A comparative study of the hyperfine structures of neutral nitrogen

792

J. Phys. Chem. 1994,98, 792-799

A Comparative Study of the Hyperfine Structures of Neutral Nitrogen Oxides: DFT vs CISD ResultsLeif A. Eriksson,+Jian Wang, Russell J. Boyd,' and Sten Lunellt Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4J3 Received: September 16, I993@

The electronic and hyperfine structures of the three smallest neutral nitrogen oxides, NO, NO2, and NO3,are investigated by means of ab initio and density functional (DFT) methods. The emphasis of the present work is on the performance of the D F T approach and the computation of hyperfine parameters. The geometries of the different species are optimized a t up to the ab initio MP4(SDTQ) level as well as local and gradient corrected D F T levels, after which the isotropic hyperfine coupling constants are calculated using configuration interaction and D F T techniques. The results are compared with experimental and previous theoretical data. The DFToptimized geometries of NO and NO2 are in very good agreement with experiment. For the NO3 radical, a statistical analysis is performed for the singly and doubly 170-substitutedisotopomers a t temperatures ranging from 0 to 273 K. It is proposed that by performing ESR measurements a t very low temperatures, and with a controlled degree of substitution, a unique ground-state configuration should be detectable.

1. Introduction

2. Method

Nitrogen oxides (ionic and neutral; closed shell and open shell) are primary and secondary reaction products from exhaust fumes formed by the utilization of fossil-based fuels in, e.g., motor vehicles. The smallest of the nitrogen oxides, NO, makes up over 98% of the NOx compounds immediately released in the exhaust fumes and is known to be thermodynamically unstable toward the termolecular disproportionation reaction 3 N 0 NO2 + NzO. Once released, a large fraction is also believed to react immediately with surrounding oxygen molecules to form N0z.I The possible mechanisms for these reactions have been investigated in great detail p r e v i o ~ s l y ,and ~ - ~review papers on this class of compounds are also a ~ a i l a b l e . ~ The environmental impacts of nitrogen oxides are substantial, mainly due to the enormous quantities released through traffic each year. Besides the fact that most NOx compounds are toxic to inhale-which constitutes a severe problem in highly trafficked cities-they also affect the environment through acidification and eutrophication of the soils and waters. Due to the polluting powers of these compounds, a large number of biological, medical, and chemical studies are presented each year, discussing various aspects of these problems. However, in some cases the equilibrium geometries of these molecules are still an open question, as well as what the actual reaction mechanisms are. In the present communication, we have chosen to investigate the three smallest mononitrogen oxides, NO, NO*, and NO3, by means of ab initio Merller-Plesset perturbation theory (MPPT),5 configuration interaction (CI), and density functional theory (DFT). Particular emphasis is put on the isotropic hyperfine (hf) structures of these compounds, here for the first time calculated using DFT. The results are compared with results from electron-spin resonance (ESR) experiments and CI calculations using all single and the most important double excitations6 from a single reference determinant (CISD). We also propose a plausible model for the determination of the true ground-state of NO3, based on a vibrational analysis of the partially I7Osubstituted species.

All geometry optimization calculations at the unrestricted ab inito HF, MP2, and MP4(SDTQ) levels were performed using the programs GAUSSIAN 907 and GAMESS.8 For these calculations, the 6-3 1G(d) basis set9was employed. Theoptimized structures were used as input for subsequent CISD calculations, in order to accurately determine the isotropic hyperfine coupling constants of the different compounds. In the CI calculations, we used uncontracted, even-tempered Gaussian basis sets (ETG),'O consisting of up to 14 s-functions and 7 p-functions, to which d-functions were augmented on both the N and 0 atoms, with the exponents 0.85. The CISD calculations were performed utilizing the MELDF-X program suite.lI Between 450 000 (NO2) and 1 350 000 (NO3) spin-adapted configurations were included in the calculations, corresponding to energy thresholds 5 1.O X au for the selection of the configurations maintained in the CI expansions. The three molecules were also investigated by means of spinunrestricted density functional theory, using the linear combination of Gaussian type orbitals-density functional theory (LCGTO-DFT) program deMon.l2 The geometry optimization calculations were performed using basis sets of DZP and TZP quality. For the orbital parts, the (5211/411/1) and (7111/ 4 11/ 1*) basis sets were employed,13combined with the auxiliary bases (5,2,;5,2) and (4,4;4,4), respectively, for the fitting of the charge density and the exchange and correlation potentials. The twocontractedorbitalbasissets(51111/2111/1)(IGLO-1I)and (5111111/211111/11) (IGLO-III)14J5 were also employed in conjunction with the (5,2;5,2) auxiliary basis (IGLO: individual gauge of localized orbitals.1sa) Thelocal densityapproximation (LDA) is that of Vosko, Wilk, and Nusair,l6 to which gradient corrections due to Beckel7 or Perdew and Wang18 for the exchange functional and to Perdew for the correlation part,19 were included. These will be termed "BP" and "PWP" in the text. For the DFT-ESR calculations, a routine by Malkin et aZ.14 was employed, and the DFT hyperfine calculations were performed on the optimized DFT, UHF, UMP2, UMP4(SDTQ), and experimental geometries, along with some other geometries previously reported in the literature.

Present address: Department of Physics, University of Stockholm, Box 6730, S-113 85 Stockholm, Sweden. f Department ofQuantum Chemistry, Uppsala University, Box 518, S-75 1 20 Uppsala, Sweden. Abstract published in Aduance ACS Abstracts, December 15, 1993.

3. Results

-+

0022-365419412098-0792$04.50/0

3.1 NO. N O has a 2111/2 ground state, where the unpaired electron can be thought of as occupying an antibonding ?r-orbital, 0 1994 American Chemical Society

