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An ab initio study of the CO–N2 complex
17 July 2002
Chemical Physics Letters 360 (2002) 565–572 www.elsevier.com/locate/cplett
An ab initio study of the CO–N2 complex J. Fiser a
a,*
, R. Pol ak
b
Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, 128 40 Prague 2, Czech Republic b J. Heyrovsk y Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, 182 23 Prague 8, Czech Republic Received 12 March 2002; in final form 22 May 2002
Abstract The interaction energy and van der Waals intermolecule bond length of several structures of the CO–N2 complex are calculated by the supermolecule CCSD(T) and MP4 methods using aug-cc-pVXZ (X ¼ D,T,Q) basis sets extended by a set of midbond functions centered in the middle of the vdW bond. The most stable structures are found to be two distorted T-shaped configurations with the N atom pointing towards the C–O bond. This conclusion is compatible with the results of high-resolution infrared, microwave and millimeter studies. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Besides both diatomic subunits of the CO–N2 van der Waals (vdW) complex being important atmospheric constituents, the vdW species itself is closely related to the isoelectronic CO and N2 dimers which are longstanding subjects of experimental and theoretical studies of weak intermolecular interactions. CO–N2 is formed by a barely polar and a nonpolar molecule similarly to, e.g., CO–H2 [1] and CO–D2 [2,3] whose spectra manifest large-amplitude motion. The CO–N2 complex was first detected spectroscopically in the infrared (IR) region of the CO stretching vibration in 1996 [4,5]. The observations of Kawashima and Nishizawa [4] indicate that the orientation of the CO subunit relative to the N–N
*
Corresponding author. Fax: +4202-2491-9752. E-mail address: fi[email protected] (J. Fiser).
bond is nearly perpendicular. However, it was difficult to determine the angle between the C–O and N–N bonds unambigously from their IR data. Based on a high resolution IR study, Xu and McKellar [5] assigned four connected subbands to K ¼ 0, 1, and 2 states in CO–orthoN2 , and one unconnected subband to a K ¼ 1 state in CO– paraN2 (the quantum number K refers to the projection of the angular momentum of the N2 or CO subunit, respectively, onto the intermolecular axis). The authors found the effective intermolecule separation for the ground state of CO–N2 to [5] and explained the simplicity of the be 4.025 A IR spectra by postulating that the N2 subunit undergoes a relatively free internal rotation within the complex. This assumption was confirmed by recent IR [6], microwave [7,8] and millimeter wave studies [7] for various CO–N2 isotopomers. With the aid of the analysis of the 14 N quadrupole hyperfine structure, the ro-vibrational and rotational spectra provide important information about the
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CO–N2 interactions. These studies revealed that the structure of the CO–N2 complex is nearly Tshaped, being very sensitive to both the 14 N nuclear spin modification and rotational state [6–8]. There exists quite a number of theoretical treatments of the CO–CO [9–14] and N2 –N2 interactions [15–19]. Van der Pol et al. [9] represented the CO–CO potential as a sum of the firstorder exchange-repulsion term, and damped multipole expansions for the second-order exchange and penetration phenomena. Their calculation predicted a T-shaped structure for the global minimum and several local minima differing slightly in energy, separated by very low barriers [9]. Meredith and Stone [11] improved this potential by extending the multipole expansion of the dispersion energy. Unfortunately, both potentials failed to explain the observed transition in the CO dimer spectrum recorded by Vanden Bout [21]. Rode et al. [12] calculated the CO–CO interaction energies using the fourth-order Møller–Plesset perturbation theory (MP4), the coupled cluster method restricted to single, double and connected triple excitations (CCSD(T)), and the symmetryadapted perturbation theory (SAPT). The authors explained the relatively large differences between the MP4 and CCSD(T) results by means of a diagrammatic analysis of correlation effects, supported by an evaluation of the fifth-order contributions to the electrostatic interaction energy using a 6s4p3d1f basis set. They pointed out that the CCSD(T) method is not sufficiently reliable for ðCOÞ2 , since it lacks certain important fifth-order contributions. Similar conclusions partly apply also to the N2 dimer [12] where, however, the MP4 results are in better agreement with the SAPT ones. On the contrary, Pedersen et al. [13] found several facts in favor of the CCSD(T) model even in case of ðCOÞ2 . They claim that the supermolecule CCSD(T) treatment is sufficient for calculating the interaction energy for the CO dimer, though they do not explicitly show that the fifth-order correlation terms become negligible in their large aug-cc-pVXZ (X ¼ T,Q) basis sets extended by midbond functions. Contrasting the number of theoretical studies on the CO and N2 dimers [9–19], there is very little theoretical work on CO–N2 . The molecular me-
chanics for clusters (MMC) approach of Franken and Dykstra [20] applied to a whole list of clusters suggested two T-shaped conformers for CO–N2 . The aim of this Letter is to calculate interaction energies of the CO–N2 complex using the CCSD(T) and MP4 methods with large augmented correlation-consistent basis sets. The results are expected to provide information on the basic properties of the potential energy surface needed in further possible investigations of the CO–N2 species, and to shed some light on the justification and the usefulness of the supermolecule CCSD(T) approach in ab initio studies of the CO–N2 , CO– CO and N2 –N2 vdW complexes.
