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On the influence of confinement effects on electric properties: An ab initio study
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Chemical Physics Letters 449 (2007) 314–318 www.elsevier.com/locate/cplett
On the influence of confinement effects on electric properties: An ab initio study Anna Kaczmarek b c
a,*
, Robert Zales´ny b, Wojciech Bartkowiak
c
a Faculty of Chemistry, Nicolaus Copernicus University, Gagarina 7, 87-100 Torun´, Poland Department of Physical Chemistry, Collegium Medicum, Nicolaus Copernicus University, Kurpin´skiego 5, 85-950 Bydgoszcz, Poland Institute of Physical and Theoretical Chemistry, Technical University of Wrocław, Wyb. Wyspian´skiego 27, 50-370 Wrocław, Poland
Received 15 August 2007; in final form 25 October 2007 Available online 30 October 2007
Abstract Confinement effects on the electric properties of model system are investigated. The system under consideration is 1-cyanoethyne molecule surrounded by helium atoms arranged in the shape of armchair nanotube. Decreasing the diameter of the helium nanotube corresponds to the simulation of the confinement effects on the enclosed system. The decomposition of the interaction energy and interaction-induced electric properties reveals that dispersion terms plays an important role in the interaction between subsystems. It is observed that the absolute values of dipole moment and first-order hyperpolarizability decrease as a function of the tube radius. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction The properties of a confined molecule can differ significantly from those of gas phase and those of bulk. Simultaneously, the weak interaction between the enclosed molecule and the cavity allows to finely tune the characteristics of the host system. Therefore the confinement effects have recently stimulated a considerable interest as a source of information about the properties and applications of different kinds of systems of chemical importance. These include catalytic effects in zeolites, chemical reactivity in carbon nanotubes or porous carbon materials, electric properties in crystal, endohedral fullerene complexes, molecules under the high pressure or the storage properties of the solid media among others [1–8]. The influence of the valence compression on the in-crystal properties according to Fowler can be threefold: the crystal field effects including electrostatic Coulomb interaction between the subsystems, the overlap compression referring to the interaction of the guest molecule with the *
Corresponding author. Fax: +48 56 654 24. E-mail address: [email protected] (A. Kaczmarek).
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.10.085
occupied orbitals of the host and the charge transfer effects involving the interaction of the confined molecule with the virtual orbitals of the cavity [8,9]. All those effects together can have substantial influence on the stability of the system, geometry of the confined molecule, its electrooptical properties, absorption and emission spectra, dielectric constants, hardness and softness, hyperfine coupling, magnetic properties and the reaction mechanisms [8–20]. Hence, numerous contributions considering these phenomena are available in the literature starting from Michels et al. [21] and Sommerfeld and Welker [22] works and including valuable reviews [23]. Nevertheless, in a great majority of those pieces the confining cavity is modeled in terms of the simple model potentials such as harmonic oscillator potential. This approach permits to reproduce a general aspects of the confinement effects, however it appears to be oversimplified. Thus, regarding chemical issues mentioned above one can see the necessity of modeling the guest molecules embedded in less simplified model systems such as helium cages [24] or real chemical species like fullerenes [4,5,20], carbon nanotubes [2,3] or zeolites [7] of the non-uniform charge distribution.
A. Kaczmarek et al. / Chemical Physics Letters 449 (2007) 314–318
The aim of the present, introductory, study is to analyze the impact of the confinement in the nanotube-like cage onto the electric properties of the small model molecule. Since the size of the carbon nanotubes is prohibitive to the high level ab initio calculations necessary to describe the subtle effects in model molecule arising from the weak interaction with the cage, the cavity is represented here by the nanotube-like helium structure. The influence of the decreasing tube radius on the dipole moment, polarizability and the first-order hyperpolarizability of the guest molecule is investigated. 2. Technical details The system under consideration consists of 1-cyanoethyne molecule placed along the axis of helium nanotube. The structure of the helium tube is analogous to the singlewall armchair carbon nanotubes. Therefore the chirality of the helium tubes of increasing diameters is denoted as (2, 2), (3, 3) and (4, 4), respectively. The length of the helium nanotubes can be characterized by the number of the helium layers along the tube. Here the nanotubes built of five helium layers corresponding to the ˚ were chosen to minimize the influence of length of 9.7 A the tube edges on the interaction with the confined molecule and simultaneously to ensure the size of the system that is affordable for ab initio calculations. Investigated structures are presented in Fig. 1. Substantial part of calculations have been performed using GAUSSIAN 03 suite of programs at the MP2 level of theory. Dunning correlation-consistent basis sets augmented with the set of diffuse functions (aug-cc-pVXZ,
315
