The experimental energies Tv and rotational constants ... ues of Tv and Bv taken directly from [34] were used. ...... D. B. Kokh, A. B. Alekseyev, and R. J. Buenker,.
Optics and Spectroscopy, Vol. 99, No. 5, 2005, pp. 719–730. Translated from Optika i Spektroskopiya, Vol. 99, No. 5, 2005, pp. 750–761. Original Russian Text Copyright © 2005 by Alekseyev.
MOLECULAR SPECTROSCOPY
Ion-Pair States of I2, Br2, IBr, and ICl V. A. Alekseyev Institute of Physics, St. Petersburg State University, Peterhof, St. Petersburg, 198504 Russia Received June 25, 2004; in final form, January 12, 2005
Abstract—The experimentally determined energies and rotational constants of the vibrational levels v = 0–20 of the Ion-Pair states Ω = 0+, Ω = 1 of the I2, Br2, IBr, and ICl molecules are modeled. The model used includes three diabatic states, which correlate to X+(3P, 1D) + Y–(1S0). These states are coupled by the spin–orbit interaction, which is assumed to be independent of the internuclear distance. For IBr and ICl, as well as for the ungerade states of I2 and Br2, satisfactory results are obtained. The model is less applicable to the gerade states of I2 and Br2, which is possibly results from the retainment of the asymptotic JAJB coupling of the angular momenta at equilibrium internuclear distances. © 2005 Pleiades Publishing, Inc.
Te(1(3P1))
–
Te(1(3P2))
≈
E(X+(3P1))
–
E(X+(3P2))
≡ 2α1,
where E is the energy of the ion state. The I2 and Br2 molecules, as well as the ICl and IBr molecules, belong to this case (see Figs. 1 and 2, respectively). The Cl2 molecule corresponds to the opposite case, in which the positions of the states are mainly determined by the value of ∆ΠΣ (Fig. 1). The states 0–(3P1) and 2(3P2) retain their 3Π character. The energies of these states depend only on the diagonal elements of the SO-coupling Hamiltonian, and, approximately, the difference between their electronic energies is given by Te(0–(3P1)) – Te(2(3P2)) ≈ 2α1 (Fig. 1). The IP states of halogens have been the subject of much experimental investigation. Studies performed up
Br2
4
α1 = 1570 Òm–1 Cl α2 = 348 Òm–1 2 α + = 498 Òm–1 α0+ = 1920 Òm–1 0 + 1 0+ 0 ( D2)
(1D2) 1 2
0– 1 0+
1 0– 0+
0
3
0+ 1 3 Σ
0+
2
0 1 2
+
g
u
2 1 0+
0– 1
Π0+
0+ 0– 1
+
0 1 0–
0+ 1 2
1.93 α1
0+ 2 1
3
2
3
u g
–2
Π1
Σ
u
2 1 0+
1 0+
Π0+
0–
3
+
0
3
1
Π1
1.96 α1
≈ E(X+(3P0)) – E(X+(3P2)) ≡ 2α0,
α1 = 3545 Òm–1 α0+ = 3225 Òm–1
I2
1.73α0+ 1.96 α1 2.07 α0+ 1.87 α1
Te(0+(3P0)) – Te(0+(3P2))
5
1.74 α0+ 1.88 α1 1.91 α0+ 1.96 α1
The energetically lowest group of Ion-Pair (IP) states of a halogen molecule contains six states, correlating to the limit X+(3PJ) + Y–(1S0) (X, Y = F, Cl, Br, I): Ω = 0+(3P2), 1(3P2), 2(3P2), 0–(3P1), 1(3P1), and 0+(3P0) (the terms are designated according to Hund’s case c). In a homonuclear molecule, the number of such states is doubled. Due to spin–orbit (SO) coupling, the states 0+(3P2), 0+(3P0), 1(3P2), and 1(3P1) are of mixed 3Π~3Σ character, where 3Π and 3Σ are the diabatic states corresponding, respectively, to the perpendicular and parallel orientation of the double occupied orbital X+ with respect to the axis of a molecule. The relative positions of the 3Π~3Σ states are determined by the SO coupling and by the parameter ∆ΠΣ = Te(3Π) – Te(3Σ), where Te is the equilibrium electronic energy of the diabatic state. If ∆ΠΣ is small with respect to the energy of the SO coupling, then
until 1987 are reviewed in [1]. The I2 molecule was the most studied. At present, experimental data are available for all states correlated to I+(3PJ) + I–(1S0) and to the higher-lying limit I+(1D2) + I–(1S0). The basic spectroscopic parameters of the IP states of I2 [2–31], Br2
(2Te – (Te(1(3P1)) + Te(1)3P2))/α1
INTRODUCTION
2
g
Fig. 1. Energies of the IP states of I2, Br2 and Cl2 in units of α1 = (E(X+ 3P1) – E(X+ 3P2))/2. The energy is counted off from the average energy of the 1g(3P1) and 1g(3P2) states. The states of I2 and Br2 in the ranges around α1 = –1 and α1 = 1 correlate to X+ 3P2 and X+ 3P1, 0, respectively. For the Cl2 molecule, the diabatic energies Te(3Π) and Te(3Σ) are shown.
0030-400X/05/9905-0719$26.00 © 2005 Pleiades Publishing, Inc.
