Determination of the energies and spectroscopic constants of the low

On the basis of these calculations we conclude that the lowest ]E; and )nu states of Alz lie within 200 cm-I of one another, with the 311u state being...
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2774

J . Phys. Chem. 1988, 92, 2774-2781

Determination of the Energies and Spectroscopic Constants of the Low-Lying Electronic States of AI,, AI,', and AI,K. K. Sunil and K. D. Jordan* Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received: December 7, 1987)

Configuration interaction and coupled-cluster calculations are carried out on the low-lying electronic states of AI,, AI2+, and A12-. On the basis of these calculations we conclude that the lowest ]E; and )nustates of Alz lie within 200 cm-I of one another, with the 311ustate being the more stable. AI2+is predicted to have a 'Eg+ground state, with a 211ustate lying 0.54 eV higher in energy. Six anion states of A12 are predicted to be bound, with the ground-state anion being 4E;.

Introduction The nature of the bonding in metal clusters has attracted considerable interest over the past few years. Recent experimental advances have led to a variety of new techniques for preparing and characterizing metal clusters and interaction of molecules with such clusters. Theoretical techniques have also advanced to the point that they can yield quantitative information on the properties of clusters as well as important qualitative information as to the nature of the bonding. Aluminum clusters have attracted both experimentall and theoretica12q3interest. However, even for AI, there remain uncertainties as to the order of the first few electronic states of the neutral and charged species. For Al, itself the assignment of the electronic ground state remains unresolved. The three states which have been proposed for the ground state of AIz distribute the two bonding electrons as follows: ( 3 ~ 0 , )(lZg+), ~ 3pug3pa, (311,),and (3puJ2 (%,). Ginter et a1.: who examined the gas-phase emission spectrum of AI,, concluded that the ground state is 32;, while Douglas et al. concluded, based on matrix isolation s t u d i e ~that , ~ ~it~is lZg+. There are two theoretical studies which predict a 31Tuground stateiq8 and two which give a 32; ground ~ t a t e . ~ There J~ are three other studies on Al, which assume the ground state to be 3Zg-.11-13 The goals of the present study are (1) to calculate the energies of the low-lying electronic states of Al, sufficiently accurately to resolve (if possible) the ambiguity in the assignment of the ground state of the neutral dimer, (2) to provide accurate spectroscopic constants to aid in the interpretation of subsequent experimental investigations of the low-lying electronic states, and (3) to characterize the low-lying states of the A12+and AI2-. The data on the ions should prove valuable in interpreting photoelectron spectra of A12 and photodetachment spectra of A12-. While this paper was in preparation, we received a preprint from Bauschlicher and co-workers14who (using the multireference CI method) carried out a detailed study of the relative energies of the 3Z ,; 3311u, and lXg+ states of A12. The main conclusions reached by these authors concerning the relative energies of these states are similar to those reached in the present investigation. Our study ( I ) Ruatta, S. A,; Hanky, L.; Anderson, S . L. Chem. Phys. Lett. 1987,

137, 5. (2) Cox, B. N . ; Bauschlicher, C. W. Surf. Sci. 1982, 115, 15. (3) Pacchioni, G.;Plavsic, D.; Koutecky, J. Ber. Bunsen-Ges Phys. Chem. 1983, 87, 503. (4) Ginter, D.; Ginter, M.; Innes, K. Astrophys. J . 1963, 139, 365. (5) Douglas, M.; Hauge, R.; Margrave, J. J . Phys. Chem. 1983,87,2945. (6) Abe, H.; Kolb, D. Eer. Bunsen-Ges Phys. Chem. 1983, 87, 523. (7) Pacchioni, G.Theor. Chim. Acta 1983, 62, 461. (8) Basch, H.; Stevens, W.; Krauss, M. Chem. Phys. Left. 1984, 109,212. (9) Upton, T. H. J . Phys. Chem. 1986, 90, 754. (IO) Lamson, S.; Messmer, R. Chem. Phys. Lett. 1983, 98, 72 (11) Leleyter, M.; Joyes, P. J . Phys. B 1980, 13, 2165. (12) McLean, A. D.; Liu, B.; Chandler, G. S . J . Chem. Phys. 1984, 80, 5 130. (13) Fox, D. J.; Schaefer, H. F. J . Chem. Phys. 1983, 78, 328. (14) Bauschlicher, C. W.; Partridge, H.; Langhoff, S. R.; Taylor, P. R.; Walch, S. P. J . Chem. Phys. 1987, 86, 7007.

0022-3654/88/2092-2774SOl S O / O

differs from that of Bauschlicher et al. in that we examine the utility of the coupled-cluster and many-body perturbation theory methods as well as the multireference C I method for calculating the state separations. In addition, we present a more detailed analysis of the spectroscopic constants of the 31T,, and 32; states and we also determine the properties of the ions.

Computational Details In an attempt to determine the order of closely spaced electronic states, it is essential that, as far as possible, biases are avoided in both the atomic basis set and the method for including electron correlation effects. This requires the use of flexible basis sets and the recovery of a high percentage of the electron correlation associated with the valence electrons. It is also important to establish that basis set superposition errors do not make a large contribution to the calculated energy differences and to determine the possible role of correlation effects involving the core electrons. In the present investigation multiple-reference single plus double excitation configuration interaction (SDCI), Mdler-Plesset (many-body) perturbation theory, and coupled-cluster methods are used to include the effects of electron correlation. The SDCI calculations utilize orbitals determined from multiconfiguration self-consistent field (MCSCF) ~alculations.'~~'~ The perturbation theoretical and coupled-cluster calculations" employ orbitals determined from spin-unrestricted Hartree-Fock calculations. In order to follow the convergence of the perturbation calculations the results obtained at second, third, and fourth order (referred to as MP2, MP3, and MP4, respectively) are presented. The MP4 results containing only the contributions of double and quadrupole excitations are referred to as MP4(DQ), while those including the effects of single or both single and triple excitations are designated MP4(SDQ) and MP4(SDTQ), respectively. Three sets of coupled-cluster results, coupled-cluster doubles (CCD), CCD + S(CCD), and CCD + ST(CCD), are presented. The latter two methods include the effects of single and triple excitations computed by using the fourth-order perturbation theory expression for such terms, but replacing the first-order doubles wave function in this expression with the CCD doubles wave function.'* This allows for the inclusion to all orders in perturbation theory certain terms involving single and triple excitations. (1 5) (a) A description of the program used to perform the CI calculations is described in: Lischka, H.; Shepard, R.; Shavitt, I. Inr. J . Quantum Chem. Symp. 1981, 15, 91. (b) A description of the program used to perform the MCSCF calculations is given in: Shepard, R.; Simons, J.; Shavitt, I. J . Chem. Phys. 1982, 76, 543. (16) The CI and MCSCF calculations used integrals over symmetryadapted contracted Gaussian-typeorbitals evaluated using the ARGOS program of R. Pitzer, Ohio State University. (1 7) The perturbation theory and coupled-cluster calculations were performed using the GAUSSIAN 82 program: Binkley, J. S.; Whiteside, R . A,; Krishnan, R.; Seeger, R.; DeFrees, D. J.; Schlegel, H. B.; Topiol, S.; Kahn, L. R.; Pople, J. A,, Carnegie-Mellon University. (18) Raghavachari, K. J . Chem. Phys. 1985,82, 4607. Lee, Y. S.; Kucharski, S. A.: Bartlett, R. J. J . Chem. Phys. 1984, 81, 5906.