2. Computational details Keeping the CO and N2 bond lengths fixed at their experimental monomer values (2.132 and 2.074 a0 [22], respectively), the CO–N2 potential energy surface depends on four coordinates, cf. Fig. 1: the distance between the centers of mass of the monomers, R Rc:m: ; and the angles hN ; hC and /. hN and hC are angles between the vector R pointing from the center of mass of N2 to that of CO, and the vectors rNN and rCO pointing from N1 to N2 and from C to O in the subunits N2 and CO, respectively, varying from 0° to 180°. / is the dihedral angle between the two plaines defined by the vectors R, rNN , and R, rCO . Because of symmetry, / may be restricted to the range between 0° and 180°. Considering the previous results [9–20] on the isoelectronic family of the vdW dimers, and with respect to our preliminary MP2, MP4 and CCSD(T) computations, we calculated the interaction energies for 11 selected structures of CO–N2 by the CCSD(T) method as a function of R. In all calculations we used the aug-cc-pVXZ (X ¼ D,T,Q) basis sets (denoted avxz) with four frozen core orbitals. Since the use of midbond functions contributes to the efficiency of the basis set for vdW molecules [13,23], we augmented the avxz sets with the set of 3s3p2d1f1g midbond functions (denoted MB) centered in the middle of the vdW bond. The exponents of midbond functions, originally devised for the CO dimer, were taken from [13]: they are 0.9, 0.3 and 0.1 for the s and p
J. Fiser, R. Polak / Chemical Physics Letters 360 (2002) 565–572
Fig. 1. Coordinates describing the CO–N2 complex.
functions, 0.6 and 0.2 for the d and 0.3 for the f and g functions. The distance R was varied in the vicinity of a minimum with a step size of 0.1 a0 . For comparison, we also report the results of the MP4/avxz and MP4/avxz + MB (x ¼ t,q) calculations for the two lowest energy structures together with the two other T-shaped configurations. The basis set superposition error was corrected using the counterpoise method of Boys and Bernardi [24]. All calculations were carried out with the MO L P R O 2000.1 program package [25].
3. Results and discussion The interaction energies and vdW bond lengths for different structures, three T-shaped, seven canted (C-shaped) and one X-shaped structure, and various basis sets are summarized in Table 1. Leaving aside the crossed (X) and C7 geometries,
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which are significantly less stable than the other arrangements, it is seen from the table that for all remaining nine structures the interaction energies calculated using the avtz(+MB) basis set lie within a very narrow interval of around 8 cm1 . This value is much lower than the range (31 cm1 ) of interaction energies for the characteristic points on the ðCOÞ2 potential energy surface [13] obtained at nearly the same level of accuracy. In Table 1, we also present the CCSD(T)/avtz data for the structures obtained by the shift of the end point of R from the center of mass of CO to the center of the C–O bond in a given configuration. Such a displacement changes the interaction energy and the corresponding vdW bond length R Rb , defined as the distance between the centers of the N–N and C–O bonds, only very slightly (at most by 1.3 cm1 and 0.1 a0 , respectively). In supermolecule studies of intermolecule potentials of vdW molecules some authors use fixed optimized monomer bond lengths (e.g., in [17,18]) instead of experimental ones. Table 1 shows that the corresponding deviation from interaction energies calculated with experimental bond lengths is lower than 1.2 cm1 for all configurations studied. The CCSD(T) and MP4 calculations using the largest avqz and avqz + MB basis sets were performed only for four structures C1 ; T1 ; T2 and T3 with geometries optimized within the avtz(+MB) calculations. Similarly to the CO dimer [13], the CCSD(T)/ avxz series converges to the complete basis set limit from above and for avxz + MB from below. An analogous conclusion applies also for the MP4 interaction energies, though these values are higher than the CCSD(T) ones by about 7–13 cm1 , i.e, they amount to 10% of the interaction energy. Because the disparity between the results of both methods was found to be more significant (about 25% of the interaction energy) for the CO dimer [13], one may tentatively conclude that the supermolecule CCSD(T) method for calculating vdW interactions [12–14] puts up a better performance in the case of CO–N2 compared to CO–CO. With respect to very small differences between the energies of various structures calculated within a given approximation, we did not perform the usual next step after calculating the ab initio points
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Table 1 Interaction energies (in cm1 , the first row) and equilibrium vdW bond lengths (in bohr, the second row) for the selected structures of the CO–N2 complex Basis set
T1 hN ¼ 90° hC ¼ 0° / ¼ 0°
T2 90° 180° 0°
T3 0° 90° 0°
C1 45° 45° 0°
C2 45° 135° 180°
C3 60° 150° 180°
C4 60° 30° 0°
C5 75° 165° 180°
C6 75° 15° 0°
C7 30° 45° 0°
X 90° 90° 90°
CCSD(T)
avdza
)87.9 8.5 )116.4 8.3 )98.7 8.4 )98.7 8.3 )97.5 8.4 )109.1 8.3 )103.3 8.4 )107.5 8.3
