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Theoretical study of low-lying electronic states and emission spectra of the excimer ions NaHe+ and NaNe+
Journal of Molecular Structure (Theochem) 490 (1999) 189–200 www.elsevier.nl/locate/theochem
Theoretical study of low-lying electronic states and emission spectra of the excimer ions NaHe 1 and NaNe 1 A.I. Panin a,*, A.N. Petrov b, Y.G. Khait b a
Department of Chemistry, Quantum Chemistry Group, St. Petersburg State University, 2 University Avenue, Old Petergof, 198904 St. Petersburg, Russia b Russian Scientific Center Applied Chemistry, 14 Dobrolyubova Avenue, 197198 St. Petersburg, Russia Received 22 January 1999; accepted 15 March 1999
Abstract Ab initio CASSCF calculations using the Gaussian basis set aug_cc_pVTZ developed by Dunning et al. specifically for calculations at the post-HF level was carried out for excimer NaHe 1 and NaNe 1 ions in order to determine spectroscopic characteristics and radiation lifetimes of their low-lying excited electronic states. The computed emission spectra of these were shown to be in good agreement with the experimental ones observed recently by Hammer et al. (Hyperfine Interactions 88 (1994) 151). Contrary to the earlier popular opinion, all excited electronic states of these ions, correlating with the limit Rg 1 1 Na, and not just the radiating singlet states turn out to be bound. It was shown that under the conditions of an experiment similar to the one performed by Hammer et al., the excited triplet NaHe 1(1 3S 1) states can decay as a result of their interactions with a high-energy Ar 1 beam. This decay has to be accompanied with the formation of the ions He 1, which in turn can interact with sodium atoms to yield the radiating NaHe 1(2 1S 1) states again and, thus to maintain the emission observed in the experiment. q 1999 Elsevier Science B.V. All rights reserved. Keywords: NaHe 1; NaNe 1; Low-lying electronic states; Potential curves; Radiative transitions; Emission spectra; Ab initio calculations
1. Introduction The main characteristic property of an excimer species is that one of its excited electronic states is bound, whereas its ground state is repulsive or quite weakly bound and dissociates at normal conditions. As a result, if the only channel of decay of the excited state is its radiative transition to the ground state, then a population inversion may be achieved, and such species are of interest for possible lasers. The idea of using excimers as an active laser media was first suggested in 1960 [1], and a number of excimer lasers * Corresponding author. E-mail address: [email protected] (A.I. Panin)
of different types are presently known [2]. These lasers may serve as sources of radiation in the region 130–1000 nm with a possibility to tune a radiation frequency in a rather broad interval. Recently considerable attention has been focused on searching for excimer species with radiative transitions in the shorter (VUV or even XUV) wave region. In 1985, on the basis of purely qualitative estimations, Basov et al. [3] predicted, in particular, that diatomic molecular cations consisting of a rare gas ion (Rg 1) and an alkali atom (M) are excimers, and that these species may be expected to decay by the emission of radiation in the region from 70 nm (in the case of NaHe 1) to 250 nm (for LiXe 1). In accordance with the model calculations performed later by Mantel and
0166-1280/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(99)00098-6
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Langhoff [4] the radiative electronic transitions in such systems should be expected in the region 60– 190 nm. These predictions were recently confirmed by Hammer et al. [5], who performed an experimental study of emission spectra of mixtures of rare gases with alkali-metal vapor excited by a high power Ar 1 beam and actually observed narrow emission continua in the region 60–200 nm. The equilibrium internuclear distances and dissociation energies of the ground states of such ions were evaluated by many authors (see Ref. [3]). In particular, the potential curves of the ground states of the ions RgM 1 (M Li, Na; Rg He, Ne) were recently calculated by Bililign et al. [6] using the ab initio methods. However, to the best of our knowledge, no reliable information about spectroscopic characteristics of excited electronic states of such ions is available. Hammer et al. [5] assigned the observed transition wavelengths of 65.76 and 79.40 nm to the cations NaHe 1 and NaNe 1, respectively, on the basis of the model estimations of Mantel and Langhoff [4]. The purpose of this article, is a theoretical investigation of low-lying states of the cations NaHe 1 and NaNe 1 to determine their spectroscopic characteristics and radiative transition probabilities and eventually to compare the emission spectra predicted by theory with the experimental ones from Ref. [5]. We will show that in contrast to the assumptions of Basov et al. [3], all excited electronic states of these ions, correlating with the limit Rg 1 1 Na, and not just the radiating singlet 2 1S 1 states turn out to be bound. Moreover, the binding energies of the excited triplet states 1 3S 1 correlating with the mentioned limit will be shown to exceed essentially with those of the 2 1S 1 states correlating with the same asymptote. Population of such triplet states should certainly lead to a decrease in efficiency and radiation power of possible lasers and may essentially affect a possibility of generation of laser radiation. From this viewpoint, a study of possible mechanisms of the triplet states decay appears to be important. In this work it will be shown that under the conditions of the experiment similar to that performed by Hammer et al. [5] the NaHe 1(1 3S 1) state should decay as a result of its interaction with a high-energy Ar 1 beam. This decay has to be accompanied with the formation of the ions He 1, which can interact again with sodium atoms to yield the radiating NaHe 1(2 1S 1) states and,
thus, to maintain the emission. Analogous processes are also likely to occur with the triplet states of both NaNe 1 and other cations RgM 1 studied by Hammer et al. [5].
2. Potential energy curves of low-lying states of the cations NaHe 1 and NaNe 1 In accordance with the available atomic data [7,8] and the Wigner–Witmer rules [9] the most stable asymptotes of NaHe 1 and NaNe 1 are the limits He(1 1Sg) 1 Na 1(1 1Sg) and Ne(1 1Sg) 1 Na 1(1 1Sg) correlating with the ground 1 1S 1 states of these molecular ions. As should be expected, owing to the small polarizabilities of the rare gas atoms, the 1 1S 1 states prove to be only weakly bound (see Ref. [6] and references therein). The second dissociation limit Na(1 2Sg) 1 He 1(1 2Sg) of the ion NaHe 1 lies 19.44 eV above its lowest limit [7] and correlates with its excited 2 1S 1 and 1 3S 1 states. In the case of NaNe 1, the second dissociation limit Na(1 2Sg) 1 Ne 1(1 2Pu) with its energy of 16.42 eV [7] correlates with the four excited states 2 1S 1, 1 1P, 1 3S 1, and 1 3P. It is clear that the radiative transitions 2 1S 1 ) 1 1S 1 (in NaHe 1) and 2 1S 1, 1 1P ) 1 1S 1 (in NaNe 1) must be accompanied by a charge transfer from rare gas atoms on sodium atom and their energies have to be close to the above-mentioned energies of the second dissociation limits of the molecular ions relative to their most stable asymptotes. Among the excited states of the ions NaHe 1 and NaNe 1, correlating with their higher-lying dissociation limits, there also exist both bound and weakly bound states. These states, however, appear to be unrelated to the emission observed by Hammer et al. [5] and will not be considered below. We will restrict our consideration to the electronic states, which correlate only with the two outlined lowest dissociation limits. All electronic structure calculations were performed in the framework of the CASSCF method [10], using the Gamess program [11] with the Gaussian basis set aug_cc_pVTZ developed by Dunning et al. [12] specifically for calculations at the post-HF level. To test the accuracy of the ab initio study performed in the present work, the ionization potentials (IPs) and excitation energies of the lowest excited
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200
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Table 1 Comparison of the computed IPs and excitation energies (in eV) of the lowest excited states of the atoms He, Ne, and Na and their cations with experimental data from Ref. [7] Species
He He 1 Ne Ne 1 Na Na 1
State
1 1Sg 1 3Sg 1 2Sg 2 2Sg 1 1Sg 1 1Pu % 1 3Pu 1 2Pu 1 2Sg 1 2Sg 1 2Pu 1 1Sg 1 3Pu
Ionization potential
Excitation energy
This work
Experimental
This work
Experimental
24.53
24.58
21.00
21.59
5.00
5.14
0.00 19.88 0.00 40.85 0.00 17.66 0.00 27.12 0.00 2.01 0.00 32.68
0.00 19.82 0.00 40.81 0.00 16.73 0.00 26.91 0.00 2.10 0.00 32.85
electronic states of both the atoms Na, He, Ne and their cations were calculated. For helium and its cation the full CI method was used, whereas for sodium, neon and their cations, we used the CASSCF method with the active space [2s2p3s3p4s4p] 8–9 for Na and Na 1, and the active space [2s2p3s3p4s3d] 8–9 for Ne and Ne 1. The results of these calculations are given in Table 1. As seen from the table, in the case of the atoms He, Na and their cations, the computed IPs and the excitation energies are in good agreement with the experimental data from Ref. [7]. However,
in the case of neon and its cation, the agreement proves to be essentially worse (e.g. the computed excitation energy of the averaged state Ne(1 1Pu % 1 3Pu) is higher by 0.93 eV than the experimental value 16.73 eV [7]). Thus, the CASSCF method and the chosen basis set may be expected to be sufficiently good for a correct description of both the ground and the low-lying excited electronic states of NaHe 1 (see also below), whereas in the case of NaNe 1, one may hope to obtain only qualitatively reliable results. Calculations of the potential energy curves of the
Fig. 1. Potential energy curves of the electronic states of NaHe 1 correlating with its two lowest dissociation limits.
