Ab initio multireference study of the BN molecule - Computational

Ab inifio multireference

study of the BN molecule

J. M. L. Martin*) and Timothy J. Lee NASA Ames Research Center, Moffett Field, California 9403.5lOGO

Gustavo E. Scuseriab) Department of Chemistry and Rice Quantum Institute, Rice University, Houston, Texas 77251-1892

Peter R. Taylo@ ELORET

Institute, Palo Alto, California 94303

(Received 16 June 1992; accepted 22 July 1992) The lowest ‘Bf and 311 states of the BN molecule have been studied using multireference configuration interaction (MRCI) and averaged coupled-pair functional (ACPF) methods and large atomic natural orbital (ANO) basis sets, as well as several coupled cluster methods, Our best calculations strongly support a 311 ground state, but the c IX+ state lies only 381 f 100 cm-r higher. The a ‘2+ state wave function exhibits strong multireference character and, consequently, the predictions of the perturbationally-based single-reference CCSD(T) coupled cluster method are not as reliable in this case as the multireference results. The best theoretical predictions for the spectroscopic constants of BN are in good agreement with experiment for the X 311 state, but strongly suggest a misassignment of the fundamental vibrational frequency for the a ‘1;+ state.

I. INTRODUCTlON Boron nitride, the &phase of which scratches diamond, r has attracted considerable interest in materials science. (See the introduction to Ref. 2 for some relevant references.) As it has proven to be very difficult to obtain pure &BN as a solid film, theoretical and experimental attention has been directed at trying to understand the growth process of thin BN films from the plasma involved in chemical vapor deposition techniques,3 and thus to understand the structural and thermochemical properties of B,, N,, and B,N, clusters. Structures and total atomization energies are currently available for boron clusters up to B4 inclusive (Ref. 4 and references therein). The B,N and B,N2 molecules have very recently been characterized experimentally’ following theoretical predictions of their structure.2’6 In addition, interest has recently been aroused by buckminsterfullerene analogs involving B and N atoms.7 Given its small size, it is somewhat surprising that a complete experimental characterization of the BN molecule has not been performed. Spectroscopic work is limited to the pioneering study of Douglas and Herzberg,’ work by Thrush,9 and Mosher and Frosch,” and two more recent papers by Bredohl, Dubois, Houbrechts, and Nzohabonayo (BDHN). 1’*12The data from these articles are limited both in abundance and accuracy, hence the BN molecule is a natural candidate for ab initio study. Until recently, the most complete ab initio study available was the work of Karna and Grein ( KG),i3*14 who ‘)Permanent address: Limburgs Universitair Centrum, Departement SBG, Universitaire Campus, B-3590 Diepenbeek, Belgium and University of Antwerp (UIA), Institute for Materials Science, Department of Chemistry, Universiteitsplein 1, B-2610 Wilrijk, Belgium. b)Camille and Henry Dreyfus Teacher-Scholar. “Mailing address: NASA Ames Research Center, Mail Stop RTC-230-3, Moffett Field, California 94035-1000. J. Chem. Phys. 97 (Q), 1 November

1992

studied a large number of valence states at the MRD-CI (Ref. 15) level with a (9@ld)/[%3pld] basis set.16’17 Their computational work was instrumental in correcting the origina111~‘2measurements of the bond distance for the lowest 311 state. Like the previous self-consistent field (SCF) level work of Verhaegen et aZ.,‘* they concluded that the molecule has a 311 ground state. However, they also found that the energy difference between the 1 311 and 1 *Zf states is only on the order of 0.1 eV, which brings it within the uncertainty of the calculations at this level of theory. Since the triple-zeta plus polarization basis set employed by Karna and Grein is rather small, Martin et al. lg undertook a new study using Pople-type (spdf ) basis sets (Ref. 20 and references therein) and the quadratic configuration interaction method QCISD(T),21 which is closely related to the CCSD( T) (Ref. 22) augmented coupled cluster method. At the highest level of theory, these authorslg found the 1 311 state to be the ground state, but the computed transition energy (100 cm-‘) fell within the uncertainty of the calculations. Hence the problem of identifying the ground state was not really resolved. By combining the experimental fundamental frequencies and r. distances with the computed anharmonicities and rotation-vibration couplings, the authors finally obtained a “self-consistent” set of spectroscopic constants for both states. At the level of theory considered, the computed quantities were still not of spectroscopic accuracy. One aim of the present study is a definitive determination of the ground state of BN; another is to obtain spectroscopic constants with an accuracy comparable to the experimental ones. Finally, it is likely that the small singlet/triplet separation, even smaller than for the isoelectronic C2 molecule, should be a sensitive test of the performance of different computational methods.

