J . Phys. Chem. 1994,98, 1073-1079
1073
Theoretical Study of Bound States of Ar-NO Thomas Schmelz and Pave1 Rosmust Fachbereich Chemie der Universitat Frankfurt, 60439 Frankfurt, F.R.G.
Millard H. Alexander* Department of Chemistry and Biochemistry, University of Maryland, College Park, Maryland 20742 Received: September 13, 1993; In Final Form: November 22, 1993” W e have used a recent a b initio C E P A (coupled electron pair approach) potential energy surface as well as two earlier potential energy surfaces to calculate the energies of the lowest rovibronic levels of the nearly degenerate electronic ground states (2A’ and 2A”) of the Ar-NO(X211) complex. W e use a variational approach which includes the rotation-vibration, electron angular, and spin momenta coupling but neglects nuclear spin coupling. The results allow a direct comparison with measurements of high-resolution rotational spectra in the vibrational ground state. Predictions are also made for the positions of bound rovibronic levels ( J = l / 2 ) , which have not yet been observed.
1. Introduction There has been considerable recent interest, both experimental as well as theoretical, in dimers of noble gases with diatomic radicals. A particularly successful example is the ArOH complex, in which the sophisticated ab initio potential energy surfaces (PES’s) of Degli-Esposti and Werner’ have been used to interpret the high-resolution spectroscopy of the satellite lines in the region of the 0-0 band of the A + X transition in the isolated OH molecule.2“ These ab initio PES’s provided an invaluable template for the subsequent development of phenomenological PES’s both for the ground7J [ArOH(X211)] and excited9-” [ A r O H ( A W ) ] electronic states of the complex. The cylindrical degeneracy of a molecule in a II electronic state is lifted by approach of a spherical atom, giving rise to two states whose wave functions 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 PES’s.I2-l7 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.ls The Ar NO(X2II) system has long been a prototype for the interaction between a molecule in a (2II) electronic state and a spherical atom. Molecular-beam studies have been reported of total integral19 and differential20 cross sections as well as integral21-23 and differe11tiaP-2~ cross sections for inelastic scattering into individual final states. Mills, Western, and Howard28329 have used molecular-beam electronic resonance to investigate the pure rotational transitions of the ArNO(X211) van der Waals complex. Some years ago Nielson, Parker, and Pack (NPP)14 used the local density method of Gordon and Kim3oto model the A’ and A” PES’sfor Ar-NO(X211). These PES’s were then modified by Joswig, Andresen, and Schinke (JAS),22as suggested earlier by NPP,14 to obtain better agreement with the integral cross section data. Another empirical PES has been extracted by Casavecchia, Lagan& and Volpi (CVL)20 from total differential scattering measurements. More recently, a fully ab initio calculation of the A’ and A” Ar-NO PES’s has appeared,3I determined with the correlated electron pair (CEPA-1) method of Meyer.32-34 These PES’s were then used in calculations of differential and integral inelastic scattering cross sections.31
+
Present address: Dtpartement de Chimie, Universitt de Marne la Vallte, 93160 Noisy le Grand, France. e Abstract published in Advance ACS Abstracts, January 1, 1994.
