Accurate multireference configuration interaction calculations on the lowest 1 and 3 electronic states of C 2,CN, BN, and BO

Transcription

1 Accurate multireference configuration interaction calculations on the lowest 1 and 3 electronic states of C 2,CN, BN, and BO Kirk A. Peterson a) Department of Chemistry, Washington State University and the Environmental Molecular Sciences Laboratory, Pacific Northwest Laboratory, K1-90, Richland, Washington Received 21 June 1994; accepted 27 September 1994 Using a series of correlation consistent basis sets from double to quintuple zeta in conjunction with large internally contracted multireference configuration interaction CMRCI wave functions, potential energy functions have been computed for the X 1 g and a 3 u states of C 2 and the 1 1 and 1 3 states of CN, BN, and BO. By exploiting the regular convergence behavior of the correlation consistent basis sets, complete basis set limits have been estimated that led to accurate predictions for the electronic excitation energies, dissociation energies, equilibrium bond lengths, and harmonic vibrational frequencies. The 1 1 states of CN and BO are predicted to be the electronic ground states of these species with predicted equilibrium excitation energies (T e ) to the low-lying a 3 states of cm 1 and cm 1, respectively. A 3 ground state of BN is predicted with an excitation energy to the low-lying a 1 state of just cm 1. Identical calculations on the singlet triplet splitting of C 2 yielded a prediction of 778 cm 1 for T e, which was just 62 cm 1 above the experimental value. Accurate equilibrium bond lengths and fundamental frequencies are also predicted for BN, BO, and the a 3 state of CN. Dipole moment functions have been computed by CMRCI for the ground and excited electronic states of the three heteronuclear diatomics, and these have been used to derive accurate microwave and infrared transition probabilities for these species. A dipole moment in v 0 of 5.42 D is calculated for the X 1 state of BO, which should lead to an intense microwave spectrum. While the X 3 ground state of BN is predicted to have a very weak infrared spectrum, this species should be observable in the microwave region since the predicted 0 is 1.98 D. Both the microwave and infrared spectra of X 1 CN should be of moderate intensity American Institute of Physics. I. INTRODUCTION The accurate calculation of excited electronic states of molecules can be an extremely challenging task for ab initio electronic structure methods. In contrast to ground state wave functions, which are typically dominated by a single configuration state function CSF, excited electronic states often exhibit strong multiconfiguration effects. Thus, a proper treatment of these systems often requires a multireference method of electron correlation, such as multiconfiguration self-consistent field MCSCF or multireference configuration interaction MRCI. The latter technique, which accounts for both nondynamical and dynamical correlation effects, has been shown to be extremely accurate, especially when full valence complete active space CAS reference functions are used. 1 3 In addition to the n-particle basis or correlation treatment, the requirements of the 1-particle basis set can also be severe for the accurate modeling of excited states. Particularly when energy differences, i.e., excitation energies, are being computed, basis sets which are well balanced with respect to different states are extremely important if accurate results are to be obtained. The recent development of the correlation consistent family of contracted Gaussian basis sets 4 6 has provided a means for systematically extending basis sets to the complete basis set CBS limit. Recent benchmark calculations on various diatomic and triatomic species 7 12 have demonstrated very regular convergence when using a series of correlation consistent basis sets for a variety of molecular properties and spectroscopic constants. The present work is an extension of these previous benchmark studies to the treatment of excited electronic states. Accurate determination of the ground electronic states of the 12-electron isoelectronic series, which includes C 2, BN, CN, and BO, has historically been a very difficult task. Similar calculations of isoelectronic BeO will appear in a separate study. In each case the two lowest electronic states, 1 and 3 1 g and 3 u for C 2, are separated by only a few tenths of an ev, and both theory and experiment have had difficulty in discerning which is the lowest state. Thus, the accurate calculation of the positions and properties of these states should provide a sensitive test of both the correlation method and one-particle basis set. Of these four species, extensive experimental and theoretical studies have only been previously carried out for the C 2 molecule T e 716 cm Both singlet and triplet band systems have been observed experimentally, and more recent theoretical work on the low-lying states of C 2 have included accurate MRCI Ref. 21 and coupled cluster calculations. 23 These calculations were able to reproduce the experimental electronic excitation energy to within a few hundredths of an ev cm 1. For the other members of this series, the spectroscopic properties are much less well characterized. For the BN radia Electronic mail address: ka peterson@pnl.gov 262 J. Chem. Phys. 102 (1), 1 January /95/102(1)/262/16/$ American Institute of Physics