)79.5 7.6 )110.7 7.4 )97.8 7.5 )98.0 7.6 )98.7 7.5 )107.6 7.4 )101.7 7.4 )106.8 7.4
)90.0 8.0 )121.3 7.8 )104.1 7.8 )105.3 7.8 )103.6 7.9 )114.2 7.8 )109.5 7.8 )113.4 7.8
)91.8 8.3 )120.6 8.0 )103.8 8.1 )103.2 8.1 )103.4 8.1 )113.8 8.0 )109.0 8.1 )113.3 8.0
)81.2 7.7 )111.3 7.5 )96.7 7.6 )97.6 7.7 )97.2 7.6 )106.3 7.5
)80.8 7.6 )111.9 7.5 )97.8 7.5 )98.2 7.6 )98.4 7.5 )107.5 7.4
)89.2 8.4 )117.5 8.2 )100.6 8.3 )100.3 8.2 )99.9 8.3 )110.7 8.2
)80.0 7.6 )111.0 7.4 )98.1 7.5 )98.3 7.6 )98.8 7.5 )108.0 7.4
)88.1 8.5 )116.4 8.3 )99.1 8.4 )98.9 8.3 )98.1 8.4 )109.3 8.3
)82.9 8.5 )107.8 8.3 )93.9 8.4 )91.9 8.3 )94.2 8.4 )102.3 8.3
)59.0 7.2 )94.2 6.9 )78.2 7.1 )77.9 7.1 )78.9 7.1 )89.5 7.0
)106.2 8.4 )117.0 8.3 )110.7 8.4 )115.4 8.3
)109.7 7.4 )119.4 7.3 )113.4 7.4 )118.1 7.3
)116.0 7.8 )127.0 7.7 )121.4 7.8 )125.9 7.7
)114.5 8.1 )125.2 8.0 )119.6 8.1 )124.7 8.0
avdz + MBa avtza avtzb avtzc avtz + MBa avqza; d avqz + MBa ;d MP4
avtza avtz + MBa avqza; d avqz + MBa ;d
a
The monomer bond lengths are kept fixed at their experimental values. R is the distance between the centres of mass of the monomers. The monomer bond lengths are kept fixed at their experimental values. R joins the centres of bonds of the monomers. c The monomer bond lengths are optimized at the avtz level. R is the distance between the centres of mass of the monomers. d The value of R is taken from the corresponding optimized avtz data. b
J. Fiser, R. Polak / Chemical Physics Letters 360 (2002) 565–572
Method
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on the potential energy surface, i.e., their fitting to an analytical expression, which would be rather difficult and time demanding. Instead, we confined ourselves only to an analysis of the two lowest energy configurations, i.e., T3 and C1 on the CCSD(T) and MP4 interaction energy surfaces. This procedure consisted in additional calculations in the vicinity of the presumed minima in both the avtz and avtz + MB basis sets by using step size of 2°; 2° and 5° for hN ; hC and /, respectively, and 0.1 a0 for the distance R. In the collection of investigated points also the structures obtained by shifting the end point of the vector R towards the C or O atom were calculated with the same basis sets and step size. The minimum on the interaction energy surface was determined from two- or threedimensional fits related to various coordinates. Finally, the CCSD(T) and MP4 interaction ener-
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gies in the avqz(+MB) basis sets were calculated for the lowest energy arrangements obtained in the previous step. Optimizing hC and R in the C1 and T3 configurations (with other coordinates fixed) leads to the roughly 45° canted parallel (Cd1 ) and distorted T3 (denoted Td3 ) arrangements, respectively, whose interaction energies (DE) and geometric parameters are listed in Table 2. A comparison of these values with those for the C1 and T3 configurations presented in Table 1 indicates that a small distortion from C1 ðDhC ¼ hC hdC 4°, cf. Table 2) increases the magnitude of DE for the Cd1 geometry only slightly (at most 1.5 cm1 ). The pertinent increase of DE for Td3 is larger (3–4 cm1 , DhC 6° relative to T3 , C closer to N). It is noteworthy that these findings are almost independent of the method and basis set used. The displacement of the end point of R to
Table 2 Interaction energies DE (in cm1 ) and geometric parameters (in bohr or degrees) for the three most stable structures, distorted C1 and two T3 configurations of the CO–N2 complex; / ¼ 0° in all cases Structurea
Method
Basis set
DE
R
rC b
hdN
hdC
Cd1
CCSD(T)
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)104.6 )115.2 )110.0 )114.8 )115.6 )126.2 )120.9 )125.8
8.0 7.9 8.0 7.9 7.9 7.9 7.9 7.9
1.218 1.218 1.218 1.218 1.066 1.218 1.066 1.218
45 45 45 45 45 45 45 45
48 50 48 50 50 50 50 50
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)107.5 )117.9 )112.8 )117.4 )119.3 )130.1 )125.2 )129.4
7.9 7.8 7.9 7.8 7.8 7.7 7.8 7.7
1.066 1.066 1.066 1.066 1.218 1.066 1.218 1.066
0 0 0 0 0 0 0 0
84 84 84 84 83 84 83 84
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)108.3 )118.5 )113.7 )118.2 )120.6 )131.0 )126.4 )130.6
7.9 7.8 7.9 7.8 7.8 7.8 7.8 7.8
1.066 1.066 1.066 1.066 1.066 1.066 1.066 1.066
20 20 20 20 18 20 18 20
70 70 70 70 72 70 72 70
MP4
Td3
CCSD(T)
MP4
Ti3
CCSD(T)
MP4
a
The monomer bond lengths are kept fixed at their experimental values. rC is the distance of the end point of the vector R from the C atom. c rC is taken from the corresponding optimized avtz(+MB) data. b
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the center of the C–O bond in the Cd1 structure requires only 0:6 cm1 ; moreover, the Td3 structure characterized with the ‘leg’ pointing at the center of the C–O bond is stable with respect to a small variation of the angle hN . It is of general interest to show the effect of the BSSE correction on the interaction energy for the given cluster. This information is provided by Fig. 2 displaying contour plots of the corrected and uncorrected interaction energy surface. It is seen from this figure that the corresponding minima differ both in the vdW bond length (DR 0:2 a.u.) and in their depths; the BSSE correction amounts to 49% of the interaction energy.
