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Electron impact excitation of molecules: Calculation of the cross section using the similarity function method and ab initio data for electronic structure
Chemical Physics Letters 436 (2007) 308–313 www.elsevier.com/locate/cplett
Electron impact excitation of molecules: Calculation of the cross section using the similarity function method and ab initio data for electronic structure S. Adamson a, V. Astapenko b,*, M. Deminskii c,d, A. Eletskii d, B. Potapkin L. Sukhanov c,d, A. Zaitsevskii c,d b
d
c,d
,
a M. Lomonosov Moscow State University, Moscow, Russia Moscow Institute for Physics and Technology, Moscow, Russia c Kintech, Moscow, Russia Russian Research Center, Kurchatov Institute, Moscow, Russia
Received 9 November 2006; in final form 15 January 2007 Available online 25 January 2007
Abstract Electron impact excitation cross sections of dipole-allowed transitions in diatomic molecules have been evaluated using the similarity function method allowing one to express cross sections through the oscillator strength of the transition and some universal function of the reduced incident electron energy. The necessary characteristics of the molecular terms involved and transition dipoles were determined by ab initio electronic structure calculations using the multireference configuration interaction technique or relativistic multipartitioning many-body perturbation theory. The results of specific calculations performed for CO and NO are in a reasonable agreement with the available experimental data and the results of comprehensive quantum-mechanical calculations. 2007 Elsevier B.V. All rights reserved.
1. Introduction Electron impact excitation of electronic states of molecules is a basic process playing the important role in a low temperature plasma containing molecules. Due to difficulties inherent to the many body problem the cross section of this process calculated using the comprehensive quantum mechanical approach can deviate drastically from the measured one (see e.g. [1]). One more obstacle hindering reliable calculations relates to the lack of exact information about the potential energy curves of molecules in excited states as well as the relevant electronic transition strengths. A notable progress in developing ab initio methods for calculation of above-mentioned characteristics of molecules has been reached recently [2–6]. However, the correct *
Corresponding author. E-mail address: [email protected] (V. Astapenko).
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.01.057
evaluation of the electron impact excitation cross section of molecules is still unattainable and requires further considerable simplifications. In this connection one should mention the usage of the semi-classical Grizinsky approach [7] and the impact parameter method [8–12] to the molecular excitation problem. One more the effective simplified approach to this problem that has manifested itself quite well in description of excitations of an optically allowed transitions of atoms [13] is based on using a similarity function. In accordance with this approach, the cross section is represented through an universal dimensionless function of the reduced energy of the incident electron and a combination of parameters related to the atomic particle under consideration. This approach permits one to describe effectively the process of the electron impact dissociation of a molecule that occurring through the formation of an excited repulsive state of the molecule [14]. In the present Letter, this approach is utilized for treatment of the electron impact excitation of electronic states
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
2. General relations According to the similarity function method the cross section of excitation of an atom into an optically allowed state is represented as follows [13] rif ¼
2pe4 ðDEif Þ2
ð1Þ
fif uðxÞ;
where e is the charge of electron, fif is the oscillator strength of the transition under consideration i ! f; u(x) is a universal dimensionless similarity function independent on both the sort of the atomic particle and the type of the transition; x = e/DEif is the ratio of the incident electron energy e to the excitation energy DEif. The most appropriate form of the similarity function u(x) providing the correct energy dependence of the cross section near the threshold (x 1) and at high energies (x 1) is [13] pffiffiffiffiffiffiffiffiffiffiffi ln½1 þ a x 1 uðxÞ ¼ : ð2Þ xþb The best agreement with experimental data is reached at the magnitudes of the fitting parameters a 0.5, b 3. Generalization of the expression Eq. (1) for the case of excitation of molecules results in the following expression for the cross section of vibrational and electronic transition i,ni ! f,nf [14] rif ðni ; nf Þ ¼