720
ALEKSEYEV
ICl 1
1
0+
0+
IBr
1.82 α0+
1.82 α1
0
1.83 α0+
0.5
1.82 α1
(2Te – (Te(1(3P1)) + Te(1)3P2))/α1
1.0
α1 = 3545 Òm–1 α0+ = 3225 Òm–1
–0.5
–1.0
0+ 1 2
0+ 1 2
Fig. 2. Energies of the IP states of ICl and IBr (see the legend to Fig. 1).
[32–46], Cl2 [47–59], ICl [60–68], and IBr [69–73] are listed in Tables 1 and 2. Some states were repeatedly studied. As a rule, the discrepancies do not exceed the experimental errors. Both tables contain the results of the most recent studies. The potentials of the gerade IP states are remarkably similar. Thus, the spread in the vibration frequencies ωe does not exceed 3% (Table 1). The exception is + 0 g (1D2). The characteristic feature of the ungerade + states is an increase in ωe and ωexe for the states 0 u (3P0) + and 1u(3P1) as compared to 0 u (3P2) and 1u(3P2). The limiting case of this trend is the Cl2 molecule, for which the difference in quanta amounts to ~10%, and, in addition, ∆Gv (the energy difference between the vibrational levels v and v – 1) is not a smooth function of the vibrational quantum number but, rather, changes randomly from level to level. The spectroscopic parame– ters of the 3Πu states 2u(3P2) and 0 u (3P1) show no anomalies (Table 1). Therefore, we can conclude from the above that the particular features of the 3Πu~3Σu + states 0 u (3P2, 0) and 1u(3P2, 1) are determined by the configuration interaction. Owing to the dominant role of the electrostatic interaction, the IP states of the homonuclear halogens present that comparatively rare case in which the potential curves of gerade and ungerade states are similar. The exchange interaction manifests itself only in differences in the electronic energies, which are insignificant in comparison with the dissociation energy. The gerade and ungerade Ω(3P2) states lie approximately symmetrically above and below some average energy (Fig. 1). Similarly to the energies of the IP states of ICl and IBr
(Fig. 2), the average energies Te(Ωgu(3P2)) = (Te(Ωg(3P2)) + Te(Ωu(3P2)))/2 of the states of I2 differ by no more than 100 cm–1. A similar result is obtained upon averaging the energies of the IP states of Br2 and Cl2. Quantum mechanically, the energy gap between the Ωg(3P2) and Ωu(3P2) states arises due to the transparency of a barrier separating the equivalent X+X– and X−X+ configurations. A particular feature of this case is X–X+ requires coherent that the transition X+X– tunneling of two electrons [1]. It is also remarkable that, for all the homonuclear halogens, 3 3 Te(3Π2u) − Te(3Π2g) ≈ Te( Π0– ) – Te( Π0– ) ≈ 1500 cm–1 u g (Table 1). The states Ω(3P2) of the heteronuclear molecules are located in a narrow energy interval ~100 cm–1 (Fig. 2, Table 2). The same is also true for the IP states of the RgX (rare gas–halogen) molecules, which correlate to the limit Rg+(2P3/2) + X–(1S0). According to the data of [74–76], for the XeI, KrF, and XeCl molecules, Te(3/2(2P3/2)) – Te(1/2(2P3/2)) = 110, –120, and –90 cm–1, respectively. The exception is XeF (∆Te = –795 cm–1 [77]), which is explained by the interactions with covalent states. The heteronuclear halogens have precisely the same number of IP states as homonuclear ones. The absence of the center of inversion is compensated by the presence of two dissociation limits X+ + Y– and X– + Y+. Since the dissociation energy of the IP states of all the halogen molecules is approximately the same, then Te(X–Y+) – Te(X+Y–) ≈ IP(Y) – EA(X) – IP(X) + EA(Y) ≈ 1.7 and 3.0 eV for IBr and ICl, respectively (here IP is the ionization potential and EA is the electron affinity energy). Attempts to detect the states X–Y+ are mentioned in the literature on spectroscopy of halogens; however, no positive results have been obtained. The states X+Y–(Ω) and X−Y+(Ω' = Ω) are coupled by an optical dipole transition. In the homonuclear molecules, – + such coupled states are X+ X g (Ω) and X– X u (Ω' = Ω), which correlate to the same dissociation limit, and the dipole moment is proportional to the internuclear distance (µ(R) ≈ eR [78]). Under coherent laser excitation, a large dipole moment favors the appearance of such macroscopic radiation effects as stimulated Raman scattering and amplified spontaneous radiation [79]. X–Y+(Ω' = Ω) and The optical transitions X+Y–(Ω) – + X− X u (Ω' = Ω) imply coherent motion X+ X g (Ω) of both electrons along the molecular axis. Due to this intriguing feature, the search for such transitions deserves further efforts. As compared to homonuclear molecules, the IP states of heteronuclear molecules have been less studied. Thus, the states 0+(3P0) and 1(3P1) are known only for ICl, IBr (Table 2), and ClF ([80] and references therein). As in the case of ungerade states, the vibraOPTICS AND SPECTROSCOPY