2 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO 263 cal, 3 3 and 1 1 bands have been observed experimentally. 13,24,25 While the triplet system was well resolved and led to an accurate determination of the bond length and vibrational frequency in v 0, 24,25 the singlet system was very weak, 24 and the question has been raised that there may have been a rotational misassignment affecting the derived values of r 0 and From the available experimental observations, the concensus is that the ground state is 3, however this conclusion is mainly based on spectroscopic observations in a Ne matrix. 27 There have been several theoretical studies previously carried out for the BN molecule. 26,28 33 Accurate studies include the multireference double excitation configuration interaction MRD-CI work of Karna and Grein, 29 which considered a total of 23 valence states, and the recent MRCI calculations of Martin et al. 26 The latter study considered just the 1 1 and 1 3 states and used large atomic natural orbital ANO basis sets. Both the earlier work of Karna and Grein T e 800 cm 1 and the more accurate study of Martin et al. T e cm 1 predicted that the 3 state was slightly lower than the 1 state. Spectroscopic investigations of the CN ion have been limited to observations of the singlet band systems. Thus, while the spectroscopic constants of the 1 1 state are fairly well determined, the question of the identity of the electronic ground state has not been answered definitively. Early ab initio calculations had placed the 3 state lower than the 1 by as much as 0.4 ev. The more accurate MRD-CI calculations carried out by Bruna, Peyerimhoff, and Buenker 38,39 predicted that the singlet state was actually lower than the triplet by about 1050 cm ev. At about the same time, Murrell et al. 40 carried out polarization CI POL-CI calculations that concluded the two states were nearly degenerate. More recently, the complete active space self-consistent field CASSCF method was used by Roos et al. 41 to calculate the potential energy curves of the 1, 3, and 3 states. Employing a [6s4p2d] contracted basis set, the CASSCF wave functions yielded a 1 ground state, with the 3 and 3 states lying higher by and cm 1, respectively. In contrast to the other members of this isoelectronic series, the BO ion has received very little attention. Spectroscopic study has been limited to the observation of the 1 1 bands, 13 which yielded estimates of the spectroscopic constants of these states. Theoretical study of this species appears to be limited to the ab initio work of Karna and Grein, 42 where the positions and properties of many lowlying singlet, triplet, and quintet states of BO were calculated by the MRD-CI method with a contracted [5s3p1d] basis set. Their work indicated that the 1 state is the ground state of this ion with a calculated excitation energy to the 3 state of 2400 cm ev. In the present study, spectroscopic properties have been calculated for the 1 1 and 1 3 states of C 2,CN, BN, and BO using large internally contracted MRCI CMRCI wave functions. By using the series of correlation consistent basis sets from cc-pvdz to cc-pv5z, accurate estimates of complete basis set limits for electronic excitation energies, dissociation energies, equilibrium bond lengths, and harmonic frequencies have been derived for each species and electronic state. By extrapolating to the CBS limit, the uncertainties associated with incomplete 1-particle basis sets are essentially removed; the remaining error is that inherent to the correlation method. The high accuracy of the CMRCI wave functions used in the present work is demonstrated by comparison to experimental results for C 2, as well as by analogy to recent CMRCI studies of first row hydrides and diatomics. 7 9,43 Conventional uncontracted MRCI calculations have also been carried out to calibrate the effects of the internal contraction on the calculated spectroscopic constants. Furthermore, selected atomic excitation energies have been computed to assess the accuracy of both the n-particle and 1-particle basis sets. From the CMRCI dipole moment functions, microwave and infrared intensities have also been calculated for CN, BN, and BO to guide future spectroscopic studies of these species. In Sec. II the methodology used in the current work is outlined, while in Secs. III A and III B the results from the potential energy and dipole moment function calculations, respectively, are discussed for each species. Conclusions are then presented in Sec. IV. II. COMPUTATIONAL DETAILS The correlation consistent polarized valence basis sets of Dunning and co-workers, 4 6,44 denoted by cc-pvxz, where x D double zeta, T triple zeta, Q quadruple zeta, and 5 quintuple zeta, have been used in the present work. More specifically, these are comprised of generally contracted 45 sets of [3s2p1d] cc-pvdz, [4s3p2d1f] cc-pvtz, [5s4p3d2f1g] cc-pvqz, and [6s5p4d3 f 2g1h] ccpv5z. The correlation consistent basis sets are constructed by adding shells of correlating s, p,d,... functions to the atomic Hartree Fock HF orbitals, with each function within a shell contributing a nearly equal amount to the correlation energy in a singles and doubles CI calculation on the atoms. Furthermore, the polarization functions, which increase from (1d) cc-pvdz to (4d3 f 2g1h) cc-pv5z are left uncontracted and the (sp) sets increase in size from (9s4p) cc-pvdz to (14s8p) cc-pv5z. In this manner the correlation consistent basis sets are well defined with respect to increases in size and accuracy and might be expected to converge systematically to both the Hartree Fock and CBS limits. In order to assess the effect of additional diffuse functions, especially for the excited electronic states where they could be more important, the augmented correlation consistent sets of Kendall et al., 5 denoted by aug-ccpvxz x D,T,Q, were also used. The augmented sets are constructed from the regular cc-pvxz basis sets by the addition of a single diffuse function in each angular momentum symmetry optimized for the atomic anions. In each case only the pure spherical harmonic components of the polarization functions were used. By systematically increasing the quality of the oneparticle basis set, regular convergence behavior toward the complete basis set limit is generally observed when correlation consistent basis sets are used. 7 11,46 48 Very often the basis set dependence is well described by a simple exponential function of the form