0021-9606/92/216549-08$006.00

@ 1992 American institute of Physics

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Martin et al.: Ab initio study of BN

6550

II. COMPUTATIONAL

METHODS

In the first calculations, aimed at an initial n-particle space calibration, the basis set used was a [5s3p2d] contraction of the Huzinaga-Dunning triple-zeta plus two polarization functions (TZ2P ) basis set. *6,23(Polarization exponents were taken from Ref. 19.) The other two basis sets used, denoted [4321] and [54321] throughout the paper, are [4s3p2dlf ] and [5s4p3d2f lg] generally contracted atomic natural orbital basis sets.24 These sets are based on van Duijneveldt’s ( 13~8~) primitive set,25 augmented with even-tempered 6d4f2g polarization sets, and contracted as described in Ref. 24. The ratio /3 between members of the even-tempered sequences aflk is 2.5 in all cases. The innermost primitive d, f, and g exponents are, respectively, 4.94, 2.37, and 1.14 for B and 9.88, 4.74, and 2.28 for N. For the 311 state, density matrix averaging was performed such that the complete active space SCF (CASSCF) orbitals obey full symmetry and equivalence restrictions.26 Both multireference configuration interaction (MRCI) and the multireference averaged coupledpair functional method (ACPF) (Ref. 27) were considered. Except for some calculations in which a larger g-in11 reference space was considered, the reference occupations were generated in an g-in-8 (full-valence) CASSCF calculation. Those occupations giving rise to CASSCF configurations with coefficients larger than a given threshold were retained as reference occupations for the multireference calculations. The resulting wave functions are denoted, e.g., as MRCI(O.05) or ACPF(O.05) for a threshold of 0.05. Additionally, single-reference calculations were performed using the CCSD(T) method,22928which is a coupled cluster method including all single and double excitations29 with a quasiperturbative estimate for the effect of triple excitations22, and the QCISD(T) method.21 It has been shown that these methods, especially CCSD( T), yield results very close to full CI even for problematic molecules30v3’at a fraction of the computational cost of an MRCI or ACPF calculation. Furthermore, for the a ix+ state, some full coupled cluster with all single, double, and triple excitations (CCSDT) (Refs. 32 and 33) were carried out. The spectroscopic constants were obtained by fitting a fifth-order regular polynomial to eleven points around the “self-consistent” r, (see Ref. 19) with a step size of 0.01 bohr. Our previous study showed that using a SimonsParr-Finlan34 expansion compromised the determinacy of the higher-order constants. Using the fourth-order portion of the fitted polynomial, a Dunham analysis35 was carried out. All multireference energy calculations were performed using the MOLECULE-SWEDEN program system36 running on the NASA Ames Central Computing Facility CRAY Y-MP/864. The CCSD(T) calculations for the 1 ‘g+ state employed the TITAN closed-shell coupled cluster program37 on the CCF Cray Y-MP, those for the 1 311 state and the atoms were performed using the open-shell CCSD( T) program of Scuseria28 on the NASA Ames Computational Chemistry Branch CONVEX C-210. (No

density matrix averaging was performed in the SCF calculations used to define the orbitals for the coupled cluster calculations.) The full CCSDT calculations were performed on an IBM RW6000 Model 550 computer at Rice University.” Finally, the unrestricted Hartree-Fock (UHF) QCISD( T) calculations were done using GAUSSIAN 90 (Ref. 38) on the CONVEX C-210. Ill. RESULTS AND DISCUSSION A. Initial one-particle