Vol. 99
No. 5
2005
OPTICS AND SPECTROSCOPY
Table 1. Spectroscopic constants (cm–1) of the IP states of I2 , Br2 , and Cl2 State I2
I2, g
No. 5 2005
Br2, g
ël2, u
Cl2, g
Te ωe ωexe Be αe × 10–5 Re (Å) Te ωe ωexe Be αe × 10–5 Re Te ωe ωexe Be αe × 10–4 Re Te ωe ωexe Be αe × 10–4 Re Te ωe ωexe Be αe × 10–4 Re Te ωe ωexe Be αe × 10–4 Re
0+(3P2) [2–4]a 41028.6b 94.99 0.1092 0.020715 4.374 3.5812 [13–15] 41411.8 101.39 0.2048 0.019969 5.449 3.6475 [32, 33] 49928.4 134.46 –0.0853 0.04238 1.063 3.1750 [37, 38] 49777.9 150.83 0.4182 0.04187 1.61 3.1940 [47, 48] 58486 ~235
0+(3P0) [5] 47217.4 96.31 0.4125 0.020541 1.73 3.5963 [16–20] 47026.1 104.19 0.2149 0.020803 5.713 3.5736 [34] 53899.8 154.70 0.6790 0 .039905 1.413 3.2720 [39] 53101.7 152.8 0.42 0 .04260 1.413 3.1667 [48, 49] 59931 ~284
1(3P2) [6, 7] 41621.4 94.75 0.1725 0.019590 2.78 3.6830 [21] 40821.0 105.02 0.2258 0.020420 5.531 3.6071 [32] 50212.1 125.615 –0.1428 0.04118 5.59 3.2209 [40, 41] 49930.4 151.74 0.4252 0.04220 1.543 3.1823 [47] 58537 ~240
1(3P1) [8, 9] 48280.3 107.96 0.2640 0.019905 6.15 3.6534 [22, 23] 47559.1 106.60 0.2151 0.02134 6.03 3.5280 [35] 53257.6 164.16 0.7588 0.04001 1.749 3.2677 [42] 52641.6 153.86 0.4286 0.042496 1.700 3.1707 [50] 59611 ~312
2(3P2) [6, 7] 41787.8 100.63 0.2102 0.018587 4.52 3.7807 [24–27] 40388.3 103.96 0.2077 0.020528 5.17 3.5975 [32] 50452.7 147.87 0.4 0.03796 1.121 3.3548 [43, 44] 48930.4 150.86 0.3911 0.042515 1.494 3.1699 [47] 59056.1 248.61 1.041 0.1186 0.106454 0.1188 0.10685 0.103862 5.63 2.8512 3.0095 2.8488 3.009 3.0468 [51, 52] [52, 53] [52, 54] [52, 55] [56, 57] 57819.4 59356.1 57572.2 59295.6 57295.7 251.95 256.98 252.24 256.64 252.34 1.03181 1.203 1.01946 1.20536 1.0071 0.116556 0 .11503 0.116623 0.114314 0.116674 6.403 7.58 6.4345 7.507 6.466 2.8761 2.8951 2.8752 2.9042 2.8747
[50] 59726.1 248.72 1.002 0.103864 6.1 3.0468 [58] 57978.8 252.41 1.00638 0.116848 6.466 2.8725
0+(1D2) 1(1D2) 2(1D2) [11] [12] [12] 51706.1 54706.2 54263.0 131.00 105.29 108.48 0.516 0.2302 0.2304 0.021950 0.019327 0.021679 9.25 4.971 6.729 3.4790 3.7076 3.5007 [16] [30] [31] 55409.9 53216.3 54489.6 97.10 106.93 108.26 0.1936 0.2078 0.2450 0.018159 0.021421 0.021319 3.362 5.458 6.537 3.8250 3.5217 3.5301 [36] 57929.7 212.23 1.5683 0.046565 4.500 3.0290 [45] [46] 65512.0 60288.1 140.86 154.48 0.3620 0.3998 0.037085 0.043655 0.9526 1.5264 3.3941 3.1283
[59] 68446.3 256.746 1.0202 0.119221 6.539 2.8438 721
Note: a The table contains the data from the most recent studies. b The values of some parameters are rounded.
0–(3P1) [10] 48646.5 102.34 0.2180 0.019590 4.664 3.7804 [28, 29] 47085.8 104.06 0.1936 0.020922 5.059 3.5635 [35] 53479.7 148.62 0.4 0.03802 1.3 3.3521 [42] ~52000
ION-PAIR STATES OF I2, Br2, IBr, AND ICl
Vol. 99
Br2, u
Parameter
722
ALEKSEYEV
Table 2. Spectroscopic constants (cm–1) of the IP states of ICl and IBr State
Parameter
ICla Te ωe ωexe Be αe × 104 Re IBr Te ωe ωexe Be αe × 104 Re
0+(3P2)
0+(3P0)
1(3P2)
1(3P1)
2(3P2)
[60–63]b 39059.5 165.69 0.288 0.058029 2.274 3.2553 [69, 70] 39487.8 119.43 0.2055 0.029852 7.40 3.4067
[62–64] 44924.4 184.16 0.7439 0.057925 2.32 3.2582 [70, 71] 45382.6 128.8 0.363 0.029852 7.404 3.3937
[62, 67] 39103.7 170.31 0.4706 0.056707 1.899 3.2930 [69, 70] 39507.8 122.09 0.2546 0.029372 8.24 3.4344
[65–67] 45552.8 184.85 0.6737 0.058962 2.378 3.2294 [70] 45996.0 128.5 0.3188 0.030621 8.69 3.3636
[62] 39061.8 173.63 0.5572 0.054786 2.054 3.3502 [72, 73] 39456.6 123.06 0.2813 0.028598 6.77 3.4806
Notes: The energies and vibrational parameters of the higher-lying states 0+(1D2) and 1(1D2) are also known: Te = 51199.9, 51615.6; ωe = 160.72, 184.03; and ωexe = 0.193, 0.596 [67, 68].; b See the note to Table 1.
