Stretch-Bender Calculations of the Rovibronic Energies in the X̃2B1 Electronic Ground State of NH2

Journal of Molecular Spectroscopy 211, 16–30 (2002) doi:10.1006/jmsp.2001.8480, available online at http://www.idealibrary.com on

Stretch-Bender Calculations of the Rovibronic Energies in the X˜ 2 B1 Electronic Ground State of NH2 Alexander Alijah∗,1 and Geoffrey Duxbury† ∗ Fakult¨at f¨ur Chemie, Universit¨at Bielefeld, 33615 Bielefeld, Germany; and †Department of Physics and Applied Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom Received May 7, 2001; in revised form October 29, 2001

The extended stretch-bender Hamiltonian, incorporating spin–orbit coupling and overall rotation, has been used to calculate the spin-vibronic structure of the X˜ 2 B1 state of NH2 up to the barrier to linearity of this state. A detailed comparison has been made with experimental measurements of these rovibronic states, the majority of which are due to Vervloet and his collaborators. We have shown that, in order to fit the variation of the vibronic spin–orbit coupling constant over the whole of this energy regime, the effective linear molecule spin–orbit coupling constant, ASO , must be increased from the earlier value of 50 cm−1 of Ch. Jungen, K.-E. J. Hallin, and A. Merer (Mol. Phys. 40, 65–94 (1980)) to 61.6 cm−1 . Evidence has also been provided for the large quenching of the spin–orbit coupling as the molecule bends, reflected in the large valuee of gK = 6 cm−1 . The pattern of calculated spinrovibronic levels, including the effects of spin uncoupling, is in good agreement with that measured C 2002 Elsevier Science experimentally.  Key Words: Renner–Teller; stretch-bender; electronic angular momentum; spin–orbit interaction; angular momentum quenching; rovibronic energies; NH2 .

state of the linear molecule. The first comprehensive treatment of electronic angular momentum effects in NH2 was by Jungen et al. (6, 7) (JHM), who made use of the Renner–Teller theoretical model developed by Jungen and Merer (8) to calculate the vibronic, electronic angular momentum, and rotational behavior in NH2 and to provide a detailed explanation of the experimental data then available. The JHM model was restricted mainly to a discussion of the effects of the large-amplitude bending motion in the two half states, although it did include some discussion of the effects of vibrational resonances involving the stretching vibrations. The main reason this pseudo-one-dimensional approach works well is that the A˜ 2 A1 ← X˜ 2 B1 electronic transition in NH2 , as in other similar dihydrides, is dominated by long progressions in the bending vibration. This results from the very large change in bond angle between the well-bent 2 B1 state, which has geometry similar to that of the ground state of water of about 105◦ , and the excited 2 A1 state, which is almost linear. Since there is a very small change in equilibrium bond length between the two states, the Franck–Condon structure observed in the electronic spectrum almost entirely reflects the bond angle change, and hence an absence of progressions in the stretching vibrations. The lack of direct information about the stretching vibrations results in the absence of directly interpretable information about the stretching part of the potential energy surface. Instead this information must be derived either from the observation of accidental stretch–bend resonances, or in the case of the ground state by infrared spectroscopy or laser excited emission spectra. In their original paper, since there was little information about the higher levels of the ground electronic state below

1. INTRODUCTION

The NH2 free radical has played an important role in aiding our understanding of vibronic coupling in small polyatomic molecules, in particular of the Renner–Teller effect. Although the original paper by Renner (1) laid the groundwork for much of our understanding of vibronic coupling associated with the breaking of the axial symmetry of linear triatomic molecules by the excitation of their bending vibrations, the first molecule to which these ideas were applied was NH2 . The electronic spectrum was detected by Herzberg and Ramsay in 1952 (2). However, the analysis of its electronic spectrum by Dressler and Ramsay (3) owed much to the development by Pople and Longuet-Higgins (4) of a model for Renner–Teller coupling based upon ideas from Renner’s original paper. Although shortly afterward the NCO radical provided a better example of Renner’s original concept, and a detailed analysis of the application of Renner’s theory was carried out by Dixon (5), NH2 has continued to play an important part in our understanding of the complex role which electronic angular momentum plays in open-shell molecules in which large-amplitude bending motion occurs. The reason NH2 continues to be an excellent vehicle for testing theories of Renner–Teller coupling is that it has lent itself to study by a wide range of experimental techniques. As a result experimental information is available about a wide range of rovibronic levels in the 2 A1 and 2 B1 half states between which electronic transitions are observed, and which correlate with a 2 u 1 Present address: Departamento de Qu´ımica, Universidade de Coimbra, 3004-535 Coimbra, Portugal.

0022-2852/02 $35.00  C 2002 Elsevier Science All rights reserved.

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THE X˜ 2 B1 STATE OF NH2

the barrier to linearity, JHM assumed that the expectation value of electronic angular momentum, L z , was unquenched as the molecule bent and could be set equal to  = 1. In the intervening years the large number of data available for higher levels of the 2 B1 state, many of which have been provided by Vervloet and his collaborators (9–11), have allowed a test of the quenching mechanism proposed by Jungen and Merer (8) to be carried out. The increased number of vibrational resonances measured in both the ground and the excited state has also allowed more information about the stretching states to be provided. Finally, photodissociation experiments carried out on ammonia by the groups in Bristol (12, 13), Bielefeld (14), and JILA (15, 16) have provided a wealth of information on the states with very high rotational angular momentum about the linear molecule axis, in particular in the ground electronic state. It is the purpose of the present paper to examine the effect of large amplitude motion and vibronic coupling at rovibronic levels of NH2 which lie below the barrier to linearity. It has recently been shown by Tarczay and his colleagues (17) that the barrier to linearity in water is very similar to that of the ground electronic state of NH2 derived by JHM and calculated by Gabriel et al. (18). In Table 1 a summary of the equilibrium structures and barrier heights is given. In view of the recent interest shown in the calculation of the effects large amplitude motion in water on the structure of the rovibrational energy levels of its ground electronic state, it is interesting to inquire what are the parallels with NH2 . In water it has long been appreciated that the effects of largeamplitude motion manifest themselves even in the rotational energy levels of the zero point vibrational state, resulting in a large number of terms in the Watson-type Hamiltonian (19) which is often used to fit the experimental spectroscopic data. This is true also for NH2 , as can be seen from two recent rotational spectroscopic studies of its ground state. These comprise a highresolution Fourier transform far-infared study of NH2 , NHD, and ND2 by Morino and Kawaguchi (20), and a microwave terahertz spectroscopic study of NH2 by M¨uller and his colleagues (21). In the latter study 41 rotational, centrifugal distortion, and fine structure constants were needed to fit the experimental data. In TABLE 1 Equilibrium Internuclear Distances, Bond Angles, and Barriers to Linearity of the Electronic Ground States of H2 O and NH2 and the Equivalent Singlet State of CH2 Molecule

H2 Oa

NH2 b,c

CH2 d

Electronic state Barrier height to linearity r0 valence angle, α 0

X˜ 1 A1 11127 35 cm−1 0.957 84 104.508

X˜ 2 B1 11774 1.034 103.4

a˜ 1 A1 8666 1.099 104

a

Tarczay et al., Ref. (17). This work. c Jungen et al., Ref. (6). d Gu et al., Ref. (40). b

17

the higher vibrational states of water it has been found that the effects of centrifugal distortion become extremely large, and the Watson Hamiltonian approach is no longer appropriate. As a result the direct solution of the full vibration–rotation Hamiltonian, including the effects of large-amplitude motion on the potential energy surface, is now often the method of choice. Similar effects of large-amplitude motion may be seen in NH2 where, in his study of the electronic ground state of NH2 , Vervloet (9) showed that the centrifugal distortion effects on the rotation and spin– rotation structure become so extreme that the use of the Watson type Hamiltonian was abandoned for the highest, v2 = 7, state below the barrier to linearity. In the present study we wish to demonstrate that the approach to these calculations started by Barrow et al. (22) and by Jungen and Merer (8) allows the fitting of a large number of experimental energy levels using a restricted number of parameters. These parameters in turn are directly related to the potential energy surfaces of the two coupled electronic states and to the effects of the electronic angular momentum derived from the parent 2 u state of the linear nuclear configuration. 2. STRETCH-BENDER VIBRONIC AND ROVIBRONIC ENERGY LEVEL CALCULATIONS

