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Electronic states and potential energy surfaces of rhodium carbide (RhC)
12 December 1997
Chemical Physics Letters 280 Ž1997. 423–429
Electronic states and potential energy surfaces of rhodium carbide ž RhC/ Hang Tan a , Muzhen Liao b, K. Balasubramanian a
a
Department of Chemistry and Biochemistry, Arizona State UniÕersity, Tempe, AZ 85287-1604, USA b Department of Chemistry, Tsinghua UniÕersity, Beijing 100084, China Received 29 August 1997; in final form 15 October 1997
Abstract Potential energy curves and spectroscopic constants of 23 electronic states of the rhodium carbide ŽRhC. have been studied using the complete-active-space multi-configuration self-consistent field ŽCASMCSCF. followed by first-order configuration interaction ŽFOCI. calculations. Multi-reference singles q doubles configuration interaction ŽMRSDCI. were used to determine the properties of low-lying electronic states. The nature of chemical bond formation in different states is discussed in terms of their wave function and Mulliken populations. q 1997 Elsevier Science B.V.
1. I. Introduction Transition metals have played an important role in heterogeneous catalysis. In recent times, the rhodium catalyst has been studied extensively in several contexts such as hydrogenation w1–4x, oxide adsorption w5–7x, decomposition w8x, and exchange reactions w9x. Rhodium complexes have also been studied from the standpoint of its selective activities in organic reactions w10–13x. The use of rhodium as an active and selective center in catalytic reactions is becoming increasingly important. Since most of these reactions on the rhodium surface are concerned with the interaction between rhodium and carbon, the potential curves, the equilibrium structure, as well as the details of both the ground and excited states are thus attracting great attention. The experimental optical spectra and the dissociation energies of RhC have reported previously w14,22x. Shim and Gingerich w15,22x have studied the ground state and the lowest-lying excited
states of RhC by using ab initio Hartree–Fock ŽHF. and configuration–interaction ŽCI. calculations. In this study, we use a complete-active-space m u lti-c o n fig u ra tio n se lf-c o n siste n t fie ld ŽCASMCSCF. technique followed by first-order configuration interaction ŽFOCI. calculations to determine the potential curves and their corresponding electronic configurations of up to 23 states. Multireference singles q doubles configuration interaction ŽMRSDCI. computations were used to determine the spectroscopic constants as well as the dissociation energies for the low-lying states. The Mulliken populations and vibrational frequencies were also computed.
2. Method of calculations Relativistic effective core potentials ŽRECP. for the rhodium atom which retained the outer 4d 8 5s1
0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 9 6 - 2
424
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
shells in the valence space were taken from Ross et al. w16x. For the carbon atom, the RECPs were taken from Pacios and Christiansen w17x which retained the outer 2s 2 2p 2 shells. The optimized Ž3s3p4d. valence gaussian basis set for the Rh atom was taken from w16x, while the Ž4s4p. gaussian basis set for the carbon atom was taken from Ref. w17x and contracted to Ž3s3p.. The carbon basis set was supplemented with one set of 3d functions generated from Dunning and Hay w18x, with a d s 0.75. The RhC molecule was computed here in the C 2v point group with the z axis chosen as the C 2 axis. The orientation is relevant for the purpose of describing the orbitals and the active space. Since the ground states of rhodium and carbon atoms, obtained from Moore’s atomic data tables w19x, are 4 F and 3 P, respectively, the possible spin multiplicities of the
electronic states of RhC should be doublet, quartet, and sextet. Consequently, we carried out the CASMCSCF calculations followed by FOCI calculations for three roots for each electronic spatial symmetry of all the spin multiplicities of doublet, quartet, and ˚ sextet and bond lengths varying from 1.3 to 8.0 A. Thirteen outer electrons of RhC were distributed in all possible ways among the active space. Both CASMCSCF and FOCI calculations described here were carried out with the active space of five a 1 , two b 2 , two b 1 , and one a 2 orbitals. The choice of active space is consistent with the standard description of valence orbitals for rhodium Ž4d 8 5s1 . and carbon Ž2s 2 2p 2 .. The MRSDCI calculations were carried out for the low-lying states, in which single and double excitations were allowed. Reference configurations
Fig. 1. Potential energy curves of RhC Žmultiplicitys 2..
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
were chosen from the CASMCSCF calculations with coefficients G 0.07. The CASMCSCF calculation included up to 3526 configuration spin functions ŽCSF.. The FOCI calculations included up to 386082 CSFs, while the MRSDCI up to 1.3 million CSFs. All CASMCSCFrCI calculations were made by using one of the author’s modified version of ALCHEMY II codes w20x to include RECPs w21x.