tional frequencies in the states 0+(3P0) and 1(3P1) are somewhat greater than in 0+(3P2) and 1(3P2) (Table 2). For the ClF molecule, this difference achieves 30%, and ∆Gv varies irregularly from level to level. The state 0–(3P1) of the heteronuclear halogens has never been observed experimentally. Taking into account the energetic closeness of the Ω(3P2) states, one can expect that Te(0–(3P1)) – Te(1(3P1)) ~ ±100 cm–1. We note in this – – connection that the states 0 g (3P1) and 0 u (3P1) of I2 are located symmetrically with respect to the states 1g(3P1) and 1u(3P1) (Fig. 1) and, therefore, Te(1gu(3P1)) ≈ – Te( 0 gu (3P1)). The electronic structure of halogens has been studied theoretically ([81–83] and references therein). The existing ab initio methods do not directly include the relativistic SO coupling effect. As a rule, the SO coupling is taken into account in terms of the perturbation theory. The energy of the SO coupling results from the motion of an electron near a nucleus. This part of the wave function of an electron differs little from the atomic term, which gives grounds to consider the SO coupling as a parameter independent of the internuclear distance. In this study, we attempted to calculate the diabatic states of I2, Br2, ICl, and IBr using available experimental data on the 3Π~3Σ states of these molecules and assuming the SO coupling to be independent of the internuclear distance. Previously, this method was applied to the modeling of the rovibronic structure of the strongly perturbed IP states of ClF [80].
constructed from the diabatic potentials by solving the equation of the configuration interaction on the grid of values of internuclear distance. In the determinant form, the equation for the configuration interaction can be written in the form (Λ = 1Σ or 1Π) EΠ – Ea
α ΠΣ
α ΠΛ
α ΠΣ
EΣ – Ea
α ΣΛ
α ΠΛ
α ΣΛ
EΛ – Ea
= 0,
where EΠ, EΣ, and EΛ are the energies of the diabatic states with the inclusion of the diagonal contribution D from the SO coupling ( E SO ). If the triplet X+(3P) satisfies the interval rule E(3P0) – E(3P1)/E(3P0) – E(3P2) = 1/3, then αΠΣ(0+) = (2)1/2αΠΣ(1) and E SO = (2/3)αΠΣ(1) for D
3Π 1
and E SO = 0 for 3Σ0+, 3Σ1, and 3Π0+. These relations are valid for Cl2 and, to a lesser degree, for Br2. D
The SO coupling operator is a one-electron operator [84]. In view of the symmetry requirements, the transiπg and σu πu are allowed, and the onetions σg electron integrals 〈σg |HSO |πg〉 and 〈σu |HSO |πu〉 are approximately equal. For the gerade states, the transitions between the following configurations are possible: σ g π u π g σ u ( Σg ) 2
4
2
Πg σ g π u π g σ u
2 3
1
3
4
3
2
Σg σ g π u π g σ u , 0
1
DETAILS OF CALCULATIONS The model includes the diabatic states 3Π, 3Σ, and 1Σ (or 1Π for the Ω = 1 states). The adiabatic potentials are
Σg σ g π u π g σ u
3
2
2
4
2
3
4
4
2
(1)
Πg σ g π u π g σ u 2
3
4
1
Σg σ g π u π g σ u .
1
OPTICS AND SPECTROSCOPY
2
4
4
0
Vol. 99
No. 5
2005
ION-PAIR STATES OF I2, Br2, IBr, AND ICl
723
For the ungerade states, the possible transitions are given by Πu σ g π u π g σ u 1
3 2 3 3 2 Σu σ g π u π g σ u
3
3
= Y 0, 0 + Y 1, 0 ( v + 1/2 ) + Y 2, 0 ( v + 1/2 ) + …, 2
(3)
Bv = Y0, 1 + Y1, 1(v + 1/2) + Y2, 1(v + 1/2)2 + …. (4) Note that Table 1 contains only basic spectroscopic parameters. In the calculations, we took into account all terms of series (3) and (4) for which the values of Yk, l are known from experiment. For the strongly perturbed state 0+(3P0) of the Br2 molecule, the experimental values of Tv and Bv taken directly from [34] were used. The solution of the vibrational Schrödinger equation for the potentials yields the energies of the levels without taking into account the so-called radial interaction caused by the motion of nuclei [84], H rad = 〈F 1 v 1 |T |F 2 v 2 〉. ad
N
ad
ad
ad
Here, F n and v n are the electronic and vibrational adiabatic wave functions, respectively, and T N is the operator of the kinetic energy of the nuclei. For levels with close values of the vibrational quantum number, Hrad can attain several tens of reciprocal centimeters. Due to strong SO coupling in iodine- and bromine-containing molecules, the vibrational levels satisfying this criterion considerably differ in energy. The calculations show that the effect of the radial interaction on the energies of the levels v ' = 0–19 is negligibly small. In the case of ClF, the shift of some levels attains 30 cm–1 [80]. OPTICS AND SPECTROSCOPY
Vol. 99
1
2 4 3 1 Πu σ g π u π g σ u
T v = T e + Gv
ad
2
3