In the present paper we use the recent ab initio PES’s to calculate the energies and wave functions for the bound rovibronic levels of the ArNO(X211) complex. We make use of our earlier method for the determination of rovibronic energy levels of Renner-Teller systems which includes explicitly the electron spin,35but the kinetic energy operator is expressed in Jacobi rather than internal coordinates. 2. Ab Initio Calculations and Potential Energy Surfaces The Af and A” PES’s were determined using the coupled electron pair (CEPA- 1) method.32-34 This method, which is sizeconsistent and faster than multireference configuration interaction methods, has been used with considerable success to provide accurate PES’s for interactions of open-shell fragments with a noble gas target, in particular NH-He,36 OH-He,37 and OHAr.1 Weused the recent augmentedcorrelation-consistentvalence triple-{ (avtz) basis of Dunning and co-workers38J9 (1 ls6p3d2f contracted to 5s4p3d2f for N and 0 and 16slOp3d2f contracted to 6s5p3d2f for Ar). This avtz atomic orbital basis contained a total of 142 contracted Gaussian functions. All calculations were carried out with the MOLPRO suite40 of ab initio programs. The geometry of the Ar-NO system will be described by the three Jacobi coordinates r (the N O bond length), R (the distance between the Ar atom and the center of mass of the N O molecule), and 0, the angle between R and 3, with 0 = 0’ corresponding to linear ArNO. CEPA calculations of the A’ and A” Ar-NO PES’s were carried out at 11 values of 0 (0 = 0, 20, 40, 60, 80, 90, 100, 120, 140, 160, and 180’) and 9 values of the centerof-mass separation R (5.5,6, 6.5, 7, 7.5,8, 8.5, 9, and 10 bohr). The N O bond distance was held to r = 2.1746 bohr, the experimental equilibrium bond length.41 Additional calculations at 0 = 90’ were carried out for R = 5 bohr. Figure 1 presents contour plots of the A’ and A” PES’s. The Af’ PES possesses the lower minimum, which occurs at R = 7.09 bohr and 0 = 73’. The minimum on the A‘ PES, at R = 6.98 bohr and 19= 95”, lies only 1.3 cm-I higher in energy. The lowest panel of Figure 1 illustrates the splitting ( VA,- V A ~between ) the two PES’s. The experimental estimate of the energy difference ~ be seen at the minima of the PES’s ( T e )is -1.5 ~ m - l .As~ can in the lower panel of Figure 1, this estimate represents an oversimplification: the calculated splitting varies quite significantly in the region of the minima in the A‘ and A” PES’s. Another crucially important parameter in bent-bent RennerTeller molecules is the barrier to linearity.Is This is the height
OQ22-3654/94/2098-1073%04.50/Q 0 1994 American Chemical Society
-
.
1074 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 9.0 r
Schmelz et al.
-
‘ 2
~
8.5
8.0 L--.
4e a!
7.5
7.0 6.5 6.0 5.5 9’0 8.5 8.0
n
4e a!
7.5
7.0
”
”
”
h
20
e e
6.0
I
A’
5.5
9.01 --8.5
--A
I
h
20
e Di
6.5
6.0, J.J
0 20 40 60 80 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0
e (degrees) n
4 e a!
0 20 40 60 80 1 0 0 1 2 0 1 4 0 1 6 0 180 0 (degrees) Figure 1. (Upper and middle panels) Contour plots of the A’ and A” ArNO CEPA PES’s. The zero of energy is taken to be the minimum on the A” PES, which occurs at an energy of -79.1 cm-l with respect to the
separated species. In both panels the contours are equally spaced at 10-cm-’ intervals. The minimum contour shown in 10 cm-I, and the maximum, 200 cm-I. The positions of the minima are indicated by X’s. The minimum on the A’ PES lies 1.3 cm-I above the minimum on the A!’ PES. (Lower panel) Contour plot of the splitting V A-~VA,,between the two CEPA PES’s. The minimum solid contour is 2 cm-I, and the maximum, 20 cm-I. The dashed contour indicates a splitting of -2 cm-I. The X’s indicate the position of the minima on the A’ and A” PES’s. of the minimum for the collinear cuts in the PES’s (e = Oo and 0 = 180’) measured with respect to the minimum point on the PES. For rotation of the Ar atom about the 0 end, we predict this barrier to be 22.0 and 20.5 cm-I for the A” and Af PES’s, respectively. For rotation of the Ar atom about the N end the comparable barrier heights are predicted to be 19.5 and 18.0 cm-I, respectively. The extreme smallness of these barriers to rotation indicates that the A r N O complex will be very floppy. In the subsequent determination of the rovibronic wave functions, it is convenient to fit the set of CEPA energies to a relatively simple functional form for both adiabatic PES’s separately. We have chosen an expansion in terms of reduced rotation matrix namely
/=0
Here L is the number of Jacobi angles 0 for which the potential has been calculated. The vector A(R) can be expressed as
A ( R ) = P-’.B(R)
(2)
where
P,, = &(COS
e),
(3)
Figure 2. Contour plots of the A’ (upper panel) and A” (lower panel) ArNO PES’s,predicted by the local-density calculation of ref 14. The zero of energy is taken to be the minimum on the A” PES, which occurs at an energy of -32.5 cm-1 with respect to the separated species. The witions of the minima are indicated by X’s. The minimum on the A’ PES lies only 1 cm-Labove the minimum on the A” PES. In both panels the solid contours are equally spaced at 10-cm-l intervals. The minimum solid contour is 10 cm-l, and the maximum solid contour, 200 cm-I. The dashed contour indicates an energy of 2 cm-I above the minimum on the A“ PES. and
COS
pI= e) (4) with e k denoting the individual angles at which ab initio points were calculated and 3
4
-‘/J
1
+ tanh(R)] [ c c i 2 i ) R - 2 i ] i= 1
In addition to the CEPA and PES’s, we have carried out calculations using two other sets of PES’S. The first was the original local density PES’S of Pack and co-workers,I4 (NPP). The second set of PES’Swas the modification of the original N P P PES’S by Schinke and co-workersZ2(JAS), which was based on ~J~ suggestions by NPP.I4 The analytic fit to these P E S ’ S ~was reexpanded in the form of eqs 1-5. Figures 2 and 3 present contour plots of the N P P and JAS PES’s. In addition to the theoretically determined PES’S described above, there exists the empirical ArNO PES of Casavecchia, Lagan& and Volpi (CLV),2O extracted from total differential scattering experiments. This potential suffers from three major shortcomings: First, CLV were unable to obtain information on the splitting between the A’ and Af’ PES’S and, hence, assumed that the two were equal. Second, the experimental data were fit to an expansion which included just the isotropic (I = 1) term in eq 1 and thelargest anisotropic term [ 1 = 3, the component whose angular behavior is governed by the Pz(cos e) Legendre polynomial]. Thus the CLV PES is symmetric with respect to a 180’ rotation of the NO molecule and ignores the small differences in the behavior of the interaction when the Ar atom approaches from the two ends of the N O molecule. Finally, the published parameters of CLVZoyield a significant (=35 cm-I) discontinuity in the calculated PES in the long-range region.
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1075
Bound States of Ar-NO
TABLE 1: Dependence on Ar Distance of Various Matrix Elements of the Electronic Orbital Angular Momentums
g
8.0
R
ALz2 b
Lx
7.5
10 9
-4.182 -4.182 -4.178 -4.177 -4.171 -4.176
-1.37(-5)i -4.77(-5)i -1.07(-4)i -2.93(-4)i -1.22(-3)i -4.76(-3)i
h
2 7.0
a:
8
6.5 6.0 5.5’
”
”
”
”
’
9.0
‘ \ 8.r 6 8.5
4
-.
8 . 0 r 8.0
h
ea:
7.5
7 6 5
LZ 0.999 86i 0.999 86i 0.999 82i 0.999 69i 0.999 05i 0.995 08i
T-shaped ArNO geometry (8 = 90°), with the NO molecule lying along the z-axis and the y-axis lying perpendicular to the ArNO plane. Atomic units used throughout. (*A1L;1*A’)- (2A’1L~~2A’’), (*A1L,12A”);power of 10 in parentheses. (2A1Lz12A”).
the variational calculation of the energies of the lower rovibronic states of ArNO.