As we have seen, the bent A˜ 2 A1 and X˜ 2 B1 states of NH2 , between which electronic transitions are observed, correlate with a 2 u state of the linear conformation and hence may be considered to be derived from a 2 u state split by strong electrostatic interaction. Rotation about the a-axis, which corresponds to the axis of the linear framework, leads to a dynamic coupling of the A˜ 2 A1 and X˜ 2 B1 states for all values of K (K a ) apart from zero. In addition the highly excited bending levels show evidence for the existence of strong stretch–bend coupling. The first three-dimensional variational calculation of the rovibronic levels of NH2 was carried out by Gabriel et al. (18) They calculated accurate ab initio potential energy surfaces for the A˜ 2 A1 and X˜ 2 B1 states of NH2 . Following this they interpolated between the points on the grid using expansion functions based upon the product of functions of the bond angle and Simons– Parr–Finlan (23) functions for the stretching motion. The resultant surfaces were used in their variational calculations. In their paper they presented experimental and calculated spin-split rotational levels in the (0, 4, 0) state of X˜ 2 B1 of NH2 . They also gave calculated rovibronic states for both the 2 A1 and 2 B1 states for N = 0 and 1 and K a = 0 and 1 up to 18 000 cm−1 . An alternative surface was calculated by Jensen et al. (24) for the X˜ 2 B1 state, and rovibrational states were calculated for N = 0 and 1 and K a = 0 and 1 up to 8500 cm−1 , neglecting Renner–Teller coupling. In the present work we have used the stretch-bender model of Duxbury et al. (25) which we used for CH2 (26), with the extensions necessary to calculate the effects of overall rotation and spin–rotation coupling which are described in the previous paper by Alijah and Duxbury (27). The analytic semirigid-bender

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ALIJAH AND DUXBURY

potentials for the 2 A1 and 2 B1 states of NH2 are derived from those of JHM (6) as described below. The bond stretching occurs relative to the semirigid-bender reference configuration, as was described previously (25–27). The essence of the stretch-bender model is that we can use, as an intermediate stage of the calculation, “primitive” eigenvalues and eigenvectors which have the following properties:

potential,

Renner–Teller coupling occurs only between vibronic states of A1 and B1 symmetry having the same values of v1 or v3 but with differing values of v2 . Anharmonic coupling occurs only between vibronic states belonging to one particular electronic state.

The ρ-dependent stretching frequencies in Eq. [3] are related to the ρ-dependent stretching force constants, frrs (ρ) and frra (ρ). In the present work we have followed the method of achieving a physically sensible angular dependence, used in our study of CH2 , by using the following variant of Badger’s relationship (28) for bond length and bond strength,

Thus at the “primitive level,” at which the large interaction matrix is constructed, the anharmonic coupling is diagonal in electronic state and off-diagonal in stretching vibrational basis states, whereas the Renner–Teller coupling is off-diagonal in electronic state but diagonal in the stretching basis states. As noted above, the starting point for our calculation of the vibronic energies of NH2 was the semirigid-bender potential used by Hallin et al. (6, 7) in their extensive treatment of the vibronic, spin, and rotational fine structure effects in NH2 . The potential functions for bending in the ground and excited states, V0b (ρ), have been based on those of a linear harmonic oscillator plus a Lorentzian perturbation, V1 (ρ), with additional small correction terms for potential shaping, V2 (ρ), and for residual stretch–bend interaction. These are as described by Barrow et al. (22) and Jungen et al. (6, 7). The bending potential may be written as V0b (ρ) = V1 (ρ) + V2 (ρ)

[1]

where V1 (ρ) =

H f (ρ 2 − m 2 )2 [ f m 4 + (8H − f m 2 )ρ 2 ]

and ρ is the bending angle, m is the equilibrium bending angle, H is the barrier height, and f = ∂ 2 V /∂ρ 2 is the harmonic oscillator force constant near equilibrium. We have chosen the form of the additional correction terms used by Jungen, et al. (6)

V2 (ρ) =

  ρ k4 ρ 2 (ρ 2 − m 2 )3 2 π + c sin 1 (ρ 2 + m 2 )2 m    4 2 ρ 3ρ + m 4 × cos π . 5 m ρ4 + m4

[2]

   1 1 1 + x 1 v1 + Vc (ρ) = ω1 (ρ) v1 + 2 2     1 1 + ω3 (ρ) v3 + 1 + x3 v3 + . 2 2 

y

y f rr (ρ) =

f rr , [r 0 (ρ)]3

[4]

where y = s or a. In the most general version of the stretch-bender, in which the asymmetric stretching motion is included, the values of frrs and frra are chosen so that ω1 (ρm ) and ω3 (ρm ) are equal to the value (cm−1 ) of the harmonic stretching frequencies at the supplement to the exterior bond angle, m. Although the form of the angle-dependent force constant defined in Eq. (4) has the correct limiting behavior, since hcω1 (ρm ) and hcω3 (ρm ) tend to zero as r 0 (ρ) tends to infinity, we have found it necessary to make an additional correction to the angle-dependent stretching frequencies calculated for very high values of ρ, as it tends to the limiting value of 180◦ . In the absence of any correction, the calculated stretching frequencies have a point of inflection, producing an unphysical correction to the effective stretchingdependent potential energy curves. We have therefore used linear extrapolation of the hcω1 (ρm ) and hcω3 (ρm ) functions, choosing this extrapolation to be the tangent to the potential energy curves at the point of inflection. In our study of the 2 B1 state we have considered only the excitation of the symmetric stretching vibration. We have therefore fixed the value of ω3 to that which fits the experimentally observed frequency, in order to obtain the correct zero-point vibrational energy shift and to enable the calculation of the vibrational dependence of the inertia defect. Since the reference bond length, r 0 (ρ), is chosen to try to follow the minimum of the potential energy surface for symmetrical displacements, it is convenient to express this as an analytic function of ρ. In calculating the derivatives of the potential with respect to the change in bond angle, we discovered that the coefficients a and b in the ground state function used by Jungen et al. for NH2 (6), namely r 0 (ρ) = r 0 + aρ 2 + b(tan(ρ/2))2 ,

The major part of the remainder of the effective bending potential, Veff (ρ), in the stretch-bender approach is associated with the stretching contributions, Vc (ρ), to the effective bending

[3]

[5]

resulted in a function which varied too slowly near linear geometry and too rapidly at large values of ρ. As a result we

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THE X˜ 2 B1 STATE OF NH2

and co-workers (25–27), this shows that the assumed change of the effective bond length in the A˜ 2 B1 state with angle is too small. When the linear molecule bond length is chosen to be the slightly shorter value given by Gabriel et al. (18), the calculated Fermi-resonance matrix elements (b) are now similar in size to those in the excited state of CH2 and similar to those derived by Dressler and Ramsay(3). As in the case of CH2 (25), at small values of v2 the slope of the graph in both states is almost linear when plotted against [(v2 + 1)(v2 + 2)]1/2 , as one would expect in the harmonic oscillator limit. The single parameter, Badger-type force constant function does not possess sufficient flexibility in shaping the potential to permit both the stretching and bending intervals to be fitted simultaneously. However, as in the case of CH2 (26), it was found that the effect of adding the angle-dependent vibrational frequency when creating the effective bending potential is equivalent to that of making a minor change to the effective bending force constant. An empirical correction, which depends on the degree of excitation of the stretching vibrations, was therefore made to the bending force constant, so that the effective vibrationally dependent force constant for the excitation of the stretching vibrations, v1 and v3 , up to two quanta of the stretching vibrations is:

600

2 B , (a) 1 2 A , (a) 1

500

2 B , (b) 1 2 A , (b) 1

400

Fermi resonance matrix element/cm-1

19

300

200

100

f eff (v1 , v3 ) = f + (δ11 − δ21 )v1 + δ21 v12 + (δ13 − δ23 )v3 + δ23 v32 . 0 0

5

10

15

[(v2+1)(v2+2)]1/2 FIG. 1. The calculated Fermi resonance matrix elements between levels 0, v2 , 0 and 1, v2 − 2, 0 of the A˜ and X˜ electronic states of NH2 plotted ˚ against the expected function for a harmonic oscillator basis: (a) rlin = 1.000 A; ˚ (b) rlin = 0.996 A.

have increased the value of coefficient a and decreased that of b. A further test of the assumed analytical function for bondlength variation is the calculation of the matrix elements of the Fermi resonance coupling between the 0, v2 , 0 and 1, v2 −2, 0 for the K = 0, vibronic levels, of X˜ 2 B1 and A˜ 2 A1 electronic states of NH2. This is plotted in Fig. 1 for two cases, (a) the linear ˚ used by JHM (6), and (b) the linear molecule bond length of 1 A ˚ used by Gabriel et al. (18). The molecule bond length of 0.996 A matrix elements of the lower state derived with either linear bond length are similar to those calculated for the equivalent state of ˜ 2 B1 state calCH2 (26). However, the matrix elements for the A ˚ are very small, and much smaller than culated using rlin = 1 A those deduced by Dressler and Ramsay (3) from their analysis of the Fermi resonance effects in the vibronic levels. Since the anharmonic coupling depends upon the bond length variation, as described by Duxbury and Jungen (29) and by Duxbury

[6]