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Fig. 2. Potential energy curves of the electronic states of NaNe 1 correlating with its two lowest dissociation limits.
three lowest electronic states of NaHe 1 and the five lowest states of NaNe 1 were carried out by the CASSCF method in the C2v subgroup of the C∞v point group. Preliminary calculations showed that the leading configuration describing the excited 1,3 1 S states of NaHe 1 is 1
1s2Na ls2 2s2 1p4 3s1 4s1
and that for the excited 1,3S 1 and 1,3P states of NaNe 1, the leading configurations are 1s2Na 1s2Ne 1s2 2s2 1p4 3s2 4s1 2p4 5s1
2
and 1s2Na 1s2Ne 1s2 2s2 1p4 3s2 4s2 2p3 5s1 ;
3
respectively. Taking into account these results, the potential energy curves of the cations NaHe 1 and NaNe 1 were calculated by the CASSCF method using the active spaces [2s–7s, 1p–3p] 8 and [3s– 8s, 2p–4p] 8, respectively. These active spaces generate 18 153 singlet and 28 248 triplet configuration state functions (CSFs) of the symmetry A1 and 17 640 singlet and 28 344 triplet CSFs of the symmetry B1 (B2) in the C2v group classification. The calculated potential energy curves of all the
Table 2 Computed spectroscopic parameters of the lowest electronic states of NaHe 1 and NaNe 1 Ion
State
Te (eV)
De (eV)
˚) Re (A
v e (cm 21)
v exe (cm 21)
Be (cm 21)
NaHe 1
1 1S 1 1 3S 1 2 1S 1 1 1S 1 1 3S 1 1 3P 1 1P 2 1S 1
0.00 18.424 18.841 0.00 14.937 15.071 15.153 15.372
0.037 1.011 0.603 0.077 0.879 0.833 0.751 0.710
2.421 3.536 3.904 2.546 3.738 3.730 3.809 3.927
138.4 239.4 206.9 98.1 127.8 123.3 119.9 119.6
21.00 1.13 1.12 5.73 0.29 0.28 0.30 0.35
0.844 0.396 0.325 0.243 0.113 0.113 0.109 0.102
NaNe 1
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200 Table 3 Vibrational energy levels (cm 21) of the two lowest 1S 1 states of NaHe 1 and radiation lifetimes (ns) of the levels of the state 2 1S 1, calculated using Eqs. (4) and (6) v
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1S 1
2 1S 1
Ev
Ev
t v(4)
t v(6)
v
Ev
t v(4)
t v(6)
65 162 227 265 285 295 300 310
104 309 512 712 909 1103 1294 1481 1665 1846 2023 2197 2366 2532 2694 2852 3005
2.16 2.19 2.23 2.27 2.32 2.37 2.42 2.47 2.52 2.59 2.65 2.72 2.80 2.88 2.97 3.06 3.17
2.19 2.24 2.28 2.33 2.38 2.43 2.49 2.55 2.61 2.68 2.75 2.83 2.91 3.00 3.10 3.21 3.33
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
3153 3297 3436 3569 3698 3821 3938 4050 4155 4255 4348 4434 4514 4588 4654 4715 4769
3.29 3.42 3.56 3.73 3.90 4.11 4.33 4.60 4.89 5.25 5.61 6.07 6.54 7.12 7.78 8.57 9.32
3.46 3.60 3.75 3.93 4.12 4.34 4.59 4.88 5.20 5.58 5.97 6.47 6.98 7.60 8.32 9.17 9.75
considered electronic states of NaHe 1 and NaNe 1 are shown in Figs. 1 and 2. As seen from these figures, the ground 1 1S 1 states of both the cations are only weakly bound, whereas all their excited states are strongly bound, with the triplet states potential curves lying lower than the corresponding singlet ones. The last result does not seem to be surprising, however, because the excited states created mainly by the leading configurations (1)–(3) obey the Hund rules. The main spectroscopic parameters of the considered states of NaHe 1 and NaNe 1, calculated in the present work, are given in Table 2. The equilibrium ˚ ) and the dissociation energy distance (Re 2.42 A (De 0.037 eV) of the ground NaHe 1(1 1S 1) state, presented in the table, are in reasonable agreement ˚, with the results of the calculations (Re 2.43 A De 0.028 eV) performed by Bililign et al. [6] by using the QCISD(T) method. In the case of NaNe 1, the values of De, obtained in the present work (0.077 eV) and in Ref. [6] (0.072 eV), are also in ˚ agreement, but the equilibrium distance Re 2.55 A from Table 2 is certain to be raised too high in ˚ obtained by Bilicomparison with the value of 2.48 A lign et al. [6]. This comparison confirms that the calculations of NaHe 1, performed in the present
193
work, appear to be sufficiently reliable and that in the case of NaNe 1 one may hope only on qualitatively correct results. 3. Lifetimes of excited electronic-vibrational states of the cations NaHe 1 and NaNe 1 and their theoretical emission spectra In the framework of the dipole approximation, the lifetime of an excited bound electronic (b)-vibrational (v2)-rotational (J2) state ubv2 J2 l with respect to its radiative decay may be evaluated using the formula [13] 4 X ^ 1 J1 lu2 ; t21 * Ebv2 J2 2 Eav1 J1 3 ukbv2 J2 uduav bv2 J2 3c3 a1 ;v1 ;J1 4 where Ebv P2 J2 is the energy of the state ubv2 J2 l, and the symbol * implies the summation (and/or integration) over all the bound (and/or unbound) vibrational–rotational states uav1 J1 l of the lower-lying electronic states a with their energies Eav1 J1 ! Ebv2 J2 . In the case of the diatomic molecule, the transition matrix element of a dipole operator may be written (in the framework of adiabatic approximation) as follows: ^ 1 J1 lu2 ukxbv J RuD Ruxav J lu2 SbJ ;aJ ; ukbv2 J2 uduav 2 2 1 1 2 1 5 where xav1 J1 R and xbv2 J2 R are the radial parts of the nuclear wave functions of the states uav1 J1 l and ubv2 J2 l, D R is the function of the electronic transition moment, and SbJ2 ;aJ1 is the so-called Henhley– London factor [13]. In practice, the use of Eq. (4) is complicated by the need of summation over a large number of states determined with only finite precision. Instead, an approximate expression for radiation lifetime that does not require a knowledge of vibrational–rotational states uav1 J1 l may be used [14–17] 4 Z∞ 3 t21 U RD2 Ruxbv2 J Ru2 dR; 6 bv2 J2 3c3 0 where U(R) is the difference between the adiabatic potentials of the electronic states involved in the radiative transition. In accordance with the completeness criterion suggested in Ref. [13], the accuracy of
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Fig. 3. Computed electric dipole transition moment in NaHe 1 between the states 1 1S 1 and 2 1S 1.
