Molecule-doped rare gas clusters - Wales Group

MOLECULAR PHYSICS, 2000, VOL. 98, NO. 4, 219± 229

Molecule-doped rare gas clusters: structure and stability of ArnNO (X 2P 1=2, 3=2), n #- 25, from new ab initio potential energy surfaces of ArNO F. Y. NAUMKIN1 and D. J. WALES2* 1 Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada 2 University Chemical Laboratories, Lens® eld Road, Cambridge CB2 1EW, UK

(Received 4 September 1999; revised version accepted 5 October 1999) High level ab initio calculations carried out for the 2A 0 and 2A 00 states of ArNO(X 2P ) predict a crossing near the T-shape con® guration, with the 2A 0 minimum being slightly deeper. Spin± orbit coupling is included through a model treatment and results in two potential energy surfaces with similar topologies, nearly parallel to each other and close to the averaged non-relativistic surface. These results are used to construct a DIM-like model for Arn NO clusters. The lowest energy cluster structures are found to resemble those for Arn‡1 with NO lying in the surface. The set of major magic numbers (structures of pronounced stability) is also the same as for the Arn‡1 clusters, and is emphasized further by the detachment of NO, which requires a larger energy than for detachment of a single Ar atom. The relations of the di erence between the two dissociation energies and of the Arn NO(1=2 !3=2) excitation energy to the magic numbers are discussed. 1.

Introduction

Van der Waals complexes of molecules with rare gas atoms have been studied by both experimentalists and theoreticians for decades. The weak interactions have been characterized in terms of binding energies and equilibrium con® gurations, scattering cross-sections, microwave and infrared spectra, etc. Related processes and phenomena in bulk matter are represented, e.g., by photoinduced dynamics of molecules embedded in rare gas solid matrices, including dissociation, recombination, transition spectra, etc. The intermediate size range, i.e., clusters, has received less attention. One such system of interest is the NO molecule in an argon environment. The ArNO complex has been investigated, in particular, by means of photoionization spectroscopy [1], an IR-UV double resonance technique [2], electronic resonance spectroscopy [3], and total integral and di erential cross-section scattering [4, 5] (see [2] for a recent review). Also Rydberg excitation of NO has been studied in the Ar solid matrix [6]. Arn NO clusters have been probed by multiphoton ionization and photodissociation [7]. There are a few ground state Ar± NO potential energy surfaces (PESs) available. The recent ab initio PES by Alexander [8] was calculated using the correlated elec* Author for correspondence. e-mail: [email protected]

tron pair approximation (CEPA) and was used successfully to describe the experimental di erential and integral cross-sections for inelastic scattering and the collisional transfer of rotational energy [2]. The empirical PES of Casavecchia et al. [5] was constructed by ® tting the experimental scattering data to a two-eventerms Legendre expansion. This PES exhibits signi® cantly stronger binding between Ar and NO in both the linear and T-shape geometries, in accord with the spectroscopic data [9, 10]. The present work investigates the Ar± NO(X 2P ) interaction potential at a higher level of theory to check the empirical predictions. We also investigate Arn NO clusters for n µ 25 in terms of their structure and stability. 2.

Theory

2.1. The model potential The electronic structure of NO, with its singly occupied molecular p orbital, can be interpreted within a valence-bond (VB) framework in terms of a single half-occupied atomic p orbital orthogonal to the molecular axis (z). For Ar± NO di erent orientations of this orbital with respect to the ArÐ NÐ O plane (xz) produce two electronic states: 2A 0 (for p ˆpx ) and 2A 00 (p ˆpy). In an Arn NO cluster, the orientation of this p orbital depends upon the interplay of interactions in all

Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals/tf/00268976.html

220

F. Y. Naumkin and D. J. Wales

the ArNO units, and thus on the cluster structure. Hence, each Ar atom is associated with a di erent mixture of 2 A 0 and 2 A 00 states. Therefore a simple additive form for the Arn NO potential is, strictly speaking, an oversimpli® cation:

" ‡X# X n

V Arn NO

6 ˆ

i

n

V Ari NO

jˆ 6 i

V Ari Arj :

…1†

However, the additivity with respect to Ar atoms can be preserved within a matrix formalism similar to the diatomics-in-molecules method [11] by treating NO as an e ective monoatomic species, whose states are determined by the orientation of the half-occupied p orbital. We can represent the ArNO potential in the basis set associated with the ® xed px and py orientations: VArNO

