- Email: [email protected]
Preliminary calculations on the Na–N2 complex
Chemical Physics Letters 392 (2004) 187–191 www.elsevier.com/locate/cplett
Preliminary calculations on the Na–N2 complex Edmond P.F. Lee
a,b
, Timothy G. Wright
c,*
a
b
School of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, UK Department of Applied Biology and Chemical Technology, Hong Kong Polytechnic University, Hung Hom, Hong Kong c Department of Chemistry, University of Sussex, Falmer, Brighton BN1 9QJ, UK Received 28 April 2004; in final form 13 May 2004 Available online 9 June 2004
Abstract High-level, RCCSD(T), calculations are performed on the molecular complex formed between a Na(2 S) atom and a N2 (X1 Rþ g) molecule, using large basis sets. The complex is found to have a linear global minimum, with a De value of only 24 cm1 . The zeropoint energy is estimated to be around 16 cm1 , suggesting that this is a very floppy complex. In addition, a T-shaped saddle-point lies only 7.5 cm1 above the potential energy minimum. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The Na–N2 complex is an example of a collision pair, and the optical excitation of this species has been used in the study of the collision dynamics between Na and N2 (see, for example, the recent [1]). The interpretation of such experiments is benefited by accurate potential surfaces for both states involved in the excitation; the upper state is often one of the surfaces associated with the Na(2 P) + N2 (X1 Rþ g ) asymptote. In previous work [2], we have tackled the Naþ –N2 cationic complex, and have noted that the sodium basis set must be selected to be able to describe the 2s and 2p orbitals of Naþ . We note that for the neutral complex, the emphasis is more on being able to describe the expected weak, long-range interactions, and as such both the basis set and the detailed treatment of electron correlation must be considered: particularly the role of the core electrons. We note that Habitz [3] has calculated surface cross-sections for the Na–N2 species, for both the ground and excited states. These calculations were performed using a Hartree–Fock method, which had been modified to include correlation approximately by additions to the one-electron operator; however, no minimum was obtained. *
Corresponding author. Fax: +44-1273-677196. E-mail addresses: [email protected] (E.P.F. Lee), t.g.wright@ sussex.ac.uk (T.G. Wright). 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.05.060
More recently, Jungen et al. [4] (in work closely related to that of [1]) reported the results of CEPA-0 calculations on both the ground and excited states of Na–N2 – these results have also been used recently in further studies of collisions of Na [5]. In the present work, we report the results of our investigations into the levels of theory that are required to be able to describe the ground state Na–N2 complex accurately. Our aims were: (i) to identify whether there is a minimum on the surface; and (ii) to determine an accurate value of the binding energy, if such a minimum exists.
2. Calculational details Calculations were performed on both the linear NaNN and the C2v T-shaped orientations. Optimized geometries and harmonic vibrational frequencies were obtained using the Gaussian suite of programs [6], and MP2, QCISD and CCSD(T) methods were employed. Analytical gradients were used for the MP2 and QCISD optimizations; analytic second derivatives were employed to obtain MP2 vibrational frequencies. All other calculations employed numerical methods. The Gaussian calculations all employed unrestricted electronic wavefunctions, but spin contamination was minimal, with hS 2 i values being extremely close to 0.75, as
188
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
expected for this weakly interacting species. In addition, RCCSD(T) numerical optimizations were performed using MO L P R O [7]. A total of eight basis sets were employed, and these are described below, together with the orbitals which were frozen in the correlation treatments. The logic of these basis sets is that the size of the basis set is systematically increased from triple-f valence (6-311+G) through to quatruple-f valence, whilst exploring the effect of including the 1s electrons in the correlation treatment. The effect of freezing the 2s and 2p electrons is briefly explored. In the cases when the 1s orbitals are correlated, tight functions have been added into the basis set in order to describe these orbitals in the correlation treatment. Whilst the basis sets for N are standard, the ones for Na have been built up from the standard cc-pVXZ basis sets (X ¼ T, Q and 5), and adding core and diffuse functions, as required, to match the N basis set. The standard basis sets were obtained from the Gaussian basis set order form (GBSOF) [8]. We emphasise that the demands on a basis set for a neutral sodium atom are different to those on a basis set for Naþ , and so the basis sets