High-level theoretical study of the NO dimer and tetramer: Has the tetramer been observed?

The ground-state properties of (NO) 2 and (NO) 4 have been investigated using multireference second-order perturbation theory (MRMP2) and include a two-tier extrapolation to the complete basis set (CBS) limit. For the NO dimer the MRMP2(18,14)/CBS predicted structure, binding energy (with respect to 2NO; D e = 3.46 kcal/mol), and dipole moment ( u e = 0.169 D) are in excellent agreement with experimental measurements ( D e = 2.8–3.8 kcal/mol; u e = 0.171 D). Additionally, three of four intermolecular anharmonic MRMP2(18,14)/CBS-estimated frequencies (143 cm − 1 , 238 cm − 1 , 421 cm − 1 ) are in excellent agreement with recent gas-phase experimental measurements (135 cm − 1 , 239 cm − 1 , 429/428 cm − 1 ); however, the predicted value of 151 cm − 1 for the out-of-plane torsion ( A 2 ) is elevated compared to recent experimental estimates of 97–117 cm − 1 . Our ﬁnding that this infrared-forbidden vibration is also predicted to have an extremely low Raman activity (0.04 Å/amu at the MP2/QZ level of theory) conﬂicts with Raman measurements of a strong intensity for a low frequency band; however, these studies were performed for low temperature solid and liquid phases. Probing the possibility of the presence of higher order clusters we investigated the stability of (NO) 4 and discovered three isomers, each resembling pairs of dimers, that were stable to dissociation to 2(NO) 2 , with the lowest-energy isomer ( C i structure) having a predicted binding energy almost identical to that of the dimer. Computed vibrational frequencies of the C i isomer indicate that the 12 highest-frequency modes resemble barely shifted NO dimer-combined bands while the 13th highest-frequency mode at ∼ 100 cm − 1 is exclusive to (NO) 4 . Moreover, this tetramer-unique vibration is infrared inactive but has a very high predicted Raman activity of some 24 Å/amu. Guided by the theoretical results, we reexamined and reassigned experimental Raman and infrared data going back to 1951 and determined that our best predictions of vibrational frequencies of (NO) 2 and (NO) 4 are consistent with experimental observations. We thus postulate the existence and observation of (NO) 4 .


I. INTRODUCTION
The nitric oxide molecule has been a focus of interest in recent times, so much so that it was chosen as Molecule of the Year in 1992 1 due to its significant roles in biological processes that include blood vessel dilation, antimicrobial defense, and messaging between neurons. 2 Accordingly, there have been significant efforts to design novel NO-releasing systems that have therapeutic properties 3,4 including anticancer activity. 5 Theoretical chemistry studies have successfully assisted in the design of a new form of these potential prodrugs 6 and it was an in-progress investigation of the synthesis mechanism of N-bound diazeniumdiolates 7,8 (a class of NO-releasing molecules) that hinted at the apparent stability of (NO) 4 and led us to reinvestigate the NO dimer.
