MOLECULAR PHYSICS, 1999, VOL. 96, NO. 9, 1295± 1304
Rydberg excitations in rare gas clusters: structure and electronic spectra of Arn (3 < n < 25)
- -
F. Y. NAUMKIN1 and D. J. WALES2* 1 Department of Chemistry, University of Toronto, Toronto, Ontario, Canada M5S 3H6 2 University Chemical Laboratories, Lens® eld Road, Cambridge CB2 1EW, UK (Received 13 September 1998; accepted 27 November 1998) Likely candidates are located for the global potential energy minima of Arn ( 3 < n < 25) clusters using the diatomics-in-molecules (DIM) approach. The favoured geometries are found to be di erent from the structures of Ar+n and correspond to the trimer Ar3 bound to the surface of an Arn - 2 core via a common atom. The Arn - 2 core is usually only slightly distorted from its own global potential minimum, although in a few cases it corresponds to a nearby local minimum. Therefore, the `magic’ sizes of the excimer systems are predicted to di er from those of the ions and correlate instead with the stability of Arn - 2. The predicted electronic photoabsorption and emission spectra of Arn , and photoexcitation spectra of Arn are discussed in terms of experimental data. Global potential energy minima for neutral Arn up to n = 55 with the Aziz potential are summarized also; the structure is the same as for the Lennard-Jones potential except at n = 21 where the stabilities of the two lowest LennardJones minima are reversed.
1. Introduction Electronic excitations of a rare gas atom produce high energy Rydberg states Rg . Corresponding states also exist for rare gas diatomic molecules Rg2 and larger clusters Rgn . This interpretation is supported by the fact that the excimer clusters can be treated as Rg+n ionic cores with an added Rydberg electron. Rgn structures are involved in exciton formation and dynamics in condensed rare gases (see, e.g., [1± 3] and references therein). Such processes usually have been simulated [4, 5] in terms of atomic or diatomic cores which carry the excitation, using an approximate `averaged-isotropic’ interatomic potential, which may not be a good representation of the true anisotropic interactions. The spectral properties of Rgn are responsible for the emission spectra observed from the surface [5] and the bulk [6, 7] of rare gas crystals. DIM studies [8± 10] of the ® rst representatives of such a family, namely the rare gas trimers Rg3, established the bound character of their lowest electronic states (both singlet and triplet), with a linear symmetric equilibrium structure similar to that of the corresponding ions Rg+3 . The Rg3 ! Rg + Rg2 dissociation energy was found to be about 0.1 eV, i.e., around half that of Rg+3 ! Rg + Rg+2 (at least for Rg = Ar and Kr), in * Author for correspondence. e-mail:
[email protected]
accord with the relation between the D e values for Rg2 and Rg+2 . Recent non-relativistic DIM calculations [11] for triplet states of Ar3 con® rmed these ® ndings. However, another DIM model [12]predicted that the Ar- Ar2 binding may be cancelled out by spin± orbit coupling when the latter e ect is included implicitly through relativistic Ar2 potentials within the non-relativistic formulation. This prediction di ers from results [8± 10] with spin± orbit coupling included directly through the atomic values, where the binding is only slightly reduced. In the present work we ® nd that the likely global potential energy minima for Arn are di erent from the global minima predicted [13± 15] for the corresponding ions Ar+n . Whereas the latter minima generally are based on solvated ion cores the global minima for Arn are all related to the global minima of the neutral Arn - 2 clusters (or a low-lying local minimum) where one atom of an Ar3 unit lies in the surface and the other two atoms protrude. This result can be interpreted in terms of the repulsive character of the di use Rydberg electron. 2. Theory 2.1. The DIM model Within the DIM method [16], electronic energies of an n -atomic system are evaluated using the known energies of its atomic and diatomic fragments in the states which correlate asymptotically with those of interest for the
Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online Ñ 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/mph.htm http://www.taylorandfrancis.com/JNLS/mph.htm
1296
F. Y. Naumkin and D. J. Wales
molecule. We consider here the Rydberg states of Arn originating from the excitation of one Ar atom to the 1, 3 ( 3p 54s ) P state. The relevant diatomic states are then the lowest excited 1, 3 S +u , g and 1, 3P u , g terms of Ar2, and the ground X 1S +g term of Ar2 . In terms of symmetry, the description is formally equivalent to that for rare gas ionic species, except for the multiplicity, because the additional half-® lled 4s orbital has spherical symmetry and zero orbital angular momentum. In the absence of spin± orbit coupling the singlet and triplet states can be considered separately in a similar way; if it is included, these sets are coupled in the united basis set [8± 10]. Here we neglect spin± orbit coupling and focus on the triplet states because they have lower energies and dominate the lowest lying relativistic metastable states (e.g. 1u for Ar2 ). Such states have lifetimes in the microsecond range and therefore should be detected experimentally more easily than the singlet states which undergo radiative transition to the ground state a thousand times faster. We can use the DIM procedures developed for ionic clusters simply by replacing the atomic and diatomic ions with the Rydberg atom and diatom. Here we employ the approach outlined in our previous application [17]. In short, the basis set for Arn is formed by 3n products W a c of atomic wavefunctions, with the excitation located on each Ar atom a and the half-occupied p orbital of the ionic core having three possible orientations c . The linear combinations of such products with appropriate symmetry properties are treated within the valence-bond framework as approximate eigenfunctions for the diatomic fragments. For example, w ( 1 S) w ( 1S) Ö 3 and ( 1/ 2) [w ( Pc ) w ( 1 S) w ( 1S) w ( 3 Pc ) ] ( with w ( 1S) and w ( 3 P c ) being the atomic wavefunctions of Ar and Ar ) are the wavefunctions of the X 1 S +g state of Ar2 and of the 3S +u , g (for c = z ) or 3 P u , g (for c = x , y ) states of the Ar2 dimer oriented along the z axis. For an arbitrary orientation of the dimer with respect to the chosen quantization axis, the wavefunctions transform according to the usual rotational properties of p -orbitals. Inclusion of spin± orbit coupling would involve additional rotational transformations associated with spin functions. The DIM method splits the total Hamiltonian into components which correspond to diatomic ( a b ) and atomic fragments [16]: ^
H
^
=
Ha b a >b
^
- ( n - 2) a
Ha
.