Hyperfine Structures of Neutral Nitrogen Oxides leading to a bond order of 2.5. The experimental NO bond distance is 1.151 At the HF/6-311G and HF/6-311G(d) levels, bond distances of 1.152 and 1.1 17 A have been reported,2’ indicating that the inclusion of polarization functions leads to a too large shortening of the bond. This has also been noted by BairdandTay1orZ2inaUHFstudyusingthe4-31Gand4-31GN* bases. In the latter basis set, polarization functions were included on the nitrogen atom only, which led to a decrease from 1.15 to 1.12 A. Using the somewhat larger basis set 6-31G(d), a bond length of 1.127 A has been reported at the unrestricted HF level.9 Using the same basis set (6-31G(d)), and including electron correlation in the form of second-order perturbation theory, the bond length is improved to RNO= 1.143 A (Table 1). At the MP4(SDTQ) level RNO= 1.147 A. This can be compared with full CI (FCI) benchmark calculations by Bauschlicher and Langh0ff,~3in which a DZP basis set was employed, yielding the somewhat elongated bond length RNO= 1.75 A. Also when DFT theory is applied, a certain method- and basisset-dependent variation in bond length can be noted. At the LDA level, and using an extended DZP basis (“DZP2”), Andzelm and Wimmer report a bond length of 1.167 A.24 When the smaller DZP and larger TZP basis sets are employed within the LDA approximation, we obtain in the present work R N=~1.161 Aand RNO= 1.158 A, respectively (cf. Table 1). Including nonlocal corrections, the bond lengths are found to increase slightly. At the LDA-BP/DZP level RNO= 1.173 A, whereas with the T Z P basis we obtain RNO = 1.161 A when the Becke-Perdew corrections are included. At the LDA-PWP level, the two bases instead generate very similar geometries, with RNO= 1.172 f 0.001 A. In general, the DFT results seem to be more consistent than the ab initio ones, and are found to be in good agreement with experimental data. Particularly encouraging are the results observed when the IGLO basis sets are employed. The smaller of these (IGLO-11) yields the bond length RNO= 1.155 A at the LDA level, whereas the larger one (IGLO-111) gives the very accurate value RNO= 1.151A. Including the nonlocalcorrections, the bonds again become slightly elongated (Table 1). The DFT calculated dipole moments range from 0.141 D (LDA/TZP) to 0.266 D (LDA/IGLO-111), to be compared with the experimental value 0.159 D.Z5 For NO, it is found experimentally that the molecule will not exhibit any ESR signal, unless the degeneracy of the .rr-orbitals is broken.26 The reason for this is t h e j couplings. At very low temperatures, the molecule is in a 2111/2 state, which does not have a direct hyperfine structure. As the temperature is raised above ca. 20 K, the 2II3/2 state becomes sufficiently populated, which has a clearly observable hyperfine pattern. With the present DFT method, we compute a state average of the density and are thus unable to separate the j = l / 2 and j = 3/2 states. Thus, we are at the present stage not able to determine accurately enough the hyperfine structure of the N O radical. This is, however, not a problem for the other molecules studied in the present work. 3.2 NOz. The NO2 molecule, with its 17 valence electrons, is known to have a bent structure, with an experimental O N 0 bond angleof 133.8’ (133.g0),and the twoNO bonds being 1.197AZ7 (1.194Azs). Theelectronicgroundstateis%AI (C%symmetry), and the unpaired electron is mainly localized on the nitrogen atom. At the HF level, the bond lengths are as usual slightly underestimated, and the O N 0 bond angle is somewhat overe~timated.2292~When electron correlation is included in the form of perturbation theory of up to fourth order, the NO bonds become significantly longer (Table 1). Employing MCSCF, CISD, and CCSD techniques in conjunction with DZP and TZP basis sets, values between 1.189 and 1.220 A have been reported for the bond lengths and 133.2-134.9’ for the bond angles, re~pectively.~& Within ~ ~ these ranges we also find the global

The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 793 minimum from MRD-CI/DZP and extrapolated FCI/DZP calculations.33 At the DFT level, the structure is modified slightly when nonlocal corrections are included, analogous to the situation for NO. At the LDA/DZP2 level, RNO= 1.208 A 2 4 or 1.209 A34 and LON0 = 133.5’.24,34 When nonlocal corrections were introduced in the form of a generalized gradient approximation, Seminario and Politzer found the structure RNO= 1.255 A and LON0 = 132.3°.34 At the LDA level, we find in the present workthat the DZPandTZP basis sets yieldverysimilar structures that are in satisfactory agreement with experiment. Including the BP nonlocal corrections, R Nincreases ~ by ~ 0 . 0 1 A 5 whereas the O N 0 angle only undergoes minor modifications. Using the IGLO-I11 bases, the values 1.194 A and 134.0’ (LDA) are obtained, again in very close agreement with experiment. At the BP and PWP levels, the N O bonds increase somewhat whereas the angle decreases by at the most 1’. The dipole moment of NO2 is measured to be 0.316 D.35 This value is reproduced very well using MCSCF theory, whereas the application of CISDT overestimates the dipole moment slightly31 ( k = 0.444 D). With the DFT approach, p = 0.256 f 0.017 D. The density functional treatments thus seem to underestimate the charge separation within the molecule, although the optimized geometries overall are highly accurate. The best dipole moment is obtained at the LDA-PWP/IGLO-I11 level. The barrier toward inversion, passing through the linear 211u transition-state structure, is rather high. At the vibrationally uncorrected MPPT and LDA-BP/DZP levels it is around 40 kcal/mol. The high barrier is also reflected in the large imaginary frequency corresponding to the reaction coordinate. At the MP2/ 6-3 1G(d) level this is 1435 cm-1. This is in very good agreement with previouslyreported MCSCFand CISDT calculations, where the ZPE corrected barrier is determined to be 38 kcal/mol, and the imaginary frequency of the linear transition state is 1458 cm-l (MCSCF/QZPP) and 1417 cm-I (CISDT/QZPP).31 The hf structure of NO2 is known experimentally to be (liso(14N) = 151.3-153.4MHza11dai,(~~O)=2X(56.942.2)MHz (absolute ~ a l u e s ) . ~ ~ JThere 8 - ~ ~is to our knowledge only one previous theoretical study reported in the literature. In the work by Lund and Thuomas,36 experimental ESR measurements were combined with U H F optimizations and hf calculations using the spin annihilated wave function (“UHFAA”). The results obtained for the NO2 molecule give approximately 75% of the 14Ncoupling and less than 50% of the measured 1 7 0 couplings. Since the hyperfine structure of a radical system is highly geometrydependent (see, e.g., rep7), accurately determined geometries and good basis sets are essential for a satisfactory reproduction of the experimental hyperfine parameters. The discrepancy between the theoretical and experimental results of ref 36 can be related to the much too short RNObonds obtained at the U H F level, in combination with a basis set that gives an insufficient description of the “s-character” of the unpaired electron, and a lack of electron correlation in the theoretical treatment. The results from the CISD and DFT hf calculations performed in the present work are shown in Table 2. All calculations performed on the various optimized C%structures yield the hf couplings 120-166 MHz for the nitrogen and -25 to -55 MHz for the two oxygens. Both nitrogen and oxygen are highly anisotropic nuclei and have previously been shown to be computationally difficult atoms in terms of accurate isotropic hf coupling constants.4@-4*Instead of van D~ijneveldt’s~~ basis sets, used in our previous papers,M*42we have here chosen to work with a series of uncontracted even-tempered Gaussian bases (ETG), as described by Schmidt and Ruedenberg,Io in the CISD calculations. This basis offers the possibility of a systematicstudy of the convergence properties with variation in basis set size. We find that the agreement with experiment is not so good when a small basis (8s,4p) is used. As the basis set is increased, the

794 The Journal of Physical Chemistry, Vol. 98, No. 3, 1994

Eriksson et al.