Fig. 2. CCSD(T)/avtz contour plots of the interaction energy surface of CO–N2 (without (a) and with (b) the BSSE correction) as functions of the vdW bond length R and the angle hC between the vdW and CO bond. The contours are separated by 5 cm1 .
Another type of a distortion from the T3 configuration was also investigated, namely a concerted internal rotation of the two subunits which could be regarded as an interconversion between the T3 and C1 conformers. Both the CCSD(T) and MP4 methods using the avqz basis set suggest that the corresponding structure (denoted Ti3 ), with DhC ¼ jDhN j ¼ 20° relative to the T3 conformer, is by about 1 cm1 lower in energy than the Td3 structure (cf. Table 2), and the barrier for the transition from Td3 to Ti3 located at the point (R ¼ 7:9 a0 ; hdN ¼ 6°; hdC ¼ 84°; / ¼ 0°; rC ¼ rO ¼ 1:066 a0 ) is very low ( 1 cm1 ). It is worth mentioning that the BSSE correction for the Ti3 configuration calculated by the both methods (with the avqz basis set) is nearly the same and amounts to 17% of the interaction energy, while in using the avtz basis it is markedly higher (41%). The gross features of the CO–N2 structure emerging from our ab initio calculations are in agreement with observations based on the IR study [4], suggesting CO–N2 to have a nearly T-shaped structure with N–N pointing at the C–O bond. Comparison of our results with microwave and millimeter wave studies of CO–N2 [7,8], taking into account hyperfine interactions, further highlights certain aspects of topological behavior of the potential energy surface. FTMW data [7] established that the CO–orthoN2 complex in its ground state (K ¼ 0) has an approximate T-shaped O-bonded structure (a distorted T2 structure in our notation), with N2 undergoing a nearly free internal rotation. In contrast, a lower K ¼ 1 CO–orthoN2 state has the T3 arrangement with the N2 subunit somewhat more localized than in the K ¼ 0 state. The spacing between the K ¼ 0 and K ¼ 1 levels of the CO–orthoN2 complex was determined to be 3:53 cm1 [5,7]. From the three CO–paraN2 states studied in Ref. [8], the lowest state, corresponding to the K ¼ 0 levels of the first excited vdW vibrational N2 bending state, has approximately a T3 arrangement. The ground state of CO– paraN2 is assessed to lie about 3–4 cm1 above the ground (K ¼ 0) state of CO–orthoN2 [5]. Xu and J€ager [8] determined the vdW bond lengths for the two lowest CO–N2 states having a T3 structure (i.e., K ¼ 0; mb;N2 ¼ 1; paraN2 and K ¼ 1; orthoN2 ) to be 7.918 and 7.874 a0 , respectively. Table 2 shows that
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the CCSD(T)/avqz(+MB) estimates, RðTd3 Þ ¼ RðTi3 Þ ¼ 7:9 ð7:8Þa0 , are in very good agreement with the spectroscopic data. The vdW bond length for the T-shaped O-bonded configuration of the ground (K ¼ 0) state of CO–orthoN2 was spectroscopically determined to be slightly shorter, i.e., 7.608 a0 [5,8]. Our results identifying the salient features of the interaction in N2 –OC differ from those obtained by the MMC approach of Franken and Dykstra [20] in two points. Their calculation favors the N2 – OC conformer (distorted T2 , O pointing at N–N, R ¼ 7:15 a0 ) to the N2 –CO structure (distorted T1 , C pointing at N–N, R ¼ 8:3 a0 ). Further, the MMC model predicts a significant discrepancy in stability of the T1 and T2 conformers (61 cm1 ), while our CCSD(T) and MP4 calculations render only a small deviation (0.7 and 2.7 cm1 , respectively), cf. Table 1. A special attention deserves the difference in stability between the ðCOÞ2 and CO–N2 complexes. The CCSD(T)/avqz + MB calculations of Pedersen et al. [13], and our results obtained with a similar model [26], lead to the ðCOÞ2 interaction energy of )133.5 cm1 [13] (all electrons correlated) and )132.8 cm1 [26] (four frozen core orbitals), corresponding to a 45° canted parallel structure of the global minimum. It is to be noted that the structure is an analogy to the C1 arrangement of the CO–N2 complex. Compared with the above given numbers, the CCSD(T) interaction energy ()118.2 cm1 ) for the Ti3 configuration of CO–N2 (cf. Table 2) is noticeably lower. We conclude that our supermolecule CCSD(T) and MP4 calculations support the notion of considerable nonrigidity of the CO–N2 complex, associated with multiple, nearly equally deep minima on the potential energy surface, separated by low barriers of about cm1 height. This feature, experimentally recognized earlier [4–7], is more pronounced than that in the case of ðCOÞ2 [27–33]. Our ab initio computations predict two distorted T3 structures of CO–N2 to be the most stable configurations. A comparison with recent theoretical results on ðCOÞ2 [13] suggests that the floppy CO–N2 complex is less strongly bound than the CO dimer.
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Acknowledgements This work was supported by the Grant Agency of the Czech Republic (Grants No.’s 203/00/0600 and MSM 113100001). The time allocation in the Czech Academic Supercomputer Centre is greatly acknowledged.