used in the present work are different to those used in our previous work (detailed in [9]). Basis 1: Standard 6-311+G(2d) basis set for both Na and N. The three 1s orbitals were kept frozen. The number of basis functions was 89. Basis 2: For Na, the 6-311+G basis set was employed, which was augmented with four d functions, with f ¼ 2:5, 0.5, 0.1 and 0.02. The standard 6-311+G(3d) basis set was employed for N. The three 1s orbitals were kept frozen. The number of basis functions was 109. Basis 3: For Na, the 6-311+G basis set was augmented with additional core s and p functions to help describe the correlation of the 1s, 2s and 2p orbitals: fs ¼ fp ¼ 40:0 and 13.0, two further p orbitals with f ¼ 3:0 and 1.0 and four d orbitals with f ¼ 25:0, 2.0, 0.05 and 0.01. For N, the standard 6-311+G(2d) basis set was augmented with tight functions: two s (f ¼ 16:201 and 5.952), two p (f ¼ 44:489 and 11.871) and one d (f ¼ 14:2). The tighter functions were included to describe better the core electrons, and in this case all electrons were correlated. The tight functions for N were taken from the standard cc-pCVTZ basis set. The number of basis functions was 139. Basis 4: For Na, the standard cc-pVTZ basis set was augmented with a number of tight functions, designed to help describe the correlation of the 1s, 2s and 2p orbitals: three s (f ¼ 40:0, 10.0 and 1.362), three p (f ¼ 25:0, 3.0 and 0.75), two d (f ¼ 25:0 and 1.25) and two f (f ¼ 25:0 and 1.25); in addition, a set of diffuse spdf functions were added, with fs ¼ 0:00794, fp ¼ 0:005, fd ¼ 0:0159 and ff ¼ 0:035. For N, the standard augcc-pCVTZ basis set was employed. All electrons were
correlated when this basis set was employed. The number of basis functions was 204. We also looked at the effect of freezing the 2s and 2p electrons, and denote this by 4(val), to indicate that only the valence electrons are correlated. Basis 5: For Na, the standard cc-pVQZ basis set was augmented with a number of tight functions, designed to help describe the correlation of the 2s and 2p orbitals: two s (f ¼ 1:5 and 0.6), two p (f ¼ 2:5 and 0.625), two d (f ¼ 2:0 and 0.8), one f (f ¼ 1:5) and one g (f ¼ 1:5). In addition diffuse spdfg functions were added with fs ¼ 0:006448, fp ¼ 0:0044732, fd ¼ 0:02435; ff ¼ 0:0518 and fg ¼ 0:06888. The standard aug-cc-pVQZ basis set was employed for N. All electrons except the 1s electrons were correlated. The number of basis functions was 238. Basis 6: For Na, the standard cc-pVQZ basis set was employed, together with tight functions to help describe the correlation of the 1s, 2s and 2p orbitals: four s (f ¼ 35:0, 15.0, 1.5 and 0.6), three p (f ¼ 20:0, 2.5 and 0.625), three d (f ¼ 20:0, 2.0 and 0.8), two f (f ¼ 20:0 and 1.5) and two g (f ¼ 20:0 and 1.5). In addition the same spdfg diffuse functions employed in Basis 5 were added. For N, the standard aug-cc-pCVQZ basis set was employed. All electrons were correlated, and the number of basis functions was 362. Basis 7: For Na, the standard cc-pVQZ basis set was employed, together with the tight functions from Basis 5 to help describe the correlation of the 2s and 2p electrons. In addition, a double augmentation with diffuse functions was made: s (f ¼ 0:006448 and 0.00249792), p (f ¼ 0:0044733 and 0.001491), d (f ¼ 0:02435 and 0.012175), f (f ¼ 0:0518 and 0.0259) and g (f ¼ 0:06888 and 0.0275552). For N, the standard d-aug-cc-pVQZ basis set was employed. The three 1s orbitals were kept frozen in the correlation treatments, and the number of basis functions was 353. Basis 8: For Na, a cc-pV5Z basis set was employed, which was augmented with tight functions to help describe the correlation of the 2s and 2p orbitals : three s (f ¼ 15:0, 1.75 and 0.7), three p (f ¼ 4:5, 1.5 and 0.5), two d (f ¼ 2:0 and 0.8), an f (f ¼ 1:6) a g (f ¼ 1:6) and an h (f ¼ 1:6). In addition, diffuse spdfgh functions were added with fs ¼ 0:00790071, fp ¼ 0:00576124, fd ¼ 0:01, ff ¼ 0:03, fg ¼ 0:05 and fh ¼ 0:05. For N, the standard aug-cc-pV5Z basis set was employed. The three 1s orbitals were kept frozen, and the number of basis function was 434.
3. Results and discussion The results of the geometry optimizations of the linear ground state (X2 Rþ ) and the C2v T-shaped structure with 2 A1 symmetry, are given in Tables 1 and 2, respectively. As may be seen, the geometry obtained is
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
189
Table 1 Optimized geometry and harmonic vibrational frequencies for the linear X2 Rþ structure of Na–N2 Method
Basis set
Bond lengths (A)
Harmonic vibrational frequencies (cm1 )
QCISD MP2 MP2 QCISD CCSD(T) RCCSD(T) RCCSD(T)
1 2 3 2 2 4(val)a 4
NaN ¼ 4.935; NaN ¼ 4.923; NaN ¼ 5.071; NaN ¼ 5.007; NaN ¼ 4.850; NaN ¼ 5.146; NaN ¼ 5.061;
14(r); 14(r); 13(r); 13(r); 16(r);
a
NN ¼ 1.098 NN ¼ 1.116 NN ¼ 1.111 NN ¼ 1.100 NN ¼ 1.106 NN ¼ 1.104 NN ¼ 1.101
28(p); 18(p); 25(p); 19(p); 17(p);
2384(r) 2170(r) 2180(r) 2380(r) 2319(r)
Only the valence electrons are correlated – see text.
