The performance of the semi-empirical AM1 method on small and nanometre-sized N2O clusters

Large N2O clusters containing up to 177 molecules were simulated using the semiempirical AM1 method.

Simulated spectra of different cluster sizes show excellent agreement with experimental spectra.

The vibrational band shape of the strong N–N stretching vibration is more strongly influenced by shape and size than that of the N–O stretching vibration.

Stabilization energies and spectral band shapes confirm a crystal like structure of N2O clusters generated in supersonic jet expansions.

The AM1 results are carefully checked against experiment and high-level ab initio methods.

Overall, AM1 reproduces experimental results (structure and vibrational frequencies) and MP2 dissociation energies of small N2O clusters far better than any other quantum mechanical method studied here, although AM1 is not explicitly calibrated for the (N2O)n van der Waals system.

The good performance of AM1 allows us to simulate N2O clusters built from hundreds of molecules, a size range neither accessible by ab initio nor by continuum methods.


Clusters containing up to several thousands of molecules represent intermediate states of matter between the gas and the condensed phase.

In the past, oligomers like dimers, trimers, and tetramers as well as small particles with diameters ranging from nanometres to a few micrometres were studied extensively.1

It is now well known that even clusters built out of one billion molecules do not match the dynamic properties of the bulk phase in every detail, although their microscopic structure is often the same.

Some examples are clusters of N2O,2–9 CO2,8,10–14 SF6,15–17 and H2O.18–21

The theoretical treatment of such large clusters is restricted to model potentials and typically involves molecular dynamics, Monte Carlo and exciton approaches.

In the less studied intermediate size regime, more rigorous quantum chemical methods may come into play, although high-level ab initio approaches are still restricted to small clusters.

In this work we validate the exceptionally good performance of the semiempirical AM1 method22,23 on N2O clusters, which allowed us to study their structural, energetical and dynamical properties up to 177 monomer units and to compare them with experimental and theoretical results obtained previously.

N2O clusters belong to the class of van der Waals clusters and a large experimental data set is available for them.

The dimer, with its slipped-antiparallel structure is by far the best characterized complex.24–27

Later the trimer28 and tetramer29 were studied in the region of the v101 combination band and the assignments were supported by ab initio calculations of the rotational constants.

The trimer structure can be derived from the dimer by placing a third N2O molecule diagonally over the other two, which are tilted against each other along the torsion coordinate.28

The tetramer is basically a sandwich structure of two dimers where all N2O molecules are parallel to each other and the oxygen atoms of adjacent molecules are pointing in opposite directions.29

Only recently the first observation of a van der Waals mode in (N2O)2 was reported by Havenith and co-workers.30

Infrared spectra of large N2O clusters were reported in supersonic jets via FTIR spectroscopy2–5 and vibrational predissociation,6 in collisional cooling cells,7 diffusion traps8 and in free-standing argon crystals.31

Electron diffraction studies on large clusters of the structurally related N2O9 and CO210 molecules indicate that their microscopic structure is similar to that in the crystalline solid.

However, the v001 (N–N-stretch) cluster band2–5 is blue shifted by about 10 cm−1 relative to the TO-mode of the crystalline solid,8,12–14,32–35 even for large clusters with up to 109 molecules per cluster,11 whereas the cubic structure of the crystalline solid is adopted at a cluster size as small as ≈30–40 molecules.10

It is worth noting that the cubic structure of the large clusters is quite different from the parallel alignment of the molecules in the dimer, trimer, and tetramer, which is probably the reason for the opposite v100 (N–O-stretch) vibrational band shifts of oligomers and large clusters.4,5

Despite the experimental interest there are few theoretical studies on N2O and its clusters.

The N2O monomer was studied at various levels of theory.36–42

Dutton et al43. presented HF, MP2, and MP3 calculations of the dimer, confirming previous results by Nxumalo et al.44

Kudoh et al45. compared DFT results with matrix spectra and Miller and co-workers used the calculated rotational constants at the MP2 level of theory to assign their spectra to the N2O trimer28 and tetramer.29

However, up to now no systematic ab initio study of small N2O clusters is available at different levels of theory employing different basis sets.

Miller et al. simulated a cluster containing 55 molecules using semiclassical Monte Carlo simulations.6

Molecular dynamics simulations were used to describe the orientational disorder in crystalline nitrous oxide.46–50

Only recently Signorell demonstrated the applicability of the exciton model to clusters with spatial dimensions between 1 and 10 nm,51 though it was noted that the vibrational band shape of N2O clusters is only in part controlled by resonant dipole–dipole coupling.

The major focus of the paper is to demonstrate the good performance of the AM1 method and to compare the results with systematic quantum chemical calculations at different levels of theory.