Although NO is a well-understood diatomic, the properties of its dimer, (NO) 2 , have been difficult to elucidate experimentally and (apparently) proved formidable to predict theoretically. A recount of relevant historical elements of experimental and computational studies of the dimer is appropriate here; we note two recent theoretical investigations that a) E-mail: ivanicj@mail.nih.gov. provide some of this information. 9, 10 Although a side-by-side C 2v structure ( Figure 1) was first inferred in 1961, 11 accurate experimental estimates of the long N−N distance (2.33 ± 0.12 Å) and dipole moment (0.171 D) were only obtained in 1981. 12 Subsequent determinations of dimer structural parameters have varied to some extent [13][14][15] (Table I), most notably the N−N distance with measurements ranging from 2.24 to 2.33 Å. Experimental determinations of the 0-K binding energy (D 0 ) with respect to dissociation to two monomers have also been somewhat diverse (Table II) with estimates varying between 1.8 and 2.3 kcal/mol; [16][17][18][19][20][21] the only direct measurement yielding 1.99 ± 0.01 kcal/mol. 21 However, without a doubt the most inconsistent experimental findings (or some interpretations of them as this article hopes to show) have originated from detections and assignments of the four lowest intermolecular vibrational frequencies. These modes are illustrated in Figure 1. While the two totally symmetric A 1 bands have been designated within the two ranges 166-188 cm −1 and 260-273 cm −1 from 1951 22 to 1984, [23][24][25] the asymmetric B 2 bend mode had been assigned at 489 cm −1 in 1976, 23 reassigned to ∼200 cm −1 in 1977 24 and 1984, 25 then to 243 cm −1 in 1998, 26 and finally to 429 cm −1 and 428 cm −1 in more recent gas-phase studies. 27,28 The infrared inactive A 2 J. Chem. Phys. 137, 214316 (2012)  torsion mode has been consistently assigned at ∼100 cm −1 since 1977. [24][25][26][27] Interestingly, the most recent, and perhaps most reliable, gas-phase estimations 27,28 of the intermolecular frequencies deviate somewhat from all previous measurements which were performed at low temperature solid and liquid phases, [22][23][24][25] including in an Ar matrix. 26 Of previous theoretical studies, 9,10,[29][30][31][32][33][34][35] only five 9,10,30,32,35 have predicted dimer structures in line with experimental results and two 9,32 have obtained reliable estimates of the electronic binding energy (D e ). Only the study of Sayos et al. 9 has obtained reasonable predictions of structure, binding energy, and vibrational frequencies; however, only the two computed A 1 frequencies were in reasonable agreement with previous experiments while the A 2 mode (190 cm −1 ) was nearly double that of previous assignments (∼100 cm −1 ). [24][25][26][27] The computed harmonic B 2 frequency of 418 cm −1 , while close to the most recent gasphase determinations of 429 cm −1 and 428 cm −1 , 27, 28 differs from previous assignments of 489 cm −1 , 23 ∼200 cm −1 , 24, 25 and 243 cm −1 . 26 In order to properly gauge theoretical methods, accurate and concordant experimental data are required; this does not seem to be the case for the intermolecular vibrational frequencies of the NO dimer. Since reliable predictions of electronic binding energies are a vital test of any theoretical technique, accurate vibrational frequencies are necessary to determine zero-point-energies.
Here, we have reinvestigated the properties of the NO dimer with multireference second-order perturbation theory (MRMP2) that included a two-tier extrapolation to the complete basis set (CBS) limit. The MRMP2/CBS optimized structure, electronic binding energy, and dipole moment are in very good agreement with experimental results. Additionally, three of four intermolecular anharmonic estimated frequencies are within 8 cm −1 of the most recent gas-phase experimental measurements; 27, 28 however, the predicted A 2 frequency of 151 cm −1 is 34 cm −1 higher. Our finding that this infrared-forbidden A 2 vibration is also predicted to have an extremely low Raman activity (0.04 Å/amu) conflicts with Raman measurements of a strong intensity for a low fre-quency band at ∼100 cm −1 . These results led us to reexamine the experimental data going back to 1951 and to investigate the stability of (NO) 4 . Our findings indicate the possibility that, along with the dimer, the tetramer may have also been observed in previous spectroscopic studies.

II. COMPUTATIONAL DETAILS
The computations in this study utilized the correlationconsistent cc-pVnZ, 36 aug-cc-pVnZ, 37 and cc-pCVnZ 38 basis sets; hereafter referred to as nZ, AnZ, and CnZ, respectively, where n = D, T, Q. Properties were determined using singlereference and multireference second-order perturbation theories.