2pe4
f ðni ; nf Þu½xðni ; nf ÞH 2 if
½DEif ðni ; nf Þ
ðe DEif ðni ; nf ÞÞ:
ð3Þ
Here ni,f is the vibrational quantum number of the lower (upper) electronic state of a molecule, Ei(f)(ni(f)) is their energy, x(ni, nf) = e/DEif(ni, nf), DEif(ni, nf) is the relevant transition energy, fif(ni, nf) is the oscillator strength of the electronic and vibrational transition, H(x) is the Heaviside step function, u(x) is the similarity function Eq. (2). The main interest for the kinetic applications is the excitation cross section summed over the vibrational states of the upper term X rif ðni Þ ¼ rif ðni ; nf Þ: ð4Þ nf
3. Ab initio calculations of molecular parameters The above-mentioned parameters of molecules involving relatively light atoms can be evaluated with the sufficient accuracy by ab initio multireference configuration interaction method, taking into account single- and double excitations (MR SDCI), as is done in the standard program packages (see for example [3] and several others). In the case of molecules containing heavy atoms, reliable data
can be obtained by the relativistic version of the intermediate effective Hamiltonian method [2]. This approach implies the description of relativistic effects (including spin–orbit interactions) within the frame of the core pseudo potential model and the construction of many-electron relativistic effective Hamiltonians using the second-order multi-partitioning perturbation theory [4]. This approach has been applied to calculate the dependencies of the potential energies and the relevant matrix element of dipole moment as a function of internuclear distance for an electronic transition of InI. Discretization of the valence shell Hamiltonian was performed using the quadruple zeta quality basis set of contracted Gaussians augmented with diffuse functions. The effective Hamiltonians were built in model spaces including about 5 · 103 Slater determinants. The calculation of the necessary molecular integrals and generating the set of one-particle wavefunctions have been performed by solving state-averaged MCSCF equations using the program package COLUMBUS [3]. The effective Hamiltonians and the transition density matrices necessary for estimating the dipole moment matrix elements were calculated using the diagram technique (see [4] and the references in) and the codes described in [2]. Fig. 1 presents the singlet terms of GeS molecule calculated by the MR SDCI method. Fig. 2 shows the internuclear distance dependencies for the dipole moment of electronic transitions InI X [1]0+ ! A [2]0+ (a) and GeSX1 R+ ! 11P (b). These data have been used for evaluating the relevant electron impact excitation cross section (see Fig. 3). In contradiction, with InI in the case of GeS the spin– orbit effects are not so pronounced and can be neglected. In this case, the calculations were performed within the frame of MR SDCI method using the spin-averaged relativistic pseudo-potential [5]. The valence part of the wavefunction is represented in the quadruple zeta basis set of contracted Gaussians. The active space including four valence r-and two p-molecular orbitals correlates with the valence s and p atomic orbitals of Ge and S at the dis1 +
XΣ 1 1Π 1 2Π 1 1Δ 1 1Φ 1 + 2Σ
-13.35 -13.40 -13.45 -13.50 -13.55
E, a.u.
of a diatomic molecule using the parameters of electronic molecular state and transitions calculated by ab initio methods of the electronic structure theory.
309
-13.60 -13.65 -13.70 -13.75 -13.80 -13.85 -13.90 -13.95 3
4
5
6
7
R(Ge-S), a.u.
Fig. 1. Singlet terms of GeS molecule calculated by the MR SDCI method.
310
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313 3
a
1
b InI
GeS
X [1] 0 + → A [2] 0 +
2
1 1 X S+ → 1 Π
D(R), a.u. 0.5
D(R), a.u 1
0 0
4
6
2
4
6
0
8
10
R, a.u.
R, a.u.
c
2
' 2
+
X Σ -B Σ
+
1
MgH
D(R), a.u. 0 2
+
2
X Σ -A Π
2
4
6
8
10
R, a.u. Fig. 2. Internuclear distance dependence of the dipole moment matrix element for the transitions: InI X [1]0+ ! A[2]0+ (a), GeS X 1R+ ! 11P (b), MgH X 2 (+) R ! A2P and X 2R(+) ! B 0 2R(+) (c), calculated by ab initio approaches.
0.04
a
b
ni = 1
0.2
ni=0
ni =1
0.03
0.15
ni =2 2 0.02
σ if , A Å
σif , Å 2
ni = 0
ni=2
0.1
0.01
0.05 0
0
20
40
60
80
100
0
ε, eV
10
20
30
40
50
60
ε, eV
d
c 4
3
σif , Å2
0
0.4
n i =2
ni =0 0.3
ni =2
σif , Å2
2
ni=1
0.2
ni =1 1
0 0
0.1
20
40
60
80
ε, eV
0
ni =0
0
10
20
30
40
ε, eV
Fig. 3. The incident electron energy dependencies of the excitation cross section calculated for the transitions: InI (X[1]0+ ! A[2]0+) (a); GeS (X1R+ ! 11P) (b); MgH (X2R(+) ! A2P) (c); MgH (X2R(+) ! B 0 2R(+)) (d) for various vibrational quantum numbers ni of the ground state.