0 1988 American Chemical Society

The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 2775

Low-Lying Electronic States of A12, AI2+,and A12-

-

-

TABLE I: Enerev Differences (eV) for AI and Its Ions'Vb A1(3s23p) A1(3s3p2) (4P)

HF MP2 MP3 MP4(DQ) MP4(SDQ) MP4(SDTQ) CCD C C D STC exptld

+

2.39 3.27 3.42 3.45 3.46 3.49 3.45 3.52 3.59

(2.39) (3.12) (3.31) (3.36) (3.36) (3.38)

-

A1+(3s2) Al'(3s3p) (3P) 3.52 4.39 4.52 4.55 4.56 4.57 4.57 4.60 4.61

(3.52) (4.21) (4.38) (4.44) (4.44) (4.44)

-

-

A1(3s23p) A1+(3s2)

Alt(3s2) AI2+ (3s)

A1(3s23p) AI-( 3s23p2)

5.61 (5.61) 5.80 (5.82) 5.88 (5.90) 5.89 (5.91) 5.90 (5.92) 5.91 (5.93) 5.88 5.92 5.98

17.52 (17.52) 18.47 (18.27) 18.63 (18.48) 18.67 (18.55) 18.67 (18.55) 18.69 (18.55) 18.69 18.73 18.82

0.02 (0.02) 0.33 (0.34) 0.38 (0.40) 0.37 (0.39) 0.39 (0.39) 0.42 (0.42) 0.35 (0.36) 0.39 (0.40) 0.46

'The excitation energies and IP's were calculated with the [ 1 2 ~ 9 p 3 d l f / 6 ~ 5 p 3 d land f l [ 1 2 ~ 9 p 4 d l f / 6 ~ 5 p 4 d lbasis f l sets. These basis sets were augmented with diffuse 40.02) and p(0.013) functions for the calculations of the EA'S. bThe results obtained from the calculations including 2p correlation effects are given first, and those ignoring 2p correlation are given in parentheses. 'In the C C D ST approximation, the effects of the single and triple excitations are evaluated with the C C D doubles wave function. dThe experimental excitation and ionization energies are from ref 3 2 a n d the experimental EA is from ref 33.

+

The starting basis sets for our investigation were the [ 12s9p/6s4p] and [ 12s9p/6s5p] contracted Gaussian basis sets of McLean et al.I9 To these two d functions with exponents of 0.60 and 0.1 5 were added. As a result of a series of exploratory calculations, the 6s5p2d basis set was expanded to [12~9p3dlf/6~5p3dlfl, with d exponents of 1.20, 0.30, and 0.075, and a f function with an exponent of 0.24. This basis set was used for the majority of the calculations on the neutral and cationic species. Limited calculations were also performed with basis sets containing an additional s function of exponent 0.45 and with two f functions rather than one. To estimate the effect of correlation of the 2p core electrons, calculations were also performed in which excitations from the 2p space were included and with a tight d d function (ad= 6.0) added to the basis set to describe 2p correlation effects. For the calculations on the anions additional diffuse s ( a , = 0.02) and p ( a p= 0.013) functions were added to the 6s4p2d, 6s5p3dlf, and 6s5p4dlf basis sets. The s components of the Cartesian d functions and the p components of the f functions were deleted from the various basis sets.

-

Results and Discussion A . Atomic Properties. For an accurate determination of the properties of the dimer, it is essential that the basis set and the method used to incorporate the effects of electron correlation be capable of giving accurate values for the excitation energies, ionization potentials (IP), and the electron affinity (EA) of the atom. This is most readily seen from the perspective of valence-bond theory, whereby the wave function of the molecule is described in terms of the various configurations of the separated atoms. To establish that the basis sets used in this study are adequate for describing the relevant states of the atom, the first two ionization potentials, the electron affinity, and the A1(3s23p; Al'(3s3p; 3P) excitation 2P) A1(3s3pZ;4P) and A1+(3s2;'S) energies were computed by using the perturbation theory and coupled-cluster methods. Two sets of calculations were performed, one allowing only for correlation effects involving the valence electrons and the other allowing also for correlation effects involving the 2p core electrons. The calculations of the excitation energies and ionization potentials were performed with the 6s5p3dlf and 6s5p4dlf basis sets, and those of the EA were camed out with a 7s6p3dlf and 7s6p4dlf basis sets including the above-mentioned diffuse s and p functions. For the determination of the EA, the energies of both the anion and neutral species were calculated with the enlarged basis sets. The results of these calculations are summarized in Table I. The perturbation theory calculations converge well for the atomic energy differences, with the MP3, MP4(DQ), MP4(SDTQ), and coupled-cluster results being quite close to one another. Correlation effects involving the 2p core electrons are found to be most important for those processes in which the

-

-

(19) McLean, A. D.; Chandler, G. S. J. Cbem. Pbys. 1980, 81, 5906. Huzinaga, S. Approximate Atomic Functions I& Department of Chemistry Report, University of Alberta, Edmonton, Alberta, Canada, 1971.

-

- -

occupation of the 3s orbital changes. For example, the A1(3s23p) A1(3s3pZ)(4P) and the A1+(3s2) Al'(3s3p) (3P) excitation energies and the A1+(3s2) A1+2(3s) ionization energy are underestimated by 0.2-0.3 eV in the valence space calculations. When 2p correlation effects are included the errors are significantly smaller, the CCD ST(CCD) energy differences falling within 0.09 eV of experiment for all cases considered. Single reference SDCI calculations give atomic energy separations within 0.03 eV of the CCD + ST(CCD) results. It is expected that most of the error in the C I and CCD ST(CCD) values for the atomic separations is due to limitations in the basis sets, the neglect of correlation effects involving the 2s electrons, and the neglect of relativistic effects rather than the CCD ST(CCD) or C I approximations themselves. B. 3Z9and %,, States. In agreement with previous theoretical work, we find that the % ; and %, states are considerably more stable than the IZg+ state. Hence, we first focus our attention on the energy separation between these two triplet states. The 3n,/32; state separations obtained at various levels of theory are summarized in Table 11. Selected total energies are given in Table 111. A bond length of 2.779 A, that determined in the theoretical study of Basch et a1.,8 was employed for the 311ustate. This bond length is 0.062 A longer than our CCD ST(CCD) optimized value (2.717 A), which should be quite close to the experimental value. The results in column A and the C I results in column D are from calculations using the experimenal bond length of 2.466 A for the 32; state. All other separations listed in Table I1 are obtained by usin energies for the 3Z; state calculated at a bond length of 2.480 , that determined from our CCD + ST(CCD) calculations. The use of the experimental bond length (2.466 A) rather than the CCD ST(CCD) value of 2.480 A value in the valence space CCD ST(CCD) calculations on the 32; state causes the energy to be too high by 18 cm-], while the use of R = 2.779 A rather than R = 2.717 A for the 311ustate causes its energy to be too high by 58 cm-'. (These results were obtained from CCD ST(CCD) calculations performed with the 6s5p3dlf basis set.) ( i ) 311,,/32gSeparation: MCSCF and CZ Results with the 6s4p2d and 6s5p3dlf Basis Sets. The results obtained with the 6s4p2d basis set, for which we examined several different methods for including electron correlation effects, are considered first. These are summarized in column A of Table 11. MCSCF calculations, in which each state is described by the two configurations needed to ensure proper dissociation (...50g2a, and ...5o,2rg for 311uand ...2ru2and ...2ap2 for % ,) place the 311ustate 2594 cm-' below the 32; state. This order of the two triplet states is found for all other calculations performed here with the exception of the first-order C I (FOCI) calculationsZodiscussed below. The 311,/3Z; separation is decreased to 1049 cm-' in the two-reference SDCI calculations (reference space I) utilizing the two-configuration MCSCF orbitals. The MCSCF configuration space for