3 264 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO A x A Be Cx, 1 where x is the cardinal number of the basis set 2, 3, 4, and 5 for DZ, TZ, QZ, and 5Z sets, respectively and A( ) corresponds to the estimated CBS limit as x for the spectroscopic constant A(x). The use of the cardinal number x in Eq. 1 is a direct consequence of the shell structure of the basis sets. Using this procedure, estimated complete basis set limits for E e, D e, and r e have been determined for each method and electronic state. Complete basis set limits for the electronic excitation energies T e were obtained as differences between the derived CBS E e values. For the harmonic frequencies, which are generally not well represented by Eq. 1, the cc-pv5z values represent our best estimates of the CBS limit. The orbitals for the CMRCI calculations were taken from CASSCF calculations with a full valence active space 8 electrons in 8 orbitals. The core orbitals were fully optimized but constrained to be doubly occupied. For the calculations on C 2, which were carried out in D 2h symmetry, the CASSCF calculations consisted of 264 ( 1 g ) and 296 ( 3 u ) CSFs, while the CN, BN, and BO calculations C 2v symmetry resulted in and CSFs. Furthermore, for the 3 state calculations, the degenerate x and y states were state-averaged to produce orbitals for the CMRCI of the correct spatial symmetry. In addition to carrying out separate CASSCF calculations for the 1 and 3 states as described above, average orbitals ( 1 3 x 3 y ) have also been computed via a stateaverage procedure with equal weights. Typically, stateaveraged orbitals are used in the calculation of electronic transition moments, where orthogonal orbitals greatly facilitate their construction. In the present case, however, they have been used to assess their effects on CMRCI excitation energies and spectroscopic constants in comparison to those obtained with separately optimized orbitals. Multireference CI wave functions were calculated using the internally contracted MRCI method of Werner and Knowles, 54,55 which significantly reduces the number of variational parameters over a conventional uncontracted MRCI while retaining the high intrinsic accuracy of this method. All single and double excitations were taken with respect to the CASSCF reference functions, and the configurations with two electrons in the external orbital space were internally contracted. The 1s-like core orbitals were not correlated frozen core approximation. With the largest basis set used in this work cc-pv5z, the total number of variational parameters in the CMRCI was for the X 1 g state of C 2 and for the 1 states of the other three species; the sizes of the triplet state calculations were very similar. These can be compared to uncontracted MRCI calculations which would have resulted in about 5 and 10 million variational parameters, respectively. The extremely large size of the MRCI expansions has forced many previous workers to utilize various reference selection or restriction techniques with these species instead of the full valence CAS reference function used here. This can have deleterious effects on the accuracy of the results. In the present work, the effect of higher excitations has been estimated by the multireference analog of the Davidson correction. 56,57 These results are designated by CMRCI Q. The CASSCF and CM- RCI calculations carried out in this work employed the MOLPRO92 suite of ab initio programs. 58 To assess the effects of internally contracting the configurations with two electrons in the external orbital space, conventional uncontracted MRCI calculations were carried out with the COLUMBUS program suite. 59 The molecular orbitals and reference spaces were identical to those used in the CMRCI work in order to ensure an accurate comparison between CMRCI and MRCI. For C 2, both the cc-pvtz and cc-pvqz basis sets were employed. The cc-pvqz basis set calculations resulted in 2.9 million CSFs for the triplet state MRCI; the MRCI/cc-pVQZ singlet state results, where the CI expansion involved just over 1.7 million CSFs, were taken from Ref. 9. The MRCI/cc-pVQZ calculations were prohibitively large for the heteronuclear diatomics of this series, hence only the cc-pvtz basis sets were used in these cases. However, even with this relatively modest basis set, the resulting CI expansions totaled 0.87 and 1.5 million CSFs for the singlet and triplet states, respectively. Potential energy functions PEFs for each state were calculated by fitting 7 8 computed energy values typically over the range 0.3a 0 r r e 0.4a 0 to polynomials of sixth degree in the internal displacement coordinate r r r e. Spectroscopic constants were then calculated from the PEF derivatives by the usual Dunham analysis. 60 Since the CI wave functions are not size consistent, dissociation energies for each state have been computed by the supermolecule approach (r 50a 0 ), which yields more accurate bond energies for this method than summing the respective atomic energies for the dissociated limit cf. Ref. 9. Electric dipole moment functions, (r) s, have also been calculated for CN, BN, and BO at the CMRCI level of theory. In each case the dipole moments were computed as expectation values and those for the molecular ions were translated to the center of mass. After fitting the dipole moments to fourth order polynomials in r r e, where r e was the CMRCI equilibrium bond length for each respective basis set, rotationless dipole moment vibrational matrix elements were derived from these (r) s using numerical vibrational wave functions calculated 61 from the CMRCI PEFs. III. RESULTS AND DISCUSSION A. Calculated atomic splittings As a first step in ascertaining the accuracy of both the correlation methods and 1-particle basis sets used in the following molecular calculations, the atomic splittings for B( 4 P 2 P), N( 2 D 4 S), and O( 1 D 3 P) were calculated using full valence CAS reference functions and orbitals optimized separately for each state. Table I displays the results for CMRCI and CMRCI Q, which are also compared to full CI FCI results and experiment. 62 The FCI calculations were carried out using the method of Knowles and Handy 63,64 as implemented in the MOLPRO program. For both the B( 4 P 2 P) and O( 1 D 3 P) splittings, the CMRCI results

