J . Phys. Chem. 1989, 93, 3989-3992 drocarbons of similar molecular weights possessing an increasing number of a electrons (cyclohexane, cyclohexene, 1,4-cyclohexadiene). In all cases, we have observed complex excitation spectra with a main intermolecular progression of about 20 cm-I, of the same order of magnitude as those observed for anisole or phenol-benzene complexes. In this latter case, the low frequencies observed in the excitation spectrum have been assigned to the stretching (50 cm-I) and bending (16 cm-I) modes of the a-hydrogen-bounded phenol. Thus, the value of the intermolecular frequencies does not seem to depend strongly on the nature of the interaction (H bonding, r a or a-g interaction) involved in the dimer.
Conclusion In this work, we have observed the different relaxation channels depending on the excess energy in the anisole-benzene excited complex: resonance fluorescence, vibrational redistribution from intramolecular modes of anisole to the intermolecular bath modes, and vibrational predissociation. We have performed a direct
3989
measurement of the binding energy of the anisolebenzene complex (1 360 cm-l in the ground state and 1700 cm-I in the excited state). The calculated values obtained with a Buckingham type atomatom potential are in very good agreement with the experimental data. For such a complex involving weak van der Waals interactions, these calculations seem to be appropriate for the determination of binding energies and equilibrium geometries. The present calculations also show that the Coulombic interactions, characterized by the net atomic charges on each molecule, seem to be prevailing to reproduce the stabilization of the excited state relative to the ground state. The vibrational progressions observed in the fluorescence excitation and emission spectra have been tentatively assigned to slipping motions of the two aromatic rings.
Acknowledgment. The authors are highly indebted to Dr. A. Tramer for helpful suggestions and discussions on this work. Thanks are also due to Dr. J. A. Beswick and J. P. Flament for their help in the theoretical part of this study. Registry No. Anisole, 100-66-3;benzene, 7 1-43-2.
Spectroscopic Properties of Low-Lying Electronic States of Rh2 K. Balasubramanian*yt and Dai-wei Liao Department of Chemistry, Arizona State University, Tempe, Arizona 85287- 1604 (Received: September 19, 1988)
Complete active space MCSCF (CASSCF) followed by multireference configuration interaction calculations including the spin-orbit term are carried out on low-lying electronic states of Rh2. The spectroscopic properties (Re,T,, we) of 36 bound electronic states including the spin-orbit term and 30 electronic states without the spin-orbit term are reported. The Mulliken population analysis of the CI natural orbitals reveals that most of the states arise predominantly from the 4d85s' atomic configuration of the rhodium atom.
1. Introduction Theoretical and experimental investigations of heavy-metal clusters and their ions have been pursued by many investigators.I-" Experimental spectroscopic investigations of transition-metal dimers are carried out with the supersonic jet expansion methods as well as matrix-isolation methods. The electronic states of jet-cooled clusters are probed by laser-induced fluorescence (LIF) methods, resonant two-photon ionization and dissociation methods, and photoelectron spectroscopic methods. In addition, Lineberger and co-workers8 have employed electron-detachment methods to study the first-row transition-metal dimers. In a recent experimental investigation, Taylor et aL6 have carried out resonance two-photon ionization spectroscopy of jet-cooled Pt,, which confirmed Bala~ubramanian's~ earlier predictions on the electronic states of Pt,. Similar experimental investigations are being carried out by Morse and co-workers on Pd, and other second- and third-row transition-metal dimers. Theoretical investigations of transition-metal dimers and trimers are very challenging due to the electron correlation problem and the large number of open-shell molecular electronic states that arise from the incomplete d shells of the metal atoms. Furthermore, for the heaiier dimers, such as second- and third-row transition-metal dimers, relativistic effects"I2 are known to be very significant. The spin-orbit effects are particularly expected to be nonnegligible due to the presence of two heavy atoms in the second- and third-row dimers. Consequently, high-level calculations that include both electron correlation and spin-orbit effects such as complete active space M C S C F (CASSCF) followed by 'Alfred P. Sloan Fellow; Camille and Henry Dreyfus Teacher-Scholar.
