J. Phys. Chem. 1993,97, 11408-1 1414
11408
Interaction Potentials for Dimer and Trimer Complexes with Molecular Nitrogen Kenneth A. Franken and Clifford E. Dykstra' Department of Chemistry, Indiana University-Purdue University at Indianapolis, 402 North Blackford Street, Indianapolis, Indiana 46202 Received: May 6, 1993'
Molecular nitrogen is nonpolar and has a relatively small dipole polarizability, making for particularly weak intermolecular interactions. The characterization of nitrogen interaction potentials is important for simulations, and in this study, we present an explicit model potential built around electrical interaction elements. The polarizability of nitrogen is found to be a small contributor to interaction in pure N2 clusters but is relatively more important in mixed clusters. The generally weak attachment of N2 to other species makes for a tendency to have attractive interactions a t more than one site of the partner. In the complex with water, for instance, nitrogen bonds at the oxygen and a t the protons, and the potential surface exhibits low barriers for several interconversion processes. Likewise, nitrogen is predicted to attach in two ways to acetylene.
Introduction Quite a number of weakly bonded complexes of nitrogen have been studied spectroscopically.1-21 The interest in nitrogen as a weak bonding partner is, in part, that it is nonpolar and does not contain hydrogen, which is in contrast to the hydrogen halides, water, and small organic molecules often the focus of molecular cluster studies. Along with information about hydrogen-bonded complexes, the nature of molecular nitrogen's interactions may reveal or test notions about the elements of weak interaction. In addition, the study of weakly bound nitrogen complexes offers a testing ground for nitrogen-nitrogen pair potentials that are necessary for liquid nitrogen simulation. A number have been devised alread~.22-3~ In this report, we first provide the characterization of nitrogen needed for the molecular mechanics for clusters (MMC) mode1,35,36 and from that we extract an explicit nitrogen potential in a concise form. MMC is a scheme for generating weak interaction potential energy surfaces of two or of many interacting species. It is based on an extensive electrical analysis. Via polarization energetics,it implicitly includesthe share of N-body, non-pairwise additive terms that arise from electrical interaction. We report results of calculations on a variety of nitrogen-containing complexes from which certain common features emerge. The surfaces are quite shallow, especially in comparison to clusters with two polar monomers, and as well, there are multiple minima for most small complexes. A comparison of the MMC results with the limited experimental and ab initio data that are available is also presented.
those from prior ab initio calculations37bwhich employed large, extended basis sets. The electrical interaction in MMC is evaluated by solving the mutual nonlinear polarization equations for a collection of any number of interacting species. The nonadditivity of the polarization energetics means that the MMC potential implicitly includes 3-body and up to N-body terms, where N is the number of molecules. The nonelectrical contributions to the interaction energy are modeled collectively in MMC via simple atom-atom LennardJones potentials. For every A-B pair of molecules in a system, there is an augmenting potential:
where the sums are over the centers, usually atoms, located in each molecule. This Lennard-Jones form is probably the simplest form of a potential which can give the steric features of an intermolecular interaction, though other functional forms in Ru may prove more accurate, especially in regions where molecules are closer than their equilibrium separation. These potentials are simply added to the electrical interaction potential in MMC, and they are strictly 2-body terms, although exchange repulsion and dispersion contributions may give rise to many-body terms.3842 In addition to representing the molecular "shape", the augmenting potentials also serve to correct for any lingering defects in the electrostatic portion of the potential caused by truncation of the multipole expansion,small errors in the ab initio computed moments, etc. The c and d parameters in eq 1 are Theoretical Approach forced to be transferable on a molecular level. The same c and The molecular mechanics for clusters (MMC) a p p r o a ~ h , ~ ~ , ~d ~values for Nz will be employed regardless of N2's bonding partner; however, the c and d parameters for nitrogens in other which was used in this study, combines electrical interaction molecules are not restricted to the same values. This transferenergies with atom-atom Lennard-Jones potentials to generate ability distinguishesthe model from a potential fitting procedure, potential surfaces over the intermolecular degrees of freedom of even though the c and d parameters are empirical at the current a weakly bound cluster. The MMC representation for molecular stage of MMC development. nitrogen, which means the selection of a set of electrical response In this study, the c parameters and d parameters for Nz were properties and the selection of Lennard-Jones parameters, was obtained by successive adjustment of their values in the course the first step in this study. The set of electrical response properties of evaluating the structures of N r H C l , Nz-HF, N2-HCN, and of nitrogen that we used corresponded to the same truncation Nz-Nz. The parameter adjustment was done to minimize the point used for otherdiatomics. Of course, because of the symmetry differences between ( 1) the equilibrium monomer-monomer of nitrogen, certain of the properties are zero. So, in these model separation distances calculated from MMC for the first three calculations, N2 was represented by its permanent quadrupole dimers and spectroscopicallydetermined values, and (2) MMC moment and by the dipole and quadrupole polarizabilities at the and ab initio Nz-Nz stabilities. The stabilization energies of equilibrium bond length. The values of these pr0perties3~8were three Nz-N2 structures were evaluated during the parameter selection for direct comparison with the ab initio values of BBhm *Abstract published in Advonce ACS Absrracrs, October 1, 1993. ~~
~
~~
~
~
0022-3654/93/2097- 1 1408304.00/0
0 1993 American Chemical Society
Potentials for Dimer and Trimer Complexes
The Journal of Physical Chemistry, Vol. 97, No. 44, 1993 11409
TABLE I: Structural Parameters and Stabilities of Dimers
a
MMC calculated results/experimental results: stability (cm-I) R,, (A) other structural data 26 3.316 T-shaped 52 3.401 T-shaped 94 3.748 T-shaped 3.865 T-shaped ( ~ 6 8 . 3 ' ) ~ ~ 177 distorted T-shape ( 4 4 9 , 3.784 NflC oxygen pointing into N-N bond NrCO 116 4.387 T-shaped, carbon pointing into N-N bond 227 linear NrHCCH 4.753 linear20 4.816 NrHCCH 245 3.845 parallel NrHCN 273 4.624 linear 4.682 linear or nearly linear7 linear NrHCl 302 4.201 4.218 linear4 NrHF 452 3.578 linear 3.576 linear3
4 ' 0 t'
of Nitrogen cluster HeN2 NeN2 ArN2
and Ahlrichs.28 The c and d values obtained for Nz are 5.4 and 1440 au, respectively. The center of mass separation distances based on microwave spectra are 3.576 A (calculated from atom separations) for Nz-HF,~4.217 for N z - H C ~ ,and ~ 4.682 for NzHCN.' N o correction for zero-point averaging was made to these values because we anticipated that the on-average lengthening due to stretching would be partially offset by an on-average contraction due to bending; that is, we assumed that the difference between the equilibrium and averaged separation distances for these three linear complexes would be small. The MMC equilibrium separations that compare with these three target values are 3.578, 4.201, and 4.624 A, respectively. The corresponding differences from the spectroscopic values are 0.002, -0.016, and -0.058 A.