calibration

In a first set of calculations, the geometries were kept frozen at the “best estimate” r, geometries” and singlepoint calculations performed, to assess the behavior of T, with different levels of electron correlation using the [5s3p2d] basis set. The results are presented in Table I. At a selection threshold of 0.05, the difference between MRCI and ACPF is improbably large. This is however mainly caused by the fact that the ACPF calculations for both states yield coefficients for several nonreference configurations that exceed 0.05. After extending the reference space with the configurations that exceed the threshold [such calculations are denoted, e.g., ACPF( 0.05a) throughout the article], the value for T, is quite close to that obtained at the MRCI +Q level, that is, after applying the multireference analog39 of the Davidson correction.40 The need to expand the ACPF reference space has been described previously, for example, in Ref. 41. A similar effect is seen at a threshold of 0.025. Here the ACPF calculations for the 311 state actually produce nonreference coefficients not just in excess of 0.025, but in excess of 0.05. The difference between MRCI + Q and ACPFa is larger in this case, though still acceptable. More surprisingly, even at a threshold of 0.01 the ACPF calculation requires additional reference occupations. Including them enhances the agreement with MRCI+Q, but there is still a 100 cm-’ difference. Finally, the CAS reference calculations (that is, zero selection threshold) yield MRCI+Q and ACPF values in very good agreement with each other. This observation can be useful when determining potential curves, as MRCI +Q energies tend to be too noisy to yield reliable higher-order constants. One conclusion is evident throughout; in this basis set, at least, the ground state is 311. The effect of diagonal Darwin and mass-velocity corrections42 on T, is to enlarge it by - 10 cm-‘. It is interesting to examine the performance of singlereference methods for the present problem, the results for which are found in Table II. Both QCISD and CCSD predict a 311 ground state, but grossly overestimate T,; connected triple excitations seem to be very important. QCISD(T) predicts the states to be essentially isoenergetic, but if the spin-symmetry-broken UHF solution is used as a reference for the ‘g+ state, the predicted T, is too high. [The restricted Hartree-Fock (RHF) solution is UHF-unstable; (.S2) exceeds 1.30 for the UHF solution.] The RHF-based CCSD(T) method predicts the wrong ground state. A major part of the problem here is a (40) (50) near-degeneracy, which in the ‘2+ state introduces a

J. Chem. Phys., Vol. 97, No. 9, 1 November

1992

Martin et al.: Ab initio study of BN TABLE

I. Multireference

results for initial n-particle calibration in the [5s3p2d] basis set. Energies and r, values are in Eh and cm-‘,

Threshold

‘x+

MRCI

0.05 0.05’ 0.025 0.025’ 0.01 0.01’ 0 0.05 0.05’ 0.025 0.025’ 0.01 0.01’ 0 0.05 0.05. 0.025 0.025’ 0.01 0.01’ 0

‘Il

T,

6551

MRCI+Q

ACPF

MRCI + RELb

-19.239

153

-79.250 049

-79.248768 -79.248 602

-19.275 079

-79.242

123

-79.250

- 19.249 322 -19.248 815 - 79.248 994 - 19.248 919 -19.249 081 - 79.252 965 -79.251 270 -79.253 646 - 19.25 1 509 -79.252 363 -19.251 293 - 19.25 1622 921 586 949 591 140 508 558

-19.218 053

511

- 79.243 448

--79.250 148

-19.244 -19.240

102 239

-79.250 -19.252

168 663

- 79.243 693

-79.253

581

- 79.245 092

-79.252 828

-‘79.245 153 238

-19.252804 574

345

674

361

588

362

579

- 79.279 380 -79.280 034 - 19.276 209 -19.219

665

-79.281067 -79.281 129 248 354 370 372

respectively.

ACPF+RELb

- 19.284 704 -19.284 -19.285 -i’9.284 -79.284 -19.284 -19.285 - 79.289 -19.287 -19.289

536 259 752 931 913 014 006 251 675

-79.287487 -79.288376 - 79.287 212 -19.281601 944 596 969 600 156 518 568

‘Nonreference occupations whose CI/ACPF coefficient exceeded the threshold included in the reference space (see text). b+ REL indicates energies including Darwin and mass-velocity terms.

second configuration with a CASSCF coefficient >0.3. The CASSCF natural orbital occupation numbers for (4~) and (5a) likewise approach 1.7 and 0.3, respectively. The 9-t diagnostic values43 for ‘X+ and 311 are 0.08 and 0.04, respectively. By comparison, the 9-t diagnostic for the X ‘Xg* state of the isoelectronic C2 molecule is 0.04.3’ As single excitations from the doubly excited ( la)2(2a)2(3a)2( 1~)~(5a)* state are triple excitations with respect to the reference state, it is understandable that a quasiperturbative mechanism for dealing with triple excitations [such as in CCSD(T)] will have trouble reproducing their effect. Watts and Bartlett4 observed a similar phenomenon in their study of the isoelectronic C2 molecule, where CCSD (T) and full CCSDT differed by - 500 cm-’ for T,. A few trends being established, we will now discuss the larger basis set and potential curve computations.

TABLE II. Single-reference results for initial n-particle calibration in the [532] basis set. Energies and T, values are in Eh and cm-‘, respectively.’ QCISD

QCISD(T)

‘I+

-79.228

b

-79.224907 -19.238315

'II T, b

2149 2956

582

-79.249

151

-19.231021 -19.249230 17

CCSD -79.215

CCSD(T) 438

-79.248

525

-79.236 'Xl0

-79.241869

4513

-144

2678

‘RHF reference function used for CC calculations and singlet state QCI calculations, except as noted. UHF reference functions used for triplet state QCI calculations. bSpin-symmetry-broken UHF reference.