0022-3654/89/2093-3989.$01.50/0
CI and relativistic CI (RCI) methods need to be employed to even obtain a semiquantitative description of the electronic states of these species. The objective of the present investigation is systematic calculations of electronic states of Rh,, including electron correlation and relativistic effects. The theoretical studies on Rh2 to date are those of ShimI3 and Norman and Kolari;14 as referred to by Morse in his review, Shim reports a single-point SCF/valence CI calculation on the electronic states of Rh,. Norman and Kolari carried out SCF-Xa-SW calculations. A single-configuration S C F treatment followed by valence-level CI is not adequate for transition-metal dimers since electron correlation effects are quite (1) Morse; M. D. Chem. Reu. 1986, 86, 1049. (2) Baetzold, R. C.; Hamilton, J. F. Prog. Solid Stare Chem. 1983, 15, 1. (3) Geusic, M. E.; Morse, M. D.; Heath, J. R.; Smalley, R. E. J . Chem. Phys. 1985, 82, 2293. (4) Muetterties, E. L. Science 1977, 196, 839. (51 Ozin, G. A. Card. Rev.-Sci. Ena. 1977, 16, 191. (6) Taylor, S.;Lemire, G. W.; Hamriik, Y . M.; Fu, 2.;Morse, M. D. J . Chem. Phvs.. in Dress. (7) Baiasubramanian, K. J. Chem. Phys. 1987, 87, 6573. op$!
~ ~ ~ ~ p , l , "" df , ;~ ~; :~C ;; 6i ~ ~~ ~~R ~' , ~9 ~~~ ~~
1988, 88, 3786, (9) Balasubramanian, K. J . Chem. Phys. 1988, 89, 6310. (10) Balasubramanian, K.; Pitzer, K. S.Adu. Chem. Phys. 1987,67,287. (11) Pitzer, K. S. Acc. Chem. Res. 1979, 12, 271. (12) Pitzer, K. S. Int. J . Quantum Chem. 1984, 25, 131. (13) Shim, I. Mat.-Fys. Medd.-K. Dan. Vidensk. Selsk. 1985, 41, 147 (as referred to in ref 1). (14) Norman, J. G., Jr.; Kolari, H. J. J . Am. Chem. SOC.1978, 100, 791.
0 1989 American Chemical Society
3990
The Journal of Physical Chemistry, Vol. 93. No. 10, 1989
large. Cocke and GingerichIs have obtained a thermodynamic De of Rh, using the third-law method as 2.92 f 0.22 eV. Matrix-isolation spectroscopic studies have been carried out on Rh,. Bands in the region 21 3-460 nm have been observed. The theoretical investigations carried out to date on related species are on RhH,I8 RhH,.19 RhHC0,2s RhC0,26 and Pd,.9 In the present investigation, we carry out complete active space MCSCF (CASSCF) followed by multireference configuration interaction and relativistic CI calculations on 36 electronic states of Rh, including the spin-orbit coupling and 30 electronic states without the spin-orbit term. Our CI calculations included up to 105 000 configurations. The nature of bonding in these states and Mulliken population analyses are considered. Section 2 describes the method of investigation, and section 3 comprises Results and Discussions.
2. Calculational Methodology First, a qualitative enumeration of the possible electronic states of Rh2 arising from the low-lying atomic states of the atom is considered. The ground state of the rhodium atom is a 4F9,2state arising frum the 4d85s' electronic configuration.20 The other spin-orbit components of the 4 F states are only 1530-3473 cm-' above the ground state. The excited 2D3i2,5/2 states arising from the 4d9 configuration are only 3309-5658 cm-I. There are also ,F, 4P, 2P. and 4 F low-lying excited states. The correlation of 4F + 4F ground-state atoms with molecular states in the absence of spin-orbit coupling yields 112 A-s states. Since the spin-orbit coupling is nonnegligible for Rh, these 1 12 A-s states would be split apart into a large number of w-w states. This simple enumeration of electronic states from just the ground-state rhodium atoms illustrates the complexity of the problem. An exhaustive study of all the electronic states of Rh2 even from the 4F + 4F combination would be impossible since many of the electronic states would be upper roots of states of the same symmetry. In this investigation, we first seek three to four roots of each symmetry without the spin-orbit term and then introduce the spin-orbit coupling. This in itself is an ambitious study. The orbitals for configuration-interaction calculations are generated with the complete active space M C S C F (CASSCF) method. In this method, the orbitals are optimized in a complete configuration space generated by distributing active (valence) electrons in all possible ways. For Rh,, the 18 electrons were distributed in all possible ways among the orbitals that correlate into the 4d and 5s orbitals of the rhodium atom at infinite separation. The CASSCF calculations were actually carried out in the DZhpoint group. The 18 outer electrons of Rh2 were distributed in all possible ways among 12 orbitals in the CASSCF. Separate CASSCF calculations were carried out for different spatial and spin symmetries. State-averaged CASSCF calculations were carried out. All calculations described here were carried out with the relativistic effective core potentials for the rhodium atom that retained the outer nine electrons in the valence space. We employ the analytical Gaussian potentials generated by La John et aI.,l We start with the 3s3p4d valence Gaussian basis set optimized by these authors for the ground state of the rhodium atom. The two large-exponent p and d functions were contracted, which resulted in a 3s2p3d basis set. As we will show later, the p functions do not play a significant role for Rh,. The addition of more diffuse s and d functions also does not change the Re and o,too much. In earlier investigations on Pd? as well as RhH2,I9 it has been shown that the effect of f-type functions is somewhat small.