+

+

+

+

x

+ +

+

~

~~~

~~

~

(20) Schaefer, H. F.; Klemm, R. A,; Harris, F E. Pbys. Reu. 1969, 181,

137.

2776 The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 TABLE 11:

-

a,,

Sunil and Jordan

’2; Excitation Energy (cm-’) of All“

basis set procedure

A

FV M C S C F 2-ref SD-CI multiref SD-CI first-order CI UHF MP2 MP3 MPWQ) MP4(SDQ) MP4(SDTQ) CCD C C D S(CCD) C C D + ST(CCD)

176 1049 309 -42 2926 380 557 1023 909 381 1333 1215 63 1

+

B

C

D

E

F

G

H

2978 284 469 957 822 216 1269 1115 473

2986 240 422 919 775 158

295 1 264 447 942

2969 90 376 933 800 110 1160 1023 370

136 2882 367 538 997 882 357

2905 287 447 916 791 248

2960 270 452 948 804 203

1248 1092 453

“The results in column A and the CI result with basis D were obtained for R ( ) Z i ) = 2.466 A. All other results are for R(3Z;)3 = 2.48 A. A bond length of 2.779 8, was used for all calculations on the 311, state. The basis sets used to obtain the results in columns A-H are A, 12s9p2d/ 6 ~ 4 p 2 d( a d = 0.60, 0.15); B, 1 2 ~ 9 ~ 2 d / 6 ~ 5( a~d 2=d0.60, 0.15); C, 1 2 ~ 9 ~ 3 d / 6 ~ 5( a~d 3=d 1.20, 0.30, 0.078); D, 1 2 ~ 9 ~ 3 d l f / 6 ~ 5 ~(ar 3 d= l f 0.24); E, D S ( a , = 0.45); F, E but with two f functions (ar= 0.48, 0.12); G, D + d (ad = 6.0); H, E d (ad = 6.0); correlation effects involving the 2p electrons included.

+

TABLE 111: Total Energies (nu) of AI2 for Selected Calculations state approximation 3z,3nu (R = 2.466 A) (R = 2.779 A) basis set procedure -483.82443 2-ref MCSCF“ -483.822 59 6s4p2d FV-MCSCF -483.825 83 -483.826 63 -483.89047 -483.891 88 S D C I (ref II)b -483.892 18 S D C I (ref 111)‘ -483.89079 -483.823 99 -483.825 89 6s5p3dlf 2-ref M C S C F -483.90055 -483.901 17 SDCI (ref 11) DThe2-ref MSCSF calculations include all configurations generated within the valence space by permitting one- and two-electron promotions from the ...2 x 2 and ...2?r; (3Z;) and ...2?r,4cg and ...2ag4uu (TIu)configurations. bThe S D C I (ref 11) results were obtained from multireference single-plus-double excitation CI calculations in which the reference space was taken to correspond to the 2-ref M C S C F configurations and the orbitals were taken to be the 2-ref MCSCF orbitals. The MO’s derived from the Is, 2s, and 2p A O s were frozen. ‘The S D C I (ref 111) calculations differ from the SDCI (ref 11) calculations in that the MO’s and reference configurations are from the FV-MCSCF calculations.

each state was then enlarged to include all configurations which can be generated within the 3s3p space (that is, the M O s derived from the 3s and 3p AO’s) via single and double excitations from the two “proper-dissociation” configurations. These MCSCF calculations were followed by SDCI calculations using these configurations as the references (reference space 11). The energy separation between the 311uand 3 X ; states is decreased to 403 and 309 cm-’ in these extended MCSCF and SDCI procedures, respectively. We also performed full-valence (FV) MCSCF calculations, which allow all possible arrangements of the six valence electrons among the eight valence MO’s as well as multireference SDCI calculations using the FV-MCSCF configurations as references (reference space 111) and the FV-MCSCF orbitals. The FV-MCSCF and corresponding SDCI calculations again place the 311, state below the 3 X g - state, with separations of 176 and 306 cm-’, respectively. From these results, it is seen that the expansion of the reference space from I1 to I11 has little effect on the state separation as determined from multireference SDCI calculations. We have also carried out calculations using the FOCI method,” in which only single excitations are permitted external to the valence space. These calculations used reference space I11 described above and employed the corresponding MCSCF orbitals. The FOCI calculations reverse the order of the two lowest states, placing the 38; state 42 cm-’ below the 311ustate. This is in contrast to the FOCI calculations of Basch et aL8 which give a 311uground state (with the 32; state lying 324 cm-’ higher in energy). The fact that the FOCI calculations of Basch et al. give the opposite ordering of the two triplet states than do those

+

performed here may be due to errors associated with the local potential utilized by these authors to replace the core electrons on the A1 atoms. In any case, the large discrepancy between the results of our FOCI and SDCI values for the 3n,/3Z, splitting leads us to conclude that the FOCI procedure cannot be trusted to give reliable separations between levels which are close in energy. Caveats concerning the use of the FOCI procedure have also been raised in other investigations.21v22 GVB-CI calculations of Uptong place the 32; state below the 311ustate. Upton’s calculations employed a basis set roughly comparable in quality to our 6s4p2d basis set. Thus, it appears that the reason that they give a 32; ground state is that the treatment of the electron correlation effects was not very extensive. Our FOCI calculations which recover only a small portion of the valence space correlation energy, also place the 32; state below the 311ustate. A detailed analysis of the effects of various extensions of the basis set was carried out through a series of perturbation theory and coupled-cluster calculations. However, before discussing these, we first consider the results of our largest CI calculations using reference space I1 and the 6s5p3dlf basis set. These calculations, including 217 100 and 234 316 spin- and symmetry-adapted configurations for the 311uand 32; states, respectively, give a %,, ground state, with the 311u/3X; separation decreased to 136 cm-’. When dealing with such a small separation it is especially important to consider the various sources of error in the calculations. This is done in the following two sections of the paper. ( i i ) Effects of Basis Set Expansion and Core Correlation on the 311u/32;Separation. The importance of the various basis set modifications and the inclusion of correlation effects involving the 2p electrons were examined by using many-body perturbation theory and coupled-cluster theory which are less CPU time demanding than the multireference C I calculations with reference space 11. For some of the basis sets both MP4(SIYTQ) or CCD ST(CCD) calculations were performed; for others only one of these approaches was considered. The effects of various basis set extensions on t h e 311u/329separation obtained from calculations correlating only valence electrons are considered first. From examination of the MP4(SDTQ) results presented in Table I1 it is seen that the use of the 3d rather than 2d basis set lowers the 32; state relative to the 311u state by about 109 cm-’. The inclusion of the fourth, tight d function (a = 6.0) proves unimportant for the 3 1 1 , / 3 X , separation, decreasing it by only 7 cm-l a t the CCD + ST(CCD) level of treatment. The addition of a single f function (a = 0.24) t o the 6s5p3d basis set increases the MP4(DQ) separation by 32 cm-I while decreasing the MP4(SDTQ) separation by 45 cm-I, showing

+

(21) Sunil, K. K.; Jordan, K. D.; Raghavachari, K. J . Phys. Chem. 1985, 89, 457.

(22) Rama Krishna, M. V.; Jordan, K. D. Chem. Phys. 1987, 11.5, 423.

The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 2777

