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Bound rovibronic levels of the HeN2+(A2Π) complex
SPECTROCHIMICA ACTA PART A Spectrochimica Acta Part A 53(1997)113331138
ELSEVIER
Bound rovibronic levels of the HeN,f(A211) complex T. Schmelz a, P. Rosmus a,*, A. Berning lyb, H.J. Werner b a Laboratoire
de Chimie
b Institut
Theorique.
fir Theoretische
UniversitP de Marne-la-Vake, 2, rue de la Butte verte (Bat. F-93166 Noisy-le-Grand, France Chemie, Universitiit Stuttgart, D-70569 Stuttgart, Germany
M2).
Received13November1996;receivedin revisedform 10December 1996
Abstract We have used ab initio MRCI (multireferenceinternally contracted configuration interaction) potential energy functions to compute the energiesof the bound rovibronic levels of the T-shapednearly-degenerateelectronically excited HeN2(A211). In these calculations a variational approach which includes the rotation-vibration, the electronic angular and spin momenta couplings has been employed. The wavenumbersfor the large amplitude bending and stretching modeshave beencalculatedto lie around 60 and 75 cm- ‘, respectively.For J= l/2, three stateswith excited bendingand one statewith excited stretchingmodewere found to be bound. The weak interaction betweenHe and the N$ (A2H,) state leadsto a spin flip. The inverted doublet in N: (A211,) becomesa normal doublet in the complex. This unusual effect is attributed to large anisotropiesand to a large difference potential betweenthe A’ and A” components,which leadsalso to the largestknown parity splittingsin Renner-Teller weakly bound complexes.0 1997Elsevier ScienceB.V. Keywords:
Electron spin couplings;Potential energy functions; Rovibronic levels
1. Introduction
In weakly bound complexes with an open shell diatom in an electronically degenerate state and a rare gas, the structure of rovibronic levels depends on the interactions among the angular momenta associated with electron orbital, electron spin and vibration-rotation motions. Due to a sensitive
* Corresponding author. Tel.: + 33 1 64737304; fax: + 33 1 64737320; e-mail: [email protected] ’ Present address: Max-Planck Gesellschaft, Arbeitsgruppe Quantenchemie an der Humboldt UniversitPt. D-101 17, Deutschland.
interplay of various effects, the interpretation of the structures of the rovibronic levels in such complexes still remains to one of the most challenging spectroscopic problems. The theory of the weakly bound complexes formed from atoms and open shell diatomic molecules has been developed in recent years [l51. Particular attention has been paid to the quantum numbers that are conserved and the angular momentum coupling cases that may be observed. The transitions from free internal rotor quantum numbers to near-rigid bender quantum numbers with increasing anisotropy of the potential energy functions (PEFs) have been studied [ 11. The model
1386-1425/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO584-8539(97)0002
1-4
Hamiltonians used in these studies have the great advantage of tracing the dominant contributions of the coupling phenomena for a certain structure of the rovibronic levels. On the other hand. in particular for cases with several competing effects, they might not be appropriate for calculations of the energy positions of the rovibronic levels, and more complete Hamiltonians and accurate potential energy functions are needed. When no experimental data are available one can predict the spectra by using ab initio PEFs and performing variational calculations using rather general formulations of the Hamiltonian. Even though the accuracy of such an approach cannot reach the accuracy needed in high resolution experiments, it provides a reliable basis for analysing the spectra. Recently, we have developed such an approach for triatomic complexes [6], which includes the rotation-vibration, electronic angular and spin couplings. This method has been successfully applied to complexes such as ArNO [7], HeHF+ [S], HeHCl+ [9] and HeOz [lo], in which the Renner-Teller effect plays a dominating role. Particularly for ionic diatoms with larger interaction energies than in neutral van der Waals complexes various limiting cases depending on the shapes of the potential energy functions have found in theoretical calculations. For instance, in HeN,C (X2x) [l l] and NeN,t (X2x) [12] the rare gas atom can almost freely move around the diatom, whereas in HeHF + (X*II) [8] and HeHCl + (X’II) [9] one finds a near-rigid linear structure with a pattern of rovibronic levels very similar to that of the strongly bound RennerTeller systems. The HeO’ (X*x) [ 11,121 complexes exhibits two equivalent minima lying between the linear and T-shaped structures. In all these ionic complexes, the bending parts of the potential energy functions have very different anisotropies and barriers to linearity. The resulting differences of the bending frequencies as well as different spin-orbit couplings and rotational constants of the diatoms influence the strengths of various coupling phenomena. Recently, the potential energy functions for the HeNf (A211) complex have been generated using accurate multiconfiguration-reference configura-
tion interaction (MRCI) wavefunctions [I 31. The computed potentials have been fitted to analytical function and used in quantum scattering calculations for electronically inelastic transitions between the individual rovibrational levels of the A%,, and X’C, states of N,’ In the present work we will use these potential energy functions to calculate the bound rovibronic states of the HeN: (A’II) complex. Together with the similar data published previously for the electronic ground state complex [l 11, they should prove useful in future analysis of the electronic transition HeN,I (A?II)-HeN’(X’Z).