As will be seen later, the interaction between the N2O subunits is dominated by dispersion forces.

Dipole–dipole interactions, arising from the small permanent dipole moment of N2O (0.16 D, ref. 52), play only a minor role.

N2O has three normal modes: v100 (mostly N–O-stretch), v001 (mostly N–N-stretch) and v010 (the doubly degenerate bending vibration).

In the following we will concentrate on the two stretching vibrations v100 and v001, but we will discuss other modes as well where experimental data exist.

Because the frequencies of intermolecular van der Waals modes are more than one order of magnitude smaller than those of the intramolecular stretching vibrations, the principal nature of the stretching fundamentals does not change upon cluster formation and we can still describe the cluster vibrations as, e.g., N–O-stretch vibrations.

The paper is organized as follows: After a short section describing the computational details we present the ab initio results of the N2O monomer.

The systematic analysis lays the footing for discussing the results on the dimer, trimer, and tetramer.

The semiempirical AM1 method is found to describe the structure, energetics and dynamics of oligomeric N2O clusters well, without any further parametrization.

In section 4, the AM1 calculations are extended to nanometre-sized clusters containing up to 177 molecules per cluster and compared with experimental and theoretical results obtained earlier4,5.

Computational details

All ab initio and most semiempirical calculations were carried out using the Gaussian 98 program package.53

Semiempirical PM3, restricted Hartree–Fock (HF), hybrid density functional (B3LYP), and second-order Møller–Plesset (MP2) perturbation theory calculations were applied to the monomer, dimer, trimer, and tetramer.

Basis sets ranging from 3-21G to double- and triple-ζ quality were used.

For the DFT calculations the default pruned (75,302) grid of the Gaussian 98 package was used.

The MP2 calculations include only valence orbitals, i.e., the frozen core approximation was used.

Geometry optimizations with the semiempirical Austin model 1 (AM1)22,23 were carried out using the Gaussian 98 program package.

For the calculation of harmonic force constants and infrared intensities of clusters containing more than 79 molecules as well as the geometry optimization of (N2O)177 the MNDO9754 program package was used.

In case of the dimer, trimer, and tetramer only the experimentally found isomers were calculated (Fig. 1a–c), which correspond to the most stable structures at all levels of theory.28,29,43–45

Although basis set superposition errors (BSSE) may amount to up to 50% of the van der Waals binding energy in these systems when using medium-size basis sets,43,44 a reordering of the isomers is only observed for small basis sets.

Energies given in this paper are not BSSE corrected.

The initial structures of large (N2O)n clusters were derived from the cubic structure of crystalline N2O.55

This choice is justified by experiments, which showed that the clusters adopt the structure of the crystalline solid for cluster sizes as small as 30–40 molecules (see above).10

Starting from a central N2O molecule a spherical section with radius r was cut out of the crystal structure, thereby generating approximately spherical particles with radii between 0.4 and 1.3 nm (Fig. 1d).

In contrast to CO2 and because of the small permanent dipole moment of N2O, orientational disorder has to be considered.4,5,46–50,56–60

Therefore we generated several clusters of the same size with random orientation of the individual N2O molecules and averaged all properties over the different clusters.

Here, orientation means that a N2O molecule is either pointing in one direction or turned over by 180°, thus fixing the alignment in space to the one found in the crystalline solid.

As expected, the effect of averaging is strongest for smaller clusters and less pronounced for larger clusters, where the number of N2O molecules is large enough for good statistics even within a single cluster.

We calculated spherical (N2O)n clusters with n = 13, 19, 43, 55, 79, 135, and 177 molecules per cluster and averaged over N = 20, 16, 15, 10, 4, 3, and 1 structures, respectively.

In order to break the high symmetry of the cubic structure the N2O molecules were randomly displaced by a few tenths of an Å from the position in the crystalline solid, ensuring a C1 symmetry of the initial geometry for the geometry optimization.

Columnar clusters (Fig. 1e) were built using a base of 2 × 2 unit cells and a length of 2, 3 and 4 unit cells, corresponding to cluster sizes of 63, 88 and 113 molecules per cluster, respectively.

Energetic and spectral properties of columnar clusters were averaged over three orientational distributions for each cluster size.



Table 1 shows the calculated bond lengths rNN and rNO as well as the calculated dipole moment μ of N2O at different levels of theory for a range of basis sets.

The experimental values are given in the first row for comparison.52,61

Both semiempirical methods, AM1 and PM3, reproduce the experimental atomic distances quite well.

The HF calculations use rather modest basis sets and underestimate the N–N distances, as one would expect for a purely covalent system.

The 3-21G basis set was included to have a measure of how well a simple ab initio method, applicable to clusters beyond ten molecules, describes the system.