( 1)
The action of H^ on each basis function can be represented as a linear combination of basis functions by constructing the representation matrices for each fragment:
^
Ha W g
c
= [d
a g E(
3
Pc ) + ( 1 - d
1
) E ( S) ]W
a g
g
(with d a b = 1 for a = b , or 0 otherwise) and, for inciding with the a b axis, ^
Ha b W g
= [( d +d
( 2)
c
z
co-
c
a g
+d -
a g Ec
b g
) E +c ( R a b ) + ( 1 - d
(Ra b )W b
c
+d
-
b g Ec
a g
)(1 - d
(R a b )W a
c
b g
) E 0 ( R a b ) ]W g
c
( 3)
,
with E c = 12[E ( L u) E ( L g ) ] ( L = S or P for c = z or 0 1 + x , y ) and E = E ( X S g ) , R a b being the distance between the atoms a and b . The resulting representation matrix for H^ is then diagonalized to give the energies and wavefunctions of the complete system. The distortion of the 4s orbital in Ar due to its interaction with Ar is approximately accounted for in the Ar2 potential which includes this perturbation. We note that the inverse antisymmetrizer operation formally appears in [17], equation (4), although only in combination with the antisymmetrizer. While such an inverse does not exist by itself, the results remain valid, namely that explicit antisymmetrization of diatomic fragment wavefunctions with respect to permutations of electrons between atoms appears to be unnecessary in constructing the Hamiltonian matrix, at least for the class of systems under study. This result is shown for the total system in [17], equation (7), the analogue of which for a diatomic fragment is obtained by applying the fragment antisymmetrizer A^ a b to [17], equation (4), resulting in 2
^
^
Aa b Ha b W
a b
2
= =
^
^
Ha b Aa b W
^
Aa b E(Ra b
+
a b
=
E(Ra b
)W
a b
.
^
)Aa b W
a b
( 4)
As input, we have used the empirical Ar2 potential of Aziz [18] (® tted to numerous experimental properties of the system, including spectral data) and ab initio results, including extensive con® guration interaction [19], for the four states of Ar2 correlating with Ar( 1 S) + Ar ( 3p54s3 P) . For the lowest excited 3 S +u state these calculations agree with experimental dissociation energies [20] within 10% , and it is important for consistency to use data of uniform accuracy for the whole set of electronic states involved, as independent data for higher energy states could complicate accurate prediction of, e.g., photoabsorption spectra. The 3P u, g states undergo sharp avoided crossings with higher lying states of the same symmetry at distances close to the equilibrium separation for the lowest state of Ar2 . Since the equilibrium distance for Ar3 is expected to be slightly larger than this, as in the case of ions, the relevant parts of the 3P u, g potentials beyond equilibrium are associated with the electronic con® guration 3p5 4s of Ar . The 3S +g potential is the result of more extensive avoided crossings and may have a larger contribution
Structure and electronic spectra of Arn clusters from the 3p5 4p con® guration. In the present work we assume that the 3p54p con® guration can be neglected for this state, too, in the separation range of interest. We also neglect the 3-body interactions for the neutral Arn - 1 subsystems on the basis of their weak in¯ uence for the same size range in Ar+n and Arn Cl2 clusters [13, 21]. Hereafter, we use the term `neutral’ to describe a system without the Rydberg excitation. 2.1. Global optimization The same global optimization approach was used as for our two previous studies of Ne+n and Arn Cl2 clusters using the DIM model [17, 21]. This `basin-hopping’ or Monte Carlo minimization [22] technique has been investigated also for atomic and molecular clusters bound by empirical potentials [23± 27]. The potential energy E is transformed to give a new surface: ~ E ( X)
= min f
E ( X) g
,