2. Potential
energy functions
The cylindrical degeneracy of a molecule in a II electronic state is lifted by the approach of a spherical atom, giving rise to two states whose wavefunctions have A’ and A” reflection symmetry in the triatomic plane. Hence, the description of the interaction of a II state molecule with a closed-shell atom (or molecule) involves two electronic PEFs. In such a case, the nuclear and electronic motion cannot be treated separately, because of the coupling of the rotational-vibrational and electronic angular momenta, which is called Renner-Teller coupling. The details of the very extensive MRCI computations have been described in Ref. [13]. As a coordinate system for the two-dimensional parts of the PEFs with fixed rNN = 2.11 bohr the Jacobi coordinates have been used. The PEFs of the A’ and A“ states were expressed as V‘4JR ‘9 = ;[Y,(R,
0) -
@I
(1)
V.4@, ‘3 = ;[vo(R,
0) + V2/;(R,e)]
(2)
Jf*(R,
where the angular dependence of the average and difference potentials V, and V?, respectively, is expressed as V,(R, 19) = i d$+I=0
‘(cos @f(R)
(3)
T. Schmek et al. /Spectrochimica Acta Part A 53 (1997) 1133-1138 Table MRCI
1 equilibrium
geometries,
dissociation
energies
and barrers
to linearity
in HeNz+(A’H)
Component
R, (bohr)
0, (“1
D, (cm-‘)
Barrier
A’ A”
4.725 5.572
90 90
297.5 136.9
222.6 62.0
Here L is the number of Jacobi angles 9 for which the potential has been calculated and p; = dj,+ ‘- ‘(cos 0)
(4)
The vectors aj(R) can be expressed as a’(R) = (P’)-‘.b’(R)
11.X
to linearity
(cm-‘)
Another crucially important parameter in bentbent Renner-Teller molecules is the barrier to linearity. Also these values (222.6 cm - ’ A’; 62.0 cm - ’ A”) deviate considerably for both components. The difference potential V(A’) - V(A”) is
(5)
where Pjk = df,+ i - ‘(cos 0,)
(6)
with 13~denoting the individual angles at which ab initio points were calculated. The radial functions b;(R) are of the form B;(R) =exp[-c;(R-R:)]
[
i
I=0
- i[ 1 + tanh(y)]
C&R’
1
[/$r &R-“1
(7)
The parameters in these functions are obtained by fits of the radial potentials for each angle ek. Some characteristic properties of the potentials are given in Table 1, their explicit forms can be obtained from the authors on request. Fig. 1 presents the contour plots for the A’ and A” components. The A’ PEF possesses the lower minimum, which occurs at R = 4.725 bohr and 0 = 90”. The minimum on the A” PEF, at R = 5.572 bohr and 0 = 90” lies 160.6 cm-’ higher in energy. This value is extremely large for a van der Waals complex. For instance, in ArNO this difference was calculated to be only 1.3 cm-’ [7], in good agreement with the experimental estimate of 1.5 cm-’ [14]. It is important to notice that the energy difference between the A’ and A” minima is more than two times larger than the spin-orbit constant in N,+ (A’II,) of - 74 cm - ‘. It is also very unusual that the equilibrium bond lengths in both components differ by as much as 0.84 bohr.
Fig. 1. Contour plots of the adiabatic potential energy functions of the A’ (upper part) and A’ (lower part) of the HeN,t (‘II) complex for r = 2.11 bohr. The energies (in cm ’ relative to the dissociation asymptote) are indicated at the contour lines.