X = D, T) have been used in all calculations [25,26]. To confirm the convergence of the results with respect to the basis set size, calculations for the medium-size system have been also performed with the aug-cc-pVQZ basis set. Boys and Bernardi counterpoise procedure has been used to correct for the basis set superposition error (BSSE) [27]. Moreover, verification of the reliability of the MP2 electric properties has been accomplished at the CCSD level of theory. Interaction energy decomposition have been carried out according to the hybrid variational–perturbational scheme [28,29] as implemented in GAMESS-US [30,31]. In this approach the interaction energy at the MP2 level of theory can be decomposed in the following way: ð10Þ
ð2Þ
HF DEMP2 ¼ eel þ eHL ex þ DE del þ eMP ; ð10Þ eel
ð1Þ
where denotes the electrostatic interaction energy for the SCF monomers, eHL ex is the Heitler–London exchange repulsion, DEHF del contains the effects of the mutual deformation of the electron densities of interacting monomers and ð2Þ eMP represents the correlation corrections to the SCF interaction energy and incorporates the dispersion contribution ð20Þ ð21Þ edisp and corrections to the electrostatic eel;r , deformation ð2Þ and exchange energy eexch-del . All the calculations within this approach are performed in the dimer-centered basis set (DCBS), therefore the interaction energy components are automatically BSSE-corrected. The electric properties: dipole moment, polarizability and first order hyperpolarizability have been obtained within the finite field approach [32]. Due to the axial symmetry of the system under consideration only the diagonal component of all properties along the symmetry axis has
Fig. 1. The structure of the investigated systems: 1-cyanoethyne molecule inside the (2, 2), (3, 3) and (4, 4) helium nanotubes: side and top view.
316
A. Kaczmarek et al. / Chemical Physics Letters 449 (2007) 314–318
been investigated. The decomposition of the interactioninduced electric properties have been carried out likewise according to the variational–perturbational treatment [33]. 3. Results and discussion The theory level of choice is determined by the convergence of the results with the increasing size of the basis set as well as with the amount of the electron correlation included in the treatment. The detailed analysis of the former problem has been performed for the system with (3, 3) helium nanotube at the MP2 level. The interaction energy in the series of Dunning basis sets takes on the values of 0.93, 1.02 and 1.04 kcal/mol for the basis set cardinal number X = 2, 3 and 4, respectively. Therefore already aug-cc-pVTZ gives the result of acceptable accuracy. The convergence of investigated electric properties with respect to the basis set size in the case of the (3, 3) complex was also analyzed. Increase of the basis set size has only small influence on the electric properties of the helium tube both in MCBS and in DCBS basis set. More significant effects are observed for the confined molecule, but still the aug-cc-pVTZ basis set can be considered as the one of reliable size and quality. Moreover, aug-cc-pVDZ gives also the qualitatively correct results, therefore it can be used for the purpose of the variational–perturbational energy decomposition, for which the application of the saturated basis set is still prohibitive. It has been shown that in some particular cases MP2 method may give unreliable results for higher order electric properties [34]. Therefore, additionally the calculations for 1-cyanoethyne molecule have been performed at the CCSD/aug-cc-pVTZ level of theory to confirm the general tendencies obtained for MP2 properties (see footnotes to Table 1). Moreover, CCSD/aug-cc-pVDZ interactioninduced properties for the smallest complex under consideration have been found to be Dlz = 0.0015 a.u., Dazz = 2.34 a.u., Dbzzz = 11.7 a.u. Comparison of those CCSD values with the MP2 results indicates that already at the level of second-order perturbation theory the qualitative trends for electric properties of the investigated system can be well reproduced. Therefore for the purpose of the current study MP2 level calculations are assumed to be conclusive. Table 1 presents interaction energy and electric properties calculated at the MP2/aug-cc-pVTZ level of theory for the sequence of complexes of increasing diameter. (2, 2) ˚ repels the confined helium nanotube of the radius of 2.8 A molecule. The minimum on the interaction energy curve occurs somewhere around the (3, 3) tube diameter, where the attraction between monomers is the strongest. The further increase of the diameter of the helium tube corresponds to the weaker attraction characteristic for the long-range region. The contributions to the interaction energy calculated according to the variational–perturbational scheme at the MP2/aug-cc-pVDZ level of theory are summarized in
Table 1 BSSE-corrected interaction energy DE (kcal/mol) and interaction-induced electric properties (a.u.) of 1-cyanoethyne molecule confined in the (n, n) armchair helium nanotubes calculated at the MP2/aug-cc-pVTZ level of theory Tube chirality ˚) Tube radius (A
(2, 2) 2.7629
(3, 3) 4.0682
(4, 4) 5.3902
Interaction energy DE
2.618
1.016
0.330
MCBSa MCBS DCBSb DCBS
0.0000 1.4911 0.0001 1.4907 1.4889 0.0017
0.0000 1.4911 0.0000 1.4909 1.4724 0.0184
0.0000 1.4911 0.0000 1.4909 1.4706 0.0202
MCBS MCBS DCBS DCBS
26.98 67.35 27.50 67.33 92.50 2.33
41.15 67.35 41.18 67.33 106.68 1.83
54.84 67.35 54.84 67.35 120.31 1.88
0.0 26.9 0.3 26.7 40.4 13.3
0.1 26.9 0.1 26.5 28.4 2.0
0.7 26.9 1.1 28.1 27.1 2.1
Dipole moment He nanotube HCCCNc He nanotube HCCCN He nanotube + HCCCN Dl Polarizability He nanotube HCCCNc He nanotube HCCCN He nanotube + HCCCN Da
First-order hyperpolarizability He nanotube MCBS HCCCNc MCBS He nanotube DCBS HCCCN DCBS He nanotube + HCCCN Db
a Calculations performed in MCBS, corresponding to the isolated monomer properties. b Calculations performed in DCBS, corresponding to the monomer properties in the basis set supplemented by the functions from the adjacent monomer. c At the CCSD/aug-cc-pVTZ level of theory for isolated HCCCN molecule, respectively, lz = 1.5002 a.u., azz = 64.37 a.u. and bzzz = 19.5 a.u.