where X is the vector of nuclear coordinates and min signi® es that an energy minimization is performed starting from X. Interwell dynamics are accelerated on the transformed surface because there are no barriers and large atomic displacements can be used to sample the con® guration ~ space. The function E ( X) resembles a multi-dimensional staircase where each step corresponds to the basin of attraction surrounding a particular minimum (the set of geometries where geometry optimization leads to that minimum). A similar approach has been used to sample the torsional space of biomolecules [28, 29]. Canonical Monte Carlo (MC) sampling was used to ~ explore the E surface. For each cluster size one run of 2000 MC steps was performed from a random starting point. Two short runs of 200 steps each were also initiated using the lowest minima for Arn - 1 and Arn + 1 as seeds. The maximum step size for the displacement of any Ar Cartesian coordinate was dynamically adjusted to give an acceptance ratio of 0.5 for a temperature corresponding to 0.001 E h . Final values for the maximum displacement typically were around 1.8 a 0 . To restrict the con® guration space to bound clusters we reset the coordinates to those of the current minimum in the Markov chain at each step. Although no systematic attempt was made to optimize the temperature of the canonical sampling we expect to have found good candidates for the global potential minima at each size. In every case the Lowest minimum can be described as an Ar3 excimer sharing a surface atom with the global minimum neutral Arn - 2 structure (or occasionally a nearby local minimum). The other two atoms of the trimer stick out of the surface like a handle. Some convergence problems were caused by this unusual arrangement, since the
1297
resulting potential energy surfaces appear to be rather ¯ at with respect to the trimer orientation around these minima. The bending modes of the e ective Ar3 unit with respect to the main body of the cluster are unusually soft. In view of these results a more e cient future optimization strategy would be to fuse one trimer atom into the surface of the lowest energy minima found for Arn - 2 for each possible surface site, and simply relax these structures. 3. Results and discussion 3.1. Structures, stabilities, and excitation distributions The lowest energy non-relativistic state of Ar3 is con® rmed (cf. [8± 10]) to have a linear D1 h structure similar to that of the Ar+3 ion with 3S +u symmetry, as for Ar2 . The equilibrium distance r e between the central and Ê longer than that for terminal atoms is about 0.08 A the dimer due to a signi® cant contribution from the ground state interaction, which is repulsive at such distances. This increase in r e is smaller than that for the ionic species, probably due to the contribution of the 3 + S g state of Ar2, which is less repulsive than the corresponding 2S +g state of Ar+2 . As a consequence, Ê for Ar3 turns out to be slightly shorter r e = 2. 52 A Ê [13± 15]), even though the than for Ar+3 (2.57± 2.61 A opposite is true for the corresponding diatomics. Inclusion of spin± orbit coupling leaves r e essentially unchanged [8± 10]. Thus, addition of the Rydberg electron appears to shrink the Ar+3 core slightly while extending the Ar+2 core. The Ar± Ar2 binding energy of about 0.1 eV is half that of Ar± Ar+2 . The corresponding values from previous work [11] employing di erent diaÊ . tomic input are nearly the same: 0.1 eV and r e = 2. 54 A Rather di erent values were obtained in a study involving signi® cantly less attractive Ar2 potentials [12], Ê . namely 0.01 eV and r e = 2. 59 A The central atom of Ar3 bears about 78% of the excitation (compared with 81% [11] and 69% [12] in previous work), which corresponds to roughly half the delocalization (with about 50% of the charge on the central atom) observed for Ar+3 [13± 15]. This feature explains the structure of Ar4 , with the fourth, almost completely neutral atom, forming an angle of about 113ë with the nearest terminal and central atoms of the trimer, compared with less than 90ë for Ar+4 [13, 14]. The interaction of the fourth Ar with Ar3 can be represented approximately by the sum of Ar± nearest terminal (neutral) atom and Ar± central (excited) atom components. The latter (diabatic, without the excitation transfer) interaction varies from the half-sum of the 3 S +u and 3 S +u potentials for the collinear location of the fourth atom to about the half-sum of the 3P u and 3P g potentials for the T-shaped ( C2v) geometry of the Ar± Ar3 system. Both of these averaged interactions have a
1298
F. Y. Naumkin and D. J. Wales Table 1. Dissociation energies (in eV) of Arn clusters neglecting spin± orbit coupling. The values marked in bold are the sizes for which we expect special stability.
Figure 1. The ground state Ar2 potential (solid) and diabatic (without the excitation-transfer) Ar-Ar ( 3P) interaction potentials V L = 12( 3L u + 3 L g) , L = S + or P (dashed).