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Chemical Physics Letters 360 (2002) 565–572 www.elsevier.com/locate/cplett
An ab initio study of the CO–N2 complex J. Fiser a
a,*
, R. Pol ak
b
Department of Physical and Macromolecular Chemistry, Faculty of Science, Charles University, 128 40 Prague 2, Czech Republic b J. Heyrovsk y Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, 182 23 Prague 8, Czech Republic Received 12 March 2002; in final form 22 May 2002
Abstract The interaction energy and van der Waals intermolecule bond length of several structures of the CO–N2 complex are calculated by the supermolecule CCSD(T) and MP4 methods using aug-cc-pVXZ (X ¼ D,T,Q) basis sets extended by a set of midbond functions centered in the middle of the vdW bond. The most stable structures are found to be two distorted T-shaped configurations with the N atom pointing towards the C–O bond. This conclusion is compatible with the results of high-resolution infrared, microwave and millimeter studies. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Besides both diatomic subunits of the CO–N2 van der Waals (vdW) complex being important atmospheric constituents, the vdW species itself is closely related to the isoelectronic CO and N2 dimers which are longstanding subjects of experimental and theoretical studies of weak intermolecular interactions. CO–N2 is formed by a barely polar and a nonpolar molecule similarly to, e.g., CO–H2 [1] and CO–D2 [2,3] whose spectra manifest large-amplitude motion. The CO–N2 complex was first detected spectroscopically in the infrared (IR) region of the CO stretching vibration in 1996 [4,5]. The observations of Kawashima and Nishizawa [4] indicate that the orientation of the CO subunit relative to the N–N
*
Corresponding author. Fax: +4202-2491-9752. E-mail address: fi[email protected] (J. Fiser).
bond is nearly perpendicular. However, it was difficult to determine the angle between the C–O and N–N bonds unambigously from their IR data. Based on a high resolution IR study, Xu and McKellar [5] assigned four connected subbands to K ¼ 0, 1, and 2 states in CO–orthoN2 , and one unconnected subband to a K ¼ 1 state in CO– paraN2 (the quantum number K refers to the projection of the angular momentum of the N2 or CO subunit, respectively, onto the intermolecular axis). The authors found the effective intermolecule separation for the ground state of CO–N2 to [5] and explained the simplicity of the be 4.025 A IR spectra by postulating that the N2 subunit undergoes a relatively free internal rotation within the complex. This assumption was confirmed by recent IR [6], microwave [7,8] and millimeter wave studies [7] for various CO–N2 isotopomers. With the aid of the analysis of the 14 N quadrupole hyperfine structure, the ro-vibrational and rotational spectra provide important information about the
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CO–N2 interactions. These studies revealed that the structure of the CO–N2 complex is nearly Tshaped, being very sensitive to both the 14 N nuclear spin modification and rotational state [6–8]. There exists quite a number of theoretical treatments of the CO–CO [9–14] and N2 –N2 interactions [15–19]. Van der Pol et al. [9] represented the CO–CO potential as a sum of the firstorder exchange-repulsion term, and damped multipole expansions for the second-order exchange and penetration phenomena. Their calculation predicted a T-shaped structure for the global minimum and several local minima differing slightly in energy, separated by very low barriers [9]. Meredith and Stone [11] improved this potential by extending the multipole expansion of the dispersion energy. Unfortunately, both potentials failed to explain the observed transition in the CO dimer spectrum recorded by Vanden Bout [21]. Rode et al. [12] calculated the CO–CO interaction energies using the fourth-order Møller–Plesset perturbation theory (MP4), the coupled cluster method restricted to single, double and connected triple excitations (CCSD(T)), and the symmetryadapted perturbation theory (SAPT). The authors explained the relatively large differences between the MP4 and CCSD(T) results by means of a diagrammatic analysis of correlation effects, supported by an evaluation of the fifth-order contributions to the electrostatic interaction energy using a 6s4p3d1f basis set. They pointed out that the CCSD(T) method is not sufficiently reliable for ðCOÞ2 , since it lacks certain important fifth-order contributions. Similar conclusions partly apply also to the N2 dimer [12] where, however, the MP4 results are in better agreement with the SAPT ones. On the contrary, Pedersen et al. [13] found several facts in favor of the CCSD(T) model even in case of ðCOÞ2 . They claim that the supermolecule CCSD(T) treatment is sufficient for calculating the interaction energy for the CO dimer, though they do not explicitly show that the fifth-order correlation terms become negligible in their large aug-cc-pVXZ (X ¼ T,Q) basis sets extended by midbond functions. Contrasting the number of theoretical studies on the CO and N2 dimers [9–19], there is very little theoretical work on CO–N2 . The molecular me-
chanics for clusters (MMC) approach of Franken and Dykstra [20] applied to a whole list of clusters suggested two T-shaped conformers for CO–N2 . The aim of this Letter is to calculate interaction energies of the CO–N2 complex using the CCSD(T) and MP4 methods with large augmented correlation-consistent basis sets. The results are expected to provide information on the basic properties of the potential energy surface needed in further possible investigations of the CO–N2 species, and to shed some light on the justification and the usefulness of the supermolecule CCSD(T) approach in ab initio studies of the CO–N2 , CO– CO and N2 –N2 vdW complexes.