0
60
I/
30
3
5
9
11
R /Lhr
Fig. 2. Contour plot of the difference potential V(A”) - V(A’). The energies (in cm - ‘) are indicated at the contour lines.
displayed in Fig. 2. It shows a very regular shape, with strongly increasing values for structures in which the He closely approaches the diatom.
3. Computations of the rovibronic energy levels
The variational calculations of the bound rovibronic energy levels were carried out using an approach closely related to the method of Carter et al. [15] for bound Renner-Teller systems which includes the electron spin explicitly. However here, in contrast to the prior work, the kinetic energy operator is expressed in Jacobi rather than in internal coordinates. The explicit form of the Hamiltonian is H=H,+H,+
+
v
cot qrIJI,
(8)
+ rI,rI,)
Gu,R2
[ PIR2 1II.”
ifi 2$+cotB --
2
(9)
Here ,u, are the reduced masses, II = .I f I, c S is the rotational-vibrational angular momentum. with J, L and S designating the total, electronic and spin angular momenta of the HeN; complex. In the present application we have kept r fixed so that all derivative terms involving r in Eq. (8) are subsequently suppressed. Following the procedure described in Ref. [15], we have obtained eigenvectors and eigenvalues for the Hamiltonian of Eqs. (8)-(10). The basis set consisted of products of 45 Morse eigenfunctions for the stretching mode and 30 Legendre functions for the bending motion. The details describing the necessary angular momentum matrix elements and the stepwise optimisation of the basis functions have been described in Ref. [15]. In the present application we have used the value for the spin-orbit constant of the diatom in the A%, state (A = 74 cm-’ [16]). The geometry dependence of the operators of the electronic angular momentum has not been considered. The eigenvalues of the Lz and Lt operators were taken to be those for a II state, and all contributions from the L, and L., operators have been neglected. Previous experience has shown [15,17] that all these assumptions are rather well justified for both strongly [17] and weakly [7] bound Renner-Teller systems.
4. Results and discussion
Before discussing the results for HeN,+, it is important to mention which accuracy can be reached for the very small energy differences existing in such complexes if calculated from good ab initio potentials. For ArNO, there are experimental values available for the rotational transitions in the vibronic ground state. Our approach yielded, for example, for the J= 5/2 o( - ) to J = 3/2 o( + ) transition a value of 0.674 cm - ‘, whereas the experimental value was determined to be 0.577 cm-’ [14]. In the J= l/2 o level we have
T. Schmelz et al. /Spectrochimica Acta Part A 53 (1997) 1133-l 138 Table 2 Rovibronic levels (J= l/2, e) in HeN$(A’fI) cub.us)”
Energy (c-l)
(0.0) (03 0) (1.0) (1.0) (0, 1) (0, 1)
0.0 3.089 58.849 62.625 73.306 16.373 106.594 107.016 113.061 113.993
GTO) (330)
(LO) (3,O)
d u,, designates the bending, V, the stretching modes of the complex
calculated a parity splitting of 0.034 cm-‘, whereas the experimental value is 0.020 cm - ’ [14]. Even though from the point of view of the purely ab initio calculations of such small energy differences such an accuracy is pleasing, it is not sufficient for high resolution spectroscopic studies. Nevertheless, the structures of the rovibronic levels are correct, which is useful for the interpretation of the experimental data in such complexes. The results obtained from the two-dimensional calculations of the large amplitude modes for J = l/2 and J = 3/2 in HeN,f are given in Tables 2 and 3. The interaction of the He atom with the NC (A211) diatom removes the orientational 2j + 1 degeneracy of the diatomic rotational states (see 2j+ 1 states for each rovibronic state in Tables 2 Table 3 Rovibronic levels (J = 3/2, e) in HeN’(A’fl) (v,, v,)
Energy (cm - 1)
(Vb v,)
Energy (cm- ‘)
a 0) (0.0) (0.0) a 0) (l,O) (1%0) (1%0) (l>O) (0. 1) a 1)
0.909 2.600 6.188 9.204 59.496 61.512 65.680 68.928 73.863 76.378
(0. 1) al) (2.0) (3.0)
79.828 82.040 106.481 106.555 107.888 108.988 115.182 115.346 121.985
GO) (3,O) (3,O)
CO) CL0)
“ vb designates the bending, v, the stretching modes of the complex.