4 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO 265 TABLE I. Calculated atomic splittings cm 1 compared to experiment. a Method Basis set B( 4 P g 2 P u ) N( 2 D u 4 S u ) O( 1 D g 3 P g ) FCI cc-pvdz cc-pvtz CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. b c d e a Values in parentheses are from all-electron calculations with correlation consistent core-valence basis sets. See the text. b Reference 62. The experimental values have been adjusted for spin orbit effects, which are not included in the ab initio calculations. c The experimental 4 P 1/2 2 P 1/2 separation including spin orbit splittings is estimated to be x cm 1, where x is an unknown constant since no intersystem combinations have been observed. d The experimental 2 D 5/2 4 S 3/2 separation including spin orbit splittings is cm 1. e The experimental 1 D 2 3 P 2 separation including spin orbit splittings is cm 1. agree very well with the FCI ones. While the differences between CMRCI and FCI do change slightly as the basis set is extended from cc-pvdz to cc-pvtz, the deviations are just 25 and 2 cm 1 for boron and oxygen, respectively, with the cc-pvtz basis. The Q correction CMRCI Q results in substantially worse agreement with FCI for B( 4 P 2 P), while it has a negligible effect for O( 1 D 3 P). In contrast, for the N( 2 D 4 S) splitting the FCI values lie closer to the CMRCI Q results; with the cc-pvtz basis set the deviations from FCI amount to 123 CMRCI and 69 cm 1 CMRCI Q. The convergence with respect to extending the basis set is also observed to vary among the three atomic cases shown in Table I. For B( 4 P 2 P), the splitting increases as the basis set size increases, while for both N( 2 D 4 S) and O( 1 D 3 P) the separation decreases. The rate of convergence is also very different, as judged by the difference between the ccpvdz and cc-pv5z CMRCI values; 670 cm 1 for B, 2480 cm 1 for N, and 1385 cm 1 for O. Complete basis set limits, obtained by fitting the total energies to Eq. 1, are also shown in Table I. The difference between the estimated CBS limit and experiment is the intrinsic error in the chosen correlation treatment. For the CMRCI method, the intrinsic errors are calculated to be 250, 212, and 12 cm 1 for the B( 4 P 2 P), N( 2 D 4 S), and O( 1 D 3 P) splittings, respectively. The CMRCI Q method yields values of 487, 59, and 11 cm 1, respectively. Part of these errors are due to an inadequate treatment of just the valence electron correlation problem, which can be approximately corrected for by comparison to the FCI results obtained with the cc-pvtz basis set. FCI calibration changes the magnitude of the CMRCI CMRCI Q errors to , 89 10, and cm 1. While the remaining deviations between theory and experiment for the N and O splittings after FCI calibration are relatively small, especially for CMRCI Q, the intrinsic error in the boron splitting is still large 250 cm 1. One of the remaining sources of error is the neglect of correlating the 1s core electrons. By carrying out an analogous procedure as in the valence electron calculations but with all the electrons correlated, all major sources of error should be approximately taken into account. For calculations on the B( 4 P 2 P) splitting with all the electrons correlated, the newly developed correlation consistent core-valence basis sets, 65 denoted cc-pcvxz, were used. These new sets were constructed by adding additional sets of functions optimized for 1s correlation to the standard cc-pvxz basis sets. Groups of functions were added in a correlation consistent fashion, e.g., the cc-pcvdz set has an additional set of tight s and p functions, while the cc-pcvtz has two sets of tight s and p functions and a tight d function. The largest set, cc-pcv5z, adds additional 4s4p3d2 f 1g uncontracted functions to the standard cc-pv5z basis. Specific details of the construction of these sets can be found elsewhere. 65 Results for the B( 4 P 2 P) splitting using the cc-pcvxz basis sets and correlating all the electrons CMRCI, CMRCI Q, and FCI are also shown in Table I in parentheses. Inspection of these results indicates that an all electron CMRCI/cc-pCVTZ calculation underestimates the analogous FCI splitting by about 196 cm 1, while CMRCI Q overestimates it by 166 cm 1. Extrapolation of the cc-pcvxz total energies to the CBS limits via Eq. 1 results in calculated splittings with intrinsic errors of 158 CMRCI and 224 CMRCI Q cm 1. Calibration, however, by the FCI/cc-pCVTZ result yields deviations from experiment of just 45 CMRCI and 60 CMRCI Q cm 1. Thus, in this manner the effect of 1s correlation on the state separation in the atomic B( 4 P 2 P) system is estimated to be 229 cm 1. B. Potential energy functions and spectroscopic constants Spectroscopic constants calculated from the CASSCF, CMRCI, and CMRCI Q PEFs are shown in Tables II V for C 2, CN, BN, and BO, respectively. For these results, CASSCF orbitals optimized separately for each state have been used, except for the electronic excitation energy T e, where the results using both separate denoted sep and stateaveraged denoted avg orbitals are shown. State-averaged orbitals in the CMRCI were also used in the calculation of potential energy functions, but the spectroscopic constants were nearly identical to the separate orbital calculations and are not explicitly shown here. Furthermore, only the results using the cc-pvxz basis sets are shown in Tables II V; the spectroscopic constants obtained using the aug-cc-pvxz sets were very similar to those obtained from the standard ccpvxz sets, especially for those larger than DZ. While the effect on the B( 4 P 2 P) splitting of correlating the 1s electrons was much larger than expected, its magnitude was still less than 1% of the total splitting. Thus, in the following molecular calculations, where core core and core valence correlation effects are expected to be of even lesser importance than in the atomic cases, only valence electron correlation has been considered.