De
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0.7054 0.8174 0.8322 0.8595 0.8983 0.9393 0.9825 1.0348 1.0761 1.1286 1.1790 1.2393 1.2965 1.3713 1.4149 1.4685 1.5218 1.5797 1.6384 1.7119 1.7647 1.8183 1.8792 1.9487
a b c d
Figure 2. The Ar± Ar3 interaction energy for the trimer ® xed at its equilibrium along the z axis, with circles representing two of its atoms, as a function of the position of the fourth atom relative to the trimer centre. The contours start from - 30 h with a step of - 1 mE h .
Ê and 6 A Ê (® gure 1), while shallow minimum between 5 A the ground state Ar2 potential has its minimum at Ê . Crossing of the two near-spherical `valleys’ 3.76 A determines the position of the fourth Ar, slightly displaced (® gure 2) due to the actual distribution of the excitation. The long range nature of these atom± atom interactions implies a much weaker atomic binding energy D D e = D e ( n ) - D e ( n - 1) 0. 015 eV (table 1) compared with Ar3 ! Ar + Ar2 dissociation, determined mainly by the other, deeper, minimum of the diabatic Ar± Ar ( 3S + ) interaction at a shorter distance (® gure 1).
a
n
D e( Arn ) is for Arn ! D e ( Arn ) = D e ( Arn ) D D e( Ar2 ) = D e( Arn ) D D e ( Ar3 ) = D e ( Arn ) D
D
De
b
0.1120 0.0148 0.0274 0.0388 0.0410 0.0432 0.0523 0.0413 0.0525 0.0504 0.0603 0.0571 0.0748 0.0436 0.0536 0.0533 0.0580 0.0587 0.0735 0.0528 0.0537 0.0609 0.0695
D
D e ( Ar2 )
c
0.1120 0.1145 0.1171 0.1188 0.1219 0.1222 0.1277 0.1294 0.1303 0.1293 0.1380 0.1337 0.1324 0.1347 0.1354 0.1359 0.1410 0.1390 0.1396 0.1401 0.1418 0.1431 0.1420
D
D e( Ar3 )
d
0.0148 0.0298 0.0439 0.0478 0.0531 0.0625 0.0570 0.0699 0.0687 0.0776 0.0832 0.0965 0.0640 0.0763 0.0767 0.0818 0.0877 0.1005 0.0804 0.0817 0.0907 0.1006
Ar + ( n - 1) Ar. D e ( Arn - 1 ) . D e ( Ar2 ) - D e ( Arn - 2 ) . D e ( Ar3 ) - D e ( Arn - 3 ) .
-
Throughout the text we take D e to be the dissociation energy for complete separation of all the atoms. The interplay of the two atom± atom interactions discussed in the paragraph above also produces a saddle point at the linear geometry of the Ar± Ar3 system, unlike the well found for Ar+4 [30, 31]. The potential barrier at linearity, which actually separates the same permutational isomer from itself, is very low (about 0.004 eV), so that the fourth Ar atom will probably undergo large amplitude vibration. A previous result [12] for D D e of Ar4 is an order of magnitude smaller than our value, again because of the more attractive Ar- Ar interaction used in the present work. The previous result suggested that stable Arn clusters (asymptotically correlating with the 3p5 4s excited Ar ) might not exist for n > 3. The presence of the Rydberg electron causes qualitatively di erent structures for Arn compared with those for Ar+n , which are characterized by neutral atoms surrounding the ionic core. In fact, most of the lowest minima that we have found can be obtained from the global minimum of Arn - 2 modelled by a Lennard-Jones potential, with one surface atom forming part of an Ar3 trimer which protrudes out of the cluster (® gure 3). For n < 25 only Ar19 , Ar20, Ar23 and Ar24 are exceptions to
Structure and electronic spectra of Arn clusters
1299
this rule, but the Arn - 2 fragments still correspond to low energy local minima of the Lennard-Jones potential. To check whether the Aziz potential for Ar ever gives a di erent global minimum structure from the LennardJones potential the lowest ten or so Lennard-Jones minima were relaxed under the Aziz Ar potential (after suitable distance scaling). Every Lennard-Jones minimum relaxed in a few steps to a minimum with essentially identical structure. Some reordering in energy was observed between the two potentials, but the global minimum structure changes only for n = 21, where the two lowest Lennard-Jones minima swap places. The Ar21 global minimum for the Aziz Ar potential is a bicapped double icosahedron with a di erent arrangement of the two caps, and forms the neutral core of Ar23 , as shown in ® gure 3( u ) . In fact n = 21 is the only crossover between the two potentials for n < 55Ð the energies for the Aziz global minima of Arn up to n = 55 are collected in table 2 since they do not appear to have been given before. Hence it appears that only Ar19 , Ar20 and Ar24 in the present size range exhibit Arn - 2 cores that are local rather than the global Arn - 2 minimum for the Aziz potential. A longer run of 5000 MC steps for Ar19 failed to locate a lower minimum. In any case, the following discussions would not change signi® cantly if these structures are only local rather than the global minima for Arn . For Ar25 we checked to see if the position of the Ar3 unit in the surface is really the optimal site. In fact the second-lowest minimum has Ar3 lying along the threefold axis of the Ar23 triple icosahedron, while in the global minimum it lies along the radius vector of a local double icosahedron. The energy di erence between these minima (® gure 3( w )) is only 0.002 eV. All the Arn global minima will be made available from the Cambridge Cluster Database (CCD) [32]. Further description of the Arn - 2 cores will be omitted since they should be familiar already; global minima for the Lennard-Jones potential also are available from the CCD. The Ar3 core remains almost linear in the larger clusters, bending by only around 1ë . In most cases, it protrudes at an angle to the surface of the neutral subsystem. For n = 9, 11, 15, 21, and 22 the trimer is perpendicular to the surface, and the predicted global minima are rather symmetrical. For instance, Ar9 and Ar15 have C5v symmetry, while Ar11 , Ar21 , and Ar22 exhibit C2v symmetry. The terminal atom remote from Figure 3. Lowest minima found for Arn in the present work. The shading indicates how the excitation is distributed: darker shading means more excitation. Parts ( a )± ( w ) correspond to n = 3- 25; for n = 25 the second-lowest minimum is shown also (on the left).