2. Computational details Keeping the CO and N2 bond lengths fixed at their experimental monomer values (2.132 and 2.074 a0 [22], respectively), the CO–N2 potential energy surface depends on four coordinates, cf. Fig. 1: the distance between the centers of mass of the monomers, R Rc:m: ; and the angles hN ; hC and /. hN and hC are angles between the vector R pointing from the center of mass of N2 to that of CO, and the vectors rNN and rCO pointing from N1 to N2 and from C to O in the subunits N2 and CO, respectively, varying from 0° to 180°. / is the dihedral angle between the two plaines defined by the vectors R, rNN , and R, rCO . Because of symmetry, / may be restricted to the range between 0° and 180°. Considering the previous results [9–20] on the isoelectronic family of the vdW dimers, and with respect to our preliminary MP2, MP4 and CCSD(T) computations, we calculated the interaction energies for 11 selected structures of CO–N2 by the CCSD(T) method as a function of R. In all calculations we used the aug-cc-pVXZ (X ¼ D,T,Q) basis sets (denoted avxz) with four frozen core orbitals. Since the use of midbond functions contributes to the efficiency of the basis set for vdW molecules [13,23], we augmented the avxz sets with the set of 3s3p2d1f1g midbond functions (denoted MB) centered in the middle of the vdW bond. The exponents of midbond functions, originally devised for the CO dimer, were taken from [13]: they are 0.9, 0.3 and 0.1 for the s and p
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Fig. 1. Coordinates describing the CO–N2 complex.
functions, 0.6 and 0.2 for the d and 0.3 for the f and g functions. The distance R was varied in the vicinity of a minimum with a step size of 0.1 a0 . For comparison, we also report the results of the MP4/avxz and MP4/avxz + MB (x ¼ t,q) calculations for the two lowest energy structures together with the two other T-shaped configurations. The basis set superposition error was corrected using the counterpoise method of Boys and Bernardi [24]. All calculations were carried out with the MO L P R O 2000.1 program package [25].
3. Results and discussion The interaction energies and vdW bond lengths for different structures, three T-shaped, seven canted (C-shaped) and one X-shaped structure, and various basis sets are summarized in Table 1. Leaving aside the crossed (X) and C7 geometries,
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which are significantly less stable than the other arrangements, it is seen from the table that for all remaining nine structures the interaction energies calculated using the avtz(+MB) basis set lie within a very narrow interval of around 8 cm1 . This value is much lower than the range (31 cm1 ) of interaction energies for the characteristic points on the ðCOÞ2 potential energy surface [13] obtained at nearly the same level of accuracy. In Table 1, we also present the CCSD(T)/avtz data for the structures obtained by the shift of the end point of R from the center of mass of CO to the center of the C–O bond in a given configuration. Such a displacement changes the interaction energy and the corresponding vdW bond length R Rb , defined as the distance between the centers of the N–N and C–O bonds, only very slightly (at most by 1.3 cm1 and 0.1 a0 , respectively). In supermolecule studies of intermolecule potentials of vdW molecules some authors use fixed optimized monomer bond lengths (e.g., in [17,18]) instead of experimental ones. Table 1 shows that the corresponding deviation from interaction energies calculated with experimental bond lengths is lower than 1.2 cm1 for all configurations studied. The CCSD(T) and MP4 calculations using the largest avqz and avqz + MB basis sets were performed only for four structures C1 ; T1 ; T2 and T3 with geometries optimized within the avtz(+MB) calculations. Similarly to the CO dimer [13], the CCSD(T)/ avxz series converges to the complete basis set limit from above and for avxz + MB from below. An analogous conclusion applies also for the MP4 interaction energies, though these values are higher than the CCSD(T) ones by about 7–13 cm1 , i.e, they amount to 10% of the interaction energy. Because the disparity between the results of both methods was found to be more significant (about 25% of the interaction energy) for the CO dimer [13], one may tentatively conclude that the supermolecule CCSD(T) method for calculating vdW interactions [12–14] puts up a better performance in the case of CO–N2 compared to CO–CO. With respect to very small differences between the energies of various structures calculated within a given approximation, we did not perform the usual next step after calculating the ab initio points
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Table 1 Interaction energies (in cm1 , the first row) and equilibrium vdW bond lengths (in bohr, the second row) for the selected structures of the CO–N2 complex Basis set
T1 hN ¼ 90° hC ¼ 0° / ¼ 0°
T2 90° 180° 0°
T3 0° 90° 0°
C1 45° 45° 0°
C2 45° 135° 180°
C3 60° 150° 180°
C4 60° 30° 0°
C5 75° 165° 180°
C6 75° 15° 0°
C7 30° 45° 0°
X 90° 90° 90°
CCSD(T)
avdza
)87.9 8.5 )116.4 8.3 )98.7 8.4 )98.7 8.3 )97.5 8.4 )109.1 8.3 )103.3 8.4 )107.5 8.3
