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The response of a molecule to an external electric field: predicting structural and spectroscopic change
Chemical Physics Letters 370 (2003) 14–20 www.elsevier.com/locate/cplett
The response of a molecule to an external electric field: predicting structural and spectroscopic change Meredith J.T. Jordan *, Keiran C. Thompson School of Chemistry, University of Sydney, NSW 2006, Australia Received 19 December 2002; in final form 19 December 2002
Abstract The accuracy of expanding the response of a molecule to an external electric field, E, as a power series in the field is investigated in the model hydrogen-bonded complex, ClH:NH3 . Even at field strengths large enough to cause dramatic structural change in the complex, both the structure and vibrational frequencies are quantitatively predicted using only terms linear in E. These results suggest that knowledge of the zero-field molecular potential energy and dipole moment surfaces may be sufficient to accurately model the interactions of molecules in a wide range of external electric fields. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction A common method for predicting environmental effects on molecular structure and property is to use electric fields to model long-range solvation interactions [1]. Determining how structure changes as the field (or the nature of the solvent) changes is computationally intensive and is not currently feasible for large molecules or at high levels of ab initio theory. However, the response of a molecule to an external electric field may also be expanded as a power series in the field [2]: DV ðRÞ ¼ la ðRÞEa 1=2aab ðRÞEa Eb . . . ;
ð1Þ
where DV ðRÞ is the change in energy due to the applied field E, lðRÞ is the dipole moment of the *
Corresponding author. Fax: +61-2-9351-3329. E-mail address: [email protected] (M.J.T. Jordan).
molecule at zero-field and aðRÞ its zero-field polarisability. The subscripts imply the summation convention. Note also the explicit dependence of the molecular properties on the internal coordinates, R, of the molecule. Eq. (1) can be thought of as a perturbation of the molecular energy by the field E and changes to, for example, the equilibrium geometry of the molecule or its spectroscopic characteristics may be calculated on this perturbed molecular energy surface. Thus a knowledge of the zero-field molecular potential energy surface (PES), dipole moment surface, polarisability surface, etc. can be used to model molecular behaviour as a function of external field. This approach has been used previously, at first order, in reduced dimension, to model field effects in the BrH: NH2 CH3 hydrogen-bonded complex [3], the methanesulfonic acid–dimethyl sulfoxide complex [4] and complexes of water and formic acid [5]. Molecular properties have also been explicitly
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00045-9
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
calculated as a function of external electric field for a number of hydrogen-bonded complexes, inþ cluding ½H3 CH2 N:H3 NCH3 [6], FH:NH3 [7,8], ClH:NH3 [7–9], BrH:NH3 [7,8], FH:NCH [8], ClH:NCH [8], BrH:NCH [8], ClH:NðCH3 Þ3 [10], BrH:NðCH3 Þ3 [10], ClH:pyridine [11], FHF [12], CNH:NCH [13,14], and in small dihydrogen bonded systems [15]. These studies have focussed on hydrogen-bonded systems because the high polarisability of the hydrogen bond means that large structural and spectroscopic changes can be induced by an external field. The accuracy of Eq. (1) in approximating the response of a molecule to an electric field has been examined in the ClH:NH3 hydrogen-bonded complex. In this complex, an external field can induce significant structural change. As the field increases, the nature of the hydrogen bond changes from a traditional hydrogen bond, where the Cl–H distance is only slightly elongated relative to the covalent bond length in HCl, to a protonshared hydrogen bond, where both Cl–H and H–N distances are elongated. At sufficiently large fields, a Cl :NHþ 4 ion-pair hydrogen bond [7] is formed in which the N–H distance is slightly longer than the covalent bond length in NHþ 4. These structural changes correlate with both vibrational and nuclear magnetic resonance spectroscopic parameters [7,9,11,13,14,16]. The ClH:NH3 complex has been studied in two dimensions, corresponding to the RClH and RNH bond lengths. Structural and vibrational spectroscopic characteristics, calculated using first- and second-order terms in Eq. (1), have been compared to results calculated from exact PESs, as reported in [7].