TABLE 1: Optimized Geometries (A and degrees), Absolute Energies (au), and Dipole Moments (D) for the NO, NOz, and NO3 Radicals, Using the Different Methods As Outlined in the Text' method

RNO/RN(Y

LONO'

E (au)

dipole moment

ref

NO HF/6-3 1G(d) MP2/6-3 1G(d) MP4(SDTQ)/6-3 1G(d) FCI/DZP LDA/DZP2 LDA/DZP LDA-BP/DZP LDA-PWP/DZP LDA/TZP LDA-BP/TZP LDA-PWP/TZP LDA/IGLO-I1 LDA/IGLO-111 LDA-BP/IGLO-I11 LDA-PWP/IGLO-111 exP

1.127 1.143 1.147 1.175 1.167 1.161 1.173 1.173 1.158 1.161 1.171 1.155 1.151 1.166 1.164 1.151

HF/6-31G(d) MP2/6-3 1G(d) MP4(SDTQ)/6-3 1G(d) MCSCF/DZP MCSCF/TZP CCSD/DZP CISD/QZPP MRD-CI/DZP LDA/DZP2 LDA-GGA/DZP2 LDA/DZP LDA-BP-DZP LDA/TZP LDA-BP/TZP LDA/IGLO-I1 LDA/IGLO-111 LDA-BP/IGLO-111 LDA-PWP/IGLO-111 exP

1.165 1.216 1.231 1.220 1.216 1.212 1.174 1.206 1.208 (1.209) 1.255 1.202 1.217 1.202 1.217 1.199 1.194 1.211 1.212 1.197 (1.194)

NOz, bent 136.1 133.7 133.3 133.2 133.4 133.2 135.1 133.4 133.5 132.3 133.7 133.5 133.7 133.3 133.7 134.0 133.7 132.9 133.8 (133.9)

-129.247 -129.564 -129.580 -129.479

88 46 82 33

0.21 1 0.072

-128.917 -129.911 -130.025 -128.960 -129.946 -130.068 -128.953 -128.975 -129.976 -130.086

43 69 52 27 33 16 00 89 23 84

0.201 0.149 0.145 0.141 0.192 0.138 0.182 0.266 0.215 0.210 (0.159)

-204.031 49 -204.568 59 -204.589 05 -204.264 61 -204.269 76 -204.593 39 -204.652 64 -204.41 1 80

0.68 1 0.188

-203.609 -205.121 -203.678 -205.194 -203.664 -203.700 -205.225 -205.297

0.260 0.265 0.261 0.245 0.239 0.267 0.260 0.272 [0.316]

10 75 20 55 31 29 94 84

9 23 24

0.310 0.310 0.444

30,31 30,31 30 31 33 24 (34) 34

27 (28) [35]

NOz, linear HF/6-3 1G(d) MP2/6-3 1G(d) MP4(SDTQ)/6-3 1G(d) LDA-BP/DZP

1.174 1.214 1.229 1.218

180.0 180.0 180.0 180.0

-203.972 -204.500 -204.521 -205.057

25 07 32 17

0.0 0.0 0.0 0.0

78 51 31 34 98 26 21 40 70

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

NO3 D3h

HF/6-3 1G(d) MP2/6-3 1G(d) MP4(SDTQ)/6-3 1G(d) MCSCF/DZP QRHF-CCSD/DZP FSMRCCSD/DZP LDA-BP/DZP LDA/IGLO-I11 LDA-P WP/ IGLO-I1I exP Cw(1L2S) HF/6-31G(d) MP2/6-3 1G(d) MP4(SDTQ)/6-3 1G(d) MCSCF/DZP QRHF-CCSD/DZP FSMRCCSD/DZPb

cb-(1S2L) HF/6-3 lG(d) MP2/6-3 1G(d) QRHF-CCSD/DZP FSMRCCSD/DZPb

1.197 1.255 1.269 1.256 1.236 1.246 1.253 1.231 1.249 1.240

120.0 120.0 120.0 120.0 120.0 120.0 120.0 120.0 120.0 120.0

-278.765 -279.535 -279.554 -279.094 -279.626 -279.562 -280.292 -278.383 -280.667

1.335/ 1.175 1.389p.216 1.397p.224 1.351/1.224 1.351/ 1.206 1.286/1.226

114.2 113.4 113.3 114.7 114.4 116.0

-278.816 -279.504 -279.533 -279.096 -279.631 -279.561

07 47 37 60 10 50

47 48 44

1.l62/ 1.235 1.189/1.258 1.198/ 1.266 1.206/ 1.266

127.4 127.1 126.4 124.0

-278.790 -279.509 -279.630 -279.571

41 70 23 75

48 44

47 48 44

49,50

All results are from this work, unless indicated otherwise. Not complete optimization.

agreement improves significantly, and with the (14s,7p,ld) basis we obtain over 90% of the nitrogen coupling and approximately 75% of the oxygen couplings. These calculations are all performed on the HF/6-31G(d) optimized structure. At the experimental

geometry together with the (14s,7p,ld) basis the agreement is slightly better for the nitrogen coupling but worse for the two oxygens. The DFT-ESRcalculations also provide a consistent picture

Hyperfine Structures of Neutral Nitrogen Oxides

The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 795

TABLE 2: Isotropic Hyperfiie Coupling C O I I S ~ U (MHz) ~ S Obtained on the Different Geometries of NO2 and NO3 As Indicated' method

ai(N)

aim(O1)

LDAIDZP LDAiTZP LDA-BP/DZP LDA-BP)TZP LDA/IGLO-I1 LDA/IGLO-111 LDA-BP/IGLO-I11 LDA-PWP/IGLO-111 LDA/IGLO-111 LDA/IGLO-I11 LDA/IGLO-I11 LDA-BP/IGLO-111 LDA-PWP/IGLO-I11 CISD/8s4p CISD/ 1OsSpld CISD/ 10s6pld CISD/12s6pld CISD/ 14s7pld CISD/14s7pld exP

154.62 158.17 161.28 158.84 160.27 155.01 163.97 165.49 159.48 147.70 159.35 162.23 164.75 123.09 135.95 137.39 136.36 138.77 140.95 151.3-1 53.4

-33.91 -33.76 -37.21 -34.55 -33.12 46.69 -39.10 -52.38 40.87 -57.08 -45.69 44.10 -59.22 -34.97 -41.96 -23.48 43.29 43.98 -33.71 59.6-62.2

4h

aiw(02) NO2 (bent) -33.91 -33.76 -37.21 -34.55 -32.12 46.69 -39.10 -52.38 40.87 -57.08 -45.69 -44.10 -59.22 -34.97 -41.96 -23.48 -43.29 43.98 -33.71

geometry

ai403)

LDA-BP/DZP HF/6-3 1G(d) exP exP exP HF/6-31G(d) HF/6-31G(d) MP2/6-31G(d) HF/6-31G(d) HF/6-31G(d) eXP

ref

20 20 20

20 36,38, 39

No3

LDA-BPIDZP LDA-BPjDZP LDA-BP/DZP LDA/IGLO-111 LDA/IGLO-I11 LDA-PWP/IGLO-111 LDA-PWP/IGLO-111 LDA-PWP/IGLO-I11

-1 1.28 -12.06 -1 1.23 -14.18 -13.58 -8.69 -8.54 -8.62

-1.15 -3.1 1 -1.66 0.45 0.1 1 -13.00 -13.70 -13.04

-1.15 -3.11 -1.66 0.45 0.11 -13.00 -13.70 -13.04

-1.15 -3.11 -1.66 0.45 0.11 -13.00 -13.70 -13.04

Cb( 1L2S) LDA-BP/DZP LDA-BP/DZP LDAIIGLO-111 LDA-PWP/IGLO-I11 LDA-PWP/IGLO-111 CISD/ 14s6pld

-1 1.53 -10.89 -13.25 -8.04 -8.29 -7.52

-8.90 4.95 -2.72 -27.75 -26.77 -17.25

-2.58 -1.48 -0.56 -8.74 -8.34 -4.39

-2.58 -1.48 -0.56 -8.74 -8.34 -4.39

HF/6-3 1G(d) CCSD/DZP HF/6-31G(d) HF/6-3 1G(d) CCSD/DZP HF/6-31G(d)