sociation limit. The model space includes all configurations which can be created from the chosen active space. To
reproduce the excited electronic states accurately, the molecular orbitals (MO) for molecular ion GeS+ were used
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
in calculations. The correctness of the choice of the calculation scheme is confirmed by agreement of the obtained results with available experimental and theoretical data (see [6] and the references cited there). Note the difference in the shape of the internuclear distance dependence D(R) of the dipole moment matrix element in the above-treated cases: monotonous increase of D(R) for the transition InI (X [1]0+ ! A [2]0+), and its monotonous decrease for GeS(X1R+ ! 11P). 4. Calculation of the excitation cross sections The oscillator strength of an electronic-vibrational transition at fixed vibrational quantum numbers is calculated by the following formula: fif ðni ; nf Þ ¼
2 gf 2 DEif ðni ; nf ÞjDðni ; nf Þj ; 3 gi
ð5Þ
where Dðni ; nf Þ ¼
Z
þ1
Wi ðni ; RÞDðRÞWf ðnf ; RÞdR
ð6Þ
1
is the matrix element of the projection of the dipole moment calculated for the nuclear wavefunctions of the initial and final state of the transition under consideration; gi,f are the statistical weights of the ground and excited states. The Eq. (5) is represented in atomic units. The nuclear wavefunctions Wi(f) of the ground state X1R+ and excited state 11P of GeS molecule were calculated in a standard manner within the framework of the Morse potential model (see details in [14]). Fig. 3b presents the incident electron energy dependence of the excitation cross section for the transition GeS(X1R+ ! 11P) calculated for various vibrational quantum numbers ni = 0, 1, 2 of the ground state. As is seen the cross section magnitude far from the threshold slightly decreases as the parameter ni rises. By contrast, near the threshold e = 4 ‚ 4.5 eV the magnitude of the cross section increases as the parameter ni rises. Such a behavior of the cross section far from the threshold is caused by the decreasing dependence of the dipole moment of the transition under consideration on the internuclear distance (see Fig. 2b). At this transition, the equilibrium point of the excited term is notably displaced (about 10%) with respect to that of the ground state. It results in a considerable contribution of high vibrational quantum numbers nf into the total cross section Eq. (4). The dependence of the cross section on the vibrational quantum number nf at a fixed incident electron energy e is a non-monotonous function. Such a behavior has been reported in the case of electron impact excitation of allowed triplet transitions of H2 (see [12] and works cited therein). The number of peaks in this function depends on the vibrational quantum number of the initial state. The maximum magnitude of the partial cross section for (ni = 0) corresponds to (nf)max = 9. For ni = 1 two peaks occur at (nf)max = 4,16; for ni = 2 there exist three peaks
311
at (nf)max = 2, 10, 24. Fig. 3b also implies that the increase of the vibrational quantum number ni is followed by some displacement of the maximum in the cross section toward the lower incident electron energy. In an opposite case, when the magnitude of the transition dipole moment increases as the internuclear distance rises, the excitation cross section rises as the vibrational quantum number of the lower term increases. An example of such a situation is the transition of InI molecule (Fig. 2a). In this case the change in the internuclear distance as a result of the transition does not exceed one percent. The excitation cross section calculated for this transition is shown in Fig. 3a. Five components make a notable contribution into the sum Eq. (4). As is seen, the excitation cross section rises smoothly as the vibrational quantum number of the ground state increases. The difference between maximum cross sections of InI and GeS is explained by a difference between the values of their transition dipole moments in the vicinity of the nuclear equilibrium positions. One can see from the Fig. 2a,b that the transition dipole moment of GeS at re = 4 a.u. is higher than that of InI at re = 5.2 a.u. Therefore, the maximum value of the GeS excitation cross section exceeds the relevant quantity for InI. Fig. 3c,d show the excitation cross section for the transitions MgH X2R(+) ! A2P and X2R(+) ! B 0 2R(+), respectively, evaluated within the frame of the above-outlined approach with the usage of the quantum mechanical electronic structure calculations [15]. The characteristics of two transitions differ qualitatively. The dipole moment of the transition X 2R(+) ! A2P decreases monotonically as the internuclear distance rises (Fig. 2c). On the contrary, the internuclear distance dependence of the dipole moment of the transition X 2R(+) ! B 0 2R(+) has a broad maximum near the equilibrium internuclear distance of the excited term MgH B 0 2R(+) (Fig. 2c). Besides the first transition is characterized by a relatively small change in the value of the internuclear distance (less 3%), while that in the second case is about 50%. The data presented in Fig. 3c imply a slight decrease in the excitation cross section of the transition X2R(+) ! A 2P far from the threshold as the vibrational number of the ground state of the molecule rises. By contrast, Fig. 3d demonstrates a sharp increase in the excitation cross section for the transition X 2R(+) ! B 0 2R(+) as ni rises. Therefore, the dependence of the excitation cross section on the vibrational quantum number of the lower electronic state is determined by the mutual arrangement of the relevant terms of the transition. Thus in the case of a large change in the equilibrium internuclear distance the excitation cross section can depend strongly on the initial vibration quantum number ni. 5. Comparison with experiment and previous calculations Some notion about the accuracy of the developed approach can be obtained from the comparison of the cal-
312
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
a
0.8
1
b 3
3
0.6 0.1
σif , Å 2
2
2
0.4
σif . Å
2
1 0.01
1
0.2
1E-3 0
5
10
15
20
25
30 0
ε, eV
c
10
20
30
40
50
ε, eV
0.04
d
0.06
0.05
0.03
0.04
σif , Å
2 0.02
1
σif . Å2 0.03 1
0.02
0.01
0.01 0.00 0
10
20
30
40
50
60
ε, eV
0.00 0
10
20
30
40
50
ε, eV
Fig. 4. Comparison of the energy dependence of electron impact excitation cross section of molecules with the experimental data and calculation results: (a) transition CO (X1R+ ! A1P), 1 – present calculation, 2 – experiment [16], 3 – calculation by the multi-channel Schwinger method [17]; (b) transition NO (X 2P ! A2R+), 1 – the present work, black squares – experiment [19], 2 – experiment [20], 3 – calculation by the distorted wave method [21]; (c) transition NO X 2P ! D2R+ , 1 – present calculation, black squares – experiment [19]; (d) transition NO X2P ! C 2Pr 1 – present calculation, black squares – experiment [19].
