754
J . Phys. Chem. 1986, 90, 754-759
Low-Lying Valence Electronic States of the Aluminum Dimer T. H. Upton Corporate Research-Science Laboratories, Exxon Research and Engineering, Annandale. New Jersey 08801 (Received: May 31, 1985; In Final Form: September 3, 1985)
A detailed ab initio theoretical characterization of a selected set of low-lying states of the aluminum dimer is presented. character, to be almost degenerate. The lowest of these is the 3X; state, We find the two lowest states, of 3Z; and with a calculated vibrational frequency (354 cm-I), bond length (2.51 A), and bond energy (1.33 eV) that are all in excellent agreement with experimental values reported for this state. Further, an optically accessible 3Z; state is found to be at 16938 cm-', in close agreement with gas-phase emission results of 17 269 cm-I. Properties for this excited state agree with experiment in detail as well. The 311ustate is only 486 cm-I above the 32;. This energy difference is below the computational limits of the calculations, and thus we cannot make a firm ground-state assignment on energetic grounds alone. We find no support for recent experimental suggestions that the ground state is of Z': character, and are unable to assign a reported 3-eV absorption features to optically accessible (with respect to the 32J valence excited states.
I. Introduction The electronic properties of the aluminum dimer have long been thought to be relatively well understood. Recently however, the dimer has been the subject of renewed interest, and uncertainties about the original assignments have emerged. Most of the original analysis was based on information about the ground and optically accessible excited states obtained from gas-phase emission studies almost two decades ago.' The dimer has been examined more recently by using matrix isolation t e c h n i q ~ e s ,and ~ , ~ the results are not in agreement with the early gas-phase work. Douglas et aL2 reported absorption spectra for the aluminum dimer in Kr matrices possessing a characteristic 14 300-cm-I absorption that they interpreted as having originated from a 'Eg+ ground state, in contrast to the earlier 3Zg-assignment.I Abe and K ~ l b on ,~ the other hand, reported optical spectra for the dimer in Ar, Kr, and Xe matrices in which this feature (and any other low-energy feature) was absent. Both studies reported absorptions at around 3 eV possessing vibronic structure (w, = 238 and 245 cm-I, respectively). A few calculations characterizing the ground state have been published and these are in disagreement as well. The most detailed calculations are the very recent results of Basch et aL4who obtain a 311uground state, and find no singlet states within 0.5 eV. Leleyter and Joyce report a 32g-ground state, with the some 2500 cm-' above this5 The local density calculations of Lamson and Messmer,6 as well as the pseudopotential calculations of Pacchioni,' lead to 311uground states. Basch et al. also characterized a number of low-lying states of the system and concluded that the lowest excited state lies only 324 cm-' above the ground state. Upon even cursory examination of the bonding interactions possible between ground-state aluminum atoms, it is not surprising that unambiguous assignment of the optical spectra has proven difficult, or that the theoretical studies are in disagreement over the state ordering. The s2p1valence configurations of two ground-state AI atoms may be combined to produce a total of eight singlet and triplet states with u4r2configurations, eight singlet and triplet u5r' states, and individual singlet and triplet u6r0states. All of these states are expected to lie within 1-4 eV of the ground state. In addition to these, there are lowlying states that dissociate to excited atom and ionic limits. This rich array of states breaks into manifolds of Z, II,and A symmetry, and it is essential that perturbations between members of these individual manifolds be taken into account when attempting to (1) D. Ginter, M. Ginter, and K. Innes, Astrophys. J., 139, 365 (1963). (2) M. Douglas, R.Hauge, and J. Margrave, J . Phys. Chem., 87, 2945 (1983). (3) H.Abe and D. Kolb, Ber. Bunsenges. Phys. Chem. 87, 523 (1983). (4) H . Basch, W. Stevens, and M. Krauss, Chem. Phys. Lett., 109, 212 (1984). ( 5 ) M. Leleyter and P. Joyes, J . Phys. B 13, 2165 (1980). (6) S . Lamson and R. Messmer, Chem. Phys. Lert., 98, 72 (1983). (7) G. Pacchioni, Theor. Chim. Acta, 62, 461 (1983).