For single-reference closed-shell computations (dimer and tetramers) restricted Hartree-Fock (RHF) 39 calculations were followed by second-order Møller-Plesset perturbation theory (MP2) corrections. 40 For single-reference open-shell computations (monomer) restricted open-shell Hartree-Fock 41 calculations were followed by second-order Z-averaged perturbation theory (ZAPT) corrections. 42,43 Geometries were optimized and Hessians computed seminumerically using analytic gradients incorporated in parallel implementations of the MP2 44 and ZAPT 45 methods. Raman activities 46 and infrared intensities 47 of vibrational modes of the dimer and tetramers were computed at the MP2 level of theory. In all cases the 1s orbitals were kept doubly occupied (core). Complete-active-space self-consistent-field (CASSCF) 48,49 wave functions were optimized using parallelized versions 50 of a determinant-based full configuration interaction code 51 and a second-order orbital optimization program. 52 Initial guess active spaces were constructed from virtual valence orbitals 53 which are generated post-RHF. Final MRMP2 energies were obtained using a parallel direct determinant implementation 54 of the method by Hirao. 55 Note that the Hirao method is identical to the Kozlowski-Davidson MROPT1 method 56 if in the latter method the barycentric definition of the zeroth-order energy is used and their effective Hamiltonian is not diagonalized. Optimized  geometries and corresponding Hessians at the MRMP2 level of theory were determined using an efficient, fully numerical code. 57 Anharmonic vibrational frequencies were computed at the MRMP2(2,2)/QZ level of theory using the vibrational self-consistent-field (VSCF) method that includes a secondorder perturbation theory correction to the vibrational energy (PT2-VSCF). 58 The quartic force field approximation 59 was used to reduce the total number of energy evaluations needed to obtain the anharmonic surface, which was generated using normal modes expressed in internal coordinates. 60 The 1s orbitals were kept doubly occupied (core) when using the nZ and AnZ basis sets but included in the perturbation treatment when using the CnZ basis sets. MRMP2/CBS energies were determined by separate extrapolations of the CASSCF and correlation energies. CASSCF energy extrapolations used n = D(2), T(3), Q(4) energies with the exponential form 61 while the correlation energies were extrapolated using n = T(3), Q(4) energies with the inverse-power form 62 The permanent dipole moment (μ e ) and zz-component of the polarizability (α zz where the z-axis corresponds to the C 2 rotation axis) were computed for (NO) 2 using the finite-field approach. [63][64][65][66] Energies (E) were calculated for applied external fields (F) along the z-axis having strengths of 0, ±0.001, ±0.002 au and E(F) fitted to a quartic. The properties μ e and α zz contribute to E(F) as We have verified at the MP2 level that dipole moments determined using this procedure match those computed analytically from response densities to at least four decimal places. All computations were performed using the GAMESS package. 67

A. Electronic structure
The electronic structure of the NO dimer has been discussed in detail previously 9, 10, 30, 32, 69 so we present only a brief description here. Although considered to be complex, the binding in the dimer can be concisely described as two weakly interacting monomers. In the C 2v point group, each monomer has an unpaired (radical) electron that can exist in the two degenerate π * antibonding (N 2px -O 2px ) and (N 2py -O 2py ) orbitals; however, when two NO monomers are placed side by side ( Figure 1) a weak σ -type bond is formed between the two in-plane monomer π * orbitals. Since the corresponding σ * -type orbital and out-of-plane π * orbitals are not much higher in energy, significant non-dynamic (valence) multireference character is observed (σ and σ * occupations of ∼1.5 and ∼0.5, respectively). Additionally, it has been found that extensive recovery of dynamic correlation is also required to obtain even a favorable binding energy. 9, 10, 32 The previous multireference results discussed here 9 utilized two references: (i) a (2,2) active space containing the σ and σ * orbitals and (ii) an (18,14) active space that contains all valence orbitals except the oxygen 2s. These references have also been used in the MRMP2 computations presented here. The molecular orbitals comprising the (2,2) and (18,14) active spaces have been illustrated and discussed in detail previously. 9 We were actually able to perform MRMP2 computations using fully optimized reaction space (FORS 49 or full-valence CASSCF (22,16)) references but found that the oxygen 2s orbitals had occupations very close to two. Concerned that the 2s orbitals, if active, would mix even slightly with the 1s core orbitals, we assessed that the CASSCF (18,14) reference was more suitable for the MRMP2 calculations.