However, HF/3-21G completely fails to give a reasonable agreement with the experimental N–O distance.

Upon inclusion of electron correlation at the MP2 level, the N–O distance improves, but the N–N distance is overestimated.

CCSD(T) calculations perform significantly better.

No attempt was made to reach the basis set limit for the monomer, as we are aiming at cluster calculations.

The B3LYP functional gives good structural results which are largely insensitive to the size of the basis set.

The difficulties in calculating the electric dipole moment μ of nitrous oxide (Table 1) were discussed in detail by several groups.39–41

The direction of μ (|μ| = 0.16 D52) was experimentally determined to be in agreement with a +NNO charge distribution, corresponding to a negative sign of the dipole moment.62

Our results confirm those of a systematic study of Frisch and Del Bene at various levels of theory.39

The dipole moments in Table 1 were calculated at the optimized geometries of the individual methods.

MP2 predicts the wrong sign, because it overestimates the contribution of the resonance structure over .

All other methods predict the correct direction of μ, but only CCSD(T) reproduces its magnitude reasonably well.

Semiempirical and Hartree–Fock methods overestimate the contribution of and, hence, the magnitude of the dipole moment, though the sign is correct.

Table 2 summarizes the structural parameters for the N2O dimer, which is planar with C2h symmetry (see Fig. 1a). rc.m. is the center-of-mass distance of the N2O subunits and θ the small angle between the molecular N2O axis and the center-of-mass distance rc.m..

A, B and C are the usual rotational constants.

Experimental values obtained from high-resolution infrared spectra of the v101,24v100,25 and v001 bands26 are included for comparison.

First of all, the intramolecular N–N and N–O bond lengths (not shown) are barely influenced upon cluster formation, as one would expect for a weakly bound van der Waals cluster, though the N–O- and N–N-distances are systematically increasing or decreasing, respectively, by one or two thousandths of an Å.

MP2 calculations give the best results for the intermolecular distance and angle.

The rotational constants are in good agreement with the experimental values as well, because small errors in the intramolecular distances have only little influence on the rotational constants.

AM1 is the next best method, despite an underestimation of the angle θ by about 5°.

The latter is also the reason for the deviation of the rotational constant A, because the A-axis lies approximately along the intermolecular center-of-mass distance.

PM3 fails to describe the intermolecular distance.

The deviations for the HF calculations are not as large, but still off by about 0.1 Å.

As for the B3LYP calculations, the results are again independent of the size of the basis set, but rc.m. is consistently overestimated by about 0.3 Å, causing the B and C constants to be too small.

Considering that the interaction between the N2O molecules is dominated by dispersion forces this is not unexpected.

Overall AM1 and MP2 describe the geometric properties of the N2O dimer best.

Table 3 summarizes the calculated and experimental28,29 rotational constants of (N2O)3 (C1 symmetry, Fig. 1b) and (N2O)4 (D2d symmetry, Fig. 1c).

Intramolecular bond lengths do not change much upon cluster formation (compare Table 2) and are not reported.

For the comparison, one should note that the experimental values represent vibrationally averaged rotational constants.

This is especially important for the trimer having a floppy and flat potential with respect to the angle φ of the top N2O molecule lying across the other two.28

Thus the calculated equilibrium values may differ substantially from the experiment.

The A-constant is sensitive to the angle θ of the two N2O molecules forming the dimer unit and in particular to the angle φ of the top N2O molecule with respect to the A-axis.

Thus we expect A to have the largest deviations from the experimental values, which is confirmed by Table 3.

Agreement with the experimental values is best at the MP2 level.

AM1 overestimates the A constant.

At PM3 level, the trimer structure studied here is not a minimum on the potential surface.

Similar to the dimer B3LYP underestimates the B and C constants, or, in other words, the intermolecular separation is calculated to be too large.

In case of the more spherical tetramer the differences between the methods are smaller.

Here, AM1 agrees well with the experiment.

PM3 overestimates the intermolecular separation of the N2O molecules and, hence, gives rotational constants much smaller than the experimental values.

The close match of the HF/6-31+G(d) calculation with the experiment is probably fortuitous, considering the previous results for the monomer, dimer and trimer.

Fig. 1c shows a small tilting of the oxygen end of each N2O molecule toward the center-of-mass, which is more pronounced at the MP2 level and results in nearly identical values for all three rotational constants.

B3LYP, again, overestimates the intermolecular separation, because it does not include dispersion explicitly.

Overall MP2 and AM1 best describe the geometric properties of small N2O clusters.

However, AM1 does not reproduce floppy intermolecular angles well, while MP2 overestimates the intramolecular N–N distance.

B3LYP and PM3 completely fail to describe the cluster geometries.