0022-3654/86/2090-0754$01.50/0
determine properties of the lowest states. As a first step in studying bonding and nucleation pathways for larger aluminum clusters, we report an analysis of the low-lying valence state manifold in the aluminum dimer using perfect pairing generalized valence bond (GVB) and extensive configuration interaction (CI) procedures. We find two nearly degenerate candidates for the ground state, of which the lowest is the 32gstate, in agreement with assignments in the early work of Ginter et al.' Further, we find an optically accessible 3Z,,- excited state at an energy very close to that reported by Ginter, and properties for both states in substantial agreement with those experimental results. The remaining ground-state candidate in our calculations is the 311ustate, which is found to be only 486 cm-I above the 32; state. Since such a small excitation is outside the limits of accuracy of the calculations, and since there are new experimental questions concerning the optical spectrum, we consider the possibility that either state may be the true ground state in the following sections. The energetic positions of most of the higher excited states may be understood in terms of a simple analysis of energy expressions characterizing each state. We devote the following section to such an analysis, and present the results of the calculations in section 111.
11. Low-Lying Valence Electronic State Character In this section we discuss in some detail the types of states that can form through the coupling of ground-state 2P, aluminum atoms, and to a lesser extent, states that arise from the coupling of excited atoms and ions. We qualitatively determine their relative energetic placement and evolution as a function of bond distance. As in the Introduction, we group the states according to their overall valence configurations. A . States Arising from a a4r2Valence Occupation. Bringing together two 2P, AI atoms such that the half-filled 3p orbitals are oriented perpendicular to the bond axis can produce a total of eight states, of which four are singlets and four triplets. It is further possible, through excitations from the filled 3s orbital, to produce an additional set of four quintet states; however, these are not expected to be low-lying-for aluminum and were not treated in the calculations. Independent of spin, four unique couplings of 3p K orbitals are possible, and they may be written as follows
{ x ~ r+l hxrl
(Ib)
bprl {xtxrl- bprl
(2a)
blxrl +
(2b)
where we have retained the local orbital character present at large distances to clarify the state origins. Terms in brackets represent determinantal wave functions. Core and sigma orbitals are left as implicit in each wave function, as are the spin functions. With
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 755
Electronic Properties of Aluminum Dimer a singlet spin coupling, these states take the term designations
'2; (la)
IA; ( l b )
lZg+ (2a)
]Ag+ (2b)
TABLE I: Energy Expressions for Covalent u4ir*States'
term
energy
while for triplets, they become 3Z9-( l a )
3A; ( l b )
32,,+(2a)
3Au+ (2b)
The local or "valence bond" description of these states is evident from the above depictions of these states, and they may be expressed equivalently in terms of symmetry (molecular) orbitals as (singlets) (ruyrgxl
- ("uxRgY1
(la)
-
(1b)
cI(*u>'~uxl Czbgvrgxl
C I ( h u 2 l + (ruy?l- C 2 h g 2 1 + bgy211 C I h l l 2 l - I~,,Zl1 - c2I(7rgx2l - b g y Z l 1
(2a) (2b)
where each term in brackets again represents a determinantal wavefunction. For triplets, Cl("uy"ux1
Iruyrgxl
- czhgyrgx1
(la)
+ I*u*rgyl
(lb)
( ~ u y ~ g+ yl (.lruyrgyl
(ruxrgxl
- (ruxrgxl
(2a) (2b)
The simple valence bond descriptions shown above include only covalent contributions, and are precisely equivalent to the symmetric orbital representations above when all c, are set to 2-1/2. In general, near the equilibrium separation for each state there will be substantial ionic contributions to the wave function, and these will lead to c1 > c2. As two atoms are brought together in any of the configurations of (1) or (2), the initial interatomic electronic interaction will be between the 3s orbitals. To a first approximation, this interaction is expected to be repulsive, as both 3s orbitals are filled (leading to 40, and 40, orbitals). The empty 3p, orbital is low-lying however, and mixes substantially with the 3s orbitals and reduces the repulsive interactions. The result is a weak bond at long bond distances between orbitals that may be viewed as 3s 3p, and 3s - 3p, "lobe" orbitals on each center. Overlap of the bonding lobes is maximized in this manner (stabilizing the 4ag) while the lobe orbitals forming the 40, orbital are directed away from one another to minimize antibonding character. Each of the states defined in configurations (1) and (2) shares this arrangement of u orbitals, differing only in the occupation and spin coupling of the r orbitals. To determine how potential energy curves for these states should appear, and how they are related to one another, we examine energy expressions for them. These are listed in Table I. Of particular interest is the form of the one-electron portion of the energy in each case. Comparing this part of the energy to the usual expressions for the one-electron energy of a two-electron bond,