culated data with experiments and results of comprehensive quantum mechanical calculations that is presented in Fig. 4. The cross section for the transition CO (X1R+ ! A1P) (Fig. 4a) was determined using the calculation results [18] for the internuclear distance dependence of the dipole moment of the transition. The relative change in the equilibrium internuclear distance of the states involved into the transition is quite large (about 10%), therefore the sum Eq. (4) includes a large number of components (up to 20) corresponding to various vibrational quantum numbers nf of the excited term. As is seen, the disagreement between the cross sections calculated in the present work and the experimental data does not exceed 50%. At the same time the comprehensive calculation [17] overestimates the maximum magnitude of the cross section about twice that is inherent to the Born approximation used in [17]. At the near threshold energy region both approaches provide a good agreement with the experiment. The cross section of transitions NO (X2P ! A2R+), X2P ! D2R+ and X2P ! C 2Pr evaluated using the quantum chemical calculations [22] is compared with experimental and theoretical data in Fig. 4b–d, respectively. As is seen, for this case the similarity function method provides a good agreement with experimental data. At the same time the distorted wave
method [21] overestimates the cross section as much as an order of magnitude. 6. Conclusions The development of quantum chemical methods for calculation of parameters of diatomic molecules permits one to use the similarity function approach [13] for evaluation of the electron impact excitation cross section for electronic-vibrational states of molecules. The performed comparison shows that the disagreement between the calculated and experimental data does not exceed that for results of more justified but much more complicated calculations. Therefore, the considered approach can provide reliable and quite simple evaluation of the electron impact excitation cross section when quantitative data about the relevant terms are available. References [1] J. Tennyson, J.D. Gorfinkiel, I. Rozum, C.S. Trevisan, N. Vinci, Radiat. Phys. Chem. 68 (2003) 65. [2] A. Zaitsevskii, R. Ferber, Ch. Teichteil, Phys. Rev. A 63 (2001) 042511.
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313 [3] H. Lischka, R. Shepard, I. Shavitt, et al., COLUMBUS: an ab initio electronic structure program, release 5.6, 2000. [4] A. Zaitsevskii, R. Cimiraglia, Int. J. Quantum Chem. 73 (1999) 395. [5] A. Nicklass, M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys. 102 (1995) 8942. [6] A. Dutta, S. Chattopadhyaya, K. Das, J. Phys. Chem. A 105 (2001) 3232, and references therein. [7] M. Cacciatore, M. Capitelli, Chem. Phys. 55 (1981) 67. [8] M.J. Redmon, B.C. Garrett, L.T. Redmon, C.W. McCurdy, Phys. Rev. A 32 (1985) 3354. [9] B.C. Garrett, L.T. Redmon, C.W. McCurdy, M.J. Redmon, Phys. Rev. A 32 (1985) 3366. [10] A. Laricchiuta et al., Chem. Phys. Lett. 329 (2000) 526. [11] R. Celiberto, R.K. Janev, A. Laricchiuta, M. Capitelli, J.M. Wadehra, D.E. Atems, Atom. Data Nucl Data 77 (2001) 1. [12] A. Laricchiuta, R. Celiberto, R.K. Janev, Phys. Rev. A 69 (2004) 022706.
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[13] A.V. Eletskii, B.M. Smirnov, JETP 84 (1983) 1639 (in Russian). [14] V.A. Astapenko, A.V. Eletskii, Rus. J. Chem. Phys. 23 (2004) 3 (in Russian). [15] R.P. Saxon, K. Kirby, B. Liu, J. Chem. Phys. 69 (1978) 5301. [16] M.J. Mumma, E.J. Stone, E.C. Zipf, J. Chem. Phys. 54 (1971) 2627. [17] Q. Sun, C. Winstead, V. McKoy, Phys. Rev. A 46 (1992) 6987. [18] D. Sundholm, J. Olsen, P. Jorgensen, J. Chem. Phys. 102 (1995) 4143. [19] M.J. Brunger, L. Campbell, A.G. Middleton, et al., J. Phys. B 33 (2000) 809. [20] V.V. Skubenich, M.M. Povch, I.P. Zapesochnyi, High Energ. Chem. 11 (1977) 92. [21] L.E. Machado, A.L. Monzani, M.-T. Lee, M.M. Fujimoto, in: C.W. McCurdy, T. Rescigno (Eds.), Proceedings of the International Symposium on Electron- and Proton-Molecule Collisions and Swarms, Lawrence Livermore Press, Berkeley, CA, 1995, p. H32. [22] R. de Vivie, S.D. Peyerimhoff, J. Chem. Phys. 89 (1988) 3028.