This form allows flexibility of adjustments for v1 = 1 and 2. We have also added small correction terms δk41 and δk42 to the k4 anharmonic constant for v1 = 1 and 2. These corrections have been used for all the calculations presented in this paper and the following one. In order to determine the parameters of the stretch-bender potential we have used a nonlinear least squares fitting procedure to fit the vibronic origins in the X˜ 2 B1 state and the A˜ 2 A1 state having values of K from 0 to 6. These correspond to the vibronic states for which the most comprehensive sets of rotational levels are available. The iterative procedure used followed that described by Jungen et al. (6, 7). The determination of the bond length variation followed the route outlined earlier, and the supplements to the bond angles and barrier heights were derived using the observed transition frequencies and B value variations in both. Finally, the minor correction terms described earlier were allowed to vary in the final least squares adjustment process. It was found that different sets of least squares minimizations gave a similar quality of fit, but with somewhat different combinations of the minor parameters. The set chosen and given in Table 2 employed a smaller number of parameters in the excited electronic state and was used for the calculation of the rovibronic levels in both the X˜ 2 B1 states and in the region where the higher lying X˜ 2 B1 levels interact strongly with those of the A˜ 2 A1 state. The estimated errors of the minor parameters given in this table are derived from the changes in these

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ALIJAH AND DUXBURY

TABLE 2 Parameters Used to Model the Rotational and Fermi Resonance Structure of the X˜ 2 B1 and A2 A1 States of NH2 and the Renner–Teller Coupling between Them A˜ 2 A1 state Bond length variation r (ρ = 0) a, coefficient of ρ 2 b, coefficient of tan2 ρ f0 m H k4 c1 frrs x1 ω3 x3 δ11 δ21 δk41 X˜ 2 B1 state Bond length variation r (ρ = 0) a, coefficient of ρ 2 b, coefficient of tan2 ρ f0 m H k4 c1 frrs x1 ω3 x3 δ11 δ21 δk41 δk42 Spin-orbit coupling parameter, ASO Variation of L z , g K

0.996 A˚ 0.000 A˚ 0.1063 A˚ 13249.59 ± 160 cm−1 rad−2 35.814 deg 729.9 cm−1 −0.35 ± 50 cm−1 rad−4 −53.01 ± 5 cm−1 8.9093 aJ A˚ 2 −0.0025 3586.0 cm−1 −0.0033 17.07 cm−1 0.00 cm−1 131.94 ± 130 cm−1 rad−4

0.996 A˚ 0.0146 A˚ 0.0185 A˚ 33596.53 ± 130 cm−1 rad−2 77.6 deg 11773.67 cm−1 374.1 ± 40 cm−1 rad−4 154.22 ± 40 cm−1 8.3955 aJ A˚ 2 −0.01440 3301.1 cm−1 −0.0093 505.55 ± 100 cm−1 −781.93 ± 250 cm−1 765.98 ± 400 cm−1 rad−4 −793.14 ± 300 cm−1 rad−4 61.6 cm−1 6.0 cm−1

Note. The errors on the parameters varied in the final least squares adjustment were derived by the procedure outlined in the text.

parameters which occurred when the fitting procedure was repeated using slightly different starting conditions. 3. VIBRATIONAL LEVELS OF THE X˜ 2 B1 STATE WHICH LIE BELOW THE ONSET OF THE A˜ 2 A1 STATE

(i) Vibrational Resonance Effects In Table 3 the band origins of the K = 0 vibronic levels of the X˜ 2 B1 state of NH2 calculated using the stretchbender model (STRB) are given. It can be seen that they are in good agreement both with the experimentally determined values (9, 24) and with the values calculated using the instantaneous axis method by Gabriel et al. (18). The good agreement between the STRB calculations and the previous calculated and

experimentally determined values shows that the size of the anharmonic coupling matrix elements is correct for energy levels corresponding to comparatively low excitation of the bending vibration (up to v2

= 8). This is important since in these rotationless vibronic levels, with K = 0, the Renner–Teller coupling is absent. The newer experimental data (9, 12) on the high-lying vibronic states of the X˜ 2 B1 electronic state, which lie below the barrier to linearity, are of great importance in quantifying several parts of the model for the vibronic interaction. In this region only nonresonant coupling between levels of the A˜ 2 A1 and X˜ 2 B1 states can occur. This means that there are no resonant interstate interference contributions to L z , the only resonant interactions are those caused by Fermi-resonance coupling between the ground state levels. As a result the vibronic levels and values of the vibronic spin-orbit constants, ASO v,K , are an excellent diagnostic both for the anharmonic coupling within the X˜ state and for the quenching of the electronic angular momentum as a function of the bending angle. The subtle variation in the size of the predicted and observed values of ASO v,K is described in detail in the following sections. (ii) Quenching of Orbital Angular Momentum In the original studies (7, 30) of the effects of orbital angular momentum in NH2 , the approximation was made either that L z =  = 1 or that its variation was small. At the time this TABLE 3 Observed and Calculated Band Origins of the X˜ 2 B1 State of NH2 v1 , v2 , v3 0, 0, 0 0, 1, 0 0, 2, 0 1, 0, 0 0, 3, 0 1, 1, 0 0, 4, 0 1, 2, 0 2, 0, 0 0, 5, 0 1, 3, 0 2, 1, 0 0, 6, 0 1, 4, 0 2, 2, 0 0, 7, 0 1, 5, 0 2, 3, 0 0, 8, 0

obsa 0 1497.32 2961.24 3219.37 4391.35 5785.55 6151.95 6335.15 7140.35 7564.63 7804.54 8451.45 8942.59 9227.14 9716.9 10286.3 10609 10948

calcb

calcc

0 1497.051 2961.538 3220.285 4392.275 4701.422 5786.492 6150.733 6337.594 7139.817 7565.588 7792.891 8446.903 8942.486 9221.739 9705.793 10276.84 10620.682 10931.124

0 1498.7 2962.6 3215.9 4391.8 4706.06 5785.3 6154 6328.62 7140.2 7566.6 7808.3 8451.9 8943.4 9223.2 9717.1 10282.5 10619.82 10946.2

o-cb

o-cc

0 0.269 −0.298 −0.915 −0.925

0 −1.38 −1.36 3.47 −0.45

−0.942 1.217 −2.444 0.533 −0.958 11.649 4.547 0.104 5.401 11.107 9.46 −11.682 16.876

0.25 −2.05 6.53 0.15 −1.97 −3.76 −0.45 −0.81 3.94 −0.2 3.8 −10.82 1.8

Note. All parameters are given in cm−1 . Vervloet, Ref. (9); Burkholder et al., Ref. (33); McKellar et al., Ref. (34); and Merienne-Lafore and Vervloet, Ref. (35). b This work, Table 2. c Gabriel et al., Ref. (18). a

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THE X˜ 2 B1 STATE OF NH2

21

FIG. 2. The spin–orbit splittings as a function of v1 and v2 for the vibronic levels of the X˜ 2 B1 state of NH2 , with K = 1, up to the barrier to linearity: (a) no Fermi resonance, g K = 0, ASO = 50 cm−1 (if g K is set to 1.2, the change is not perceptible at the resolution of this graph); (b) Fermi resonance included, ASO = 61.6 cm−1 , g K = 6.

was a valid approximation, since there was little information about the size of the spin–orbit coupling at high-lying vibronic levels of the X˜ 2 B1 ground electronic state. However, as a result of the recent investigations of the high-lying levels of this state summarized above, it has become possible to make a test of the possible quenching of the orbital angular momentum as a function of the bending angle. This was discussed in paper I (27). In making a comparison between our calculations and the ex-

perimental values of the effective vibronic spin–orbit coupling parameters parameters we have chosen to fix the value of the atomic spin–orbit coupling constant, ASO , and to allow the electronic angular momentum to depend upon the bending angle. Hence the effective vibronic spin–orbit coupling parameter may be written as

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ASO v,K = ASO L z v,K .

[7]

22

ALIJAH AND DUXBURY

TABLE 4 Mean Term Values and Doublet Splittings (cm−1 ) for the Angular Momentum Levels, with N = K a , of the X˜ 2 B1 Electronic State of NH2 Term value

Doublet splitting

v1 , v 2 , v 3

Ka

obsa

0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0 0, 0, 0

0 1 2 3 4 5 6 7 8 9 10 11 12

0 34.28 116.29 244.36 417.91 635.61 895.95 1197.2 1537.6 1915.39 2328.79 2776.08 3255.55

0.00 34.11 115.72 243.12 415.85 632.69 892.21 1192.84 1532.94 1910.86 2324.95 2773.63 3255.37

0 34.3 116.3 245.3

0.00 0.17 0.57 1.24 2.06 2.92 3.74 4.36 4.66 4.54 3.84 2.45 0.18

−0.25 −0.53 −0.8 −1.07 −1.33 −1.58 −1.8 −1.99 −2.18 −2.33 −2.46 −2.58

−0.27 −0.52 −0.77 −1.01 −1.23 −1.44 −1.62 −1.78 −1.93 −2.06 −2.18 −2.28

0.2 0.34

0.02 0.00 −0.02 −0.06 −0.10 −0.14 −0.18 −0.21 −0.25 −0.27 −0.28 −0.30

0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0 0, 1, 0

0 1 2 3 4 5 6 7 8 9 10 11 12

1497.32 1533.85 1622.5 1761.22 1948.75 2183.2 2462.1 2783.61 3144.94 3544 3978.64 4446.91 4946.77