+
Ebl(414r+ 4rdJ = (1
+ SI,Z)-I(~+ ~ I 2hi1S,,) I
and antibond,
EaY4141 - 4141) = (1 - S1,2)-"
(XrYI/XIYr))
"Definition of symbols: SI,"Y = (xl/yr); hl,"Y = (xl/h/y,); J I , X Y = (xtvr/l/r,,/xLYr) = (xlxl/Yrx,); Kl," = (xlxl/1/rl2/YrYr) =
conclude that the lag+ component will fall below the lZg+ state. Similarly, the 32,- falls below the component. For the antibonding states, the order is not quite so clear and we must compare specific matrix elements. From simple electrostatic and thus considerations we know that (xryr/xpI)> (xpr/ylxr), 3Au+ will be below 32u+.In addition, (xIyr/xpI) must be larger than K(xLY,) so that IZ; must lie below 3A;. These relations will hold true at most values of R, assuming that (a) the one-electron portions of the energy do not vary significantly from state to state (within bonding or antibonding groups); (b) that the states are predominantly covalent in character; (c) that mixing between states of the same symmetry does not preferentially stabilize a given state. With these minor qualifications, we obtain the final anticipated order, '2,'
> 3Au > 'Zu> 'Zg+ > lA, > 32;
(3)
The above discussion characterizes the states arising from configurations 1 and 2, but there are additional states that merit consideration: 2IA;
c , ( ~ , y ~ u+x c21rgxrgyl l
(4a)
I&-
(ruyrgxl + I~,,~gyl
(4b)
2IZ,+
~l((~lIx4
ZIAg+ I&+
]A,+
+ (..,21l + c2((rgx21+ bgv211
(5a)
CIhux4
- (Tly211+ c2(1..gx21 - (.,2l1
(5b)
(ruyrgyl
+(~ux~gxl
(5c)
- blIxr*A
(5d)
I, . , .I
and 2374-
C1buyruxl
3K
(ruy*gxl
-
+ CZ(Tgxrgyl
(4a)
(~uxrpyl
(4b)
In terms of local orbitals, it becomes evident that these states are derived from ionic limits. Singlet and triplet states labeled as (4a,b) above share a common spatial character in terms of local orbitals,
- 2hl$lr)
we see that all of the expressions in Table I are of analogous forms, regardless of the configuration (( 1) or (2) above) from which they originated. The one-electron energy dominates the total energy, and as a result we immediately conclude that the 3Z[, 'Ag, and lZg+states should have bonding potential energy curves, while 3Au, )Z,+, and IZ,-will have antibonding (repulsive) potential energy curves. Disregarding the small u-bonding contributions, all curves will thus show a bond order of one (or minus one). The two-electron portion of the energy is more complex, but a comparison of the terms allows all of the states in this system to be ordered in energy. Considering first the bonding states, we
(XLYr/XLYr).
IXLd
+ (XrYrl
(4a)
(4b) and the remaining singlet states (5a-d) are also related to one another, (XLVI~ - (XrYrl
(XIXI1+
brxrl + CVPIl + lYrYrl (XIXI1+ (xrxrl- bPIl - lYrYr1 IXlXll - Ixrxrl + CVLYII - lYrYrl
(5a) (5b) (5c)
(54 fx1~11 - (xrxrl- CVIYI) + CVrYrI Energy expressions for these states are given in Table 11. Using
756 The Journal of Physical Chemistry, Vol. 90, No. 5, 1986 TABLE 11: Energy Expressions for Ionic u47* States term energy
Upton TABLE IV. Spectroscopic Constants for AI, Electronic States term 32;
T,, cm-l 0 (0)
ITg+
TABLE III: Energy Expressions for 3rIu
+ h,," - h,,""SlF - hl;zS,,"x + J,,"' - K I P + ( I + S l , " s l ~ } - ~ ( h+l ~h," + hlySl,"+ h,,"S,," + J,," + K,,"} {I
-
Sl,""Sl,"}-l{hly
(X,Xl/Z,Z,)
IZ,+
u 5 d and u6 States
(X,Z,/XIZI)I
the same procedure followed earlier leading to (3), we may order the ionic states in energy as (bonding states) 2IL:,+ > 2IA, > 23Z,(6a) and (antibonding states)
3053 (2600) (4065) '5, 4457 In, 5263 2'2,' 6173 ' 2 - 15164d 3& 16662d 3Zuc 17396d 32; 16938 (1 7269) (17701) 13Z,- 24422 32432 'Au 26848 30937
w,, cm-' 354 (350) (344) (333) 28 1 (2701 244
248
275 (278) (297) 459 162 325 315
R,, A 2.51 (2.47) (2.57) (2.55) 2.72 (2.78) 2.96 (3.02) (3.10) 2.57 2.72 2.62
D., eV 1.33 (1.55 f -0.2)' (1.03)b (I.15)C 1.27 (1.19iC 0.95 (0.71)b (0.69)' 0.78 0.68 0.56
2.62 (2.57) (2.67)' 2.52 4.03 2.68 2.69
2.50",' 1.57e 0.58' -4.Y -3.7g
'From ref 12. *From ref 5 . From ref 4. Vertical. e With remect to 2Pu-4P, limits. "With respect to 2P,-zP, limits. gWith respect to IS,-'D, limits.