-11.07 -10.31 -13.30 -8.56 -8.78 -8.93

3.29 4.06 3.32 0.34 -0.97 4.00

-15.89 -11.00 -8.71 -28.74 -23.99 -17.07

-16.37 -13.33 -8.84 -28.74 -23.99 -17.07

HF/6-3 1G(d) CCSD/DZP HF/6-31G(d) HF/6-31 G(d) CCSD/DZP HF/6-3 1G(d)

HF/6-31G(d) CCSD/DZP

48

HF/6-3 1G(d) HF/6-31G(d) CCSD/DZP

48

48 48

cb-(1S2L) LDA-BP/DZP LDA-BP/DZP LDA/IGLO-111 LDA-PWP/IGLO-111 LDA-PWP/IGLO-111 CISD/ 14s6pld

10-16 (3 x -42) The values listed are obtained using CISD and DFT techniques as specified in the text. 1 MHz = 0.357 G.

exP a

48 48 36.38

in terms of vibronic interactions between the Z A ' ~ground of the hf structure of NOz. In general, the agreement with state and the excited 2E' state. ESR spectra of the radical in experiment is better than at the CISD level. In particular, the sodium nitrates3 indicate a threefold symmetry axis and I7Ohf calculations using the uncorrected LDA approximation and the IGLO-I11 basis set yield almost experimental accuracy. One splittings of -15 G (-42 MHz). The values are, however, associated with a high degree of uncertainty, and the possibility thing to note is the general tendency of the DFT-calculated hf structures to slightly overestimate the positive nitrogen coupling. of the molecule having CzUsymmetry is not ruled out. It is, furthermore, concluded that a more detailed analysis at temCompared with the smaller IGLO-I1 basis, IGLO-I11 apparently peratures below 10 K is necessary to elucidate this point.s3 leads to a more well-balanced description of the spin density on Interactions with thecrystal latticemay alsobe important. Other the nitrogen vs the oxygen nuclei. Using the nonlocal PWP measurements of NO3 in, e.g., SF6 at 110 K predict the molecule corrections, the couplings become numerically larger than at the to be completely symmetric,38and in the paper by Lund and LDA level, whereas the BP calculations yield I7Ocouplings that Thuomas:6 it is speculated that the differences between thevarious are too small. 3.3 NOJ. 3.3.1 Electronic and Hyperfine Structure. The observations are due to interactions with the matrices. Nelson and co-workersS4have shown that both the D 3 h and the Cb largest of the neutral mononitrogen oxides, NO3' ('NO3 refers structures would produce qualitatively similar electronic spectra, here to the "non-peroxy" form, as opposed to the O N 0 0 peroxy whereas Chantry et aLsshave concluded that the radical has Cb radical.), has turned out to pose a particularly interesting and or possibility C, symmetry, i.e, that there is no C3 axis present. difficult problem. Different computational methods predict the ground-state structure to be of either D3h or CzV~ y m m e t r y , ~ ~ ~On ~ the ~ ~other ~ hand, the doubly charged species Nos2-has been found experimentally to have either a planar structure of D3h whereas analyses of the vibrational spectra give a symmetric D3h symmetrys6or a nonplanar structure of C3 symmetry, whereas s t r u c t ~ r e . ~In~ thevibrational -~~ spectra certain anomalities have, the CO3- anion (isoelectronic with Nos) is known to have a Cb however, been observed that cannot be related to a completely structure.s7 symmetric geometry.49 An explanation of the observed pheThe ground-state geometries and properties of NO3 obtained nomena has been provided by Hirota et al.szand by Weaver et

796 The Journal of Physical Chemistry, Vol. 98, No. 3, 1994

in the present work are presented in Table 1, along with results from previous calculations. At the HF/DZP level, the NO3 radical is found to have a CZ,distorted structure with one short and two long which is also supported by single-point MP4 c a l c u l a t i ~ n sas ~ ~well as more sophisticated MCSCF2,47 and CCSD48treatments. The electronicground state is 2B2.From a H F study of the isotope shifts in the vibrational spectra of IsNand 180-substitutedN03,46the C2, structure is also concluded to be the most plausible. The molecule has one long NO bond ( ~ 1 . 3 5A) and two short ones (e1.21 A) and an ONO’ bond angleof approximately 114’ (will bedenoted “1L2S” throughout the text, and may be regarded as an 02N-0 conformer). In the symmetric D3h structure, which is found to be the electronic ground-state structure using HF/“DZ”,36CCSD,44and MPPT techniques (Table l ) , the N O bonds are found to be N 1.24 A and the O N 0 angles are 120’. This is also the structure determined through analyses of the vibrational ~ p e c t r a . ~At ~~~’ the HF/6-31G(d) level, the bond lengths are generally slightly shorter. A third minimum has also been reported, also of C2, symmetry (2B2 state), but with one short (1.20 A) and two long (1.26 A) NO bonds and an ONO’ bond angle of 126’ 48 (denoted “ 1S2L” or ’ON-02” analogous to the 1L2S O2N-0 conformer described above). Interesting to note is that the two studies where CCSD techniques are employed with the DZP basis set predict different ground-state structures. Kaldor’s Fock space multireference CCSD (FSMRCCSD) calculations yield the symmetric D3h structure as a minimum on the potential whereas the quasirestricted Hartree-Fock CCSD (QRHF-CCSD) calculations by Bartlett et ~ 1 . generate 4~ a CzU-(lL2S)ground state-in agreement with H F theory. The D3h structure is at the QRHFCCSD/DZP level found to be a second-order saddle point, and the second stationary point of C2, symmetry, C2,-(1S2L), is a transition-statestructureconnecting two C2,-(1L2S) minima. We will return to this aspect below. The computed energy differences between the various minima and saddle points are very small, in particular when the more sophisticatedtechniquesareadopted. At the ZPEcorrected HF/ 6-31G(d) level (present work), the C2,-( 1L2S) structure is approximately 29 kcal/mol more stable than the D3h structure; employing QRHF-CCSD/DZP8 or MCSCF/DZP7 techniques, the difference is only 3 kcal/mol and 1.4 kcal/mol, respectively. At the MP4(SDTQ)/6-31(d)(MP2/6-31G(d)) level, the energy differenceis instead 13 (19) kcal/mol in favor of the D3h structure; a value that at the FSMRCCSD/DZP level has decreased to less than 1 kcal/m01.4~ The second Cz, minimum (1S2L), finally, lies only 0.6 kcal/mol above the first one at the QRHF-CCSD/ DZP whereas the difference is 15 kcal/mol at the ZPEcorrected HF/6-31G(d) level. Using MP2 theory, this conformer is actually slightly more stable than the C2,-( 1L2S) form. The DFT geometry optimization calculations on NO3 consistently predict a structure of D3h symmetry. The fact that the MPPT and DFT techniques generate the (erroneous) D3h geometry as the ground-state equilibrium structure can be related to the incorrect spin-unprojected wave functions/densities used for the generation of the potential surfaces. When we regard the spin-projected energies (PMP2, PMP3, etc.) at the two MP4(SDTQ)-optimized geometries, we instead find that the CZ,(1L2S) geometry is slightly more stable than the D3h form. The NO3 radical is thus a case with a significant amount of spin contamination, whereby the unrestricted wave functions are poor representatives as approximate eigenfunctionsfor the system. At the UHF level, the spin contamination has not yet become large enough to distort the results ( (S2) = 0.78), whereas at the MPPT level the contamination is built up successively through the use of the unprojected wave function in the perturbation expansions to generate the first and higher order wave functions,thus favoring the D3h structure.