1496.75 1533.18 1621.44 1759.45 1946.20 2179.79 2458.13 2779.03 3140.31 3539.88 3975.74 4446.04 4949.05

1498.7 1535 1623.3 1761.2

0.57 0.67 1.06 1.77 2.55 3.41 3.97 4.58 4.63 4.12 2.90 0.87 −2.28

−0.28 −0.67 −1.01 −1.4 −1.68 −1.94 −2.16 −2.39 −2.55 −2.72 −3.28 −2.93

−0.36 −0.70 −1.03 −1.33 −1.60 −1.83 −2.04 −2.22 −2.37 −2.50 −2.61 −2.70

0.25 0.55

0.08 0.04 0.02 −0.07 −0.08 −0.11 −0.12 −0.17 −0.18 −0.22 −0.67 −0.23

0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0 0, 2, 0

0 1 2 3 4 5 6 7 8 9 10 11 12

2961.26 3000.64 3097.43 3248.86 3452.63 3705.3 4002.57 4337.34

2961.17 3000.57 3097.34 3248.75 3452.79 3706.49 4006.60 4349.20 4751.42 5166.32 5624.64 6117.93 6643.57

2962.6 3001.6 3097 3248.3

0.09 0.07 0.09 0.11 −0.16 −1.19 −4.03 −11.85

−0.42 −0.89 −1.35 −1.76 −2.1 −2.35 −2.37

0.32 0.7

0.08 0.09 0.05 0.02 −0.01 0.00 0.14

8.33 3.86 −0.81 −6.30

−2.96 −3.13 −3.28 −3.37

−0.50 −0.97 −1.40 −1.78 −2.09 −2.35 −2.51 −2.31 −2.84 −3.00 −3.10 −3.18

0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0 0, 3, 0

0 1 2 3 4 5 6 7 8 9 10

4391.98 4435.30 4543.20 4711.75 4937.27 5215.15 5540.46 5907.01 6352.98 6791.05 7273.43

4391.8 4434.1 4539.1 4704.3

−0.61 −0.84 −1.45 −2.57 −4.56 −8.10 −14.72 −28.61

−0.58 −1.23 −1.83 −2.34 −2.69 −2.86 −2.79

−0.74 −1.40 −1.97 −2.43 −2.78 −3.02 −3.09 −2.91 −3.42 −3.60

0.44 0.95

5174.65 5628.5 6117.12 6637.26 4391.37 4434.46 4541.75 4709.18 4932.71 5207.05 5525.74 5878.4

calcb

calcc

obs-calcb

obsa

calcb

a

Dressler and Ramsay, Ref. (3), Vervloet, Ref. (9), Ross et al., Ref. (10), Dixon et al., Ref. (12) This work c Gabriel et al., Ref. (18). b

 C 2002 Elsevier Science

calcc

obs-calcb

−0.12 −0.13 −0.18 −0.19 0.16 0.18 0.14 0.09 0.09 0.16 0.30

THE X˜ 2 B1 STATE OF NH2

23

TABLE 5 Comparison of Observed and Calculated Spin–Split Rotational Levels in the (0, 4, 0) State of the X˜ 2 B1 State of NH2 F1 /cm−1

F2 /cm−1

N Ka Kc

Obsa

Calcb

Calcc

Obsa

Calcb

Calcc

000

5785.57

5786.49

5785.31

101 111 110

5806.68 5830.31 5836.13

5808.06 5831.63 5837.76

5806.49 5828.91 5834.86

5806.73 5831.2 5837.05

5808.06 5832.49 5838.63

5806.49 5829.56 5835.52

202 212 211 221 220

5847.95 5866.83 5884.31 5953.46 5954.4

5850.12 5868.77 5887.18 5956.46 5957.51

5847.78 5865.4 5883.22 5948.95 5949.98

5848.05 5867.34 5884.91 5955.34 5956.26

5850.14 5869.26 5887.67 5958.24 5959.27

5847.77 5865.77 5883.59 5950.31 5956.26

303 313 312 322 321 331 330

5907.59 5920.94 5955.71 6017.13 6021.66 6141.58 6141.66

5910.77 5923.74 5960.44 6021.65 6026.69 6147.24 6147.36

5907.3 5919.49 5954.91 6012.7 6017.6 6132.28 6132.39

5907.68 5921.35 5956.21 6018.48 6023.01 6144.16 6144.29

5910.83 5924.09 5960.8 6022.92 6027.9 6149.75 6149.87

5907.26 5919.7 5955.13 6013.68 6018.54 6134.22 6134.34

a

Vervloet, Ref. (9). This work. c Gabriel et al., Ref. (18). b

The calculated value of L z v,K takes into account the quenching of electronic orbital angular momentum, since L z v,K = v, K |(ρ)|v, K ,

[8]

(ρ) =  − u(ρ)

[9]

where

and where u(ρ) is approximated as u(ρ) = g K /A(ρ).

[10]

Although u(ρ) is isotopically independent the value of g K will be isotopically dependent, since the value of A(ρ) for ND2 will be smaller than that for NH2 by a factor of ∼0.56. When we analyze the behavior of the experimental and calcu−1 lated values of ASO v,K we find that with a value of ASO = 50 cm , and the value of g K = 1.2 cm−1 which was used originally (6, 7), the value of ASO v,K for the levels just below the barrier to linearity is underestimated by about 20%; see Fig. 2a. However, if a value of ASO = 61.6 cm−1 is used, together with a value of the quenching parameter of g K = 6 cm−1 , good agreement can be achieved both for the 0, 0, 0 vibrational level and for the 0, 7, 0 and 0, 8, 0 levels, as is shown in Fig. 2b. The remaining deviation is probably associated with the analytic form chosen to represent

the angular momentum quenching, since this is only the leading term in the expansion first proposed by Jungen and Merer (8), Eq. (8) above. The value of ASO we have now derived is much closer than the previous one, derived by Jungen et al. (7), to the value of 70 cm−1 obtained by equating ASO with |2ASO | of the A3  state of NH. Similar conclusions were drawn by Alijah and Duxbury in their study of the analogous electronic states of PH2 (31), and by Jungen et al. in their study of H2 O+ (6, 7). In fact the value of g K which we have derived is very close to that found for H2 O+ , g K = 5.7 cm−1 , if we reverse the signs of g K given in Ref. (7), since contamination with higher sigma states of the linear molecule should lead to a decrease in the matrix element of L z as described by Jungen and Merer on pp. 16 and 17 of Ref. (8). (iii) Rovibronic and Spin–Rovibronic Energy Levels The second of the two original papers of Jungen et al. (7) described the interpretation of the rotational structure of the spectrum of NH2 , including the asymmetry effects, the spin uncoupling, and various resonances. Subsequently there have been two sets of calculations of rotation–vibration term values in the 2 B1, state. In the first of these, by Jensen et al. (24), an ab initio calculation was made of the rotation–vibrational energies in the electronic ground state using the MORBID approach. First of all an ab initio three-dimensional potential energy surface was calculated, and then the energies were derived using the MORBID method for a closed shell molecule. This neglected the coupling to the 2 A1 state and restricted the calculation to term values with

 C 2002 Elsevier Science

24

ALIJAH AND DUXBURY

TABLE 6 Observed and Calculated Rotational Term Values (cm−1 ) for Most of the (0, v 2 , 0) and the (1, v 2 , 0) Vibronic States of the X˜ 2 B1 State of NH2 . That Lie below and up to the Barrier to Linearity 0, 0, 0

000

obsa

obsa

F1

F2

calc F1

0

0, 1, 0

calc F2

(o-c)av.

(o-c)df.

obsa

obsa

(b)

(c)

F1

F2

calc F1

0

1497.32

calc F2

(o-c)av. (b)

(o-c)df. (c)