'Xu+> 'A, > 3Z[ (6b) They are classified as bonding and antibonding, as was done before, by noting the sign of the overlap terms. Since they are ionic however, there will be a net -1/R attraction regardless of the bonding or antibonding character. At no point though, would we expect the lowest of these states to be competitive with the lowest of the covalent states. Dissociative limits for these states are not all the same. The cationic component of each state is necessarily the Al+ lS, with a 3s2valence configuration. As a result, the triplet states (4a) and (4b) above must have a triplet anionic limit, in particular the ground 3PgAl- state.8 The l Z + and IA states dissociate to the excited singlet anion-cation ID,-'S, limit.8 The combination of covalent and ionic states given above defines the full set of valence singlet and triplet states that may be formed from four 3s electrons and two 3p, electrons. As shown below, further states of the same symmetries arise through promotion of a 3s electron to a 3p, orbital. B. States Arising from a6 and u5a1Occupations. A few states arise from configurations in which all six valence electrons occupy u orbitals, or in which one of the six occupies a a orbital instead. In the former case, the 2P, A1 atoms are brought together with the half-filled 3p orbitals on each center directed along the molecular axis (producing L: states) while in the latter one of the 3p orbitals is directed perpendicular to the axis (producing II states). In both cases, the 3p orbitals may be coupled to into either a singlet or a triplet. The lowest energy states of either symmetry, the 'Zg+ and the 311uare not comparable in an obvious way. In the first, a two-electron U, bond is formed from overlapping 3p orbitals, but in the second there are two orthogonal one-electron bonds (both u, and a,). Energy expressions for these two states are listed in Table 111, expressed in terms of local orbitals. There, it is apparent that the one-electron terms are of identical form (noting 3p, parity). There are exchange terms present in the 311uexpression that are absent in the other state that will stabilize it preferentially, offsetting the weaker 3p, overlap. As a result, we must expect these two states to be comparable in energy, with the 311uslightly more stable than the lZg+.The 'II, state and especially the antibonding 38,+state will be higher in energy. An ionic lZ,+ state is possible from the u6 configuration, dissociating to the IS,-'D, limit. Similarly, both singlet and triplet ionic II states will occur, with ID, and )P, anionic limits (IS, cationic limits), respectively. The bonding states may be placed in energy relative to the c4a2 states by comparing these expressions with the low-lying states
represented in Table I. In particular, the 3Z; and 311uenergy expressions are comparable in every term, differing only by the interchange of a 3p, orbital in 311ufor a 3p, in 3Z;. Here, the relative energies of the states depends even more delicately on the difference in overlap between these two orbitals and it is only possible to conclude that they will be very close in energy. C. States Arising from 3s3p2-3s23p Atom Limits. The 3s3p2 4P, state of the A1 atom is 3.61 eV above the ground state ( J ~ e i g h t e d ) . The ~ 3s3p2 2S, and 2P, states are above the ionization threshold for the ground-state atom (the 3s3p2 2D, state is not known experimentally, but should be below the 2S, state). Thus we would not expect states arising from a combination of 3s23p and 3s3p2 atoms to compete for the ground state. On the other hand, a few of these states are likely to contribute to the optical spectrum in the 1-4-eV range, so they merit consideration. Procedures for determining the character of these states are the same as those used above, and thus need not be repeated here in detail. A very large number of states are possible (144 singlet, ' but the triplet, and quintet states with a 4ug24 ~ " occupation), vast majority may be thought of as being derived from the a4r2 or u5r1states discussed before through a 3s 3p excitation, and we adopt this approach in classifying them. The full spectrum of possible states is summarized in Table V. Each of the states discussed in earlier sections possesses u orbitals resulting from the combination of 3s2 orbitals on each A1 atom, and the lowest states possess a 4u,2 4 ~ occupation. 2 The lowest excitations will be of 4 ~ , , 5ug character, in which the 5a, is composed largely of 3p, orbital character. Triplet states formed in this way will dissociate variously to each of the excited 3s3p2 limits, while singlets correlate with the 2D,, 2P,, and 2S, excited atoms. Thus, from the eight a4a2states in Table I, 16 additional states may be immediately derived of which 12 will be singlets or triplets (four
(8) G. Herzberg, 'Spectra of Diatomic Molecules", Van Nostrand, New York, 1955.