3. Results and discussion Qualitatively, RhŽ 4 F. q CŽ 3 P. will generate states such as SqŽ 2., Sy, P Ž3., DŽ3., F Ž2., and G, with spin multiplicities of 2 through 6. Some of these states are likely to be higher in energy and thus may not be contained within the first three roots of the
425
FOCI calculations. Based on the molecular orbitals of RhC, four low-lying configurations are found, i.e. 1s 2 2 s 2 3 s 11p 4 1d 4 , 1s 2 2 s 2 3 s 2 1p 41d 3, 1s 2 2 s 2 1p 4 2 p 1 1d 4 , and 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d3 , from which 2 Sq 4 , 2D4 , 2 P4 , 4 F, 4 P, 2 F, 2 P ŽII.4 are obtained respectively, wherein the electronic states enumerated within the braces are all generated from the same configuration mentioned above. The ground state is predicted to be 2 Sq, while the three lowest excited states are 2D, 2 P, and 4 F. Figs. 1–3 show the computed potential energy curves of RhC dissociating into RhŽ 4 F. q CŽ 3 P.. The curves have been generated at the FOCI level. As seen from the figures, all computed states converge into the same dissociation limit. The prediction of the 2 Sq ground state and the 2 P as the lowest excited state is in good agreement with Refs. w14,15x.
Fig. 2. Potential energy curves of RhC Žmultiplicitys 4..
426
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
Another excited state, 2D, was predicted with energy close to 2 P. No relevant data were obtained experimentally due to the fact that the excitation from 2 Sq to 2D is forbidden. Generally, all sextet states are much higher than the doublet and quartet states, while the doublet states are relatively lower compared to the quartet states. From this standpoint, it can be seen that the bond formation and electron correlation energies seem to overcome the spin exchange stabilization energy. In addition, it is interesting to see that the 4 F and 4 P states are considerably lower than other quartet states. Table 1 shows the spectroscopic properties of RhC computed at the FOCI level. Besides 23 states assigned therein, we present experimental and theoretical data available in Refs. w14,15,22x. As seen from Table 1, our equilibrium bond lengths for all
˚ the states characterized before, namely R e s 1.575 A ˚ for B 2 P, and 1.705 A˚ for for X 2 Sq , 1.629 A E 2 Sq are in good agreement with 1.613, 1.655, and ˚ derived experimentally. We have also com1.687 A pared our results to the observed spectroscopic systems such as B 2 P y X 2 Sq and E 2 Sqy X 2 Sq. Our results for transition energies viz., Te s 13313 and 23598 cmy1 agree well with the experimental data of 10243 and 21439 cmy1 , respectively. The computed vibrational frequencies, viz. ve s 1102 cmy1 for X 2 Sq, 937 cmy1 for B 2 P, and 950 cmy1 for E 2 Sq are also in very good agreement with the experimental results 1050, 939, and 928 cmy1 , respectively. All these evidences are reasonable to support that other data should have comparable accuracy. As can be seen from Table 1, there are some other electronic states which are yet to be observed.
Fig. 3. Potential energy curves of RhC Žmultiplicitys 6..
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
427
Table 1 FOCI spectroscopic properties of RhC d State
˚. R e ŽA FOCI
experimental
previous theory
FOCI
experimental
previous theory
FOCI
experimental
Previous theory
X 2 Sq A2D B2 P a4 F b4 P C2F D 2 P ŽII. E 2 Sq ŽII. c 4D d 4 Sq e 4DŽII. F 2DŽII. G 2DŽIII. f 4 P ŽII. 6 q S 6 D g 4DŽIII. 6 F 6 P 6 P ŽII. 6 q S ŽII. 6 DŽII. 6 y S
1.575 1.648 1.629 1.675 1.706 1.720 1.740 1.705 1.725 1.826 1.809 1.835 1.747 1.777 1.833 1.884 1.835 1.946 1.959 1.939 2.057 1.972 1.997
1.613
1.715a
0 12240 13313 15677 17804 20192 23043 23598 25900 28749 31622 31960 33260 34572 35704 36725 36775 37056 39327 46333 47038 50014 54910
0
0
1050
821b
10243 c
9797
1102 981 937 898 892 856 902 950 907 767 755 733 1074 641 672 597 736 612 582 604 577 654 598
Te Žcmy1 .
1.655
1.687
21439
ve Žcmy1 .