Electron impact excitation of molecules: Calculation of the cross section using the similarity function method and ab initio data for electronic structure S. Adamson a, V. Astapenko b,*, M. Deminskii c,d, A. Eletskii d, B. Potapkin L. Sukhanov c,d, A. Zaitsevskii c,d b
d
c,d
,
a M. Lomonosov Moscow State University, Moscow, Russia Moscow Institute for Physics and Technology, Moscow, Russia c Kintech, Moscow, Russia Russian Research Center, Kurchatov Institute, Moscow, Russia
Received 9 November 2006; in final form 15 January 2007 Available online 25 January 2007
Abstract Electron impact excitation cross sections of dipole-allowed transitions in diatomic molecules have been evaluated using the similarity function method allowing one to express cross sections through the oscillator strength of the transition and some universal function of the reduced incident electron energy. The necessary characteristics of the molecular terms involved and transition dipoles were determined by ab initio electronic structure calculations using the multireference configuration interaction technique or relativistic multipartitioning many-body perturbation theory. The results of specific calculations performed for CO and NO are in a reasonable agreement with the available experimental data and the results of comprehensive quantum-mechanical calculations. 2007 Elsevier B.V. All rights reserved.
1. Introduction Electron impact excitation of electronic states of molecules is a basic process playing the important role in a low temperature plasma containing molecules. Due to difficulties inherent to the many body problem the cross section of this process calculated using the comprehensive quantum mechanical approach can deviate drastically from the measured one (see e.g. [1]). One more obstacle hindering reliable calculations relates to the lack of exact information about the potential energy curves of molecules in excited states as well as the relevant electronic transition strengths. A notable progress in developing ab initio methods for calculation of above-mentioned characteristics of molecules has been reached recently [2–6]. However, the correct *
Corresponding author. E-mail address: [email protected] (V. Astapenko).
0009-2614/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.01.057
evaluation of the electron impact excitation cross section of molecules is still unattainable and requires further considerable simplifications. In this connection one should mention the usage of the semi-classical Grizinsky approach [7] and the impact parameter method [8–12] to the molecular excitation problem. One more the effective simplified approach to this problem that has manifested itself quite well in description of excitations of an optically allowed transitions of atoms [13] is based on using a similarity function. In accordance with this approach, the cross section is represented through an universal dimensionless function of the reduced energy of the incident electron and a combination of parameters related to the atomic particle under consideration. This approach permits one to describe effectively the process of the electron impact dissociation of a molecule that occurring through the formation of an excited repulsive state of the molecule [14]. In the present Letter, this approach is utilized for treatment of the electron impact excitation of electronic states
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
2. General relations According to the similarity function method the cross section of excitation of an atom into an optically allowed state is represented as follows [13] rif ¼
2pe4 ðDEif Þ2
ð1Þ
fif uðxÞ;
where e is the charge of electron, fif is the oscillator strength of the transition under consideration i ! f; u(x) is a universal dimensionless similarity function independent on both the sort of the atomic particle and the type of the transition; x = e/DEif is the ratio of the incident electron energy e to the excitation energy DEif. The most appropriate form of the similarity function u(x) providing the correct energy dependence of the cross section near the threshold (x 1) and at high energies (x 1) is [13] pffiffiffiffiffiffiffiffiffiffiffi ln½1 þ a x 1 uðxÞ ¼ : ð2Þ xþb The best agreement with experimental data is reached at the magnitudes of the fitting parameters a 0.5, b 3. Generalization of the expression Eq. (1) for the case of excitation of molecules results in the following expression for the cross section of vibrational and electronic transition i,ni ! f,nf [14] rif ðni ; nf Þ ¼
2pe4
f ðni ; nf Þu½xðni ; nf ÞH 2 if