Low-Lying Electronic States of A12, AI2+, and Alzthat this f function is particularly important for correlation effects involving single and triple excitations. The replacement of the single f function by two f functions (af= 0.48, 0.24) decreases the MP4(SDTQ) separation by another 58 cm-’. In this case, much of the effect is already present at the MP2 approximation. Calculations performed with a 7s5p3d3f basis set (af= 0.96,0.24, 0.06) (not included in Table 11) give nearly the same state separation as does the 7s5p3d2f basis set. However, the use of the three f basis set is more important if the three d functions are replaced by four. Comparison of the results obtained with 6sSp4d2f and 6s5p4d3f basis sets (not included in the Table) shows that the MP3 separation is 40 cm-’ larger with the latter basis set. (Higher order calculations with the 6s5p4d3f basis set were not performed due to the large amount of CPU time which would be required.) CCD ST(CCD) calculations with the 6s5p4dlf basis set show that correlation effects involving the 2p electrons decrease the separation between the two triplet states by about 83 cm-l. Correlation effects involving 2p electrons make nearly the same contribution to the 311u/3Z; separation in the CCD and CCD ST(CCD) approximations, indicating that single and triple excitations are relatively unimportant for describing the effect of core correlation on the state separation. (iii) Corrections to the CZ Value of the 311u/3Z; State Separation. In the previous section we estimated the effects of expansion of the basis set and the inclusion of core correlation on the 311u/3Z - separation. In this section we combine these with estimates of other sources of error in the CI calculations to obtain a revised estimate of the separation between the two triplet states. As mentioned previously, the use of the nonoptimal bond length for the 311ustate introduces a 58-cm-’ error in the CCD + ST(CCD) energy of this state. It is expected that a similar error would occur in the multireference C I energy for this state due to the use of a bond length of 2.779 A. The effect of size consistency errors on the splitting between the two triplet states is apparently quite small. This follows from the observations that the application of a generalized Davidson correctionz3to the multireference C I energies causes only a few cm-’ change in the state separation and that adoption of reference space I11 rather than I1 does not appreciably alter the splitting calculated with the 6s4p2d basis set. Valence space basis set superposition errors (BSSE), estimated with the 7s5p3dlf basis set, lower the energy of the 311ustate relative to the 3Z; state by 24 cm-1.24 The net effect of BSSE on the separation is lowered to 12 cm-l when correlation of the 2p core is included. These results are from MP4(SDTQ) calculations on an aluminum atom, in the presence of the basis functions of the other atom. The calculations are performed with u and A orientations of the 3p orbital and at “bond lengths” corresponding to those of the 3Z; and 311ustates. The resulting energies are compared with those obtained from calculations on the isolated A1 atom to estimate the superposition errors. Assuming that the effects of the various basis set extensions described in the previous section are additive, valence space calculations with the 7s5p4d3f basis set in place of the 6s5p3dlf basis set would lower the 3Z; state relative to the 311ustate by about 12 cm-’. The net effect of the various corrections (for BSSE, truncation of the atomic basis sets, inclusion of correlation of the 2p core electrons, and the adjustment due to the slight error in the bond length of the 311ustate) is to decrease the 311u/3Z; splitting by about 45 cm-l (ignoring, for the time being, zero-point vibrational corrections). Application of this “correction factor” to the C I splitting obtained with the 6s5p3dlf basis set gives a splitting of about 90 cm-I. Relativistic effects and correlation effects involving the 2s electrons are expected to be relatively

+

+

(23) Langhoff, S.R.; Davidson, E. R. Int. J. Quantum Chem. 1974,8,61. (24) The estimates of the BSSE are obtained from calculations On the atom in the presence of the basis functions of the other atom located at R = 2.48 or 2.716 A. To determine the BSSE in the 3Z*state calculations are performed with the A orientation of the 3p orbital, and to determine that of the ’nustate calculations were performed with both u and K orientations of the 3p orbital.

unimportant for the 311u/3Z; state separation. Perhaps the greatest source of error in estimation of the magnitude of the various corrections is the use of the MP4(SDTQ) and CCD ST(CCD) procedures. As will be discussed below, the effects of single and triplet excitations are probably overestimated in these approximations. If the estimates of the corrections due to the inclusion of correlation of the core electrons, expansion of the basis set, and elimination of BSSE are made using the MP4(DQ) or CCD procedures, the 311u/3Z; splitting is actually increased from the SDCI value to about 170 cm-’. The true state separation probably lies between 60 and 200 cm-’. The inclusion of the zero-point energies increases the splitting by about 35 cm-l, making it even more likely that the ground state is 311u. Our conclusions concerning the order and separation between the two lowest triplet states of A12 are in agreement with those reached by Bauschlicher et (iu) Errors in the MPI(SDTQ) and CCD ST(CCD) Approximations. For a given basis set the 311u/3Z; separations obtained from multireference CI calculations using reference space I1 should be very close to the results that would be obtained from calculations including all possible excitations of the valence electrons. This was shown to be true for the 6s4p2d basis set. Thus, we believe that comparison of the multireference CI, coupled cluster, and perturbation theoretical results provides a test of the suitability of the latter two procedures for predicting the separation between two closely spaced states as in A12. For the two basis sets, A and D in Table 11, for which multireference C I calculations were performed, the MP4(SDTQ) 311u/3Z; separations are only about 70 cm-’ larger than the C I values. On the other hand, the separations obtained in the CCD ST(CCD) approximation are about 320 cm-’ larger than the C I results. We note also that the 311u/3Z; separation is about 300 cm-’ larger in the CCD approximation than in the MP4(SDTQ) approximation. Since the coupled-cluster approximations should yield more accurate results than the “corresponding” Moller-Plesset approximations (Le., CCD vs MP4(DQ) and CCD + ST(CCD) vs MP4(SDTQ)), we are led to conclude that (1) the fairly good agreement between the MP4(SDTQ) and the multireference SDCI values of the 311u/3Z; energy difference is fortuitous and (2) the discrepancy between the SDCI and CCD ST(CCD) separation is due to a deficiency of the latter. There are several factors which could cause an error in the separation obtained from the CCD ST(CCD) approximation. These include (1) the small spin contaminations associated with the use of U H F orbitals, (2) the neglect in the CCD + ST(CCD) procedure of certain types of interactions involving single and triple excitations, and (3) the neglect of certain contributions due to higher order classes of excitations. With regard to (3) we note that the C I calculations carried out with reference space I1 allow for all excitations through quadrupole excitations as well as for selected quintuple and hextuple excitations with respect to the H F configuration. The CCD ST(CCD) calculations do not include the quintuple excitations and include hextuple excitations only to the extent that they enter through the TZ3terms. A coupled-cluster procedure allowing for the use of two reference configurations (...27r: and ...21r: for 3Z; and 5ug2rUand 5aU2rg for 311,)would probably include the most important contributions of these higher order excitations. It is well documented that the U H F procedure introduces spin contamination into the wave function, which often worsens convergence of the perturbation series for the energies.25 Cases are also known in which spin contamination introduces sizable errors into the energies obtained from CCD + ST(CCD) calculations.22 The use of spin-projected procedures2swould provide a means of addressing this issue. We expect that the major limitation of the CCD ST(CCD) procedure is factor (2) listed above, namely that the ST(CCD) correction does not allow for mixing between different triple excitations, between different Single excitations, or between Single