2. Methods As in [7], all calculations have been performed using second order many body Møller–Plesset perturbation theory (MP2) [17–20] and the Dunning correlation-consistent polarised valence double zeta basis set augmented with diffuse s, p and d functions on non-hydrogen atoms (aug0 ccpVDZ) [21–23]. All calculations were done by freezing the s and p electrons below the valence
15
shells in the Hartree–Fock molecular orbitals. The ClH:NH3 complex was optimised at zero-field under the constraint of C3v symmetry yielding an and an equiequilibrium NH distance of 1.02 A librium HNH angle of 107.0°. The parameters specifying the NH3 geometry were held fixed at these equilibrium values and two-dimensional potential energy, molecular dipole and molecular polarisability surfaces were generated by varying the RClH and RNH bond lengths. The zero-field PES was the same as that used in [7] and the molecular property surfaces were generated analogously, using interpolation and extrapolation techniques described previously [7,16,24]. The vibrational spectroscopic and structural properties of the ClH:NH3 complex were characterised by the anharmonic frequencies describing proton and heavy-atom motion and by the expectation values of the bond lengths, hRClH i and hRClN i. These properties were calculated as previously [7,23], treating the ClH:NH3 complex as a collinear, pseudotriatomic system. Vibrational eigenfunctions and eigenvalues were calculated using the discrete variable representation [25] of a two-dimensional basis set constructed as a direct product of onedimensional tridiagonal Morse functions [26], each optimised to the appropriate one-dimensional Schr€ odinger equation. The results quoted below used up to 250:120 primitive:contracted basis functions in each coordinate. Vibrational eigenvalues were converged to better than 0:01 cm1 , and bond length expectation values . The rewere converged to better than 0.0005 A sults obtained at first and second order in Eq. (1) were compared to exact results for ClH:NH3 computed on PESs calculated in the presence of 0.0025, 0.0055, and 0.0100 a.u. external electric fields. As in [7], electric fields were uniform and applied along the ClH:N hydrogen-bonding axis. Because of the larger vibrational basis sets used, the results quoted here are slightly more accurate than those previously reported in [7]. All calculations reported here were carried out on the computing facilities in the School of Chemistry at the University of Sydney. Ab initio data points for the molecular dipole moment and polarisability surfaces were generated using GA U S S I A N 98 [27].
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M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
3. Results and discussion When an external electric field is applied along the ClH:N direction, the field preferentially stabilises geometries in which there is charge separation along this axis. A proton-shared structure is stabilised relative to a traditional hydrogen-bonded structure and, at sufficiently high fields, proton transfer occurs. The PESs generated in this study have been calculated at field strengths of 0.0025, 0.0055, and 0.0100 a.u. for illustrative purposes. These fields have been shown to correspond, respectively, to a traditional hydrogen-bonded complex, a proton-shared hydrogen-bonded com-
plex and a hydrogen-bonded ion pair [7]. The fields chosen, therefore, encompass dramatic structural change in the complex. Approximating the response of a molecule to an electric field by Eq. (1) is appealing because a knowledge of the zero-field molecular dipole moment and molecular polarisability surfaces can be used to determine molecular properties in a range of different environments. In this case, because of the symmetry imposed on the ClH:NH3 complex and the direction of the external electric field, the only relevant components of the dipole and polarisability are lz and azz , where z is the ClH:NH3 symmetry axis. These are plotted, in Figs. 1 and 2,
Fig. 1. The absolute value of the lz component (a.u.) of the ClH:NH3 molecular dipole moment vector as a function of the RClH and RNH bond lengths (a.u.). Contours are at intervals of 0.05 a.u.
Fig. 2. The azz component (a.u.) of the ClH:NH3 molecular polarisability tensor as a function of the RClH and RNH bond lengths (a.u.). Contours are at intervals of 10 a.u.