ˆ

0@

V A 0 cos2 ¿ ‡V A 00 sin2 ¿

…V A 0 ¡V A 00 †sin ¿ cos ¿

…V A 0 ¡V A 00 †sin ¿ cos ¿ V A 0 sin2 ¿ ‡V A 00 cos2 ¿

1A ;

…2†

with V A 0;A 00 ˆV A 0 ;A 00 …R;³ † being the 2 A 0 and 2 A 00 state potentials for the Ar position relative to the mid-point of the NO bond, determined by spherical coordinates R;³;¿. Here we assume the NÐ O distance to be ® xed, e.g., at equilibrium. However, it could be treated as a parameter so long as the interaction with Ar remains weak, so that excited states of NO need not be included. The total cluster Hamiltonian matrix is then (using the 2 £2 unit matrix E)

" ‡ X# ˆX n

HArn NO

i

n

VAri NO

E

jˆ 6 i

V Ari Arj :

…3†

Here we de® ne the zero of energy for non-interacting Ar atoms and NO. Diagonalization of H yields potential energy surfaces of two electronic states of the system, which therefore can be obtained analytically by solving the quadratic eigenvalue equation. Even though spin± orbit splitting, D, between the 2P 3=2 and 2P 1=2 states of NO is relatively small (around 120 cm¡1 ), it may a ect signi® cantly the associated ArNO potentials due to their close non-relativistic energies, which are degenerate for the linear geometry. We use the following model for the relativistic case. The above splitting is determined mostly by the halfoccupied p orbital, as for a system containing a rare gas ion or a halogen atom. The relevant spin± orbit interaction matrix, HSO, which is added to the nonrelativistic Hamiltonian matrix within the `atoms-inmolecules’ approach [12], couples states corresponding to the px;y and pz orientations (see, e.g., [13, 14]). In our

case the pz orientation means an excited state of NO, and any such state is separated from the ground state by more than 5 eV (at least near the ground state equilibrium geometry). The o -diagonal elements in HSO due to pz have values of a fraction of D and will not a ect the relativistic potentials signi® cantly. A similar prediction 2 was con® rmed [15] for the analogous Ar± I¡ 2 ( P 1=2 ; 3=2) interactions, associated with even stronger spin± orbit coupling and smaller gaps between the non-relativistic states. Thus, a suitable approximation is to consider only the part of HSO associated with px;y , which leads, for a given spin projection (and with i ˆp ¡1 ), to HSO

0

i¯

¡i¯

0

… †

ˆ

;

…4†

with ¯ being a characteristic parameter of the spin± orbit interaction. To reproduce the splitting for the isolated NO molecule, we have ¯ ˆ12 D. Finally, the relativistic Hamiltonian matrix is obtained, using H of equation (3), as Hrel ˆH ‡HSO ; …5† and diagonalization produces potentials for the Arn NO(2P 1=2; 3=2 ) clusters. The results are analytical, as before: V 1=2 ;3=2

V0

ˆV 0 ‡‰V 1 ‡V 2 ¨ f…V 1 ¡V 2 †2 1=2 ‡4V 212 ‡D2g Š=2;

X X ˆX …† ‰ ‡ ˆX ‰ … †¡ … †Š ˆ

n

n

i

jˆ 6 i

n

V 1…2†

i

V Ari Arj ;

V A 0…A 00 † i cos2 ¿i

n

V 12

i

V A0 i

V A 00…A 0†…i †sin2 ¿i Š;

V A 00 i cos ¿i sin ¿i ;

…6†

and in the case of a single Ar atom reproduce the expressions given previously (see, e.g., [5]). As above, spherical polar coordinates are employed for the Ar atom relative to an origin at the mid-point of the NO bond. Note that in this case ¿ ˆ0 or 908, depending on which non-relativistic state is lower in energy for a given position of Ar. Spin± orbit coupling requires a de® nite value for the total angular momentum projection, which implies mixing of the px and py based states with equal weights. This e ect may therefore lead to averaging the 2A 0 and 2 00 A potentials, and it is common to use the mean value (see [8] and references therein). Such averaging has been 2 shown to occur in Ar± I¡ 2 ( P 1=2 ; 3=2), where spin± orbit terms are much larger [15]. The above formalism allows for a direct check of this approximate treatment in the case of NO and for an investigation of the e ect of