+

+

+

+

+

+

+

(25) Schlegel, H. B. J . Chem. Phys. 1986, 84, 4530. Handy, N. C. Knowles, P. J. Somasundram, K. Theor. Chim.Acta. 1985, 68, 87.

2778

The Journal of Physical Chemistry, Vol. 92, No. 10, 1988

TABLE I V SpectroscopicConstants of A120-b state method' ref T,,cm-' 'II. CCD+ST 0 CCD + S 0 0 CCD 9 486 GVB-CI FOCI 8 0 7 0 CI CI 14 0 517 3Z; CCD+ST 1150 CCD + S 1309 CCD 9 GVB-CI 0 8 324 FOCI 7 636 CI CI CI

CI expt

CCD + ST GVB-CI

'E,+

0

11 12 12 14 28

CI

165 2338 3053 4065 2600 2326

9 8 I 14

CI CI CI

Re, A 2.717 2.720 2.718 2.72 2.779 2.768 2.725 2.480 2.487 2.485 2.51 2.551 2.540 2.57 2.488 2.491 2.493 2.466 2.969 2.96 3.10 3.02 2.954

De,eV

1.35 1.25 1.24 1.27 1.19 1.12 1.40 1.29 1.11 1.08 1.33 1.15 1.05 1.03 1.44 0.93 1.38 1.50 1.06 0.95 0.69 0.71

Sunil and Jordan

we, cm-'

wexe, cm-l

284.97 284.36 284.75 28 1 270

1.652 1.602 1.544

E,, cm-I 0.1692 0.1689 0.1691

ae,cm-'

0.0144 0.0178 0.0108

0.0017 0.0086 0.0031

0.2031 0.2020 0.2022

0.0015 0.0015 0.0015

-0.0117

0.2054

0.0012

wcy,, cm-I

0.0012 0.0012 0.0012

1.569

271 355.15 348.75 349.92 3 54 333

2.045 2.043 1.997 1.979

344 344.60 208.79 343 3 50 232 244 193

2.175 2.100 2.022

216

The various coupled cluster spectroscopic constants are determined from calculations employing the 6s5p3dlf basis set and correlating only the valence electrons. bThe CCD ST(CCD) values of ye are 311u,2.5 X 10"; ' Z i , -5.0 X lo-' cm-I. CTheeffects of the single and triple excitations in the CCD S and CCD ST approximations are evaluated with the CCD doubles wave function.

+

+

+

and triple excitations. Such terms, which are included in the multireference CI calculations and which would appear first in fifth-order perturbation theory, could prove quite important for determining the splitting between the 311uand 32; states of Al,. Comparison of the CCD, CCD S(CCD), and CCD ST(CCD) results shows that the triple excitations contribute far more to the state separation than do the single excitations, leading us to speculate that the main deficiency of the CCD ST(CCD) procedure for the 311u/32;separation of Al, is the neglect of coupling between the triple excitations. It is not possible to "separate out" the contributions of such terms in the unitary group C I approach utilized here. Their importance would be more readily established by means of fifth-order perturbation theory or CCSDT calculations, either of which would be very demanding in terms of CPU time. It should also be noted the spin contamination problem should be less severe in the CCSDT than in the CCD + ST(CCD) procedure. C. Spectroscopic Constantsfor the 32g,311u,and It;,+ States of All. To determine of the spectroscopic constants of the lowest 32,, 311u,and lZg+ states of Al,,the CCD + ST(CCD) procedure, correlating only the valence electrons and using the 6s5p3dl f basis set, is adopted. Even though this approach introduces a few hundred wavenumber error in the separations between the electronic states, it should prove reliable for predicting the spectroscopic constants. Calculations were performed for a range of bond lengths for the 3Z; and 311ustates. The calculated points were fit to cubic spline functions and the resulting potential energy curves used together with the Numerov-Cooley algorithm26to compute the vibrational energy levels and wave functions. The points were chosen with the object of obtaining accurate results for the first 10 vibrational levels. This was accomplished in an iterative manner, with about 15 points being calculated initially, and successive points being chosen so as to obtain convergence on the first 10 vibrational levels. The final spectroscopic results were obtained from curves generated from about 30 calculated points.27 AG values were determined from the vibrational energy levels, and the first four AG values were back-averaged to obtain the vibrational constants. the spectroscopic analysis was first carried out keeping all vibrational constants through wgz,. However, since

+

+

+

(26)Cooky, J. W.Math. Comput. 1961, I S , 363. (27)The energies at all the calculated points are available from the authors upon request.

w,ge was

found to be very small for both triplet states and because only the constants through w a e were determined from the experimental data for the 3 X ; state,28w g , was set equal to zero in the spectroscopic analysis used to determine the constants given in Table IV. The first three E, values, obtained from expectation values of R-, over the vibrational wave functions, were used to compute Re,E,, a,,and ye, which are also summarized in Table IV along with the dissociation energies. The bond lengths reported in Table IV are those determined from the B,'s. These differ only slightly from those associated with the minima in the potential energy curves. In this table we have also included our results for the ]Eg+state, the results of other theoretical studies, and experimental results, when available. Calculations were performed at fewer internuclear separations for the IXg+ state than for the two triplet states, and a less detailed analysis of the spectroscopic constants was performed in this case. The computed properties for the ' 2 ; state are in excellent agreement with experimental data.28 An exception is w y , for which our calculations give a positive value in contrast to the experimental analysis. The experimental vibrational constants were determined from a least-squares analysis of the first 17 AG values, while we used only the first four values in our analysis of the theoretical data. To determine whether the use of different numbers of AG values could be responsible for the sign difference between the theoretical and experimental values of ade,the procedure used to obtain the theoretical values of the spectroscopic constants was also applied to the first four experimeqtal AG values. However, this did not introduce appreciable changes in the "experimental" spectroscopic constants. In particular, the sign of w y , remained negative. There are several other possible causes for the opposite signs of the experimental and theoretical values of wy,. Firstly, errors in the experimental AG's could cause the experimentally derived w a , to be in error. We believe that this is unlikely. More likely, errors in the calculated AG's are responsible. There are several sources of errors in the calculated vibrational levels. These may be divided into two categories: (1) those inherent in the CCD + ST(CCD) calculations and (2) those introduced due to the use of a finite grid of internuclear separations, the fitting of these points, and the evaluation of the energy levels of the resulting potential energy curve. We believe that the former source of ~~~

~~

(28)Huber, K.;Herzberg, G. Constants of Diatomic Molecules, Van Nostrand: New York, 1979

The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 2779

Low-Lying Electronic States of A12, Alz+, and A l i TABLE V Properties of the AI2 Ions Calculated by Using the CCD + ST(CCD) Method' IPf EA, eV

species A12+

Alt'

state/confign

vert adiab D., eV R.. A

2 2 g + ( 4 ~ , 2 5 ~ g ) 6.19 211y(4u,22nu) 6.46 411,(4a,5a,2n,) 8.60 42;(4u,2n,2) 9.14 4 Z [ ( 5 ~ 2 ~ 2 ) 1.44 2nu(2Tu 83 ) 2n,(5u822n,) 0.85

w.

cm-'

5.92 6.46

1.37 0.83

3.208 2.192

169 216

1.50 0.93 0.86

2.45 1.87 1.81

2.558 2.461 2.720

335 355 287