B. Geometry and binding energy
A summary of earlier computational (top) and experimental (bottom) studies is given in Table III. The first six listed theoretical studies used single-reference theories and it is clear that of these only the MP2 method predicts a N-N distance (2.19 Å) that is somewhat close to the experimental range (2.24-2.33 Å). This may be slightly fortuitous since the coupled cluster with singles and doubles and perturbative triples excitation [CCSD(T)] method, which treats electron correlation more rigorously, obtains a poorer prediction for the N-N separation of 2.11 Å. Of the single-reference theories, the CCSD(T) method recovers the best electronic binding energy, D e , of 0.39 kcal/mol; however, its estimation (including counterpoise-corrected) is still far (relatively) from the experimental window of 2.8-3.8 kcal/mol. As discussed earlier, it is difficult to pin down a reliable experimental electronic binding energy since there has been no real congruence in the measurements of the intermolecular vibrational frequencies or of the 0-K binding energy. Our experimental range for D e of 2.8-3.8 kcal/mol has been computed by adjusting the D 0 values in Table II using zero-point vibrational energies of (NO) 2 (6.54 kcal/mol computed from a recent determination of vibrational frequencies 27 ) and NO (2.72 kcal/mol 70 ).
Although the density functional theory (DFT) methods improve the RHF geometry slightly, they predict considerably different binding energies (DFT-B3LYP: −3.00 kcal/mol; DFT-PLAP1: 8.18 kcal/mol) that are far from experimental Of the earlier multireference studies, a CASSCF (18,14) result shows that inclusion of only non-dynamic correlation gives rise to, perhaps surprisingly, an unbound dimer. Previous multireference configuration interaction (MRCI), average coupled pair functional (ACPF), and complete-activespace second-order perturbation theory (CASPT2) studies clearly demonstrate that rigorous simultaneous inclusion of non-dynamic and dynamic correlation effects are required to predict even a reasonable geometry and binding energy. It is hard to conclude which of the MRCI+Q, ACPF, or CASPT2 studies is definitive; they are all highly accurate in some ways. Certainly, the counterpoise and core-correlation corrected ACPF binding energy of 3.3 kcal/mol falls within the experimental range. An earlier MRCI(10) + Q/TZ computation predicted an accurate binding energy of 3.46 kcal/mol but this was performed at the CASPT2(18,14)/6-311G(2d) geometry; 9 it is difficult to know how (or whether) geometry optimization would affect this result.
Our results are shown in the middle portion of Table III. The MP2/QZ geometry is similar to that of the MP2/TZ2Pf obtained previously; however, when using the ZAPT flavor of open-shell second-order perturbation theory for the monomer, a very reasonable binding energy is obtained. In fact, the MP2(ZAPT)/QZ binding energy of 4.23 kcal/mol is almost identical to that of the CASPT2(18,14)/QZ counterpoisecorrected binding energy (4.32 kcal/mol) and not too far off the experimental range of 2.8-3.8 kcal/mol. While no singlereference theory will dissociate the dimer properly, it is useful to have such a cheap method that predicts very good relative energies, particular for the study of much larger systems that interact with or release NO and/or (NO) 2 , e.g., diazeniumdiolates. Binding energies of (NO) 2 at the MRMP2 level of theory were determined by optimizing the geometries of two monomers (in C 2v ) separated by a very large distance. The MRMP2(2,2)/QZ dimer optimized geometry has a rather long N-N distance (2.583 Å) but the predicted binding energy of 2.47 kcal/mol is quite good. Therefore, we can conclude that the MP2/QZ and MRMP2(2,2)/QZ computations should provide reasonably accurate energetics when applied to larger systems such as (NO) 4 to be discussed later. Since the MRMP2 (18,14) method is expected to be the most reliable of those used here, we have investigated the effects of going from n = D, T, Q for the nZ, AnZ, and CnZ basis sets. As expected, the MRMP2(18,14)/nZ optimized geometries gradually improve when going from DZ to TZ to QZ with the N-N distance decreasing from 2.455 to 2.385 to 2.339 Å. However, the binding energy is very slightly lowered from 2.54 to 2.53 kcal/mol when going from DZ to TZ, but improves to 2.99 kcal/mol for QZ. The reason for the DZ-TZ similarity is not immediately obvious. Very similar geometry and binding energy trends are noted when using the CnZ basis sets and the inclusion of core correlation makes little difference to either the geometry or binding energy. In fact, the results appear to become very slightly poorer when including core correlation; this notion will be discussed further below. Perhaps, the most surprising results are observed when using the AnZ basis sets. The N-N distance increases from 2.383 Å (ADZ) to 2.436 Å (ATZ) and then decreases substantially to 2.326 Å (AQZ) while the binding energy continually lowers from 3.25 to 3.14 to 2.68 kcal/mol. We have determined that the MRMP2 (18,14)/ATZ optimized geometry is distorted due to an intruder state appearing almost exactly where the true minimum would be. An intruder state also appears when using the AQZ basis set but at a structure not very near the minimum. We have not seen any indication that intruder states are present near the minimum geometries when using the nZ and CnZ basis sets. In order to not distract from the current discussion, we do not discuss the natures of the intruder states further here.