The states 1Πg(u), correlating to X+(1D) + X–(1S), have the same configurations as the 3Πg(u) states. The configurations 3Σ and 1Σ are not coupled by any one-electron + transition. Indeed, the trial calculations for the 0 u states of I2 revealed no strong interaction between the configurations 3Σ and 1Σ. In any case, if αΣΛ > 500 cm–1 (for comparison, αΠΣ ~ 3000 cm–1, see Table 3), it is impossible to reconstruct experimental data with reasonable accuracy. In the subsequent calculations, αΣΛ was assumed to be zero. The diabatic potentials were approximated by a Morse function. The energies and wave functions of the levels of the adiabatic potentials were calculated with the Automatic Vibrational Level Finder (AVLF) module from the program LEVEL 7.4 [85]. The wave functions were used for the calculation of the rotational constants. We restricted ourselves to the calculations of the levels v ' = 0–19. The experimental energies Tv and rotational constants Bv were calculated by the formulas
ad
4
No. 5
2005
Σu σ g π u π g σ u . 1
4
4
1
(2)
RESULTS AND DISCUSSION +
0 u - and 1u States of I2 and Br2 The calculated potential curves for these molecules are shown in Figs. 3 and 4, and the corresponding spectroscopic constants and parameters of the SO coupling are presented in Table 3. The diabatic energies were determined from fitting between the calculated and experimental adiabatic potentials. In the Br2 molecule, 3 Te( Π0+ ) – Te(3Π1u) = 1064 cm–1, which agrees well u
with the expected value E SO (3Π1) = (2/3)αΠΣ(1) = 1047 cm–1. The outer branches 3Σ1u and 3Π1u divide the energy gap between the potentials 1u(3P1) and 1u(3P2) into three equal parts, which ensures the correct asymptotics (Fig. 4). The states I+(3P0) and I+(3P2) are shifted with respect to the state I+(3P1) due to the interactions between the states I+(3P0) and I+(1S0) and between states I+(3P2) and I+(1D2); therefore, state 3Σ1u proves to be 3 considerably higher than state Σ0+ (Fig. 3). D
u
Figure 5 illustrates the accuracy of the calculations. + + The vibrational energies of states 0 u (3P2) and 0 u (3P0) + are reproduced within ±4 cm–1. For the 0 u (1D2) state, agreement with experiment is somewhat worse. In the range of high energies where the accuracy of the Morse + potential is insufficient, the potential 0 u (1D2) is sensitive to the behavior of the repulsive branch Π0+ (Figs. 3 u and 4). The trial calculations showed that the use of the Rydberg–Klein–Rees (RKR) potentials makes it possible to increase the accuracy by an order of magnitude. 3
+
The potential of the state 0 u (3P0) of the I2 molecule is characterized by a low frequency and large anharmonicity of vibrations (Table 1). This is explained by a 3 strong interaction between Π0+ and 1Σu. For the Br2 u molecule, this interaction leads to an inflection of the + inner branch of the 0 u (3P0) potential (Fig. 4). In the I2 molecule, the interaction between 3Π1u and causes a shift in the energy of the states 1u(1D2) by ~800 cm–1 (Fig. 3). The states 3Π1u and 1Πu have the same configuration of the molecular orbitals (the isoconfigurational SO coupling [84]). As was expected, the matrix element 〈1Πu |HSO |3Π1u 〉 is approximately 21/2 times smaller than 〈3Σ1u |HSO |3Π1u 〉 (Table 3). The state 1Πu of the Br2 molecule is located considerably 1Π u
724
ALEKSEYEV
Table 3. Spectroscopic constants (cm–1) of the diabatic potentials of the IP states and parameters of the SO coupling Te
ωe
ωexe
Re (Å)
αΠΣ (αΠΛ)
3Σ
43164
103.8
0.20
3.464
3248 (2410)
3Π
45925
100.6
0.21
3.765
1Σ
49639
109.0
0.17
3.265
3Σ
45049
102.6
0.19
3.481
3Π
45169
100.6
0.21
3.795
1Π
53932
105.1
0.20
3.702
3Σ
45724
102.2
0.19
3.526
3Π
43230
103.2
0.19
3.677
1Σ
54778
96.7
0.20
3.826
3Σ
46109
107.6
0.23
3.503
3Π
42920
104.1
0.20
3.635
1Π
52515
105.6
0.21
3.519
3Σ
50885
152.0
0.42
3.097
3Π
52904
148.3
0.40
3.351
1Σ
57107
158.0
0.36
2.901
3Σ
51236
150.8
0.41
3.104
3Π
51840
148.3
0.40
3.356
3Σ
52338
153.5
0.42
3.155
3Π
50528
150.9
0.43
3.206
3Σ
52225
155.0
0.49
3.169
3Π
49804
151.8
0.40
3.183
3Σ
41895
185.0
0.46
3.136
3Π
41881
172.5
0.55
3.340
3Σ
43425
179.5
0.46
3.134
3Π
41067
172.6
0.51
3.341
3Σ
42624
127.9
0.23
3.295
3Π
42062
122.2
0.32
3.483
3Σ
43967
126.2
0.29
3.298
3Π
41381
124.7
0.25
3.477
a
State +
I2 ( 0 u )
I2 (1u)
+
I2 ( 0 g )
I2 (1g)
+
Br2 ( 0 u )
Br2 (1u) +
Br2 ( 0 g ) Br2 (1g) ICl (0+) ICl (1) IBr (0+) IBr (1)
3454 (2515)
2407 (2675)
2896 (2515)
1831 (1277)
1499
1393
1085 2927 2998 2935 2970
Note: a The energies of the diabatic states include the diagonal contribution from the SO coupling.