"The calculations on the cation and anion states used the 6sSp3dlf and 7s6p3dlf basis sets, respectively. To obtain the IP's and EA's, calculations using these basis sets were also performed on 311uAI,. errors, that is, those inherent in the CCD + ST(CCD) approximation, is likely to be the more important. In addition to the previously mentioned neglect of coupling between various triple and single excitations, the CCD ST(CCD) approximation is expected to lead to an increasing error as the internuclear separation is increased. This is due to the fact that at large internuclear separations the ...2 r g 2configuration should also be included as a reference. Whether this latter factor is important over the relatively small range of internuclear separations spanned (in a classical sense) by the first five vibrational levels is not known. and Bauschlicher et al.I4 have all reported Upton: Basch et R,'s, Dis, and W ~ for S the 3Zg-,311u,and 'Z states of Al,. Basch et al. also reported w d e values. Pacchioni 7 also considered these three states but did not report any vibrational constants. McLean et and Leleyter and Joyes" have reported properties for the 3Z; state only. With regard to the vibrational constants, the latter authors determined only we while the former also calculated w d e . We are unaware of any previous theoretical determinations of rotational constants other than Be. The calculations of Basch et al., Leleyter and Joyes, and Pacchioni yield smaller De's and larger Re's than do the present CCD ST(CCD) calculations. In addition, the w,)s reported by Basch et al. are lower than those determined in the present study. All of these authors used smaller basis sets and recovered a smaller percentage of the electron correlation energy than do the present calculations. McLean et al.'* showed that single-reference SDCI calculations including the Davidson correctionz3give properties for the 3Z; state in good agreement with experiment (and hence with our CCD ST(CCD) predictions). Without the Davidson correction, the D, and wevalues obtained from the single-reference SDCI calculations have large errors. Upton's calculations give w i s , Dis, and R,)s for the %,,, 3Z ,; and IZg+ states in fairly good agreement with those obtained in the present calculations, while giving significantly different values of the state separations. The CCD ST(CCD) calculations performed in the present study recover far more of the valence space electron correlation than do any of the other published calculations with the exception of those of Bauschlicher et al. Thus, it is not surprising that excellent agreement is found between the w,)s, De)s, and Re's determined from these two sets of calculations. For the 3Z; state the CCD ST(CCD) values of we and Re are slightly closer to experiment than are the multireference C I values. On the other hand, the multireference CI calculations give a De about 0.05 eV closer to the experimental value. D. Positive and Negative Ions of AI2. We have also carried out CCD ST(CCD) calculations on several low-lying electronic states of the positive and negative ions of A12. The results, summarized in Table V, were obtained by using the 6s5p3dlf and 7s6p3dlf basis sets for the cation and anion, respectively. The additional s and p functions in the latter basis set are the same as those used to the calculation of the atomic EA. To determine the I P S and EA's, calculations were performed on the neutral AI2 molecule using the 6s5p3dlf and 7s6p3dlf basis sets, respectively. ( i ) Cation States. The two candidates for the ground state of A12+ are and ,nu,formed respectively by ejection of an electron from the 27r, and 5ug orbitals of the 311uground state of A12. The CCD + ST(CCD) approximation yields vertical

+

+

+

+

+

+

+

ionization energies of 6.19 eV (2Zg+)and 6.46 eV (2n,).The corresponding adiabatic IPSare 5.92 and 6.46 eV. Removal of an electron from the 4a, orbital gives rise to relatively low-lying 4110 and 42,-cation states, lying 2.4-3 eV above the 2Zg+ground state. The De calculated for the state (1.39 eV) is slightly greater than that for the 311ustate of the neutral molecule. Our value for the first I P is in good agreement with the 5.8 and 6.0-6.4-eV values determined experimentally by Hanley et al.29a and Upton et al.,29brespectively. On the other hand, our De for the ground-state cation is appreciably larger than the 0.90 f 0.30-eV value reported by Hanley et al. At least three other theoretical studies have examined cation sates of Al,. Upton, using the GVB-CI method, obtained an adiabatic I P (311, 2Zg+)of 6.02 eV and a De of 1.5 eV for the 2Zg+ion.30 Both of these values are within 0.2 eV of our CCD + ST(CCD) results. Given the facts that our calculations use a more flexible basis set and more thoroughly treat electron correlation effects, our values of the I P and EA should be closer to the exact nonrelativistic values. Theoretical properties for the 2Z8+ cation state have also been reported by Leleyter and Joyes." However, these differ significantly from the results of the present study, presumably due to the use of a small basis set. The other theoretical results for A12+ are those of McLean et al.,lz who considered only the state. The adiabatic IP (for formation of 211uA12+)obtained by these authors is close to the present result. Although the ,Zg+ state of the cation is more stable than the 211, state, the latter has the shorter bond length and higher vibrational frequency. These trends are similar to those found in the 311uand 3Z; states of the neutral molecule and follow from simple MO considerations. The larger De for the 2Zg+cation than the 211ucation, on the other hand, is due in part to the chargeinduced dipole term in the interatomic potential. According to our calculations, the u component of the polarizability of A1 is about 7 a: larger than the ir component. (ii) Anion States. There are six possible low-lying anion states of Al,. In Table IV we have included CCD + ST(CCD) results and 27r: (,TI,) anion states, for the 5ug2ir? (IZ;), 5u:2iru (,nu), which can be treated by using single configuration reference wave functions. The other three possible states Z ,(,; 2Zg+,and 2Ag) derive from the 5ug2iru2configuration and cannot be described by single determinental wave functions employing real orbitals. For this reason we have also carried out C I calculations on all six possible low-lying anion states. The CI results will be discussed later in this section. The 42; state is predicted by the CCD + ST(CCD) calculations to be the ground state of the anion, lying over 0.6 eV below either of the 211ustates. The calculated EA (1.50 eV) for formation of the 4Z; anion is 0.4 eV greater than that of Leleyter and Joyes" and about 3.3 times larger than that of the A1 atom. The calculated De (2.45 eV) of the ground state anion is much larger than that of AI2 and A12+. Leleyter and Joyes have argued that the strong bond of A1,- is responsible for the fact that the secondary emission of Alz-, produced upon Ar+ bombardment of aluminum foil, exceeds that of Al-. The two anion states are found to be bound by about 0.9 eV and to be very close in energy, lying 0.86 eV ( 5 ~ 2 2 and ~ ~ ) 0.93 eV (2~:) below the 311uground state of the neutral molecule. These results differ significantly from those of previous theoretical studies. Upton had predicted that the lowest ,II, anion is bound by only 0.09 eV,30while Leleyter and Joyes predicted that it is slightly unbound." It appears that the failure of the previous studies to predict strongly bound anion states is due to inadequate flexibility of the basis sets employed in these studies. The bond length and vibrational frequency calculated for 42&A1,- are intermediate between those of the 311uand 3Z; states of the neutral molecule, being somewhat closer to the former. Our CCD + ST(CCD) calculations give a considerably shorter bond