5 266 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO TABLE II. Calculated spectroscopic constants of the X 1 g and a 3 u states of C 2 compared to experiment. E e D e T e cm 1 r e e e x e B e e State Method Basis set hartrees kcal/mol sep avg Å cm 1 cm 1 cm 1 cm 1 X 1 g CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. a a 3 u CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. a a The experimental dissociation energy was taken from Ref. 66. All other values are from Ref X 1 g and a 3 u states of C 2 As is well known, the ground state of C 2 has a large amount of multireference character. In the current CMRCI calculations, the Hartree Fock configuration, 1 2 g 1 2 u 2 2 g 2 2 u 1 4 u in D 2h symmetry, has a coefficient of only The next largest configuration, which arises from the 2 2 u 3 2 g double excitation, has a CI coefficient of 0.34 with several others being greater than 0.1. On the other hand, the a 3 u state, which arises from the single excitation 1 4 u 3 g, is relatively well described by a single configuration HF coefficient of 0.92 and only a few other configurations have CI coefficients greater than 0.1. Primarily due to its higher bond order, the singlet state lies lower in energy than the triplet by 716 cm The CASSCF and CMRCI results for the X 1 g state of C 2 have been presented previously in Ref. 9, but have been included in Table II for completeness. The spectroscopic constants calculated by CMRCI for the ground state are in excellent agreement with experiment, 13 especially with the cc-pvqz or cc-pv5z basis sets. Extrapolation to the CBS limits via Eq. 1 yields results for CASSCF nearly identical to the cc-pv5z values and only minor improvements over cc-pv5z for CMRCI. The spectroscopic constants calculated by CMRCI Q are very similar to the CMRCI ones; the bond length is slightly longer and the harmonic frequency is a little smaller. The CMRCI/cc-pV5Z value of D e is estimated to be just 0.8 kcal/mol below the CBS limit. As discussed in Ref. 9, inclusion of core correlation 0.9 kcal/ mol and corrections for the difference between CMRCI and conventional MRCI 0.3 kcal/mol were predicted to increase D e by an additional 0.6 kcal/mol, which resulted in a prediction of kcal/mol. The relatively small uncertainty associated with this prediction arises in part from the excellent agreement between the small basis set FCI and MRCI results calculated by Bauschlicher and Langhoff. 21 This prediction for D e can be compared to the spectroscopic value recently determined by Urdahl et al. 66 of kcal/mol. D 0 was converted to D e using the experimental vibrational constants. Some of the error in our earlier prediction actually arose from an underestimation of the effects of 1s correlation. Recent calculations by Pradhan et al. 67 and Woon et al. 65 indicate that the contribution to D e from 1s correlation is closer to 1.5 kcal/mol, which changes our prediction for D e to kcal/mol. This is now in excellent agreement with the accurate experimental value of Urdahl et al. 66

6 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO 267 TABLE III. Calculated spectroscopic constants of the X 1 and a 3 states of CN compared to experiment. a E e D e T e cm 1 r e e e x e B e e State Method Basis set hartrees kcal/mol sep avg Å cm 1 cm 1 cm 1 cm 1 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. b a 3 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS a The dissociation energies are calculated in a supermolecule approach relative to C ( 2 P) N( 2 D) for the X 1 and C ( 2 P) N( 4 S) for the a 3. b Reference 13. The results for the a 3 u state of C 2 are very similar in accuracy to those for the ground state. In this case the effect of the Q correction on the spectroscopic constants is even smaller than for the ground state, e.g., the values of r e calculated by CMRCI and CMRCI Q are nearly identical and the e s differ by only a few cm 1. For the excitation energy T e, the use of state-averaged orbitals appears to yield better agreement with experiment than the use of separately optimized orbitals at both the CASSCF and CMRCI levels. For example, the value of T e calculated by CMRCI/cc-pV5Z with separate orbitals for each state is higher, and thus further from experiment, by over 75 cm 1 than when average orbitals are employed. However this is somewhat fortuitous, since the averaging just raises the total energy of the singlet state more than that of the triplet. Much