'\'
'
"
'
'
'1'
i ~
3.9
3.8
R 3.7
3.6
3.5 0
90
180
e b 4.0
3.9
3.8
R 3.7
3.6
Results and Discussion Mixed Dimers. Table I presents the key structural information and calculated stabilities of the nitrogen dimer complexes where the partner species was a rare-gas atom or a linear molecule. The rare gas-nitrogen clusters are all T-shaped, and the stabilities, of course, are small. The T-shaped arrangement can be rationalized as the optimum location for a rare-gas atom to be polarized by the mostly quadrupolar charge field of the nitrogen. Recent microwave spectra of ArN2have confirmed the T-shaped structure but with apparent wide-amplitude bending leading to anon-averageangle0f68"~~ (or 22O fromT-shaped). Thecenterof-mass separation distance obtained from the microwave spectra rotationalconstants is 3.865 A.z1 Consistent with wide-amplitude motion, this average distance is considerably longer than the MMC equilibrium value of 3.748 A. In fact, the calculated Ar-N2 potential surface, shown in Figure la, shows onlya slight energetic difference between the MMC equilibrium structure and the observed on-average structure. A one-dimensional vibrational analysis along the Ar stretching coordinate was carried out with the Numerov-Cooley method43 and numerical anharmonic vibrational wavefunctions were obtained. For the ground vibrational state, the calculated on-average separation distance using the MMC potential is 0.08 1 8, longer than the equilibrium, and the resulting bond length of 3.829 A is then only 0.036 A less than the experimental value.21 Averging over the bending vibration should yield a further increase in the distance, though the size of the change should be much less than for the stretch. Another result of the recent study of Ar-Nz was the conclusion that the upper limit to the dipole moment of the complex is about 0.01 Da2' The MMC value a t equilibrium is larger, 0.021 D, but vibrational excursions that involve greater separation distances imply a smaller on-average value. This is revealed by the dipole moment surface in Figure lb. With the complex fixed a t the
3.5 0
90
180
e Figure 1. (a) Calculated potential energy surface of ArN2. R is the separation between the nitrogen mass center and the argon atom in angstroms and 0 is the angle in degrees between the line from argon to the nitrogen mass center and nitrogen-nitrogen bond. X indicates the location of the calculated equilibrium structure, and + indicates the location of the on-average structure determined from microwave spectroscopy.2*(b) Calculated magnitude of the dipole moment in debye of ArN2 as a function of the separation distance and angle 0.
experimental on-average structure, the calculated dipole moment is 0.015 D. A more strongly bound complex is that of nitrogen and carbon monoxide. The potential energy surface as a function of the in-plane orientation angles of CO and N2 (Figure 2) shows two local minima. They correspond to the two possible T-shaped structures with carbon monoxide pointing at the N-N bond. T-shaped structures are mostly a consequence of quadrupole quadrupole interaction between the two molecules. The lowest energy structure has the oxygen pointing a t the N-N bond in a somewhat distortedT-shapeand hasastabilityof 177 cm-l. Higher in energy, a t 116 cm-1, is the structure with the carbon pointing at the N-N bond. Isoelectronic with CO is acetylene, and in its complex with nitrogen, a minimum in the potential energy is found for the linear arrangement of the monomers (i.e., HCCH-NN). The reason the minimum for this structure is linear rather than T-shaped is that the sign of the quadrupole moment of acetylene is opposite that of carbon monoxide. Here, the quadrupolequadrupole interaction favors a head-to-tail arrangement. In
Franken and Dykstra
11410 The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
a
360
45
e 2 180
0 0
45 45
180
90
el
45
el
Figure 2. Sliceof the potential energy surface of the N2-CO cluster.The angles are the in-plane twist angles of N2 (01) and CO (82) about their mass centers, and the separation distance has been relaxed (optimized) throughout. The lowest energy structure is in the vicinity of 81 = 90' and 82 = 160' or 200'. A second minimum occurs at 81 = 90°, 82 = 0'. Potential energy countours are in IO-cm-1 steps.
TABLE II: Calculated Dipole Moments of Nitrogen Complexes at the Calculated Ehuilibrium Structures cluster diwle moment (DI diwle enhancement (DP HeN2 0.005 0.005 0.008 0.008 NeN2 0.021 0.021 ArN2 0.147 0.147 NrHCCH (linear) 0.112 0.1 12 N2-HCCH (parallel) 3.29 0.25 N2-HCN 2.13 0.29 NrHF 1.37 0.22 NrHCl 2.00 0.11 N2-H20 (oxygen-in) 2.00 0.12 N2-H20 (proton-in) 0 The enhancement is the change relative to the partner molecule's permanent dipole moment value used in the MMC calculations.