-

(29) (a) Hanley, L.; Ruatta, S. A.; Anderson, S. L. J . Chem. Phys. 1987, 87, 260. (b) Upton, T. H.; COX,D.M.; Kaldor, A. In Physics and Chemistry of Small Clusrers; Jena, P., Ed.; Plenum: New York, 1987; p 755. (30) Upton, T. H. J . Chem. Phys. 1987, 86, 7054.

2780 The Journal of Physical Chemistry, Vol. 92, No. 10, 1988 TABLE VI: Electron Affinities (ev), Bond Lengths (A), and Spectroscopic Constants (em-’) of AI2- Determined with the Configuration Interaction Method and the 7s6p2d Basis Setn EA state vert adiab R,b we We& WCY, 4 42[ 1.20 1.24 2.577 323.3 1.545 0.109 0.1882 1.27 1.31 2.588 320.0 3.289 0.585 0.1870 22; 0.57 0.61 2.578 318.8 0.960 -0.159 0.1885 0.66 0.70 2.588 310.4 2.110 0.269 0.1866 0.56 0.58 2.620 301.5 1.097 -0.072 0.18 14 0.67 0.68 2.640 289.6 0.410 -0.266 0.1792 1211, 0.54 0.59 2.688 252.9 3.283 0.336 0.1736 0.62 0.63 2.658 188.2 5.323 2.872 0.1765 22gc 0.26 0.27 2.660 285.4 1.220 -0.031 0.1760 0.46 0.46 2.690 271.7 1.418 -0.028 0.1733 2211, 0.14 0.19 2.588 379.6 4.938 0.228 0.1873 0.46 0.48 2.622 383.3 7.034 0.369 0.1817

103~~ 1.25 1.16 1.34 1.42 1.27 1.21 -0.40 -5.59 1.28 1.37 1.95 2.04

OThe first set of entries for each state are obtained from the CI calculations, and the second set of entries are from the Davidson-corrected CI results. bThe bond lengths are determined from the minima of the potential energy curves. length for 42; than did the calculations of Leleyter and Joyes. (211,) anion state has a bond length and vibrational The 5~822.n~ frequency close to those of the 311ustate of the neutral, while the 2r: (211u)anion state has a bond length and vibrational frequency close to those of the 3 Z i state of the neutral. These results suggest that the 211uanions may be viewed as being formed by the addition of an electron to a relatively diffuse orbital of the appropriate neutral species (5gg in the case of 311uand 2 r u in the case of 32;). Since the 311u(5ug2r,) state of the neutral is predicted (in the CCD + ST(CCD) approximation) to be 0.06 eV more stable than the 32; (2~:) state, these results imply that the (2ru3)anion lies about 0.99 eV below its parent 3 Z i state of the neutral (27:) anion state, at R = 2.717 molecule. Calculations on the 2nu A, the equilibrium bond length of the 311ustate of the neutral molecule, “collapsed” onto the lower lying ,nustate, preventing determination of the vertical EA for formation of the 211u(2rU3) anion. The errors in the description of the ,nuanion states around their respective potential minima due to the use of only a single reference configuration in the CCD + ST(CCD) calculations are not known. The SDCI calculations on the anion states were performed using the 7s6p2d basis set. Except for the ,TIu states, these calculations used the minimum number of configurations required to construct proper eigenfunctions of spin and spatial symmetry (one for 42; and two for 2Ag, 22,,-,and 22g+) and used the corresponding oneor two-configuration MCSCF orbitals. For each of the two 211u states, a proper eigenfunction can be constructed with only a single Slater determinant. However, to allow for the mixing between the 2ru3and 5u227ru configurations, the anion energies were obtained from two-reference SDCI calculations. These calculations were performed using one-configuration MCSCF orbitals for the 42; state due to difficulties encountered in extracting the second root from two configurational (5c22ru, 27,’) MCSCF calculations on the anion states. To determine the EA’s, single-reference SDCI calculations with the 7s6p2d basis set were also carried out on 311uAl, at its equilibrium geometry. In order to obtain adiabatic EA’s and spectroscopic constants for the anion states about 20 points were calculated on each potential energy curve. These were then spline fit and used in a procedure analogous to that described previously for the neutral molecule to determine the spectroscopic constants. The EA’s and other properties determined from the CI calculations are summarized in Table VI. The CI calculations show that all six anion states are bound, Le., that they lie energetically below the ground 311ustate of the neutral molecule. In agreement with the coupled-cluster calculations, the most stable anion state is found to be the 42; state. The EA (for formation of the 42, anion) obtained from the SDCI calculations is about 0.25 eV smaller than the CCD ST(CCD) value. Even greater differences between the EA’S calculated by

+

Sunil and Jordan the two methods are found for the 211ustates. Part of the differences between the EA’s obtained with the two methods is due to the use of a smaller basis set for the CI calculations. However, we believe that a more important factor is that the CI calculations recover a smaller portion of the correlation energy than do the CCD + ST(CCD) calculations. Part of the discrepancy between the two sets of EA’s is removed when the Davidson correction23 is made to the C I results. The ,nustates have the opposite ordering in the CCD ST(CCD) and SDCI calculations, with the 5u;2ru configuration lying lower in energy in the CI calculations and the 27rU3configuration lying lower in the CCD + ST(CCD) calculations. Neither of the two sets of calculations can be considered definitive with regard to the ordering of the 211ustates. The coupled-cluster calculations are based on a single-reference configuration and hence cannot adequately describe the states at those bond lengths at which there is strong mixing between the 2ru3and 5ug22r, configurations. The SDCI calculations, while treating these two configurations on an equal footing, suffer from other limitations, the principal ones being the use of molecular orbitals appropriate for the 42; anion state, the use of a smaller basis set, and the recovery of only a relatively small portion of the valence space correlation. Relatively poor agreement is found between the C I and CCD + ST(CCD) values of the bond lengths and the vibrational frequencies of the ’IIUstates. For example, the 2 r U 3(2IIJ state is predicted in the CCD + ST(CCD) procedure to have a 2.461-A bond length, as compared to the SDCI value of 2.588 A. The application of the Davidson correction to the CI results worsens the agreement between the CI and CCD + ST(CCD) values of Re and we for the 211ustates. It is not known which of the three sets (CCD + ST(CCD), CI, or CI-DC) of spectroscopic constants for the 211states should be the most reliable. CI calculations with larger reference spaces using the 7s6p3dl f basis set should resolve this. The CI spectroscopic constants for the ’Xi, +,and ’Ag anion states should be more reliable than those for the 9II, states. This is supported by the observation that fairly good agreement is found between the CI and CCD + ST(CCD) values of the bond lengths and frequencies for the 42gstate. The inclusion of the Davidson correction results in relatively small changes in the values of we, Be, and R, calculated for the 42;, 22;, ’22,and 2Ag states. Also, except for the 211, states, the SDCI and SDCI-DC values for a , are fairly close. For most of the states there are large differences between the SDCI and SDCI-DC values of w,xe and w a e , indicating that CI calculations based on larger reference spaces are needed to obtain accurate values of these higher order spectroscopic constants. The CCD ST(CCD) calculations give vertical detachment energies of 1.50 and 1.60 eV for the Al~-(~2;) A12(311u)and Al,-(4Z,-) A12(3Z,-) processes, respectively. Hence, it should be possible to determine the 311u/3Z; splitting from the photodetachment spectrum, provided that the ions can be prepared sufficiently cold and the resolution sufficiently high. Unpublished photodetachment spectra of A12-from the Lineberger group3’ have a pronounced peak at 1.6 eV excitation energy, extending about 1 eV to lower energies. The latter feature is consistent with detachment from excited anion states. This preliminary data does not provide any information on the 311u/32;state separation.