Ab initio PES study of Arn NO, n 4 25 the Ar± NO interactions on this mixing. In particular, if the averaged Ar± NO potential is a su ciently good description, then the equality in expression (1) would be restored and the simple additive representation of V Arn NO would be a valid approximation. 2.2. Global optimization The same global optimization approach was used as ¤ for our previous studies of Ne‡ n , Arn Cl2 and Arn clusters [ ] using the DIM model 16± 18 . In this `basin-hopping’ or Monte Carlo minimization [19] technique the potential energy E is transformed to give a new surface: ~ …X†ˆmin fE…X†g; E

where X is the vector of nuclear coordinates and min signi® es that an energy minimization is performed starting from X. Canonical Monte Carlo (MC) sampling was used to ~ surface. For each cluster size ® ve runs of explore the E 5000 MC steps were performed from random starting geometries. The maximum step size for the displacement of any Ar Cartesian coordinate was dynamically adjusted to give an acceptance ratio of 0.5. The runs were conducted at temperatures corresponding to 0:01 Eh and 0:005 Eh for the ground state relativistic and non-relativistic potentials, respectively. The coordinates were reset to those of the current minimum in the Markov chain at each step. No systematic attempt was made to optimize the temperature of the MC sampling, since the lowest minima found in all ® ve runs for each size were usually the same. The lowest minimum was always found by a majority of the runs. The global minima for the ground state relativistic potential were relaxed using the relativistic ® rst excited state; the changes involved are very small and so we did not conduct separate global optimization runs for this state. All the results will be made available from the Cambridge Cluster Database [20]. 3.

Results and discussion

3.1. ArNO ab initio potential energy surfaces The ArNO(2A 0 ; 2 A 00) potential energy surfaces (PES) have been calculated ab initio with the open-shell coupled-cluster RCCSD-T method [21, 22] and the spdf (aug-cc-pVTZ) basis set [23, 24] using the Molpro package [25]. The results have been corrected for the basis set superposition error using the standard counterpoise method [26]. The NÐ O bond length was ® xed at Ê [27]. Calculations the ground state value r0 ˆ1:154 A have been carried out for 9 equidistant polar angles between ³ ˆ0 and 1808, the data being listed in table 1. Figure 1 shows both PESs interpolated radially with cubic splines and angularly with Legendre polynomials. The 2A 0 PES exhibits three minima, the lowest for the T-

221

shape con® guration (with Ar slightly nearer to the O atom) and two almost equally deep local minima for the two linear con® gurations, with a slightly stronger binding (larger De and shorter Re ) for the Ar± ON geometry (table 2). The 2 A 00 PES has a well only for the distorted T-shape con® guration (signi® cantly shifted towards the N end of NO), and saddles for both linear con® gurations. The 2 A 0 state is found to be more strongly bound by about 2.4 cm¡1 , and the two states cross between the linear and T-shaped geometries. These results are consistent with the very close approach of the two PESs predicted in previous calculations [8], although there the 2 A 0 global minimum was found to lie above that of 2 A 00 by 1.3 cm¡1 and the surfaces did not cross. The present results predict the Ar± NO binding to be generally stronger (by 20± 30% in De ), with a deeper well for the Ar± ON geometry than for Ar± NO [8, 28]. To improve the accuracy further and check these predictions we have employed ArN and ArO potentials calculated [29, 30] at the same level of theory using spdf and spdfg (aug-cc-pVQZ [23, 24]) basis sets. A correction was applied within the anisotropic atom± atom model [31, 32] assuming a small di erence between the perturbations of these pair interactions on formation of the NO molecule, calculated for the di erent basis sets:

ˆV effArN ‡V effArO spdf ;new eff V eff ˆV spdfg …7† ArX ArX ‡…V ArX ¡V ArX †; eff V ArX (X ˆN, O) are the e ective atom± atom V ArNO

where interactions within the ArNO complex. These interactions can be represented in terms of their components parallel and perpendicular to the NO axis, similar to the ArN2 and ArO2 cases [29, 30]: V ArX

ˆV k…RX †cos2 ³X ‡V ?…RX †sin2 ³X ;

…8†

with RX being the ArÐ X distance and ³X the angle between the ArX and NO axes. Such a correction has been shown [33] to reproduce the direct ab initio results accurately for the total system, obtained with the larger basis set. The topologies of both the 2 A 0 and 2 A 00 surfaces are unchanged by the above correction, which mainly increases their binding (table 2). The behaviour of the two surfaces near the T-shape con® guration is also preserved (® gure 2), with the gap between the minima increasing to 5.7 cm¡1. The De values increase by 12± 18% (twice as much for the T-shape compared with the linear con® gurations, as expected) and Re shortens Ê , with ³e una ected. The global minima by 0.06 §0.01 A become more stable along the bending coordinate, as does the well near the N end for the 2A 0 state, while