Table 2 Optimized geometry and harmonic vibrational frequencies for the T-shaped C2v 2 A1 structure of Na–N2 Method
Basis
Bond lengths (A)
Harmonic vibrational frequencies (cm1 )
Erel (cm1 )a
QCISD QCISD MP2 MP2 CCSD(T) RCCSD(T)
1 2 2 3 2 4
NaN ¼ 5.622; NaN ¼ 5.589; NaN ¼ 5.499; NaN ¼ 5.611; NaN ¼ 5.476; NaN ¼ 5.082;
12 i(b2 ); 10(a1 ); 2383(a1 ) 16 i(b2 ); 11(a1 ); 2380(a1 ) 16 i(b2 ); 12(a1 ); 2170 (a1 ) 8 i(b2 ); 12(a1 ); 2181(a1 ) 31 i(b2 ); 13(a1 ); 2319(a1 )
26.5 13.2 16.5 17.5 15.4 14.4
a
NN ¼ 1.098 NN ¼ 1.100 NN ¼ 1.116 NN ¼ 1.111 NN ¼ 1.106 NN ¼ 1.102
Relative to the corresponding linear structure given in the previous table.
weakly bound, as demonstrated by the long Na–N bond length and the low intermolecular vibrational frequencies; the low intermolecular vibrational frequencies also indicate that the surface is flat both along the Na N2 intermolecular stretch coordinate, and along the intermolecular bending coordinate. It comes, therefore, as little surprise to find that the T-shaped saddle point lies very close in energy above the linear minimum, with a very low imaginary bending vibration. Note that when the 2s and 2p electrons on Na are frozen, then there is no electron correlation of the Na electrons. We find that the NaN bond length increases compared to when these electrons are by around 0.1 A active. This suggests that the polarization of the 2s and 2p electrons of Na is of some importance in the interaction, but around the minimum, this effect is not large (although at shorter interaction energies this effect is likely to increase). As noted above, Jungen and colleagues [1,4,5] have performed a number of calculations on Na–N2 employing the CEPA method, those calculations did not
correct for basis set superposition error and employed a modified aug-cc-pVTZ basis set. The results suggest [10] a linear geometry with the NaN bond length being which is a little longer than the value obtained 5.8 A, in the present work. Table 3 gives the binding energy obtained using the RCCSD and RCCSD(T) methods, and using the larger basis sets, and employing the RCCSD(T)/4 optimized linear geometry. It is clear that there is consistency within the RCCSD and the RCCSD(T) methods, separately, but that triples have a significant effect upon the binding energy. It is also noteworthy that, despite the different BSSEs for the basis sets, the counterpoise-corrected binding energy is consistent. The weakness of the interaction means that the BSSE is inevitably a significant percentage of the binding energy, but with the largest basis set, the BSSE has been reduced to ca. 30%, and is only 8.3 cm1 , which is the equivalent of only 0.3 cm1 per non-core electron. With the consistency achieved for the binding energy calculated with basis sets 5–8 at the RCCSD(T) level, we are confident that the value of 24 cm1 is reliable.
Table 3 Calculated binding energies and BSSEs (cm1 ) at the RCCSD and RCCSD(T) levels of theory, employing the larger basis sets, and the RCCSD(T)/4optimized geometrya Basis set
RCCSD b
DEe (CP) BSSE(N2 ) BSSE(Na) BSSEtotal a b
RCCSD(T)
4(val)
4
5
6
7
8
4(val)b
4
5
6
7
8
12.9 14.4 0.2 14.6
13.0 11.0 17.5 28.5
15.4 6.6 17.5 24.1
15.2 5.0 5.2 10.2
16.0 8.9 36.4 45.3
15.8 1.7 6.3 8.0
20.2 0.2 15.2 15.4
21.2 11.6 18.7 30.4
23.9 7.1 17.7 24.8
23.9 5.2 5.5 10.7
24.6 9.4 37.1 46.5
24.3 1.9 6.4 8.3
Ee {RCSD(T)/6 } ¼ )271.67984021 Eh ; Ee {RCCSD(T)/7} ¼ )271.49790001 Eh ; Ee {RCCSD(T)/8 } ¼ )271.52306393 Eh . Only the valence electrons are correlated – see text.
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Similarly, the T-shaped structure is a saddle point at all levels of theory; consistently calculated to lie very close in energy above the linear minimum. The energy gap between the linear and the T-shape is calculated to be 14 cm1 at the RCCSD(T)/4 level, and this decreases to 9.5 cm1 at the RCCSD(T)/8//RCCSD(T)/4 level of theory, and then to 7.5 cm1 once the full counterpoise correction has been applied. Although the intermolecular stretch is likely to be very anharmonic, it seems unlikely that the potential energy surface is able to support more than one quantum of the stretch above the zero-point level. In addition, given the barrier to the T-shaped structure of 7.5 cm1 , and a bending vibration of 20 cm1 , then the zero-point energy suggests that N2 is likely to be close to freely rotating in the Na–N2 complex. Given the weakness of the binding of the Na–N2 complex, it seems that it will have a low concentration in most systems, with probably a low-pressure environment or molecular beam being required in order to stabilize it. It is plausible that it will be formed in the upper atmosphere, but given that it requires a threebody collision to form, and given that it is so weakly bound, the steady-state concentration is likely to be very small. Also, its possible formation by electron–ion recombination Naþ –N2 þ e ! Na–N2
ð1Þ
is not likely to yield much of the neutral product, since the excess energy from the neutralization process is likely to lead to dissociation of the product; potentially a three-body collision may stabilize the product, but in low-pressure regions these will be infrequent. Another possibility would involve a higher cationic complex, such as Na(N2 )þ 2 , where one of the ligands could take some of the excess energy away, leading to stabilization of the product complex þ
Naþ ðN2 Þ2 þ e ! Na–N2 þ N2
may provide a source of Naþ (via Naþ –L) at lower altitudes than uncomplexed Na would do. Recently, the involvement of Naþ –ligand complexes has been explored in some detail as a route to sporadic sodium layers [12,13], and within the model employed, the requirement to estimate the electron recombination rate constants for reactions such as Naþ –L þ e ! Na þ L
ð3Þ
was notable. The dynamics of the recombination process will be dictated by both the ionic, Naþ –L, and the neutral, Na–L, surfaces, and the overlaps between these. For Naþ –N2 we have calculated a potential energy surface using high-quality ab initio methods already [2], but to our knowledge, no such surface exists for Na–N2 . As noted above, part of the aim of the present work was to identify the position of the potential energy minimum; the position of any saddle point(s); and to explore the likely levels of theory required. The results herein lead us to conclude that the RCCSD(T) method is the one of choice: the interaction is weak, and so the inclusion of the majority of the electron correlation energy is required, but single-reference methods are adequate, at least close to the minimum. We note that Habitz [3] has indicated that an ionic curve arising from Naþ –N 2 may cross the Na–N2 (X2 Rþ ) curve, and in this region, the RCCSD(T) method is likely to fail, and multireference methods would have to explored, if this region were important to a particular problem. Table 3 indicates that a quadruple-f basis set is required, but that the 1s electrons may be frozen in the correlation treatment. Note that if consistency between the Naþ –N2 and the Na–N2 results is required, then the 2s and 2p electrons of NaðþÞ need to be correlated, since these are valence for Naþ . Thus, we conclude that for the neutral species, a calculation of approximately RCCSD(T)/aug-cc-pVQZ quality is required.