(9) C. Moore, 'Atomic Energy Levels", Vol. I, National Bureau of Standards, Washington, D.C., 1971, NSRDS-NBS 35, p 1.
TABLE V Possible States from Limits Considered limit states
-
-
The Journal of Physical Chemistry, Vol, 90, No. 5, 1986 757
Electronic Properties of Aluminum Dimer
0.0
1-
'i 0
m
0.5
1
1.01
4
1.5
" ' " * " " "3.0"
* ' * ' I m '
2.0
"
Bond distance
"
"
i
"
4.0
"
"
5.0
1.5
"""""""'""'"""""""'""""""" 2.0 3.0 4.0 5.0
(A)
Bond Distance
Figure 1. Potential energy curves for the lowest singlet and triplet states of AI,. Curves were obtained from cubic spline fits to data obtained from C1 calculations carried out at 0.3-A intervals. The absolute position of the zero in energy is -483.76986 a.u.
are quintets). It is worth noting that the set of states so derived includes terms with symmetries that are the same as every entry in Table 11. Each term in Table I1 may be obtained in two ways: either via an ionic limit, or through coupling of ground and excited atoms. The antibonding states in Table 11, in fact, each have ~ bonding counterparts derived via singlet 417~ 5 1 7 excitations from the three lowest states of Table I. The lowest triplet states of the symmetries given in Table I1 will dissociate to the lowest energy allowed limit, the excited atom 2Pu-4P, limit, rather than ions. The singlet states cannot dissociate to this limit. The A states 4 the 2Pu-zP, may be formed from the *D,-'S, ions (at ~ 6 . eV'O), or the unknown 2Pu-ZD,pair.* The 2' states atoms (at ~ 7 . eV9), 0 are most likely formed from the ID,-'S, limit or the nearly degenerate zSg-zPuatomic limit (at ~ 6 . 4 2eV). Other limits at similar energies are also possible, implying that great care is required in examining the large R behavior of these states. A similar approach may be adopted to identify most of the 5ug excitation from the excited states of II symmetry. A 417, bonding states produces I q 3 1 1 g states. The remaining II states result from coupling 3s23pand 3s3p2atoms in which all 3p orbitals are perpendicular to the molecular axis. Again, the rich array of dissociative limits for these states suggests the large number that are possible and urges caution in defining their limiting behavior. The possible states are summarized in Table V.
-
-
1 3 3 2 u
111. Summary of Results In the last section, three candidates were identified as possible ground states for the AI dimer: the 32;, 311u, and I2,' states. In Figure 1, we show potential energy curves for these states. The relative state separations are as described before: the 32; and 311uare extremely close in energy, and the 'Zg+is somewhat higher. The 3Z; - 311uenergy difference (486 cm-I) is smaller than what we estimate to be the limit of accuracy of the computational procedure used. This precludes a firm ground-state assignment based on calculated total energies. Ginter et al.,' however, concluded that the AIz ground state was of 32[ character, and our calculated 32; properties are in excellent agreement with those experimental results. The calculated vibrational frequency for (10) The ID, limit is not known experimentally, but its position is estimated by obtaining the IP,-'D, energy splitting from calculations on the anion and adding this excitation energy to the experimental excitation energy for the )P,
6.0
(8)
Figure 2. Potential energy curves for the six u47*covalent states. Plots were obtained as in Figure 1.