1137
and 3). The rovibronic states up to 122 cm-’ relative to the J = l/2 zero point level were calculated to be bound. For J= l/2, three bending modes and only one stretching mode were calculated to exist within this energy region. The assignments were made by the inspection of the contour plots of both vibrational components coupled by the Renner-Teller effect. Surprisingly, the J = l/2 states were calculated to lie below the J = 3/2 states, even though the A state of NT forms an inverted doublet (A = - 74 cm-‘). In our approach we cannot easily trace this effect in the final wavefunctions, since we have only the total J and the parity as attributes of the individual rovibronic states. For open shell complexes Dubernet et al. [l] showed that such a spin flip can be caused by large spin decoupling effects (Fig. 8 in Ref. [l]). The spin decoupling has been detected in many diatomics [16], but to our knowledge the HeN,‘(A’II) complex is the first example of this effect in van der Waals molecules. The quenching of the spin-orbit splitting in the rovibronic ground state of the complex (Table 3) can be interpreted as being mainly caused by the Renner-Teller effect and also electron spin decoupling accompanying the extremely large differences in the anisotropies of the A’ and A” components in the complex. It has been known that such anisotropies also lead to large parity splittings. In ArOH [18,19] and ArNO [14] the parity splittings in the low rovibronic levels are in the range of about 0.01-0.2 cm ~ ‘. For He02 [lo] we have obtained parity splittings in the range of 0.2-1.4 cm- ‘. In the present case of HeN,t we found the largest differences (up to 1.7 cm ~ l; Table 4) known so far for van der Waals complexes. The dominant contribution to this splitting comes from the mixing of the A’ and A” component of the electronic wavefunctions which is a direct manifestation of the Renner-Teller effect.
5. Conclusions
Using two-dimensional MRCI potential energy functions of the HeN,t(‘II) complex bound rovibronic levels have been calculated. It has been
Table
4
Some calculated parity splittings” (e-j’) in the rovibronic levels of HrN+(A’Il) (l.h. I.,)
J= l/2 (cm~ ‘)
J=3/2
(0, 0)
- 1.656 0.767
~ 1.451 - 1.520 1.294 - 0.099 -0.129 - 1.589 0.225 0.023 - 1.323 - 0.697 1.373 -0.201
(0, 01 (0.0) (0.0)
(LO) (2.0) (2.0) (2.0) (0, 1) (0, 1) (0. 1) (0, 1)
-0.275 - 1.655 - 1.723 1.308
a The parity e corresponds to (-l)-“‘. ( _ , )J+ IQ,
(cm-~‘)
the parity f to
found that the weak interaction of N: with He causes a spin flip of the inverted doublet of the diatom. The calculated structures of energy levels in HeN,t (‘C +) (free rotor case) and HeN,+ (A2H) (Renner-Teller near-degenerate T-shape structure) should prove useful in identifying and interpreting the electronic A-X transition in this complex.
Acknowledgements
This study has been supported by the EC grant ERFMRXCT960088 and the Deutsche Forschungsgemeinschaft. We thank Marie-Lise Dubernet for helpful discussions.