the approximate formula (6) is determined by proximity to zero of the expression X gbv2 J2 1 2 *ukbv2 J2 uav1 J1 l2 : 7 v1 ;J1
In the present work the radial parts of the vibrational–rotational wave functions as well as energies and lifetimes of the rovibrational states created by a fixed ab initio adiabatic potential were calculated by using the program set [18] implementing numerical solution of the Schro¨dinger equation for nuclear subsystem. Although the ground 1 1S 1 state of NaHe 1 is quite weakly bound (its well depth is only 0.037 eV), this state has eight bound vibrational levels, whereas the excited 2 1S 1 state of this ion has 34 bound levels. The vibrational energy levels of both these electronic states are presented in Table 3. On the basis of the data provided in this table, it is easy to estimate that in the case of NaHe 1 wavelengths of all bound–bound radiative transitions must be observed in the region 63.8– 65.9 nm. Table 3 also lists the lifetimes of the vibrational levels of the NaHe 1(2 1S 1) state, calculated by using both Eqs. (4) and Eq. (6) for the case J2 0, J1 1. The condition J2 0 seems to be not too restrictive, because lifetimes are weakly dependent on the rotational quantum number [17]. As seen from Table 3, Eqs. (4) and (6) lead to very close results. This is not surprising, however, because our calculations have showed that gbv2 J2 from Eq. (7)
are very close to zero for all (2 0,1,…,33. Consequently, it may be concluded that Eq. (6) is sufficiently reliable and that, as it was expected, the quality of the constructed CASSCF wave functions seems to be satisfactory for the goals of the present work. As seen from Fig. 3, the electronic transition Table 4 Vibrational energy levels (cm 21) of the three lowest singlet states of NaNe 1 and radiation lifetimes (ns) of the levels of the 1 1P and 2 1S 1 states, calculated using Eq. (6) v
0 1 5 10 15 17 20 25 30 35 40 45 50 53 55 60 61
1 1S 1
2 1S 1
Ev
Ev
tv
Ev
tv
47 138 417 623 735 761
57 174 636 1197 1738 1949 2259 2758 3230 3672 4079 4444 4760
2.21 2.22 2.27 2.34 2.42 2.46 2.52 2.65 2.82 3.05 3.37 3.86 4.45
60 179 649 1221 1775 1991 2309 2823 3313 3778 4211 4608 4962 5150 5265 5512 5554
5.68 5.72 5.84 6.03 6.25 6.35 6.50 6.81 7.19 7.68 8.35 9.30 10.71 11.42 10.06 3.23 2.49
1 1P
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200
195
Fig. 4. Computed electric dipole moments for transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 in NaNe 1.
moment in NaHe 1 between the two lowest 1S 1 states is a rapidly decreasing function of the internuclear distance. As a result, higher-lying vibrational levels which are characterized by larger internuclear distances must have (in accordance with Eq. (6)) larger radiation lifetimes. It is the tendency that can be seen from the data of Table 3. When R(Na–He) ! 0, the transition moment goes to that of Al 1 between 1 1Pu and 1 1Sg atomic states. In the case of the considered electronic states of NaNe 1 only the electronic transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 are allowed by the selection rules. The electric dipole moments of these transitions as functions of the internuclear distance R(Na–Ne) are shown in Fig. 4. When R(Na–Ne) ! 0, the dipole moments under discussion tends to zero, because for the united atom Sc 1 the corresponding values should be equal to zero just due to the selection rules. There exist 17 bound vibrational levels within the potential well of the ground 1 1S 1 state of this cation, whereas its 2 1S 1 and 1 1P states have 50 and 61 such levels, respectively. The energies of these levels and their radiation lifetimes calculated by using Eq. (6) for the case J2 0 and J1 1 are presented in Table 4.
On the basis of the data provided in this table, it is not difficult to determine that wavelengths of all bound– bound electronic–vibrational transitions in NaNe 1 have to lie in the range between 75.7 and 79.5 nm. Adiabatic potential Ua(R) of a ground electronic state determines both a finite set of bound vibrational–rotational states uan 1J1l and a continuous set of states ua1J1 l dissociating into the products Na 1(1 1Sg) 1 Rg(1 1Sg) with the kinetic energy 1 of their relative motion. The radiation from the upper state ubv2 J2 l to the continuum of the states ua1J1 l is characterized by continuous distribution of photons with their energies hn obeying the energy conservation law Ebv2 J2 1 1 hn. The energy distribution in a radiation spectrum is described by the so-called “continuous line contour” function gn2 1 [2], gv2 1 < ukv2 u1l2
8
determining a probability density for spontaneous radiation from the vibrational level v2 to the lowerlying continuum of the states with their energies 1 ^ 1=2 d1. The probability function is normally, weakly dependent on the rotational quantum numbers,
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Fig. 5. Theoretical distribution of photon energies in spontaneous radiation spectrum corresponding to the transitions 2 1S 1 ) 1 1S 1 in NaHe 1.
which are therefore omitted from R Eq. (8). If the ground state is repulsive then gv2 1 d1 1. If the ground state is bound then this integral is equal to the probability as a result of radiation from the state ubv2 J2 l, the system will be found in one of the lower-lying states belonging to the continuous spectrum. The probability that after such a radiation, the system will be found in a bound state is equal to P
vbv2 J2
v1 ;J1
^ 1 J1 lu2 ukbv2 J2 uduav
kXbv2 J2 uD R2 uXbv2 J2 l
;
9
where the sum goes over all the bound states uav1 J1 l of the ground electronic state a. The calculations based on Eq. (9) showed that, as a result of spontaneous radiation from the zero vibrational level of the 2 1S 1 state of NaHe 1, the cation will be found in a bound state with probability 83%. For the first vibrational level this probability is 32%, and for the rest of the levels it changes from 10 to 30%. In the case of the cation NaNe 1, this probability is greater than 90% for zero, first and second vibrational levels of the 2 1S 1 and 1 1P states, greater than 50% for their third and fourth ones, and that varies from 10 to 40% for the remaining levels. To calculate the energy distribution in the emission spectra of the cations, we also need some information
Fig. 6. Theoretical distribution of photon energies in spontaneous radiation spectrum corresponding to the transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 in NaNe 1.
about the vibrational level populations. However, as the radiation was observed by Hammer et al. [5] in conditions that were far from the equilibrium ones, such information seems to be unavailable. Assuming that all the vibrational levels are populated identically, the overall energy distribution (up to normalization) may be written as follows: g hn ,
X
t21 v2 gv2 1
10
b;v2
where 1 Ebv2 J2 2 hn and the summation goes over all the radiating excited states (in the case of NaNe 1 b 2 1S 1, 1 1P and contributions from the 1 1P term which were taken into account with weight 2). The emission spectra of the cations NaHe 1 and NaNe 1, calculated by using the energy distribution function (10), are presented in Figs. 5 and 6. In these figures the regions, where bound–bound transitions occur are marked off by vertical dotted lines. There are three intensive peaks in the emission spectrum of NaNe 1: the first and third ones originate mostly from the transition 1 1S 1 ( 1 1P, whereas in the central peak a contribution of transition 1 1S 1 ( 2 1S 1 proves to be dominating. Positions of the most intensive peaks in the computed spectra, 65.8 (65.76) nm for NaHe 1 and 78.7 (79.40) nm for NaNe 1, agree well with the observed transition wavelengths [5] given in parentheses.
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197
Fig. 7. Relative positions of the low-lying energy levels of the linear triatomic dication ArNaHe 11 corresponding to (a) infinite separation of ˚. Ar 1 and NaHe 1 and (b) R(Ar–Na) 5 A
4. Role of argon cations in a process of the NaHe 1(1 3S 1) state’s evolution In the course of the reaction Na 12 Sg 1 He1 12 Sg ! NaHe1 21 S1 ; 13 S1 11 the yield of the cations NaHe 1 in their nonradiating triplet 1 3S 1 and radiating singlet 2 1S 1 states may be expected to be in the proportion 3–1. These cations being of great interest for possible short wavelength lasers, a study of the possible mechanisms of the triplet NaHe 1(1 3S 1) state’s evolution appears to be of importance, because population of such states will
certainly lead to a decrease in efficiency and radiation power of possible lasers. Taking into account the conditions of the experiment performed by Hammer et al. [5], in this section we will consider a possibility of decay of the NaHe 1(1 3S 1) state as a result of its interaction with the ions Ar 1(1 2Pu). Relative positions of low-lying energy levels of the system Ar 1 1 NaHe 1 (at infinite separation), calculated on the basis of the available atomic data [7,8], are presented in Fig. 7(a). When Ar 1 approaches NaHe 1, the energy levels increase due to Coulomb repulsion, the shifts of the levels being different depending on the state of the cation NaHe 1. In the case of its ground 1 1S 1 state, the Mulliken charge on
Table 5 Excitation energies (eV) of the three lowest states of Ar 1 calculated by the CASSCF method in the active space [3s3p4s3d] 7 by using the contracted atomic basis set [17s12p5d/4s3p1d] State 2
1 Pu 1 2Sg 1 2Pg
Configuration 2
5
3s 3p 3s 13p 6 3s 23p 44s 1
DE [17s12p5d/4s3p1d]
DE Experimental [8]
00.00 13.42 17.37
0.00 13.42 17.12
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Fig. 8. Cuts through the 3 2P PES of the linear structure of ArNaHe 11 at different R(Ar–Na) as functions of the internuclear distance R(Na–He).