ð2Þ
Any Na–N2 so-formed could potentially be ionized by incoming solar radiation. Taking into account the binding energy of Naþ –N2 from [2] of 2770 cm1 , and estimating the zero-point vibrational energy from the vibrational energy levels reported in that work; the binding energy of Na–N2 and estimating the zero-point contribution from the harmonic vibrational frequencies reported herein; and finally, employing the ionization energy of Na as 41449.44 cm1 [11]; then the ionization energy of Na–N2 may be calculated to be 38840 cm1 (4.82 eV). This suggests that any Na–N2 formed may be ionized by 257.5 nm radiation, rather than the 241.3 nm radiation required for Na atoms. Given the fall-off in the intensity of shorter wavelengths as the atmosphere is penetrated by solar radiation, this suggests that Na–N2 (and by implication other Na–L complexes) will be ionized lower in the atmosphere than Na atoms, and so
4. Conclusions High-level ab initio calculations have shown that the Na–N2 complex has a linear minimum, with a binding energy, De of 24 cm1 . A T-shaped saddle-point lies only 7.5 cm1 higher in energy. The harmonic vibrational frequencies suggest that the complex is extremely floppy along the bending direction, probably with the N2 being close to freely rotating. It is suggested that a calculation of approximately RCCSD(T)/aug-cc-pVQZ quality is required if one wishes to calculate an accurate surface for this species. Acknowledgements The authors are grateful to the EPSRC for the award of computer time at the Rutherford Appleton Labora-
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
tories under the auspices of the Computational Chemistry Working Party (CCWP), which enabled these calculations to be performed. E.P.F.L. is grateful to the Research Grant Council (RGC) of the Hong Kong Special Administration Region for support. We also acknowledge correspondence with Prof. Dr. M. Jungen (Basel).
References [1] R. Goldstein, J. Grosser, O. Hoffmann, V. Schumann, D. W€ oßner, M. Jungen, M. Lehner, J. Chem. Phys. 114 (2001) 2144. [2] P. Sold an, V. Spirko, E.P.F. Lee, T.G. Wright, J. Chem. Phys. 111 (1999) 3420. [3] P. Habitz, Chem. Phys. 54 (1980) 131. [4] M. Jungen, Helv. Chim. Acta 84 (2001) 1459. [5] M. Jungen, M. Lehner, R. Guerout, J. Stalder, Phys. Chem. Chem. Phys. 6 (2004) 1666. [6] M.J. Frisch et al., GA U S S I A N 03, Gaussian Inc., Pittburgh, 2003. [7] MO L P R O is a package of ab initio programs written by H.-J. Werner and P.J. Knowles, with contributions from others.
191
[8] EMSL Basis Set Library {Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1/29/01, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, WA 99352, USA, and funded by the US Department of Energy. The Pacific Northwest Laboratory is a multi-program laboratory operated by Battelle Memorial Institute for the US Department of Energy under contract DE-AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information.}. [9] P. Soldan, E.P.F. Lee, T.G. Wright, J. Chem. Soc., Faraday Trans. 94 (1998) 3307. [10] M. Jungen, personal communication. [11] M.W. Chase, Jr., NIST-JANAF Themochemical Tables, fourth edn., J. Phys. Chem. Ref. Data, Monograph 9, 1998. [12] S.C. Collins, J.M.C. Plane, M.C. Kelley, T.G. Wright, P. Soldan, T.J. Kane, A.J. Gerrard, B.W. Grime, R.J. Rollason, J.S. Friedman, S.A. Gonzalez, Q. Zhou, M. Sulzer, J. Atmosph. Solar-Terr. Phys. 64 (2002) 845. [13] S.E. Daire, J.M.C. Plane, S.D. Gamblin, P. Soldan, E.P.F. Lee, T.G. Wright, J. Atmosph. Solar-Terr. Phys. 64 (2002) 863.