222

F. Y. Naumkin and D. J. Wales Table 1. ArNO(2P

x;y

) ab initio energies (in ·Eh). ³

Ra =A

3.00 3.25 3.50 3.75 4.00 4.25 4.50 5.00 5.50 6.00 7.00 8.00

³

ˆ08

19 767.3 7139.4 2167.9 358.5 7209.9 7324.8 7294.0 7172.8 793.6 752.3 718.6 77.6

22:58

458

67:58

908

112:58

1358

157:58

1542.2 46.5 7374.1 7415.8 7348.0 7265.4 7195.6 7104.7 757.9 733.4 712.7 75.6

2665.1 457.8 7235.0 7375.9 7341.4 7268.1 7200.0 7107.8 759.4 734.1 712.4 74.3

6876.5 2070.6 335.4 7204.5 7311.6 7281.5 7221.8 7123.0 767.9 739.0 715.1 77.7

12 036.2 3 908.4 926.8 754.9 7303.6 7309.5 7252.0 7140.5 776.5 743.2 715.2 75.2

14 412.4 4 690.8 1 148.8 714.6 7314.0 7328.4 7268.0 7148.3 780.3 745.3 716.7 77.6

A 00 state 2296.9 1557.5 319.8 77.5 7302.5 7347.5 7418.6 7397.4 7371.4 7335.8 7290.1 7257.1 7215.4 7189.8 7114.8 7101.7 762.6 756.1 735.9 732.5 713.3 712.5 75.3 75.9

2413.9 365.6 7264.8 7383.2 7341.1 7265.9 7197.4 7105.8 758.2 733.5 712.6 75.2

5419.9 1409.4 43.1 7329.2 7362.3 7301.0 7228.5 7122.7 766.8 738.1 714.2 76.1

10 975.7 3 428.3 713.2 7147.3 7342.0 7324.7 7257.7 7140.8 776.2 743.1 715.7 76.4

14 412.4 4 690.8 1 148.8 714.6 7314.0 7328.4 7268.0 7148.3 780.3 745.3 716.7 77.6

2

16 683.8 6 084.0 1 853.8 294.3 7201.3 7301.6 7273.4 7162.7 789.3 750.3 718.2 77.9

9951.2 3561.3 992.8 51.0 7233.7 7274.1 7237.2 7140.5 778.2 744.6 716.4 77.0

3485.1 1140.1 68.8 7259.1 7308.0 7267.8 7209.9 7117.5 765.5 737.7 714.0 75.8

A 0 state

1808

2

3.00 3.25 3.50 3.75 4.00 4.25 4.50 5.00 5.50 6.00 7.00 8.00 a

19 767.3 7139.4 2167.9 358.5 7209.9 7324.8 7294.0 7172.8 793.6 752.3 718.6 77.6

14 501.1 5 075.3 1 394.0 90.0 7288.7 7337.0 7286.5 7163.3 788.3 749.4 717.7 77.0

6288.8 1840.4 203.5 7298.7 7381.9 7332.7 7257.8 7139.7 775.6 742.7 715.5 76.1

Distance from Ar to the middle of the NÐ O bond. ¯

Table 2. Equilibrium parametersa (in meV, A , and degrees) for the Ar± NO potentials. Linear (N end) A 0…2A 00†, CEPA [8, 28] 2 0 2 00 A … A ), RCCSD-T 2 0 2 00 b A A 2

… †

1/2 (3/2) 2 00 1 2 0 2… A ‡ A † Empir. (scatter. ) [5] Empir. (scatter. ) [34, 4] Exp. (spectrosc. ) [9, 10, 3] a

De

Re

7.40 8.86 9.93 9.93 9.93 8.0

4.32 4.27 4.27 4.27 4.10

Linear (O end)

T-shaped De

9.65 (9.81) 11.71 (11.41) 13.86 (13.15) 13.16 (13.03) 13.09 13.8 11 §0.5 c 13 º §2

Re ; ³e

De

Re

3.7, 94 (3.8, 73) 3.67, 95.1 (3.75, 69.5) 3.60, 95.1 (3.69, 69.4) 3.62, 93.9 (3.62, 90.8) 3.62, 92.7 3.60, 90 3.86 3:71; º85c

7.09 9.22 10.39 10.39 10.39 8.0

4.10 4.05 4.05 4.05 4.10

Here the Ar position is speci® ed relative to the NO centre-of-mass. modi® ed using the model of equations (7) and (8) and more accurate ArX potentials. c zero-point corrected D0 and vibrationally averaged R0 ; ³0 . b

the minimum for Ar± ON is slightly more destabilized with respect to bending. The observed PES topologies can be interpreted, at least in part, using the atom± atom model of equations