½DEif ðni ; nf Þ
ðe DEif ðni ; nf ÞÞ:
ð3Þ
Here ni,f is the vibrational quantum number of the lower (upper) electronic state of a molecule, Ei(f)(ni(f)) is their energy, x(ni, nf) = e/DEif(ni, nf), DEif(ni, nf) is the relevant transition energy, fif(ni, nf) is the oscillator strength of the electronic and vibrational transition, H(x) is the Heaviside step function, u(x) is the similarity function Eq. (2). The main interest for the kinetic applications is the excitation cross section summed over the vibrational states of the upper term X rif ðni Þ ¼ rif ðni ; nf Þ: ð4Þ nf
3. Ab initio calculations of molecular parameters The above-mentioned parameters of molecules involving relatively light atoms can be evaluated with the sufficient accuracy by ab initio multireference configuration interaction method, taking into account single- and double excitations (MR SDCI), as is done in the standard program packages (see for example [3] and several others). In the case of molecules containing heavy atoms, reliable data
can be obtained by the relativistic version of the intermediate effective Hamiltonian method [2]. This approach implies the description of relativistic effects (including spin–orbit interactions) within the frame of the core pseudo potential model and the construction of many-electron relativistic effective Hamiltonians using the second-order multi-partitioning perturbation theory [4]. This approach has been applied to calculate the dependencies of the potential energies and the relevant matrix element of dipole moment as a function of internuclear distance for an electronic transition of InI. Discretization of the valence shell Hamiltonian was performed using the quadruple zeta quality basis set of contracted Gaussians augmented with diffuse functions. The effective Hamiltonians were built in model spaces including about 5 · 103 Slater determinants. The calculation of the necessary molecular integrals and generating the set of one-particle wavefunctions have been performed by solving state-averaged MCSCF equations using the program package COLUMBUS [3]. The effective Hamiltonians and the transition density matrices necessary for estimating the dipole moment matrix elements were calculated using the diagram technique (see [4] and the references in) and the codes described in [2]. Fig. 1 presents the singlet terms of GeS molecule calculated by the MR SDCI method. Fig. 2 shows the internuclear distance dependencies for the dipole moment of electronic transitions InI X [1]0+ ! A [2]0+ (a) and GeSX1 R+ ! 11P (b). These data have been used for evaluating the relevant electron impact excitation cross section (see Fig. 3). In contradiction, with InI in the case of GeS the spin– orbit effects are not so pronounced and can be neglected. In this case, the calculations were performed within the frame of MR SDCI method using the spin-averaged relativistic pseudo-potential [5]. The valence part of the wavefunction is represented in the quadruple zeta basis set of contracted Gaussians. The active space including four valence r-and two p-molecular orbitals correlates with the valence s and p atomic orbitals of Ge and S at the dis1 +
XΣ 1 1Π 1 2Π 1 1Δ 1 1Φ 1 + 2Σ
-13.35 -13.40 -13.45 -13.50 -13.55
E, a.u.
of a diatomic molecule using the parameters of electronic molecular state and transitions calculated by ab initio methods of the electronic structure theory.
309
-13.60 -13.65 -13.70 -13.75 -13.80 -13.85 -13.90 -13.95 3
4
5
6
7
R(Ge-S), a.u.
Fig. 1. Singlet terms of GeS molecule calculated by the MR SDCI method.
310
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313 3
a
1
b InI
GeS
X [1] 0 + → A [2] 0 +
2
1 1 X S+ → 1 Π
D(R), a.u. 0.5
D(R), a.u 1
0 0
4
6
2
4
6
0
8
10
R, a.u.
R, a.u.
c
2
' 2
+
X Σ -B Σ
+
1
MgH
D(R), a.u. 0 2
+
2
X Σ -A Π
2
4
6
8
10
R, a.u. Fig. 2. Internuclear distance dependence of the dipole moment matrix element for the transitions: InI X [1]0+ ! A[2]0+ (a), GeS X 1R+ ! 11P (b), MgH X 2 (+) R ! A2P and X 2R(+) ! B 0 2R(+) (c), calculated by ab initio approaches.
0.04
a
b
ni = 1
0.2
ni=0
ni =1
0.03
0.15
ni =2 2 0.02
σ if , A Å
σif , Å 2
ni = 0
ni=2
0.1
0.01
0.05 0
0
20
40
60
80
100
0
ε, eV
10
20
30
40
50
60
ε, eV
d
c 4
3
σif , Å2
0
0.4
n i =2
ni =0 0.3
ni =2
σif , Å2
2
ni=1
0.2
ni =1 1
0 0
0.1
20
40
60
80
ε, eV
0
ni =0
0
10
20
30
40
ε, eV
Fig. 3. The incident electron energy dependencies of the excitation cross section calculated for the transitions: InI (X[1]0+ ! A[2]0+) (a); GeS (X1R+ ! 11P) (b); MgH (X2R(+) ! A2P) (c); MgH (X2R(+) ! B 0 2R(+)) (d) for various vibrational quantum numbers ni of the ground state.