7.0
3. Determination of Rovibronic Energies and Wave Functions
6.5
6.0 5.5’ I 0 20 40 60 80 100120140160180 ’
’
’
0 (degrees)
Figure 3. Contour plots of the A’ (upper panel) and A“ (lower panel) ArNO PES’S, predicted by the JAS modification22 of the local-density N P P PES’s.14 The zero of energy is taken to be the minimum on the A” PES, whichoccursatanenergyof-124.3 cm-I withrespecttotheseparated species. The positions of the minima are indicated by X’s. In both panels the contours are equally spaced at 10-cm-Lintervals with a maximum at 200 cm-’. The minimum contour on the A” PES is 10 cm-’. Since the minimum on the A’ potential lies 26.4 cm-I above the minimum on the A” potential, the minimum contour on the A‘ PES is 30 cm-I.
The variational calculations of the bound rovibronic energy levels of ArNO were carried out using an approach closely related to our method35for bound Renner-Teller systems which includes the electron spin explicitly. However, in contrast to this prior work, here the kinetic energy operator is expressed in Jacobi rather than internal coordinates. The explicit form of the Hamiltonian is45346
H = H,
+ H,+
V
9.0 8.5
4
8.0
h
7.5
2 7.0
a:
6.5 6.0 5.5’ I 0 20 40 60 80 100120140160180 0 (degrees) ’
’
’
’
200
160 -
40/ n
,
U
9
(
Y
5
6
7
8
9
1 0 1 1 1 2
R (bohr)
Figure 4. (Upper panel) Contour plot of the CLV ArNO PES.2o The zero of energy is taken to be the minimum on the PES,which occurs at an energy of -1 11.6 cm-’ with respect to the separated species. The contoursareequllyspacedat 10-cm-1 intervals,andthemaximumcontour shown is 200 cm-l. The position of the minima is indicated by the X. (Lower panel) Plots of the CLV ArNO PES for 8 = Oo and B = 90°, computed as prescribed in ref 20. The discontinuity at long range, mentioned in section 2, is clearly apparent.
The upper panel of Figure 4 displays a contour plot of the CLV PES. The discontinuity a t long range, discussed in the preceding paragraph, is illustrated in the lower panel of Figure 4. Since even the lowest rovibronic wave functions of the A r N O complex sample this discontinuity, the CLV PES was not used further in
+
Here p1 = mN-l + mAI-l and p2 = mo-I (mN + mAr)-l. Also, 11= J + L + S is the rotational-vibrational angular momentum, with J, L, and S designating the total, electronic, and spin angular momenta of the ArNO molecule. As discussed above, in the present application we have kept r fixed so that all derivative terms involving r in eq 8 are subsequently suppressed. Following closely the proceduredescribed in ref 35, weobtained vibrational eigenvalues and eigenvectors for the Hamiltonian of eqs 6-8. The basis set consisted of products of Morse eigenfunctions for the stretching mode and Legendre functions for the torsional motion of the rigid NO moiety. The interested reader is referred to our earlier paper35 for technical details of the necessary angular momentum matrix elements and the stepwise optimization of the basis functions. Our method permits full three-dimensional variational calculations to be performed on virtually any open-shell triatomic Renner-Teller molecule. We have tested our method for weakly bound complexes such as ArOH and 02+He:7 intermediate bound structures such as HF+He,47 and also for classical Renner-Teller molecules such as NH247 and H20+.47*48 For the latter two molecules we obtained results identical to those reported previously based on the more usual expansion of the triatomic Hamiltonian in internal coordinates. In the present application we have further neglected the variation of the spin-orbit constant of NO (A = 123.2 cm-I 41) on the approach of the Ar atom. For the comparable ArOH(X211)van der Waals complex, the recent spectroscopic study of
Schmelz et al.
1076 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994
l
1
.......
......-:: __.. .......
.......
........