(6) and (7). For the 2A 00 state, the Ar± O interaction is

isotropic and determined by the 3P state only, and hence there is no minimum for the Ar± ON geometry. For the 2 0 A state, V ArO is given by equation (8) with V k ˆV …3 P †

Ab initio PES study of Arn NO, n 4 25

223

(a)

(b)

Figure 1. Ab initio Ar± NO PES for the (a) 2A 0 and (b) 2A 00 states. z and x are coordinates of Ar relative to the centre of the NO molecule lying along the z axis. Contours are given in steps of 0.5 meV.

and V ? ˆV …3 å ¡†, the energy increasing with bending (due to weaker binding for the 3å ¡ state), leading to a well for the linear ArÐ OÐ N con® guration. For unperturbed N the isotropic Ar± N interaction leads to no well near the N end in this model, consistent with the ab initio result for the 2 A 00 state, but not for the 2A 0 state. The well obtained for Ar± NO in the 2A 0 state can be rationalized in terms of the electron density shift associated with formation of the strong NÐ O bond. In the VB picture this shift may a ect the halfoccupied px orbital on N facing the doubly occupied p orbital on O more weakly, while pulling the py;z orbitals of N towards the centre of NO. Bending in the xz plane would then increase repulsion between Ar and the px electron, leading to a well near the N end, unlike bending in the yz plane. The slightly shallower global minimum on the 2A 00 PES is consistent with such a shift, increasing the repulsion of Ar in the T-shape con® guration from the electron density of the py orbital. The model using unperturbed ArX potentials predicts the 2 00 A state to be more strongly bound due to a larger

contribution from the more attractive ArO(3P ) component. Finally, the global minimum for the T-shape con® guration on each PES is a consequence of the sum of the two Ar± X interactions rather than (mainly) one for either linear con® guration. As a consequence of the PES crossing, the lowest energy at any given point in the absence of spin± orbit coupling is a function with two minima near the T-shape con® guration, separated by a low barrier, and saddles for the linear con® gurations (® gure 3). Spin± orbit coupling splits the non-relativistic surfaces in the linear geometry and in the crossing region by D, and shifts them more weakly in other places. As a result, both relativistic surfaces have wells in the T-geometry which are slightly shallower and less resistant to bending (® gure 4). Their well depths are equal within 1% and their positions almost coincide and approach a T-shape con® guration with the Ar atom at the mid-point of the NÐ O bond (table 2). The associated mixing of the original (non-relativistic) PES topologies produces very shallow wells for the linear con® gurations for the

224

F. Y. Naumkin and D. J. Wales coupling. In particular, the De and ³e values for the Twell on the averaged PES are between those for the 1=2 and 3=2 PES, while Re is practically the same (® gure 2 and table 2). The actual mix for the 1/2 state in this geometry is º52% 2 A 0 + 48% 2 A 00. Comparison with empirical Ar± NO PESs [4, 5, 34] ® tted to reproduce measured scattering cross-sections shows that the present ground state surface is closest to the empirical PES [5] with the deepest well for the T-shape con® guration (table 2). The slight underestimation of the empirical De value is consistent with the further increase in binding expected for still larger basis sets. However, our ab initio PES is signi® cantly deeper for both linear con® gurations, and this e ect is expected to increase with the size of the basis. Hence, the empirical PES may overestimate the anisotropy of the Ar± NO interaction. This situation resembles that found for ArN2 [29]. On the other hand, the 2-even-term Legendre polynomial expansion employed in [5] for De and Re could not re¯ ect di erences in their values for ³ ˆ0 and 1808, as well as deviation of ³e from 908, thus missing these contributions to the anisotropy.

Figure 2. Angle dependence of the equilibrium parameters of the non-relativistic mean (dotted), 2A 0 (solid) and 2A 00 (dashed) Ar± NO PESs, and of the relativistic PESs for the 1/2 (bold solid) and 3/2 (bold dashed) states.

ground 1/2 state (originally 2A 00 for these geometries), and reduces the corresponding well depths for the excited 3/2 state (originally 2 A 0). The new surfaces do not cross and lie nearly parallel to each other (® gure 2), separated by about D. In particular, they are both close to the averaged non-relativistic PES (® gure 5), which is the approximate relativistic PES in the limit where the Ar± NO interaction is much weaker than spin± orbit

3.2. Arn NO cluster structures and stabilities The above ArNO potentials were used within the model of section 2.1 to describe Arn NO clusters. We employed the Aziz potential [35] for Ar2 and included the Axilrod± Teller term to account for three-body nonadditivity in Ar3 subsystems. The predicted global minima for Arn NO (® gure 6) generally are based on the the lowest minima of Arn‡1 , with NO playing the role of a surface atom. This result is di erent from, e.g., Arn Cl2 clusters, where Cl2 is `solvated’ by Ar atoms when n is su ciently large [17], and may be understood in terms of the binding between the molecule in question and Ar relative to that between

Figure 3. Minimal energy non-relativistic Ar± NO `PES’ formed by taking the lowest of the 2A 0 and 2A 00 energies at each point from the ab initio data corrected with the more accurate ArN and ArO potentials. Coordinates and contours are as for ® gure 1.