1300
F. Y. Naumkin and D. J. Wales
Table 2. Energies of global minima for the Aziz Ar potential up to Ar55 in units of the pair well depth (® rst column) and eV. The only structure which di ers from the Lennard-Jones global minimum occurs at n = 21 (see the core of Ar23 in ® gure 3( u )). n
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
E
- 1.0000 - 3.0000 - 6.0000 - 9.0716 - 12.5501 - 16.3388 - 19.5523 - 23.7317 - 27.8963 - 32.0721 - 37.0616 - 43.2227 - 46.5673 - 50.8574 - 55.1332 - 59.4123 - 64.3297 - 70.2357 - 74.4701 - 78.6947 - 83.5026 - 89.2262 - 93.4191 - 98.0712 - 103.6326 - 108.0152 - 112.5248
E / eV
- 0.0123 - 0.0370 - 0.0741 - 0.1120 - 0.1549 - 0.2017 - 0.2413 - 0.2929 - 0.3443 - 0.3959 - 0.4574 - 0.5335 - 0.5748 - 0.6277 - 0.6805 - 0.7333 - 0.7940 - 0.8669 - 0.9192 - 0.9713 - 1.0307 - 1.1013 - 1.1531 - 1.2105 - 1.2791 - 1.3332 - 1.3889
n
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
E
- 117.8238 - 122.3342 - 127.3373 - 133.0321 - 137.8623 - 142.6889 - 148.1232 - 153.8232 - 158.6521 - 164.8648 - 170.8959 - 175.7298 - 180.6099 - 186.0534 - 191.7526 - 196.6530 - 202.3013 - 208.8638 - 213.7664 - 219.4590 - 226.0114 - 230.9949 - 236.9895 - 243.5883 - 250.1832 - 256.7962 - 263.4297
E / eV
- 1.4543 - 1.5100 - 1.5717 - 1.6420 - 1.7016 - 1.7612 - 1.8283 - 1.8986 - 1.9582 - 2.0349 - 2.1093 - 2.1690 - 2.2292 - 2.2964 - 2.3668 - 2.4273 - 2.4970 - 2.5780 - 2.6385 - 2.7087 - 2.7896 - 2.8511 - 2.9251 - 3.0066 - 3.0880 - 3.1696 - 3.2515
the neutral subsystem is slightly closer to the central atom of the triatomic core than in isolated Ar3, and the other terminal atom is further away; the di erence Ê between the two distances increases from zero to 0.10 A Ê and between n = 3 to 11 and then varies between 0.05 A Ê . 0.08 A The dissociation energy D e of Arn increases with n in parallel to that of the ground state clusters (® gure 4) whose structures have been optimized using the pairwise additive approximation with the same Aziz potential [18] (table 2). The shift in D e for the two systems, of about 0.75 eV, is determined mainly by the di erence between the D e values of Ar2 and Ar2 . The atomic binding energy D D e exhibits peaks for odd n from 9 to 15 and for n = 21 and 25, with weak features for n = 17 and 19. All these sizes show peaks in the second di erences D 2D e ( n ) = D D e ( n ) - D D e ( n + 1) , which are most prominent for n = 9, 15, and 21. Thus the most stable Arn clusters correspond to Arn - 2 subsystems based on the pentagonal bipyramid, icosahedron, double icosahedron and triple icosahedron ( n = 9, 15, 21 and 25), as expected. Comparison with D D e and D 2 D e of the ground state clusters (® gure 4) con® rms the above
Figure 4. (a ) Total dissociation energy D e and ( b ) atomic binding energy D e ( n ) - D e ( n - 1) and second di erence D 2 D e = D D e ( n ) - D D e ( n + 1) as functions of the cluster size for Arn (solid) and Arn (dashed).