Preliminary calculations on the Na–N2 complex Edmond P.F. Lee
a,b
, Timothy G. Wright
c,*
a
b
School of Chemistry, University of Southampton, Highfield, Southampton SO17 1BJ, UK Department of Applied Biology and Chemical Technology, Hong Kong Polytechnic University, Hung Hom, Hong Kong c Department of Chemistry, University of Sussex, Falmer, Brighton BN1 9QJ, UK Received 28 April 2004; in final form 13 May 2004 Available online 9 June 2004
Abstract High-level, RCCSD(T), calculations are performed on the molecular complex formed between a Na(2 S) atom and a N2 (X1 Rþ g) molecule, using large basis sets. The complex is found to have a linear global minimum, with a De value of only 24 cm1 . The zeropoint energy is estimated to be around 16 cm1 , suggesting that this is a very floppy complex. In addition, a T-shaped saddle-point lies only 7.5 cm1 above the potential energy minimum. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The Na–N2 complex is an example of a collision pair, and the optical excitation of this species has been used in the study of the collision dynamics between Na and N2 (see, for example, the recent [1]). The interpretation of such experiments is benefited by accurate potential surfaces for both states involved in the excitation; the upper state is often one of the surfaces associated with the Na(2 P) + N2 (X1 Rþ g ) asymptote. In previous work [2], we have tackled the Naþ –N2 cationic complex, and have noted that the sodium basis set must be selected to be able to describe the 2s and 2p orbitals of Naþ . We note that for the neutral complex, the emphasis is more on being able to describe the expected weak, long-range interactions, and as such both the basis set and the detailed treatment of electron correlation must be considered: particularly the role of the core electrons. We note that Habitz [3] has calculated surface cross-sections for the Na–N2 species, for both the ground and excited states. These calculations were performed using a Hartree–Fock method, which had been modified to include correlation approximately by additions to the one-electron operator; however, no minimum was obtained. *
Corresponding author. Fax: +44-1273-677196. E-mail addresses: [email protected] (E.P.F. Lee), t.g.wright@ sussex.ac.uk (T.G. Wright). 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.05.060
More recently, Jungen et al. [4] (in work closely related to that of [1]) reported the results of CEPA-0 calculations on both the ground and excited states of Na–N2 – these results have also been used recently in further studies of collisions of Na [5]. In the present work, we report the results of our investigations into the levels of theory that are required to be able to describe the ground state Na–N2 complex accurately. Our aims were: (i) to identify whether there is a minimum on the surface; and (ii) to determine an accurate value of the binding energy, if such a minimum exists.
2. Calculational details Calculations were performed on both the linear NaNN and the C2v T-shaped orientations. Optimized geometries and harmonic vibrational frequencies were obtained using the Gaussian suite of programs [6], and MP2, QCISD and CCSD(T) methods were employed. Analytical gradients were used for the MP2 and QCISD optimizations; analytic second derivatives were employed to obtain MP2 vibrational frequencies. All other calculations employed numerical methods. The Gaussian calculations all employed unrestricted electronic wavefunctions, but spin contamination was minimal, with hS 2 i values being extremely close to 0.75, as
188
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
expected for this weakly interacting species. In addition, RCCSD(T) numerical optimizations were performed using MO L P R O [7]. A total of eight basis sets were employed, and these are described below, together with the orbitals which were frozen in the correlation treatments. The logic of these basis sets is that the size of the basis set is systematically increased from triple-f valence (6-311+G) through to quatruple-f valence, whilst exploring the effect of including the 1s electrons in the correlation treatment. The effect of freezing the 2s and 2p electrons is briefly explored. In the cases when the 1s orbitals are correlated, tight functions have been added into the basis set in order to describe these orbitals in the correlation treatment. Whilst the basis sets for N are standard, the ones for Na have been built up from the standard cc-pVXZ basis sets (X ¼ T, Q and 5), and adding core and diffuse functions, as required, to match the N basis set. The standard basis sets were obtained from the Gaussian basis set order form (GBSOF) [8]. We emphasise that the demands on a basis set for a neutral sodium atom are different to those on a basis set for Naþ , and so the basis sets used in the present work are different to those used in our previous work (detailed in [9]). Basis 1: Standard 6-311+G(2d) basis set for both Na and N. The three 1s orbitals were kept frozen. The number of basis functions was 89. Basis 2: For Na, the 6-311+G basis set was employed, which was augmented with four d functions, with f ¼ 2:5, 0.5, 0.1 and 0.02. The standard 6-311+G(3d) basis set was employed for