939
928
a
Previous theoretical value of 1.572 at the HF level Žfrom Refs. w15,22x.. Previous theoretical value of 1026 at the HF level Žfrom Refs. w15,22x.. c Another value without the consideration of vibrational perturbation is 9463 cmy1 Žfrom Ref. w14x.. d All experimetal results from Ref. w14x. b
For example, the D 2 P y X 2 Sq transition should be strong as it is dipole-allowed. Our calculations place this transition in the 23043 cmy1 region. Table 2 shows the spectroscopic properties of the low-lying states of RhC obtained from the MRSDCI
Table 2 MRSDCI spectroscopic properties of RhC State
X 2 Sq B2 P A2D a4 F
MRSDCI
MRSDCIqQ
Re ˚. ŽA
Te Žcmy1 .
ve Žcmy1 .
Te Žcmy1 .
1.552 1 609 1.636 1.657
0 11454 13217 16153
1026 1023 1066 819
0 11457 13212 15846
computations. The MRSDCI results are relatively more accurate compared to the FOCI results in that the former technique considers higher-order electron correlations to a greater extent. Compared to the results from FOCI, the B 2 P and A2D states are reversed in stability at the MRSDCI level, but the energy difference therein is also very small, as seen from Table 1, suggesting that they are nearly-degenerate states. We employed the Davidson correction subsequent to the MRSDCI computations. The MRSDCI q Q values are almost equal to the FOCI and MRSDCI results for the states such as X 2 Sq, B 2 P, etc., which were generated from the first roots of the CI calculations. We thus conclude that the Te values presented in Table 1 exhibit reasonable accuracy in that higher-order electron correlation effects do not change these results too much.
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
428 Table 3 FOCI spectroscopic properties of RhC State 2
q
X S A2D B2 P a4 F b4 P C2F D 2 P ŽII. E 2 Sq ŽII. c 4D d 4 Sq e 4DŽII. F 2DŽII. G 2DŽIII. f 4 P ŽII. 6 q S 6 D g 4DŽIII. 6 F 6 P 6 P ŽII. 6 q S ŽII. 6 DŽII. 6 y S
Electronic configuration Žpercentage of contributions. 1s 2 2 s 2 3s 1 1p 4 1d 4 1s 2 2 s 2 3s 2 1p 4 1d3 1s 2 2 s 2 1p 4 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 4s 1 1p 4 1d4 1s 2 2 s 2 3s 1 4s 1 1p 4 1d 3 1s 2 2 s 2 3s 1 1p 3 2 p 1 1d4 1s 2 2 s 2 1p 4 2 p 2 1d 3 1s 1 2 s 2 3s 1 1p 4 2 p 2 1d3 1s 1 2 s 2 3s 2 4s 1 1p 4 1d3 1s 2 2 s 1 3s 1 1p 4 2 p 1 1d4 1s 2 2 s 2 3s 1 1p 4 2 p 2 1d2 1s 2 2 s 1 3s 1 1p 4 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 1 3s 1 4s 1 1p 4 2 p 1 1d1 2d2 1s 2 2 s 2 3s 1 1p 2 2 p 2 1d 4 1s 2 2 s 2 3s 1 4s 1 1p 3 2 p 1 1d3 1s 2 2 s 2 3s 1 4s 1 1p 3 2 p 1 1d3
Ž83. Ž78. Ž80. Ž83. Ž76. Ž29., Ž38., Ž57., Ž82. Ž35., Ž58. Ž28., Ž42., Ž55., Ž44., Ž90. Ž54., Ž42., Ž37., Ž21., Ž37., Ž34. Ž34.
We calculated the dissociation energy for the process RhC Ž X 2 Sq . ™ Rh Ž 4F . q C Ž 3P . as D 0 s 6.12 eV at the FOCI level. This is in good agreement with the experimental value of 6.01 eV and a revised value reported in Ref. w22x. Table 3 shows the leading configurations contributing to the CI wave functions of 23 electronic states of RhC. The X 2 Sq, A2D, B 2 P, a4 F, b 4 P, c 4D, and 6D are well represented by their leading reference configurations in that these configurations make contributions of more than 76%. The other electronic states are considerably more complex. For example, the E 2 Sq ŽII. state of experimental interest is 57% 1s 2 2 s 2 4s 1 1p 4 1d 4 , 9% 1s 2 2 s 2 5s 1 1p 4 1d 4 , while other configurations make less important contributions. Table 4 shows the Mulliken population analysis of the electronic states of RhC. As can be seen from Table 4, the total Rh Mulliken populations for all mentioned states are less than 9.0, implying charge transfer from rhodium to carbon, resulting in RhqCy
1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 5s 1 1p 4 1d 4
Ž27. Ž18. Ž 9.