sociation limit. The model space includes all configurations which can be created from the chosen active space. To
reproduce the excited electronic states accurately, the molecular orbitals (MO) for molecular ion GeS+ were used
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
in calculations. The correctness of the choice of the calculation scheme is confirmed by agreement of the obtained results with available experimental and theoretical data (see [6] and the references cited there). Note the difference in the shape of the internuclear distance dependence D(R) of the dipole moment matrix element in the above-treated cases: monotonous increase of D(R) for the transition InI (X [1]0+ ! A [2]0+), and its monotonous decrease for GeS(X1R+ ! 11P). 4. Calculation of the excitation cross sections The oscillator strength of an electronic-vibrational transition at fixed vibrational quantum numbers is calculated by the following formula: fif ðni ; nf Þ ¼
2 gf 2 DEif ðni ; nf ÞjDðni ; nf Þj ; 3 gi
ð5Þ
where Dðni ; nf Þ ¼
Z
þ1
Wi ðni ; RÞDðRÞWf ðnf ; RÞdR
ð6Þ
1
is the matrix element of the projection of the dipole moment calculated for the nuclear wavefunctions of the initial and final state of the transition under consideration; gi,f are the statistical weights of the ground and excited states. The Eq. (5) is represented in atomic units. The nuclear wavefunctions Wi(f) of the ground state X1R+ and excited state 11P of GeS molecule were calculated in a standard manner within the framework of the Morse potential model (see details in [14]). Fig. 3b presents the incident electron energy dependence of the excitation cross section for the transition GeS(X1R+ ! 11P) calculated for various vibrational quantum numbers ni = 0, 1, 2 of the ground state. As is seen the cross section magnitude far from the threshold slightly decreases as the parameter ni rises. By contrast, near the threshold e = 4 ‚ 4.5 eV the magnitude of the cross section increases as the parameter ni rises. Such a behavior of the cross section far from the threshold is caused by the decreasing dependence of the dipole moment of the transition under consideration on the internuclear distance (see Fig. 2b). At this transition, the equilibrium point of the excited term is notably displaced (about 10%) with respect to that of the ground state. It results in a considerable contribution of high vibrational quantum numbers nf into the total cross section Eq. (4). The dependence of the cross section on the vibrational quantum number nf at a fixed incident electron energy e is a non-monotonous function. Such a behavior has been reported in the case of electron impact excitation of allowed triplet transitions of H2 (see [12] and works cited therein). The number of peaks in this function depends on the vibrational quantum number of the initial state. The maximum magnitude of the partial cross section for (ni = 0) corresponds to (nf)max = 9. For ni = 1 two peaks occur at (nf)max = 4,16; for ni = 2 there exist three peaks
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at (nf)max = 2, 10, 24. Fig. 3b also implies that the increase of the vibrational quantum number ni is followed by some displacement of the maximum in the cross section toward the lower incident electron energy. In an opposite case, when the magnitude of the transition dipole moment increases as the internuclear distance rises, the excitation cross section rises as the vibrational quantum number of the lower term increases. An example of such a situation is the transition of InI molecule (Fig. 2a). In this case the change in the internuclear distance as a result of the transition does not exceed one percent. The excitation cross section calculated for this transition is shown in Fig. 3a. Five components make a notable contribution into the sum Eq. (4). As is seen, the excitation cross section rises smoothly as the vibrational quantum number of the ground state increases. The difference between maximum cross sections of InI and GeS is explained by a difference between the values of their transition dipole moments in the vicinity of the nuclear equilibrium positions. One can see from the Fig. 2a,b that the transition dipole moment of GeS at re = 4 a.u. is higher than that of InI at re = 5.2 a.u. Therefore, the maximum value of the GeS excitation cross section exceeds the relevant quantity for InI. Fig. 3c,d show the excitation cross section for the transitions MgH X2R(+) ! A2P and X2R(+) ! B 0 2R(+), respectively, evaluated within the frame of the above-outlined approach with the usage of the quantum mechanical electronic structure calculations [15]. The characteristics of two transitions differ qualitatively. The dipole moment of the transition X 2R(+) ! A2P decreases monotonically as the internuclear distance rises (Fig. 2c). On the contrary, the internuclear distance dependence of the dipole moment of the transition X 2R(+) ! B 0 2R(+) has a broad maximum near the equilibrium internuclear distance of the excited term MgH B 0 2R(+) (Fig. 2c). Besides the first transition is characterized by a relatively small change in the value of the internuclear distance (less 3%), while that in the second case is about 50%. The data presented in Fig. 3c imply a slight decrease in the excitation cross section of the transition X2R(+) ! A 2P far from the threshold as the vibrational number of the ground state of the molecule rises. By contrast, Fig. 3d demonstrates a sharp increase in the excitation cross section for the transition X 2R(+) ! B 0 2R(+) as ni rises. Therefore, the dependence of the excitation cross section on the vibrational quantum number of the lower electronic state is determined by the mutual arrangement of the relevant terms of the transition. Thus in the case of a large change in the equilibrium internuclear distance the excitation cross section can depend strongly on the initial vibration quantum number ni. 5. Comparison with experiment and previous calculations Some notion about the accuracy of the developed approach can be obtained from the comparison of the cal-
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S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313
a
0.8
1
b 3
3
0.6 0.1
σif , Å 2
2
2
0.4
σif . Å
2
1 0.01
1
0.2
1E-3 0
5
10
15
20
25
30 0
ε, eV
c
10
20
30
40
50
ε, eV
0.04
d
0.06
0.05
0.03
0.04
σif , Å
2 0.02
1
σif . Å2 0.03 1
0.02
0.01
0.01 0.00 0
10
20
30
40
50
60
ε, eV
0.00 0
10
20
30
40
50
ε, eV
Fig. 4. Comparison of the energy dependence of electron impact excitation cross section of molecules with the experimental data and calculation results: (a) transition CO (X1R+ ! A1P), 1 – present calculation, 2 – experiment [16], 3 – calculation by the multi-channel Schwinger method [17]; (b) transition NO (X 2P ! A2R+), 1 – the present work, black squares – experiment [19], 2 – experiment [20], 3 – calculation by the distorted wave method [21]; (c) transition NO X 2P ! D2R+ , 1 – present calculation, black squares – experiment [19]; (d) transition NO X2P ! C 2Pr 1 – present calculation, black squares – experiment [19].