HF/3-21G, the simplest and fastest pure ab initio method used here fails for both intra- and intermolecular parameters.


Stabilization energies of the clusters are crucial for an understanding of their formation.

Table 4 shows the dissociation energies D0 of the dimer, trimer and tetramer for fragmentation into monomers.

In order to obtain D0 electronic energies were corrected by zero-point energy (ZPE), except for AM1 and PM3, which include the zero-point motion implicitly in the parametrization.63

For the trimer and tetramer at MP2/aug-cc-pVDZ level ZPE was estimated from the MP2/6-31+G(d) results, with an expected error on the order of 1 kJ mol−1.

B3LYP deviates systematically from all other calculations in giving much smaller dissociation energies for all cluster sizes consistent with the overestimated intermolecular distance.

DFT methods are well known for their deficiencies in calculating bond energies of weakly bound complexes.64–67

BLYP and B3LYP functionals tend to underestimate the attractive dispersion interaction (see ref. 64 and references therein) due to an erroneous asymptotic behavior at low densities.66,67

Recently Zhang et al66. and Wesolowski et al67. suggested that the PW91 exchange and correlation functionals are possible alternatives to describe weakly bound complexes.

To verify this for the N2O system, we have included PW91 dissociation energies into Table 4.

They are indeed larger, but fall short of the MP2 results by roughly a factor of two.

At the MP2 level the major contribution to the binding energy comes from electron correlation (see Table 4), reflecting the dominant part of dispersion interactions.

For the aug-cc-pVDZ basis the SCF part is even repulsive at the minimum geometry and the trimer and tetramer are only held together by correlation effects.

This effect may become even more pronounced for larger basis sets.

The cohesive energy in N2O clusters appears to be largely additive.

As we go from the dimer to the trimer and tetramer the dissociation energy increases by a factor of about three and five to six, respectively, in line with the growing number of pairs.

Dipole contributions are almost negligible.

Quadrupole interactions are more sizeable, but dispersion forces dominate the direct neighbor interaction.

The coincidence between MP2/aug-cc-pVDZ and RHF/3-21G is due to basis set superposition, which behaves qualitatively like dispersion.

With increasing basis set size the dissociation energy at the HF level drops below the MP2 values.

Similarly, PM3 describes the clusters as too weakly bound.

Except for the dimer, the agreement of the AM1 energies with the MP2 values is surprisingly good, despite a lack of polarization and diffuse functions and the fact that electron correlation is taken into account only implicitly in the parametrization.

Moreover, the reference data for the AM1 parametrization did not include polarizabilities23,63 which are essential for describing dispersion interactions.

The AM1 values for the parallel and perpendicular components of the polarizability tensor in N2O are 4.99 and 0.36 Å3, respectively, and thus smaller than the experimental values52 of 5.33 and 2.16 Å3, so that the interactions involving polarizabilities will be underestimated by AM1.

The good agreement for the total interaction energies (see above) is thus fortuitous, but together with the performance for the cluster geometry, it suggests that the AM1 method may be applied to larger clusters, where high level ab initio calculations are not feasible.

It is interesting to note that the related PM3 method fails in both categories, thus emphasizing the limited applicability of PM3 on nitrogen containing systems.

Vibrational frequencies

Vibrational frequencies are most easily calculated at the harmonic level.

It is common to apply a simple scaling factor to correct for the anharmonicity and for errors in the electronic structure approach.

For our purpose of simulating large clusters, it is sufficient if a common scaling factor can be applied to all vibrations, or, at least, to the same type of vibrations, for example stretching vibrations.

Table 5 summarizes the calculated harmonic frequencies for the N2O monomer at different levels of theory.

The experimental anharmonic values are given for comparison and the values in parentheses are the ratios of the experimental anharmonic and calculated harmonic frequencies.

MP2 calculations perform best in terms of a common scaling factor and also in terms of absolute frequencies.

The basis set size mostly affects the bending vibration ω010, but the aug-cc-pVTZ basis set is sufficient to reproduce all experimental fundamentals within 1%.

The density functional calculations require scaling factors which vary only by a few percent between the three fundamentals.

In contrast the scaling factors for the HF frequencies vary by 8% or even more (HF/3-21G).

Both semiempirical approaches have difficulties in describing the bending vibration of N2O. It is underestimated by about 50 cm−1, indicating that the potential energy surface is too flat in the bending coordinate.

However, AM1 and PM3 give comparable scaling factors for the two stretch fundamentals.

When discussing the infrared spectra of nanometre-sized N2O clusters we will compare the band-shapes of simulated and experimental spectra (see section 4.2).

Calculated infrared absorption intensities should reflect the experimentally observed intensities.