the 329state is 354 cm-', almost concident with the experimentally measured value of 350 cm-'.I The l2,+ and 311ustates are found to have much lower fundamental frequencies of 244 and 281 cm-', respectively. The calculated bond distance for the 32g-state is 2.5 1 8,, and the bond energy (De) is 1.33 eV (compared to accepted values of 2.47 8, and 1.55 eV, respectively12). Small discrepancies in these values are a result of the computational approach used (see Appendix). Analogous computed values for the '2,' state are 2.96 8, and 0.96 eV, and for the 311uwe find 2.12 A and 1.27 eV. The trend in bond lengths is as expected: bonds increase monotonically in length as the P component of the bonding is replaced by 3p, character. A similar line of reasoning may be applied to the trend in the vibrational frequencies. Thus we detect few surprises in the calculated state properties. Also shown in Figure 1 are two other low-lying states. The IAg has a configuration identical with the 3Z; state (see section ILA), while the 'nuis the singlet complement of the 311ustate. The change in coupling has almost no effect on bond lengths in either case, but weakens force constants by a small amount (see Table IV). The splitting between pairs of states is a measure of twice the exchange energy between the half-filled orbitals (see Tables I and 111). These energies are very similar (4460 and 4740 cm-' for 2 - A and II-II pairs, respectively, at Re). As discussed in the last section, the energy expressions for the 311uand 32; states are identical in form, with possible differences only in matrix element magnitude. That the matrix elements are also of similar size confirms the very small splitting between the two states at their respective minima. Turning now to the excited-state spectrum, we show potential 7~ energy curves in Figure 2 for each of the covalent 1 7 ~ states (characterized in Table I). The state order and bonding character is as predicted by the simple energy expressions of Table I and in eq 3. Careful examination shows that the splittings between the states is not exactly as Table I would predict, reflecting perturbation of the absolute energetic positions by other states of the same symmetry. We note for example, that while each of the antibonding states in Figure 2 is derived predominately from the orbital configurations defined in eq 1 and 2, there are substantial contributions to each from mixing of states formed through 517, excitations out of the three bonding states also triplet 4u, in the figure. States predominately derived from such excitations correlate with excited 3s2p2 atom limits (see last section) and in the 2.5-3.5-8, region are higher in energy than the antibonding
-
ion."
(11) H. Hotop and W. Lineberger, J . Phys. Chem. ReJ Data, 4, 539 (1975).
(12) K. Huber and G. Herzberg, "Constants of Diatomic Molecules", Van Nostrand, New York, 1979.
758
-* F
The Journal of Physical Chemistry, Vol. 90, No. 5, 1986
4.01
3.0
" 7i 2.0
-
1
1.0
were also considered by Basch et al., and for the most part the agreement between the two studies is remarkably good. Few of the excited states are accessible by allowed optical transitions from the 32; state. Of those listed, only the 32; state reported by Ginter et al.,I the 311ustates, and the second 32; shown in Figure 3 might be observed at low energies. While other 32,or 311ustates may be derived from the limits considered in this study (see Table V), none were found to be in the 1-4-eV range. The computed excitation energy for the 32g- 32; transition is in very good agreement with that reported experimentally in the gas-phase study. We find little in the calculated spectrum, on the other hand, that is readily related to the rare gas matrix spectral features reported by Douglas et aL2 or Abe and Kolb.3 As stated above, we disagree with the reassignment of the ground state suggested by Douglas et al. (to ]Eg+). Basch et al. calculated a 311u 311gtransition in this energy range, and their assignment is consistent with a %,, ground state. Abe and Kolb3 also carried out matrix isolated absorption experiments and made special efforts to examine impurity features (particularly A120). In disagreement with Douglas et aL2 (as well as Ginter et al.]), they report no low-energy absorptions. In both matrix studies, an absorption with vibronic structure (238 and 245 cm-I) was observed at about 3 eV. Abe and Kolb did not attempt assignment of this feature, while Douglas et al. attributed it to the IZgt transition. We find an allowed 32; z3Z,,- transition at almost exactly this energy, but are reluctant to assign it to the observed spectral feature. First of all, we believe that some matrix shift must be associated with the experimental finding, and more importantly, our calculated vibrational frequency (459 cm-l) is almost twice the observed progression. The only other allowed state in Table IV derived from valence occupations is the Z3II, state, but its very long bond length and weak force constant preclude its being associated with the well-resolved vibronic structure seen in both experimental spectra. For other reasons, Basch et al. were reluctant to make this assignment as On the basis of the calculated state character, we conclude that the higher energy matrix-isolated spectral feature likely does not result from a transition to a state with purely valence orbital (3s,3p) occupations. We find no excited states that possess both the proper vibronic character and a plausible energetic position. The lowest energy transitions outside of the valence states are likely to be singlet or triplet 4uu 60, transitions, where the 6u, is predominately of 4s character. As the lowest energy atomic limit for such transitions is only 3.14 eV above the ground-state limit, it is possible that the 3 eV feature from both studies may arise from such states. The most interesting feature associated with this system that has emerged from theoretical study is the apparent near-degeneracy of the ground state. There appears to be no clear means by which to reconcile the correct ordering based on the calculations alone; there are compelling reasons arising from experiment-theory comparisons to accept either ordering of the two lowest states. Confusion about the optical spectrum for A12 is conceivably resolvable if it is assumed that the ground state in the gas phase is 'E; (leading to a 32; 32; dominated spectrum) while in a Kr matrix the 311ustate is stabilized (producing the 311u 311g transition computed by Basch et al. and observed by Douglas et al.). This explanation suffers from at least one apparent inconsistency: none of the matrix optical spectra reported by Abe and Kolb (including Kr) show the low-energy 311u 311gtransition. Resolution of uncertainty over the true ground state and any matrix induced changes in the state order are likely keys to understanding the apparently contradictory experimental findings and is, we believe, a suitable focus for more detailed experimental study.