225

(a)

(b)

Figure 4. Relativistic Ar± NO PES for the (a) 1/2 and (b) 3/2 states. Coordinates and contours are as for ® gure 1.

Figure 5. The averaged non-relativistic Ar± NO PES as the mean of the 2A 0 and 2A 00 surfaces. Coordinates and contours are as for ® gure 1.

two Ar atoms. As De (Ar2) ˆ12.3 meV is only slightly less than the De (ArÐ NO) value for the T-shape con® guration, and exceeds the binding energy for the linear con® gurations (unlike for ArCl2 ), the Arn NO cluster would gain less binding with NO inside the Ar shell. In other words, the angularly averaged Ar± NO interaction represented by its isotropic component has

and 11.0 meV for the 2 A 0 and 2A 00 states, respectively, and therefore is less attractive than the Ar2 potential, the opposite of the Ar± Cl2 case (see [17]). Spin± orbit coupling appears to have little e ect upon the global minima. Both Ar atoms in Ar2 NO occupy the T-shaped positions relative to NO, and in Ar3 NO the third Ar atom De

ˆ9:6 meV

226


Figure 6. Lowest-energy minima found for Arn NO showing the results for n ˆ1- 25 for the relativistic ground state and the results for n ˆ18, 21, 23 and 25 for the nonrelativistic ground state where the structure is discernibly di erent.

lies closer to the oxygen atom. The same tilt in the orientation of NO with respect to the rest of the cluster holds for most of the larger sizes. The NO molecule lies in the equator of the Ar6 NO pentagonal bipyramid, but in an axial site in the Ar7 NO capped pentagonal bipyramid. It lies on the principal rotation axis in Ar8 NO, Ar9 NO and Ar10 NO, but in Ar11 NO it lies in the open face. The familiar double, triple and quadruple polyicosahedral motifs appear at Ar18 NO, Ar22 NO and Ar25NO [20]. The NO molecule never appears as a three-coordinate cap or enclosed in argon atoms, but instead prefers to sit in the surface. Usually it adopts a roughly tangential orientation with respect to the radius vector from the centre of mass. For the ground state relativistic potential the only deviation in structure from the global minimum of Arn‡1 occurs for Ar20 NO [20], where the double icosahedron is capped di erently from Ar21 . Only four global minima of the ground state nonrelativistic potential are noticeably di erent from their counterparts with the relativistic potential. Ar18 NO is still based upon a double icosahedron, but the oxygen atom of NO sticks a little further out of the surface. For Ar21NO the un® lled site of the triple icosahedron is di erent from that obtained with the relativistic potential, while for Ar23NO and Ar25 NO the NO molecule occupies a di erent site of the same framework. Table 3 lists the dissociation energies De of the global minima for the process Arn NO !nAr + NO along with the atomic binding energies D De ˆDe …n †¡De …n ¡1†. The De values increase with the cluster size, almost linearly for larger n (® gure 7), and the D De values exhibit

227


Table 3. Dissociation energies (in eV) of ArnNO global minima. The bold values mark the sizes expected to have special stability. SO stands for spin± orbit coupling. X state (no SO) a

n

De

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

0.0139 0.0392 0.0749 0.1134 0.1529 0.1987 0.2371 0.2880 0.3401 0.3923 0.4422 0.5112 0.5533 0.6050 0.6591 0.7113 0.7619 0.8275 0.8805 0.9339 0.9876 1.0503 1.1013 1.1577 1.2173

a b c

D De

b

0.0253 0.0357 0.0385 0.0395 0.0458

0.0384 0.0509 0.0521 0.0522

0.0499 0.0690

0.0421 0.0517 0.0542

0.0521 0.0507 0.0656

0.0529 0.0535 0.0536 0.0628

0.0510 0.0564 0.0595

1/2 state De

0.0131 0.0385 0.0731 0.1107 0.1502 0.1958 0.2346 0.2863 0.3380 0.3881 0.4403 0.5091 0.5507 0.6015 0.6555 0.7069 0.7592 0.8263 0.8776 0.9305 0.9841 1.0489 1.0992 1.1550 1.2156