0
b
45
~~~
addition, the quadrupole-quadrupole interaction is attractive for a side-by-side arrangement, and in fact, another minimum energy structure was found with that arrangement. The energy of this conformation is only 18 cm-' below that of the linear structure, and that is too small relative to the likely accuracy of MMC to be certain that it is the lower energy form. As shown by the values in Table 11, both forms have a dipole moment of at least 0.1 D. The nitrogen-acetylene complex has been observed spectroscopically,20 and the spectroscopic data show a linear complex with a center of mass separation of 4.816 A. As with the Ar-N2 cluster, the experimental center-of-mass separation is considerably longer than that predicted by MMC, 4.753 A. The discrepancy between the MMC and experimentalZocenter of mass separations has much to do with the effects of vibrational averaging. We carried out a one-dimensional vibrational analysis along the normal mode stretching coordinate for this complex, using the Numerov-Cooley meth0d.~3 The on-average centerof-mass separation for the ground vibrational state is 4.819 A, and that is within 0.003 A of the experimental value. We may expect some shortening of the on-average separation distance due to vibrational averaging over the bending modes. If this is small, then the MMC picture is in good accord with the experimentally derived result.20 We examined interconversion and inversion of the nitrogenacetylene complex. The barrier for interconversion from the linear to the parallel geometry is only 20 cm-1. The transition state is planar and represents a concerted bend (Le., nitrogen rotated counterclockwise by 5 2 O from the linear arrangement and acetylene clockwise by 45') from the linear form. The parallel
e2
0
-45 -45
0
45
el
Figure 3. (a) Torsional vibrational potential for the N r H C N cluster. The angles are the in-plane twist angles of N2 (01) and HCN (02) about their mass centers. The equilibrium structure is in the vicinityof 01 = ' 0 and 02 = 0'. The potential energycontours are in 30-~m-~ steps. The ellipticform of the contours around the minimum shows that a concerted (N2 clockwise and HCN counterclockwise) torsion is a lower energy distortionthan an opposed torsion where the ends of the two partners are as a function of the pulled away from each other. (b) Optimized orientation angles. The contours are in 0.03-A steps. Comparison with the energy contours shows that a concerted distortion that raised the potential energy only 20 cm-l would be accompanied by a contraction in the separation distance of 0.12 A.
configuration can rearrange via a twist out of plane and return to an equivalent structure. By symmetry, the potential for this process must reach a maximum when the two molecules are perpendicular. The calculated energy of this crossed-structure :orresponds to a barrier height of 90 cm-1. HF and H C N form linear complexes with nitrogen,3.ss7 and this is partially a consequence of dipolequadrupole interaction. We may anticipate certain of the vibrational averaging effects From a comparison of the two potential surfaces. Given in Figures 3a and 4a are potential energy surface slices for N r H C N and N r H F . These slices give the MMC potential energy as a function If the two in-plane monomer orientation angles, but with the ieparation distance relaxed (optimized) throughout. H C N as :he partner to nitrogen gives rise to a softer bending potential :han does HF. We may expect wider amplitude excursions for :orsions in the ground state for N2-HCN than for N r H F . Then, f we examine how the optimized separation distance varies with
The Journal of Physical Chemistry, Vol. 97, No. 44, 1993 11411
Potentials for Dimer and Trimer Complexes
a
a
45
b
02
0 _--Rcom ---
#el
------.__
-45 -45
0
45
01 b
-45
0
-
45
tJ1
Figure 4. (a) Torsional vibrational potential for the N r H F cluster. The angles are the in-plane twist angles of N2 (el) and H F (02) about their mass centers. The equilibrium structure is at 81 = Oo and 02 = Oo, and the potential energy contours are in 30-cm-l steps. (b) Optimized as a function of the orientation angles. Contours are in steps of 0.03 A.
the angles (Figures 3b and 4b), it becomes clear that torsional vibrational motions imply a shorter than equilibrium on-average separationdistance in both cases. This will oppose the lengthening due to averaging over the monomer-to-monomer stretching. Legon and Millen have already considered this interplay of vibrational averaging effects for a series of complexes.4 It is important for modeling because this partial balancing of effects means the onaverage separation distances of N r H F , N r H C l , and N2-HCN should be similar to equilibrium values, even though this is not generally true for weak complexes. The N2-H20 cluster is a particularly interesting species, and with MMC we find two minimum energy structures. The equilibrium or global minimum structure obtained is shown in Figure 5 along with the secondary minimum structure. In the equilibrium structure, the oxygen end of water is pointing into the nitrogen-nitrogen bond, and the calculated stability is 383 cm-1. Less stable, but by only 37 cm-l, is a conventional hydrogen bonding structure where one OH bond of water is directed toward the end of the nitrogen molecule. It is this secondary structure that has been observed experimentally by Leung et a1.l’ Their center-of-mass separation distance (which we have calculated from the structural parameters they reported) is 3.88 %L, whereas the MMC equilibrium value is 3.78 A.