1s 2 2 s 2 4s 1 1p 3 2 p 1 1d 4
Ž35.
2 s 2 3s 2 1p 4 2 p 2 1d3 1s 2 2 s 2 3s 1 4s 1 1p 4 1d3 1s 1 2 s 1 3s 2 1p 4 2 p 1 1d 4 1s 2 2 s 1 3s 2 1p 4 2 p 2 1d 2
Ž21. Ž20. Ž 4. Ž31.
1s 2 2 s 2 4s 1 1p 3 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 1 3s 1 4s 1 1p 4 2 p 1 1d2 2d1 1s 2 2 s 1 3s 2 1p 2 2 p 2 1d 4
Ž 6. Ž42. Ž37. Ž21. Ž32.
polarity. The overall populations for rhodium in the 2 q S and 2 P states near their equilibrium structures are 4d7.909 5s 0.856 and 4d 8.239 5s 0.113 , respectively, indicating that the 2 Sq arises from the Ž4d 8 5s1 . atomic configuration, while 2 P arises from Ž4d9 ., which is similar to the conclusion drawn in Ref. w15x. Fig. 4 shows the calculated electron difference density maps for the 2 Sq ground state at the minimum bond length with respect to the separated atoms
Table 4 Population analysis for the low-lying electronic states of RhC State 2
q
X S B2 P A2D a4 F b4 P C2F E 2 Sq ŽII. D 2 P ŽII.
Gross population Rh
C
Rh Žs. Rh Žp. Rh Žd. C Žs.
C Žp.
8.900 8.688 8.981 8.751 8.741 8.705 8.676 8.704
4.100 4.312 4.019 4.249 4.259 4.295 4.324 4.296
0.856 0.113 1.612 0.845 0.831 0.841 0.154 0.803
2.122 2.344 2.094 2.292 2.304 2.334 2.343 2.337
0.135 0.335 0.082 0.288 0.235 0.204 0.332 0.169
7.909 8.239 7.287 7.618 7.675 7.660 8.191 7.732
1.924 1.920 1.876 1.915 1.915 1.919 1.945 1.916
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
429
Acknowledgements This research was supported by the US Departm e n t o f E n e rg y u n d e r G ra n t N o . DEFG0286ER13358.
References
Fig. 4. Difference map of electron density of RhC.
in order to show the nature of bonding. As seen from Fig. 4 there is enhancement of electron density outside the sigma bond. The overlap of the dp orbital of rhodium with the pp orbital of carbon leads to the formation of two p bonds in orthogonal directions. Thus the over bond order is three between rhodium and carbon. The gain in electron density supports the Mulliken population analysis data. Furthermore, the difference map reveals p bonding in RhC consistent with the high dissociation energy suggesting multiple bonding.
4. Conclusion We obtained the potential energy curves and spectroscopic properties such as R e , Te , and ve for 23 electronic states of RhC. We compared our results with experiment for three of the observed states, while the spectroscopic properties of several other states are predicted. Our computed De agrees quite well with experiment. Mulliken populations and charge density map were used to analyze the bonding nature.
w1x F. Gassner, E. Dinjus, W. Leitner, Organometallics 15 Ž1996. 2078. w2x S.B. Vorgin, A.I. TripolX skii, G.D. Zakumbaeva, Theor. Exp. Chem. 31 Ž1995. 321. w3x I. Palinko, Appl. Cata. A Gen. 126 Ž1995. 39. w4x W. Leitner, E. Dinjus, F. Gaßner, J. Organ. Chem. 475 Ž1994. 257. w5x H. Cordatos, R.J. Gorte, Appl. Cata. B Environ. 240 Ž1995. 485. w6x S.B. Vorgin, A.I. TripolX skii, G.D. Zakumbaeva, Theor. Exp. Chem. 31 Ž1995. 99. w7x R. Dumplemann, N.W. Cant, D.L. Trimm, Cat. Lett. 32 Ž1995. 357. w8x Z. Pei, S.D. Worley, J. Phys. Chem. 98 Ž1994. 5135. w9x S. Garcia-Granda, P. Lahuerta, J. Latorre, J. Chem. Soci. Dalton Transac. 4 Ž1994. 539. w10x K. Khumtaveeporn, H. Alper, J. Organ. Chem. 60 Ž1995. 8142. w11x M.J. Atherton, J. Fawcett, G.C. Saunders, J. Chem. Soc. 15 Ž1996. 3215. w12x S.D. Perera, B.L. Shaw, J. Chem. Soc. 4 Ž1995. 633. w13x P.C.H. Li, M. Thompson, Analyst 120 Ž1995. 2529. w14x K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure, Vol. 4, Constants of Diatomic Molecules ŽVan Nostrand–Reinhold, New York, 1979.. w15x I. Shim, K.A. Gingerich, Surf. Sci. 156 Ž1985. 623. w16x R.B. Ross, J.M. Powers, T. Atashroo, W. Ermler, L.A. LaJohn, P.A. Christiansen, J. Chem. Phys. 101 Ž1994. 2148. w17x L.F. Pacios, P.A. Christiansen, J. Chem. Phys. 82 Ž1985. 2664. w18x T.H. Dunning, Jr., P.J. Hay, in: Ed. H.F. Schaefer, III, Methods of Electronic Structure Theory ŽPlenum Press, New York, 1977. p. 1. w19x C.E. Moore, Tables of Atomic Energy Levels ŽNational Bureau of Standards, Washington, DC, 1971.. w20x The major authors of ALCHEMY II are B. Lengsfield, B. Liu, Y. Yoshmine. w21x K. Balasubramanian, Chem. Phys. Lett. 127 Ž1986. 585. w22x I. Shim, K.A. Gingerich, J. Chem. Phys. 81 Ž1984. 5937.