structural interpretation of Arn as Arn - 2 + (perturbed) Ar2 , as the major peaks for Arn occur at n = 7, 13, 19 and 23, and have similar relative heights compared with those for Ar9 , Ar15 , Ar21 and Ar25 . This result may explain the origin of the `magic’ sizes for Arn which are di erent from those of Arn (with n = 13, 16, 19, 22, and 25 [13, 14]). For all the sizes studied here almost all of the excitation remains in the trimer, with 78± 80% concentrated on the central atom. The trimer atom shared with the surface of the neutral subsystem is excited slightly less than the unshared terminal atom (7± 11% versus 11± 13% ). For the most prominent `magic’ sizes, n = 9, 15, and 21, the excitation in the central atom is larger (® gure 5). The less pronounced features at n = 11, 13, and 19 are marked by a maximum excitation of the terminal atoms. 3.2. Electronic spectra 3.2.1. Photoabsorption Due to the similar electronic structure of Ar+3 and Ar+3 , the electronic spectra also match. There are three
Structure and electronic spectra of Arn clusters
Figure 5. Size dependence of the charge distribution in the Ar3 core: central atom (solid), unshared terminal atom (dotted) and terminal atom shared with the surface of the Arn - 2 unit (dashed). 3
1301
Figure 6. Linear symmetric ( D 1 h ) cuts of the Ar3 potential energy surfaces correlating with the Ar ( 3P) + 2Ar( 1S) asymptote.
S states and three 3P states, the latter being degenerate
due to axial symmetry, and in each of the two 3L sets, there are two 3 L u states and one 3 L g state [8± 10]. However, near r e for the ground state the 3S states lie below the 3P states (® gure 6), unlike Ar+3 , as a consequence of the relative energies for the diatomic species. The higher lying states, particularly 3 P , exhibit features corresponding to avoided crossings. The set of singlet states is very similar to the triplet states. Figure 7 shows the calculated electronic spectra of Arn for vertical excitations from the lowest to higher lying triplet states. The spectra cover the interval between 1eV and 3 eV, with most transitions concentrated between 2 eV and 2.5 eV. The ® rst three higher lying states (as well as the lowest state) with transition energies between 1eV and 2 eV and (at least up to n = 12) and the highest state with transition energy of 2.7± 2.8 eV are relatively isolated from the rest of the spectra and are associated with the corresponding states of Ar3 . This result implies stability of the spectral widths with respect to the cluster size. Most of the other states correspond to excitation transfer to other atoms which constitute the neutral subsystem for the lowest state. Such a pattern of states is similar to that found [14] for Ar+n , though for the ionic system the highest density of states is in the middle of the spectrum, while it is in the upper half for the excimers. The initial lowest states for these transitions and the two nearest higher lying states are non-degenerate and correlate with the three 3S states of Ar3 , while many other states are close to double degeneracy, with energy di erences in pairs of the order of 1meV. The near degeneracy is closer for the more symmetrical clusters, particularly Ar9 and Ar15 , and resembles the situation for Ar3 with its doubly degenerate 3 P states. The separations between the
Figure 7. Photoabsorption spectra of Arn clusters for the vertical transitions from the lowest excited state equilibrium con® guration to the higher lying triplet states.
pairs decrease with increasing n , resulting in a high density of states between 2 eV and 2.5 eV. 3.2.2. Fluorescence Table 3 contains the calculated vertical energies for transitions from the lowest excited states of Arn in their equilibrium geometries to the ground state of Arn. The atomic non-relativistic 3 P excitation energy of 11.6067 eV is used here. After a signi® cant drop from Ar2 to Ar3 , the transition energies vary weakly with n , re¯ ecting the excitation distribution of the distorted Ar3 core. The overall trend is a slight reduction of the transition energy with increasing n , interrupted by a few weak maxima, in particular n = 13 and 19, which correspond to `magic’ Arn , even though their equilibrium structures are di erent from those of Arn . Also, `magic’ Ar9 and Ar15 are marked by shallow minima in the transition energy. This behaviour is determined mainly by the increase in ground state energy for the
1302
F. Y. Naumkin and D. J. Wales
Table 3. Vertical transition energies (in eV) for emission from Arn clusters and photoexcitation of Arn clusters (without spin± orbit coupling). n
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
E [Arn ( lowest
triplet) !
10.0563 9.5433 9.5370 9.5311 9.5240 9.5183 9.5153 9.5056 9.5099 9.5152 9.5058 9.5342 9.5245 9.5032 9.5069 9.5059 9.5050 9.5193 9.5105 9.5062 9.5051 9.5044 9.5074 9.5014
Arn ]
E [Arn
!