.............. ...............-
O+ -
e-
0-
et
1.50 1.25 -
-..:::::::
................. ..... .... . . -.: ................-_
bl
CEPA
.......- ot 0Ot 0-
NPP
1.00-
bx
0.75 -
-
0.50-
e+ P
v
?! w“
’ asymmetric top asymmetric top
- 0.2
(T-shaped)
( not T-shaped) asymmetric top ( not T-shaped)
with
O+ 0-
TABLE 2: Energies (in cm-l) of Lower Bend-Stretch Levels of Positive Parity for the ArNO Complex” Yb
us
CEPAb
NPPE
JASd
0
0
1
0
0
1
2
0
1
1
0
2
0.000 0.223 5.350 5.631 10.200 10.848 15.548 16.426 14.370 14.600 17.622 18.765
3
0
0.000 1.635 6.779 1 1.205 16.715 17.989 25.119 27.325 30.762 3 1.909 46.108 47.618 36.018 37.789
2
1
0
3
1
2
4
0
2
2
0.000 0.245 10.951 11.780 15.996 16.914 22.331 24.269 27.131 29.937 31.887 33.682 36.594 37.606 39.132 41.877 43.798 47.491 48.282 49.174 51.433 52.471 54.004 55.199
O+ 0-
0.25 1
e-
et
49.660 50.445
*
Lester and co-workers6 indicated that the difference between the spin-orbit splittings in the complex and the isolated OH radical was less than 10%. Similarly, the geometry dependence of the expectation values of the operators for the electronic angular momentum has not been considered; all values were set to the those appropriate to the isolated NO(X211)molecule. This latter approximation is well justified, as shown recently for H20+.48 Further justification is given in Table 1, which lists calculated values of some representative L electronic matrix elements for perpendicular (T-shaped) approach of the Ar atom to the N O molecule. As can be seen, the changes are negligibly small. Ultimately, as more precise PES’s become available, then it will be necessary to include the geometry dependence of the L and L-S which are contained in the Hamiltonian of eq 6. To guide the reader in the subsequent discussion of the calculated rovibronic levels of the ArNO complex, Figure 5 presents a correlation diagram of the lower rovibronic levels. This illustrates the level splitting which arises from the deviation
e+
-
et
O+
o+ 0-
0-
ee+
Figure 6. Calculated positions of the lowest three (J = ‘/z, 3/2, 5/2) rotational levels in the K = stack for the ArNO complex as predicted by our calculations with the CEPA,” NPP,I4 and JASz2 PES’s. The solid bars indicate the experimental excitation energies from ref 29 for the transitions indicated by the vertical arrows. Note (see Table 4) that the CEPA calculations put the J = I/zo(+) state ~ 0 . cm-’ 1 above the J = 3 / 2 0 ( - ) state, in contrast to experiment.
6.0
The upper and lower entries correspond respectively to the e and o labeled levels (see Figure 5). CEPA PES’s of ref 31. Electron gas PES of ref 14; only fully bound levels are given. Modified electron gas PES of ref 22; only those low-lying levels are given for which a clear correspondence to the CEPA levels could be found.
e -
e
A’ and A” Figure 5. Schematic correlation diagram for the rotational levels with K = l / 2 for the ArNO complex, with rotational constants A = 1.70 cm-I, E = 0.0740 cm-I, and C = 0.0710 cm-I. These values, as well as the labeling of the fine-structure levels, are taken from ref 29.
JAS
c
‘
I
8.5 7 8.0 h
0
P g 7.0
v
6.5
1 I
6*o’ 0
50
100 150 0 (degrees)
’
Figure 7. Wave functions for the (a= 0, us = 0 ) J = I/2e(+) level with energy (Table 2) of 0 cm-I. The upper panel indicates the component of the rovibronic wave function (54.1% of the total weight) associated with the A” electronic state, and thelower panel, thecomponent associated with the A’ electronic state. In both panels the maximum amplitude is normalized to unity, with a contour spacing of 0.05.