B . The ‘8+ state We can perform the greatest variety of calculations on the ‘8+ state, since it is nominally closed-shell. Computed spectroscopic constants are displayed in Table III. There is a small difference between MRCI and ACPF in the computed o, but this decreases as the reference threshold is reduced to zero. Much more striking is the large difference between the MRCI/ACPF results and those from the CCSD(T) method. Given the large ,7, diagnostic, the CCSD (T) results could be expected to show problems, but at first sight the reverse seems true. Consider the harmonic frequencies. Assuming that the cubic anharmonicity contribution is negative, as is almost universally the case, the harmonic frequency will be larger than the fundamental vibrational frequency. This is what is observed for the CCSD(T) w, values, compared to the experimentally deduced value, in both the [4321] and [54321] basis sets, but the MRCI and ACPF values are much smaller than the experimental fundamental. It is also clear that basis set effects cannot be responsible for more than a small part, say -2O%, of the discrepancy. Obviously, using the computed anharmonic constants to obtain theoretical estimates of the fundamental frequency (as EU,- 20~~) simply amplifies the apparent failure of the multireference approaches. This phenomenon is especially puzzling, given that the MRCI and ACPF results include CAS reference values, since ACPF( CAS) calculations would normally be expected to produce results essentially identical to those of a full CI calculation, although we note that Watts and Bartlett@ observed discrepancies between large basis set

MRCI and CCSD(T) calculations on C2, with the latter


1992


6552 TABLE

III. Results at various levels of theory for the cz‘LX+ state.

0, Level of theory ACPF(0.05)/[4321] ACPF(0.025)/[4321] ACPF(0.025)/[54321] MRCI(0.025)/[4321] LRCI(0.0)/[432] MRCI(0.0)/[4321] ACPF(0.0)/[432] ACPF(0.0)/[4321] QCISD/[4321] b QCISD(T)/[432] QCISD(T)/[4321] CCSD/[432] CCSD/[432 l] CCSD/[5432 1] CCSD(T)/[432] CCSD(T)/[4321] CCSD(T)/[54321] CCSDT/[432] expC

(cm-‘) 1679.4 1673.3 1685.1 1688.6 1689.4 1673.5 1679.9 1668.6 1674.6 1692.0 1677.2 1676.5 1678.9 1685.4 1699.4 1714.3 1720.1 1733.7 1678.0 vo= 1712

%s

(cm-‘)

1.2857 1.2869 1.2804 1.2849 1.2846 1.2867 1.2861 1.2873 1.2869 1.2548 1.2884 1.2866 1.2811 1.2803 1.2734 1.2752 1.2755 1.2699 1.2846 r,,= 1.283

11.87 11.48 11.38 11.15 11.15 11.50 11.28 11.64 11.40 30.28 12.70 12.68 12.50 12.25 12.37 17.70 17.61 16.44 12.38

‘MRCI(0.025)/[4321] based on an 8-in-11 CASSCF. bTne grid used for the curve is not centered near r. for this case, significantly tiecting ‘Reference 11.

method giving better agreement with experiment. One possibility is that the CASSCF active space is not large enough, and indeed the (7~) and (3~) orbitals are rather low-lying. We therefore performed calculations based on an 8-in-11 CASSCF wave function, but as can be seen from Table III this hardly affects the results. It was not feasible to use still larger CASSCF wave functions and reference spaces, but in any event it seems unlikely that they would give significantly different results. Another striking difference between the computational methods is the computed anharmonicity constants w.+xx, All multireference calculations predict a value around 11 cm-‘, while CCSD(T) predicts 17 cm-‘. The CCSD and MRCI/ACPF values are again quite close to one another, however. Halving the step size used in computing the curve hardly affects these figures, so the differences do not seem to be due to aspects of the fitting. Also puzzling is the computed bond distance, which at the MRCI/ACPF levels is consistently too large and at the CCSD(T) level is too small. The experimental :. is 1.283 A,t’ and the “best estimate”” for r, is 1.2795 A. On the decrease in r, at the other hand, a significant ACPF(0.025a) level is seen as the basis set is expanded from [4321] to [54321]. Also, whereas connected triple excitations usually lengthen a bond, they shorten it here. The effect of the Darwin and mass-velocity terms on the potential curve was investigated and found to be negligible. In the light of the T, results quoted in the previous subsection, and the very large ,7r diagnostic, it would be reasonable to expect CCSD(T) to be unreliable for this molecule, as we stated above. It is thus very surprising that this method appears to yield the correct fundamental frequency, while even expanded multireference treatments