1496.75

101 111 110

21.11 31.77 36.54

21.14 32.01 36.80

21.19 31.57 36.43

21.19 31.77 36.63

−0.07 0.22 0.14

−0.03 −0.03 −0.06

1518.45 1531.20 1536.25

1518.48 1531.50 1536.25

1518.09 1530.46 1535.63

1518.09 1530.71 1535.90

0.37 0.77 0.48

−0.03 −0.03 0.27

202 212 211 221 220

62.04 69.26 83.54 115.40 116.67

62.10 69.41 83.74 115.93 117.20

62.23 69.13 83.71 114.83 116.18

62.24 69.24 83.83 115.27 116.61

−0.16 0.15 −0.13 0.61 0.54

−0.05 −0.04 −0.09 −0.09 −0.10

1559.46 1568.44 1583.59 1621.59 1622.81

1559.52 1568.63 1583.84 1622.27 1623.49

1559.44 1567.99 1583.55 1620.51 1621.82

1559.45 1568.14 1583.70 1621.09 1622.39

0.05 0.48 0.09 1.13 1.05

−0.05 −0.04 −0.10 −0.10 −0.10

303 313 312 322 321 331 330

120.74 124.73 153.00 178.79 184.61 243.85 244.06

120.82 124.85 153.20 179.20 185.02 244.65 244.87

120.97 124.68 153.59 178.56 184.71 242.66 242.91

121.00 124.76 153.68 178.87 184.99 243.34 243.58

−0.20 0.07 −0.53 0.29 −0.03 1.25 1.22

−0.05 −0.04 −0.12 −0.10 −0.13 −0.12 −0.14

1618.30 1623.56 1653.59 1685.07 1690.75 1760.61 1760.80

1618.39 1623.71 1653.83 1685.59 1691.27 1761.65 1761.84

1618.64 1623.50 1654.39 1684.70 1690.81 1758.91 1759.13

1618.67 1623.61 1654.50 1685.12 1691.19 1759.81 1760.03

−0.31 0.08 −0.74 0.42 0.02 1.77 1.74

−0.06 −0.04 −0.13 −0.11 −0.14 −0.14 −0.14

404 414 413 423 422 432 431 441 440

195.69 197.56 243.46 262.17 277.23 330.40 331.87 417.35 417.38

195.78 197.67 243.67 262.52 277.59 331.07 332.54 418.43 418.46

195.90 197.58 244.55 262.36 278.24 329.93 331.57 415.38 415.42

195.94 197.65 244.63 262.60 278.44 330.44 332.07 416.29 416.33

−0.19 0.00 −1.02 −0.14 −0.93 0.55 0.39 2.05 2.04

−0.05 −0.04 −0.12 −0.11 −0.17 −0.15 −0.16 −0.17 −0.17

1693.28 1695.92 1744.87 1768.58 1783.53 1847.28 1848.57 1948.09 1948.12

1693.39 1696.06 1745.12 1769.03 1783.98 1848.13 1849.42 1949.46 1949.48

1693.98 1696.34 1746.72 1769.15 1785.00 1846.86 1848.34 1945.61 1945.642

1694.04 1696.43 1746.82 1769.48 1785.41 1847.55 1849.01 1946.80 1946.83

−0.68 −0.39 −1.77 −0.51 −1.45 0.50 0.32 2.57 2.56

−0.05 −0.05 −0.14 −0.12 −0.03 −0.16 −0.17 −0.18 −0.18

505 515 514 524 523 533 532 542 541 551 550

286.46 287.27 352.79 364.67 393.80 438.37 443.66 525.91 526.19 634.94 634.95

286.57 287.38 353.04 365.00 394.15 438.96 444.24 526.84 527.11 636.28 636.28

286.65 287.33 354.35 365.38 396.02 438.86 444.74 525.14 525.47 632.13 632.14

286.70 287.40 354.46 365.59 396.18 439.28 445.12 525.87 526.20 633.26 633.26

−0.16 −0.04 −1.49 −0.65 −2.12 −0.40 −0.98 −0.87 0.82 2.91 2.91

−0.05 −0.04 −0.14 −0.12 −0.20 −0.17 −0.19 −0.19 −0.19 −0.21 −0.21

1783.81 1785.02 1855.38 1871.30 1900.65 1955.43 1960.16 2056.78 2056.99

1783.93 1785.15 1855.66 1871.71 1901.07 1956.17

−0.05 −0.05 −0.15 −0.13 −0.21 −0.18

0.63 0.58

−0.20 −0.22

2181.24

1785.00 1786.05 1858.47 1873.28 1904.50 1957.24 1962.61 2057.21 2057.48 2180.54 2180.54

−1.10 −0.92 −2.89 −1.63 −3.53 −1.16

2179.82

1784.93 1785.96 1858.35 1873.00 1904.29 1956.68 1962.09 2056.25 2056.52 2179.09 2179.09

0.71

0.03

606 616 615 625 624 634 633 643 642 652 651 661 660

393.18 393.51 478.82 485.42 532.66 567.13 580.51 656.39 657.68 765.25 765.30 895.17 895.17

393.29 393.62 479.10 485.74 533.03 567.67 581.04 657.21 658.50 766.42 766.47 896.73 896.73

393.34 393.60 480.80 486.73 536.34 568.84 583.65 657.25 658.79 764.29 764.35 891.56 891.56

393.41 393.66 480.93 486.92 536.48 569.20 583.95 657.86 659.39 765.23 765.29 892.88 892.88

−0.14 −0.07 −1.90 −1.24 −3.56 −1.62 −3.02 −0.76 −1.00 1.08 1.06 3.73 3.73

−0.05 −0.05 −0.14 −0.13 −0.22 −0.18 −0.23 −0.21 −0.22 −0.23 −0.23 −0.23 −0.23

2057.94 2058.18

Note. (o-c)av. = 1/2(δ F1 + δ F2 ); (o-c)df. = 1/2(δ F1 − δ F2 ). Dressler and Ramsay, Ref. (3); Vervloet, Ref. (9); Ross et al., Ref. (10); Dixon et al., Ref. (12).

a

 C 2002 Elsevier Science

THE X˜ 2 B1 STATE OF NH2

25

TABLE 6—Continued 0, 0, 0 obsa

obsa

F1 707 717 716 726 725 734 744 743 753 752 762 761 771 770 808 818 817 827 826 836 835 845 844 854 853 863 862 872 871 881 880

0, 1, 0

F2

calc F1

calc F2

(o-c)av. (b)

(c)

515.98 516.10 620.34 623.63 691.62 742.36 808.56 812.77 917.44 917.697 1047.24 1047.24 1196.35 1196.35

516.09 516.22 620.63 623.95 692.01 742.86 809.32 813.51 918.50 918.75 1048.64 1048.64 1198.11 1198.11

516.10 516.25 622.90 625.60 719.01 748.22 811.56 816.58 918.94 919.27 1046.36 1046.37 1192.09 1192.09

516.16 516.18 622.74 625.78 719.32 748.45 812.08 817.07 19.75 920.08 1047.50 1047.51 1193.60 1193.60

−0.10 −0.05 −2.33 −1.90 −3.10 −5.72 −2.88 −3.69 −1.38 −1.45 1.00 1.00 4.38 4.38

−0.05 −0.18 −0.45 −0.13 −0.20 −0.27 −0.23 −0.25 −0.25 −0.25 −0.25 −0.25 −0.25 −0.25

654.89 654.94 777.16 778.70 868.21 883.45 927.79 981.93 992.56 1091.49 1092.49 1220.98 1221.03 1370.26 1370.26 1536.73

655.00 655.05

654.94 654.96 777.47 781.38 874.96 883.95 928.29 987.58 1000.25 1092.46 1093.46 1223.63 1223.69

655.00 655.02 780.276 781.57 875.15 888.66 937.10 988.04 1000.64 1096.89 1098.19 1224.63 1224.70 1371.24 1371.24 1533.79 1533.79

−0.03 0.01 −1.40 −2.62 −6.64 −2.35 −4.40 −5.53 −7.54 −2.21 −2.36 −2.51 −2.53 0.31 0.31 4.74

−0.05 −0.03

779.02 868.63

982.64 993.25

1222.26 1222.31 1371.86 1371.86 1538.65

1532.11 1532.11

(o-c)df.

obsa

obsa

F1

F2

obs F2

calc F1

calc F2

(o-c)av. (b)

(o-c)df. (c)

calc F2

(o-c)av.

(o-c)df.

−0.14 −0.23 −0.25 −0.30 −0.28 −0.28 −0.62 −0.62 −0.24

0, 2, 0 obs F1

calc F1

1, 0, 0 calc F2

(o-c)av.

(o-c)df.