Eriksson et al. The observed and computed hyperfine structures of NO3 are listed in Table 2. The central nitrogen atom has a 14Nisotropic coupling of 10-16 MHz (absolute ~ a l u e ) , 3 with ~ * ~very ~ ~ low ~ a n i ~ o t r o p y .The ~ ~ only reported value on 170is, as mentioned above, -42 MHz (-15 G),53which, however, is highly uncertain. The present CISD/ 14s6pld calculationsonthe two HF/6-3 1G(d)optimized CZ,structures of the NO3 surface predict the nitrogen to have an isotropic hf coupling of ca. -8 MHz, irrespective of structure, whereas the hf parameters for the oxygens differ significantly. Analyzing the hf structure of the nitrogen atom alone will thus not reveal the true geometry of the species. For the CZ,-(1L2S) minimum, the two oxygens with short bonds have hf couplings of only -4.39 MHz, and the oxygen with the elongated NO bond attains a hf coupling of -17.25 MHz (CISD/ 14~6pld//HF/6-3lG(d)). The rotational average is-1 2.95 MHz, which when corrected for the fact that we seem to obtain only 75% of the oxygen couplings using the ETG basis sets (cf.NO$, is adjusted to -17.3 MHz. The second CZ, minimum, on the other hand, generates an oxygen hf structure such that the two oxygens with elongated NO bonds obtain couplings of -17.07 MHz, whereas the hf structure of the oxygen atom with the shortened bond is +4.00 MHz. The rotational average is -10.05 MHz, which after correction becomes -13.4 MHz. All these results do, however, give far too low values to account for the observed couplings of aiso(3 170)= 4 2 MHz. At the LDA-BP/DZP level (D3h symmetry), the optimized bond length is 1.253 A and the 14N hyperfine coupling is -1 1.3 MHz, in very good agreement with experiment. Using the local spin density approximation and the “TZPP” basis set IGLOIII,lS the bond length is shortened to 1.231 A and the value of ai,(14N) is changed to -14.2 MHz, whereas at the optimized LDA-PWP/IGLO-111 level RNO= 1.249 A and ai,(14N) = -8.7 MHz. Employing the LDA-BP/DZP, LDA/IGLO-111, and LDA-PWP/IGLO-I11 methods, wealso performed hf calculations on the three stationary points reported by Bartlett et al. (QRHFCCSD/DZP)48as well as on the stationary points optimized at the HF/6-31G(d) level. Again it is found that the nitrogen hf coupling does not change much among the three structures, the value being around -10.8, -13.4, or -8.4 MHz for the three methods used, respectively. The oxygen hf couplings, on the other hand, range from near zero (LDA and LDA-BP) to values between 0 and -28 MHz (PWP). In a recent studyS8it was concluded that, for most heteroatomcontaining compounds, the nonlocal exchange functional corrections by Perdew and Wang18 are required for an accurate description of isotropic hyperfine couplings of these atoms (the anisotropic values are usually well described at all DFT levels examined thus far). On the basis of these results, we are thus inclined to trust the LDA-PWP calculations to a greater extent thantheLDAor LDA-BPones. ForthetwoCastructures(IL2S and 1S2L), the results are in qualitative agreement with the findings from the CISD//HF calculations, although we observe 1 7 0 couplings that are approximately 10 MHz more negative than the CISD ones. Thus, at the CCSD/DZP (HF/6-31G(d)) CZ,-(1L2S)minima, we observe the oxygen couplings 2 X -8.3 (-8.7) and 1 X -26.8 (-27.8) MHz, respectively. At the C%(1S2L) minima, the corresponding numbers are 1 X -1.0 (0.3) and 2 X -24.0 (-28.7) MHz, respectively, whereas at the D 3 h minimum the oxygen couplings are approximately 3 X -1 3 MHz. The hf structures of the three minima on the potential surface are thus significantly different. However, the differences between the hf structures of the HF- and CCSD-optimized geometries are not too substantial. 3.3.2 ‘’0Substitution Effects. We now turn to the final objective of the present communication: to conduct an analysis of vibrational stabilization of preferred conformationsin partially 170-substitutedNO3. In the analysis, we have assumed theground state to be the CZ,-(1L2S) structure, in accordance with a majority

Hyperfine Structures of Neutral Nitrogen Oxides

*

*

01

01

\

The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 797

E (kcal/mol)

*

I1

01

I C2" DISTORTION P A T H W A Y

03'

(a)

Figure 1. Pseudorotation about the C3 axis of the NO3 radical, connecting

the three minima of Cb(112S)-symmetry.

E (kcal/mol)

I Figure 2. Schematic potential energy surface of the planar NO3 radical, based on results from UHF and QRHF-CCSD calculations. The minimum points (I) represent the global CW( 1L2S) minima, the central

point (111) is the D3h structure (second-ordersaddle point), and points (11) represent the transition structures of C%-(1S2L) symmetry. Also indicated are the &and C,reaction paths (a and b, respectively),discussed in the text. of the theoretical results, and have used the ZPE computed for the different isotopomers of this conformer. The analysis is performed at the HF/6-31G(d) and MP2/6-31(d) levels. It should be pointed out that the commonly used scale factor of ca. 90% has not been applied to the frequencies in these calculations. The reason for this is that we have found in a previous a p p l i ~ a t i o n ~ ~ that the ZPE values obtained from the unscaled frequencies are in significantly better agreement with ZPEvalues from corrected calculations, than if a 90% scaling would be imposed. By labeling one of the oxygens in the C2,-( 1L2S) structure, we see in Figure 1 that there will be a statistical ratio of 2 to 1 in favor of this atom being in a short bond position. In accordance with the findings by Bartlett et we can furthermore draw a schematic potential surface, connecting the different stationary points as shown in Figure 2. The three minima (I) represent the three stable CzU-(1L2S)structures of Figure 1, the central point (111) is the DV structure, and the midpoints on the lines connecting the C b minima (11) are the corresponding C2"-( 1S2L) structures also found on the surface. The two types of distortional pathways denoted a and b are shown more explicitly in Figure 3. In a, we pass from the C2,(1L2S) minimum, over the D3h structure (which is a saddle point along this C b reaction coordinate), and down into the minimum represented by an adjacent C2,-( 1S2L) structure. The second pathway (b) proceeds from one C2"-(1L2S) minimum to another, by way of a Cz,-( 1S2L) transition state. It can thus be regarded as a C, reaction coordinate with a saddle point of C2,-(1S2L) symmetry. Due to the symmetry of the system, each of the distortion pathways is, of course, threefold degenerate. In C2, symmetry, the (1S2L) structures will thus represent local minima, although their true characters are saddle points on the global potential hypersurface. Furthermore, the D3h structureis a saddle point along the Cb symmetry pathway but represents a transition point between a global minimum and a saddle point on the global surface. The QRHF-CCSD calculations have accordingly shown

>

I C s DISTORTION PATHWAY

(b) Figure 3. Two distortional pathways a and b of Figure 2, shown more explicitly. Each energy separation (verticalaxes) representsca. 15kcal/ mol at the ZPE-corrected UHF/6-31G(d) level and ca. 1 kcal/mol from

the QRHF-CCSD/DZP calculations of ref 48.