smaller differences are observed between the two different choices of orbitals at the CMRCI Q level of theory. Figure 1 a depicts the excitation energies calculated by CMRCI and CMRCI Q asa function of the correlation consistent basis set used separately optimized orbitals and compares them to experiment. Also included in this figure are the results using the aug-ccpvxz basis sets. While large differences are observed between the cc-pvdz and aug-cc-pvdz results, the effect of additional diffuse functions on T e is negligible by the ccpvqz level. Using the cc-pv5z basis set and separately optimized orbitals state averaged results in parentheses, CMRCI overestimates the excitation energy by cm 1, while CMRCI Q is too small by cm 1. Saturation of the one-particle basis set, as estimated by using Eq. 1, yields T e values sep results of 981 CMRCI and 443 cm 1 CMRCI Q, which differ from experiment by 265 and 273 cm 1, respectively. These values can also be compared to the results of Bauschlicher and Langhoff 21 who used a [5s4p3d2f1g] ANO basis set in conjunction with uncontracted, reference-selected MRCI wave functions. Their values of 654 MRCI and 142 MRCI Q cm 1 are in fair agreement with the present cc-pvqz results and in better agreement with experiment. Coupled cluster calculations including the approximate effects of triple excitations, CCSD T, have also been used by Watts and Bartlett 23 to investigate the first three excited electronic states of C 2. Their all-electrons correlated CCSD T result for the excitation energy of the a 3 u state with the cc-pvqz basis set, 885 cm 1, is nearly identical to the CMRCI value shown in Table II.

7 268 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO TABLE IV. Calculated spectroscopic constants of the X 3 and a 1 states of BN compared to experiment. a E e D e T e cm 1 r e e e x e B e e State Method Basis set hartrees kcal/mol sep avg Å cm 1 cm 1 cm 1 cm 1 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. b r X 3 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. c r a The dissociation energies are calculated in a supermolecule approach relative to B( 2 P) N( 2 D) for the a 1 and B( 2 P) N( 4 S) for the X 3. b Reference 24. c Reference 25. After extrapolating to the CBS limit, the remaining errors in the CMRCI spectroscopic constants can be attributed to deficiencies in the correlation treatment, i.e., effects of higher excitations, core core and core valence correlation, and perhaps errors due to the internal contraction of the double excitations including the restriction to the first order interacting space. In Ref. 9, uncontracted MRCI calculations were carried out that demonstrated the internal contraction had only a small effect on the spectroscopic constants and dissociation energy of C 2. The results of similar calculations on the a 3 u state, together with those for the ground state, are shown in Table VI. Just as in the ground state, the contraction error in the computed spectroscopic constants is nearly negligible. For the excitation energy, the difference between CMRCI and conventional MRCI is computed to be cm 1 for the cc-pvtz basis set, which increases by just 12.3 cm 1 to cm 1 with the cc-pvqz basis. Probably most of this difference is not due to the internal contraction itself but to the restriction of the resulting CI problem to just the first order interacting space of the reference function. When the difference between CMRCI and MRCI is applied to the CMRCI/CBS limit T e 981 cm 1, the resulting prediction of 778 cm 1 differs from experiment by just 62 cm 1. In the calibration study of Bauschlicher and Langhoff, 21 their small basis set MRCI result for T e was larger than the FCI value by 81 cm 1. Thus, most of the remaining error in our calculations is probably due to neglected higher excitations, rather than the neglect of core correlation. 2. X 1 and a 3 CN At first, the X 1 and a 3 states of CN might be expected to be qualitatively different than those of isoelectronic C 2. In this case the ground state does not dissociate to ground state atoms; the lowest allowed asymptote for the X 1 is C ( 2 P) N( 2 D). The a 3 state, however, can dissociate to C ( 2 P) N( 4 S). On this basis alone it might then be expected that the 3 state lies lower than the 1 state. However, inspection of the calculated CASSCF orbitals and CMRCI dipole moment function see below, indicates that near r e both states correlate with C( 3 P) N ( 3 P).