culated data with experiments and results of comprehensive quantum mechanical calculations that is presented in Fig. 4. The cross section for the transition CO (X1R+ ! A1P) (Fig. 4a) was determined using the calculation results [18] for the internuclear distance dependence of the dipole moment of the transition. The relative change in the equilibrium internuclear distance of the states involved into the transition is quite large (about 10%), therefore the sum Eq. (4) includes a large number of components (up to 20) corresponding to various vibrational quantum numbers nf of the excited term. As is seen, the disagreement between the cross sections calculated in the present work and the experimental data does not exceed 50%. At the same time the comprehensive calculation [17] overestimates the maximum magnitude of the cross section about twice that is inherent to the Born approximation used in [17]. At the near threshold energy region both approaches provide a good agreement with the experiment. The cross section of transitions NO (X2P ! A2R+), X2P ! D2R+ and X2P ! C 2Pr evaluated using the quantum chemical calculations [22] is compared with experimental and theoretical data in Fig. 4b–d, respectively. As is seen, for this case the similarity function method provides a good agreement with experimental data. At the same time the distorted wave
method [21] overestimates the cross section as much as an order of magnitude. 6. Conclusions The development of quantum chemical methods for calculation of parameters of diatomic molecules permits one to use the similarity function approach [13] for evaluation of the electron impact excitation cross section for electronic-vibrational states of molecules. The performed comparison shows that the disagreement between the calculated and experimental data does not exceed that for results of more justified but much more complicated calculations. Therefore, the considered approach can provide reliable and quite simple evaluation of the electron impact excitation cross section when quantitative data about the relevant terms are available. References [1] J. Tennyson, J.D. Gorfinkiel, I. Rozum, C.S. Trevisan, N. Vinci, Radiat. Phys. Chem. 68 (2003) 65. [2] A. Zaitsevskii, R. Ferber, Ch. Teichteil, Phys. Rev. A 63 (2001) 042511.
S. Adamson et al. / Chemical Physics Letters 436 (2007) 308–313 [3] H. Lischka, R. Shepard, I. Shavitt, et al., COLUMBUS: an ab initio electronic structure program, release 5.6, 2000. [4] A. Zaitsevskii, R. Cimiraglia, Int. J. Quantum Chem. 73 (1999) 395. [5] A. Nicklass, M. Dolg, H. Stoll, H. Preuss, J. Chem. Phys. 102 (1995) 8942. [6] A. Dutta, S. Chattopadhyaya, K. Das, J. Phys. Chem. A 105 (2001) 3232, and references therein. [7] M. Cacciatore, M. Capitelli, Chem. Phys. 55 (1981) 67. [8] M.J. Redmon, B.C. Garrett, L.T. Redmon, C.W. McCurdy, Phys. Rev. A 32 (1985) 3354. [9] B.C. Garrett, L.T. Redmon, C.W. McCurdy, M.J. Redmon, Phys. Rev. A 32 (1985) 3366. [10] A. Laricchiuta et al., Chem. Phys. Lett. 329 (2000) 526. [11] R. Celiberto, R.K. Janev, A. Laricchiuta, M. Capitelli, J.M. Wadehra, D.E. Atems, Atom. Data Nucl Data 77 (2001) 1. [12] A. Laricchiuta, R. Celiberto, R.K. Janev, Phys. Rev. A 69 (2004) 022706.
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[13] A.V. Eletskii, B.M. Smirnov, JETP 84 (1983) 1639 (in Russian). [14] V.A. Astapenko, A.V. Eletskii, Rus. J. Chem. Phys. 23 (2004) 3 (in Russian). [15] R.P. Saxon, K. Kirby, B. Liu, J. Chem. Phys. 69 (1978) 5301. [16] M.J. Mumma, E.J. Stone, E.C. Zipf, J. Chem. Phys. 54 (1971) 2627. [17] Q. Sun, C. Winstead, V. McKoy, Phys. Rev. A 46 (1992) 6987. [18] D. Sundholm, J. Olsen, P. Jorgensen, J. Chem. Phys. 102 (1995) 4143. [19] M.J. Brunger, L. Campbell, A.G. Middleton, et al., J. Phys. B 33 (2000) 809. [20] V.V. Skubenich, M.M. Povch, I.P. Zapesochnyi, High Energ. Chem. 11 (1977) 92. [21] L.E. Machado, A.L. Monzani, M.-T. Lee, M.M. Fujimoto, in: C.W. McCurdy, T. Rescigno (Eds.), Proceedings of the International Symposium on Electron- and Proton-Molecule Collisions and Swarms, Lawrence Livermore Press, Berkeley, CA, 1995, p. H32. [22] R. de Vivie, S.D. Peyerimhoff, J. Chem. Phys. 89 (1988) 3028.