)79.5 7.6 )110.7 7.4 )97.8 7.5 )98.0 7.6 )98.7 7.5 )107.6 7.4 )101.7 7.4 )106.8 7.4
)90.0 8.0 )121.3 7.8 )104.1 7.8 )105.3 7.8 )103.6 7.9 )114.2 7.8 )109.5 7.8 )113.4 7.8
)91.8 8.3 )120.6 8.0 )103.8 8.1 )103.2 8.1 )103.4 8.1 )113.8 8.0 )109.0 8.1 )113.3 8.0
)81.2 7.7 )111.3 7.5 )96.7 7.6 )97.6 7.7 )97.2 7.6 )106.3 7.5
)80.8 7.6 )111.9 7.5 )97.8 7.5 )98.2 7.6 )98.4 7.5 )107.5 7.4
)89.2 8.4 )117.5 8.2 )100.6 8.3 )100.3 8.2 )99.9 8.3 )110.7 8.2
)80.0 7.6 )111.0 7.4 )98.1 7.5 )98.3 7.6 )98.8 7.5 )108.0 7.4
)88.1 8.5 )116.4 8.3 )99.1 8.4 )98.9 8.3 )98.1 8.4 )109.3 8.3
)82.9 8.5 )107.8 8.3 )93.9 8.4 )91.9 8.3 )94.2 8.4 )102.3 8.3
)59.0 7.2 )94.2 6.9 )78.2 7.1 )77.9 7.1 )78.9 7.1 )89.5 7.0
)106.2 8.4 )117.0 8.3 )110.7 8.4 )115.4 8.3
)109.7 7.4 )119.4 7.3 )113.4 7.4 )118.1 7.3
)116.0 7.8 )127.0 7.7 )121.4 7.8 )125.9 7.7
)114.5 8.1 )125.2 8.0 )119.6 8.1 )124.7 8.0
avdz + MBa avtza avtzb avtzc avtz + MBa avqza; d avqz + MBa ;d MP4
avtza avtz + MBa avqza; d avqz + MBa ;d
a
The monomer bond lengths are kept fixed at their experimental values. R is the distance between the centres of mass of the monomers. The monomer bond lengths are kept fixed at their experimental values. R joins the centres of bonds of the monomers. c The monomer bond lengths are optimized at the avtz level. R is the distance between the centres of mass of the monomers. d The value of R is taken from the corresponding optimized avtz data. b
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Method
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on the potential energy surface, i.e., their fitting to an analytical expression, which would be rather difficult and time demanding. Instead, we confined ourselves only to an analysis of the two lowest energy configurations, i.e., T3 and C1 on the CCSD(T) and MP4 interaction energy surfaces. This procedure consisted in additional calculations in the vicinity of the presumed minima in both the avtz and avtz + MB basis sets by using step size of 2°; 2° and 5° for hN ; hC and /, respectively, and 0.1 a0 for the distance R. In the collection of investigated points also the structures obtained by shifting the end point of the vector R towards the C or O atom were calculated with the same basis sets and step size. The minimum on the interaction energy surface was determined from two- or threedimensional fits related to various coordinates. Finally, the CCSD(T) and MP4 interaction ener-
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gies in the avqz(+MB) basis sets were calculated for the lowest energy arrangements obtained in the previous step. Optimizing hC and R in the C1 and T3 configurations (with other coordinates fixed) leads to the roughly 45° canted parallel (Cd1 ) and distorted T3 (denoted Td3 ) arrangements, respectively, whose interaction energies (DE) and geometric parameters are listed in Table 2. A comparison of these values with those for the C1 and T3 configurations presented in Table 1 indicates that a small distortion from C1 ðDhC ¼ hC hdC 4°, cf. Table 2) increases the magnitude of DE for the Cd1 geometry only slightly (at most 1.5 cm1 ). The pertinent increase of DE for Td3 is larger (3–4 cm1 , DhC 6° relative to T3 , C closer to N). It is noteworthy that these findings are almost independent of the method and basis set used. The displacement of the end point of R to
Table 2 Interaction energies DE (in cm1 ) and geometric parameters (in bohr or degrees) for the three most stable structures, distorted C1 and two T3 configurations of the CO–N2 complex; / ¼ 0° in all cases Structurea
Method
Basis set
DE
R
rC b
hdN
hdC
Cd1
CCSD(T)
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)104.6 )115.2 )110.0 )114.8 )115.6 )126.2 )120.9 )125.8
8.0 7.9 8.0 7.9 7.9 7.9 7.9 7.9
1.218 1.218 1.218 1.218 1.066 1.218 1.066 1.218
45 45 45 45 45 45 45 45
48 50 48 50 50 50 50 50
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)107.5 )117.9 )112.8 )117.4 )119.3 )130.1 )125.2 )129.4
7.9 7.8 7.9 7.8 7.8 7.7 7.8 7.7
1.066 1.066 1.066 1.066 1.218 1.066 1.218 1.066
0 0 0 0 0 0 0 0
84 84 84 84 83 84 83 84
avtz avtz + MB avqzc avqz + MBc avtz avtz + MB avqzc avqz + MBc
)108.3 )118.5 )113.7 )118.2 )120.6 )131.0 )126.4 )130.6
7.9 7.8 7.9 7.8 7.8 7.8 7.8 7.8
1.066 1.066 1.066 1.066 1.066 1.066 1.066 1.066
20 20 20 20 18 20 18 20
70 70 70 70 72 70 72 70
MP4
Td3
CCSD(T)
MP4
Ti3
CCSD(T)
MP4
a
The monomer bond lengths are kept fixed at their experimental values. rC is the distance of the end point of the vector R from the C atom. c rC is taken from the corresponding optimized avtz(+MB) data. b
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the center of the C–O bond in the Cd1 structure requires only 0:6 cm1 ; moreover, the Td3 structure characterized with the ‘leg’ pointing at the center of the C–O bond is stable with respect to a small variation of the angle hN . It is of general interest to show the effect of the BSSE correction on the interaction energy for the given cluster. This information is provided by Fig. 2 displaying contour plots of the corrected and uncorrected interaction energy surface. It is seen from this figure that the corresponding minima differ both in the vdW bond length (DR 0:2 a.u.) and in their depths; the BSSE correction amounts to 49% of the interaction energy.