2961.17

obs F1

obs F2

3219.37

calc F1

000

2961.24

101 111 110

2982.37 2997.75 3003.08

2982.41 2998.16 3003.52

2982.64 2997.65 3003.14

2982.64 2998.02 3003.51

−0.25 0.12 −0.02

−0.04 −0.03 −0.07

3240.14 3250.43 3255.17

3240.18 3250.65 3255.42

3238.59 3248.67 3253.61

3238.59 3248.87 3253.80

1.57 1.77 1.59

−0.03 −0.03 −0.06

202 212 211 221 220

3023.46 3034.75 3050.72 3096.39 3097.55

3023.53 3035.00 3051.03 3097.29 3098.45

3024.32 3035.17 3051.63 3096.34 3097.59

3024.33 3035.38 3051.85 3097.15 3098.39

−0.83 −0.40 −0.86 0.10 0.01

−0.05 −0.04 −0.10 −0.09 −0.10

3280.37 3287.25 3301.46 3332.20 3333.48

3280.43 3287.40 3301.66 3332.71 3333.99

3279.69 3286.30 3301.10 3331.35 3332.76

3279.70 3286.41 3301.21 3331.78 3333.18

0.71 0.97 0.40 0.89 0.77

−0.05 −0.03 −0.09 −0.08 −0.09

303 313 312 322 321 331 330 404 414

3082.49 3089.51 3121.22 3159.94 3165.36 3248.09 3248.25 3157.63 3161.42

3082.58 3089.70 3121.50 3160.61 3166.03 3249.45 3249.61 3157.74 3161.59

3084.07 3090.65 3123.41 3161.03 3166.86 3248.08 3248.27 3160.03 3163.48

3084.11 3090.81 3123.56 3161.57 3167.39 3249.30 3249.49 3160.07 3163.61

−1.56 −1.13 −2.13 −1.03 −1.43 0.08 0.05 −2.37 −2.04

−0.06 −0.04 −0.13 −0.12 −0.13 −0.13 −0.13 −0.07 −0.05

3337.99 3341.72 3369.84 3394.54 3400.40 3457.72 3457.87 3411.49 3413.21

3338.07 3341.84 3370.04 3394.94 3400.79 3458.50 3458.64 3411.59 3413.32

3338.43 3341.90 3371.24 3395.28 3401.71 3458.17 3458.44 3413.29 3414.83

3338.46 3341.98 3371.32 3395.58 3401.98 3458.82 3459.09 3413.34 3414.90

−0.41 −0.16 −1.34 −0.69 −1.25 −0.39 −0.51 −1.78 −1.60

−0.05 −0.04 −0.11 −0.09 −0.12 −0.13 −0.13 −0.05 −0.04

 C 2002 Elsevier Science

3217.33

26

ALIJAH AND DUXBURY

TABLE 6—Continued 0, 2, 0

1, 0, 0

obs F1

obs F2

calc F1

calc F2

(o-c)av.

(o-c)df.

obs F1

obs F2

calc F1

calc F2

(o-c)av.

(o-c)df.

413 423 422 432 431 441 440

3213.28 3243.55 3258.08 3334.78 3335.88 3451.72 3451.74

3213.57 3244.12 3258.63 3335.87 3336.96 3453.48 3453.51

3217.11 3246.08 3261.73 3336.63 3337.89 3452.02 3452.04

3217.26 3246.54 3262.11 3337.57 3338.82 3453.61 3453.63

−3.76 −2.48 −3.56 −1.77 −1.93 −0.21 −0.21

−0.15 −0.12 −0.17 −0.16 −0.16 −0.17 −0.17

3458.82 3476.53 3491.12 3542.86 3543.98 3629.29 3629.06

3459.04 3476.87 3491.44 3543.50 3544.61 3630.26 3630.05

3462.41 3479.30 3495.78 3545.81 3547.58 3629.49 3629.54

3462.49 3479.54 3495.97 3546.31 3548.08 3630.39 3630.43

−3.52 −2.72 −4.60 −2.88 −3.54 −0.17 −0.43

−0.13 −0.11 −0.12 −0.14 −0.14 −0.08 −0.10

505 515 524 523 533 532 542 541 551 550

3248.09 3249.95 3346.46 3375.47 3442.98 3447.05 3560.35 3560.56 3704.25 3704.25

3248.23 3250.11 3346.97 3375.98 3443.92 3447.97 3561.83 3562.04 3706.34 3706.35

3251.44 3253.07 3350.79 3381.95 3447.24 3451.96 3563.42 3563.64 3705.57 3705.57

3251.54 3253.19 3351.18 3382.25 3448.01 3452.67 3564.713 3564.93 3707.48 3707.48

−3.33 −3.10 −4.27 −6.38 −4.18 −4.79 −2.98 −2.99 −1.23 −1.23

−0.05 −0.05 −0.13 −0.21 −0.17 −0.20 −0.19 −0.19 −0.19 −0.19

3500.52 3501.25 3577.32 3606.82 3649.11 3652.74 3738.93 3737.23 3842.03 3842.03

3500.63 3501.36 3577.65 3607.16 3649.68 3653.241 3739.65 3738.05 3843.34 3843.34

3503.97 3504.58 3582.52 3614.12 3655.16 3661.48 3739.76 3740.13 3844.55 3844.55

3504.03 3504.65 3582.73 3614.26 3655.57 3661.86 3740.48 3740.85 3845.66 3845.67

−3.43 −3.31 −5.14 −7.21 −5.97 −8.68 −0.83 −2.85 −2.42 −2.42

−0.05 −0.04 −0.12 −0.19 −0.16 −0.13 0.00 −0.11 −0.20 −0.20

calc F2

(o-c)av.

(o-c)df.

−0.08

0, 4, 0 obs F1

obs F2

calc F1

1, 2, 0 calc F2

(o-c)av.

(o-c)df.

obs F1

obs F2

000

5785.57

5786.352

101 111 110

5806.68 5830.31 5836.13

5806.73 5831.20 5837.05

5808.03 5831.61 5837.63

5808.03 5832.47 5838.5

−1.32 −1.29 −1.48

−0.05 −0.02 −0.05

202 212 211 221 220

5847.95 5866.83 5884.31 5953.46 5954.40

5848.05 5867.34 5884.91 5955.34 5956.26

5850.35 5869.08 5887.15 5956.48 5957.489

5850.36 5869.57 5887.65 5958.26 5959.254

−2.36 −2.24 −2.79 −2.97 −3.04

−0.08 −0.02 −0.10 −0.09 −0.09

6212.50

6212.59

6283.08 6284.23

6285.11

303 313 312 322 321 331 330

5907.59 5920.94 5955.71 6017.13 6021.66 6141.58 6141.66

5907.68 5921.35 5956.21 6018.48 6023.01 6144.16 6144.29

5911.45 5924.57 5960.60 6022.01 6026.87 6147.35 6147.46

5911.51 5924.92 5960.96 6023.27 6028.08 6149.86 6149.97

−3.84 −3.60 −4.82 −4.83 −5.13 −5.74 −5.74

−0.05 −0.06 −0.14 −0.10 −0.14 −0.06 −0.12

6270.42 6276.98 6308.59 6345.58

6270.47

6431.37

6432.64

404 414 413 423 422 432 431 441 440 000

5983.65 5992.09 6049.33 6101.02 6113.61 6228.16 6228.88 6389.27 6389.30 7140.35

5983.80 5992.42 6049.83 6102.10 6114.74 6230.26 6230.93 6392.41 6392.44

5989.33 5997.51 6056.97 6108.37 6121.88 6236.91 6237.68 6399.41 6399.43 7139.88

5989.43 5997.79 6057.28 6109.37 6122.77 6238.86 6239.62 6402.46 6402.48

−5.66 −5.40 −7.55 −7.30 −8.15 −8.68 −8.75 −10.09 −10.08

−0.06 −0.04 −0.19 −0.09 −0.24 −0.15 −0.11 −0.09 −0.08

6344.04

6344.13

6629.63 7564.63

6631.34

101 111 110

7161.44 7191.41 7197.44

7161.49 7192.90 7198.96

7161.61 7191.96 7198.209

7161.61 7193.45 7199.70

−0.15 −0.55 −0.74

−0.04 0.01 −0.02

7585.39 7603.46

202 212 211

7202.81 7227.75 7245.86

7202.89 7228.62 7246.81

7204.21 7229.45 7248.16

7204.23 7230.30 7249.02

−1.37 −1.69 −2.25

−0.06 −0.03 −0.08

7625.81 7639.48 7656.23

calc F1 6147.588

6186.94

 C 2002 Elsevier Science

6346.23

7604.03 7624.90 7656.64

6169.17 6183.66 6189.27

6169.17 6184.02 6189.63

6210.98 6221.28 6238.10 6281.28 6282.62

6211.00 6221.48 6238.31 6282.07 6283.39

1.56

1.67

−0.10

6270.78 6276.88 6310.32 6346.23 6352.50 6431.20 6431.41

6270.83 6277.03 6310.47 6346.83 6353.02 6432.38 6432.59

−0.36

−0.01

−0.62

−0.04

0.22

−0.09

6346.69 6349.79 6404.46 6431.75 6448.25 6520.29 6521.72 6632.60 6632.64 7562.25

6346.81 6349.92 6404.60 6432.20 6448.61 6521.18 6522.62 6634.15 6634.18

−2.66

0.03

−2.92

−0.17

7583.98 7601.77 7607.70

7583.98 7602.29 7608.22

7626.18 7639.40 7657.19

7626.20 7639.69 7657.49

1.71 −0.04 −0.84 0.93 −0.90

THE X˜ 2 B1 STATE OF NH2

27

TABLE 6—Continued 0,5,0

1,3,0

obs F1

calc F2

calc F1

calc F2

(o-c)av.

(o-c)df.

obs F1

obs F2

calc F1

calc F2

(o-c)av.