Table 2. The main term for short range interaction energy at the SCF level is the exchange contribution with the value 13.22 kcal/mol for (2, 2) complex. For the tube with the largest diameter the delocalization and electrostatic contribution, being of the same order of magnitude, highly prevail the vanishing exchange term. Second-order corrections to the interaction energy are non-negligible. The ð20Þ absolute values of dispersion contribution Dedisp to interacHF tion energy comparable to the DE interaction energy for small and even much more pronounced for large diameter tubes. The confinement of the model molecule in the helium nanotube-like cage influences its electric properties. In the case of the dipole moment the confinement effects are moderate (see Table 1). Nevertheless, clearly the presence of the cavity causes the decrease of the absolute value of the dipole moment of the guest molecule. Those quenching effects are more pronounced for the polarizability: azz for the 1-cyanoethyne molecule is diminished by more than 2 a.u. for the smallest radius tube with respect to the free guest molecule polarizability. Moreover, the confinement
A. Kaczmarek et al. / Chemical Physics Letters 449 (2007) 314–318
317
Table 2 Decomposition of the interaction energy and interaction-induced properties for 1-cyanoethyne molecule confined in (n, n) helium nanotube calculated according to the variational–perturbational scheme at the MP2/aug-cc-pVDZ level of theory ð10Þ
eel DE (2, 2) (3, 3) (4, 4) Dl (2, 2) (3, 3) (4, 4)
2.719 0.040 0.001 0.0273 0.0266 0.0242
Da (2, 2) (3, 3) (4, 4)
0.89 2.16 2.10
Db (2, 2) (3, 3) (4, 4)
7.6 0.6 0.4
eHL ex
DEHF del
ð20Þ
DEHF
ð21Þ
ð2Þ
edisp
eel;r
eex-del
6.070 1.074 0.310
0.084 0.004 0.001
1.117 0.034 0.004
0.0016 0.0021 0.0021
0.0039 0.0008 0.0003
13.219 0.215 0.003
1.641 0.064 0.016
8.859 0.111 0.015
0.0183 0.0017 0.0000
0.0141 0.0060 0.0025
0.0051 0.0187 0.0218
0.0039 0.0009 0.0002
3.78 3.13 0.08
2.09 2.95 0.05
2.58 1.98 1.98
0.57 0.10 0.02
0.06 0.02 0.02
0.65 0.06 0.01
4.4 1.2 0.3
2.0 2.2 2.0
6.5 1.7 0.3
32.2 75.2 161.1
8.7 79.2 160.6
15.9 3.5 0.9
DEMP2 3.822 0.933 0.321 0.0011 0.0183 0.0202 2.60 1.92 1.94 16.0 1.9 1.1
All energy contributions in (kcal/mol) and electric properties in (a.u.). For explanation of the symbols see text.