-
W
2
Upton
1
'3.7"-
2.0
3.0
4.0
Bond Distance
(w)
5.0
6.0
Figure 3. Potential energy curves for selected states dissociating to excited atom and ionic limits. Plots were obtained as in Figure 1 and the zero in energy is the same as used there.
states of Figure 2. The state mixing produces the inflection points (avoided crossings) in the antibonding curves shown. Finally, we comment on the shape of the curves in Figures 1 and 2 at long bond distances ( R > 3.5 A). The bonding curves of Figure 2 show a pronounced bend at R = 3.5 A that is suggestive of a crossover from a minimum at longer R. The antibonding curves possess a small minimum at these distances that coincides with those of the bonding curves. This long R minimum reflects the weak bonding of the 3s orbitals described in section 1I.A that is unique to the u4r2states. It is only visible after the overlap of the r orbitals has dropped off sufficiently for the bonding-antibonding interactions between them to essentially vanish. Consistent with this, the curves in Figure 1 show that this anomaly is not present for the ru or u6 12,' states. Shown in Figure 3 are excited states that correlate with either ionic or excited-state limits. Only a selection of the lowest energy states are shown. There are four with symmetries in Table 11, and one arising from an excited u5u1(2311u) configuration. As anticipated in the last section, the calculations show the 'A,, 32u+, and states to be mixtures, with large contributions near Re from the ionic components but dissociating to states consistent with a 4u,-5ug excitation from lower states. As noted earlier, for triplet states this excitation correlates with the 4P-2P limit, which is lower than the AI+-Al- limit, and as a result all three of the triplet states dissociate to the excited neutral limit some 3.27 eV (3.61 eV experimentally*) above the ground-state atom limit. Two 311ustates derive from the excited atom limit, with u 3 r 3and u 5 r 1occupations, respectively. The lower of the two is shown, which is seen to have a small minimum at long R , and an avoided crossing with the upper state at shorter bond distance. The singlet states both approach higher limits. The IEU+state dissociates to the ID,-'S, ionic limit of u6 character (and shows an avoided crossing at large R ) while the ]Au approaches the 2Pu-2P, atomic limit. IV. Discussion Spectroscopic constants for all of the states examined are gathered in Table IV. Experimental values from Ginter et al.' are shown in parentheses for the 32.gand the observed 32; excited state. In each case, the only significant discrepancies are in the calculated bond lengths (see Appendix). With the exception of the ordering, our results compare favorably with those calculated by Basch et aL4 and Leleyter and Joyes5 (bracketed quantities in Table IV). Certain of the higher excited states in the table
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Acknowledgment. The author is grateful to Don Cox for a number of useful discussions. Appendix: Computational Details
In attempting to determine electronic-state properties for such a large group of states, the emphasis is necessarily on establishing a computational procedure that is as free from bias toward any
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J. Phys. Chem. 1986, 90, 759-763 one of the states as possible. As a result, it is unlikely that a truly optimal description of any given state will be achieved, but it is hoped that the relative energetic position of each state will be properly defined. The computational procedure used was as follows. All of the data reported here were the result of configuration interaction calculations. Three different orbital bases were employed to produce the results given above: one set was used in calculations on all u4a2 states, a second basis was generated for the u4a2states, and a third basis for the u6aostate. This procedure was followed since it was found that attempts to represent states of all different overall occupations from within a single basis required prohibitively large calculations. The u4a2 basis was constructed from several different selfconsistent wave functions. The Ne core orbitals were taken from GVB calculations for the 32; state in which correlation effects were included in the 3s orbital pairs. The valence u basis consisted of the four GVB natural orbitals used to represent these pairs. The virtual u basis was taken from GVB calculations on the u6 IXg+ state in which the 5ug bond and the 3s orbital pairs included correlation effects. Four orbitals were taken: the 5ug and 5uu natural orbitals as well as the localized 3s natural orbitals. The valence xu orbitals were obtained from the lZg+wave function, while the agorbitals came from a calculation in which a,,and ag orbitals were half-filled (a mixture of 3Auand 3;5u+). Two other xu and ag orbitals were obtained in both x and y directions by Schmidt orthogonalization. The final basis thus consisted of four orbitals each of ug and u,, symmetry, and three each of a,,,uyand * g x m symmetry, in addition to the N e core. The u6 basis was of the same size and was obtained in a similar manner. The N e core orbitals and four valence u GVB natural orbitals came from the 'Zg+wave function. The valence x orbitals were natural orbitals describing angular correlation of the 3s electron pairs (four orbitals). The four u natural orbitals from the 32; calculation served as virtual orbitals for this calculation. As before, eight additional a orbitals were obtained by orthogonalization. The u5a1basis was derived mostly from GVB calculations on the jIIUstate. The Ne core, u natural orbitals from the 3s pairs,