(2’)

4,

0.016 0.016 0.016 0.015 0.015 0.016 0.015 0.016 0.015 0.025 0.019 0.019 0.017 0.017 0.017 0.017 0.017 0.016 0.016

16 03 20 68 71 15 82 21 95 48 33 06 52 26 51 07 27 92 77

-79.276 134 10 -79.276 445 61 - 79.289 666 96 - 79.268 27 1 42 - 79.270 066 12 - 79.25 1045 45 - 79.270 560 90 -79.256 280 56 -79.276 816 13 -79.253 585 05 - 79.256 679 04 - 79.277 086 91 -79.222 016 58 -79.241 391 51 -79.254 077 23 -79.255 649 16 - 79.275 842 60 - 79.289 48 1 25 - 79.252 274 04

these results.

fail. One way to shed more light on this matter is to perform full CCSDT calculations. In this way we can assess the higher-order contributions of connected triple excitations. The full CCSDT calculations in this work were performed in a smaller [432] basis set obtained by deleting the f functions from [4321]. For comparison, CCSD(T) and QCISD (T) calculations in this smaller basis set have also been carried out, as well as MRCI(O.0) and ACPF(O.0) calculations. The results are again presented in Table III. (Note that a couple of points at the beginning of the curve had to be omitted because of convergence problems; this has no perceptible effect on the computed spectroscopic constants as we checked for the CCSD results.) They show unambiguously that the CCSD(T) method breaks down here. The computed CCSDT spectroscopic constants are all in line with the CCSD and multireference calculations. Very peculiarly, the QCISD(T) method, presumably benefiting from a fortunate error compensation, yields results almost identical to CCSDT, except for ae. QCISD itself, however, yields anomalous results, unlike CCSD. This is a consequence of the well-known fact22 that the (T) correction to QCISD (T) corrects for terms, starting at fifth order in perturbation theory, that are neglected in QCISD but not in CCSD. The QCISD(T) results in the [4321] basis set are, except again for a0 in excellent agreement with the best multireference calculations. If we assume additivity of basis set effects at the ACPF( 0.025a) level to correct the ACPF( 0.0)/[432 l] values, we can predict spectroscopic constants for the ‘Z+ state. In this way we find we= 1686 cm-‘, wpxe= 11.3 cm-‘, a,=O.O161 cm-‘, and r,= 1.280 A. The vibrationally averaged bond distance is then 1.284 A, in excellent agreement with experiment. Greater difficulties are pre-


1992

Martin et al.: Ab initio study of BN TABLE

6553

IV. Results at various levels of theory for the X %I state.

Level of theory ACPF(0.05)/[4321] ACPF(O.O25)/I4321] ACPF(0.025)/[54321] ACPF(0.0)/[4321] CCSD/[4321] CCSD(T)/[4321] expt’

0, (cm-‘) 1458.2 1472.5 1486.2 1477.0 1566.8 1494.9 v,= 1496

%% (cm-‘)

a, (cm-‘)

13.47 12.96 12.74 12.66 12.63 12.95

1.3432 1.3414 1.3343 1.3411 1.3252 1.3371 r,= 1.329

0.018 0.017 0.017 0.017 0.016 0.017

T, (cm-‘) 31 81 80 56 53 60

-79.278 40169 -79.278 766 45 - 79.29 1 624 08 -79.278 916 66 -79.261 586 59 -79.275 113 98

498 509 429 461 4432 -160

‘Reference 12.

sented by vo, which is predicted to be 1662.8 cm-‘, 50 cm - ’ lower than the experimentally deduced value.’ ’ It is possible that this value is underestimated somewhat by the multireference methods, but it can be argued that this effect will be small, as follows. As the ACPF wave function is expanded by the inclusion of more configurations, the fundamental frequency increases. Hence it is reasonable to expect the ACPF value to converge from below. On the other hand, comparison of the CCSD and CCSDT results indicates that as the excitation level in the coupled cluster frequency treatment is increased, the fundamental decreases-it converges from above. The excellent agreement between the ACPF and CCSDT values then suggests that the fundamental frequency is well converged at these levels of treatment. It seems very unlikely that our prediction (corrected for basis set incompleteness) can be in error by 50 cm-‘. Consequently, it seems probable that the experimental value deduced by BDHN is in error. A misassignment of the numbering of the rovibrational bands is one possible explanation. From the CCSD(T) energies and the atomic energies for B and N at the same level of theory,45 we can compute the atomization energy for the singlet state, which is 97.1 and 101.8 kcal/mol in the [4321] and [54321] basis sets, respectively. Using the correction formula recently proposed by one of the authors& AE ,,,~=AE+aAn,+bAn,+cAn,~i,,