Although the MRMP2 (18,14) optimized geometries and binding energies are very reasonable when using the QZ and CQZ basis sets, the not insignificant improvements from T to Q prompted us to compute MRMP2(18,14)/CBS properties for the nZ and CnZ sets. The intruder state problem precluded the AnZ sets from CBS extrapolations. It is our assessment that the best predictions of the NO dimer properties are from the MRMP2(18,14)/CBS(nZ) computations. At this level the N-N and N-O distances are found to be 2.311 Å and 1.158 Å, respectively, while the NNO angle is 95.2 o . These parameters are certainly within the experimental ranges, although the N-N distance is slightly longer than the most recent measurements of 2.26 Å 14 and 2.28 Å. 15 Since there has been some variation in experimental measurements of the N-N distance (Table I) we are confident that our geometry is of high accuracy. The associated electronic binding energy is predicted to be 3.46 kcal/mol and this value falls well within the experimental window. The CBS(CnZ) optimized geometry and binding energy are very similar to those of the CBS(nZ) predictions with the former having a slightly longer N-N distance (2.319 Å) and slightly smaller binding energy (3.34 kcal/mol).

C. Dipole moment and zz-component of the polarizability
While the MRPT(18,14)/CBS(nZ) results appear highly accurate for the NO dimer with regards to the optimized geometry and electronic binding energy, further evidence for the validity of the method is confirmed by precise predictions of other properties. As far as we are aware, there is only one experimentally measured value of the dipole moment of the dimer: μ e = 0.1712 D. 12 Of the previous studies summarized in Table III, only the CASPT2(18,14)/QZ computation predicts a reliable dipole moment of 0.179 D, 9 noting that this computation was performed at the CASPT2(18,14)/6-311G(2d) geometry. All of the dipole moments determined in this work place the nitrogen atoms at the negative end of the dipole and the oxygen atoms at the positive end. The MP2/QZ dipole moment of 0.505 D is three-fold exaggerated and the MRMP2(2,2)/QZ dipole moment of 0.745 D is over four times the experimental value. While these computed values might initially appear inaccurate, some consolation is due since they do recover the correct direction of the dipole and the actual dipole moment itself is quite small. For the MRMP2 (18,14) level of theory, dipole moment trends are similar for all basis sets -there is a large increase in going from DZ to TZ followed by a slight decrease when going from TZ to QZ. Interestingly, the MRMP2(18,14)/QZ value of 0.178 D is more reliable than the MRMP2(18,14)/CQZ value of 0.190 D. Our best prediction of the dipole moment is that of the MRMPT2(18,14)/CBS(nZ) with u e = 0.169 D; this estimate is essentially identical to the experimental value. When using the CnZ basis sets, the CBS prediction of 0.176 D is not quite as good but still excellent. It is our feeling that the core-correlation corrected CBS(CnZ) results are generally slightly less reliable than those of the CBS(nZ). There are two plausible reasons for this: (i) treatment of core correlation using second-order perturbation theory may not be completely reliable and (ii) the core-correlating basis functions may be ever-so-slightly lacking compared to the valence-correlating functions.