References [I] M.L. Dubemet. D. Flower. J.M. Hutson. J. (‘hem. Ph>\ 94 (1991) 7602. [2] W.M. Fawzy. J.T. Ilougen. J. Mol. Spectrosc. I37 I 1989) 154. [3] B.J. Howard. C.M. Wcstrrn. P.D. Mills. Faraday Discuss. Chem. Sot. 73 (1982) 129. [4] W.H. Green Jr., M.I. Lester. J. Chem. Phys. Y6 (1992) 2573. [5] C. Chakravarty, D.C. Clary, J. Chem. Phys. 94 (1991) 4149. [6] T. Schmelz. PhD Thesis, University of Frankfurt, 1994. [7] T. Schmelz, P. Rosmus, M.H. Alexander, J. Phys. Chem. 98 (1994) 1073. [8] T. Schmelz, P. Rosmus, J. Chem. Lett. 220 (1994) 117. [9] D. Simah, DEA report. University of Marne la Vallee. 1995. [lo] T. Schmelz, E.A. Reinsch, P. Rosmus, J. Phys. Chem. 99 (1995) 15580. [I l] S. Miller, J. Tennyson. B. Follmeg. P. Rosmus. H.J. Werner, J. Chem. Phys. 88 (1988) 2178. [12] T. Schmelz and P. Rosmus, submitted; see [5]. [13] A. Berning, H.J. Werner, J. Chem. Phys. Chem. 100 (1994) 1953. [14] P.D.A. Mills, CM. Westernand, B.J. Howard, J. Phys. Chem. 90 (1986) 3331. [15] S. Carter, N.C. Handy, P. Rosmus, G. Chambaud, Mol. Phys. 71 (1990) 605. [16] K.P. Huber and G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, NY. 1979. [17] M. Brommer, B. Weiss, B. Follmeg, P. Rosmus. S. Carter, N.C. Handy, H.J. Werner, P.J. Knowles, J. Chem. Phys. 98 (1993) 5222. [18] Y. Oshima, M. Ida. Y. Endo. J. Chem. Phys. 95 (1991) 7001. [19] M.T. Berry, M.R. Brustein, M.1. Lester, Ch. Chakravarty. D.C. Clary. Chem. Phys. Lett. 178 (1991) 301.
ELSEVIER
Bound rovibronic levels of the HeN,f(A211) complex T. Schmelz a, P. Rosmus a,*, A. Berning lyb, H.J. Werner b a Laboratoire
de Chimie
b Institut
Theorique.
fir Theoretische
UniversitP de Marne-la-Vake, 2, rue de la Butte verte (Bat. F-93166 Noisy-le-Grand, France Chemie, Universitiit Stuttgart, D-70569 Stuttgart, Germany
M2).
Received13November1996;receivedin revisedform 10December 1996
Abstract We have used ab initio MRCI (multireferenceinternally contracted configuration interaction) potential energy functions to compute the energiesof the bound rovibronic levels of the T-shapednearly-degenerateelectronically excited HeN2(A211). In these calculations a variational approach which includes the rotation-vibration, the electronic angular and spin momenta couplings has been employed. The wavenumbersfor the large amplitude bending and stretching modeshave beencalculatedto lie around 60 and 75 cm- ‘, respectively.For J= l/2, three stateswith excited bendingand one statewith excited stretchingmodewere found to be bound. The weak interaction betweenHe and the N$ (A2H,) state leadsto a spin flip. The inverted doublet in N: (A211,) becomesa normal doublet in the complex. This unusual effect is attributed to large anisotropiesand to a large difference potential betweenthe A’ and A” components,which leadsalso to the largestknown parity splittingsin Renner-Teller weakly bound complexes.0 1997Elsevier ScienceB.V. Keywords:
Electron spin couplings;Potential energy functions; Rovibronic levels
1. Introduction
In weakly bound complexes with an open shell diatom in an electronically degenerate state and a rare gas, the structure of rovibronic levels depends on the interactions among the angular momenta associated with electron orbital, electron spin and vibration-rotation motions. Due to a sensitive
* Corresponding author. Tel.: + 33 1 64737304; fax: + 33 1 64737320; e-mail: [email protected] ’ Present address: Max-Planck Gesellschaft, Arbeitsgruppe Quantenchemie an der Humboldt UniversitPt. D-101 17, Deutschland.