higher than the state 3Π1u and has no marked effect on the energies of the lower levels 1u(3P2) and 1u(3P1). It seems that the interaction between 3Π2u and 1∆2u causes a small shift of the state 2u(3P2) but does not appreciably affect the rovibronic structure of this state. We note in this connection that the basic spectroscopic
parameters of the 3Πu states 2u(3P2) and 0 u (3P1) are the same (Table 1). As was expected, the values of ωe, ωexe, –
and Re calculated for the states Π0+ and 3Π1u proved to 3
u
be close to the corresponding experimental values of these parameters for the 3Πu states. In turn, the spectroOPTICS AND SPECTROSCOPY
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ION-PAIR STATES OF I2, Br2, IBr, AND ICl
725
E × 10 3, Òm–1 56 1Π
1(1D2)
0+(1D2)
52
1Π
0+(3P0)
1Σ
0+(3P0)
1(3P1)
1(3P1)
48 3Σ 1 3Σ 0+
44
3Π 0+
3Σ 1
3Π 1
3Σ 0+ 3Π + 0 3Π 1 + 3 0 ( P2 )
1(3P2) 0+(3P2) u states
40
0+(1D2)
1Σ
1(1D2)
3
4
g states
1(3P2) R, Å
3
4
R, Å
D
Fig. 3. Diabatic (with the inclusion of the energy E SO ) and adiabatic potentials of the IP states of I2.
scopic parameters calculated for states Σ0+ and 3Σ1u of u the I2 molecule agree satisfactorily with the experimental parameters of the isoconfigurational state 2u(1D2) (Tables 1 and 3). The consistency between the calculation results indicates that the model takes into account all the significant interactions that affect the shape and relative location of the IP potentials. 3
+
0 g and 1g States of I2 and Br2 The configurations of the states 3Πg and 3Πu differ by the electron permutation πu πg (see (1), (2)). As was noted in the Introduction, for all homonuclear halogens, Te(3Π2u) – Te(3Π2g)
g
≈ Te( Π0– ) – Te(( Π0– ) ≈ 1500 cm–1. 3
3
u
u
while, for I2, Te( Π0+ ) – Te( Π0+ ) ~ 2500 cm–1 (Table 3). 3
g
The energy gap between the states
state 2g(u)(1D2) somewhat exceeds the corresponding diabatic energy due to the interaction between 1∆2g(u) and 3Π2g(u), which, in turn, leads to a decrease in the energy of the state 2g(u)(3P2), with the ungerade states being shifted somewhat larger. The latter is supported – – by the fact that, for I2, Te( 0 u (3P1)) – Te( 0 g (3P1)) is greater than Te(2u(3P2)) – Te(2g(3P2)) by 160 cm–1 (Table 1). In view of this, Te(1∆2g) – Te(1∆2u) ≈ 400 cm–1. This value can be used for estimating the difference Te(3Σg) – Te(3Σu). Consequently, we find that the 3Πg diabat should lie ~1500 cm–1 below the 3Πu diabat, whereas, conversely, the 3Σg diabat should lie ~400 cm–1 above its ungerade counterpart. This is actually true for the Cl2 molecule (Fig. 1). However, for the Br2 mole3 3 cule, Te( Σ0+ ) – Te( Σ0+ ) is within 1500–2500 cm–1, 3
u
1 3 4 2 1u(1D2) ( σ g π u π g σ u )
1 4 3 2 1g(1D2) ( σ g π u π g σ u )
and of I2 is also close to this value (Table 1). The configurations of the states 1∆2g and 1∆2u are, respectively, the same as those of the states 3Σg and 3Σ . For the I molecule, T (2 (1D )) – T (2 (1D )) = u 2 e g 2 e u 2 225 cm–1 (Table 1). It is likely that the energy of the
g
In contrast to the ungerade states, we failed to reproduce the structure of the gerade states using for 3Πg the experimental values of ωe, ωexe, and Re for the state 2g(3P2). Moreover, the spectroscopic parameters for the 3 3 states Σ0+ and Π0+ differ appreciably from the correg
g
sponding parameters of the states
Σ0+ and
3
g
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Π0+
3
g
726
ALEKSEYEV E × 10 3, Òm–1 1(1D2) 60 0+(1D2) 58 1Σ
1(3P1) 1(3P1)
0+(3P0) 54
0+(3P0) 3Π + 0 3Σ 1
u states
0+(3P2)
2.5
3Π 0+
3Σ 0+
1(3P2)
50
3Σ 1
3Π 1
3.0
3Σ 0+
1(3P2) R, Å 2.5
3.5
3Π 1
3.0
0+(3P2) g states 3.5
R, Å
D
Fig. 4. Diabatic (with the inclusion of the energy E SO ) and adiabatic potentials of the IP states of Çr2.