Our predictions of the zz-component of the polarizability (α zz ) of (NO) 2 are given in Table III; we do not know of any previous theoretical or experimental studies of polarizabilities for the dimer. Our best estimate for α zz is 3.96 Å 3 at the MRMPT2(18,14)/CBS(nZ) level of theory. The dipole moment and zz-component of the polarizability of (NO) 2 are of interest to us as the synthesis of diazeniumdiolates (a class of NO-releasing molecule) involves the addition of two NO units to an amine group and it is not conclusively known whether the reaction path involves successive additions of individual NO units or simultaneous addition of 2NO or (NO) 2 . The dipole moment of (NO) 2 (0.171 D) is similar to that for one NO monomer (0.1574 D) 71 and, hence, approximately half that of two non-interacting NO monomers. This suggests that the nitrogen atoms in (NO) 2 are more neutral than in NO and contain more radical character, possibly altering their chemical reactivity. Additionally, our best prediction of α zz for the dimer (3.96 Å 3 ) is almost twice that of the monomer (α = 2.30 Å 3 ) 72 indicating that transfer of charge from nitrogen to oxygen (or vice versa) requires less effort in the dimer. These results suggest that while (NO) 2 may or may not be stable under room temperature conditions, an (NO) 2 coupling may enhance the reactivity of one or both nitrogen atoms. This feature may play an important role in biological pathways involving NO whereby some chemical reactions may involve NO dimers rather than individual NO radical moieties.  harmonic and anharmonic a,b ) and experimentally interpreted intermolecular vibrational frequencies (cm −1 ) of (NO) 2 . N-O stretch combinations not shown. See Figure 1 for mode labels.

D. Intermolecular vibrational frequencies
A summary of earlier computed harmonic (top) and experimentally assigned (bottom) intermolecular vibrational frequencies is given in Table IV. With regard to the experimental data, we see that assignments of the two A 1 modes are fairly consistent (166-188 cm −1 and 260-273 cm −1 ) across all the condensed-phase studies but occur at lower frequencies in more recent IR and Raman gas-phase studies (135 cm −1 and 239 cm −1 ) 27, 28 which are expected to more accurately describe an isolated NO dimer. We note that the Ar matrix study of Krim and Lacome 26 assigned the second (higher) A 1 band at 299 cm −1 , a higher frequency than all other studies. It is clearly evident that there are discrepancies in the determinations of the B 2 band which has been assigned at ∼490 cm −1 (condensed phase), 22, 23 ∼200 cm −1 (condensed phase), 24,25 243cm −1 (Ar matrix), 26 and 429 cm −1 and 428 cm −1 in gasphase studies. 27,28 The IR-inactive A 2 mode has been assigned at 97 cm −1 in two Raman studies 24, 25 as a strong intensity band, and indirectly inferred from combination bands as between 96 cm −1 and 117 cm −1 in IR experiments. 24,26,27 We note that the most recent Raman gas-phase study did not observe a band in the ∼117 cm −1 region. 28 With regard to previous computational studies (Table IV), it is clear that the single-reference methods DFT and CCSD(T) predict intermolecular frequencies in significant error from experiment, even when considering the aforementioned discrepancies in assignments. Both the multireference MR-AQCCSD and CASPT2 (18,14) methods describe the A 1 modes reasonably well. While the CASPT2 (18,14) harmonic frequency for the B 2 mode (418 cm −1 ) deviates from earlier studies, it is in good agreement with the most recent gas-phase measurements. However, for the A 2 mode the CASPT2 (18,14) harmonic frequency of 190 cm −1 is almost double that of experimental determinations.
Our results are shown in the middle portion of Table IV. We find that the MP2/QZ predicted frequencies overall are of similar quality to the earlier DFT and CCSD(T) results, being quite higher than experiment. The MRMP2(2,2)/QZ results fare slightly better, with the A 1 modes having predicted harmonic frequencies lower than the observed. The MRMP2(18,14)/nZ (n = D, T, Q) computed harmonic frequencies steadily increase up to CBS values of 176 cm −1 / 266 cm −1 for the A 1 modes and 471 cm −1 and 180 cm −1 for the B 2 and A 2 modes, respectively. These computed A 1 harmonic frequencies are (surprisingly) in good agreement with the condensed-phase measurements but slightly elevated when compared to the gas-phase studies. Similarly, the computed B 2 harmonic frequency agrees well with earlier condensed-phase experiments but is ∼40 cm −1 higher than the gas-phase studies, while the A 2 mode has a predicted harmonic frequency that is 60-80 cm −1 larger than experimentally determined. When compared to the gas-phase studies the MRMP2(18,14)/CBS harmonic frequencies compare well but are consistently elevated.