Chemical Physics Letters 280 Ž1997. 423–429
Electronic states and potential energy surfaces of rhodium carbide ž RhC/ Hang Tan a , Muzhen Liao b, K. Balasubramanian a
a
Department of Chemistry and Biochemistry, Arizona State UniÕersity, Tempe, AZ 85287-1604, USA b Department of Chemistry, Tsinghua UniÕersity, Beijing 100084, China Received 29 August 1997; in final form 15 October 1997
Abstract Potential energy curves and spectroscopic constants of 23 electronic states of the rhodium carbide ŽRhC. have been studied using the complete-active-space multi-configuration self-consistent field ŽCASMCSCF. followed by first-order configuration interaction ŽFOCI. calculations. Multi-reference singles q doubles configuration interaction ŽMRSDCI. were used to determine the properties of low-lying electronic states. The nature of chemical bond formation in different states is discussed in terms of their wave function and Mulliken populations. q 1997 Elsevier Science B.V.
1. I. Introduction Transition metals have played an important role in heterogeneous catalysis. In recent times, the rhodium catalyst has been studied extensively in several contexts such as hydrogenation w1–4x, oxide adsorption w5–7x, decomposition w8x, and exchange reactions w9x. Rhodium complexes have also been studied from the standpoint of its selective activities in organic reactions w10–13x. The use of rhodium as an active and selective center in catalytic reactions is becoming increasingly important. Since most of these reactions on the rhodium surface are concerned with the interaction between rhodium and carbon, the potential curves, the equilibrium structure, as well as the details of both the ground and excited states are thus attracting great attention. The experimental optical spectra and the dissociation energies of RhC have reported previously w14,22x. Shim and Gingerich w15,22x have studied the ground state and the lowest-lying excited
states of RhC by using ab initio Hartree–Fock ŽHF. and configuration–interaction ŽCI. calculations. In this study, we use a complete-active-space m u lti-c o n fig u ra tio n se lf-c o n siste n t fie ld ŽCASMCSCF. technique followed by first-order configuration interaction ŽFOCI. calculations to determine the potential curves and their corresponding electronic configurations of up to 23 states. Multireference singles q doubles configuration interaction ŽMRSDCI. computations were used to determine the spectroscopic constants as well as the dissociation energies for the low-lying states. The Mulliken populations and vibrational frequencies were also computed.
2. Method of calculations Relativistic effective core potentials ŽRECP. for the rhodium atom which retained the outer 4d 8 5s1
0009-2614r97r$17.00 q 1997 Elsevier Science B.V. All rights reserved. PII S 0 0 0 9 - 2 6 1 4 Ž 9 7 . 0 1 1 9 6 - 2
424
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
shells in the valence space were taken from Ross et al. w16x. For the carbon atom, the RECPs were taken from Pacios and Christiansen w17x which retained the outer 2s 2 2p 2 shells. The optimized Ž3s3p4d. valence gaussian basis set for the Rh atom was taken from w16x, while the Ž4s4p. gaussian basis set for the carbon atom was taken from Ref. w17x and contracted to Ž3s3p.. The carbon basis set was supplemented with one set of 3d functions generated from Dunning and Hay w18x, with a d s 0.75. The RhC molecule was computed here in the C 2v point group with the z axis chosen as the C 2 axis. The orientation is relevant for the purpose of describing the orbitals and the active space. Since the ground states of rhodium and carbon atoms, obtained from Moore’s atomic data tables w19x, are 4 F and 3 P, respectively, the possible spin multiplicities of the
electronic states of RhC should be doublet, quartet, and sextet. Consequently, we carried out the CASMCSCF calculations followed by FOCI calculations for three roots for each electronic spatial symmetry of all the spin multiplicities of doublet, quartet, and ˚ sextet and bond lengths varying from 1.3 to 8.0 A. Thirteen outer electrons of RhC were distributed in all possible ways among the active space. Both CASMCSCF and FOCI calculations described here were carried out with the active space of five a 1 , two b 2 , two b 1 , and one a 2 orbitals. The choice of active space is consistent with the standard description of valence orbitals for rhodium Ž4d 8 5s1 . and carbon Ž2s 2 2p 2 .. The MRSDCI calculations were carried out for the low-lying states, in which single and double excitations were allowed. Reference configurations
Fig. 1. Potential energy curves of RhC Žmultiplicitys 2..