Figure 5. Calculated equilibrium structure of the water-nitrogen cluster (a) and the structureof the secondarypotential minimum (b). Thecenterof-mass separations are 3.370 8, for a and 3.775 A for b. The angle 0 of water’s symmetry axis and the mass centers line is 4 8 O in a. Structure b, on the other hand, is planar and the calculated equilibrium angles are 81 = 9’ and 82 = - 4 9 O . (All angles are defined such that they are Oo when they coincide with a line connecting the centers of mass and clockwise rotation is positive.)
Curtiss and Eisgrubel45 carried out ab initio calculations on the water-nitrogen complex at a number of different levels. At the SCF level with an STO-3G basis, they found two minima that correspond to those obtained from MMC calculations,though with the energeticordering reversed. At higher levels of treatment, only a partial search of the surface was carried out, and it was not certain if their secondary minimum (the MMC global minimum) remained a true feature of the surface. It was clear, however, that an arrangement with oxygen pointing into the nitrogen-nitrogen bond was attractive and that the surface in the vicinity of this arrangement was relatively flat. The 37-cm-1 energy difference that we obtain from MMC is small in relation to the geometrical changes needed to go from one form to another. It seems reasonable that the arrangement with oxygen pointing into the nitrogen molecule is not the global minimum and is only a shallow secondary minimum or even a shoulder region of the surface. The key feature is the shallowness of the surface and the ease for the nitrogen-water complex to distort significantly. We have explored the pathways for the water-nitrogen complex to interconvert from one form to another, for the hydrogen-bond swapping in the secondary structure (Figure 5b), and for inversion in the structure calculated to be the global minimum. We have calculated the potential energy barriers for these different paths. Inversion of the water molecule in the structure of Figure 5a has a very small barrier of 14 cm-1, and the transition-state structure is a planar arrangement. Inversion of the nitrogen molecule in Figure 5a involves a transition state with the nitrogen molecule nearly parallel to the water Czuaxis and a barrier of 53 cm-1. We find the interchange of O H bonds in the experimentallyobserved structure (Figure 5b) has a remarkably small barrier of 18 cm-1. The saddle point for this pathway is that of a planar structure wherein the C h symmetry axis of water coincides with the nitrogen molecular axis. Calculated values of the harmonic vibrational frequencies are given in Table 111. The nitrogen dimers often have very low frequency torsional modes, i.e., modes around or less than 30 cm-l. The vibrational transition frequencies may be even lower due to anharmonicity effects. For instance, the calculated ArN2 harmonic stretching frequency of 31 cm-l differs from the one-dimensional Numerov-Cooley result of 24.7 cm-l for the n = 0 to n = 1 (anharmonic) transition frequency. The Occurrence of low-frequency vibrations is an expected manifestation of the shallowness of the surfaces. With this, is the overall conclusion that a characteristic of nitrogen’s interactions is N2 is weakly “sticky” everywhere whereas polar molecules tend to be “sticky” at certain spots. Nitrogen does not attach strongly, and it can rather freely slide, roll, and twist next to other molecules.