221 220

7333.33 7334.14

7336.31 7337.04

7337.10 7337.952

7339.98 7340.818

−3.72 −3.79

−0.10 −0.03

7709.53 7710.42

7711.53

7709.88 7711.13

7710.99 7712.23

−0.71

303 313 312 322 321 331 330

7262.83 7281.61 7317.64 7397.11 7401.01 7548.24 7548.31

7262.95 7282.27 7318.41 7399.22 7403.13 7552.04 7552.11

7266.04 7284.96 7322.32 7403.07 7407.22 7557.02 7557.10

7266.10 7285.58 7322.95 7405.12 7409.22 7560.82 7560.90

−3.18 −3.33 −4.61 −5.93 −6.15 −8.78 −8.79

−0.06 −0.05 −0.14 −0.06 −0.12 0.01 0.00

7683.92 7692.81 7726.07 7772.02 7777.64 7872.48

7772.93 7778.27 7874.25

7686.67 7695.03 7730.44 7775.43 7781.32 7875.55 7875.73

7686.72 7695.25 7730.66 7776.21 7782.06 7877.20 7877.37

404 414 413 423 422 432 431 441 440

7339.67 7352.49 7411.99 7481.18 7492.21 7634.80 7635.30 7826.20 7826.26

7339.85 7353.03 7412.71 7482.85 7493.84 7637.81 7638.31

7345.16 7358.03 7419.88 7490.08 7501.85 7647.04 7647.58 7842.36 7842.41

7345.28 7358.52 7420.41 7491.69 7503.35 7650.01 7650.54 7846.68 7846.74

−5.46 −5.52 −7.80 −8.87 −9.57 −12.22 −12.25

−0.06 −0.05 −0.19 −0.06 −0.13 −0.04 −0.05

7757.77

7757.90

7817.39 7854.41 7868.57

7817.76 7855.14 7869.29

−16.16

0.02

8089.70

7763.41 7768.02 7826.15 7861.75 7877.62 7965.33 7966.55 8096.52 8096.56

7763.49 7768.20 7826.35 7862.38 7878.16 7966.59 7967.80 8098.59 8098.64

7830.57

7684.00 7693.08

7959.72

0, 6, 0 obs F1

obs F2

calc F1

−3.35 −3.74 −3.01 −5.61 −8.67 −7.30 −8.96

−0.11 0.00 −0.03 −0.06 −0.12 0.11 −0.12 −0.05 −0.17 −0.10 −0.18

1, 4, 0 calc F2

(o-c)av.

(o-c)df.

000

8451.42

101 111 110

8472.47 8508.87 8514.99

8472.56 8511.75 8517.91

8468.97 8506.35 8512.71

8447.20 8468.97 8509.32 8515.72

3.54 2.47 2.24

−0.09 0.09 0.09

202 212 211 221 220

8513.88 8545.15 8563.65 8676.28 8676.96

8513.96 8546.82 8565.49 8681.21 8681.87

8511.78 8543.93 8563.05 8679.32 8680.02

8511.80 8545.67 8564.84 8684.34 8685.02

2.13 1.18 0.63 −3.08 −3.11

−0.06 0.07 −0.05 0.09 0.10

303 313 312 322 321 331 330

8574.28 8598.82 8635.71 8740.22 8743.46 8927.15 8927.20

8574.42 8600.08 8637.09 8743.81 8747.00 8932.94

8574.26 8599.52 8637.76 8745.76 8749.17 8938.30 8938.35

8574.33 8600.79 8639.07 8749.39 8752.74 8944.22 8944.27

0.05 −0.70 −2.02 −5.56 −5.72 −11.22

−0.06 0.00 −0.06 0.03 0.02 0.13

404 414 413 423 422 432 431 441 440 000

8652.00 8669.63 8730.57 8824.50 8833.79

8652.22 8670.67 8731.83 8827.36 8836.56 9018.57

8654.82 8673.81 8737.38 8836.27 8845.957 9033.41 9033.75 9270.66 9272.02

−2.63 −3.15 −5.62 −8.92 −9.43

9013.97 9242.53 9243.14 9716.90

8654.67 8672.78 8736.27 8833.43 8843.244 9028.75 9029.10 9265.10 9266.00 9706.39

101 111 110

9737.92 9765.17 9771.12

9737.98 9772.12

9728.18 9755.56 9761.70

202 212

9779.38 9801.68

9779.46 9806.02

9771.16 9793.50

9249.06

−2.74 −2.20

(o-c)df.

obs F1

obs F2

8942.59

calc F2

(o-c)av.

(o-c)df.

2.57

−0.06

0.40

−0.06

−0.50

−0.12

−0.25

−0.12

−2.11

−0.07

−3.80

−0.13

8938.93

8985.59

8986.47

9003.78

9003.85

9038.86 9104.83 9105.90

9039.45 9107.68

9061.58 9074.26

9074.66

9167.34

9168.65

9286.92

9289.42

−0.08 −0.01 −0.14 −0.01 −0.06

9136.38

9136.55

9201.18

9201.66

9263.07

9264.21

−15.16

0.05

−22.91

0.09

9372.75 9526.52 9526.66

9529.66

9728.18 9762.83 9769.19

9.77 9.45

−0.06 0.32

10329.24

10330.97

9771.19 9798.06

8.25 8.07

−0.06 0.22

10343.95 10364.72

10344.02

 C 2002 Elsevier Science

calc F1

8960.81 8983.05 8989.29

8960.81 8983.87 8990.12

9003.41 9020.70 9039.43 9105.08 9106.21

9003.43 9021.16 9039.90 9106.76 9107.87

9064.69 9076.40 9113.72 9171.204 9176.61 9291.17 9291.31

9064.75 9076.73 9114.06 9172.392 9177.75 9293.53 9293.67

9142.52 9149.54 9211.03 9257.80 9273.58 9381.64 9382.61 9536.64 9536.72 10273.10

9142.62 9149.82 9211.33 9258.91 9275.05 9383.46 9384.42 9539.51 9539.58

10295.103 10322.21 10328.73

10295.103 10323.7 10330.27

10338.09 10359.92

10338.12 10360.80

−0.14 −6.11

−0.06

−9.76

−0.19

−10.68

0.33

−9.99

−0.15

7.14

−0.21

5.88

−0.05

28

ALIJAH AND DUXBURY

TABLE 6—Continued 0, 7, 0

1, 5, 0

obs F1

obs F2

calc F1

calc F2

(o-c)av.

(o-c)df.

211 221 220

9819.77 9970.77 9971.32

9824.39 9979.71 9980.27

9812.20 9970.10 9971.57

9816.92 9980.21 9980.77

7.52 −0.37 −0.38

303 313 312 322 321 331 330

9840.08 9855.40 9891.58 10034.86 10037.60 10273.67

9840.23 9858.66 9895.13 10041.49 10044.15 10282.59

9834.14 9849.30 9886.85 10037.90 10040.75 10286.26 10286.29

9834.23 9852.69 9890.44 10044.68 10047.46 10295.45 10295.48

404 414 413 423 422 432 431 441 440

9918.50 9926.23 9986.15 10119.33 10127.22

9918.78 9989.31 10124.59 10132.35

10360.06 10640.22 10640.47

10648.64

9915.57 9922.87 9985.39 10126.14 10134.43 10376.91 10377.12 10667.27 10667.33

9915.77 9925.68 9988.52 10131.50 10139.62 10384.21 10384.41 10675.91 10675.95

0, 8, 0 obs F1

obs F2

000

10948.47

101 111 110

10969.55 10867.72

202 212 211 221 220

11011.03 10904.77 10919.01 11171.15 11171.82

11011.06

303 313 312 322 321 331 330

11071.53 10958.80

11071.64 10966.57

11234.64 11237.66 11566.22

11244.62 11247.55 11577.95

404 414 413 423 422 432 431 441 440

11149.37

11149.61

11069.79 11318.20 11327.04

11082.18 11326.01 11334.65

11651.45 12005.42 12005.48

12015.88

obs F1

obs F2

calc F1

calc F2

0.10 0.28 0.26

10382.95 10467.34 10468.22

10384.04

10379.50 10463.77 10464.76

5.97 6.03 4.71 −3.12 −3.23 −12.73

−0.06 0.12 0.03 0.16 0.15 0.27

10402.55 10417.26 10453.48 10529.87

10402.67 10418.02

2.97

−0.08

10477.37

10477.56

0.78 −6.86 −7.24

−0.02 0.10 0.05

10546.25 10623.23

10547.05 10613.53 10625.03

−27.08

0.45

10943.06 10943.14

10947.24

10471.09

10531.96

(o-c)av.

(o-c)df.

10380.38 10466.48 10467.45

3.55

−0.21

3.55

−0.18

10400.22 10415.74 10454.8 10530.52 10535.30 10677.12 10677.23

10400.28 10416.37 10455.45 10532.44 10537.17 10680.65 10680.75

2.36 1.59

−0.05 −0.13

−0.57

−0.16

10479.35 10489.15 10553.73 10618.29 10631.65 10768.24 10768.97 10953.87 10953.98

10479.48 10489.67 10554.28 10619.81 10633.07 10770.98 10771.70 10957.92 10958.01

−1.95

−0.06

−7.35

−0.25

−8.23

−0.38

−10.80

−0.08

Barrier to linearity calc F1

calc F2

(o-c)av.

(o-c)df.