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
were chosen from the CASMCSCF calculations with coefficients G 0.07. The CASMCSCF calculation included up to 3526 configuration spin functions ŽCSF.. The FOCI calculations included up to 386082 CSFs, while the MRSDCI up to 1.3 million CSFs. All CASMCSCFrCI calculations were made by using one of the author’s modified version of ALCHEMY II codes w20x to include RECPs w21x.
3. Results and discussion Qualitatively, RhŽ 4 F. q CŽ 3 P. will generate states such as SqŽ 2., Sy, P Ž3., DŽ3., F Ž2., and G, with spin multiplicities of 2 through 6. Some of these states are likely to be higher in energy and thus may not be contained within the first three roots of the
425
FOCI calculations. Based on the molecular orbitals of RhC, four low-lying configurations are found, i.e. 1s 2 2 s 2 3 s 11p 4 1d 4 , 1s 2 2 s 2 3 s 2 1p 41d 3, 1s 2 2 s 2 1p 4 2 p 1 1d 4 , and 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d3 , from which 2 Sq 4 , 2D4 , 2 P4 , 4 F, 4 P, 2 F, 2 P ŽII.4 are obtained respectively, wherein the electronic states enumerated within the braces are all generated from the same configuration mentioned above. The ground state is predicted to be 2 Sq, while the three lowest excited states are 2D, 2 P, and 4 F. Figs. 1–3 show the computed potential energy curves of RhC dissociating into RhŽ 4 F. q CŽ 3 P.. The curves have been generated at the FOCI level. As seen from the figures, all computed states converge into the same dissociation limit. The prediction of the 2 Sq ground state and the 2 P as the lowest excited state is in good agreement with Refs. w14,15x.
Fig. 2. Potential energy curves of RhC Žmultiplicitys 4..
426
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
Another excited state, 2D, was predicted with energy close to 2 P. No relevant data were obtained experimentally due to the fact that the excitation from 2 Sq to 2D is forbidden. Generally, all sextet states are much higher than the doublet and quartet states, while the doublet states are relatively lower compared to the quartet states. From this standpoint, it can be seen that the bond formation and electron correlation energies seem to overcome the spin exchange stabilization energy. In addition, it is interesting to see that the 4 F and 4 P states are considerably lower than other quartet states. Table 1 shows the spectroscopic properties of RhC computed at the FOCI level. Besides 23 states assigned therein, we present experimental and theoretical data available in Refs. w14,15,22x. As seen from Table 1, our equilibrium bond lengths for all
˚ the states characterized before, namely R e s 1.575 A ˚ for B 2 P, and 1.705 A˚ for for X 2 Sq , 1.629 A E 2 Sq are in good agreement with 1.613, 1.655, and ˚ derived experimentally. We have also com1.687 A pared our results to the observed spectroscopic systems such as B 2 P y X 2 Sq and E 2 Sqy X 2 Sq. Our results for transition energies viz., Te s 13313 and 23598 cmy1 agree well with the experimental data of 10243 and 21439 cmy1 , respectively. The computed vibrational frequencies, viz. ve s 1102 cmy1 for X 2 Sq, 937 cmy1 for B 2 P, and 950 cmy1 for E 2 Sq are also in very good agreement with the experimental results 1050, 939, and 928 cmy1 , respectively. All these evidences are reasonable to support that other data should have comparable accuracy. As can be seen from Table 1, there are some other electronic states which are yet to be observed.
Fig. 3. Potential energy curves of RhC Žmultiplicitys 6..