We considered anharmonicity effects by computing anharmonic frequencies at the MRMP2(2,2)/QZ level of theory (Table IV) and found that all frequencies are lowered (unsurprisingly) by 10%-20%. Scaling the MRMP2(18,14)/CBS harmonic frequencies using anharmonic/harmonic ratios from the MRMP2(2,2)/QZ results led to best estimates of 143 cm −1 , 238 cm −1 , 421 cm −1 , and 151 cm −1 for the A 1 (1), A 1 (2), B 2 , and A 2 modes, respectively (Table IV). These values for the A 1 and B 2 modes are in excellent agreement with the gas-phase experimental measurements, with differences of just 1-8 cm −1 . The IR-inactive A 2 mode has a predicted frequency (151 cm −1 ) that may be considered slightly elevated compared to the experimentally determined value of 117 cm −1 ; however, the latter was only indirectly inferred from analysis of combination bands and a more recent Raman gas-phase study did not observe a band in the ∼117 cm −1 region. 28 As such, it is difficult to assess definitively the accuracy of the predicted A 2 frequency. Regarding the B 2 mode, the commensurate values of the predicted (421 cm −1 ) and gas-phase experimental (429/428 cm −1 ) frequencies do little to explain previous assignments of between 199 cm −1 and 243 cm −1 for this mode. [24][25][26] With a focus upon the A 2 mode, condensed-phase Raman studies have assigned a frequency of 97 cm −1 , 24, 25 some 20 cm −1 less than the value determined by East et al., 27 and labeled it as a strong intensity band. 25 In contrast, the MP2/QZ computed Raman activity for the A 2 mode is essentially zero (0.04Ǻ/atomic-mass-unit) and this corresponds with no observation of a band in the ∼117 cm −1 region in the most recent gas-phase Raman study. 28 As such, we speculated that another species may be responsible for the observed highintensity Raman band at ∼100 cm −1 . As discussed in detail below, we postulate that NO tetramers, consisting of weakly bound dimers, may be present in the condensed-phases experiments.

A. Geometries and binding energies
Consideration of the size of the (NO) 4 systems as well as the stringent requirements to adequately describe (NO) 2 led to the choice of the MRMP2(4,4)/QZ level of theory, since the MRMP2(2,2)/QZ computations resulted in reasonable predictions of NO dimer geometries and binding energies. The (4,4) active space is constructed by merging two (2,2) spaces that are each analogous to the (2,2) active space used for the isolated dimer. MP2/QZ calculations were also performed to verify the results. We discovered three isomers of (NO) 4 that were stable to dissociation to 2(NO) 2 ( Figure 2 and Table V). Two isomers have C 2 structures while the most stable species has a C i geometry. It is clear that all resemble weak interactions of NO dimers. In fact, for each level of theory, predicted geometries of the constituent dimers resemble very closely the isolated dimers (Table V). Additionally, the MP2 and MRMP2 predicted binding energies of the most stable tetramer (relative to two dimers) of 4.22 and 2.54 kcal/mol, respectively, are almost identical to those for the dimer (relative to two monomers), 4.23 and 2.47 kcal/mol, respectively. Therefore, since the dimer has been observed in experiments, we feel it is likely that the tetramer has also, at least at low temperatures.

B. Intermolecular vibrational frequencies
Since the NO tetramers involve pairs of weakly bound dimers, we can make some presumptions regarding the natures of their vibrational frequencies: (i) the highest four (v 15v 18) will resemble monomer-combined bands, (ii) the next highest eight modes (v 7 -v 14 ) will resemble dimer-combined bands that are expected to occur at slightly shifted frequencies to those of the individual dimers, and (iii) the lowest six frequencies will be unique to the tetramer and probably lie in  the 0-120 cm −1 range. For clarity, we first discuss the lowestenergy C i isomer and make conjectures regarding the other species later. With regard to the dimer-combined bands (v 7v 14 ), symmetry constraints indicate that +/combinations of the dimer modes will either be Raman active or IR active, i.e., four modes will be of the A g irreducible representation while the other four will be of A u character. Inspection of the individual modes and harmonic frequencies computed at the MRMP2(4,4)/QZ level of theory combined with MP2/QZ computed Raman activities/IR intensities allowed us to identify which individual dimer modes have contributed (+/-) in each of the eight dimer-combined bands of the tetramer. Additionally, the C i isomer has a computed sixth-lowest frequency vibrational mode of A g character at 118 cm −1 that is IR inactive but has a very large MP2/QZ computed Raman activity. The v 6 vibrational modes for all species are illustrated in Figure 2.