TABLE 3: ZPE (J/mol), ZPE Differences (J/mol), and Statistical Weights for the Singly and Doubly l'O-Substituted NO3 Radicals of C2,(1L2S) Geometry. The Geometries and ZPE are Obtained at the UHF/631G(d) and UMP2/ 6-31G(d) Levels substituted position ZPE (J/mol) AZPE (J/mol) stat. wt UHF/6-3 1G(d) 01 1 40 727.1 0 2 (03) 40 645.9 81.2 2 0 1 , 0 2 (01,03) 40 476.8 2 02,03 40 395.4 81.4 1 UMP2/6-3 1G(d) 01 1 35 495.0 0 2 (03) 35 402.3 92.7 2 0 1 , 0 2 (01,03) 35 272.4 2 02,03 92.9 1 35 179.5 ~

~

the &-( 1S2L) structure to have one imaginary frequency, and the D3h geometry to have These observations arecodirmed energetically by scanning the HF/6-3 1G(d) potential surfaces along pathways a and b, although the computer codes presently available will not allow us to compute the vibrational frequencies at the different points, using the appropriate symmetry. Thus, although the C2,-( 1S2L) structure is a transition state along the C, pathway b, the programs automatically assume a Ca symmetry for the wave function and thus generate vibrational frequencies corresponding to a local Cb minimum along pathway a. In Table 3, we present the total ZPE, the statistical ratios, and the ZPE differences between the different substitutional isomers of the CZ,-(1L2S) minimum structure. As seen, the ZPE differenceisvery similar between thesingly and doublysubstituted cases. Also noticeable is that the difference in AZPE is only 10 J/mol between the HF and MP2 levels of theory, although the geometries are quite different (cf. Table 1).

798

The Journal of Physical Chemistry, Vol. 98, No. 3, 1994

TABLE 4 Calculated Abundance Ratios of the Two Sin ly Substituted Isotopomers of the CZ,-(lL2S) Structure of Nb3. In the Calculations, the Boltzmann Populations Are Calculated at Each Temperature and Multiplied by the Statistical Weights UHF/6-31G(d)

Eriksson et al.

TABLE 5 Calculated Abundance Ratios of the Two Doubly Substituted Isotopomers of the CZr(lL2S) Structure of Nos. In the Calculations, the Boltzmann Populations Are Calculated at Each Temperature and Multiplied by the Statistical Weights

UMP2/6-3 1G(d)

UHF/6-31G(d)

UMP2/6-31G(d)

temp (K)

01 subst

0 2 ( 0 3 ) subst

01 subst

0 2 ( 0 3 ) subst

temp

0 1 , 0 2 (01,03)

02,03

0 1 , 0 2 (01,03)

02,03

0 4 10 30 50 100 200 273

0.00 0.04 0.16 0.27 0.29 0.31 0.32 0.33

1.oo

0.00 0.03 0.14 0.26 0.29 0.31 0.32 0.32

1.oo 0.97 0.86 0.74 0.71 0.69 0.68 0.68

(K) 0

subst

subst

subst

subst

0.00 0.15 0.28 0.33 0.37 0.43 0.59 0.62 0.65 0.66 0.66

1.oo 0.85 0.72 0.67 0.63 0.57 0.41 0.38 0.35 0.34 0.34

0.00 0.11 0.24 0.29 0.33 0.40 0.58 0.62 0.64 0.65 0.66

1.oo 0.89 0.76 0.7 1 0.67 0.60 0.42 0.38 0.36 0.35 0.34

0.96 0.84 0.73 0.7 1 0.69 0.68 0.67

The only oxygen isotope that has nuclear spin I # 0 is 170, which thus is the only oxygen isotope that is observable by ESR. If NO3 is selectively substituted with one or two 170isotopes, different spectra will result, depending on whether the isotope is positioned in a long or short bond position (long+short vs short+short for the doubly substituted case). Assuming the temperature to be sufficiently low to effectively reduce the internal pseudorotational motion, the molecule will reside in a minimum configuration (with respect to the 170position in the C2”-(1L2S) geometry), generating a ZPE as low as possible. The ZPE will thus modify slightly the depths of the potential wells, and we will observe a unique ground-state ESR spectrum. The situation is thus analogous to that of partially deuterated hydrocarbon radical cations, where the present method of analysis has been applied with great success to explain the effects arising from different vibrational or rotational isotopomers in the observed ESR spectra (for a review, see, e.g., ref 60). If the temperature is gradually raised, the vibrational motion will increase, and we thereby start to populate also the other minimum configurations (cf. Figure 1). In the ESR measurements this is observed as an increasing overlap between the two spectra generated for the different substituted positions. As the temperature increases further, we will approach the statistical ratio, where we have a free “pseudorotational” motion, and an averaged D3h geometry is observed. From the calculated energies of the different stationary points, it has been shown above that the barrier toward pseudorotation is very small. Thus, although the true ground-state configuration may be of Cb geometry, the experimental measurements must be conducted at very low temperatures in oder to single out a unique ground-state geometry, and not an averaged D3h structure. For the Cb-(1L2S) structure, there are two different oxygen hf couplings present (cf. Table 2), representing the long-bonded (large hf coupling) and the two short-bonded (small hf couplings) positions, respectively. In the singly substituted case, we will thus observe a hf spectrum corresponding to a singleoxygen atom with hf coupling -28 or -9 MHz (LDA-PWP//HF results; -17 or -4 MHz from the CISD//HF calculations), depending on whether the 170isotope is in a long-bonded or short-bonded position. The statistical weights are 1:2 for the two cases. Performing a Boltzmann population analysis a t different temperatures, based on the differences in ZPE and including the statistical weights, we find, however, that at 0 K only the shortbond positions are populated (Table 5). This agrees with the intuitive picture, that short bonds have larger force constants. In a simple uncoupled harmonic oscillator model, the molecule will hence lower its ZPE (and thus its total energy) more by placing the heavier isotope in a short-bonded position. As the temperature is increased, thevibrational motions become sufficient to overcome the small barrier (represented by a Cb-( 1S2L) structure) between two minima, and we should start to see an onset of the isotopomer with a long-bond substitution. At, e.g., 20 K, the spectra representing 170substitution at the short-bonded 0 2 / 0 3 positions vs the long-bonded position 0 1 will be in a ratio of 0.8:0.2. At