interplay of various effects, the interpretation of the structures of the rovibronic levels in such complexes still remains to one of the most challenging spectroscopic problems. The theory of the weakly bound complexes formed from atoms and open shell diatomic molecules has been developed in recent years [l51. Particular attention has been paid to the quantum numbers that are conserved and the angular momentum coupling cases that may be observed. The transitions from free internal rotor quantum numbers to near-rigid bender quantum numbers with increasing anisotropy of the potential energy functions (PEFs) have been studied [ 11. The model
1386-1425/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SO584-8539(97)0002
1-4
Hamiltonians used in these studies have the great advantage of tracing the dominant contributions of the coupling phenomena for a certain structure of the rovibronic levels. On the other hand. in particular for cases with several competing effects, they might not be appropriate for calculations of the energy positions of the rovibronic levels, and more complete Hamiltonians and accurate potential energy functions are needed. When no experimental data are available one can predict the spectra by using ab initio PEFs and performing variational calculations using rather general formulations of the Hamiltonian. Even though the accuracy of such an approach cannot reach the accuracy needed in high resolution experiments, it provides a reliable basis for analysing the spectra. Recently, we have developed such an approach for triatomic complexes [6], which includes the rotation-vibration, electronic angular and spin couplings. This method has been successfully applied to complexes such as ArNO [7], HeHF+ [S], HeHCl+ [9] and HeOz [lo], in which the Renner-Teller effect plays a dominating role. Particularly for ionic diatoms with larger interaction energies than in neutral van der Waals complexes various limiting cases depending on the shapes of the potential energy functions have found in theoretical calculations. For instance, in HeN,C (X2x) [l l] and NeN,t (X2x) [12] the rare gas atom can almost freely move around the diatom, whereas in HeHF + (X*II) [8] and HeHCl + (X’II) [9] one finds a near-rigid linear structure with a pattern of rovibronic levels very similar to that of the strongly bound RennerTeller systems. The HeO’ (X*x) [ 11,121 complexes exhibits two equivalent minima lying between the linear and T-shaped structures. In all these ionic complexes, the bending parts of the potential energy functions have very different anisotropies and barriers to linearity. The resulting differences of the bending frequencies as well as different spin-orbit couplings and rotational constants of the diatoms influence the strengths of various coupling phenomena. Recently, the potential energy functions for the HeNf (A211) complex have been generated using accurate multiconfiguration-reference configura-
tion interaction (MRCI) wavefunctions [I 31. The computed potentials have been fitted to analytical function and used in quantum scattering calculations for electronically inelastic transitions between the individual rovibrational levels of the A%,, and X’C, states of N,’ In the present work we will use these potential energy functions to calculate the bound rovibronic states of the HeN: (A’II) complex. Together with the similar data published previously for the electronic ground state complex [l 11, they should prove useful in future analysis of the electronic transition HeN,I (A?II)-HeN’(X’Z).
2. Potential
energy functions
The cylindrical degeneracy of a molecule in a II electronic state is lifted by the approach of a spherical atom, giving rise to two states whose wavefunctions have A’ and A” reflection symmetry in the triatomic plane. Hence, the description of the interaction of a II state molecule with a closed-shell atom (or molecule) involves two electronic PEFs. In such a case, the nuclear and electronic motion cannot be treated separately, because of the coupling of the rotational-vibrational and electronic angular momenta, which is called Renner-Teller coupling. The details of the very extensive MRCI computations have been described in Ref. [13]. As a coordinate system for the two-dimensional parts of the PEFs with fixed rNN = 2.11 bohr the Jacobi coordinates have been used. The PEFs of the A’ and A“ states were expressed as V‘4JR ‘9 = ;[Y,(R,
0) -
@I
(1)
V.4@, ‘3 = ;[vo(R,
0) + V2/;(R,e)]
(2)
Jf*(R,
where the angular dependence of the average and difference potentials V, and V?, respectively, is expressed as V,(R, 19) = i d$+I=0
‘(cos @f(R)
(3)
T. Schmek et al. /Spectrochimica Acta Part A 53 (1997) 1133-1138 Table MRCI
1 equilibrium
geometries,
dissociation
energies
and barrers
to linearity
in HeNz+(A’H)
Component
R, (bohr)
0, (“1
D, (cm-‘)
Barrier
A’ A”
4.725 5.572
90 90
297.5 136.9
222.6 62.0
Here L is the number of Jacobi angles 9 for which the potential has been calculated and p; = dj,+ ‘- ‘(cos 0)
(4)
The vectors aj(R) can be expressed as a’(R) = (P’)-‘.b’(R)
11.X
to linearity
(cm-‘)
Another crucially important parameter in bentbent Renner-Teller molecules is the barrier to linearity. Also these values (222.6 cm - ’ A’; 62.0 cm - ’ A”) deviate considerably for both components. The difference potential V(A’) - V(A”) is
(5)
where Pjk = df,+ i - ‘(cos 0,)
(6)
with 13~denoting the individual angles at which ab initio points were calculated. The radial functions b;(R) are of the form B;(R) =exp[-c;(R-R:)]
[
i
I=0
- i[ 1 + tanh(y)]
C&R’
1
[/$r &R-“1
(7)
The parameters in these functions are obtained by fits of the radial potentials for each angle ek. Some characteristic properties of the potentials are given in Table 1, their explicit forms can be obtained from the authors on request. Fig. 1 presents the contour plots for the A’ and A” components. The A’ PEF possesses the lower minimum, which occurs at R = 4.725 bohr and 0 = 90”. The minimum on the A” PEF, at R = 5.572 bohr and 0 = 90” lies 160.6 cm-’ higher in energy. This value is extremely large for a van der Waals complex. For instance, in ArNO this difference was calculated to be only 1.3 cm-’ [7], in good agreement with the experimental estimate of 1.5 cm-’ [14]. It is important to notice that the energy difference between the A’ and A” minima is more than two times larger than the spin-orbit constant in N,+ (A’II,) of - 74 cm - ‘. It is also very unusual that the equilibrium bond lengths in both components differ by as much as 0.84 bohr.