Fig. 2. CCSD(T)/avtz contour plots of the interaction energy surface of CO–N2 (without (a) and with (b) the BSSE correction) as functions of the vdW bond length R and the angle hC between the vdW and CO bond. The contours are separated by 5 cm1 .
Another type of a distortion from the T3 configuration was also investigated, namely a concerted internal rotation of the two subunits which could be regarded as an interconversion between the T3 and C1 conformers. Both the CCSD(T) and MP4 methods using the avqz basis set suggest that the corresponding structure (denoted Ti3 ), with DhC ¼ jDhN j ¼ 20° relative to the T3 conformer, is by about 1 cm1 lower in energy than the Td3 structure (cf. Table 2), and the barrier for the transition from Td3 to Ti3 located at the point (R ¼ 7:9 a0 ; hdN ¼ 6°; hdC ¼ 84°; / ¼ 0°; rC ¼ rO ¼ 1:066 a0 ) is very low ( 1 cm1 ). It is worth mentioning that the BSSE correction for the Ti3 configuration calculated by the both methods (with the avqz basis set) is nearly the same and amounts to 17% of the interaction energy, while in using the avtz basis it is markedly higher (41%). The gross features of the CO–N2 structure emerging from our ab initio calculations are in agreement with observations based on the IR study [4], suggesting CO–N2 to have a nearly T-shaped structure with N–N pointing at the C–O bond. Comparison of our results with microwave and millimeter wave studies of CO–N2 [7,8], taking into account hyperfine interactions, further highlights certain aspects of topological behavior of the potential energy surface. FTMW data [7] established that the CO–orthoN2 complex in its ground state (K ¼ 0) has an approximate T-shaped O-bonded structure (a distorted T2 structure in our notation), with N2 undergoing a nearly free internal rotation. In contrast, a lower K ¼ 1 CO–orthoN2 state has the T3 arrangement with the N2 subunit somewhat more localized than in the K ¼ 0 state. The spacing between the K ¼ 0 and K ¼ 1 levels of the CO–orthoN2 complex was determined to be 3:53 cm1 [5,7]. From the three CO–paraN2 states studied in Ref. [8], the lowest state, corresponding to the K ¼ 0 levels of the first excited vdW vibrational N2 bending state, has approximately a T3 arrangement. The ground state of CO– paraN2 is assessed to lie about 3–4 cm1 above the ground (K ¼ 0) state of CO–orthoN2 [5]. Xu and J€ager [8] determined the vdW bond lengths for the two lowest CO–N2 states having a T3 structure (i.e., K ¼ 0; mb;N2 ¼ 1; paraN2 and K ¼ 1; orthoN2 ) to be 7.918 and 7.874 a0 , respectively. Table 2 shows that
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the CCSD(T)/avqz(+MB) estimates, RðTd3 Þ ¼ RðTi3 Þ ¼ 7:9 ð7:8Þa0 , are in very good agreement with the spectroscopic data. The vdW bond length for the T-shaped O-bonded configuration of the ground (K ¼ 0) state of CO–orthoN2 was spectroscopically determined to be slightly shorter, i.e., 7.608 a0 [5,8]. Our results identifying the salient features of the interaction in N2 –OC differ from those obtained by the MMC approach of Franken and Dykstra [20] in two points. Their calculation favors the N2 – OC conformer (distorted T2 , O pointing at N–N, R ¼ 7:15 a0 ) to the N2 –CO structure (distorted T1 , C pointing at N–N, R ¼ 8:3 a0 ). Further, the MMC model predicts a significant discrepancy in stability of the T1 and T2 conformers (61 cm1 ), while our CCSD(T) and MP4 calculations render only a small deviation (0.7 and 2.7 cm1 , respectively), cf. Table 1. A special attention deserves the difference in stability between the ðCOÞ2 and CO–N2 complexes. The CCSD(T)/avqz + MB calculations of Pedersen et al. [13], and our results obtained with a similar model [26], lead to the ðCOÞ2 interaction energy of )133.5 cm1 [13] (all electrons correlated) and )132.8 cm1 [26] (four frozen core orbitals), corresponding to a 45° canted parallel structure of the global minimum. It is to be noted that the structure is an analogy to the C1 arrangement of the CO–N2 complex. Compared with the above given numbers, the CCSD(T) interaction energy ()118.2 cm1 ) for the Ti3 configuration of CO–N2 (cf. Table 2) is noticeably lower. We conclude that our supermolecule CCSD(T) and MP4 calculations support the notion of considerable nonrigidity of the CO–N2 complex, associated with multiple, nearly equally deep minima on the potential energy surface, separated by low barriers of about cm1 height. This feature, experimentally recognized earlier [4–7], is more pronounced than that in the case of ðCOÞ2 [27–33]. Our ab initio computations predict two distorted T3 structures of CO–N2 to be the most stable configurations. A comparison with recent theoretical results on ðCOÞ2 [13] suggests that the floppy CO–N2 complex is less strongly bound than the CO dimer.
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Acknowledgements This work was supported by the Grant Agency of the Czech Republic (Grants No.’s 203/00/0600 and MSM 113100001). The time allocation in the Czech Academic Supercomputer Centre is greatly acknowledged.
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