sodium is about 1 1.0e, whereas for the excited state 1 3S 1 this charge is only 1 0.3e. Therefore, in considering the interaction of Ar 1 with NaHe 1(1 3S 1) the approach of Ar 1 to the cation NaHe 1 along its bond line from the Na side is the most energetically advantageous. In the course of such an approach, the Coulomb level shift will be larger for the ground state of NaHe 1 than for its excited triplet one. Relative energies of the low-lying levels of the linear system (Ar…NaHe) 11, estimated at R(Ar–Na)
˚ without considering their splitting due to the 5A electron–electron interactions, are shown in Fig. 7(b). As seen from this figure, the fifth energy level (that we are actually interested in) and the fourth level change their positions long before the argon cation enters sufficiently close to the neighborhood of NaHe 1. Thus, to describe the process of the interaction of Ar 1 with the first triplet state of ˚ one needs NaHe 1 at the distances R(Ar–Na) # 5 A the basis and calculation method which could reproduce correctly at least three lowest doublet states of the ion Ar 1. As a result of a number of calculations of low-lying doublet Ar 1 states in different standard basis sets it became apparent that a specifically optimized basis is required for a correct description of the needed Ar 1 states. To this end, we took uncontracted functions from the [17s12p5d] ANO basis set [19] and performed the CASSCF calculation using the energy functional averaged over the lowest three states of Ar 1. Coefficients of the obtained atomic orbitals were taken as contraction coefficients to produce a relatively small contracted basis [17s12p5d/4s3p1d]. The CASSCF calculations were carried out in the active space [3s3p4s3d] 7 without atomic space symmetry restrictions. Table 5 shows that in the framework of such an approach, the calculated excitation energies of the three lowest electronic states of
Fig. 9. Two dimensional representation of the PES of the NaHe 1(1 3S 1) state allowing for its interaction with the ion Ar 1 (see reaction (12)).
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200 1
the ion Ar are quite close to the experimental energies from Ref. [8]. As seen from Fig. 7(a), the lowest electronic states of the linear triatomic system (Ar…NaHe) 11, describing the interaction of Ar 1 with NaHe 1(1 3S 1) ˚ are the metastable at the distances R(Ar–Na) # 5 A states 4 2S 1, 3 2P, 1 4S 1, and 1 4P. Thus, to study the evolution of the NaHe 1(1 3S 1) state as a result of its interaction with Ar 1, one needs, in particular, to investigate the potential energy surface of the ArNaHe 11(3 2P) state. Since at the present stage only qualitative conclusions are expected, we used the basis set aug_cc_pVDZ for Na and He, the above described contracted basis [4s3p1d] for Ar 1, and a minimal active space going to [1s(He) 3s(Na)3s3p4s3d(Ar)] 9 at the limit of separated atoms (cations). In this active space the PESs of the five lowest 2P states of the system were calculated by the CASSCF method, with the energy functional averaged over the five lowest roots (the fifth root was required for a correct description of the He 1 detachment as 2 1S 1 and 1 3S 1 states of NaHe 1 have the same dissociation limit). Cuts through the 3 2P PES of the linear structure of ArNaHe 11 at different R(Ar–Na) are presented in Fig. 8 as functions of the internuclear distance R(Na–He). This figure shows that as Ar 1 approaches NaHe 1 the depth of the potential well of the NaHe 1(1 3S 1) state decreases and this state becomes finally repulsive and dissociates into ArNa 1 and He 1. A two dimensional representation of the PES of the NaHe 1(1 3S 1) state with allowing for its interaction with the ion Ar 1 is shown in Fig. 9. More careful analysis of the configuration structure of the corresponding wave functions and Mulliken charges on atoms leads to the conclusion that as a result of the outlined interaction the following reaction has to be the case: Ar1 12 Pu 1 NaHe1 13 S1 ! ArNa1 11 S1 1 He1 12 Sg :
12
Thus, under the conditions of the experiment performed by Hammer et al. [5], the triplet NaHe 1( 3S 1) states must decay with a formation of the He 1 ions, which in turn can interact with sodium atoms to yield again the radiative NaHe 1(2 1S 1) states and, thus, to increase the intensity of emission observed in [5].
199 1
Calculations of PESs of the S symmetry states of the linear system (Ar…NaHe) 11 proved to be more complicated, since to localize the required PES larger number of states is to be taken into consideration. These calculations have not added any new elements for understanding the role of Ar 1 in destroying the triplet NaHe 1(1 3S 1) state and therefore are not discussed here. Since a high energy Ar 1 beam was used in the experiment [5] the ions Ar 1 can approach NaHe 1 (1 3S 1) both along the bond Na–He line and at an angle to this line. The first case is the most energetically advantageous from a purely electrostatic viewpoint and was considered above. The second case turns out to be essentially simpler. Indeed, electrostatic arguments lead to the conclusion that in the course of Ar 1 approaching NaHe 1 at some angle to the line Na–He (say, perpendicular to this line) the energy levels of interest are forced upward due to Coulomb repulsion and they turn out to lie essentially higher than the levels of the system of isolated atomic (ionic) species. Moreover, analysis shows that already ˚ the system ArNaHe 11 becomes at R(Ar–Na) , 5 A metastable or unstable with respect to vibration of the Na–He bond. As a result, at sufficiently high energies of the Ar 1 ions the lifetime of the triplet NaHe 1(1 3S 1) state should also essentially decrease in the case, when Ar 1 approaches NaHe 1 at some angle to the Na–He bond line. 2
5. Summary Results of the ab initio calculations performed in the present work support the correctness of qualitative arguments adduced originally by Basov et al [3] concerning the excimer character of the cations NaHe 1 and NaNe 1 and prospects for using them as a possible laser media with emission in the vacuum ultraviolet region. The main parameters of the potential curves and spectroscopic characteristics of all the electronic states of these ions, correlating with their two lowest dissociation limits, were first calculated. The obtained theoretical emission spectra of the cations NaHe 1 and NaNe 1 was shown to be in good agreement with the experimental spectra observed recently by Hammer et al. [5]. Particular attention was paid to study of evolution
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of the bound non-radiating excited triplet NaHe 1(1 3S 1) state, which could take place in the conditions of the experiment performed by Hammer et al. [5]. It was shown that as a result of its interaction with a high-energy beam of the Ar 1 ions this state has to break up and that the decay of the NaHe 1(1 3S 1) state has to be accompanied with a formation of the ions He 1, which in their turn can interact with sodium atoms again to yield the radiating NaHe 1(2 1S 1) states and, thus, to maintain emission. Processes of such kind have to be taken into account in future investigations of a possibility to use the alkali metal—rare gas cations as sources of laser radiation. Acknowledgements We gratefully acknowledge the Russian Foundation for Basic Research (Grant No. 97-03-33713a) for financial support of the present work. References [1] F.G. Houtermans, Helv. Phys. Acta 33 (1960) 933. [2] K. Rhodes (Ed.), Excimer Lasers Applied Physics, 30, Springer, Berlin, 1979. [3] N.G. Basov, M.G. Voitik, V.S. Zuev, V.P. Kutakhov, Sov. J. Quant. Electron. 5 (1985) 1455.
[4] M. Mantel, H. Langhoff, Z. Phys. D15 (1990) 297. [5] J.W. Hammer, K. Petkau, T. Griegel, H. Langhoff, M. Mantel, Hyperfine Interact. 88 (1994) 151. [6] S. Bililign, M. Gutowski, J. Simons, W.H. Breckenridge, J. Chem. Phys. 100 (1994) 8212. [7] Ch.E. Moore, Analysis of Optical Spectra, National Bureau of Standards, US Government Printing Office, Washington, DC, 1970, NSRDS-NBS 34. [8] D.A. Verner, E.M. Verner, G.L. Ferland, Atomic Data Nucl. Data Tables 64 (1996) 1. [9] G. Herzberg, Molecular spectra and molecular structure, Spectra of Diatomic Molecules, 1, Van Nostrand Reinhold, New York, 1950, pp. 318–319. [10] B.O. Roos, Adv. Chem. Phys. 69 (1987) 339. [11] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S.J. Su, T. L. Windus, M. Dupius, J.A. Montgomery, J. Comput. Chem. 14 (1993) 1347. [12] R.A. Kendal, T.H. Dunning Jr., R. Harrison, J. Chem. Phys. 96 (1992) 6796. [13] L.A. Kuznetsova, N.E. Kuzmenko, Y.Y. Kuzyakov, Y.A. Plastinin, Probabilities of Optical Transitions in Diatomic Molecules, Nauka, Moskow, 1980. [14] J. Tellinghuisen, Chem. Phys. Lett. 105 (1984) 241. [15] J. Tellinghuisen, P.S. Julienne, J. Chem. Phys. 81 (1984) 5779. [16] V. I. Pupyshev, Opitic. Spectr. (in Russian) 63 (1987) 570. [17] N.E. Kuzmenko, V.I. Pupyshev, A.V. Stolyarov, Opitic. Spectr. (in Russian) 63 (1987) 756. [18] A.V. Abarenkov, A.V. Stolyarov, J. Phys.B 23 (1990) 2419. [19] P.-O. Widmark, P.-A. Malmquist, B.O. Roos, Theor. Chim. Acta 77 (1990) 291.