Arn ( lowest triplet) ] 11.7080 11.7483 11.7889 11.7816 11.7994 11.7667 11.7748 11.7744 11.7821 11.7875 11.7961 11.8136 11.7975 11.7962 11.7989 11.7997 11.8116 11.8257 11.8224 11.8097 11.8219 11.8321 11.8206 11.8291
trimer equilibrium con® guration relative to that for the dimer (by about 0.4eV), and by the relative D e values of Arn . Hence, the major variation correlates with the red shift of the emission spectrum by about 7 nm from Ar2 to Ar3 [33], with deviations of order 1 nm for larger n . The trimer shift does not change noticeably on including spin± orbit coupling [33], and ® ts the spectral components quite well at wavelengths about 10 nm longer than those for the dimer, detected [34] in electron beam excited rare gas clusters and assigned to Rg+2 - Rg2 complexes. Concentration of the excitation in the weakly distorted Ar3 core suggests that (with spin± orbit coupling included) the radiative lifetime for larger clusters should be close to that for the trimer, i.e., about 8 m s [33], compared to 3.4 m s for Ar2. This lifetime would be consistent with characteristics of the emission observed [5] near the surface of excited solid argon thin ® lms, and assigned to excimer species with a lifetime of the order of 10 m sec. Such species might perhaps be desorbed Ar3 rather than Ar2 suggested earlier [5]. This suggestion is supported by comparing the binding energies of Ar3 and Ar2 with the rest of the cluster, obtained as D D e ( Ark ) = D e ( Arn ) - D e ( Ark ) - D e ( Arn - k ) with k = 3 and 2, respectively. Desorption of Ar3 is energetically more favourable for all n studied here (table 1),
D
though the di erence between D D e( Ar2 ) and D e ( Ar3 ) decreases somewhat with increasing cluster size, being determined by the di erence between the atomic binding energies for Ar3 and Arn - 2 . Both these binding energies increase with size for small n and reach a limiting value around n = 19. D D e ( Ar3 ) has prominent local maxima at the most prominent `magic’ sizes of Arn with n = 9, 15, and 21, while D D e ( Ar2) exhibits weaker features at n = 11, 13, and 19. 3.2.3. Photoexcitation of Arn Figure 8 illustrates the calculated spectra for vertical transitions from the equilibrium con® gurations of Arn to the singlet states of Arn . These transitions are expected to be more intense than those to the triplet states when spin± orbit coupling is included, as the ground state of Arn is basically a singlet. For calculations of the electronic states of Arn we used consistent data [19] for the singlet states of Ar2 as input. The spectra cover the interval between 11.5 eV and 13 eV, which is about half that obtained for the equilibrium Arn geometries (see section 3.2.1) due to larger separations between atoms, and lie almost completely above the atomic non-relativistic 1S ! 1P transition at 11.7868 eV. The highest density of states occurs as 12 0. 2 eV, i.e., in the lower energy part of the spectra, unlike the photoabsorption spectra of Arn , which are concentrated at higher energy. The predicted distribution of the density of states correlates well with the experimental excitation spectra of Arn [1], which exhibit peaks near 12 eV and are slightly blue-shifted and spread out with increasing n . The density is reduced due to degeneracy of states for the ground state clusters with high symmetry, e.g., at n = 13 and 19. The lower limit of the transition energies varies weakly with n (table 3), while the upper limit rises signi® cantly, in agreement
Figure 8. Excitation spectra of Arn clusters for the vertical transitions from the ground state equilibrium con® guration to the excited singlet states.
Structure and electronic spectra of Arn clusters with previous results for n < 13 [12]. Such variations in the limits with the cluster size are opposite to those predicted [17] for the vertical ionization potentials of Nen clusters. Hence, the photoexcitation spectrum width of Arn increases with n . `Magic’ Ar13 , Ar19 and Ar23 are marked by weak maxima in the lowest transition energy. 