N. The three 1s orbitals were kept frozen. The number of basis functions was 109. Basis 3: For Na, the 6-311+G basis set was augmented with additional core s and p functions to help describe the correlation of the 1s, 2s and 2p orbitals: fs ¼ fp ¼ 40:0 and 13.0, two further p orbitals with f ¼ 3:0 and 1.0 and four d orbitals with f ¼ 25:0, 2.0, 0.05 and 0.01. For N, the standard 6-311+G(2d) basis set was augmented with tight functions: two s (f ¼ 16:201 and 5.952), two p (f ¼ 44:489 and 11.871) and one d (f ¼ 14:2). The tighter functions were included to describe better the core electrons, and in this case all electrons were correlated. The tight functions for N were taken from the standard cc-pCVTZ basis set. The number of basis functions was 139. Basis 4: For Na, the standard cc-pVTZ basis set was augmented with a number of tight functions, designed to help describe the correlation of the 1s, 2s and 2p orbitals: three s (f ¼ 40:0, 10.0 and 1.362), three p (f ¼ 25:0, 3.0 and 0.75), two d (f ¼ 25:0 and 1.25) and two f (f ¼ 25:0 and 1.25); in addition, a set of diffuse spdf functions were added, with fs ¼ 0:00794, fp ¼ 0:005, fd ¼ 0:0159 and ff ¼ 0:035. For N, the standard augcc-pCVTZ basis set was employed. All electrons were
correlated when this basis set was employed. The number of basis functions was 204. We also looked at the effect of freezing the 2s and 2p electrons, and denote this by 4(val), to indicate that only the valence electrons are correlated. Basis 5: For Na, the standard cc-pVQZ basis set was augmented with a number of tight functions, designed to help describe the correlation of the 2s and 2p orbitals: two s (f ¼ 1:5 and 0.6), two p (f ¼ 2:5 and 0.625), two d (f ¼ 2:0 and 0.8), one f (f ¼ 1:5) and one g (f ¼ 1:5). In addition diffuse spdfg functions were added with fs ¼ 0:006448, fp ¼ 0:0044732, fd ¼ 0:02435; ff ¼ 0:0518 and fg ¼ 0:06888. The standard aug-cc-pVQZ basis set was employed for N. All electrons except the 1s electrons were correlated. The number of basis functions was 238. Basis 6: For Na, the standard cc-pVQZ basis set was employed, together with tight functions to help describe the correlation of the 1s, 2s and 2p orbitals: four s (f ¼ 35:0, 15.0, 1.5 and 0.6), three p (f ¼ 20:0, 2.5 and 0.625), three d (f ¼ 20:0, 2.0 and 0.8), two f (f ¼ 20:0 and 1.5) and two g (f ¼ 20:0 and 1.5). In addition the same spdfg diffuse functions employed in Basis 5 were added. For N, the standard aug-cc-pCVQZ basis set was employed. All electrons were correlated, and the number of basis functions was 362. Basis 7: For Na, the standard cc-pVQZ basis set was employed, together with the tight functions from Basis 5 to help describe the correlation of the 2s and 2p electrons. In addition, a double augmentation with diffuse functions was made: s (f ¼ 0:006448 and 0.00249792), p (f ¼ 0:0044733 and 0.001491), d (f ¼ 0:02435 and 0.012175), f (f ¼ 0:0518 and 0.0259) and g (f ¼ 0:06888 and 0.0275552). For N, the standard d-aug-cc-pVQZ basis set was employed. The three 1s orbitals were kept frozen in the correlation treatments, and the number of basis functions was 353. Basis 8: For Na, a cc-pV5Z basis set was employed, which was augmented with tight functions to help describe the correlation of the 2s and 2p orbitals : three s (f ¼ 15:0, 1.75 and 0.7), three p (f ¼ 4:5, 1.5 and 0.5), two d (f ¼ 2:0 and 0.8), an f (f ¼ 1:6) a g (f ¼ 1:6) and an h (f ¼ 1:6). In addition, diffuse spdfgh functions were added with fs ¼ 0:00790071, fp ¼ 0:00576124, fd ¼ 0:01, ff ¼ 0:03, fg ¼ 0:05 and fh ¼ 0:05. For N, the standard aug-cc-pV5Z basis set was employed. The three 1s orbitals were kept frozen, and the number of basis function was 434.
3. Results and discussion The results of the geometry optimizations of the linear ground state (X2 Rþ ) and the C2v T-shaped structure with 2 A1 symmetry, are given in Tables 1 and 2, respectively. As may be seen, the geometry obtained is
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
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Table 1 Optimized geometry and harmonic vibrational frequencies for the linear X2 Rþ structure of Na–N2 Method
Basis set
Bond lengths (A)
Harmonic vibrational frequencies (cm1 )
QCISD MP2 MP2 QCISD CCSD(T) RCCSD(T) RCCSD(T)
1 2 3 2 2 4(val)a 4
NaN ¼ 4.935; NaN ¼ 4.923; NaN ¼ 5.071; NaN ¼ 5.007; NaN ¼ 4.850; NaN ¼ 5.146; NaN ¼ 5.061;
14(r); 14(r); 13(r); 13(r); 16(r);
a
NN ¼ 1.098 NN ¼ 1.116 NN ¼ 1.111 NN ¼ 1.100 NN ¼ 1.106 NN ¼ 1.104 NN ¼ 1.101
28(p); 18(p); 25(p); 19(p); 17(p);
2384(r) 2170(r) 2180(r) 2380(r) 2319(r)
Only the valence electrons are correlated – see text.
Table 2 Optimized geometry and harmonic vibrational frequencies for the T-shaped C2v 2 A1 structure of Na–N2 Method
Basis
Bond lengths (A)
Harmonic vibrational frequencies (cm1 )
Erel (cm1 )a
QCISD QCISD MP2 MP2 CCSD(T) RCCSD(T)
1 2 2 3 2 4
NaN ¼ 5.622; NaN ¼ 5.589; NaN ¼ 5.499; NaN ¼ 5.611; NaN ¼ 5.476; NaN ¼ 5.082;
12 i(b2 ); 10(a1 ); 2383(a1 ) 16 i(b2 ); 11(a1 ); 2380(a1 ) 16 i(b2 ); 12(a1 ); 2170 (a1 ) 8 i(b2 ); 12(a1 ); 2181(a1 ) 31 i(b2 ); 13(a1 ); 2319(a1 )
26.5 13.2 16.5 17.5 15.4 14.4
a
NN ¼ 1.098 NN ¼ 1.100 NN ¼ 1.116 NN ¼ 1.111 NN ¼ 1.106 NN ¼ 1.102
Relative to the corresponding linear structure given in the previous table.