+

+

-

-

Conclusions The low-lying electronic states of Alz and its ions have been studied by using flexible basis sets and with electron correlation included via coupled-cluster and multireference C I methods. The largest valence space CI calculations place the 311ustate 136 cm-’ below the 32g-state (neglecting zero-point corrections). Based on a series of MBPT and coupled-cluster calculations investigating (3 1) Lineberger, W. C., personal communication. (32) Moore, C. Atomic Energy Leuels; National Bureau of Standards: Washington, DC, 1971; Vol. I. (33) Hotop, H.; Lineberger, W. C. J . Phys. Chem. Ref Data 1975,4,539.

J . Phys. Chem. 1988, 92, 2781-2789 the effect of further expansion of the basis set, the inclusion of correlation effects involving the core electrons, and correction for basis set superposition errors and zero-point vibrational energies, it is concluded that the actual energy separation between the two lowest lying triplet states probably falls in the 90-230-cm-' range, with the 311ustate being the ground state, consistent with the conclusion reached by Bauschlicher and co-workers. A12 is found to have six bound anion states, with the ground state being 42,-. The adiabatic EA for formation of the ground-state anion for All is over 1 eV larger than that of Al atom. The other five bound anion states are doublets. There are two low-lying cation states (22g+ and 211u), with the former being about 0.5 eV more stable. The lowest IP of Alz is nearly identical with

2781

that of AI. Consequently, the dissociation energies of the ground states of AIz and AI2+ are nearly the same. Acknowledgment. This research was carried out with the support of the National Science Foundation and Alcoa Foundation. The calculations were performed on the Cray X-MP/48 at the Pittsburgh Supercomputing Center with a grant of time from Cray Research and on the Chemistry Department's Harris H 1000 superminicomputer, acquired with grants from Harris Corp. and the National Science Foundation. We thank Drs. J. Kline and P. Siska for the use of their spline fitting and Numerov-Cooley routines and Dr. K. Raghavachari for the routines used to perform the CCD ST(CCD) calculations.

+

Infrared Spectra and Ultrasonic Relaxation Spectra of LiAsF, and Macrocycles in

Meizhen Xu,+ Naoki Inoue,t Edward M. Eyring, and Sergio Petrucci*ss Weber Research Institute, Polytechnic University, Farmingdale, New York I 1 735, and Department of Chemistry, University of Utah, Salt Lake City, Utah 841 12 (Received: July 8, 1987; In Final Form: October 29, 1987)

Infrared spectra of the 8, region of LiAsF6 dissolved in the solvent 1,3-dioxolane(DXL) reveal the presence of three solute bands that are assigned to "spectroscopically free" AsF6- (I = 703 cm-I), to contact ion pairs of LiAsF6 (v = 718 cm-I), and to a combination band (v5 + 86) (8 = 675 cm-I). Addition of macrocyles, such as 18-crown-6(18C6), 15-crown-5(15C5), and 12-crown-4 (12C4), in molar ratio R = 1 leaves only the solute band corresponding to the spectroscopicallyfree AsF6species, implying segregation of Li+ in a macrocycle cage. An attempt has been made to mimic the specific coordination of 12C4 to Li' by a nonmacrocyclic solvent component such as 1,2-dimethoxyethane (DME, representing one-half of the 12C4 structure) by using mixtures of DME-DXL of various compositions. It is concluded that, in a very wide solvent composition range, the system appears spectroscopicallyas if DME were the solvent present in large excess with respect to DXL. Ultrasonic relaxation spectra in the frequency range -0.5-7400 MHz for LiAsF6 with one of the three macrocyles 18C6, 15C5, and 12C4 added to 1,3-dioxolane at t = 25 OC, in the concentration range 0.1-0.4 M, are reported. The spectra are described by the sum of two Debye relaxation processes. For the macrocycles 15C5 and 12C4 reacting with LiASF6 in DXL, the slower relaxation process has a much lower amplitude and lower relaxation frequency than for 18C6. This may reflect the smaller cavity size and greater ligand rigidity of both 15C5 and 12C4 with respect to 18C6.

Introduction The structural and dynamical study of the interaction of lithium salts with macrocyclic ligands in media of low permittivity has theoretical and practical significance. Theoretically, it is interesting to probe the structural interaction and mechanism of attack of a macrocycle on ions in electrolyte media of low permittivity in which fully dissociated ionic species are in the minority and the macrocycle interacts basically with a dipolar ion pair. This situation is quite different from that in an aqueous solution where, presumably, the anion is inert (apart from long-range electrostatic interactions with the cation) and complexation occurs between a free cation and a macrocycle. In a medium of low permittivity the nature of the anion may play a role in the complexation process if the anion must be excluded from contact with the cation by the interacting macrocycle. In this respect, the size or other peculiarities of the cavity of different macrocyles, interacting with a given lithium salt in a given solvent, may also play a role in the complexation process. From the practical point of view, since lithium salts in ethereal solutions are used in constructing secondary batteries, it is important to ionize the electrolyte to decrease the internal resistance of the cell containing the electrolyte solution. Crown ethers may, under certain conditions, fulfill this purpose by coordinating the On leave from the Department of Chemistry, University of Peking, Pekin , China. !On leave from the Department of Physics, Ehime University, Ehime, Japan. 8 Polytechnic University.

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cation and segregating the anion outside the first coordination sphere of Li+. To address as many of the above considerations as possible, we have investigated LiAsF6 in the solvent 1,3-dioxolane adding to the system the macrocyles 18-crown-6 (1 8C6), 15-crown-5 (1 5C5), or 12-crown-4 (1 2C4) in molar ratio R = [macrocycle]/ [Li'] N 1 . 1,3-Dioxolane (e = 6.95)' was chosen because no sign of ion pairs dimerization has been found, by ultrasonic relaxation, in previous work,2 thus avoiding obscuring competitive effects due to dimerization of the ion pairs. For the sake of clarity, the presentation of the results, after the Experimental Section, will be split up according to the two methods of attack, namely, infrared spectrometry and ultrasonic relaxation kinetics.

Experimental Section The equipment for the infrared (IR) spectroscopy3 and ultrasonic absorption relaxation kinetic measurements4 have been described previously. LiAsF6 (Agrichem Co., Atlanta, GA) was redried in vacuo ( 1 Torr) at t = 70 OC overnight. 1,3-Dioxolane was kept over molecular sieves (3 A) and then distilled at reduced pressure over sodium metal. The product, stored in desiccators and used within 1-2 days from distillation, was tested for absence of water by N

(1) Saar, D.; Brauner, J.; Farber, H.; Petrucci, S.Adu. Mol. Inrerucr. Relax. Processes 1980, 16, 263. (2) Onishi, S.; Farber, H.; Petrucci, S.J. Phys. Chem. 1980, 84, 2922. (3) Saar, D.; Petrucci, S.J. Phys. Chem. 1986, 90, 3326. (4) Petrucci, S.J. Phys. Chem. 1967, 71, 1174. Delsignore, M.; Maaser, H. E.; Petrucci, S . J. Phys. Chem. 1984, 88, 2405. Eggers, F.; Funck, T.; Richmann, K.; Eyring, E. M.; Petrucci, S . J. Phys. Chem. 1987, 91, 1961.

0 1988 American Chemical Society