Balasubramanian and Liao TABLE I: Spectroscopic Properties of the Electronic States of Rh2 in the Absence of the Spin-Orbit Terms state Re, T,. cm-I we, cm-l 2.28 0.0 266 2.53 4864 164 Ire 2.55 4985 176 Iru 32; 2.54 5269 165 5381 182 I'" 2.57 5862 232 2.36 3*s 5902 2.54 I48 6597 2.58 175 Ins 6772 5zg+ 313 2.36 7213 160 In,(II) 2.58 7352 182 2.61 769 1 127 7993 235 8261 110 8365 248 282 8457 8701 23 1 8933 230 9343 140 9673 161 TI,( 111) 2.60 9814 156 2.48 QS 9975 330 2.37 3ng(11) 9986 149 3n,(111) 2.44 10093 173 3A" 2.53 10809 122 2.75 'A, 11445 3ng(111) 2.57 156 11548 117 I zg+ 2.75 12498 134 'A,( 111) 2.8 1 'rm 2.80 12540 133 12681 124 IAg(W 2.79
TABLE 11: Spectroscopic Properties of Low-Lying States of Rh2, Including Spin-Orbit Effects state Re, A Te,cm-' we. cm-' 5Ag(4g) 2.26 0.0 305 5Ag(3g) sAg(2,)
54 Irg(4g) lru(4") 3~~0;) 3z;(lg)
3r"(4") 3ru(3~) 3 ~ ~ ( 2 ~ )
34(4) In"
3Ag(3g)
18) 52,+(2,) 5 z g + ( 18) 5zg+(o,+)
3q4g) 1&(2u) 3nu(1u) 3rg(3g) 3nu(2")
3nu(ou+) 3nu(oU-) srP 'ng(2g) 3ng(
3 ~ ~ 2 , ) 3A"(3")
3ng(o,+) 3ng(o;)
(15) Cocke, D. L.; Gingerich, K. A. J . Chem. Phys. 1974, 60,1958. (16) Ozin, G . A.; Hanlan, A . J . L. Inorg. Chem. 1979, 18, 1781. (17) Hanlan, A . J. L.; Ozin, G.A,, Inorg. Chem. 1977, 16, 2848. (18) Balasubramanian, K.; Liao, D.-W. J . Chem. Phys. 1988, 88, 317. (19) Balasubramanian, K.; Liao, D.-W. J . Phys. Chem. 1988, 92, 6259. (20) Moore, C.E. Atomic Energy Leuels; Circular No. 467; US. NBS: Washington. DC, 1971; Vol. 111. (21 ) La John, L. A.; Christiansen, P. A,; Ross, R. B.; Atashroo, T.:Ermler, W. C. J . Chem. Phys. 1987, 87, 2812.