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
427
Table 1 FOCI spectroscopic properties of RhC d State
˚. R e ŽA FOCI
experimental
previous theory
FOCI
experimental
previous theory
FOCI
experimental
Previous theory
X 2 Sq A2D B2 P a4 F b4 P C2F D 2 P ŽII. E 2 Sq ŽII. c 4D d 4 Sq e 4DŽII. F 2DŽII. G 2DŽIII. f 4 P ŽII. 6 q S 6 D g 4DŽIII. 6 F 6 P 6 P ŽII. 6 q S ŽII. 6 DŽII. 6 y S
1.575 1.648 1.629 1.675 1.706 1.720 1.740 1.705 1.725 1.826 1.809 1.835 1.747 1.777 1.833 1.884 1.835 1.946 1.959 1.939 2.057 1.972 1.997
1.613
1.715a
0 12240 13313 15677 17804 20192 23043 23598 25900 28749 31622 31960 33260 34572 35704 36725 36775 37056 39327 46333 47038 50014 54910
0
0
1050
821b
10243 c
9797
1102 981 937 898 892 856 902 950 907 767 755 733 1074 641 672 597 736 612 582 604 577 654 598
Te Žcmy1 .
1.655
1.687
21439
ve Žcmy1 .
939
928
a
Previous theoretical value of 1.572 at the HF level Žfrom Refs. w15,22x.. Previous theoretical value of 1026 at the HF level Žfrom Refs. w15,22x.. c Another value without the consideration of vibrational perturbation is 9463 cmy1 Žfrom Ref. w14x.. d All experimetal results from Ref. w14x. b
For example, the D 2 P y X 2 Sq transition should be strong as it is dipole-allowed. Our calculations place this transition in the 23043 cmy1 region. Table 2 shows the spectroscopic properties of the low-lying states of RhC obtained from the MRSDCI
Table 2 MRSDCI spectroscopic properties of RhC State
X 2 Sq B2 P A2D a4 F
MRSDCI
MRSDCIqQ
Re ˚. ŽA
Te Žcmy1 .
ve Žcmy1 .
Te Žcmy1 .
1.552 1 609 1.636 1.657
0 11454 13217 16153
1026 1023 1066 819
0 11457 13212 15846
computations. The MRSDCI results are relatively more accurate compared to the FOCI results in that the former technique considers higher-order electron correlations to a greater extent. Compared to the results from FOCI, the B 2 P and A2D states are reversed in stability at the MRSDCI level, but the energy difference therein is also very small, as seen from Table 1, suggesting that they are nearly-degenerate states. We employed the Davidson correction subsequent to the MRSDCI computations. The MRSDCI q Q values are almost equal to the FOCI and MRSDCI results for the states such as X 2 Sq, B 2 P, etc., which were generated from the first roots of the CI calculations. We thus conclude that the Te values presented in Table 1 exhibit reasonable accuracy in that higher-order electron correlation effects do not change these results too much.
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
428 Table 3 FOCI spectroscopic properties of RhC State 2
q
X S A2D B2 P a4 F b4 P C2F D 2 P ŽII. E 2 Sq ŽII. c 4D d 4 Sq e 4DŽII. F 2DŽII. G 2DŽIII. f 4 P ŽII. 6 q S 6 D g 4DŽIII. 6 F 6 P 6 P ŽII. 6 q S ŽII. 6 DŽII. 6 y S
Electronic configuration Žpercentage of contributions. 1s 2 2 s 2 3s 1 1p 4 1d 4 1s 2 2 s 2 3s 2 1p 4 1d3 1s 2 2 s 2 1p 4 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 4s 1 1p 4 1d4 1s 2 2 s 2 3s 1 4s 1 1p 4 1d 3 1s 2 2 s 2 3s 1 1p 3 2 p 1 1d4 1s 2 2 s 2 1p 4 2 p 2 1d 3 1s 1 2 s 2 3s 1 1p 4 2 p 2 1d3 1s 1 2 s 2 3s 2 4s 1 1p 4 1d3 1s 2 2 s 1 3s 1 1p 4 2 p 1 1d4 1s 2 2 s 2 3s 1 1p 4 2 p 2 1d2 1s 2 2 s 1 3s 1 1p 4 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 1 3s 1 4s 1 1p 4 2 p 1 1d1 2d2 1s 2 2 s 2 3s 1 1p 2 2 p 2 1d 4 1s 2 2 s 2 3s 1 4s 1 1p 3 2 p 1 1d3 1s 2 2 s 2 3s 1 4s 1 1p 3 2 p 1 1d3
Ž83. Ž78. Ž80. Ž83. Ž76. Ž29., Ž38., Ž57., Ž82. Ž35., Ž58. Ž28., Ž42., Ž55., Ž44., Ž90. Ž54., Ž42., Ž37., Ž21., Ž37., Ž34. Ž34.