Fig. 1. Contour plots of the adiabatic potential energy functions of the A’ (upper part) and A’ (lower part) of the HeN,t (‘II) complex for r = 2.11 bohr. The energies (in cm ’ relative to the dissociation asymptote) are indicated at the contour lines.
0
60
I/
30
3
5
9
11
R /Lhr
Fig. 2. Contour plot of the difference potential V(A”) - V(A’). The energies (in cm - ‘) are indicated at the contour lines.
displayed in Fig. 2. It shows a very regular shape, with strongly increasing values for structures in which the He closely approaches the diatom.
3. Computations of the rovibronic energy levels
The variational calculations of the bound rovibronic energy levels were carried out using an approach closely related to the method of Carter et al. [15] for bound Renner-Teller systems which includes the electron spin explicitly. However here, in contrast to the prior work, the kinetic energy operator is expressed in Jacobi rather than in internal coordinates. The explicit form of the Hamiltonian is H=H,+H,+
+
v
cot qrIJI,
(8)
+ rI,rI,)
Gu,R2
[ PIR2 1II.”
ifi 2$+cotB --
2
(9)
Here ,u, are the reduced masses, II = .I f I, c S is the rotational-vibrational angular momentum. with J, L and S designating the total, electronic and spin angular momenta of the HeN; complex. In the present application we have kept r fixed so that all derivative terms involving r in Eq. (8) are subsequently suppressed. Following the procedure described in Ref. [15], we have obtained eigenvectors and eigenvalues for the Hamiltonian of Eqs. (8)-(10). The basis set consisted of products of 45 Morse eigenfunctions for the stretching mode and 30 Legendre functions for the bending motion. The details describing the necessary angular momentum matrix elements and the stepwise optimisation of the basis functions have been described in Ref. [15]. In the present application we have used the value for the spin-orbit constant of the diatom in the A%, state (A = 74 cm-’ [16]). The geometry dependence of the operators of the electronic angular momentum has not been considered. The eigenvalues of the Lz and Lt operators were taken to be those for a II state, and all contributions from the L, and L., operators have been neglected. Previous experience has shown [15,17] that all these assumptions are rather well justified for both strongly [17] and weakly [7] bound Renner-Teller systems.