Theoretical study of low-lying electronic states and emission spectra of the excimer ions NaHe 1 and NaNe 1 A.I. Panin a,*, A.N. Petrov b, Y.G. Khait b a
Department of Chemistry, Quantum Chemistry Group, St. Petersburg State University, 2 University Avenue, Old Petergof, 198904 St. Petersburg, Russia b Russian Scientific Center Applied Chemistry, 14 Dobrolyubova Avenue, 197198 St. Petersburg, Russia Received 22 January 1999; accepted 15 March 1999
Abstract Ab initio CASSCF calculations using the Gaussian basis set aug_cc_pVTZ developed by Dunning et al. specifically for calculations at the post-HF level was carried out for excimer NaHe 1 and NaNe 1 ions in order to determine spectroscopic characteristics and radiation lifetimes of their low-lying excited electronic states. The computed emission spectra of these were shown to be in good agreement with the experimental ones observed recently by Hammer et al. (Hyperfine Interactions 88 (1994) 151). Contrary to the earlier popular opinion, all excited electronic states of these ions, correlating with the limit Rg 1 1 Na, and not just the radiating singlet states turn out to be bound. It was shown that under the conditions of an experiment similar to the one performed by Hammer et al., the excited triplet NaHe 1(1 3S 1) states can decay as a result of their interactions with a high-energy Ar 1 beam. This decay has to be accompanied with the formation of the ions He 1, which in turn can interact with sodium atoms to yield the radiating NaHe 1(2 1S 1) states again and, thus to maintain the emission observed in the experiment. q 1999 Elsevier Science B.V. All rights reserved. Keywords: NaHe 1; NaNe 1; Low-lying electronic states; Potential curves; Radiative transitions; Emission spectra; Ab initio calculations
1. Introduction The main characteristic property of an excimer species is that one of its excited electronic states is bound, whereas its ground state is repulsive or quite weakly bound and dissociates at normal conditions. As a result, if the only channel of decay of the excited state is its radiative transition to the ground state, then a population inversion may be achieved, and such species are of interest for possible lasers. The idea of using excimers as an active laser media was first suggested in 1960 [1], and a number of excimer lasers * Corresponding author. E-mail address: [email protected] (A.I. Panin)
of different types are presently known [2]. These lasers may serve as sources of radiation in the region 130–1000 nm with a possibility to tune a radiation frequency in a rather broad interval. Recently considerable attention has been focused on searching for excimer species with radiative transitions in the shorter (VUV or even XUV) wave region. In 1985, on the basis of purely qualitative estimations, Basov et al. [3] predicted, in particular, that diatomic molecular cations consisting of a rare gas ion (Rg 1) and an alkali atom (M) are excimers, and that these species may be expected to decay by the emission of radiation in the region from 70 nm (in the case of NaHe 1) to 250 nm (for LiXe 1). In accordance with the model calculations performed later by Mantel and
0166-1280/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(99)00098-6
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Langhoff [4] the radiative electronic transitions in such systems should be expected in the region 60– 190 nm. These predictions were recently confirmed by Hammer et al. [5], who performed an experimental study of emission spectra of mixtures of rare gases with alkali-metal vapor excited by a high power Ar 1 beam and actually observed narrow emission continua in the region 60–200 nm. The equilibrium internuclear distances and dissociation energies of the ground states of such ions were evaluated by many authors (see Ref. [3]). In particular, the potential curves of the ground states of the ions RgM 1 (M Li, Na; Rg He, Ne) were recently calculated by Bililign et al. [6] using the ab initio methods. However, to the best of our knowledge, no reliable information about spectroscopic characteristics of excited electronic states of such ions is available. Hammer et al. [5] assigned the observed transition wavelengths of 65.76 and 79.40 nm to the cations NaHe 1 and NaNe 1, respectively, on the basis of the model estimations of Mantel and Langhoff [4]. The purpose of this article, is a theoretical investigation of low-lying states of the cations NaHe 1 and NaNe 1 to determine their spectroscopic characteristics and radiative transition probabilities and eventually to compare the emission spectra predicted by theory with the experimental ones from Ref. [5]. We will show that in contrast to the assumptions of Basov et al. [3], all excited electronic states of these ions, correlating with the limit Rg 1 1 Na, and not just the radiating singlet 2 1S 1 states turn out to be bound. Moreover, the binding energies of the excited triplet states 1 3S 1 correlating with the mentioned limit will be shown to exceed essentially with those of the 2 1S 1 states correlating with the same asymptote. Population of such triplet states should certainly lead to a decrease in efficiency and radiation power of possible lasers and may essentially affect a possibility of generation of laser radiation. From this viewpoint, a study of possible mechanisms of the triplet states decay appears to be important. In this work it will be shown that under the conditions of the experiment similar to that performed by Hammer et al. [5] the NaHe 1(1 3S 1) state should decay as a result of its interaction with a high-energy Ar 1 beam. This decay has to be accompanied with the formation of the ions He 1, which can interact again with sodium atoms to yield the radiating NaHe 1(2 1S 1) states and,
thus, to maintain the emission. Analogous processes are also likely to occur with the triplet states of both NaNe 1 and other cations RgM 1 studied by Hammer et al. [5].
2. Potential energy curves of low-lying states of the cations NaHe 1 and NaNe 1 In accordance with the available atomic data [7,8] and the Wigner–Witmer rules [9] the most stable asymptotes of NaHe 1 and NaNe 1 are the limits He(1 1Sg) 1 Na 1(1 1Sg) and Ne(1 1Sg) 1 Na 1(1 1Sg) correlating with the ground 1 1S 1 states of these molecular ions. As should be expected, owing to the small polarizabilities of the rare gas atoms, the 1 1S 1 states prove to be only weakly bound (see Ref. [6] and references therein). The second dissociation limit Na(1 2Sg) 1 He 1(1 2Sg) of the ion NaHe 1 lies 19.44 eV above its lowest limit [7] and correlates with its excited 2 1S 1 and 1 3S 1 states. In the case of NaNe 1, the second dissociation limit Na(1 2Sg) 1 Ne 1(1 2Pu) with its energy of 16.42 eV [7] correlates with the four excited states 2 1S 1, 1 1P, 1 3S 1, and 1 3P. It is clear that the radiative transitions 2 1S 1 ) 1 1S 1 (in NaHe 1) and 2 1S 1, 1 1P ) 1 1S 1 (in NaNe 1) must be accompanied by a charge transfer from rare gas atoms on sodium atom and their energies have to be close to the above-mentioned energies of the second dissociation limits of the molecular ions relative to their most stable asymptotes. Among the excited states of the ions NaHe 1 and NaNe 1, correlating with their higher-lying dissociation limits, there also exist both bound and weakly bound states. These states, however, appear to be unrelated to the emission observed by Hammer et al. [5] and will not be considered below. We will restrict our consideration to the electronic states, which correlate only with the two outlined lowest dissociation limits. All electronic structure calculations were performed in the framework of the CASSCF method [10], using the Gamess program [11] with the Gaussian basis set aug_cc_pVTZ developed by Dunning et al. [12] specifically for calculations at the post-HF level. To test the accuracy of the ab initio study performed in the present work, the ionization potentials (IPs) and excitation energies of the lowest excited
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200
191
Table 1 Comparison of the computed IPs and excitation energies (in eV) of the lowest excited states of the atoms He, Ne, and Na and their cations with experimental data from Ref. [7] Species
He He 1 Ne Ne 1 Na Na 1
State
1 1Sg 1 3Sg 1 2Sg 2 2Sg 1 1Sg 1 1Pu % 1 3Pu 1 2Pu 1 2Sg 1 2Sg 1 2Pu 1 1Sg 1 3Pu
Ionization potential
Excitation energy
This work
Experimental
This work
Experimental
24.53
24.58
21.00
21.59
5.00
5.14
0.00 19.88 0.00 40.85 0.00 17.66 0.00 27.12 0.00 2.01 0.00 32.68
0.00 19.82 0.00 40.81 0.00 16.73 0.00 26.91 0.00 2.10 0.00 32.85
electronic states of both the atoms Na, He, Ne and their cations were calculated. For helium and its cation the full CI method was used, whereas for sodium, neon and their cations, we used the CASSCF method with the active space [2s2p3s3p4s4p] 8–9 for Na and Na 1, and the active space [2s2p3s3p4s3d] 8–9 for Ne and Ne 1. The results of these calculations are given in Table 1. As seen from the table, in the case of the atoms He, Na and their cations, the computed IPs and the excitation energies are in good agreement with the experimental data from Ref. [7]. However,
in the case of neon and its cation, the agreement proves to be essentially worse (e.g. the computed excitation energy of the averaged state Ne(1 1Pu % 1 3Pu) is higher by 0.93 eV than the experimental value 16.73 eV [7]). Thus, the CASSCF method and the chosen basis set may be expected to be sufficiently good for a correct description of both the ground and the low-lying excited electronic states of NaHe 1 (see also below), whereas in the case of NaNe 1, one may hope to obtain only qualitatively reliable results. Calculations of the potential energy curves of the
Fig. 1. Potential energy curves of the electronic states of NaHe 1 correlating with its two lowest dissociation limits.