Franken and Dykstra
11412 The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
TABLE I V MMC and ab Initio Calculations of Nitrogen Dimer Interaction Energies in cm-1 at Three Fixed Geometries
TABLE 111: Harmonic Vibrational Frequencies and Transition Moments of Nitrogen Dimers cluster NrHe
mode
stretch bend NrNe stretch bend NrAr bend stretch NrCO bend bend stretch bend N A C bend bend stretch/bend stretch/bend NrHF degenerate bend stretch degenerate bend NrHCl degenerate bend stretch degenerate bend NrHCN degenerate bend degenerate bend stretch NrHCCH degenerate bend (linear form) degenerate bend stretch NrHCCH (parallel form) bend out of plane bend stretch bend N r H 2 0 (proton-in) bend bend bend stretch bend bend N r H 2 0 (oxygen-in) bend stretch bend bend
freq (cm-I)
transition dipole (D)
40 16 29 20 24 31 5 14 34 47 15 23 40 54 42 87 22 1 22 56 94 14 56 55 8 41 55 30 35 52 77 28 30 67 80 178 45 75 14 102 145
0.0021 0.0036 0.0022 0.0046 0.001 1 0.0043 0.0057 0.0023 0.0099 0.010 0.016 0.01 1 0.0099
0.0087 0.23 0.042 0.51 0.33 0.028 0.33 0.77 0.25 0.03 1 0.0 16 0.0039 0.020 0.023 0.P 0.020 0.033 0.15 0.50 0.57 0.23 0.57 0.66 0.17 0.55 0.85 0.16
geometry T-shaped crossed canted-parallel at 50'
separation CPF-basis CPF-Basis dist, au MMC ACCDS IZ8 1128
8 7 7.6
88 80 91
75 46 69
86 56 84
86 61
TABLE V Structural Parameters, Stabilities, and Dipole Moments of Dimers of Nitrogen MMC Calculated ResultdAb Initio Results: stability geometry (cm-1) crossed 80 56.4 canted-parallel 91 84.3 89 T-shaped 86.0
dipole
0.0
other structural data e = 9000
0.0
planar, 6' = 59O*
0.04
planar, transition state
(A) moment (D) 3.114 3.7028 3.992 4.022* 4.167 4,2328
e = 5006
a 0 is the angle between the two N-N bonds as viewed along a line connecting the centers of mass. b 0 is the angle between the N-N bond and a line connecting the centers of mass.
energies (their basis I).28 Use of thedifferent electron correlation approaches, ACCDS and CPF, offers some idea of the convergence of the correlation effects on the stabilization energies. Three near-equilibrium geometries, for which there was specific energetic information from ref 28,were chosen, and the comparison of ab initio results and corresponding MMC energies is in Table IV. The results of the different ab initio calculations are similar, although stabilities differ by up to 15cm-l. The biggest difference between the ab initio and MMC calculations is for the crossed structure where MMC yields a stabilization energy that differs by about 20 cm-l. MMC results for (N2)2 show three nearly isoenergetic minima, and the structural information on these minima is in Table V. The lowest energy structure is a canted parallel structure with a cant angle of 59O and a stability of 91 cm-l. The T-shaped structure is only 2 cm-1 less stable. A third minimum energy a Calculated to be less than 10-6 D. structure predicted by MMC is the crossed structure, with an energy only 11 cm-l higher than the minimum energy structure; however, from the comparison with the ab initio calculations, it (Nz), Clusters. Attraction of two nitrogen molecules is may be that the stability of this structure is relatively exaggerated especially weak among the many studied weakly bound clusters. with the MMC model by up to 20 cm-l. A slice of the potential Weak interaction depends on the juxtaposition of the permanent energy surface following the two in-plane torsional angles and charge field interactions, electrical polarization, dispersion, and covering the regions of both the canted parallel and T-shaped steric or exchange repulsion. The attraction of two nitrogen structures is given in Figure 6. The advantage of the model over molecules is especially weak relative to the interaction between a b initiocalculations, of course, is the calculational easeof mapping strongly polar molecules such as HF and HCl, and there are out a surface in detail and providing a potential for simulation. differences in the juxtaposition of contributing elements between The disadvantage is in the limitations of the model, which in this the two types of complexes. There have been a number of theoretical treatments of the N2-N2 i r ~ t e r a c t i o n , ~ ~both - ~ ~ J ~ *case, ~ ~ are foremost in the nonelectrical elements. One focus of our research is the effect of polarization on the ab initio and model calculations. Most have predicted up to three energeticsof medium