4. Results and discussion
Before discussing the results for HeN,+, it is important to mention which accuracy can be reached for the very small energy differences existing in such complexes if calculated from good ab initio potentials. For ArNO, there are experimental values available for the rotational transitions in the vibronic ground state. Our approach yielded, for example, for the J= 5/2 o( - ) to J = 3/2 o( + ) transition a value of 0.674 cm - ‘, whereas the experimental value was determined to be 0.577 cm-’ [14]. In the J= l/2 o level we have
T. Schmelz et al. /Spectrochimica Acta Part A 53 (1997) 1133-l 138 Table 2 Rovibronic levels (J= l/2, e) in HeN$(A’fI) cub.us)”
Energy (c-l)
(0.0) (03 0) (1.0) (1.0) (0, 1) (0, 1)
0.0 3.089 58.849 62.625 73.306 16.373 106.594 107.016 113.061 113.993
GTO) (330)
(LO) (3,O)
d u,, designates the bending, V, the stretching modes of the complex
calculated a parity splitting of 0.034 cm-‘, whereas the experimental value is 0.020 cm - ’ [14]. Even though from the point of view of the purely ab initio calculations of such small energy differences such an accuracy is pleasing, it is not sufficient for high resolution spectroscopic studies. Nevertheless, the structures of the rovibronic levels are correct, which is useful for the interpretation of the experimental data in such complexes. The results obtained from the two-dimensional calculations of the large amplitude modes for J = l/2 and J = 3/2 in HeN,f are given in Tables 2 and 3. The interaction of the He atom with the NC (A211) diatom removes the orientational 2j + 1 degeneracy of the diatomic rotational states (see 2j+ 1 states for each rovibronic state in Tables 2 Table 3 Rovibronic levels (J = 3/2, e) in HeN’(A’fl) (v,, v,)
Energy (cm - 1)
(Vb v,)
Energy (cm- ‘)
a 0) (0.0) (0.0) a 0) (l,O) (1%0) (1%0) (l>O) (0. 1) a 1)
0.909 2.600 6.188 9.204 59.496 61.512 65.680 68.928 73.863 76.378
(0. 1) al) (2.0) (3.0)
79.828 82.040 106.481 106.555 107.888 108.988 115.182 115.346 121.985
GO) (3,O) (3,O)
CO) CL0)
“ vb designates the bending, v, the stretching modes of the complex.
1137
and 3). The rovibronic states up to 122 cm-’ relative to the J = l/2 zero point level were calculated to be bound. For J= l/2, three bending modes and only one stretching mode were calculated to exist within this energy region. The assignments were made by the inspection of the contour plots of both vibrational components coupled by the Renner-Teller effect. Surprisingly, the J = l/2 states were calculated to lie below the J = 3/2 states, even though the A state of NT forms an inverted doublet (A = - 74 cm-‘). In our approach we cannot easily trace this effect in the final wavefunctions, since we have only the total J and the parity as attributes of the individual rovibronic states. For open shell complexes Dubernet et al. [l] showed that such a spin flip can be caused by large spin decoupling effects (Fig. 8 in Ref. [l]). The spin decoupling has been detected in many diatomics [16], but to our knowledge the HeN,‘(A’II) complex is the first example of this effect in van der Waals molecules. The quenching of the spin-orbit splitting in the rovibronic ground state of the complex (Table 3) can be interpreted as being mainly caused by the Renner-Teller effect and also electron spin decoupling accompanying the extremely large differences in the anisotropies of the A’ and A” components in the complex. It has been known that such anisotropies also lead to large parity splittings. In ArOH [18,19] and ArNO [14] the parity splittings in the low rovibronic levels are in the range of about 0.01-0.2 cm ~ ‘. For He02 [lo] we have obtained parity splittings in the range of 0.2-1.4 cm- ‘. In the present case of HeN,t we found the largest differences (up to 1.7 cm ~ l; Table 4) known so far for van der Waals complexes. The dominant contribution to this splitting comes from the mixing of the A’ and A” component of the electronic wavefunctions which is a direct manifestation of the Renner-Teller effect.
5. Conclusions
Using two-dimensional MRCI potential energy functions of the HeN,t(‘II) complex bound rovibronic levels have been calculated. It has been
Table
4
Some calculated parity splittings” (e-j’) in the rovibronic levels of HrN+(A’Il) (l.h. I.,)
J= l/2 (cm~ ‘)
J=3/2
(0, 0)
- 1.656 0.767
~ 1.451 - 1.520 1.294 - 0.099 -0.129 - 1.589 0.225 0.023 - 1.323 - 0.697 1.373 -0.201
(0, 01 (0.0) (0.0)
(LO) (2.0) (2.0) (2.0) (0, 1) (0, 1) (0. 1) (0, 1)
-0.275 - 1.655 - 1.723 1.308
a The parity e corresponds to (-l)-“‘. ( _ , )J+ IQ,
(cm-~‘)
the parity f to
found that the weak interaction of N: with He causes a spin flip of the inverted doublet of the diatom. The calculated structures of energy levels in HeN,t (‘C +) (free rotor case) and HeN,+ (A2H) (Renner-Teller near-degenerate T-shape structure) should prove useful in identifying and interpreting the electronic A-X transition in this complex.
Acknowledgements
This study has been supported by the EC grant ERFMRXCT960088 and the Deutsche Forschungsgemeinschaft. We thank Marie-Lise Dubernet for helpful discussions.
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