has a dramatic effect on the first-order hyperpolarizability. Interaction-induced hyperpolarizability Db is a third derivative of the interaction energy with respect to applied electric field and therefore its components can behave differently than those of interaction energy itself. Since the helium tube possesses an inversion center its first-order hyperpolarizability should be equal zero. Thus, the response of the whole system can be considered as the response of the 1-cyanoethyne molecule modulated by the confining cage. Decreasing the tube radius one observes the significant enhancement of the absolute value of the first-order hyperpolarizability of the complex. Therefore, the response of the isolated 1-cyanoethyne molecule is weaker than the response of the same system enclosed in the helium tube. This remark is crucial from the point of view of new materials design. One should also note, that since we analyze the confinement effect by means of helium analogs of carbon nanotubes, what is considered here is not the pure effect of the compression [35]. The absolute value of bzzz contribution arising from the 1-cyanoethyne molecule interaction with the helium cage of the smallest radius is higher than 13 a.u. The interactioninduced first-order hyperpolarizabilities become smaller for larger diameter of confining helium structure. The influence of the cavity onto the guest molecule properties according to the long-range multipole–induced–multipole should vanish for the infinite tube radius. However, one has to be aware of the high sensitivity of the hyperpolarizability values to the numerical errors. Although all the calculations have been performed with the very tight energy convergence criteria, the finite field procedure used to obtain the electric properties carries the inherent errors causing inaccuracies in the b values of the order of 0.5 a.u. particularly important for the long range weak interactions. One should also notice that those errors will cumulate for the interaction-induced properties. Therefore
one should not ascribe the fluctuations in the hyperpolarizability values to the physical phenomena. Most of the important contributions to the interactioninduced polarizability of the considered complex is already included at the SCF level of theory. Only marginal contributions arise at the MP2 level for small tube, where the dispersion term cancels with the exchange–delocalization term. Also for larger complexes the contributions to interaction-induced polarizability arising from the inclusion of correlation effects are small and does not exceed 5% of the SCF value. Similarly, a cancellation of dispersion and exchange–delocalization terms causes moderate changes in the interaction-induced first order hyperpolarizability due to the inclusion of electron correlation effects. 4. Summary The present study provides analysis of the confinement effects on the 1-cyanoethyne molecule. The embedding environment is a nanotube-like helium structure that replaces model analytic potentials widely used in this context by other authors. For the smallest cavity size that corresponds to a tube ˚ significant repulsion between the subsysradius of 2.8 A tems is observed, while the minimum on the potential energy curve occurs for medium-size system of the radius ˚ . Moreover, confinement causes moderate equal to 4 A decrease of the polarizability of the 1-cyanoethyne molecule and more pronounced changes in its first-order hyperpolarizability. The response of the 1-cyanoethyne molecule enclosed in the helium cage is significantly stronger than the response of the isolated molecule, what is of importance for the design of new materials. Inclusion of the electron correlation effects has only minor influence on the electrooptical properties of confined molecule, since most of the contributions is well repro-
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duced at the SCF level. However, this is no longer true for the interaction energy. The decomposition of the interaction energy reveals the importance of the dispersion term as a contribution stabilizing the mutual interaction between the cage and the enclosed molecule. Acknowledgements Wroclaw Centre for Networking and Supercomputing and Poznan Networking and Supercomputing Center are gratefully acknowledged for the allotment of the computational facilities. Calculations have been carried out partly in ACK Cyfronet (Grant MNiSW/Sun6800/PWr/117/ 2006). Authors thank Prof Andrzej J. Sadlej for inspiring discussions and Dr. Robert Gora for access to the EDS code. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
D.A. Britz, A.N. Khlobystov, Chem. Soc. Rev. 35 (2006) 637. M.D. Halls, K. Raghavachari, Nano Lett. 5 (2005) 1861. M.D. Halls, H.B. Schlegel, J. Phys. Chem. B 106 (2002) 1921. J. Cioslowski, J. Am. Chem. Soc. 113 (1991) 4139. J. Cioslowski, E.D. Fleischmann, J. Chem. Phys. 94 (1991) 3730. E.E. Santiso, A.M. George, K.E. Gubbins, M.B. Nardelli, J. Chem. Phys. 125 (2006) 084711. R. Rungsirisakun, B. Jansang, P. Pantu, J. Limtrakul, J. Molec. Struct. 733 (2005) 239. P.W. Fowler, P.A. Madden, Mol. Phys. 49 (1983) 913. P.W. Fowler, P.A. Madden, Phys. Rev. B 29 (1984) 1035. D. Keˆdziera, A. Avramopoulos, M.G. Papadopoulos, A.J. Sadlej, Phys. Chem. Chem. Phys. 5 (2003) 1096.
