Tarana and Hor´acˇ ek, 2007]). Separation by projectors Q and P results therefore into the additional peak in the background cross section. The corresponding separation of the cross section is shown in Fig. 1 for R = 2.4 Bohr. In order to avoid this behaviour another calculation has been carried with Σ′res excluding the lowest R -matrix eigenstate (Fig. 4). The corresponding cross section separation is also plotted in Fig. 1. With increasing R (R > Rcrit ) it is necessary to include the lowest R -matrix state into Σres in order to avoid poles with the small imaginary part in the background T -matrix and to obtain smooth transition to the bound state region of R. At the SEP level of correlation 4 R -matrix poles have been calculated and additional 16 poles have been added on the SE level. The calculated cross sections and corresponding phase shifts for several values of R are presented in Fig. 2. In contrast to the SE calculation, for the three largest considered geometries, the ionic state is bound. This means that at the SEP level of correlation the crossing point of the neutral target ground state potential with the first R -matrix pole curve is shifted towards smaller values of R compared to the SE level. The results of FFR separation are shown in Fig. 2. The discussion of the lowest R -matrix pole exclusion from Σres remains valid also in the SEP calculation, but the critical geometry Rcrit , where the lowest R -matrix pole becomes important, is smaller than in the SE calculation. Calculated R -matrix poles show good agreement with previously published SEP work enabling us to compare our calculations at the CI level with previous R matrix results [Brems et al., 2002]. The discrepancies are caused by different compact basis sets used.
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14 R = 2.30 R = 2.40 R = 2.50 R = 2.60 R = 2.67 R = 2.70
(a)
90 80 70
10
σ (Å2)
60 σ (Å2)
Total Background (Σ’res) Resonance (Σ’res) Background (Σres) Resonance (Σres)
(b)
12
50 40 30
8 6 4
20 2
10 0
0 0
2
4
6
8
10
12
0
2
4
6
E (eV)
8
10
12
14
16
E (eV)
Figure 3. Cross sections (a) for several geometries calculated at the CI level of the theory for several internuclear distances (Bohr) and corresponding FFR separation (b) for internuclear separation R = 2.4 Bohr
At the CI level of the theory 4 R -matrix poles have been calculated for a range of geometries from 2.0 Bohr to 5.5 Bohr. These CI poles have been completed by 16 SE poles. The calculated cross sections and corresponding phase shifts are plotted in Fig. 3. With increasing internuclear distance the contribution of the excited configuration increases in the target eigenstates as well as in the eigenstates of HΩ,N +1 (3σg → 3σu since these orbitals become degenerated asymptotically). These excitations have not been allowed in the previous models of correlation. Double excitations allow the energy of the target state as well as the energies of the poles to be lower than in the previous models and more consistent with previous calculations of the neutral target ground state potential curves [Blomberg and Siegbahn, 1981]. The electron affinity calculated at this level of correlation (2.2 eV) is closer to the experimental value [Blondel et al., 1989] (3.40 eV) than the value obtained at the SEP level [Brems et al., 2002] (13.5 eV). The discrepancy of our results with the experimental value is caused by the correlation imbalance in the calculation of the neutral target state and the negative ion state as well as not enough large R used for calculation of the electron affinity. The incorrect value obtained at the SEP level is understandable, because the absence of double excited configurations in the wave functions causes incorrect behaviour of the potential curves for large values of R. In the SEP calculation the target ground state is represented by single closed-shell Slater determinant. In the asymptotic region of internuclear separations this gives ground state of system F+ + F− and does not take into account the contribution of the open-shell configurations. Therefore the CI level of calculation improves the description of the target considerably for large internuclear distances. The potential curves of the R -matrix poles as a functions of the internuclear distance are plotted in Fig. 4. These curves show the avoided crossing near the crossing point of the neutral target state with the lowest R -matrix pole. The crossing point where the resonance turns into the bound state is located at Rthr = 2.76 Bohr at variance with previously calculated results [Brems et al., 2002; Morgan and Noble, 1984; Ingr et al., 1999]. Comparison of the calculated position with previous works is in Tab. 1. Table 1. Positions of the crossing points in Bohr where the resonance turns into the bound state. Morgan and Noble [1984] 2.56
Ingr et al. [1999] 2.62
Brems et al. [2002] 2.41
This work 2.76
Determination of the crossing point position, which is of particular importance in the nuclear dynamics calculations, is complicated by the unbalanced correlation in independent calculations of the ionic and neutral target states. The FFR method has been again used to determine the discrete state and corresponding separation of the cross sections and eigenphases for internuclear distance R = 2.4 Bohr are plotted in Fig. 3. Again two calculations have been carried out in order to study the effect of the lowest pole inclusion into the region Σres (see Fig. 3). The internuclear distance Rcrit is near the avoided crossing point of the first and second R -matrix state.
Concluding remarks The Bonn implementation of the R -matrix method has been used in order to study the 2 Σu resonance in electron scattering off the fluorine molecule using different treatments of correlation including recently implemented
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V0 Vk=1,2,3,4 εd Vbgk=1,2,3
20
16
SE
SE(bg) SEP SEP(bg)
CI
CI(bg)
14 12
15
3p
10 E (eV)
E (eV)
free
10
8 6
5
2p
Σ’res
4 0 2 1p −5 2
2.5
3
3.5
4
4.5
5
5.5
6
R (Bohr)
Σres
0 R (Bohr)
Figure 4. R -matrix poles (a) calculated at the CI level of the theory as a function of the internuclear distance and the lowest R -matrix poles of the free particle (b) compared with poles calculated at different levels of correlation and with the corresponding background R -matrix poles
SD-MRCI approach. The FFR method has been used to separate the resonance and to determine its coupling with the background continuum. The effect of the correlation on the resonance position and width has been studied. We have shown that inclusion of the doubly excited configurations into SD-MRCI expansions of the wave functions improves the scattering calculations results for larger internuclear distances. Our results show how the position of the crossing point of the neutral target potential curve with the ionic potential curve depends on the included correlation. In contrast with previous results our calculation shows that the ion enables the autodetachment in the equilibrium internuclear separation of the neutral molecule. Results obtained from the CI calculation are suitable for construction of the non-local resonance model of the nuclear dynamics which will show the correlation effects in the DEA an VE of the fluorine molecule. This model will be a subject of the forthcoming work.
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