We calculated the dissociation energy for the process RhC Ž X 2 Sq . ™ Rh Ž 4F . q C Ž 3P . as D 0 s 6.12 eV at the FOCI level. This is in good agreement with the experimental value of 6.01 eV and a revised value reported in Ref. w22x. Table 3 shows the leading configurations contributing to the CI wave functions of 23 electronic states of RhC. The X 2 Sq, A2D, B 2 P, a4 F, b 4 P, c 4D, and 6D are well represented by their leading reference configurations in that these configurations make contributions of more than 76%. The other electronic states are considerably more complex. For example, the E 2 Sq ŽII. state of experimental interest is 57% 1s 2 2 s 2 4s 1 1p 4 1d 4 , 9% 1s 2 2 s 2 5s 1 1p 4 1d 4 , while other configurations make less important contributions. Table 4 shows the Mulliken population analysis of the electronic states of RhC. As can be seen from Table 4, the total Rh Mulliken populations for all mentioned states are less than 9.0, implying charge transfer from rhodium to carbon, resulting in RhqCy
1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 3s 1 1p 4 2 p 1 1d 3 1s 2 2 s 2 5s 1 1p 4 1d 4
Ž27. Ž18. Ž 9.
1s 2 2 s 2 4s 1 1p 3 2 p 1 1d 4
Ž35.
2 s 2 3s 2 1p 4 2 p 2 1d3 1s 2 2 s 2 3s 1 4s 1 1p 4 1d3 1s 1 2 s 1 3s 2 1p 4 2 p 1 1d 4 1s 2 2 s 1 3s 2 1p 4 2 p 2 1d 2
Ž21. Ž20. Ž 4. Ž31.
1s 2 2 s 2 4s 1 1p 3 2 p 1 1d 4 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 2 3s 1 1p 3 2 p 2 1d3 1s 2 2 s 1 3s 1 4s 1 1p 4 2 p 1 1d2 2d1 1s 2 2 s 1 3s 2 1p 2 2 p 2 1d 4
Ž 6. Ž42. Ž37. Ž21. Ž32.
polarity. The overall populations for rhodium in the 2 q S and 2 P states near their equilibrium structures are 4d7.909 5s 0.856 and 4d 8.239 5s 0.113 , respectively, indicating that the 2 Sq arises from the Ž4d 8 5s1 . atomic configuration, while 2 P arises from Ž4d9 ., which is similar to the conclusion drawn in Ref. w15x. Fig. 4 shows the calculated electron difference density maps for the 2 Sq ground state at the minimum bond length with respect to the separated atoms
Table 4 Population analysis for the low-lying electronic states of RhC State 2
q
X S B2 P A2D a4 F b4 P C2F E 2 Sq ŽII. D 2 P ŽII.
Gross population Rh
C
Rh Žs. Rh Žp. Rh Žd. C Žs.
C Žp.
8.900 8.688 8.981 8.751 8.741 8.705 8.676 8.704
4.100 4.312 4.019 4.249 4.259 4.295 4.324 4.296
0.856 0.113 1.612 0.845 0.831 0.841 0.154 0.803
2.122 2.344 2.094 2.292 2.304 2.334 2.343 2.337
0.135 0.335 0.082 0.288 0.235 0.204 0.332 0.169
7.909 8.239 7.287 7.618 7.675 7.660 8.191 7.732
1.924 1.920 1.876 1.915 1.915 1.919 1.945 1.916
H. Tan et al.r Chemical Physics Letters 280 (1997) 423–429
429
Acknowledgements This research was supported by the US Departm e n t o f E n e rg y u n d e r G ra n t N o . DEFG0286ER13358.
References
Fig. 4. Difference map of electron density of RhC.
in order to show the nature of bonding. As seen from Fig. 4 there is enhancement of electron density outside the sigma bond. The overlap of the dp orbital of rhodium with the pp orbital of carbon leads to the formation of two p bonds in orthogonal directions. Thus the over bond order is three between rhodium and carbon. The gain in electron density supports the Mulliken population analysis data. Furthermore, the difference map reveals p bonding in RhC consistent with the high dissociation energy suggesting multiple bonding.
4. Conclusion We obtained the potential energy curves and spectroscopic properties such as R e , Te , and ve for 23 electronic states of RhC. We compared our results with experiment for three of the observed states, while the spectroscopic properties of several other states are predicted. Our computed De agrees quite well with experiment. Mulliken populations and charge density map were used to analyze the bonding nature.
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