[11] S.A. Cruz, J. Soullard, Chem. Phys. Lett. 391 (2004) 138. [12] T. Sako, I. Cernusak, G.H.F. Diercksen, J. Phys. B 37 (2004) 1091. [13] A.L. Buchachenko, J. Phys. Chem. B 105 (2001) 5839. [14] O. Shameema, C.N. Ramachandran, N. Sathyamurthy, J. Phys. Chem. A 110 (2006) 2. [15] R.W. Munn, M. Malagoli, M. in het Panhius, Synth. Met. 109 (2000) 29. [16] R. Dutt, A. Mukherjee, Y.P. Varshni, Phys. Lett. A 280 (2001) 318. [17] H.E. Montgomery, Chem. Phys. Lett. 352 (2002) 529. [18] A. Banerjee, K.D. Sen, J. Garza, R. Vargas, J. Chem. Phys. 116 (2002) 4054. [19] P.K. Chattaraj, U. Sarkar, J. Phys. Chem. A 107 (2003) 4877. [20] R.W. Munn, P. Petelenz, Chem. Phys. Lett. 392 (2004) 7. [21] A. Michels, J. de Boer, A. Bijl, Physica 4 (1937) 981. [22] A. Sommerfeld, H. Welker, Ann. Phys. 32 (1938) 56. [23] W. Jaskolski, Phys. Rep. 271 (1996) 1. [24] M.G. Papadopoulos, A.J. Sadlej, Chem. Phys. Lett. 288 (1998) 377. [25] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007. [26] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 100 (1994) 2975. [27] S.F. Boys, F. Bernardi, Mol. Phys. 19 (1970) 553. [28] W.A. Sokalski, S. Roszak, K. Pecul, Chem. Phys. Lett. 153 (1988) 153. [29] S.M. Cybulski, G. Chałasin˜ski, R. Moszyn˜ski, J. Chem. Phys. 92 (1990) 4357. [30] M.W. Schmidt et al., J. Comput. Chem. 14 (1993) 1347. [31] R.W. Gora, W. Bartkowiak, S. Roszak, J. Leszczynski, J. Chem. Phys. 120 (2004) 2802. [32] H.A. Kurtz, J.J.P. Stewart, K.M. Dieter, J. Comput. Chem. 11 (1991) 82. [33] R.W. Gora, in: T. Simos, G. Maroulis (Eds.), Recent Progress in Computational Sciences and Engineering (B), vol. 7, Brill Academic Publishers, 2006, p. 1210. [34] G. Maroulis, C. Pouchan, Phys. Rev. A 57 (1998) 2440. [35] P.W. Fowler, A.J. Sadlej, Mol. Phys. 77 (1992) 709.
Chemical Physics Letters 449 (2007) 314–318 www.elsevier.com/locate/cplett
On the influence of confinement effects on electric properties: An ab initio study Anna Kaczmarek b c
a,*
, Robert Zales´ny b, Wojciech Bartkowiak
c
a Faculty of Chemistry, Nicolaus Copernicus University, Gagarina 7, 87-100 Torun´, Poland Department of Physical Chemistry, Collegium Medicum, Nicolaus Copernicus University, Kurpin´skiego 5, 85-950 Bydgoszcz, Poland Institute of Physical and Theoretical Chemistry, Technical University of Wrocław, Wyb. Wyspian´skiego 27, 50-370 Wrocław, Poland
Received 15 August 2007; in final form 25 October 2007 Available online 30 October 2007
Abstract Confinement effects on the electric properties of model system are investigated. The system under consideration is 1-cyanoethyne molecule surrounded by helium atoms arranged in the shape of armchair nanotube. Decreasing the diameter of the helium nanotube corresponds to the simulation of the confinement effects on the enclosed system. The decomposition of the interaction energy and interaction-induced electric properties reveals that dispersion terms plays an important role in the interaction between subsystems. It is observed that the absolute values of dipole moment and first-order hyperpolarizability decrease as a function of the tube radius. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction The properties of a confined molecule can differ significantly from those of gas phase and those of bulk. Simultaneously, the weak interaction between the enclosed molecule and the cavity allows to finely tune the characteristics of the host system. Therefore the confinement effects have recently stimulated a considerable interest as a source of information about the properties and applications of different kinds of systems of chemical importance. These include catalytic effects in zeolites, chemical reactivity in carbon nanotubes or porous carbon materials, electric properties in crystal, endohedral fullerene complexes, molecules under the high pressure or the storage properties of the solid media among others [1–8]. The influence of the valence compression on the in-crystal properties according to Fowler can be threefold: the crystal field effects including electrostatic Coulomb interaction between the subsystems, the overlap compression referring to the interaction of the guest molecule with the *
Corresponding author. Fax: +48 56 654 24. E-mail address: [email protected] (A. Kaczmarek).
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.10.085
occupied orbitals of the host and the charge transfer effects involving the interaction of the confined molecule with the virtual orbitals of the cavity [8,9]. All those effects together can have substantial influence on the stability of the system, geometry of the confined molecule, its electrooptical properties, absorption and emission spectra, dielectric constants, hardness and softness, hyperfine coupling, magnetic properties and the reaction mechanisms [8–20]. Hence, numerous contributions considering these phenomena are available in the literature starting from Michels et al. [21] and Sommerfeld and Welker [22] works and including valuable reviews [23]. Nevertheless, in a great majority of those pieces the confining cavity is modeled in terms of the simple model potentials such as harmonic oscillator potential. This approach permits to reproduce a general aspects of the confinement effects, however it appears to be oversimplified. Thus, regarding chemical issues mentioned above one can see the necessity of modeling the guest molecules embedded in less simplified model systems such as helium cages [24] or real chemical species like fullerenes [4,5,20], carbon nanotubes [2,3] or zeolites [7] of the non-uniform charge distribution.