(Table 3). The values of αΠΣ for the gerade states proved to be smaller by 20–30% than those for the ungerade states but, in this case, the values of αΠ∆ for the gerade and ungerade states proved to be approximately equal to each other (Table 3). A decrease in αΠΣ is compensated by a decrease in the Te(3Σg) – Te(3Πg). Possibly, this is the manifestation of a partial retainment of the asymptotic JAJB coupling of the angular momenta in the range around Re. The model of JAJB coupling with respect to the IP states of halogens and halogenides of rare gases was considered in [86, 87].
therefore, 〈3Σg |HSO |3Πg〉 Ⰶ 〈3Σu |HSO |3Πu〉. However, the calculated values differ by no more than 30%. The range of the internuclear distance in which the passage to the JAJB coupling occurs can be determined from studying the hyperfine structure of molecular transitions. The criterion of this passage is the degree of closeness of the magnitude of the molecular quadrupole-interaction constant eQq to its magnitude in an atom. Such investigations were performed for the tran+ + 3 B( Π0+ ) and 1g(3P2) sitions 0 g (3P0), 0 g (3P2) u
c4( σ g π u π g σ u ) correspond, respectively, to the states 3Σ and 3Π (see (2)). The weighting coefficients c −c g g 1 4 depend on the internuclear distance. In the range
A(3Π1u) in I2 [88–90]. On the whole, the experimental values of eQq agree satisfactorily with the theoretical ones obtained in terms of the JAJB coupling scheme. Based on this, the authors of [88–90] concluded that the JAJB coupling is retained in the range around Re. As was shown above, the dominant MO configurations of the states 3Σg and 3Πg are not coupled by any one-electron transition. It seems to be possible that this explains the tendency of gerade states to retain the asymptotic JAJB coupling. At present, the hyperfine structure of transitions from ungerade IP states has not been studied.
around Re, σ g π u π g σ u and σ g π u π g σ u dominate. Formally, it follows from this that c1c3 + c2c4 Ⰶ 1 and,
In the Cl2 molecule, the difference Te(3Σg) – Te(3Πg) is much greater than the parameter of the SO coupling
The SO coupling operator is the one-electron operator [84]. According to (1), for the state 3Σu, there is only one configuration, which is coupled by one-electron transitions with the two 3Πu configurations. The same is true for the state 1Σu. The combinations c1( σ g π u π g σ u ) + c2( σ g π u π g σ u ) and c3( σ g π u π g σ u ) + 2 1
2 4
4 3
2
2
4
2
2
2
3
4
1
2
1
4
3
2
2
2
4
2
OPTICS AND SPECTROSCOPY
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ION-PAIR STATES OF I2, Br2, IBr, AND ICl ∆Gv, cm–1 (a) 4
727 +
(Fig. 1); therefore, the pairs 0 g (3P0)–1g(3P1) and 0 g (3P2)–1g(3P2) belong, respectively, to the states 3Σg and 3Πg. This is confirmed by experimental data on the radiative lifetimes of these states [91], as well as by ab initio calculations [82]. It is remarkable that, except for Te, all the remaining parameters for the 3Σg and 3Πg states are the same. In the series Cl2–Br2–I2, the difference between Re of the gerade states increases (Table 1). The most probable explanation for this is the effect of Rydberg and covalent configurations. This is primarily true for the I2 molecule. +
1 0 2
3 –4
∆Bv/Bvexp, % (b) 0.2 1 0 3
2
–0.2 0
10
20 v level
Fig. 5. Deviations (a) Gv (calc) – Gv (exp) and (b) {Bv (calc) – + Bv (exp)}/Bv (exp) for the 0 u states of the I2 molecule:
+ + + (1) 0 u (3P2), (2) 0 u (3P0), and (3) 0 u (1D2).
The equilibrium internuclear distances for the states 3Σu and 3Πu differ by ~0.25 Å (Table 3). For the gerade states, the difference in the number of electrons in the bonding and antibonding σ and π orbitals of the 3Σg and 3Πg configurations manifests itself only in the difference in the energies Te, whereas the equilibrium distances Re(3Σg) and Re(3Πg) are approximately equal. It is possible that the latter is simply a concatenation of circumstances. Explaining this remarkable similarity in the potentials of the gerade states and, in particular, the role of symmetry, requires further investigations.