(1)

where n, and n, represent the number of (T and r bonds, respectively, and npair the number of closed-shell electron pairs (0, 6, and c are fitted constants”), we obtain atomization energies of 105.3 and 105.6, respectively, with the [4321] and [54321] basis sets. The mean absolute errors for a number of reference molecules were 0.86 and 0.51 kcal/ mol, respectively.46 To this should be added some uncertainty because of the inaccuracy of CCSD(T) in this case; some of this, however, can be absorbed into the computed transition energy, as discussed below. C. The 311 state Some results for the 311state are displayed in Table IV. We list only ACPF multireference results here. Again, it is seen that the ACPF calculations tend to predict a lower w, than CCSD(T), although the effect is not as severe as for the ‘2+ state. Also, the discrepancy between ACPF and CCSD(T) for the computed r, values is nowhere near as

large. As for the transition energies, these agree very well, except for the CCSD(T) result. In view of the behavior of the CCSD(T) method for the ‘2+ state, the ACPF values of T, are undoubtedly the more reliable ones. Hence our best theoretical predictions strongly support that the ground state of BN is X ‘II, which is a departure from the X ‘Z+ ground state of the isoelectronic C2,47 BeQ4* and Ni+ (Ref. 49) molecules. Assuming basis set additivity for the [4321]-[54321] extension at the ACPF(0.025a) level in order to correct the ACPF(0.0)/[4321] results, we find a best estimate for T, of 381 cm-‘. In view of the convergence which has been established with respect to CASSCF active space, ACPF reference space, and one-particle basis set, we expect this result to be uncertain by no more than f 100 cm-‘. Assuming basis set additivity also for the ACPF w, we obtain an w, value of 1491 cm-‘, compared to an approximate experimental y. of 1496 cm-‘.i2 For the bond distance, the same approximation leads to r,= 1.330 A, in reasonable agreement with the earlier “best estimate”” of 1.3251 A and the experimental r. of 1.329 A. Finally, we can compute a value for the dissociation energy which is probably a lower bound because of residual basis set incompleteness. Combining the atomization energy of the a ‘Z+ state with our computed T, at the CCSD(T)/[4321] level [thus absorbing some of the error in the CCSD(T) method for the a ‘2+ state], and the spin-orbit coupling constant (for the X 311 state) of -25.14 cm-i,12 we find D,= 105.2 kcal/mol, to which we assign an uncertainty of about *2 kcal/mol. This is in very good agreement with the previously computed value of 106.5 f 2 kcal/mol.” Correcting for zero-point vibration with the above values of w, and w& we finally find Do = 103.1 kcal/mol.

D. Thermochemistry Using the revised spectroscopic constants for the X 311 and a ‘Z+ states, as well as the data from Kama and Grein’3*‘4 for 23 other valence states, the thermodynamic functions at 0.1 MPa have been computed essentially exactly by modified direct numerical summation using an algorithm described in detail in the Appendix of Ref. 19. The spin-orbit coupling constant, A= -25.14 cm-‘, for the X 311 state was taken from Ref. 12. Centrifugal stretching constants were computed from the familiar relationship


1992


6554

TABLE

T (JO 100 200 298.15 300 4Ml 500 600 700 800 900 loo0 1100 1200 13&I 1400 1500 1600 1700 1800 1900 2Ow 2100 2200 2300 2400 2500 2600 2700 2800 2900 3ooo 3100 3200 3300 3400 3500 3600 3700 3800 3900 4cm 4100 4200 4300 44oa 4500 4600 4700 4800 4900 5OOu 5100 5200 5300 5400 5500 5600 5700 5800 5900 6ooo

V. JANAF-style

CP

(J/K mol) 29.873 29.843 30.191 30.201 30.919 31.844 32.768 33.597 34.315 34.934 35.472 35.943 36.362 36.740 37.086 37.407 37.710 37.998 38.275 38.543 38.805 39.061 39.312 39.558 39.798 40.033 40.263 40.486 40.703 40.914 41.119 41.318 41.511 41.699 41.882 42.060 42.235 42.406 42.575 42.742 42.909 43.075 43.241 43.409 43.579 43.751 43.926 44.105 44.289 44.476 44.669 44.868 45.072 45.282 45.497 45.719 45.947 46.180 46.420 46.665 46.916

thermodynamic

so (J/K

mol)