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Fig. 2. Potential energy curves of the electronic states of NaNe 1 correlating with its two lowest dissociation limits.
three lowest electronic states of NaHe 1 and the five lowest states of NaNe 1 were carried out by the CASSCF method in the C2v subgroup of the C∞v point group. Preliminary calculations showed that the leading configuration describing the excited 1,3 1 S states of NaHe 1 is 1
1s2Na ls2 2s2 1p4 3s1 4s1
and that for the excited 1,3S 1 and 1,3P states of NaNe 1, the leading configurations are 1s2Na 1s2Ne 1s2 2s2 1p4 3s2 4s1 2p4 5s1
2
and 1s2Na 1s2Ne 1s2 2s2 1p4 3s2 4s2 2p3 5s1 ;
3
respectively. Taking into account these results, the potential energy curves of the cations NaHe 1 and NaNe 1 were calculated by the CASSCF method using the active spaces [2s–7s, 1p–3p] 8 and [3s– 8s, 2p–4p] 8, respectively. These active spaces generate 18 153 singlet and 28 248 triplet configuration state functions (CSFs) of the symmetry A1 and 17 640 singlet and 28 344 triplet CSFs of the symmetry B1 (B2) in the C2v group classification. The calculated potential energy curves of all the
Table 2 Computed spectroscopic parameters of the lowest electronic states of NaHe 1 and NaNe 1 Ion
State
Te (eV)
De (eV)
˚) Re (A
v e (cm 21)
v exe (cm 21)
Be (cm 21)
NaHe 1
1 1S 1 1 3S 1 2 1S 1 1 1S 1 1 3S 1 1 3P 1 1P 2 1S 1
0.00 18.424 18.841 0.00 14.937 15.071 15.153 15.372
0.037 1.011 0.603 0.077 0.879 0.833 0.751 0.710
2.421 3.536 3.904 2.546 3.738 3.730 3.809 3.927
138.4 239.4 206.9 98.1 127.8 123.3 119.9 119.6
21.00 1.13 1.12 5.73 0.29 0.28 0.30 0.35
0.844 0.396 0.325 0.243 0.113 0.113 0.109 0.102
NaNe 1
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200 Table 3 Vibrational energy levels (cm 21) of the two lowest 1S 1 states of NaHe 1 and radiation lifetimes (ns) of the levels of the state 2 1S 1, calculated using Eqs. (4) and (6) v
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
1 1S 1
2 1S 1
Ev
Ev
t v(4)
t v(6)
v
Ev
t v(4)
t v(6)
65 162 227 265 285 295 300 310
104 309 512 712 909 1103 1294 1481 1665 1846 2023 2197 2366 2532 2694 2852 3005
2.16 2.19 2.23 2.27 2.32 2.37 2.42 2.47 2.52 2.59 2.65 2.72 2.80 2.88 2.97 3.06 3.17
2.19 2.24 2.28 2.33 2.38 2.43 2.49 2.55 2.61 2.68 2.75 2.83 2.91 3.00 3.10 3.21 3.33
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
3153 3297 3436 3569 3698 3821 3938 4050 4155 4255 4348 4434 4514 4588 4654 4715 4769
3.29 3.42 3.56 3.73 3.90 4.11 4.33 4.60 4.89 5.25 5.61 6.07 6.54 7.12 7.78 8.57 9.32
3.46 3.60 3.75 3.93 4.12 4.34 4.59 4.88 5.20 5.58 5.97 6.47 6.98 7.60 8.32 9.17 9.75
considered electronic states of NaHe 1 and NaNe 1 are shown in Figs. 1 and 2. As seen from these figures, the ground 1 1S 1 states of both the cations are only weakly bound, whereas all their excited states are strongly bound, with the triplet states potential curves lying lower than the corresponding singlet ones. The last result does not seem to be surprising, however, because the excited states created mainly by the leading configurations (1)–(3) obey the Hund rules. The main spectroscopic parameters of the considered states of NaHe 1 and NaNe 1, calculated in the present work, are given in Table 2. The equilibrium ˚ ) and the dissociation energy distance (Re 2.42 A (De 0.037 eV) of the ground NaHe 1(1 1S 1) state, presented in the table, are in reasonable agreement ˚, with the results of the calculations (Re 2.43 A De 0.028 eV) performed by Bililign et al. [6] by using the QCISD(T) method. In the case of NaNe 1, the values of De, obtained in the present work (0.077 eV) and in Ref. [6] (0.072 eV), are also in ˚ agreement, but the equilibrium distance Re 2.55 A from Table 2 is certain to be raised too high in ˚ obtained by Bilicomparison with the value of 2.48 A lign et al. [6]. This comparison confirms that the calculations of NaHe 1, performed in the present
193
work, appear to be sufficiently reliable and that in the case of NaNe 1 one may hope only on qualitatively correct results. 3. Lifetimes of excited electronic-vibrational states of the cations NaHe 1 and NaNe 1 and their theoretical emission spectra In the framework of the dipole approximation, the lifetime of an excited bound electronic (b)-vibrational (v2)-rotational (J2) state ubv2 J2 l with respect to its radiative decay may be evaluated using the formula [13] 4 X ^ 1 J1 lu2 ; t21 * Ebv2 J2 2 Eav1 J1 3 ukbv2 J2 uduav bv2 J2 3c3 a1 ;v1 ;J1 4 where Ebv P2 J2 is the energy of the state ubv2 J2 l, and the symbol * implies the summation (and/or integration) over all the bound (and/or unbound) vibrational–rotational states uav1 J1 l of the lower-lying electronic states a with their energies Eav1 J1 ! Ebv2 J2 . In the case of the diatomic molecule, the transition matrix element of a dipole operator may be written (in the framework of adiabatic approximation) as follows: ^ 1 J1 lu2 ukxbv J RuD Ruxav J lu2 SbJ ;aJ ; ukbv2 J2 uduav 2 2 1 1 2 1 5 where xav1 J1 R and xbv2 J2 R are the radial parts of the nuclear wave functions of the states uav1 J1 l and ubv2 J2 l, D R is the function of the electronic transition moment, and SbJ2 ;aJ1 is the so-called Henhley– London factor [13]. In practice, the use of Eq. (4) is complicated by the need of summation over a large number of states determined with only finite precision. Instead, an approximate expression for radiation lifetime that does not require a knowledge of vibrational–rotational states uav1 J1 l may be used [14–17] 4 Z∞ 3 t21 U RD2 Ruxbv2 J Ru2 dR; 6 bv2 J2 3c3 0 where U(R) is the difference between the adiabatic potentials of the electronic states involved in the radiative transition. In accordance with the completeness criterion suggested in Ref. [13], the accuracy of
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Fig. 3. Computed electric dipole transition moment in NaHe 1 between the states 1 1S 1 and 2 1S 1.
the approximate formula (6) is determined by proximity to zero of the expression X gbv2 J2 1 2 *ukbv2 J2 uav1 J1 l2 : 7 v1 ;J1
In the present work the radial parts of the vibrational–rotational wave functions as well as energies and lifetimes of the rovibrational states created by a fixed ab initio adiabatic potential were calculated by using the program set [18] implementing numerical solution of the Schro¨dinger equation for nuclear subsystem. Although the ground 1 1S 1 state of NaHe 1 is quite weakly bound (its well depth is only 0.037 eV), this state has eight bound vibrational levels, whereas the excited 2 1S 1 state of this ion has 34 bound levels. The vibrational energy levels of both these electronic states are presented in Table 3. On the basis of the data provided in this table, it is easy to estimate that in the case of NaHe 1 wavelengths of all bound–bound radiative transitions must be observed in the region 63.8– 65.9 nm. Table 3 also lists the lifetimes of the vibrational levels of the NaHe 1(2 1S 1) state, calculated by using both Eqs. (4) and Eq. (6) for the case J2 0, J1 1. The condition J2 0 seems to be not too restrictive, because lifetimes are weakly dependent on the rotational quantum number [17]. As seen from Table 3, Eqs. (4) and (6) lead to very close results. This is not surprising, however, because our calculations have showed that gbv2 J2 from Eq. (7)
are very close to zero for all (2 0,1,…,33. Consequently, it may be concluded that Eq. (6) is sufficiently reliable and that, as it was expected, the quality of the constructed CASSCF wave functions seems to be satisfactory for the goals of the present work. As seen from Fig. 3, the electronic transition Table 4 Vibrational energy levels (cm 21) of the three lowest singlet states of NaNe 1 and radiation lifetimes (ns) of the levels of the 1 1P and 2 1S 1 states, calculated using Eq. (6) v
0 1 5 10 15 17 20 25 30 35 40 45 50 53 55 60 61
1 1S 1
2 1S 1
Ev
Ev
tv
Ev
tv
47 138 417 623 735 761
57 174 636 1197 1738 1949 2259 2758 3230 3672 4079 4444 4760
2.21 2.22 2.27 2.34 2.42 2.46 2.52 2.65 2.82 3.05 3.37 3.86 4.45
60 179 649 1221 1775 1991 2309 2823 3313 3778 4211 4608 4962 5150 5265 5512 5554
5.68 5.72 5.84 6.03 6.25 6.35 6.50 6.81 7.19 7.68 8.35 9.30 10.71 11.42 10.06 3.23 2.49
1 1P
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200
195
Fig. 4. Computed electric dipole moments for transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 in NaNe 1.