4. Conclusion The lowest energy structures of Arn clusters ( n < 25) determined from the DIM approach di er signi® cantly from those of their ionic counterparts Ar+n . For the excimer systems the excitation is localized in a triatomic core similar to the charge behaviour for the ions, but the other neutral atoms form substructures incorporating one end of the core and do not solvate it. This result is analogous to the behaviour of the excited states found in rare gas solids, which migrate towards the surface. Such behaviour can be interpreted in terms of the excitation distribution in the linear symmetric Ar3 core: about 80% of it is concentrated on the central atom, and this determines the interaction with additional neutral atoms. Thus, the lowest excited states of Arn ( n < 25) are associated with surface excitons. The Ar± Ar3 PES is found to be very ¯ at with respect to the rotation of Ar around the trimer in the plane, suggesting a non-rigid structure. Ar9, Ar15 , and Ar21 are particularly stable relative to adjacent sizes as a consequence of favourable interactions in the neutral subsystem, incorporating also the more weakly excited terminal atom of the core. There is also some indication of less pronounced but signi® cant relative stability for other clusters at odd-numbered sizes, particularly Ar11 and Ar13 . The simulated photoabsorption spectra for the triplet states have a constant width determined by the lowest and highest energy transitions in the triatomic core, and the density of intermediate states, corresponding mostly to the transfer of excitation to other atoms, increases with n . The predicted emission spectra for radiative transitions from the lowest excited state to the ground state of Arn exhibit a signi® cant red shift of several nm from Ar2 to Ar3 , and vary only weakly for larger clusters, suggesting possible alternative interpretations of available experimental data in terms of the trimers. The calculated photoexcitation spectra of ground state Arn clusters gradually spread out with increasing n and exhibit most features just above the atomic 1S ! 1 P transition energy, in agreement with experiments. The authors thank Professor P. J. Knowles for his comments on their previous paper [17], which have
1303
helped to clarify some aspects of theory concerning the antisymmetrizer question, and for providing a preprint of [13]. The present work was initiated during the visit of F. Y. N. to the University of Cambridge, sponsored by the INTAS organization. Financial support from the Royal Society of London is gratefully acknowledged. References
[1] Wormer , J., and Møller , T., 1991, Z. Phys. D, 20, 39. [2] Møller , T., 1991, Z. Phys. D, 20, 1. [3] Peterson, E. S., Schwartz , B. J., and Harris, C. B., 1993, J. chem. Phys., 99, 1693. [4] Buller , W. T., and Johnson, R. E., 1991, Phy. Rev. B, 43, 6118. [5] Reimann, C. T., Brown, W. L., Grosjean, D. E., and Nowakowski, M. J., 1992, Phys. Rev. B, 45, 43. [6] Nahme, H., and Schwentner , N., 1990, Appl. Phys. B, 51, 177. [7] Savchenko, E. V., Ogurtsov , A. N., Grigorashchenko , O. N., and Gubin, S. A., 1994, Chem. Phys., 189, 415. [8] Naumkin, F. Y., and Pevgov , V. G., 1989, Sov. Phys. Lebedev Inst. Rep. USA, 9, 64. [9] Naumkin, F. Y., 1992, Ph.D. thesis, General Physics Institute, Russian Academy of Sciences. [10] Naumkin, F. Y., and Pevgov , V. G., 1993, Proc. gen. Phys. Inst. Russ. Acad. Sci., 42, 92. [11] Goubert, L., Desoppere , E., Wieme, W., Polak , R., Paidarova , I., and Billing , G. D., 1995, J. phys. Chem., 99, 15479. [12] Last, I., and George, T. F., 1993, J. chem. Phys., 98, 6406. [13] Doltsinis, N. L., and Knowles, P. J., 1998, Molec. Phys., 94, 981. [14] Ikegami, T., Kondow , T., and Iwata, S., 1993, J. chem. Phys., 98, 3038. [15] Kuntz , P. J., and Valldorf, J., 1988, Z. Phys. D, 8, 195. [16] Ellison, F. O., 1963, J. Amer. chem. Soc., 85, 3540. [17] Naumkin, F. Y., and Wales, D. J., 1998, Molec. Phys., 93, 633. [18] Az iz , R. A., 1993, J. chem. Phys., 99, 4518. [19] Spiegelmann, F., and Gadea , F. X., 1984, J. de Phys., 45, 1003. [20] Killeen, K. P., and Eden, J. G., 1986, J. chem. Phys., 84, 6048. [21] Naumkin, F. Y., and Wales, D. J., 1998, Chem. Phys. Lett., 290, 164. [22] Li, Z., and Scheraga , H. A., 1987, Proc. Natl. Acad. Sci. USA, 84, 6611. [23] Wales, D. J., and Doye, J. P. K., 1997, J. phys. Chem. A., 101, 5111. [24] Doye, J. P. K., and Wales, D. J., 1997, J. chem. Soc. Faraday Trans, 93, 4233. [25] Doye, J. P. K., and Wales, D. J., 1998, Phys. Rev. Lett., 80, 1357. [26] Doye, J. P. K., and Wales, D. J., 1998, New J. Chem., 22, 733. [27] Wales, D. J., and Hodges, M. P., 1998, Chem. Phys. Lett., 286, 65.
1304
Structure and electronic spectra of Arn clusters
[28] Baysal , C., and Meirovitch, H., 1996, J. chem. Phys., 105, 7868. [29] Derreumaux , P., 1997, J. chem. Phys., 106, 5260. [30] Bohmer , H. U., and Peyerimhoff, S. D., 1989, Z. Phys. D, 11, 239. [31] Ikegami, T., and Iwata, S., 1966, J. chem. Phys., 105, 10 734.
[32] Wales, D. J., Doye, J. P. K., Dullweber , A., and Naumkin, F. Y., The Cambridge Cluster Database, URL http://brian.ch.cam.ac.uk. [33] Naumkin, F. Y., 1992, Bull. Lebedev Phys. Inst. USA, 9, 19. [34] Bondarenko, E. A., Verkhovtseva , E. T., Doronin, Y. S., and Rattner , A. M., 1991, Chem. Phys. Lett., 182, 637.