Table 6 lists the calculated and experimental intensities of the three fundamental modes of N2O. Because absolute absorption intensities involve large errors, we will concentrate on the relative intensities, which are given in parentheses.

The intensity of the N–N-stretch vibration (I001) was set to 100% for each calculation.

For MP2 and B3LYP the results are largely basis set independent.

Except for the N–O-stretch vibration, which is a factor of five too weak at the MP2 level, the absolute intensities agree within their expected uncertainty.

At the Hartree–Fock level neither the absolute nor the relative intensities are close to experiment.

The deviations are even larger when PM3 is used.

AM1 performs better, but it consistently overestimates all three intensities.

The I100/I001 ratio is within 50% of the experimental value, and thus acceptable for the simulation of infrared N2O cluster spectra in the N–O- and N–N-stretch regions.

Vibrational frequency shifts for the dimer, trimer and tetramer are summarized in Tables 7 and 8, together with the absorption intensities.

Since the van der Waals interaction is weak, the frequency shifts are small.

For the dimer and tetramer only one mode of the N–O- and N–N-stretch vibrations is infrared active, the others not being listed in the tables.

Experimental values are only available for the dimer,25,26,30 where we also added the frequencies of the two lowest van der Waals modes (Table 7).

In the dimer, stretching frequency shifts are relatively small , comparable in size and opposite in sign.

The opposite sign is reproduced by all but the MP2 calculations, where both stretches are red-shifted and the ω100 shift is only a fraction of the observed value.4,5

AM1 shows perfect agreement with the experiment, followed by B3LYP, which slightly underestimates the frequency shifts.

PM3 predicts almost no shift at all for the N–N-stretch vibration.

Hartree–Fock overestimates both shifts by at least a factor of two.

The failure of MP2 cannot be explained simply by geometry effects, because the N–N- and N–O-bond lengths are changing by the same amount in all calculations (see Table 2) and imply opposite signs of the frequency shifts.

Signorell pointed out that pure resonant dipole–dipole coupling would lead to a small blue shift of both stretch vibrations in the N2O dimer.51

Therefore other effects must play a crucial role in explaining the MP2 red shifts.

On the other hand the good B3LYP performance is inconsistent with the poor description of the interaction potential.

This is particularly true for the sensitive torsional van der Waals vibration.

All other calculations overestimate the torsional frequency, but MP2 is within the expected uncertainty for such a low frequency vibration.

Within their uncertainty, the predictions would also be consistent with the anti-geared bending vibration, but this is not the experimentally preferred assignment.30

A similar situation is observed in case of the trimer and tetramer (Table 8).

The trimer has three infrared active modes for each vibration.

If we just consider the strongest trimer vibrations, then the N–O and N–N vibrational frequency shifts have again opposite sign, even at the MP2 level.

However, the MP2 shifts are much smaller than, for example, at the AM1 or B3LYP level.

In the case of the tetramer we observe the same behavior as for the dimer, but the shift of the N–N-stretch is twice that of the N–O-stretch in all but the PM3 and HF calculations.

The cluster infrared absorption intensities reflect the pattern of the monomer calculations (compare Table 6).

In contrast to H-bonded systems, there is no intensity amplification, at best a slight attenuation, upon cluster formation, which renders detection by direct absorption more difficult.

Most important, though, is the comparison between the different methods.

The relative intensity change with increasing cluster size (for example, of the N–O-stretch vibration) is comparable for all but the MP2 calculations.

This suggests that all methods but MP2 are suitable for spectral intensities of larger N2O clusters, although the absolute intensities are off in some cases.

AM1 is one of these cases, but it agrees best with the experimental frequency shifts.

Large clusters

In the preceding sections we have demonstrated the suitability of the semiempirical Austin Model 1 (AM1) in describing the properties of small N2O clusters.

AM1 reproduces the structures, energetics, vibrational frequencies and absorption intensity changes with reasonable accuracy, but at a fraction of the cost of high level ab initio calculations.

It should be noted that none of the ab initio methods studied here describes all three categories (geometry, energetics, dynamics) equally well and fails to model at least one of them.

Obviously, there is no theoretical justification for the good performance of AM1.

It seems to benefit from fortuituos error compensation and the results obtained here cannot be transferred easily to other systems, as was recently shown by Signorell for the structurally related CO2 dimer.51

Our results encourage the application of AM1 to large (N2O)n (n > 4) clusters, although we have no proof for scalability.

One should keep in mind that the microscopic structure of oligomers and large clusters is different.

The former have a parallel arrangement of N2O molecules, whereas the latter are derived from the cubic structure of the crystalline solid, where no parallel arrangement of neighboring molecules occurs.