moment in NaHe 1 between the two lowest 1S 1 states is a rapidly decreasing function of the internuclear distance. As a result, higher-lying vibrational levels which are characterized by larger internuclear distances must have (in accordance with Eq. (6)) larger radiation lifetimes. It is the tendency that can be seen from the data of Table 3. When R(Na–He) ! 0, the transition moment goes to that of Al 1 between 1 1Pu and 1 1Sg atomic states. In the case of the considered electronic states of NaNe 1 only the electronic transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 are allowed by the selection rules. The electric dipole moments of these transitions as functions of the internuclear distance R(Na–Ne) are shown in Fig. 4. When R(Na–Ne) ! 0, the dipole moments under discussion tends to zero, because for the united atom Sc 1 the corresponding values should be equal to zero just due to the selection rules. There exist 17 bound vibrational levels within the potential well of the ground 1 1S 1 state of this cation, whereas its 2 1S 1 and 1 1P states have 50 and 61 such levels, respectively. The energies of these levels and their radiation lifetimes calculated by using Eq. (6) for the case J2 0 and J1 1 are presented in Table 4.
On the basis of the data provided in this table, it is not difficult to determine that wavelengths of all bound– bound electronic–vibrational transitions in NaNe 1 have to lie in the range between 75.7 and 79.5 nm. Adiabatic potential Ua(R) of a ground electronic state determines both a finite set of bound vibrational–rotational states uan 1J1l and a continuous set of states ua1J1 l dissociating into the products Na 1(1 1Sg) 1 Rg(1 1Sg) with the kinetic energy 1 of their relative motion. The radiation from the upper state ubv2 J2 l to the continuum of the states ua1J1 l is characterized by continuous distribution of photons with their energies hn obeying the energy conservation law Ebv2 J2 1 1 hn. The energy distribution in a radiation spectrum is described by the so-called “continuous line contour” function gn2 1 [2], gv2 1 < ukv2 u1l2
8
determining a probability density for spontaneous radiation from the vibrational level v2 to the lowerlying continuum of the states with their energies 1 ^ 1=2 d1. The probability function is normally, weakly dependent on the rotational quantum numbers,
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Fig. 5. Theoretical distribution of photon energies in spontaneous radiation spectrum corresponding to the transitions 2 1S 1 ) 1 1S 1 in NaHe 1.
which are therefore omitted from R Eq. (8). If the ground state is repulsive then gv2 1 d1 1. If the ground state is bound then this integral is equal to the probability as a result of radiation from the state ubv2 J2 l, the system will be found in one of the lower-lying states belonging to the continuous spectrum. The probability that after such a radiation, the system will be found in a bound state is equal to P
vbv2 J2
v1 ;J1
^ 1 J1 lu2 ukbv2 J2 uduav
kXbv2 J2 uD R2 uXbv2 J2 l
;
9
where the sum goes over all the bound states uav1 J1 l of the ground electronic state a. The calculations based on Eq. (9) showed that, as a result of spontaneous radiation from the zero vibrational level of the 2 1S 1 state of NaHe 1, the cation will be found in a bound state with probability 83%. For the first vibrational level this probability is 32%, and for the rest of the levels it changes from 10 to 30%. In the case of the cation NaNe 1, this probability is greater than 90% for zero, first and second vibrational levels of the 2 1S 1 and 1 1P states, greater than 50% for their third and fourth ones, and that varies from 10 to 40% for the remaining levels. To calculate the energy distribution in the emission spectra of the cations, we also need some information
Fig. 6. Theoretical distribution of photon energies in spontaneous radiation spectrum corresponding to the transitions 2 1S 1 ) 1 1S 1 and 1 1P ) 1 1S 1 in NaNe 1.
about the vibrational level populations. However, as the radiation was observed by Hammer et al. [5] in conditions that were far from the equilibrium ones, such information seems to be unavailable. Assuming that all the vibrational levels are populated identically, the overall energy distribution (up to normalization) may be written as follows: g hn ,
X
t21 v2 gv2 1
10
b;v2
where 1 Ebv2 J2 2 hn and the summation goes over all the radiating excited states (in the case of NaNe 1 b 2 1S 1, 1 1P and contributions from the 1 1P term which were taken into account with weight 2). The emission spectra of the cations NaHe 1 and NaNe 1, calculated by using the energy distribution function (10), are presented in Figs. 5 and 6. In these figures the regions, where bound–bound transitions occur are marked off by vertical dotted lines. There are three intensive peaks in the emission spectrum of NaNe 1: the first and third ones originate mostly from the transition 1 1S 1 ( 1 1P, whereas in the central peak a contribution of transition 1 1S 1 ( 2 1S 1 proves to be dominating. Positions of the most intensive peaks in the computed spectra, 65.8 (65.76) nm for NaHe 1 and 78.7 (79.40) nm for NaNe 1, agree well with the observed transition wavelengths [5] given in parentheses.
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197
Fig. 7. Relative positions of the low-lying energy levels of the linear triatomic dication ArNaHe 11 corresponding to (a) infinite separation of ˚. Ar 1 and NaHe 1 and (b) R(Ar–Na) 5 A
4. Role of argon cations in a process of the NaHe 1(1 3S 1) state’s evolution In the course of the reaction Na 12 Sg 1 He1 12 Sg ! NaHe1 21 S1 ; 13 S1 11 the yield of the cations NaHe 1 in their nonradiating triplet 1 3S 1 and radiating singlet 2 1S 1 states may be expected to be in the proportion 3–1. These cations being of great interest for possible short wavelength lasers, a study of the possible mechanisms of the triplet NaHe 1(1 3S 1) state’s evolution appears to be of importance, because population of such states will
certainly lead to a decrease in efficiency and radiation power of possible lasers. Taking into account the conditions of the experiment performed by Hammer et al. [5], in this section we will consider a possibility of decay of the NaHe 1(1 3S 1) state as a result of its interaction with the ions Ar 1(1 2Pu). Relative positions of low-lying energy levels of the system Ar 1 1 NaHe 1 (at infinite separation), calculated on the basis of the available atomic data [7,8], are presented in Fig. 7(a). When Ar 1 approaches NaHe 1, the energy levels increase due to Coulomb repulsion, the shifts of the levels being different depending on the state of the cation NaHe 1. In the case of its ground 1 1S 1 state, the Mulliken charge on
Table 5 Excitation energies (eV) of the three lowest states of Ar 1 calculated by the CASSCF method in the active space [3s3p4s3d] 7 by using the contracted atomic basis set [17s12p5d/4s3p1d] State 2
1 Pu 1 2Sg 1 2Pg
Configuration 2
5
3s 3p 3s 13p 6 3s 23p 44s 1
DE [17s12p5d/4s3p1d]
DE Experimental [8]
00.00 13.42 17.37
0.00 13.42 17.12
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Fig. 8. Cuts through the 3 2P PES of the linear structure of ArNaHe 11 at different R(Ar–Na) as functions of the internuclear distance R(Na–He).
sodium is about 1 1.0e, whereas for the excited state 1 3S 1 this charge is only 1 0.3e. Therefore, in considering the interaction of Ar 1 with NaHe 1(1 3S 1) the approach of Ar 1 to the cation NaHe 1 along its bond line from the Na side is the most energetically advantageous. In the course of such an approach, the Coulomb level shift will be larger for the ground state of NaHe 1 than for its excited triplet one. Relative energies of the low-lying levels of the linear system (Ar…NaHe) 11, estimated at R(Ar–Na)
˚ without considering their splitting due to the 5A electron–electron interactions, are shown in Fig. 7(b). As seen from this figure, the fifth energy level (that we are actually interested in) and the fourth level change their positions long before the argon cation enters sufficiently close to the neighborhood of NaHe 1. Thus, to describe the process of the interaction of Ar 1 with the first triplet state of ˚ one needs NaHe 1 at the distances R(Ar–Na) # 5 A the basis and calculation method which could reproduce correctly at least three lowest doublet states of the ion Ar 1. As a result of a number of calculations of low-lying doublet Ar 1 states in different standard basis sets it became apparent that a specifically optimized basis is required for a correct description of the needed Ar 1 states. To this end, we took uncontracted functions from the [17s12p5d] ANO basis set [19] and performed the CASSCF calculation using the energy functional averaged over the lowest three states of Ar 1. Coefficients of the obtained atomic orbitals were taken as contraction coefficients to produce a relatively small contracted basis [17s12p5d/4s3p1d]. The CASSCF calculations were carried out in the active space [3s3p4s3d] 7 without atomic space symmetry restrictions. Table 5 shows that in the framework of such an approach, the calculated excitation energies of the three lowest electronic states of
Fig. 9. Two dimensional representation of the PES of the NaHe 1(1 3S 1) state allowing for its interaction with the ion Ar 1 (see reaction (12)).