180.680 201.324 213.295 213.481 222.260 229.257 235.146 240.261 244.795 248.873 252.583 255.986 259.132 262.057 264.793 267.363 269.787 272.082 274.261 276.338 278.322 280.221 282.044 283.797 285.486 287.115 288.690 290.214 291.690 293.122 294.512 295.864 297.179 298.459 299.707 300.923 302.111 303.270 304.403 305.511 306.596 307.657 308.697 309.717 310.717 311.698 312.661 313.608 314.538 315.454 316.354 317.241 318.114 318.974 319.823 320.660 321.486 322.301 323.106 323.902 324.688

functions for BN at 0.1 MPa.

gef( T) (J/K

mol)

149.328 170.756 182.916 183.104 191.844 198.650 204.254 209.040 213.232 216.969 220.348 223.435 226.280 228.921 231.387 233.700 235.881 237.943 239.901 241.764 243.543 245.245 246.876 248.444 249.952 251.406 252.810 254.167 255.48 1 256.755 257.990 259.190 260.357 261.492 262.598 263.676 264.727 265.753 266.755 267.735 268.693 269.630 270.548 271.447 272.328 273.192 274.040 274.872 215.688 276.491 277.279 278.054 278.816 279.565 280.303 281.029 281.744 282.449 283.143 283.827 284.501

H(T) -Ho W/mol) 3.135 6.113 9.057 9.113 12.166 15.304 18.535 21.854 25.251 28.714 32.235 35.806 39.422 43.071 46.769 50.494 54.250 58.035 61.849 65.690 69.557 73.45 1 77.369 81.313 85.281 89.272 93.287 97.325 101.384 105.465 109.567 113.689 117.830 121.991 126.170 130.367 134.582 138.814 143.063 147.329 151.612 155.911 160.227 164.559 168.909 173.275 177.659 182.060 186.480 190.918 195.376 199.852 204.349 208.867 213.406 217.967 222.550 227.157 231.787 236.441 241.120

ACP

(J/K

mol)

k5Q (J/K

- 11.794 - 11.752 - Il.391 -11.381 - 10.659 -9.732 - 8.807 - 7.977 -7.258 - 6.639 -6.101 - 5.630 -5.211 -4.833 - 4.486 -4.165 -3.862 - 3.574 - 3.298 -3.031 -2.771 -2.518 -2.271 - 2.032 - 1.801 - 1.579 - 1.367 - 1.165 -0.975 -0.798 -0.633 -0.483 -0.346 -0.222 -0.113 - 0.020 0.062 0.129 0.184 0.226 0.258 0.279 0.289 0.291 0.285 0.272 0.252 0.227 0.198 0.164 0.129 0.091 0.052 0.012 - 0.029 - 0.069 -0.107 -0.147 -0.183 -0.219 -0.253


mol)

-80.591 -88.798 -93.431 -93.503 -96.685 -98.966 - 100.657 - 101.950 - 102.968 - 103.786 - 104.456 - 105.016 - 105.487 - 105.890 - 106.235 - 106.533 - 106.792 - 107.017 - 107.215 - 107.385 - 107.534 - 107.664 - 101.775 - 107.871 - 107.951 - 108.021 - 108.078 - 108.126 - 108.166 - 108.196 - 108.221 - 108.239 - 108.252 - 108.261 - 108.265 - 108.268 - 108.266 - 108.265 - 108.261 - 108.256 - 108.248 - 108.242 - 108.236 - 108.228 - 108.222 - 108.216 - 108.210 - 108.205 - 108.201 - 108.196 - 108.194 - 108.191 - 108.189 - 108.190 - 108.188 - 108.190 - 108.192 -108.193 - 108.197 - 108.200 - 108.204

1992

Agef( 77 (J/K

mol)

- 69.249 -77.209 -81.841 -81.913 - 85.233 - 87.762 - 89.777 -91.428 v-92.809 -93.985 -95.OcO -95.886 -96.668 -97.362 -97.983 - 98.544 -99.050 -99.512 -99.935 - 100.324 - 100.681 - 101.009 -101.315 - 101.598 - 101.860 - 102.107 - 102.335 - 102.549 - 102.748 - 102.935 -103.112 - 103.278 - 103.432 - 103.578 - 103.716 - 103.846 - 103.968 - 104.084 - 104.195 - 104.299 - 104.398 - 104.492 - 104.581 - 104.665 - 104.747 - 104.824 - 104.897 - 104.968 - 105.036 - 105.099 - 105.161 - 105.221 - 105.278 - 105.333 - 105.386 - 105.437 - 105.487 - 105.534 - 105.580 - 105.624 - 105.668