4 6 7 8 10 30 50 100 200 273

room temperature, the statistical ratio is virtually obtained (0.675: 0.325 instead of the statistical 0.667:0.333). For the doubly substituted species, the effects are even larger. There will be two types of spectra present, one with hf couplings 2 X -9 MHz (statistical weight 1) and one type with hf couplings 1 X -9 MHz and 1 X -28 MHz (statistical weight 2). The first type, where only the short bonds are substituted is, however, favored by the ZPE and is the only form present at 0 K. Increasing the temperature, we start populating the isomer with one long and one short bond substituted; at 7-8 K the ratio is 2:1, and at 20 K we observe an equal population of the two forms (Table 5 ) . Again, above 100 K basically the statistical ratio 1:2 is obtained, leading to a free “pseudorotation” and an averaged D3h structure. These results indicate the importance of performing the ESR experiments a t very low temperatures, and with a highly controlled degree of substitution. Furthermore, a matrix substance must be chosen such that there are no interactions leading to an unphysical predominance of a symmetric D3a geometry. Analyses of vibrational spectra conducted a t temperatures above, e.g., 50 K, will not be able to reveal a unique ground-state configuration, since we have then already passed the fast motion limit. Should the true ground-state equilibrium structure, however, be of D3h symmetry, no isotope substitution effects will be visible in the ESR spectra, even at 4 K. In such a case, the singly and doubly substituted isotopomers will give a unique spectrum of one or two isotropic couplings with values of -1 3 MHz.

4. Conclusions

In the present study, the electronic ground states of the neutral NO, NO*, and NO3radicals have been investigated with respect to their electronic and hyperfine structures. The geometries are optimized at the HF, MP2, MP4, and DFT levels, using basis sets of DZP quality or higher. The optimized structures are generally in good agreement with other theoretical results and with experiments. The barrier toward inversion is computed for the NO2 radical and found to be relatively high (-40 kcal/mol), in agreement with previous MCSCF and CI studies. The DFT computed dipole moments are generally slightly underestimated relative to experiment. For the NO3 radical, three types of stationary points are reported on the potential hypersurface. The minimum structure is, at most levels of theory, assumed to be of C, symmetry with one long and two short NO bonds. It is also shown that the second minimum structure of Cz, symmetry observed (Cb(lS2L))44,48is a saddle point connecting two global minima through a C,distortion. Considering only distortions along a Cb symmetry pathway, this is instead a local minimum, with a D3h structure as saddle point. On the global potential surface, the D3h geometry corresponds to a saddle point of second order,4*

Hyperfine Structures of Neutral Nitrogen Oxides although the DFT and spin-unprojected MPPT calculations throughout predict this to be the ground-state structure. The hyperfinestructures of the two larger species are computed using CISD and DFT techniques. In the case of NOz, the CISD calculations seem to be more basis set dependent than the DFT calculations and generally show a somewhat poorer agreement with experiment. Using different functional forms for the exchange and correlation potentials in the DFT study generates slightly modified hf structures, although the differences are much smaller than previously observed for heteroatom compounds.58 The hf calculations on the different stationary points on the NO3 potential surface (&A, CzU-(1L2S),and CzU-(1S2L)) show that the hf coupling on the nitrogen atom is highly insensitive to changes in geometry and that it is particularly well reproduced at the DFT level. The 170hf structures, on the other hand, vary significantly among the different geometries. Assuming a D3h symmetry, ais0(170) -13 MHz (LDA-PWP/IGLO-111), whereas for the Cb structures the long-bonded oxygens attain values up to -28 MHz. There is, however, a significant variation depending on computational method employed for the hf structure calculations, although a consistent pattern can be observed throughout . Assuming a C2, ground-state structure for NO3 (Cz,-( 1L2S)), a vibrational analysis is performed for the singly and doubly ''0substituted species on the basis of their HF/6-3 1G(d)- and MP2/ 6-31G(d)-computed ZPE differences. It is shown that at very low temperatures (520 K) it should be possible to single out a unique ground-state structure, whereas at higher temperatures a vibrational averaging sets in, leading to a pseudorotationally averaged D3h structure. In particular, the doubly substituted NO3 molecule exhibits a significant change in ESR spectrum between 0 K (2 X -9 MHz) and room temperature (overlap between 2 X -9 MHz and (1 X -9 MHz 1 X -28 MHz) in a ratio 1:2). At no level of theory do we get near the previously reported I7Ocouplings of 3 X -42 M H z , ~and ~ it is believed that a reinterpretation of the experimental data could be necessary.

+

Acknowledgment. The authors wish to thank the Natural Sciencesand Engineering Research Council of Canada (NSERC), the Swedish National Science Research Council (NFR), and the Killam Trust for financial support. Professors D. R. Salahub and 0. Goscinski are gratefully acknowledged for valuable discussions. We also thank the Theoretical Chemistry Group at UniversitC de MontrCal for comments and suggestions regarding the deMon program. References and Notes (1) Lindqvist, 0.; Ljungstrbm, E.; Svensson, R. Technical Report; Chalmers University of Technology: Gothenburg, Sweden, 198 1. (2) Siegbahn, P. E. M. J . Comput. Chem. 1985, 6, 182. (3) Pearson, R. G. Symmetry Rules for Chemical Reactions; Wiley: New York, 1976. (4) Laane, J.; Ohlsen, J. R. Prog. Inorg. Chem. 1980, 27, 465. (5) Mdler, C.; Plesset, M. S.Phys. Reu. 1934, 46, 618. (6) Davidson, E. R.; Bender, C. F. Chem. Phys. Lett. 1978, 59, 369. (7) Frisch, M. J.; Head-Gordon, M.; Trucks, G. W.; Foresman, J. B.; Schlegel, H. B.;Raghavachari, K.; Robb, M. A.; Binkley, J. S.;Gonzalez, C.; DeFrees, D. J.; Fox, D. J.; Whiteside, R. A.; Seeger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.; Pople, J. A. GAUSSIAN 90; Gaussian Inc.: Pittsburgh, PA, 1990. (8) Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Jensen, J. H.; Koseki, S.; Gordon, M. S.; Nguyen, K. A.; Wildus, T. L.; Elbert, S.T. QCPE Bull. 1990,10, 52. (9) See,e.g.: Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; Wiley: New York, 1986. (10) Schmidt, M. W.; Ruedenberg, K. J. Chem. Phys. 1979, 71, 3951. (1 1) MELDF-X was originally written by L. McMurchie, S. Elbert, S. Langhoff, and E. R. Davidson. It has since been modified substantially by