A. Kaczmarek et al. / Chemical Physics Letters 449 (2007) 314–318
The aim of the present, introductory, study is to analyze the impact of the confinement in the nanotube-like cage onto the electric properties of the small model molecule. Since the size of the carbon nanotubes is prohibitive to the high level ab initio calculations necessary to describe the subtle effects in model molecule arising from the weak interaction with the cage, the cavity is represented here by the nanotube-like helium structure. The influence of the decreasing tube radius on the dipole moment, polarizability and the first-order hyperpolarizability of the guest molecule is investigated. 2. Technical details The system under consideration consists of 1-cyanoethyne molecule placed along the axis of helium nanotube. The structure of the helium tube is analogous to the singlewall armchair carbon nanotubes. Therefore the chirality of the helium tubes of increasing diameters is denoted as (2, 2), (3, 3) and (4, 4), respectively. The length of the helium nanotubes can be characterized by the number of the helium layers along the tube. Here the nanotubes built of five helium layers corresponding to the ˚ were chosen to minimize the influence of length of 9.7 A the tube edges on the interaction with the confined molecule and simultaneously to ensure the size of the system that is affordable for ab initio calculations. Investigated structures are presented in Fig. 1. Substantial part of calculations have been performed using GAUSSIAN 03 suite of programs at the MP2 level of theory. Dunning correlation-consistent basis sets augmented with the set of diffuse functions (aug-cc-pVXZ,
315
X = D, T) have been used in all calculations [25,26]. To confirm the convergence of the results with respect to the basis set size, calculations for the medium-size system have been also performed with the aug-cc-pVQZ basis set. Boys and Bernardi counterpoise procedure has been used to correct for the basis set superposition error (BSSE) [27]. Moreover, verification of the reliability of the MP2 electric properties has been accomplished at the CCSD level of theory. Interaction energy decomposition have been carried out according to the hybrid variational–perturbational scheme [28,29] as implemented in GAMESS-US [30,31]. In this approach the interaction energy at the MP2 level of theory can be decomposed in the following way: ð10Þ
ð2Þ
HF DEMP2 ¼ eel þ eHL ex þ DE del þ eMP ; ð10Þ eel
ð1Þ
where denotes the electrostatic interaction energy for the SCF monomers, eHL ex is the Heitler–London exchange repulsion, DEHF del contains the effects of the mutual deformation of the electron densities of interacting monomers and ð2Þ eMP represents the correlation corrections to the SCF interaction energy and incorporates the dispersion contribution ð20Þ ð21Þ edisp and corrections to the electrostatic eel;r , deformation ð2Þ and exchange energy eexch-del . All the calculations within this approach are performed in the dimer-centered basis set (DCBS), therefore the interaction energy components are automatically BSSE-corrected. The electric properties: dipole moment, polarizability and first order hyperpolarizability have been obtained within the finite field approach [32]. Due to the axial symmetry of the system under consideration only the diagonal component of all properties along the symmetry axis has
Fig. 1. The structure of the investigated systems: 1-cyanoethyne molecule inside the (2, 2), (3, 3) and (4, 4) helium nanotubes: side and top view.