A.I. Panin et al. / Journal of Molecular Structure (Theochem) 490 (1999) 189–200 1
the ion Ar are quite close to the experimental energies from Ref. [8]. As seen from Fig. 7(a), the lowest electronic states of the linear triatomic system (Ar…NaHe) 11, describing the interaction of Ar 1 with NaHe 1(1 3S 1) ˚ are the metastable at the distances R(Ar–Na) # 5 A states 4 2S 1, 3 2P, 1 4S 1, and 1 4P. Thus, to study the evolution of the NaHe 1(1 3S 1) state as a result of its interaction with Ar 1, one needs, in particular, to investigate the potential energy surface of the ArNaHe 11(3 2P) state. Since at the present stage only qualitative conclusions are expected, we used the basis set aug_cc_pVDZ for Na and He, the above described contracted basis [4s3p1d] for Ar 1, and a minimal active space going to [1s(He) 3s(Na)3s3p4s3d(Ar)] 9 at the limit of separated atoms (cations). In this active space the PESs of the five lowest 2P states of the system were calculated by the CASSCF method, with the energy functional averaged over the five lowest roots (the fifth root was required for a correct description of the He 1 detachment as 2 1S 1 and 1 3S 1 states of NaHe 1 have the same dissociation limit). Cuts through the 3 2P PES of the linear structure of ArNaHe 11 at different R(Ar–Na) are presented in Fig. 8 as functions of the internuclear distance R(Na–He). This figure shows that as Ar 1 approaches NaHe 1 the depth of the potential well of the NaHe 1(1 3S 1) state decreases and this state becomes finally repulsive and dissociates into ArNa 1 and He 1. A two dimensional representation of the PES of the NaHe 1(1 3S 1) state with allowing for its interaction with the ion Ar 1 is shown in Fig. 9. More careful analysis of the configuration structure of the corresponding wave functions and Mulliken charges on atoms leads to the conclusion that as a result of the outlined interaction the following reaction has to be the case: Ar1 12 Pu 1 NaHe1 13 S1 ! ArNa1 11 S1 1 He1 12 Sg :
12
Thus, under the conditions of the experiment performed by Hammer et al. [5], the triplet NaHe 1( 3S 1) states must decay with a formation of the He 1 ions, which in turn can interact with sodium atoms to yield again the radiative NaHe 1(2 1S 1) states and, thus, to increase the intensity of emission observed in [5].
199 1
Calculations of PESs of the S symmetry states of the linear system (Ar…NaHe) 11 proved to be more complicated, since to localize the required PES larger number of states is to be taken into consideration. These calculations have not added any new elements for understanding the role of Ar 1 in destroying the triplet NaHe 1(1 3S 1) state and therefore are not discussed here. Since a high energy Ar 1 beam was used in the experiment [5] the ions Ar 1 can approach NaHe 1 (1 3S 1) both along the bond Na–He line and at an angle to this line. The first case is the most energetically advantageous from a purely electrostatic viewpoint and was considered above. The second case turns out to be essentially simpler. Indeed, electrostatic arguments lead to the conclusion that in the course of Ar 1 approaching NaHe 1 at some angle to the line Na–He (say, perpendicular to this line) the energy levels of interest are forced upward due to Coulomb repulsion and they turn out to lie essentially higher than the levels of the system of isolated atomic (ionic) species. Moreover, analysis shows that already ˚ the system ArNaHe 11 becomes at R(Ar–Na) , 5 A metastable or unstable with respect to vibration of the Na–He bond. As a result, at sufficiently high energies of the Ar 1 ions the lifetime of the triplet NaHe 1(1 3S 1) state should also essentially decrease in the case, when Ar 1 approaches NaHe 1 at some angle to the Na–He bond line. 2
5. Summary Results of the ab initio calculations performed in the present work support the correctness of qualitative arguments adduced originally by Basov et al [3] concerning the excimer character of the cations NaHe 1 and NaNe 1 and prospects for using them as a possible laser media with emission in the vacuum ultraviolet region. The main parameters of the potential curves and spectroscopic characteristics of all the electronic states of these ions, correlating with their two lowest dissociation limits, were first calculated. The obtained theoretical emission spectra of the cations NaHe 1 and NaNe 1 was shown to be in good agreement with the experimental spectra observed recently by Hammer et al. [5]. Particular attention was paid to study of evolution
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of the bound non-radiating excited triplet NaHe 1(1 3S 1) state, which could take place in the conditions of the experiment performed by Hammer et al. [5]. It was shown that as a result of its interaction with a high-energy beam of the Ar 1 ions this state has to break up and that the decay of the NaHe 1(1 3S 1) state has to be accompanied with a formation of the ions He 1, which in their turn can interact with sodium atoms again to yield the radiating NaHe 1(2 1S 1) states and, thus, to maintain emission. Processes of such kind have to be taken into account in future investigations of a possibility to use the alkali metal—rare gas cations as sources of laser radiation. Acknowledgements We gratefully acknowledge the Russian Foundation for Basic Research (Grant No. 97-03-33713a) for financial support of the present work. References [1] F.G. Houtermans, Helv. Phys. Acta 33 (1960) 933. [2] K. Rhodes (Ed.), Excimer Lasers Applied Physics, 30, Springer, Berlin, 1979. [3] N.G. Basov, M.G. Voitik, V.S. Zuev, V.P. Kutakhov, Sov. J. Quant. Electron. 5 (1985) 1455.
[4] M. Mantel, H. Langhoff, Z. Phys. D15 (1990) 297. [5] J.W. Hammer, K. Petkau, T. Griegel, H. Langhoff, M. Mantel, Hyperfine Interact. 88 (1994) 151. [6] S. Bililign, M. Gutowski, J. Simons, W.H. Breckenridge, J. Chem. Phys. 100 (1994) 8212. [7] Ch.E. Moore, Analysis of Optical Spectra, National Bureau of Standards, US Government Printing Office, Washington, DC, 1970, NSRDS-NBS 34. [8] D.A. Verner, E.M. Verner, G.L. Ferland, Atomic Data Nucl. Data Tables 64 (1996) 1. [9] G. Herzberg, Molecular spectra and molecular structure, Spectra of Diatomic Molecules, 1, Van Nostrand Reinhold, New York, 1950, pp. 318–319. [10] B.O. Roos, Adv. Chem. Phys. 69 (1987) 339. [11] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S.J. Su, T. L. Windus, M. Dupius, J.A. Montgomery, J. Comput. Chem. 14 (1993) 1347. [12] R.A. Kendal, T.H. Dunning Jr., R. Harrison, J. Chem. Phys. 96 (1992) 6796. [13] L.A. Kuznetsova, N.E. Kuzmenko, Y.Y. Kuzyakov, Y.A. Plastinin, Probabilities of Optical Transitions in Diatomic Molecules, Nauka, Moskow, 1980. [14] J. Tellinghuisen, Chem. Phys. Lett. 105 (1984) 241. [15] J. Tellinghuisen, P.S. Julienne, J. Chem. Phys. 81 (1984) 5779. [16] V. I. Pupyshev, Opitic. Spectr. (in Russian) 63 (1987) 570. [17] N.E. Kuzmenko, V.I. Pupyshev, A.V. Stolyarov, Opitic. Spectr. (in Russian) 63 (1987) 756. [18] A.V. Abarenkov, A.V. Stolyarov, J. Phys.B 23 (1990) 2419. [19] P.-O. Widmark, P.-A. Malmquist, B.O. Roos, Theor. Chim. Acta 77 (1990) 291.