the half-filled 5ug orbital, and the half-filled a orbitals (in each direction) were used. This set was supplemented with the 5uu natural orbital from the IZg+ calculations, as well as the x and u virtuals used in the u calculations described above. The resulting basis had the same overall composition as the previous two. In all the calculations reported here, the atomic basis set used to obtain the self-consistent wave functions was the (1 ls,7p/6s,4p) contracted Gaussian set of Dunning,13 augmented with a single 3d Gaussian of exponent { = 0.25. The configuration interaction calculations needed to define all of these needed states were complex and need not be described in detail. In brief, all excitations were allowed among valence natural orbitals, and from these, single excitations ( u u and a a) were allowed to virtual orbitals. This was done from each of the basic occupations u4a2, u6, u2a4,as well as u5x' and u3a3. A limited number of higher u x excitations were also allowed. Care was taken to maintain a consistent level of excitation into the virtual orbitals with a maximum of three electrons in virtual orbitals at any time. Test calculations were performed to ensure that the results for the same state from the three different bases did not differ drastically (typically less than 0.005 a.u. for any given state), but final results for states of a given overall occupation were taken from a calculation using the corresponding basis. Tests were also performed to guarantee that the dissociation limits were described equally well by all bases and that anticipated degeneracies were present. While this procedure wc feel does satisfy the need to avoid bias toward a particular state, it does not provide an optimal description of the lowest states. Test calculations were carried out for the 32; state using a C1 basis derived completely from GVB calculations on that state. The resulting bond distance and force constant differed little from that reported in the text, but the bond energy increased substantially to 1.55 eV, in very good agreement with the accepted experimental value.
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Registry No. AIz, 32752-94-6.
(13) T. Dunning, in "Modern Theoretical Chemistry", H. Schaefer, Ed., Plenum Press, New York, 1977.
Ab Initio Study on the Low-Lying Triplet States of Chlorobenzene Shin-ichi Nagaoka, Takeshi Takemura, Hiroaki Baba, * Division of Chemistry, Research Institute of Applied Electricity, Hokkaido University, Sapporo 060, Japan
Nobuaki Koga, and Keiji Morokuma* Institute for Molecular Science, Myodaiji, Okazaki 444, Japan (Received: June 6, 1985)
Potential energy surfaces have been calculated with the ab initio unrestricted Hartree-Fock (UHF) method and for some cases with nonorthogonal configuration interaction (CI) and complete active space self-consistent field (CASSCF) methods , symmetry. The 3(7rA,i7A*) and 3 ( ~ s , ~ sstates, * ) where for low-lying triplet states of chlorobenzene under the constraint of C the subscripts A and S mean antisymmetric and symmetric, respectively, with respect to a u, symmetry operation, are bound near the C-CI equilibrium bond length of the ground state. The 3 ( x s , u s * )and 3(7rA,us*)states are bound at much longer C-C1 bond distances, the lengthening amounting to about 0.6 A. Qualitative features of the calculated results are consistent with those suggested on the basis of phosphorescence and photoreaction properties of halogenated benzenes.
Introduction Halogen substitutions in benzene cause significant changes in its phosphorescence spectrum. Benzene exhibits a structured phosphorescence spectrum with a maximum near 370 nm in rigid-glass solution, while chlorobenzene (CB) shows a broad, structureless spectrum with a maximum near 480 nm1.1,2 ~i~ (1) Takemura, T.; Yamada, Y.; Baba, H. Chem. Phys. 1982, 68, 171.
0022-3654/86/2090-0759$01.50/0
and Chakrabarti assigned this phosphorescence for CB to an emission Originating from a triplet excimer.2 Recently we have studied the phosphorescence emission and dynamics in the triplet states of CB, p-dichlorobenzene, and p-dibromobenzene in 2methYlPentane in the temperature range 70 to 100 K by timeresolved emission ~pectroscopy.~~~ From the results of these studies (2) Lim, E. C.; Chakrabarti, S. K. Mol. Phys. 1967,23, 293.
0 1986 American Chemical Society