E × 10 3, Òm–1 0+(3P0)
48
0+(3P0)
1(3P1)
1(3P1)
3Σ 1
44
3Σ 1
3Σ 0+
3Π 0+ 3Σ 0+
3Π 0+ 3Π 1
3Π 1
0+(3P2)
0+(3P2)
40 1(3P2)
1(3P2)
IBr 3
4
R, Å D
3
ICl 4 R, Å
Fig. 6. Diabatic (with the inclusion of the energy E SO ) and adiabatic potentials of the IP states of ICl and IBr. OPTICS AND SPECTROSCOPY
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728
ALEKSEYEV
0+(3P2, 0) and 1(3P2, 1) States of ICl and IBr The structure of the IP states of the ICl and IBr molecules can be well reproduced using the model of two diabatic states 3Π and 3Σ coupled by the SO coupling. The calculated potential curves are presented in Fig. 6. Similarly to the case of ungerade states, the closeness of the calculated parameters of one of the diabatic states to their experimental values for 2(3P2) (Tables 2, 3) allows one to unambiguously attribute these diabats to 3Π and 3Σ. It is interesting that the values of T (3Σ ) – e 1 Te(3Σ0+) and Te(3Π0+) – Te(3Π1) proved to be approximately the same as those for the ungerade states of the I2 molecule (Fig. 3). The basic spectroscopic parameters ωe, ωexe, and Re of the XY IP states are somewhat different (Table 2). The difference in Re is especially noticeable. In this respect, the XY IP states are also close to the ungerade X2 states. The states Ω(3P2) of the IBr and ICl molecules are concentrated in the interval ∆Te(Ω(3P2)) ~ 100 cm–1 (Table 2, Fig. 2). Intuitively, an attempt to explain such an energetic structure in terms of the LS coupling scheme seems to be contradictory. Indeed, the difference between the energies of the initial states Te(3Π) – Te(3Σ) and the energy of the SO coupling both exceed ∆Te(Ω(3P2)) by an order of magnitude. The closeness of the energies Te(Ω(3P2)) testifies that the coupling between the electron angular momentum I+(3P2) and the axis of a molecule is weak, which can be considered an indication that the JAJB coupling is retained. Clarification of this question requires further study. In particular, it is interesting to consider the dipole moment functions for µ(R) optical transitions. The strongest σu, for transitions are the transitions σ σ* (σg 3 the homonuclear molecules) between the Π states σ1π4(3)π*3(4)σ*2 and σ2π4(3)π*3(4)σ*1, as well as between the 1Σ states σ1π4π*4σ*1 and σ2π4π*4σ*0. The effect of the configuration interaction on the behavior of µ(R) is especially significant in the range of intersection of the diabatic potentials. According to [92], µ(R) of the tranA3Π1 in the ICl molecule slowly sition β 1(3P2) decreases in the range 2.8–4.0 Å, whereas the calculations predict that the weights of the configurations 3Π and 3Σ in the range R ~ 3 Å vary considerably (Fig. 6). At present, no investigations of the functions µ(R) for other transitions in the ICl and IBr molecules have been carried out. The dipole moment functions for transitions in homonuclear molecules were studied in detail in [93, 94] (and references therein). CONCLUSIONS We showed that a comparatively simple model that includes two or three diabatic states coupled with each other by SO interaction independent of the internuclear distance fairly well reproduces the experimental ener-
gies and rotational constants of the mixed 3Π~3Σ states 0+(3P2), 0+(3P0), 1(3P2), and 1(3P1) of the I2, Br2, IBr, and ICl molecules. The interaction of the 3Π state with the higher-lying 1Σ state explains the anomalous prop+ erties of the 0 u (3P0) state of I2 and Br2. For the ungerade states of the I2 and Br2 molecules, as well as for all the states (gerade and ungerade) of the ICl and IBr molecules, the calculated diabatic potentials can be unambiguously assigned to 3Π and 3Σ on the basis of the criterion of closeness of the calculated and experimental spectroscopic parameters for the 3Π states 2(3P2) and 0–(3P1). The model is less applicable to the gerade states of I2 and Br2, which is possibly connected with the fact that, in these states, the JAJB coupling of the angular momenta is retained. ACKNOWLEDGMENTS I am grateful to Prof. T. Ishiwata for reprints of his papers and to K. Lawley for helpful discussions. I would like to thank the authors of [93] for providing a copy of their paper prior to its publication. REFERENCES 1. J. C. D. Brand and A. R. Hoy, Appl. Spectrosc. Rev. 23, 285 (1987). 2. T. Ishiwata and I. Tanaka, Laser Chem. 7, 79 (1987). 3. M. L. Nowlin and M. C. Heaven, Chem. Phys. Lett. 239, 1 (1995). 4. J. Tellinghuisen, J. Mol. Spectrosc. 217, 212 (2003). 5. T. Ishiwata, T. Kusayanagi, T. Hara, and I. Tanaka, J. Mol. Spectrosc. 119, 337 (1986). 6. T. Ishiwata and T. Yotsumoto, Bull. Chem. Soc. Jpn. 74, 1605 (2001). 7. G. W. King, I. M. Littlewood, and J. R. Robins, Chem. Phys. 56, 145 (1981). 8. S. Motohiro and T. Ishiwata, J. Mol. Spectrosc. 204, 286 (2000). 9. P. J. Jewsbury, T. Ridley, K. Lawley, and R. J. Donovan, J. Mol. Spectrosc. 157, 33 (1993). 10. S. Motohiro, S. Nakajima, and T. Ishiwata, J. Chem. Phys. 117, 187 (2002). 11. T. Ishiwata, A. Tokunaga, Y. Shinzawa, and I. Tanaka, J. Mol. Spectrosc. 117, 89 (1986). 12. S. Motohiro, A. Umakoshi, and T. Ishiwata, J. Mol. Spectrosc. 208, 213 (2001). 13. J. C. D. Brand, A. R. Hoy, A. K. Kalkar, and A. B. Yamashita, J. Mol. Spectrosc. 95, 350 (1982). 14. P. J. Wilson, T. Ridley, K. P. Lawley, and R. J. Donovan, Chem. Phys. 182, 325 (1994). 15. J. P. Perrot, M. Broyer, J. Chevaleyre, and B. Femelat, J. Mol. Spectrosc. 98, 161 (1983). 16. T. Ishiwata, J. Yamada, and K. Obi, J. Mol. Spectrosc. 158, 237 (1993). 17. J. P. Perrot, A. J. Bouvier, A. Bouvier, et al., J. Mol. Spectrosc. 114, 60 (1985). OPTICS AND SPECTROSCOPY
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Translated by V. Rogovoœ
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