8 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO 269 TABLE V. Calculated spectroscopic constants of the X 1 and a 3 states of BO compared to experiment. a E e D e T e cm 1 r e e e x e B e e State Method Basis set hartrees kcal/mol sep avg Å cm 1 cm 1 cm 1 cm 1 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS Expt. b r B a 3 CASSCF cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS CMRCI Q cc-pvdz cc-pvtz cc-pvqz cc-pv5z Est.CBS a The dissociation energies are calculated in a supermolecule approach relative to B ( 1 S) O( 1 D) for the X 1 and B ( 1 S) O( 3 P) for the a 3. b Reference 13. These values are uncertain, see the text. The bonding in CN is then directly analogous to that of C 2, and the 1 state is stabilized over the 3 due to its higher bond order. This also accounts for the nearly identical magnitudes of the leading CI coefficients for CN compared to those calculated for C 2. Calculated spectroscopic constants for the X 1 and first excited 3 state of CN are shown in Table III, where they are also compared to the available experimental results. For the ground state, which has been studied experimentally 13 the a 3 state has not been observed, the calculated CMRCI spectroscopic constants are in excellent agreement with experiment. The variation in r e and e as the basis set size is increased from cc-pvdz to cc-pv5z is also very regular and similar to that observed for C 2. It is a little surprising, however, that the calculated CMRCI values for the vibrational anharmonicity e x e differ from experiment by about 1 cm 1, which is somewhat larger than expected. The removal or addition of more energies in the fit did not lead to any appreciable change in the calculated value. Hopefully the present work will motivate new high resolution spectroscopy experiments on this species that could resolve this discrepancy. For the a 3 state of CN, the spectroscopic constants of Table III show much the same variation with basis set as the ground state. The CMRCI equilibrium separation, for instance, is nearly converged to the estimated CBS limit with the cc-pv5z basis set, and the corresponding harmonic frequency differs from the cc-pvqz value by only a couple of wave numbers. Assuming the same errors in r e as observed for the a 3 u state of C 2 at the CMRCI/cc-pV5Z level, the equilibrium separation is predicted to be Å. Applying the same type of correction to e yields 1711 cm 1, which is expected be within 15 cm 1 of the unmeasured experimental value. It is also notable that the relatively simple CASSCF method yields very accurate spectroscopic constants for both the ground and excited states of this species. As observed for C 2, however, dynamic correlation is very important for an accurate calculation of the electronic excitation energy. The calculated value of T e changes by cm 1 when dynamic correlation is included via CMRCI. Similar to the results for C 2, the choice of orbitals used in the CMRCI also has a relatively large effect on the computed T e. With the cc-pv5z basis set, the difference in the CMRCI T e s between using separate and average orbitals is just over 84 cm 1, which can be compared to 75 cm 1 for C 2. Figure

9 270 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO FIG. 1. Equilibrium excitation energies, T e , from CMRCI calculations using separately optimized orbitals with the cc-pvxz solid lines and aug-cc-pvxz dashed lines basis sets for a C 2 ; b CN ; c BN; and d BO. 1 b depicts the calculated equilibrium excitation energies calculated by CMRCI with separately optimized orbitals. The trend with respect to basis set is very similar to that observed for C 2, although the effect of the multireference Davidson correction CMRCI Q is smaller for CN. Except at the DZ level, the effect of additional diffuse functions is also nearly negligible. Using the extrapolation of Eq. 1, the complete basis set limits for T e are estimated to be 1042 and 702 cm 1 for CMRCI and CMRCI Q, respectively 957 and 715 cm 1 with average orbitals. These differ from the directly calculated cc-pv5z values by only about 40 cm 1. Uncontracted MRCI/cc-pVTZ calculations for both states of CN are also shown in Table VI. The results of these calculations indicate that the difference between CM- RCI and MRCI for T e is 161 cm 1 with this basis set. Based on the cc-pvqz results for C 2, this is not expected to increase significantly with further basis set saturation. Applying this correction to the CMRCI/CBS limit result yields a prediction for T e of cm 1, where the uncertainty is estimated from the results for C 2. Calculated dissociation energies for each state are also shown in Table III for each basis set and level of theory. The CMRCI value of D e for the X 1 state is calculated to be kcal/mol with the cc-pv5z basis set. The estimated CBS limit is just 0.6 kcal/mol larger at kcal/mol. Similar to C 2, the CMRCI Q values for D e are lower than those calculated using CMRCI by about 2 kcal/mol. A small part 0.5 kcal/mol of this could perhaps be due to a somewhat better description of the N( 2 D) state by CMRCI Q see Table I. For the a 3 state of CN, the CBS limit for D e is estimated to be kcal/mol at the CMRCI level. Assuming a similar correction as C 2 for the effect of core correlation 1.5 kcal/mol and correcting for the internal contraction in the CMRCI Table VI, the equilibrium dissociation energies of X 1 and a 3 CN are predicted to be and kcal/mol, respectively. These values are estimated to be accurate to within 1.5 kcal/mol. 3. X 3 and a 1 BN Similar to the case of CN, while the 3 state of BN dissociates to ground state atoms, B( 2 P) N( 4 S), the 1 state yields B( 2 P) N( 2 D). Furthermore, near r e the bonding of the 1 state correlates with B( 4 P) N( 4 S). This