The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 799 D. Feller, R. Cave, D. Rawlings, R.Frey, R. Daasch, L. Nitzche, P. Phillip, K. Iberle, C. Jackels, and E. R. Davidson. (12) (a) St-Amant, A.; Salahub, D. R. Chem. Phys. Lert. 1990,169,387. (b) Salahub, D. R.; Fournier, R.; Mlynarski, P.; Papai, I.; St-Amant, A.; Ushio, J. In Density Functional Methods in Chemistry; Labanowski, J., Andzelm,J., Eds.;Springer-Ver1ag;NewYork, 1991. (c) St-Amant, A. Ph.D. Thesis, Universitd de Montreal, 1991. (13) (a) Sim, F.; Salahub, D. R.; Chin, S.; Dupuis, M. J . Chem. Phys. 1991,95,4317. (b) Godbout, N.; Salahub, D. R.; Andzelm, J.; Wimmer, E. Can. J. Chem. 1992, 70, 560. See also ref 12. (14) Eriksson, L. A.; Malkin, V. G.; Malkina, 0. L.; Salahub, D. R. J. Chem. Phys. in press. (15) (a) Kutzelnigg, W.; Fleischer, U.; Schindler, M. NMR-Basic Principles and Progress; Springer-Verlag: Heidelberg, 1990; p 165,Vol 23. (b) Huzinaga, S.J. Chem. Phys. 1965, 42, 1293. Huzinaga, S.; Sakai, Y. J. Chem. Phys. 1969,50, 1371. (16) Vosko, S . H.; Wilk, L.; Nusair, M. Can. J. Phys. 1980, 58, 1200. (17) Becke, A. D.Phys. Rev. A 1988,38, 3098. (18) Perdew, J. P.; Wang, Y. Phys. Reu. B 1986, 33, 8800. (19) Perdew, J. P. Phys. Reu. B 1986, 33,8822;1986,34, 3098. (20) Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structures. Vol. 4. Constants of Diatomic Molecules; Van Nostrand; Princeton, 1979. (21) Zhu, Z. H.; Murrell, J. N. Chem. Phys. Lett. 1982, 88, 262. (22) Baird, N. C.; Taylor, K. F. Chem. Phys. Lett. 1981, 80, 83. (23) Bauschlicher, C. W., Jr.; Langhoff, S.L. J . Chem. Phys. 1987,86, 5595. (24) Andzelm, J.; Wimmer, E. J . Chem. Phys. 1992, 96, 1280. (25) Neumann, R. M. Astrophys. J. 1970,161,779. (26) (a) Atkins, P. W.; Symons, M. C. R. The Structures of Inorganic Radicals; Elsevier Publishing Co.: Amsterdam, 1967. (b) Kasal, P. H.;Gaura, R. M. J. Phys. Chem. 1982.86, 4257. (27) Harmony, M. D.; Laurie, V. W.; Kuczkowski, R. L.; Schwendeman, R. H.; Ramsay, D. A.; Lovas, F. J.; Lafferty, W. J.; Maki, A. G. J. Phys. Chem. Ref. Data 1979,8, 619. (28) Morino, Y.; Tawimoto, M.; Saito, S.; Hirota, E.: Awata, R.; Tanaka, T. J. Mol. Spectrosc. 1983, 98, 331. (29) Turner, J. A. J . Phys. Chem. 1986, 90, 6000. (30) Kaldor, U. Chem. Phys. Lett. 1990, 170, 17. (31) Xie, Y.; Davy, R. D.; Yates, B. F.; Blahous, C. P.; Yamaguchi, Y.; Schaefer, H. F. 111. Chem. Phys. 1989, 135, 179. (32) Blahous, C.P.; Yatcs, B. F.;Xie, Y. M.; Schaefer, H. F . 111.J . Chem. Phys. 1990, 93, 8105. (33) Hirsch, G.; Buenker, R. J. Can. J . Chem. 1985, 63, 1542. (34) Seminario, J. M.; Politzer, P. Int. J . Quantum Chem. Symp. 1992, 26, 491. (35) Hodgeson, J. A.; Silbert, E. E. Curl, R. F. J. Phys. Chem. 1963,67, 2833. (36) Lund, A.; Thuomas, K. A. Chem. Phys. Lett. 1976,44,569. (37) Lunell, S.; Eriksson, L. A.; Huang, M.-B. J . Mol. Strucr. (THEOCHEM) 1991, 230, 263. (38) Morton, J. R.; Preston, K. F.; Strach, S . J. J . Phys. Chem. 1979,83, 533. (39) Adrian, F. J.; Kim, B. F.; Bohandy, J. J. Chem.Phys. 1985,82,1804. (40) Shiotani, M.; Sjhvist, L.; Lund, A.; Lunell, S.;Eriksson, L. A.; Huang, M.-B. J . Phys. Chem. 1990, 94, 8081. (41) Feller, D.; Davidson, E. R. J. Chem. Phys. 1984, 80, 1006. (42) Sjijqvist, L.; Lund, A.; Eriksson, L. A.; Lunell, S.; Shiotani, M.J. Chem. SOC.Faraday Trans. 1991, 87, 1083. (43) van Duijneveldt, F. IBM Technical Report RJ945, 1971. (44) Kaldor, U. Chem. Phys. Lett. 1990, 166, 599. (45) Boehm, R. C.; Lohr, L. L. J . Phys. Chem. 1989, 93, 3430. (46) Morris, V. R.; Bhatia, S.C.; Hall, J. H., Jr. J . Phys. Chem. 1991, 95, 9203. (47) Davy, R. D.; Schaefer, H. F., 111. J . Chem. Phys. 1989, 91, 4410. (48) Stanton, J. F.; Gauss, J.; Bartlett, R. J. J . Chem. Phys. 1991, 94, 4084. (49) Ishiwata, T.; Fujiwara, L.; Nagure, Y.; Obi, K.; Tanaka, I. J. Phys. Chem. 1983,87, 1349. Ishiwata, T.; Tanaka, I.; Kawaguchi, K.; Hirota, E. J. Chem. Phys. 1985,82, 2196. (50) Weaver, A.; Arnold, D. W.; Bradforth, S.E.; Neumark, D. M.J . Chem. Phys. 1991, 94, 1740. (51) Friedl, R. R.; Sander, S . P. J. Phys. Chem. 1987, 91, 2721. (52) Hirota, E.; Kawaguchi, K.; Ishiwata, T.; Tanaka, I. J. Chem. Phys. 1991, 95, 771. (53) Reuveni, A.; Luz, Z. J. Magn. Res. 1976, 23, 271. (54) Nelson, H. H.; Pasternack, L.; McDonald, J. R. J. Chem. Phys. 1983, 79, 4279. (55) Chantry, G.W.; Horsfeld, A.; Morton, J. R.; Wiffen, D. H.Mol. Phys. 1962, 5, 589. (56) Reuveni, A.; Luz, 2. J. Magn. Res. 1976, 23, 265. (57) Top, Z. H.;Raziel, S.; Luz, Z. J. Magn. Res. 1973, 12, 102. (58) Eriksson, L. A,; Malkina, 0. L.; Malkin, V. G.; Salahub, D. R. J . Chem. Phys., accepted for publication. (59) Lunell, S.;Eriksson, L.A.; Worstbrock, L. J. Am. Chem. Soc. 1991, 113, 7508. (60) Lunell, S.; Eriksson, L. A. Int. J . Quantum Chem. Symp. 1992.26, 575.