NH--Ho) (kJ/mol) - 1.134 -2.318 -3.456 - 3.416 -4.582 - 5.602 -6.528 -7.367 -8.126 -8.821 -9.458 - 10.043 - 10.585 - 11.087 - 11.552 - 11.985 - 12.386 - 12.758 - 13.102 - 13.418 - 13.708 - 13.972 - 14.213 - 14.427 - 14.619 - 14.788 - 14.935 - 15.061 - 15.168 - 15.257 - 15.328 - 15.384 - 15.425 - 15.454 - 15.470 - 15.476 - 15.475 - 15.465 - 15.449 - 15.428 - 15.404 - 15.377 - 15.348 - 15.320 - 15.290 - 15.263 - 15.236 - 15.213 - 15.191 - 15.174 - 15.158 - 15.148 - 15.141 -15.136 - 15.138 - 15.142 - 15.152 - 15.163 - 15.180 - 15.200 - 15.223

log

Kf

221.705 108.628 71.298 70.829 51.878 40.480 32.864 27.413 23.317 20.127 17.570 15.475 13.727 12.247 10.976 9.874 8.909 8.056 7.298 6.619 6.007 5.453 4.950 4.490 4.068 3.679 3.321 2.989 2.680 2.393 2.125 1.874 1.639 1.418 1.210 1.013 0.828 0.653 0.487 0.330 0.180 0.038 -0.098 -0.227 -0.350 -0.468 -0.581 -0.689 -0.792 -0.891 -0.987 - 1.078 -1.166 - 1.251 - 1.332 -1.411 - 1.486 - 1.559 - 1.630 - 1.698 - 1.764

Martin et a/.: Ab initio study of BN

D,= ~I$/cL$.~~ The results are displayed in Table V. The tabulated functions include the heat capacity Cp, the standard entropy Se, the free energy function gef( T) and the enthalpy function H(T) s -[G(T) --H&T, -No, as well as the corresponding functions for the association reaction B(g) +N(g) +BN(g) and the natural logarithm of its equilibrium constant KY The atomic functions necessary for the latter were taken from Ref. 19, and include corrections for all electronic states below the ionization limit as well as for spin-orbit coupling. By comparing to the earlier calculation,‘9 we can assess the effect of the change in the computed T, and other spectroscopic constants. At low temperatures, a very significant effect is seen on C, which is primarily due to the change in T, This effect decreases with rising temperature, but at very high temperatures the effect of the changes in the other constants is beginning to be noticeable. The other thermodynamic functions change significantly there too, with 9 and -[G(T) ---H&T staying relatively close to the earlier values to as high as 4000 K but with H(T) --Ho changing quite substantially, differing 1.283 kJ/mol from the previous calculation at 6000 K. The errors for the other thermodynamic functions at this temperature are C, 0.709 -H&T 0.077 J/ J/k mol, Sc 0.291 J/K mol, -[G(T) K mol. These errors are still comfortably within the generally accepted limits of “chemical accuracy,” thus confirming the earlier assertion’9’so that high-quality thermodynamic functions can be computed from first principIes using present-day a& initio methods. As for the association reaction, the earlier conclusion that the molecule becomes unstable with respect to the atoms between 4100 and 4200 K is confirmed.

IV. CONCLUSIONS The lowest ‘2+ and 311 states of BN have been studied using single and multireference ab initio methods and basis sets of spdf and spdfg quality. We have established that the ground state is X 311, with the a ‘8+ state lying 381 f 100 cm-’ higher in energy. The dissociation energy of BN is predicted to be 105 f 1 kcal/mol. Because of severe multireference effects (Y, = 0.08), results obtained with the CCSD(T) and QCISD(T) methods should be considered with caution. As previously discussed,3’ neither of these methods should be expected to perform well under such extreme circumstances. Our best predicted spectroscopic constants, based on multireference wave functions, suggest an assignment problem with the experimental v. for the a IX+ state.

ACKNOWLEDGMENTS J. M. L. M. is a Senior Research Assistant of the National Fund for Scientific Research of Belgium (NFWO/ FNRS) and is additionally supported by a Fulbright/I-Iays Travel Grant and a NATO travel grant. The work performed at Rice University was supported by the National Science Foundation (Grant No. CHE-9017706). P. R. T.

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1992