In the following sections we will present and discuss the properties of large N2O clusters at AM1 level.

Geometries and energies

Earlier in the paper the construction of large spherical N2O clusters from the cubic structure of the crystalline solid, with a single N2O molecule at the center of the sphere was described.

In order to break the cubic symmetry each molecule was randomly displaced by 0.1 Å and randomly oriented along the molecular axis prior to the geometry optimization.

Fig. 2a shows the radial atom number density of a (N2O)55 cluster prior to the geometry optimization.

For easier visualization each atom position was convoluted with a Gaussian profile (γ = 1 pm). r is the distance of the atoms from the center-of-mass of the cluster.

Fig. 2a reflects the regular cubic structure of the initial geometry.

Each group of peaks corresponds to what we can call a shell of N2O molecules around the center-of-mass.

After geometry optimization with the semiempirical AM1 method (Fig. 2b) the geometry is slightly disturbed, but the cubic structure with its “shells” is largely maintained.

In a microscopic picture this is also true for the alignment of the individual N2O molecules.

Thus, the geometry optimization does not change the arrangement within the initial cubic structure, being in accord with experimental results that even N2O clusters containing only a few tens of molecules take up the cubic structure of the crystalline solid.10

At the same time we observe a contraction of the cluster by about 0.5 nm.

The contraction is a consequence of the cluster’s finite extent in space and also of the overestimated AM1 rotational constants (Tables 2 and 3).

For comparison Fig. 2c shows the initial structure of a (N2O)55 cluster with a random alignment of the molecules in space, while their position was not changed from the one in Fig. 2a.

The sharp peaks indicate the positions of the central N-atoms in each N2O molecule and “shell”.

Because of the random alignment, the positions of the nitrogen and oxygen ends are spread out.

After the geometry optimization (Fig. 2d) the atom number density is more uniformly distributed than in Fig. 2b, consistent with an amorphous structure.

The amorphous structure is about 70 kJ mol−1 less stable than the one with a crystal like alignment, consistent with the experimental observation.10

Random orientations of the molecules lead to a variation of the total stabilization energy between 10 and 20 kJ mol−1 or less than 0.5 kJ mol−1 per molecule.

It should be noted that a cluster with a crystal like head-tail orientation of the molecular dipole moments is not the most stable structure, but lies within the energy range mentioned above.

In general an orientation of the surface molecules with the oxygen-end pointing away from the center-of-mass is energetically favored.

However, the energy differences per molecule are rather small and one would not expect a preferential orientation of the surface molecules under the experimental conditions of a supersonic expansion.2–5

Therefore we did not include a Boltzmann factor while averaging over all calculated structures.

With increasing cluster size the dissociation energy per molecule (D0/n) converges to a limit of about 20 kJ mol−1 (Fig. 3).

Energies of clusters with n = 13, 19, 43, 55, 79, and 135 molecules were averaged over N = 20, 16, 15, 10, 4, and 3 structures, respectively.

The dissociation limit is close to the experimental sublimation enthalpy of 24 kJ mol−1 at 136 K.56

This is consistent with a sublimation energy of ≈23 kJ mol−1, indicating that AM1 can describe the energetics of large N2O clusters with reasonable accuracy.

Vibrational spectra

Vibrational band shapes are more sensitive to orientational disorder than energies.

On the spectroscopic level the orientational disorder relaxes the electric-dipole selection rules, resulting in a broadening of the vibrational band shapes, which is among the most significant differences between the structurally related systems N2O and CO2.46–50,56–60

This effect was experimentally observed in infrared68,69 and Raman35,70 spectra of the crystalline solids and theoretically confirmed by molecular dynamics simulations.46,48,49

The orientational disorder has no effect on the vibrational center.

However, we expect the finite cluster extent in space in combination with the orientational disorder to influence the infrared spectra of large N2O clusters.

As was mentioned earlier, we generated several cluster structures with different orientational distributions for a given cluster size.

Fig. 4a-g shows the infrared spectra in the N–O-stretch region of seven such structures of (N2O)43.

The calculated frequencies were folded with a Gaussian profile (γ = 3 cm−1) and scaled by 0.802,4,5 which is the scaling factor for the monomer N–O-stretch vibration (compare Table 5).

Vibrational analysis shows that the major source of the differences between the spectra is the orientation of the molecules at the cluster surface.

The N–O-stretch wavenumbers of surface molecules with the oxygen-end pointing to or away from the center-of-mass of the cluster differ by ≈20 cm−1.

In the inner region, where each molecule is surrounded by neighbors, the wavenumbers of a molecule pointing in one or the other direction differ only by a few cm−1, confirming the results of molecular dynamic simulations of the crystalline solid.46,48,49

The large differences between the spectra also indicate that the clusters studied here are still too small to provide a statistical orientational ensemble within a single cluster.