weakly bound, as demonstrated by the long Na–N bond length and the low intermolecular vibrational frequencies; the low intermolecular vibrational frequencies also indicate that the surface is flat both along the Na N2 intermolecular stretch coordinate, and along the intermolecular bending coordinate. It comes, therefore, as little surprise to find that the T-shaped saddle point lies very close in energy above the linear minimum, with a very low imaginary bending vibration. Note that when the 2s and 2p electrons on Na are frozen, then there is no electron correlation of the Na electrons. We find that the NaN bond length increases compared to when these electrons are by around 0.1 A active. This suggests that the polarization of the 2s and 2p electrons of Na is of some importance in the interaction, but around the minimum, this effect is not large (although at shorter interaction energies this effect is likely to increase). As noted above, Jungen and colleagues [1,4,5] have performed a number of calculations on Na–N2 employing the CEPA method, those calculations did not
correct for basis set superposition error and employed a modified aug-cc-pVTZ basis set. The results suggest [10] a linear geometry with the NaN bond length being which is a little longer than the value obtained 5.8 A, in the present work. Table 3 gives the binding energy obtained using the RCCSD and RCCSD(T) methods, and using the larger basis sets, and employing the RCCSD(T)/4 optimized linear geometry. It is clear that there is consistency within the RCCSD and the RCCSD(T) methods, separately, but that triples have a significant effect upon the binding energy. It is also noteworthy that, despite the different BSSEs for the basis sets, the counterpoise-corrected binding energy is consistent. The weakness of the interaction means that the BSSE is inevitably a significant percentage of the binding energy, but with the largest basis set, the BSSE has been reduced to ca. 30%, and is only 8.3 cm1 , which is the equivalent of only 0.3 cm1 per non-core electron. With the consistency achieved for the binding energy calculated with basis sets 5–8 at the RCCSD(T) level, we are confident that the value of 24 cm1 is reliable.
Table 3 Calculated binding energies and BSSEs (cm1 ) at the RCCSD and RCCSD(T) levels of theory, employing the larger basis sets, and the RCCSD(T)/4optimized geometrya Basis set
RCCSD b
DEe (CP) BSSE(N2 ) BSSE(Na) BSSEtotal a b
RCCSD(T)
4(val)
4
5
6
7
8
4(val)b
4
5
6
7
8
12.9 14.4 0.2 14.6
13.0 11.0 17.5 28.5
15.4 6.6 17.5 24.1
15.2 5.0 5.2 10.2
16.0 8.9 36.4 45.3
15.8 1.7 6.3 8.0
20.2 0.2 15.2 15.4
21.2 11.6 18.7 30.4
23.9 7.1 17.7 24.8
23.9 5.2 5.5 10.7
24.6 9.4 37.1 46.5
24.3 1.9 6.4 8.3
Ee {RCSD(T)/6 } ¼ )271.67984021 Eh ; Ee {RCCSD(T)/7} ¼ )271.49790001 Eh ; Ee {RCCSD(T)/8 } ¼ )271.52306393 Eh . Only the valence electrons are correlated – see text.
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Similarly, the T-shaped structure is a saddle point at all levels of theory; consistently calculated to lie very close in energy above the linear minimum. The energy gap between the linear and the T-shape is calculated to be 14 cm1 at the RCCSD(T)/4 level, and this decreases to 9.5 cm1 at the RCCSD(T)/8//RCCSD(T)/4 level of theory, and then to 7.5 cm1 once the full counterpoise correction has been applied. Although the intermolecular stretch is likely to be very anharmonic, it seems unlikely that the potential energy surface is able to support more than one quantum of the stretch above the zero-point level. In addition, given the barrier to the T-shaped structure of 7.5 cm1 , and a bending vibration of 20 cm1 , then the zero-point energy suggests that N2 is likely to be close to freely rotating in the Na–N2 complex. Given the weakness of the binding of the Na–N2 complex, it seems that it will have a low concentration in most systems, with probably a low-pressure environment or molecular beam being required in order to stabilize it. It is plausible that it will be formed in the upper atmosphere, but given that it requires a threebody collision to form, and given that it is so weakly bound, the steady-state concentration is likely to be very small. Also, its possible formation by electron–ion recombination Naþ –N2 þ e ! Na–N2
ð1Þ
is not likely to yield much of the neutral product, since the excess energy from the neutralization process is likely to lead to dissociation of the product; potentially a three-body collision may stabilize the product, but in low-pressure regions these will be infrequent. Another possibility would involve a higher cationic complex, such as Na(N2 )þ 2 , where one of the ligands could take some of the excess energy away, leading to stabilization of the product complex þ
Naþ ðN2 Þ2 þ e ! Na–N2 þ N2
may provide a source of Naþ (via Naþ –L) at lower altitudes than uncomplexed Na would do. Recently, the involvement of Naþ –ligand complexes has been explored in some detail as a route to sporadic sodium layers [12,13], and within the model employed, the requirement to estimate the electron recombination rate constants for reactions such as Naþ –L þ e ! Na þ L
ð3Þ
was notable. The dynamics of the recombination process will be dictated by both the ionic, Naþ –L, and the neutral, Na–L, surfaces, and the overlaps between these. For Naþ –N2 we have calculated a potential energy surface using high-quality ab initio methods already [2], but to our knowledge, no such surface exists for Na–N2 . As noted above, part of the aim of the present work was to identify the position of the potential energy minimum; the position of any saddle point(s); and to explore the likely levels of theory required. The results herein lead us to conclude that the RCCSD(T) method is the one of choice: the interaction is weak, and so the inclusion of the majority of the electron correlation energy is required, but single-reference methods are adequate, at least close to the minimum. We note that Habitz [3] has indicated that an ionic curve arising from Naþ –N 2 may cross the Na–N2 (X2 Rþ ) curve, and in this region, the RCCSD(T) method is likely to fail, and multireference methods would have to explored, if this region were important to a particular problem. Table 3 indicates that a quadruple-f basis set is required, but that the 1s electrons may be frozen in the correlation treatment. Note that if consistency between the Naþ –N2 and the Na–N2 results is required, then the 2s and 2p electrons of NaðþÞ need to be correlated, since these are valence for Naþ . Thus, we conclude that for the neutral species, a calculation of approximately RCCSD(T)/aug-cc-pVQZ quality is required.