D De

3/2 state

D De (NO)

c

0.0254 0.0346 0.0376 0.0395

0.0131 0.0262 0.0365 0.0383 0.0411

0.0456

0.0455

0.0388 0.0517 0.0517

0.0502 0.0521

0.0395 0.0530 0.0555 0.0566 0.0597

0.0689

0.0706

0.0416 0.0508 0.0540

0.0513 0.0523

0.0411 0.0521 0.0555 0.0566 0.0585

0.0671

0.0687

0.0648

0.0679

0.0513 0.0529 0.0536 0.0503 0.0559 0.0606

0.0519 0.0550 0.0591 0.0524 0.0588 0.0653

De

0.0131 0.0385 0.0731 0.1100 0.1494 0.1953 0.2344 0.2859 0.3380 0.3851 0.4400 0.5090 0.5504 0.6007 0.6545 0.7041 0.7590 0.8263 0.8773 0.9295 0.9839 1.0488 1.0991 1.1546 1.2155

D De

D De (NO)

0.0254 0.0346 0.0369 0.0393

0.0131 0.0262 0.0365 0.0376 0.0403

0.0460

0.0450

0.0521

0.0554

0.0689

0.0704

0.0537

0.0545

0.0673

0.0686

0.0649

0.0677

0.0390 0.0515 0.0471 0.0550 0.0415 0.0503 0.0496 0.0549 0.0510 0.0521 0.0545 0.0503 0.0556 0.0609

0.0393 0.0526 0.0536 0.0594 0.0408 0.0513 0.0538 0.0582 0.0516 0.0539 0.0586 0.0522 0.0583 0.0651

Arn NO !n Ar ‡NO. Arn NO !Arn¡1NO ‡Ar. Arn NO !Arn ‡NO:

relatively large peaks at n ˆ12; 18, 22 and 25, matching the `magic’ numbers of Arn‡1, with smaller peaks at n ˆ6 ; 9 and 15, which are associated with weaker features for Arn‡1. Here we compare our results with the data for argon clusters obtained with the same Ar2 potential. The predicted magic numbers are una ected by spin± orbit coupling and are the same for the 1/2 and 3/2 states. Our results are consistent with the magic numbers 12, 18, and 22, obtained in mass spectroscopic experiments [7] on photoionization of Arn NO clusters produced in supersonically expanded Ar ‡NO gas mixtures, while the additional detected magic number 8 deviates from nearest predicted 6 and 9. We assume that the experimental results re¯ ect the relative stabilities of the neutral clusters. The surface location of NO suggests an alternative dissociation mechanism through detachment of NO. The associated energies D De (NO) are larger than those for detachment of a single Ar atom (except for n ˆ6 and 13), due to De (ArÐ NO) for the T-shape con® guration being larger than De (Ar± Ar) and to the orientation

of NO in the cluster. The associated magic numbers coincide with those most pronounced for DDe , i.e., 6, 12, 18, and 22 (table 3), and the weak features at n ˆ9 and 15 are missing for DDe (NO) for the 1/2 state. This result supports the above mentioned set of magic numbers observed in experiments [7]. The di erences between the two D De values increase for sizes between subsequent magic numbers (® gure 8). The above variations in the dissociation energies suggest that the clusters are more likely to be formed through Arn ‡NO !Arn NO than through Arn¡1 NO ‡Ar !Arn NO and are more likely to dissociate through Arn NO !Arn¡1 NO + Ar than through Arn NO !Arn + NO. The overall process Arn ‡NO !Arn¡1 NO ‡Ar

…9†

is exothermic, and the calculated energy release, DDe (NO)¡D De , peaks at n ˆ11 ; 17, and 21 (for the 1=2 state), i.e., one size before the magic numbers

228


Figure 7. Size dependence of (a) the total dissociation energies and (b) their ® rst and second di erences for Arn NO clusters: non-relativistic data (dotted) and relativistic data for the 1/2 (solid) and 3/2 (dashed) states.

(® gure 8). For the 3/2 state, however, the ® rst two peaks are shifted to n ˆ10 and 16. Since the global minima for the relativistic 1/2 and 3/2 states appear to be practically identical, the energy difference between these structures at any given size is usually just D, the splitting for free NO. The 1/2 ! 3/2 excitation energy D ¡De…3=2†‡De …1=2† exhibits peaks at n ˆ10 ; 16, and (more weakly) 20, i.e., two sizes before the magic numbers (® gure 8), mainly as a consequence of reduced De values for the 3/2 state. 4.