316
A. Kaczmarek et al. / Chemical Physics Letters 449 (2007) 314–318
been investigated. The decomposition of the interactioninduced electric properties have been carried out likewise according to the variational–perturbational treatment [33]. 3. Results and discussion The theory level of choice is determined by the convergence of the results with the increasing size of the basis set as well as with the amount of the electron correlation included in the treatment. The detailed analysis of the former problem has been performed for the system with (3, 3) helium nanotube at the MP2 level. The interaction energy in the series of Dunning basis sets takes on the values of 0.93, 1.02 and 1.04 kcal/mol for the basis set cardinal number X = 2, 3 and 4, respectively. Therefore already aug-cc-pVTZ gives the result of acceptable accuracy. The convergence of investigated electric properties with respect to the basis set size in the case of the (3, 3) complex was also analyzed. Increase of the basis set size has only small influence on the electric properties of the helium tube both in MCBS and in DCBS basis set. More significant effects are observed for the confined molecule, but still the aug-cc-pVTZ basis set can be considered as the one of reliable size and quality. Moreover, aug-cc-pVDZ gives also the qualitatively correct results, therefore it can be used for the purpose of the variational–perturbational energy decomposition, for which the application of the saturated basis set is still prohibitive. It has been shown that in some particular cases MP2 method may give unreliable results for higher order electric properties [34]. Therefore, additionally the calculations for 1-cyanoethyne molecule have been performed at the CCSD/aug-cc-pVTZ level of theory to confirm the general tendencies obtained for MP2 properties (see footnotes to Table 1). Moreover, CCSD/aug-cc-pVDZ interactioninduced properties for the smallest complex under consideration have been found to be Dlz = 0.0015 a.u., Dazz = 2.34 a.u., Dbzzz = 11.7 a.u. Comparison of those CCSD values with the MP2 results indicates that already at the level of second-order perturbation theory the qualitative trends for electric properties of the investigated system can be well reproduced. Therefore for the purpose of the current study MP2 level calculations are assumed to be conclusive. Table 1 presents interaction energy and electric properties calculated at the MP2/aug-cc-pVTZ level of theory for the sequence of complexes of increasing diameter. (2, 2) ˚ repels the confined helium nanotube of the radius of 2.8 A molecule. The minimum on the interaction energy curve occurs somewhere around the (3, 3) tube diameter, where the attraction between monomers is the strongest. The further increase of the diameter of the helium tube corresponds to the weaker attraction characteristic for the long-range region. The contributions to the interaction energy calculated according to the variational–perturbational scheme at the MP2/aug-cc-pVDZ level of theory are summarized in
Table 1 BSSE-corrected interaction energy DE (kcal/mol) and interaction-induced electric properties (a.u.) of 1-cyanoethyne molecule confined in the (n, n) armchair helium nanotubes calculated at the MP2/aug-cc-pVTZ level of theory Tube chirality ˚) Tube radius (A
(2, 2) 2.7629
(3, 3) 4.0682
(4, 4) 5.3902
Interaction energy DE
2.618
1.016
0.330
MCBSa MCBS DCBSb DCBS
0.0000 1.4911 0.0001 1.4907 1.4889 0.0017
0.0000 1.4911 0.0000 1.4909 1.4724 0.0184
0.0000 1.4911 0.0000 1.4909 1.4706 0.0202
MCBS MCBS DCBS DCBS
26.98 67.35 27.50 67.33 92.50 2.33
41.15 67.35 41.18 67.33 106.68 1.83
54.84 67.35 54.84 67.35 120.31 1.88
0.0 26.9 0.3 26.7 40.4 13.3
0.1 26.9 0.1 26.5 28.4 2.0
0.7 26.9 1.1 28.1 27.1 2.1
Dipole moment He nanotube HCCCNc He nanotube HCCCN He nanotube + HCCCN Dl Polarizability He nanotube HCCCNc He nanotube HCCCN He nanotube + HCCCN Da
First-order hyperpolarizability He nanotube MCBS HCCCNc MCBS He nanotube DCBS HCCCN DCBS He nanotube + HCCCN Db
a Calculations performed in MCBS, corresponding to the isolated monomer properties. b Calculations performed in DCBS, corresponding to the monomer properties in the basis set supplemented by the functions from the adjacent monomer. c At the CCSD/aug-cc-pVTZ level of theory for isolated HCCCN molecule, respectively, lz = 1.5002 a.u., azz = 64.37 a.u. and bzzz = 19.5 a.u.
Table 2. The main term for short range interaction energy at the SCF level is the exchange contribution with the value 13.22 kcal/mol for (2, 2) complex. For the tube with the largest diameter the delocalization and electrostatic contribution, being of the same order of magnitude, highly prevail the vanishing exchange term. Second-order corrections to the interaction energy are non-negligible. The ð20Þ absolute values of dispersion contribution Dedisp to interacHF tion energy comparable to the DE interaction energy for small and even much more pronounced for large diameter tubes. The confinement of the model molecule in the helium nanotube-like cage influences its electric properties. In the case of the dipole moment the confinement effects are moderate (see Table 1). Nevertheless, clearly the presence of the cavity causes the decrease of the absolute value of the dipole moment of the guest molecule. Those quenching effects are more pronounced for the polarizability: azz for the 1-cyanoethyne molecule is diminished by more than 2 a.u. for the smallest radius tube with respect to the free guest molecule polarizability. Moreover, the confinement
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Table 2 Decomposition of the interaction energy and interaction-induced properties for 1-cyanoethyne molecule confined in (n, n) helium nanotube calculated according to the variational–perturbational scheme at the MP2/aug-cc-pVDZ level of theory ð10Þ
eel DE (2, 2) (3, 3) (4, 4) Dl (2, 2) (3, 3) (4, 4)
2.719 0.040 0.001 0.0273 0.0266 0.0242
Da (2, 2) (3, 3) (4, 4)
0.89 2.16 2.10
Db (2, 2) (3, 3) (4, 4)
7.6 0.6 0.4
eHL ex
DEHF del
ð20Þ
DEHF
ð21Þ
ð2Þ
edisp
eel;r
eex-del
6.070 1.074 0.310
0.084 0.004 0.001
1.117 0.034 0.004
0.0016 0.0021 0.0021
0.0039 0.0008 0.0003
13.219 0.215 0.003
1.641 0.064 0.016
8.859 0.111 0.015
0.0183 0.0017 0.0000
0.0141 0.0060 0.0025
0.0051 0.0187 0.0218
0.0039 0.0009 0.0002
3.78 3.13 0.08
2.09 2.95 0.05
2.58 1.98 1.98
0.57 0.10 0.02
0.06 0.02 0.02
0.65 0.06 0.01
4.4 1.2 0.3
2.0 2.2 2.0
6.5 1.7 0.3
32.2 75.2 161.1
8.7 79.2 160.6
15.9 3.5 0.9
DEMP2 3.822 0.933 0.321 0.0011 0.0183 0.0202 2.60 1.92 1.94 16.0 1.9 1.1
All energy contributions in (kcal/mol) and electric properties in (a.u.). For explanation of the symbols see text.