ð2Þ
Any Na–N2 so-formed could potentially be ionized by incoming solar radiation. Taking into account the binding energy of Naþ –N2 from [2] of 2770 cm1 , and estimating the zero-point vibrational energy from the vibrational energy levels reported in that work; the binding energy of Na–N2 and estimating the zero-point contribution from the harmonic vibrational frequencies reported herein; and finally, employing the ionization energy of Na as 41449.44 cm1 [11]; then the ionization energy of Na–N2 may be calculated to be 38840 cm1 (4.82 eV). This suggests that any Na–N2 formed may be ionized by 257.5 nm radiation, rather than the 241.3 nm radiation required for Na atoms. Given the fall-off in the intensity of shorter wavelengths as the atmosphere is penetrated by solar radiation, this suggests that Na–N2 (and by implication other Na–L complexes) will be ionized lower in the atmosphere than Na atoms, and so
4. Conclusions High-level ab initio calculations have shown that the Na–N2 complex has a linear minimum, with a binding energy, De of 24 cm1 . A T-shaped saddle-point lies only 7.5 cm1 higher in energy. The harmonic vibrational frequencies suggest that the complex is extremely floppy along the bending direction, probably with the N2 being close to freely rotating. It is suggested that a calculation of approximately RCCSD(T)/aug-cc-pVQZ quality is required if one wishes to calculate an accurate surface for this species. Acknowledgements The authors are grateful to the EPSRC for the award of computer time at the Rutherford Appleton Labora-
E.P.F. Lee, T.G. Wright / Chemical Physics Letters 392 (2004) 187–191
tories under the auspices of the Computational Chemistry Working Party (CCWP), which enabled these calculations to be performed. E.P.F.L. is grateful to the Research Grant Council (RGC) of the Hong Kong Special Administration Region for support. We also acknowledge correspondence with Prof. Dr. M. Jungen (Basel).
References [1] R. Goldstein, J. Grosser, O. Hoffmann, V. Schumann, D. W€ oßner, M. Jungen, M. Lehner, J. Chem. Phys. 114 (2001) 2144. [2] P. Sold an, V. Spirko, E.P.F. Lee, T.G. Wright, J. Chem. Phys. 111 (1999) 3420. [3] P. Habitz, Chem. Phys. 54 (1980) 131. [4] M. Jungen, Helv. Chim. Acta 84 (2001) 1459. [5] M. Jungen, M. Lehner, R. Guerout, J. Stalder, Phys. Chem. Chem. Phys. 6 (2004) 1666. [6] M.J. Frisch et al., GA U S S I A N 03, Gaussian Inc., Pittburgh, 2003. [7] MO L P R O is a package of ab initio programs written by H.-J. Werner and P.J. Knowles, with contributions from others.
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[8] EMSL Basis Set Library {Basis sets were obtained from the Extensible Computational Chemistry Environment Basis Set Database, Version 1/29/01, as developed and distributed by the Molecular Science Computing Facility, Environmental and Molecular Sciences Laboratory which is part of the Pacific Northwest Laboratory, P.O. Box 999, Richland, WA 99352, USA, and funded by the US Department of Energy. The Pacific Northwest Laboratory is a multi-program laboratory operated by Battelle Memorial Institute for the US Department of Energy under contract DE-AC06-76RLO 1830. Contact David Feller or Karen Schuchardt for further information.}. [9] P. Soldan, E.P.F. Lee, T.G. Wright, J. Chem. Soc., Faraday Trans. 94 (1998) 3307. [10] M. Jungen, personal communication. [11] M.W. Chase, Jr., NIST-JANAF Themochemical Tables, fourth edn., J. Phys. Chem. Ref. Data, Monograph 9, 1998. [12] S.C. Collins, J.M.C. Plane, M.C. Kelley, T.G. Wright, P. Soldan, T.J. Kane, A.J. Gerrard, B.W. Grime, R.J. Rollason, J.S. Friedman, S.A. Gonzalez, Q. Zhou, M. Sulzer, J. Atmosph. Solar-Terr. Phys. 64 (2002) 845. [13] S.E. Daire, J.M.C. Plane, S.D. Gamblin, P. Soldan, E.P.F. Lee, T.G. Wright, J. Atmosph. Solar-Terr. Phys. 64 (2002) 863.