10 Kirk A. Peterson: 1 and 3 states of C 2,CN, BN, and BO 271 TABLE VI. Calculated differences between CMRCI and conventional MRCI CMRCI MRCI for total energies, dissociation energies, and spectroscopic constants. Species Basis set E e hartrees E r 50 hartrees D e kcal/mol T e cm 1 r e Å e cm 1 e x e cm 1 e cm 1 C 2 CN BO X 1 g cc-pvtz cc-pvqz a 3 u cc-pvtz cc-pvqz X 1 cc-pvtz a 3 cc-pvtz BN a 1 cc-pvtz X 3 cc-pvtz X 1 cc-pvtz a 3 cc-pvtz yields a more favorable bond order for the singlet state in comparison to the triplet, but at a slightly higher cost in energy; this ultimately leads to a 3 ground state for BN, albeit by a small margin. Qualitatively, the leading CI coefficients for the two states of BN are similar to those observed for C 2 and CN. For BN, however, the HF coefficients are somewhat more dominant, although the configuration in the singlet state still has a CI coefficient greater than 0.3 in the CMRCI. The calculated spectroscopic constants for the X 3 and a 1 states of BN are shown in Table IV. For both states, the basis set dependence and accuracy of the BN spectroscopic constants at the CMRCI level are very reminiscent of the results for the isoelectronic molecules C 2 and CN.At the CMRCI/cc-pV5Z level, the calculated value of r e for the X 3 state Å, when corrected for zero-point vibrations using e r Å, differs by only Å from the experimental r 0 value deduced from the emission spectrum. 25 In comparison, the average deviations in the CMRCI/cc-pV5Z r e values from experiment for C 2 and CN were found to be cf. Tables II III Å. Likewise, the fundamental vibrational frequency 0 ( e 2 e x e )at the same level of theory, cm 1, is in excellent agreement with the experimental value of 1496 cm 1. Unfortunately, the experimental constants for the a 1 state appear not to be as reliable as those for the ground state. This should not be entirely unexpected since the 1 1 band of the observed c 1 a 1 transition was apparently very weak, 24 making an accurate rotational analysis difficult. As noted by Martin et al., 26 the experimentally derived 0 value for this state, 1712 cm 1, is inconsistent with high quality ab initio results. This is also the case in the present work, where CMRCI/cc-pV5Z yields a value for 0 of 1680 cm 1. A more accurate prediction can be obtained by comparison to C 2 and CN, which yields predictions for a 1 BN of e 1705 cm 1 and cm 1. These are expected to be accurate to within 15 cm 1. Using averaged coupled pair functional ACPF wave functions with reference selection and somewhat smaller basis sets than used here, Martin et al. 26 estimated 0 to be 1663 cm 1 e 1686 and e x e 11.3 cm 1, which agrees well with the present work. The experimental r 0 value of Å Ref. 24 is also affected by the apparent rotational misassignment and is somewhat too large when compared to our CMRCI/cc-pV5Z value of Å from r e Å and e cm 1. Correcting our ab initio result by the average of the errors observed for C 2 and CN at the same level of theory indicates that the actual experimental value of r 0 should probably lie closer to Å, from a predicted r e of Å. In agreement with previous experimental and theoretical work, the a 1 state is calculated to lie very close to the X 3. In addition, there is a large difference in the calculated T e values when separately optimized or state-averaged orbitals are used in the CMRCI. For instance, with the cc-pv5z basis set the difference between the CMRCI T e sep and T e avg values is nearly 181 cm 1. This can be compared to the corresponding differences for C 2 and CN of only 75 and 84 cm 1, respectively. As shown previously for C 2 Table II and CN Table III, the CMRCI Q results for T e are much less sensitive to the orbitals employed T e 9.4 cm 1 with the cc-pv5z basis set, although the magnitude of the CMRCI Q T e values are about 200 cm 1 larger than CM- RCI sep. Figure 1 c depicts the basis set dependence of the CMRCI and CMRCI Q values for T e obtained using separately optimized orbitals. Additional diffuse functions in the basis set are observed to produce a larger effect at the DZ and TZ levels. In general, however, the dependence on basis set is much less pronounced for BN than is observed for C 2 Fig. 1 a or CN Fig. 1 b. With the cc-pv5z basis set, the singlet triplet splitting is calculated by CMRCI sep to be 84.3 cm 1, which increases by only 3.5 cm 1 to 87.8 cm 1 at the estimated CBS limit. From the results of uncontracted MRCI/cc-pVTZ calculations Table VI, it is estimated that CMRCI underestimates the MRCI excitation energy by about 102 cm 1. Adjusting the CMRCI/CBS limit value by this amount yields a prediction for T e of 190 cm 1. This is expected to be accurate to about 100 cm 1 based on the results for C 2 and the rapid convergence with respect to the 1-particle basis set. The ACPF calculations of Martin et al. 26 yielded a prediction for T e of cm 1. Since