from T-shaped geometry and from the mixing of the A’ and A” electronic states. Table 2 lists the energy levels for J = ‘ / 2 and positive parity determined from the CEPA PES’s. The position of the pure rotational levels (Ob = 0 and us = 0) is illustrated schematically in Figure 6. The calculated zero-point energy for the two-
Bound States of Ar-NO
The Journal of Physical Chemistry, Vol. 98, No. 4, 1994 1077
8.5
A
I’
3 7.5 - ;
h
0
e as
7.06.5 -
__I
6.0
8*5!
h
2
0
e
6.0 L 8.5 I
1,
8.0
8.0
7.5
2 7.5 0
e
7.0 6.5 6.0
A’
0
7.0
6.5 50 100 150 8 (degrees)
b
6.0
I
50 100 150 8 (degrees)
Figure 8. Wave functions for the (ub = 1, us = 0) J = ‘/ze(+) level with energy (Table 2) of 10.951cm-I. Theupperpanelindicatesthecomponent of the rovibronic wave function (58.5% of the total weight) associated with the A” electronic state, and the lower panel, the component associated with the A‘ electronic state. In both panels the maximum amplitude is normalized to unity, with a contour spacing of 0.05. The dashed contours indicate negative values. The vibrational assignment is confirmed by the nodal pattern.
Figure 9. Wave functions for the (ub = 0,us = 1) J = I/ze(+) level with energy (Table 2) of 15.996cm-I. Theupper panelindicatesthecomponent of the rovibronic wave function (58.5% of the total weight) associated with theA”e1ectronicstate, and thelower panel, the component associated with the A’ electronicstate. In both panels the maximum amplitude is normalized tounity, with a contour spacing of 0.05. The dashedcontours indicate negative values. The vibronic assignment is confirmed by the nodal pattern.
dimensional (Rand 0) potential is 24 cm-I. Rovibronic levels lying up to 55 cm-1 above the zero-point energy are bound. The assignments in Table 2 were obtained from an inspection of the nodal structure of both Renner-Teller components of the rovibronic wave functions. As an example, Figures 7-9 present contour plots of the rovibronic wave functions for the J = ‘ 1 2 e-labeled states of positive parity for the ground (Ob = 0 and us = 0), lowest bend (l,O), and lowest stretch (0,l) states. Both the A” and A’ electronic states have almost equal weights in the final rovibronic wave function ( 5 6 5 9 % and 41-46%, respectively). The plots demonstrate nicely the large amplitude character of both rovibronic levels. Also, the coupling between the bending and stretching motion is clearly apparent even in these two lowest excited rovibronic states. At energies greater than =35 cm-I above the zero-point energy, the rovibronic levels become so strongly coupled that a completely unambiguous assignment of the nominal Ar-NO stretch (us)and bend (ub) quantum numbers is no longer possible. For example, Figure 10 shows the A” Renner-Teller component of the rovibronic wave functionfor the (ub = 3, us = 0) and (0,3) levels. Due to the small rotational constant of the NO molecule (1.672 cm-I 41) relative to the barrier to rotation of the Ar atom around the NO molecule ( ~ 2 cm-l, 0 see above), the low-lying bending states of the ArNO complex can be regarded as “harmonic oscillator-like” in that they exhibit equal spacings. Comparable energies determined for the NPPI4 and JASZ2 PES are also given in Table 2. Although the agreement between the calculated energies is good for the lowest three vibrational levels [(O,O), (l,O),and (O,l)J, for the higher levels thedifferences between the NPP, JAS, and CEPA values are so large that for some of the levels a direct comparison is impossible. These