to largeclusters. To this end, we investigated local minima for thedimer: a planar T-shaped structure, a crossed several larger (Nz), clusters; results for n = 2,3,4,9,and 18 are "X"structure, and a canted parallel structure with a cant angle given in Table VI along with results for Nz-HF and N2-HCCH of around 45O. The exact energetic ordering of these minima complexes. Previous studies51 have shown relatively small differs with the treatment used to study the interaction, but most energetic differences for the local minima of small clusters. Given studies indicate that the energies of all three forms are within that there are three nearly isoenergetic minima for the dimer of about 30 cm-1 of each other. One high level a b initio treatment nitrogen, we have not attempted an exhaustive search for global of the N2 dimer,28 for example, calculates the energies of the minima in these larger clusters. Instead, we have followed one T-shape and the crossed shape to be within 25-30 cm-l of each minimum energy structure for each different size of cluster to see other, depending upon the basis set used. Experimental evidence the effects of different treatments of electrical polarization. In is limited, but suggests a center of mass separation of approximately 3.8 A1913 and a T-shaped1 or crossed13 geometry. other dimers,35 polarization contributions account for about 10% of the total interaction energy, but for the nitrogen dimer, The absolute well depth of equilibrium nitrogen dimer structures polarization energetics are almost negligible. Does this hold for from a b initio calculations22,25,2*,29.34.47~9 varies substantially with larger nitrogen clusters? As shown by the results in Table VI, the level of correlation treatment employed and the basis set as cluster size increases, the effects of polarization are more used. We performed a limited number of ab initio calculations noticeable, yet still small. With 18 interacting monomers, on (N2)2 structures to further explore correlation effects. These polarization accounts for only 1.5% of the interaction energy or calculations were performed at the ACCDSOlevel with the basis about 9% of the electrostatic interaction energy. The polarization used by B6hm and Ahlrichs for their C P F calculations of (N2)2
The Journal of Physical Chemistry, Vol. 97, No. 44, 1993 11413
Potentials for Dimer and Trimer Complexes
HCN-HCN. The separation distances between the first and second monomer mass centers, and between the second and third are 4.6625 and 4.4195 A, respectively. The MMC potential finds this trimer to have a linear equilibrium structure and the separations are 4.594 and 4.417 A. The difference between the experimental results for this complex and the calculated values is in line with thedifferences for the dimers. Calculated stabilities of the trimer species are given in Table VI1 along with sums of pair stabilities. The difference between the sum of pair energies and the trimer energy yields the three-body term and gives an indication of the addivity of the interactions. All of the trimers show a small degree of nonadditivity, the 3-body term amounting to 2 4 % of the total energy.
e2
Appendix. Interaction Potential Functions 0 ' -90
"
"
"
1
"
' 90
0
01 Figure 6. In-plane torsional vibration potential for the nitrogen dimer. 81 and 82 represent the in-plane twist angles of the nitrogens about their mass centers for the first and second nitrogen respectively. At 81 = 82 = Oo, the geometry is linear. The complex is T-shaped at 81 = Oo, 82 = 90° and the equilibrium structure is found at 81 = 82 = 59'. The potential energy contours are in 10-cm-l steps, and the separation distance was
relaxed (optimized) throughout.
TABLE VI. Calculated Stabilities in cm-1 of Selected Nitrogen-Containing Clusters calculated stabilities at different levelsa permanent nonmutual direct cluster (Nd2 (N2)3
(Nzh (Nzh (Ndl8 Nz-HF Nz-HCCH
polarization moments electrical olarization EI + EII+ E111 + EIV E1 + EII+ E111 EI + El11 Em 91 270 526 2013 4784 452 227
91 270 526 2012 4782 449 227
90 265 518 1986 4711 333 212
70 191 412 1625 3921 -86 79
(linear) a E1 is the permanent moment interaction energy, E11 is the energy of direct polarization. El11is the nonelectrical part of the potential. Explicit exprcssions are given in the Appendix. EIVis the additional polarization energy arising from mutual polarization effects to infinite order.