Conclusion

An analytical DIM-like model potential including spin± orbit coupling has been developed for Arn NO clusters. It is additive in a matrix rather than scalar fashion. High level ab initio potential energy surfaces of the Ar± NO(X 2P ) van der Waals complex have been calculated and found to cross near the T-shape con® guration. The 2 A 0 surface has two shallow wells for both linear con® gurations and, for the equilibrium T-shape geometry, appears to lie slightly below the 2 A 00 PES, which has only a single well for the distorted T-shape con® guration. Within this model, spin± orbit coupling is shown to couple the two states strongly, producing two relativistic

Figure 8. Size dependence of the Arn NO !Arn¡1NO ‡Ar (solid) and Arn NO!Arn ‡NO (dotted) dissociation energies and (dashed) of the energy liberated in the associated reaction Arn ‡NO !Arn¡1NO ‡Ar (all for the 1/2 state), and of the Arn NO…1=2 !3/2) excitation energy (dot-dashed).

PESs of generally similar topology close to that of the averaged non-relativistic PES. The ground state PES is found to be more attractive than previous empirical surfaces for the linear geometries, suggesting a smaller anisotropy in the Ar± NO interaction. The lowest energy structures of the Arn NO clusters are found to be a ected weakly by spin± orbit coupling and closely resemble those for Arn‡1 , with NO occupying a position in the surface of the cluster. No low lying local minima with NO fully solvated were found for the range of sizes studied (n µ 25). The predicted Arn NO `magic’ numbers (12, 18, 22 and 25) reproduce those for the corresponding Arn‡1 clusters and correlate well with those found experimentally for Arn NO‡. The NO detachment channel associated with the surface location of the NO molecule is found to have a larger dissociation energy than for detachment of Ar. The De values for the two dissociation channels suggest that Arn NO may form by adding NO to Arn . Accordingly, Arn NO seem more likely to dissociate through detachment of Ar. The energy change for Arn + NO!Arn¡1 NO+ Ar is generally negative and

Ab initio PES study of Arn NO, n 4 25 exhibits peaks at the sizes preceding (immediately for the 1/2 state) magic numbers. The NO(1/2!3/2) excitation energy varies weakly with the cluster size, exhibiting maxima at magic numbers minus two. The cluster stability is a ected weakly by this excitation. As the Ar± NO‡ system is predicted [36] to have an order of magnitude larger binding energy than Ar± NO and Ar2, the ionization of Arn NO is likely to lead to a signi® cant structural change associated (particularly for larger n) with `solvation’ of the molecule in the cluster, in accord with experiments [9]. This work was been carried out in part during F.N.’s research associateship in the group of Professor F. R. W. McCourt and during his visiting assistant professorship in the group of Professor J. C. Polanyi. D.J.W. thanks the Royal Society for ® nancial support. Note added in proof

Since this paper was accepted, new work by Alexander [37] on the ArNO…A 0 ;A 00†PES has appeared. There, the PES is calculated using the unrestricted coupled-cluster method UCCSD(T) with spdf (aug-ccpVTZ) and spdfg (aug-cc-pVQZ) basis sets and extrapolated to the basis set limit. The resulting surfaces are essentially the same as those reported by us. Of particular interest is the comparison between our PES, constructed in terms of the more accurate ArN and ArO potentials (for the spdfg basis set), and the PES obtained directly with this basis. Our predictions lead to only slightly deeper (by º3%) wells for both states in the same positions, with the gap between the two minima being somewhat larger (0.70 compared to 0.56 meV). These results con® rm the reliability of our procedure. In fact, our surfaces are even closer to the basis set limit results. In the basis set limit, the global minima for both states are predicted to be deeper by less than 1 meV relative to the spdfg data. When this di erence is added the averaged (isotropic) Ar± NO potentials remain bound more weakly than the Ar± Ar potential. This result supports our prediction of surface locations for NO in Arn NO …n µ 25† clusters. References [1] Mack, P., Dyke, J. M., Smith, D. M., Wright, T. G., and Meyer, H., 1998, J. chem. Phys., 109, 4361. [2] James, P. L., Sims, I. R., Smith, I. W. M., Alexander, M. H., and Yang, M., 1998, J. chem. Phys., 109, 3882. [3] Mills, P. D. A., Western, C. M., and Howard, B. J., 1986, J. phys. Chem., 90, 4961. [4] Joswig, H., Andresen, P., and Schinke, R., 1986, J. chem. Phys., 85, 1904.

229

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