differences reflect the largedifferences in the dissociation energies predicted by the three potentials: for the lower (A”) PES the values of De are 32 cm-I for the NPP PES, 124 cm-I for the JAS PES, and 79 cm-I for the CEPA PES. Currently, the only available experimental data concerning the vibrational levels of the complex are those of Miller and C h e r ~ g .These ~ ~ authors derived a value of =40 cm-I for the stretching frequency, significantly larger than our calculated values (Table 2). Similarly, their estimate of the zero-point corrected dissociation energy (DO= 120 cm-I) is significantly higher than the values calculated for the CEPA PES’S (Do = 55 cm-I). For the OH(X211)Ar complex, the CEPA calculations, of comparable quality, by Degli-Esposti and Werner’ predict a well depth of De = 105 cm-I, which is =I 8% smaller than the best experimental estimate of the well depth (De = 127 cm-I *). If we assume a similar (2625%)error in the CEPA calculations on ArNO, we would predict a zero-point corrected dissociation energy of =95 cm-I for the ArNO complex. A quantity which deserves special attention is the parity splitting. In van der Waals complexes of noble gas atoms with diatomic molecules in closed-shell IZ electronic states, the degeneracy of rotational levels with the same J but opposite parity is lifted by Coriolis interaction^.^^ However, for the interaction of noble gas atoms with diatomic molecules in electronically degenerate states, the dominant contribution to this splitting c o m a from the mixing of the two (A’ and A’f) electronic wave functions, which is a direct manifestation of the Renner-Teller effect. Several approaches to the determination of this parity splitting, which have been used on perturbation theory, have appeared in the In the particular case of ArOH(XzII), Green and Lester7 used a perturbation expansion up through third order to
1078 The Journal of Physical Chemistry, Vol. 98, No. 4, 1994
Schmelz et al. TABLE 3 Splitting (in cm-l) between Positive and Negative Parity Levels of the ArNO Complex’ CEPAb
level
8.0 -
JASc Vb
2 7.50
J = 1/20 J = 312 o J = 512 o
0.034 0.047 0.068
J = 1/20
0.023 -0.030
&
= 0,
Us
NPPd
exW
0.003 0.003 0.004
0.020 0.029
=0
0.029 -0.01 1 0.074 Yb’1,Va”O
g 7.0-
J=3/20 J=5/20
6.5 -
-0.019 -0.025 -0.001
-0.0008 Vb
J
6.0
J = 1/20 J = 312 o J = 5/20
-0.066
J = 1/2e
J=3/2e J = 5/2e
-0.026 -0.061 0.069
J = 112 e J = 312 e J = 512 e
0.017 0.043 0.056
= 0,
Us
=1
0.225 0.388 -0.01 1
0.006
0.003 Vb
-0.003 -0.012 0.0007
= 0, Vs = 0
0.057 0.055 0.015 Vb=1,VB=o -0.0 19 -0.024 0.0004 Yb
-0.014 -0.015 -0.01 1
0.002 0.004 0.004 -0.014
-0.019 -0.016
= 0, VI = 1
J - 1/2e J = 3/2e J = 512 e
6.0
‘
0
50 100 150 0 (degrees)
Figure 10. Wave functions for the (ob = 3, us = 0) (upper panel) and (0,3) (lower panel) J = ‘/2e(+) level with energies (Table 2) of 36.594
and 43.798 cm-’, respectively. Only the components of the rovibronic wave function associated with the A” electronic state are displayed. In both panels the maximum amplitudeis normalized to unity, with a contour spacing of 0.05. The dashed contours indicate negative values. As can be seen, the nodal patterns do not allow an unambiguous assignment. determine the doubling for the lowest K = - l / 2 state. Ohshima et al.51 used second-order perturbation theory to calculate a general expression for the relative ratio of the parity doubling in the II(I = ‘ 1 2 as compared to the lI(I = 3/2 levels.50 However, Dubernet et a1.52concluded that this latter expression was incomplete and suggested that matrix elements off-diagonal in thevan der Waals stretching quantum number (v,) could play an important role in the magnitude of this parity splitting. In Table 3 we list the parity splittings for the lower three ( J = l/2, 3/2,5 / 2 ) rotational levels of the ground state ( v b = 0, v, = 0) and the first two excited (ob = 1, us = 0 and ob = 0, v, = 1) vibrational levels of the ArNO complex, as calculated using the CEPA, NPP, and JAS PES’s. For the ground state, a comparison is also made with the two available experimental values of Howard and c o - w ~ r k e r s .There ~~ is