has a dramatic effect on the first-order hyperpolarizability. Interaction-induced hyperpolarizability Db is a third derivative of the interaction energy with respect to applied electric field and therefore its components can behave differently than those of interaction energy itself. Since the helium tube possesses an inversion center its first-order hyperpolarizability should be equal zero. Thus, the response of the whole system can be considered as the response of the 1-cyanoethyne molecule modulated by the confining cage. Decreasing the tube radius one observes the significant enhancement of the absolute value of the first-order hyperpolarizability of the complex. Therefore, the response of the isolated 1-cyanoethyne molecule is weaker than the response of the same system enclosed in the helium tube. This remark is crucial from the point of view of new materials design. One should also note, that since we analyze the confinement effect by means of helium analogs of carbon nanotubes, what is considered here is not the pure effect of the compression [35]. The absolute value of bzzz contribution arising from the 1-cyanoethyne molecule interaction with the helium cage of the smallest radius is higher than 13 a.u. The interactioninduced first-order hyperpolarizabilities become smaller for larger diameter of confining helium structure. The influence of the cavity onto the guest molecule properties according to the long-range multipole–induced–multipole should vanish for the infinite tube radius. However, one has to be aware of the high sensitivity of the hyperpolarizability values to the numerical errors. Although all the calculations have been performed with the very tight energy convergence criteria, the finite field procedure used to obtain the electric properties carries the inherent errors causing inaccuracies in the b values of the order of 0.5 a.u. particularly important for the long range weak interactions. One should also notice that those errors will cumulate for the interaction-induced properties. Therefore
one should not ascribe the fluctuations in the hyperpolarizability values to the physical phenomena. Most of the important contributions to the interactioninduced polarizability of the considered complex is already included at the SCF level of theory. Only marginal contributions arise at the MP2 level for small tube, where the dispersion term cancels with the exchange–delocalization term. Also for larger complexes the contributions to interaction-induced polarizability arising from the inclusion of correlation effects are small and does not exceed 5% of the SCF value. Similarly, a cancellation of dispersion and exchange–delocalization terms causes moderate changes in the interaction-induced first order hyperpolarizability due to the inclusion of electron correlation effects. 4. Summary The present study provides analysis of the confinement effects on the 1-cyanoethyne molecule. The embedding environment is a nanotube-like helium structure that replaces model analytic potentials widely used in this context by other authors. For the smallest cavity size that corresponds to a tube ˚ significant repulsion between the subsysradius of 2.8 A tems is observed, while the minimum on the potential energy curve occurs for medium-size system of the radius ˚ . Moreover, confinement causes moderate equal to 4 A decrease of the polarizability of the 1-cyanoethyne molecule and more pronounced changes in its first-order hyperpolarizability. The response of the 1-cyanoethyne molecule enclosed in the helium cage is significantly stronger than the response of the isolated molecule, what is of importance for the design of new materials. Inclusion of the electron correlation effects has only minor influence on the electrooptical properties of confined molecule, since most of the contributions is well repro-
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duced at the SCF level. However, this is no longer true for the interaction energy. The decomposition of the interaction energy reveals the importance of the dispersion term as a contribution stabilizing the mutual interaction between the cage and the enclosed molecule. Acknowledgements Wroclaw Centre for Networking and Supercomputing and Poznan Networking and Supercomputing Center are gratefully acknowledged for the allotment of the computational facilities. Calculations have been carried out partly in ACK Cyfronet (Grant MNiSW/Sun6800/PWr/117/ 2006). Authors thank Prof Andrzej J. Sadlej for inspiring discussions and Dr. Robert Gora for access to the EDS code. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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