Therefore it is necessary to average over several orientational distributions in order to compare simulated with experimental spectra.

The bottom trace in Fig. 4 is the average of all 15 calculated structures (n = 43).

It shows a broad, smeared-out band at about 1295 cm−1 and two smaller bands of nearly equal intensity at lower frequencies.

We checked the significance of the features by block-averaging only half of the calculated structures.

Fig. 5 shows the evolution of the N–O-stretch spectrum as a function of cluster size.

The spectra were averaged over the same number of structures as for the energy calculations.

While the calculated frequencies were scaled by 0.802 (see above), no scaling or normalization was applied to the calculated intensities.

Fig. 5 also includes an experimental jet-FTIR spectrum of a 7.5% mixture of N2O in helium.4,5

With increasing cluster size the band near 1295 cm−1 (C) gains intensity relative to the weaker structures (S) at lower wavenumbers.

Thus, we can easily assign the band C to molecules in the cluster core.

A similar spectral evolution is observed in ragout-jet FTIR spectra as a function of N2O concentration,2,4,5 in agreement with our theoretical results.

It should be noted that C is very close to the N–O-stretch vibration of the crystalline solid (1292.5 cm−1).34

The weaker bands (S) can be attributed to surface molecules, pointing toward or away from the center-of-mass of the cluster or lying parallel to the surface.4,5

Indeed, these “surface modes” are spread over the full spectral region of the N–O-stretch vibration and are partly covered by the main cluster band C.

The lowest frequency bands correspond to surface molecules with the oxygen-end pointing toward the center-of-mass, while molecules pointing in the other direction show spectral features underneath C.

Overall our simulated spectra resemble the spectral features and evolution of the experimental cluster spectra nicely.

Visual inspection would indicate a cluster size of about 100 molecules per cluster for the experimental spectrum in Fig. 5.

For the N–N-stretch vibration, agreement with experiment is not as good as for the N–O-stretch band (Fig. 6).

The calculated frequencies in Fig. 6 were scaled by 0.829,4,5 which is the scaling factor of the monomer N–N-stretch vibration.

Otherwise Fig. 6 was generated in the same way as Fig. 5.

An experimental jet-FTIR spectrum of 7.5% N2O in helium4,5 is included for comparison (bottom trace).

From experiment we know that the highest frequency band around 2245 cm−1 (C) belongs to the cluster core and grows with increasing cluster size, but even under conditions favoring the formation of larger clusters C is less prominent than in the case of the N–O-band and we would not expect a similar prominent peak like in Fig. 5.

Our simulations show an increase of the band marked S2 (≈2236 cm−1) relative to C with increasing cluster size.

According to a detailed vibrational analysis both bands (C and S2) have similar contributions from the clusters interior and the effect of different orientations of surface molecules is less pronounced.4,5

It should be noted, however, that S2, the dominant band in our simulations, is rather close to the N–N-stretch frequency in the crystalline solid (2237 cm−1,34 marked with Cr in Fig. 6).

Hence, S2 reflects the dynamics of the crystalline solid rather than that of large clusters, whereas the experimental cluster band (C) is blue shifted by about 10 cm−1 with respect to the crystalline solid.

A similar blue-shift was observed for the ν001-band of nanometre-sized CO2 clusters.7,8,11–14,71

For the strong ν001 band, one should also consider shape effects due to through-space transition dipole coupling.1,51

Fig. 7 compares simulated spectra for columnar clusters with those of the spherical shapes.

In the case of the weaker ν100 band (transition dipole moment of dM = 0.130 D),72 the shape effect is not very pronounced and can be traced back to the increased surface of the columns.

Quite in contrast the differences are significant for ν001 (dM = 0.249 D),72 making this band less suitable for the testing of the AM1 model, as long as the experimental shape remains unknown.

So far we considered only spectra of clusters with the cubic structure of the crystalline solid.

Fig. 8 shows the N–N- (a) and N–O-stretch (b) spectra of an amorphous like (N2O)55 cluster (compare Fig. 2).

Both spectra were averaged over four structures with a random alignment of the molecules.

The amorphous like structure results in broad spectra with little resemblance to experimental and simulated spectra of crystal like clusters (compare Figs. 5 and 6).

We limit ourselves to four different structures, but obviously further averaging would reduce individual features even further.

Fig. 8 indicates once again that N2O clusters containing a few tens of molecules adopt the cubic structure of the crystalline solid.10

Even surface molecules do not realign much during the geometry optimization, but rather retain the initial alignment of the underlying cubic structure.