considerable disagreement between the calculated parity splittings. The CEPA PES’s yield the best agreement with the two available experimental values. For these two cases, the N P P PES’s yield parity splittings which are an order of magnitude too small; while the JAS PES’s lead to a sign reversal in one case. Table 4 compares the calculated wavenumbers for pure rotational transitions in the ArNO complex, as predicted by all three sets of PES’S. Also included is a comparison with the available experimental data of Mills et al.29 (In this comparison we neglected the tiny shifts corresponding to the various nuclear hyperfine levels.) As might have been anticipated, because the JAS PES’s predict an incorrect equilibrium geometry (compare Figures 1 and 3), they fail to describe correctly the observed rotational transitions. In contrast, both the CEPA PES’s and, particularly, the N P P PES’s yield good agreement with exper-
-0.083 0.028 -0.006 0.0003 0.043 -0.006 0.0214 0.076 0.0004 See Figure 3; For the o labeled levels the positive parity levels lie above the negative parity levles; this is reversed for the e labeled levels. CEPA PES’s of ref 31. Electron gas PES’s of ref 14, modified as described in ref 22. Electron gas PES’s of ref 14. Experimental values from ref 29. TABLE 4: Comparison of Calculated and Experimental Splittings (in cm-l) for Pure Rotational Transitions in the ArNO Complex. transition CEPAb N P F JASd expte 0.105 1.488 0.014 J = 1/20(+)+J= 3/20(-) -0.095 J = 3/2e(-)-+J=3/20(+) 0.315 0.277 1.701 0.225 J = 512 o(-) J = 312 o(+) 0.674 0.558 1.981 0.577 a See also Figure 5. CEPA PES’S of ref 31. Electron gas PES’Sof ref 14. Electron gas PES’s of ref 14, modified as described in ref 22. Experimental values from ref 29; the observed, but tiny, hyperfine splittings are here suppressed. -+
iment. The CEPA PES’s predict the J = 3/20(-) state to be lower in energy than the J = l/20(+) state, which disagrees with experiment. Mills et have shown that the positions of the J = l / 2 0 and J = 3/2e states depend critically on the deviation from T-shaped structure and on the magnitude of the RennerTeller quenching parameter, which is, of course, related to the splitting between the A’ and A” PES’s. Here, a change in the angular geometry of only a few degrees will reverse the ordering of the J = ‘120 and J = 3/20 states. On the whole, the CEPA PES’s describe well the measured values of both the parity and rotational splitting. TheNPP PES’s predict the rotational splittings somewhat more accurately but substantially underestimate the parity splitting. In earlier work,31 we showed how the CEPA PES’s gave a better description than the N P P PES’s of the available i ~ ~ t e g r a land ~ l -differential2+Z7 ~~ scattering data. The accuracy achieved in the present calculations is certainly not sufficient for high-resolution spectroscopic predictions. Indeed, the disagreements between theory and experiment were seen to be as large as 50%. Notwithstanding, the CEPA PES’s should provide a reliable template for a more refined characterization of the electronic ground state of ArNO. We feel that the development of refined PES surfaces will be most profitable only after additional spectroscopic information is obtained on excited rovibronic states of ArNO. We encourage the experimentalist to obtain this information, which will provide
Bound States of Ar-NO further tests of our ability to simulate the nuclear-electronic dynamics of this weakly bound open-shell complex.
Acknowledgment. M.H.A. is grateful to the U.S.Army Research Office for partial support of the work reported here, under Grant DAAL03-91-G-0129. T.S.and P.R.are grateful for the support of the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie.
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