1q2g) 1") lZg+(o,+)
3~
2.28 2.26 2.28 2.53 2.55 2.54 2.54 2.57 2.57 2.35 2.36 2.36 2.54 2.58 2.36 2.36 2.36 2.3 1 2.60 2.3 1 2.33 2.31 2.31 2.31 2.60 2.48 2.47 2.53 2.53 2.48 2.48 2.75 2.52 2.75
336 886 1192 6056 6177 6457 646 1 6573 6573 6662 6890 6892 7094 7616 7964 7964 7964 8457 8459 8992 9542 9612 9704 9704 10125 10507 11076 11238 1 I266 11478 11478 11948 12172 12738
265 279 266 164 176 165 165 182 182 231 231 232 148 176 313 313 313 321 186 234 28 1 235 235 235 230 156 113 173 173 156 156 122 186 117
Configuration-interaction calculations that employ a multireference configuration list were carried out following the CASSCF calculations. All configurations in the CASSCF with coefficients 1 0 . 0 7 were included as reference configurations in the C I calculations. The CI calculations included the most important single
Low-Lying Electronic States of Rhz
The Journal of Physical Chemistry, Vol. 93, No. 10, 1989 3991
TABLE 111: Weights (Percent) of Important Electronic Configurations in the CI Wave Functions of the Electronic States of Rh2 Near R e without Spin-Orbit Interaction state R, 8, electronic configuration (% of contribution)
Ir,
2.50
'rU
2.50
'2,-
2.50
'ru
2.50 2.25 2.50
'A,
'nu 'II, ',;I 'II,(ll) IA" In,(ii)
~II, 'A,(II) 3~,(11) 317,
311,,(11)
5rs 'II,(lll)
'II,(III)
2.50 2.50 2.50 2.50 2.75 2.50 2.75 2.25 2.25 2.50 2.50 2.75 2.50
311g
2.50 2.50 311,(111) 2.50 3Au 2.50 'A, 2.25 ~II,(III) 2.50 '2,' 2.75
311,(11)
lAu(III) 2.75 3rU(11)
'A,(II)
2.75 2.75
and double excitations and a complete list of single excitations as determined by the POLCI method. That is, in contrast with full MRSDCI, the possibility of two electrons in the external space was eliminated, which enabled truncation of the configuration count to a manageable number of less than 105 000 configurations for most of the states. We also calculate the spin-orbit splittings for the electronic states of Rhz using the recently developed relativistic CI method for polyatomics." The same valence Gaussian basis sets are employed to generate the spin-orbit integrals that are obtained with use of a spin-orbit operator expressed in terms of differences potentials. The natural orbitals obtained of 1 and 1 from the CASSCF/CI calculations without the spin-orbit term are used to transform the spin-orbit integrals. The transformed integrals are added to the one-electron CI matrix in the relativistic CI calculations. The RCI calculations, in general, included reference configurations corresponding to low-lying A-s states of the same Cl symmetry that could mix in the RCI. The RCI calculations are thus multireference singles + doubles CI including the spin-orbit term. The CASSCF calculations of the various electronic states included between 2000 and 3000 configurations. The CI calculations included from 40 000 to 105000 configurations; the lower numbers are for the closed-shell states while the higher numbers are for the open-shell states. The CASSCF/MRCI calculations were carried out with the present author's23modified version of ALCHEMY I1 codes,24as
+
(22) Balasubramanian, K. J . Chem. Phys. 1988, 89, 5731.
described in ref 23. The spin-orbit integrals over Gaussian basis sets were obtained with R. Pitzer's modified version of ARGOS codes. The RCI calculations were carried out by the general procedure described in ref 22.
3. Results and Discussion Table I contains our calculated spectroscopic properties of low-lying electronic states in the absence of spin-orbit effects from the CASSCF/CI methodology. Table I1 lists the spectroscopic properties of 36 electronic states of Rh2, including the spin-orbit effects. Among the electronic states that we calculated, the lowest state is a state in the absence of spin-orbit effects and a 5$(4g) state if spin-orbit coupling is included. The calculated vibrational frequency (we) and the Re of the 'A, state are almost in exact agreement with an R, = 2.28 A and we = 267 cm-' assumed by Cocke and Gingerich" in their third-law determination of the D t value of Rhz. The existence of many low-lying excited electronic states might change this Dl value of Rhz slightly. The third-law DO0 value of Rh2 is 2.92 f 0.22 eV. The dissociation energy (De)of the Rhz 5Ag state was calculated from the CASSCF/CI method by taking the difference of the energy at Reand the energy at 8.0 A. This value is 2.1 eV, which is in reasonable agreement with the predicted experimental value considering the fact that our CI space is somewhat small but (23) Balasubramanian, K. Chem. Phys. Lett. 1986, 127, 5 8 5 . (24) The major authors of ALCHEMY I1 are B. Lengsfield, B. Liu, and M. Yoshimine. (25) McKee, M. L.; Dai, C. H.; Worley, S. D. J. Phys. Chem. 1988, 92, 1056. ( 2 6 ) McKee, M. L.; Worley, S. D.J. Phys. Chem. 1988, 92, 3699.