Since the S features of the N–O-stretch vibration (Fig. 5) are reproduced by our simulations and could be assigned to surface molecules,4,5 we can conclude that the cubic structure is adopted even at the cluster surface.

A similar result was found experimentally for nanometre-sized CO2 clusters10,73.

Discussion and conclusions

The semiempirical Austin Model 1 (AM1) was used to describe the structures, energetics and dynamics of N2O and its clusters.

We compared the performance of AM1 with higher level quantum chemical methods and experimental results.

Overall AM1 gave a surprisingly good description of the monomer and small clusters (n ≤ 4).

Indeed it is the only theoretical method studied here which describes the structures, energies and vibrational frequencies with consistently acceptable accuracy.

All other methods (PM3, HF, B3LYP, and MP2) show larger deviations in at least one of the categories.

PM3 and HF cannot describe either of them.

B3LYP overestimates intermolecular distances and largely underestimates dissociation energies.

Among the ab initio methods, MP2 best reproduces the experimental geometries and can be assumed to be most reliable in the prediction of dissociation energies.

It fails to describe the frequency shift of the N–N-stretch vibration and the band strength of the N–O-stretch vibration.

Except for intermolecular angles and absolute infrared absorption intensities AM1 reproduces experimental results and MP2 dissociation energies far better than expected for a semiempirical method, which is not explicitly calibrated for such systems.

AM1 describes the relative intensity and the intensity change upon cluster formation correctly.

Therefore AM1 can be used to predict spectral changes with increasing cluster size.

The good performance of AM1 for N2O clusters must be attributed to some fortuitous error compensation, though AM1 is known to describe nitrogen-containing compounds generally better than MNDO.22,23

AM1 deviates from MNDO only in the explicit formulation of the core repulsion function that allows for a better tuning of the repulsions between atoms at about van der Waals distance.

It should be emphasized again that AM1 uses a minimal valence basis set without polarization or diffuse functions and that electron correlation is only included implicitly through the parametrization.

Moreover, intermolecular interactions were not considered during the AM1 parameterization:23 van der Waals and H-bonded clusters were not among the reference molecules, and the reference data did not include polarizabilities which are crucial for the description of dispersion interactions.63

Given these limitations, the results of semiempirical calculations must be carefully checked against experiment or high-level ab initio methods.

A careful inspection is even more important for systems not included in the parametrization and effects that have only little influence on the physical properties of the reference systems.

Therefore, in the present systematic work, we have carefully validated the use of AM1 for the modelling of N2O clusters.

However, this does not justify corresponding applications to other related systems: for example, AM1 fails to describe the vibrational frequency shifts of CO251 and is thus not suited for studying the properties of larger CO2 clusters.

It is worth noting that PM3 cannot describe N2O clusters adequately, although AM1 and PM3 have the same theoretical underpinning and differ only in the amount of experimental reference data and the optimization techniques used in the parametrization.

Based on the good results obtained for the oligomers we extended our AM1 study to nanometre-sized N2O clusters containing up to 177 molecules per cluster.

We observed an excellent agreement with experimental spectra in the case of the N–O-stretch vibration.

Weaker bands at the low frequency side (S in Fig. 5) could be assigned to molecules at the cluster surface, following an earlier vibrational analysis.4,5

The agreement between simulated and experimental N–N-stretch spectra is not as good.

This may be due to shape or size effects caused by the strong transition moment, resulting in an anomalous blue shift of the experimental main cluster band C (Fig. 6) relative to the crystalline solid.2,4–8,11–14,71

Both, the energetics and the spectral band shapes, confirm a crystal like structure of the N2O clusters.10

With increasing cluster size the dissociation energy per molecule approaches a limit of around 20 kJ mol−1, which is within 20% of the experimental sublimation enthalpy (Fig. 3).56

Oligomers and large clusters differ in their microscopic structure.

The parallel arrangement of molecules in the dimer, trimer and tetramer does not occur in the cubic structure of large clusters.

However, the good agreement between simulated and experimental results justifies the use and scalability of AM1 to simulate the properties of large N2O clusters and confirms the conclusions derived from it.4,5

All simulated spectra were scaled by the monomer scaling factors, which differ only slightly for the N–O- and N–N-stretch vibrations (Table 5).

No further frequency adjustment was made to match simulated and experimental spectra.

Experimentally, all main cluster bands (v100, v001 and v101) are blue shifted with respect to the isolated molecule.2,4,5

In contrast the v100 and v101 bands of the dimer, trimer (considering only the strongest band) and tetramer are red shifted,24,25,27 while the v001 band is again blue shifted.

AM1 reproduces the signs and magnitudes of the shifts for all but the main N–N-stretch cluster band (S2 in Fig. 6), the latter being underestimated by a factor of two, but matching the limit of the crystalline solid.