20.90

1.99

10932.05 10882.59

10928.73 11185.26

10953.98 10845.82 10850.28

10953.98 10862.69 10867.70

10997.14 10884.68 10898.74 11169.11 11169.74

10997.19 10896.23 10910.67 11182.75 11183.40

13.88

0.03

19.17

2.21

1.97

0.23

11060.18 10941.09 10969.69 11235.52 11238.81 11580.62 11580.64

11060.37 10949.94 10978.82 11245.62 11248.75 11592.54 11592.56

11.31 17.17

0.08 1.08

−0.94 −1.18 −14.50

0.12 0.05 0.18

11141.36 11015.19 11063.29 11323.12 11332.55 11670.63 11670.78 12037.15 12037.17

11141.75 11023.16 11070.87 11331.04 11340.14 11680.05 11680.21 12047.83 12047.84

7.93

0.15

8.91 −4.97 −5.50

−4.81 0.10 −0.01

−31.83

0.27

N = 0 and 1 and to vibrational states up to 0, 6, 0. The second set of calculations, by Gabriel et al. (18), were also carried out from the ab initio standpoint, using two three-dimensional potential energy surfaces and obtaining the eigenvalues via the

Carter–Handy–Rosmus–Chambaud (32) method. In this case a complete list of term values with N = 0 and 1 was computed for both electronic states. In addition some selected examples of spin–rotation splittings were also tabulated.

 C 2002 Elsevier Science

THE X˜ 2 B1 STATE OF NH2

In the previous tables we have chosen to provide two comparisons with the earlier calculations of Gabriel et al. (18). These compose a comparison with the N = K a term values of the 2 B1 state below the barrier to linearity and of the spin–rovibrational levels of the 0, 4, 0 vibrational state. We have also given a detailed comparison with sets of spin–rovibrational levels up to a maximum value of K = 6 and N = 6 for some of the lower states (20, 21, 33–35) and up to K = 4 and N = 6 for the higher levels measured by Vervloet (9). In considering what is usually referred to as “spin–rotation” interaction, but which we are mainly treating as an effect of electronic angular momentum coupling, two points must be borne in mind, the sign of the effective coupling constant, and the neglect of coupling to the B˜ 2 B2 and higher electronic states. The S 3 main contribution to ASO v,K ≡ εaa K −  K K + · · · is the cou2 2 pling between the A1 and B1 states. Since all the levels of the 2 B1 state we are considering lie below those of the 2 A1 state this results in a negative value of this parameter. As a result all the vibronic states are “inverted,” and the spin–uncoupling occurs smoothly as N increases. This contrasts with the excited state, for when some of the states are “regular” and ASO v,K is large, very rapid uncoupling can occur as discussed by Barrow, et al. (22). The effect of coupling between the X˜ 2 B1 and B˜ 2 B2 states is to generate a spin–rotation constant εbb . Since the B˜ 2 B2 state is nonresonant with any of the accessible levels of the X˜ 2 B1 state and well separated from any of the levels which lie below the barrier to linearity, εbb has little or no increase with the excitation of v2 . However, the coupling term dependent on εbb contributes to the magnitude of the splitting as a function of N , particularly the asymmetry splitting. Its effect increases with increasing values of N . The neglect of this coupling accounts for part of the difference between the observed and calculated magnitudes of the spin–rotation splitting, which is more evident in the lowest vibrational states when ASO v,K is small. In Table 4 we show a comparison between our calculated “spin-free” N = K a term values and both those derived from the experimental spin–rotation term values (12) and those calculated for N = 1 to 3 by Gabriel et al. (18). It can be seen that there is good agreement up to the highest vibrational states below the barrier. The bulk of our discussion of the effects of rotation will be concerned with a direct comparison with the available sets of experimentally derived term values. In Table 5 we show a comparison with the only other calculation of this type by Gabriel et al. (18) for spin-split rotational levels of the 0, 4, 0 state of X˜ 2 B1 NH2 up to N = 3. It can be seen that although the agreement between the observed and calculated vales of the levels with K a = 0 is slightly better in the Gabriel et al. calculations, there is systematic underestimation of the increase in the spin–rotation splitting with increasing values of K , as well as systematic underestimation of the K dependent origin of the term values plotted as a function of N .

29

We believe that the former of these is due to the use of a value of ASO = −44.6 cm−1 and a neglect of its ρ-dependence in the calculations. We will defer a discussion of the latter point to the next section. In Table 6 we show a detailed comparison between our calculated term values and those determined from experimental data, mainly from the work of Vervloet (9). It can be seen that the use of the stretch-bender model has allowed us to reproduce the systematic behavior of both the rovibrational and spin doubling over a very wide range of vibrational amplitudes, including levels with one quantum of excitation of v1 . It can be seen that the error in the calculation of the spin–rotation splitting between the F1 and F2 levels, (δ F1 − δ F2 ), is very small, whereas that in the level positions, (δ F1 + δ F2 )/2, is larger and depends on both N and K a . These discrepancies have two causes: the small systematic shifts in the calculation of the doublet splitting are due to the neglect of εbb discussed above, while the larger residuals in the level positions arise principally from the assumptions made in our current model about the form of the angular dependence of the bond-length variation. This latter point will be discussed in more detail in the conclusion. The main point to be noted is that the use of the large-amplitude stretch-bender model has allowed a good fit to these experimental data and a prediction of the position of the spin–rotation components, which have not so far been located via spectroscopic experiments. 3. CONCLUSION

The extensive experimental data which have become available since the treatment by Jungen et al. (7, 8) of the vibronic coupling in the ground state of NH2 have allowed it to be used as a test bed for extensions of their theoretical treatment. In the present paper we have attempted to show that the stretch-bender model, which may be regarded as developed from the JHM (6–8) and BDD (22) treatments, has allowed an almost complete model of the spin–rotational structure of the X˜ 2 B1 state below the barrier to linearity. Since extreme effects of centrifugal distortion arise in the normal perturbation series treatment (19); this serves to show the advantages to be gained from the use of the large amplitude model. In considering the high-angular-momentum states, with N = K a , in the X˜ 2 B1 state of NH2 , Dixon et al. (12) chose to adjust the analytical potential of Jungen et al. (7) in order to obtain a better fit to their data, but one which gives a much poorer fit to the low-angular-momentum states. In view of our use of a very slightly modified version of the JHM potential to fit a wide range of data up to K a = 7 and to calculate the N = K a term values up to N = 12 (see Table 4), we believe that it gives a very good representation of the effective bending potential in the X˜ 2 B1 state. As a result we believe that their discrepancies are more likely to be due to the form of the analytic variation of the bond length which was used, since this affects the angular dependence of the A(ρ) parameter. As A(ρ) K 2 forms the

 C 2002 Elsevier Science

30

ALIJAH AND DUXBURY

effective K -dependent term which is added to the vibration-only potential, it is very large and extremely sensitive to the angular dependence of the bond length which is assumed. In the case of the calculation by Gabriel et al. (18), which gives an excellent representation of levels having K a = 0 and 1, we believe that the systematic underestimation of the value of the vibronic origin as K increases may result from having an effective K -dependent potential which is too high in the region close to linearity. As has been pointed out several times, the fit obtained using the standard contact transformed Hamiltonian is superior for a single low-lying vibrational state, such as the zero point level. However, the developments carried out by the Tennyson group (36) show that it is possible, using a more complete model, to achieve almost the same level of agreement over the whole energy range spanned by the experimental data. It may eventually become possible to use such an approach for the two surface problem in triatomic molecules. Two groups have already applied larger scale variational calculations to the Renner–Teller problem. The first method, pioneered by Carter et al. (32), has already given considerable insight into several of the molecules in which Renner–Teller coupling occurs, such as CO+ 2 (37), NH2 , (18) and H2 O+ (38). More recently Jensen and Bunker and their collaborators have successfully extended their MORBID approach, in their RENNER program, to tackle the Renner–Teller coupling in two key molecules, CH2 (39, 40) and HO2 (41). The principal advantages of the stretch-bender approach over the variational methods mentioned above are twofold: first, it minimizes the size of the rovibronic interaction matrix so that calculations may be carried out using a desktop workstation, and second, no high-quality fit of the complete potential energy surfaces is required. This last point is important since in general such a fit is difficult to obtain. As a result of these advantages of the stretch-bender, the extension of the method to fouratom systems in degenerate electronic states should be possible shortly. REFERENCES 1. R. Renner, Z. Phys. 92, 172–193 (1934). 2. G. Herzberg and D. A. Ramsay, J. Chem. Phys. 20, 347–347 (1952). 3. K. Dressler and D. A. Ramsay, Philos. Trans. R. Soc. London Ser. A 251, 553–602 (1959). 4. J. A. Pople and H. C. Longuet-Higgins, Mol. Phys. 1, 372–383 (1958). 5. R. N. Dixon, Philos. Trans. R. Soc. London Ser. A 252, 165–192 (1960). 6. Ch. Jungen, K.-E. J. Hallin, and A. J. Merer, Mol. Phys. 40, 25–63 (1980). 7. Ch. Jungen, K.-E. J. Hallin, and A. J. Merer, Mol. Phys. 40, 65–94 (1980). 8. Ch. Jungen and A. J. Merer, Mol. Phys. 40, 1–23 (1980).

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