contributions to the energetics of mixed clusters, however, are much more noticeable. In the interaction of nitrogen with the highly polar H F molecule, polarization accounts for 26% of the total interaction energy. Polarization accounts for 6.6% of the total interaction energy in the linear N2-HCCH complex. A useful result in Table VI is that there is little difference between the infinite-order(first column) and second-order (second column) polarization energies, where second order means direct polarization and infinite order means the full mutual polarization (back polarization). Since restricting nitrogen potentials to direct polarization removes 4- through N-body terms, this result means that the electrical share of the true many-body nitrogen potential is mostly in 2- and 3-body terms. A consequence of neglecting mutual polarization effects is that an analytical potential can be easily extracted from the MMC representation. This explicit potential is given in the Appendix. Larger Mixed Clusters. The final part of this study is the investigation of mixed trimer clusters. We are particularly interested in how nitrogen will attach to a molecular pair that is more strongly bound than nitrogen's binding to either of those in the pair. Thus, we have carried out MMC calculations to locate equilibrium structures of NT(HF)2, N*-(HCN)2, and N2(HzO),. One of these trimers has been identified in the gas phase and its structure has been determined from microwave spectra. Ruoff et al.19found that nitrogen attaches to the end of the HCN dimer to form a linear or near linear complex of the form N r
Explicit interaction potentials developed in this work may be expressed in terms of the center-of-mass coordinates of the molecules. The first step is to find the potential, VA, at a point (xAYA,zA) due to all other molecules, e.g., molecule B at (XBYB,ZB). We shall define a vector ?AB and a scalar RABas XA - XB iAB
= Y A- Y B
(AI)
(ZA-ZB
RAB= V (xA - xB)' + CYA - YB)* + ( z A - zBI2 (A21 The intrinsic permanent moments of molecule B (designated by a subscript 0) are the dipole ;("), and the second, traced Cartesian moment tensor, 8fb).53 These are initially given in the local axis system of the molecule, and so they must be rotated into the laboratory frame. This can be accomplished by means of a usual geoemtrical transformation matrix, which we designate
U:
ctB = UBC'BL(0)
(A31
OB = uBefb)u;f The electrostatic potential at A due to B is ?;f,GB 3i:BeBiAB
+
R,,S
(A41
tree
~,,3
where q B is the net permanent charge of B. The field and field gradient are the first and second derivatives, respectively, of the potential with respect to position coordinates. Using the notation v;' = aVA/axi a n d c = a2VA/axiaxj, where xt denotes a component of rAB, we have
11414 The Journal of Physical Chemistry, Vol. 97, No. 44, 1993
TABLE VII: Calculated Stabilities of Mixed Nitrogen Trimers stability (an-') sum of pair stabilities (cm-')'
cluster NN-HCN-HCN NN-HF-HF Nr(H20)z
1772 2052 265 1
1739 1935 2564
Sums of stabilities of component dimer pairs in the trimer.
TABLE VIII: Augmenting Potential Parameters' molecule
center
HCCH
comb c1
C1 HI
Hz Ar HC1
com c1 H
He HF
com
Ne NZ
H F com NI
N2
x (A)
0.0 -0.601 0.601 -1.661 1.661 0.0 -0.035 965 59 1.247904 41
0.0 -0.870 615 5 0.046 184 5 0.0 -0.547 0.547
v(A)
dau)
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 10.6 10.6 0.6 0.6 10.641 21.0 0.0 1.4612 0.0 6.1 3.3395 5.4 5.4
d(au) 3000. 3230. 3230. 19.0 19.0 2698.22 7300. 6.0 180.833 0.0 1400. 479.338 1440. 1440.
a These parameters and electrical proper tie^^^ can be used in the formulas leading to eq A1 1 to generate interaction potentials for collections of these species. Center of mass.
column (or row) of OB, Oij denotes the ij element of OB,and 6, is the Kroenecker delta. After summation, wemay transform the field and field gradients at molecule A back into its local axis system with the inverse of the transformation matrix U. After this transformation, the interaction energy is calculated. The permanent moment contribution is
The energy of direct polarization is
The energy of the augmenting potential (see eq 1 in the text) is given by
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1.
where the sums in i and j run over the centers in molecules A and B, respectively. The c and d values used in this study have been reported previously but are collected in Table VIII for convenience. The total interaction energy is the sum of the permanent moment energy, the polarization energy and the augmenting potential E = E,
+ E,, + E,,,
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