Best estimates of the nine aforementioned vibrational frequencies for the C i isomer were obtained by adjusting the MRMP2(4,4)/QZ computed frequencies as follows. First, we compute the four intermolecular harmonic vibrational frequencies for two dimers separated at 100 Å at the MRMP2(4,4)/QZ level of theory, v D i [MRMP2(4,4)/QZ] (Table IV). Second, we use these with intermolecular dimer harmonic frequencies computed at the MRMP2(18,14)/CBS level of theory, v D i [MRMP2 (18,14)/QZ], to determine the scaling factors (18,14)/QZ]/v D i [MRMP2(4, 4)/QZ]. The frequencies of the four pairs of (+/-) dimercombined modes computed for the tetramer at the MRMP2(4,4)/QZ level of theory are then multiplied by the relevant scaling factor to give our best estimates. Since the v 6 vibrational mode of the tetramer (Figure 2) resembles the A 1 (1) mode of the dimer, this computed MRMP2(4,4)/QZ frequency is multiplied by the scale factor for the dimer A 1 (1) mode. The computed MRMP2(4,4)/QZ and best prediction harmonic frequencies are given in Table VI, together with Raman activities and IR intensities computed at the MP2/QZ level of theory and our interpretations (assignments) of the condensed-phase experimental data. While our predictions of tetramer vibrational frequencies do not include anharmonic effects, we feel they are a good guide to analyze the experimental observations: similar level results for the dimer at the harmonic level are only slightly higher than the gas-phase/anharmonic values.
With regards to the eight dimer-combined (+/-) modes, which are either Raman or IR active for the C i isomer, we find very good agreement between our best predictions and the observed (our assignments) condensed-phase frequencies.
Additionally, the computed Raman activities and IR intensities are in line with the experimentally observed intensities. The IR active A 2 dimer-combined band has a very low computed IR intensity and the predicted frequency of 200 cm −1 is close to the A 1 (1) dimer-combined band at 181 cm −1 ; therefore, a peak for this A 2 dimer-combined band may not be clearly discernable in the IR spectra. However, close inspection of the spectra reveals hints of a peak in the correct vicinity. 23,24 While the dimer A 2 mode has an essentially zero computed Raman activity, the Raman-active A 2 dimer-combined band (191 cm −1 ) in the tetramer has a computed activity (3.7 Å/atomic-mass-unit) that implies ease of detection. Additionally, the close-lying Raman-active A 1 (1) dimer-combined band (179 cm −1 ) has a computed Raman activity that is some four to five times higher (17 Å/atomicmass-unit). These predicted features are experimentally confirmed in all Raman spectra where the A 2 and A 1 (1) dimercombined bands are clearly discernible as neighbors and, furthermore, with the expected intensity ratios. The tetramerunique Raman-active v 6 mode has a best estimate frequency of 102 cm −1 and this is in excellent agreement with the experimentally observed at 97 cm −1 . Additionally, the computed Raman activity is very high and this corresponds well with the observed strong intensity.
These findings suggest that a (NO) 4 species has been observed, perhaps together with the NO dimer, in the condensedphase experiments. Although we have presented an analysis for the lowest-energy C i isomer, the presence of the other two TABLE VI. Selected computed (harmonic) and experimentally observed a intermolecular vibrational frequencies (cm −1 ) of the lowest-energy isomer (C i ) of (NO) 4 . Lowest five and highest four (N-O stretch combinations) frequencies not shown. See Figure 2 for (NO) 4 structures and mode 6 motions. Modes 7 to 14 resemble (NO) 2 intermolecular combination bands and are labeled as such. See Figure 1 for (NO) 2 mode labels.