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The Journal of Physical Chemistry, Vol. 93, KO. 10, 1989
TABLE IV: Mulliken Population Analysis of Electronic States of Rht
1
A, re
Ir"
3v -
3:;
3-1*
In, In
5VE+
&ll) 1-1"
In,w) 3r1, 1
A"( I I ,l
3 q 1 1 )
)re 3
~
Sra
lrI"(II1) Ing(I I I )
)ne 3n,(ll) nu(I I I ) 3A" l-1,
3rIg( I I I ) I
zg+
1A,(111)
3rU(~r) 'Ag(11)
)
1.039 0.988 0.986 0.943 0.93 1 1.045 I ,036 1.039 1.065 1.033 1.066 0.976 1.020 0.994 0.985 1.035 1.018 0.734 0.992 0.991 1.042 0.862 1.044 0.731 1.006 0.979 0.975 1.000 0.969 0.997
0.039 0.056 0.056 0.057 0.057 0.040 0.046 0.049 0.052 0.048 0.041 0.043 0.056 0.042 0.043 0.039 0.058 0.06 I 0.046 0.05 1 0.065 0.063 0.063 0.059 0.037 0.06 I 0.03 I 0.034 0.033 0.034
7.923 7.956 7.959 8.001 8.013 7.916 7.918 7.912 7.884 7.920 7.894 7.98 I 7.925 7.964 7.973 7.927 7.925 8.205 7.982 7.959 7.893 8.075 7.894 8.21 1 7.958 7.961 7.995 7.967 7.999 7.970
adequate to calculate the properties near the well (Re,T,, w e ) . The reported Re and o,values in Tables I and I1 should be somewhat more accurate than the T, values, since the adiabatic energy separations are more sensitive to higher order electron correlation effects. However, we believe that the T, values should be accurate to 10-1 5% for very low-lying states and to 20-25% for states with T, > 7000 cm-l. Moreover, the density of electronic states predicted by comparable CASSCF/CI/RCI calculations on Pt2 by K.B.' agreed well with a recent experimental investigation of Morse and co-workers on Pt2.6 We believe the present calculations on Rh, are comparable in accuracy to the earlier calculations on Pt,. As can be seen from Table 11, the spin-orbit splittings are nonnegligible for most of the electronic states of Rh,. The spin-orbit coupling also brings in applicable change in Re and we values when the splitting or spin-orbit contamination with other electronic states is large. Table 111 shows the weights of various configurations in the CI wave functions of the electronic states of Rh2. The first striking
Balasubramanian and Liao feature we note in that table is that the weight of the leading configurations for most of the electronic states is less than 40%. The only states with weights between 54% and 63% are lru, '4,,Z:' 3A,(lI), 3rg,'I, and 311g. Consequently, it is clear that a multireference MCSCF followed by multireference CI treatment is warranted for a proper description of the electronic states of Rh,. The general trend is that low-X states tend to be mixtures of many electronic configurations while the high-X states are a bit purer. The assignment of the various electronic states was made after a careful analysis of the orbitals and coefficients of the configurations in the DZhgroup. It must be noted that assignment of highly open-shell electronic states was difficult. The 5Ag state of Rh, has a leading configuration of 1ug22ug21u,8g4bU37rg27ru4 and thus corresponds to an approximate bond order of 3. The first excited ITg state has a bond order between 2 and 3. Thus, the lowest-lying states have overall bond orders between 2 and 3. Table IV shows the gross Mulliken populations of various electronic states of Rh2 considered here. The Mulliken population analyses were carried out with use of the MRCI natural orbitals. The reported gross d populations were corrected for the dXz+yz++Zz term. Although the Mulliken populations cannot be used in an absolute sense due to partitioning of the overlap populations between various centers and the possibility of basis set superposition errors, they seem to provide qualitative insight into the nature of the mixing of different atomic states of the metal atoms in the molecular states of the dimer species. Most of the low-lying electronic states of Rh, have gross s populations near unity and gross d populations near 8.0. The gross p populations are noticeably small for most of the electronic states. A few states have rather small s populations and larger d populations. They are T,, 311g(II), and 3Au. There seems to be 4d8Ss' and 4d9 mixing in these states. The electronic states with the largest s populations are sZ;gc and 'Au ( - 1.06.5). The overall findings of the Mulliken populations of the electronic states of Rh2 are very much like those of Pd,9 for which most of the electronic states have gross d populations near 9.0 and gross s populations near 1 .O. 4. Conclusion
In this investigation, state-averaged CASSCF followed by MRCI calculations are carried out on 30 electronic states of Rh,. In addition, spectroscopic properties of 36 electronic states are calculated, including the spin-orbit effects from the CASSCF/ CI/RCI method. The lowest electronic state was found to be 5.1,(4,). The De value of the 5Ag state was calculated to be 2.1 eV with the CASSCF/CI method. The assumed Re and we values in the third-law d e t e r m i n a t i ~ n 'of ~ the Do0 value of Rh2 are in excellent agreement with our calculational results.
Acknowledgment. This research was supported by the U S . Department of Energy under Grant No. DEFG02-86-ER135.58. Registry No Rh,, 12596-98-4