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
respectively, as functions of the RClH and RNH bond lengths. At this point it should be noted that accurate calculation of molecular polarisabilities is difficult and requires appropriate basis sets [28] and theoretical methods that adequately account for electron correlation [29]. Indeed this is not attempted here. Instead the polarisability, aðRÞ, has been calculated at the MP2/aug0 -ccpVDZ level of theory, consistent with the zero-field PES and the dipole moment surface and consistent with the identification of Eq. (1) as a Taylor Series. That is, while the values of a are not physically accurate, they are calculated in such a way that the expression a ¼ ol=oE remains valid, ensuring that errors
17
from such a calculation could be corrected by the inclusion of higher-order terms in Eq. (1). The approximations introduced by Eq. (1) are illustrated in Figs. 3 and 4 for the highest field considered, 0.0100 au. Panel (a) in Fig. 3 is the zero-field PES and panel (b) the exact 0.0100 a.u. field surface, both from [7]. It is apparent from Fig. 3 that the nature of the PES changes significantly in the presence of the field. Fig. 4 illustrates the difference between the exact PES at 0.0100 a.u. field and approximate PESs generated at first order (panel (a)) and second order (panel (b)) in E. The exact PES is indicated by the solid line and the approximate PES the dotted line. At both first and second order in E the approximate PES
Fig. 3. ClH:NH3 potential energy surfaces at zero-field (panel (a)) and 0.0100 a.u. field (panel (b)). Contour values are at intervals of 0.001 a.u. above the global minimum on each surface.
Fig. 4. The exact ClH:NH3 potential energy surface at 0.0100 a.u. field (solid line) and approximate surfaces generated at first order (panel (a)) and second order (panel (b)) in E (dotted lines). Contour values are at intervals of 0.001 a.u. above the global minimum on each surface.
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M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
accurately reproduces the shape of the exact PES in the chemically relevant regions of configuration space, with the second-order PES being slightly more accurate. The PESs shown in Fig. 4, however, are illustrated with respect to the global minimum on each surface. It should be noted that the minimum energy on the first-order surface is 0.0023 EH above the minimum on the exact PES and the minimum on the second-order surface is 0.000046 EH below the minimum on the exact PES. Expectation values of the Cl–N and Cl–H distances, hRClN i and hRClH i, calculated from the ground state wavefunction at the various field strengths, are reported in Table 1. It is apparent from Table 1 that even large changes in the ClH:NH3 hydrogen-bond structure can be accurately predicted using Eq. (1) to perturb the zerofield PES. At the largest field considered, 0.0100 a.u., using the first-order term in E in Eq. (1) led to in hRClN i and 0.004 A in hRClH i. errors of 0.003 A Incorporation of the second-order term in E, that is, using the molecular polarisability surface, effectively corrected these errors and led to, at worst, in hRClH i at a field strength of an error of 0.001 A 0.0100 a.u. It can also be seen from Table 1 that the hRClH i distance is more sensitive to the approximations used in Eq. (1) than the hRClN i distance. Given that the heavy-atom distance better characterises the nature of the hydrogen bond [7,9,11,14,16], these results indicate that dramatic structural change can be accurately described using only the terms linear in E in Eq. (1). Two-dimensional anharmonic frequencies calculated from approximate PESs obtained using
first- and second-order corrections in E are compared to exact results in Table 2. The frequencies have been reported as a function of vibrational state for the three different field strengths considered. The notation (i; j) represents i quanta of excitation in the heavy-atom stretching mode and j in the proton-stretching mode. It is apparent from Table 2 that errors are larger using first-order corrections and increase with the number of quanta of excitation and the vibrational frequency. It can also be seen from Table 2 that the largest errors were obtained at the largest field considered. The errors observed at field strengths of 0.0025 and 0.0055 a.u., however, were similar. The vibrational frequency correlating most strongly with hydrogen bond structure type is the proton-stretching frequency, (0,1) [7,9,11,14,16]. Although the proton-stretching fundamental occurs at significantly higher energy than the heavy-atom motion, errors associated with the approximate PESs are small, at most 6 cm1 at a field strength of 0.0100 a.u. The sensitivity of the fundamental frequencies to the procedure used to generate the PES has been previously found to be at most 7 cm1 [7]. Furthermore, errors associated with the ab initio methods used to generate the data points are likely to be at least 20 cm1 . The errors seen here in the fundamental frequencies are therefore within the intrinsic errors associated with the PESs themselves. These results indicate that changes in vibrational frequency associated with change in the hydrogen bond structure can be accurately described by the first-order term in Eq. (1).
Table 1 ) in the ClH:NH3 complex as a Expectation values hRClN i and hRClH i in the ground vibrational state of the Cl–N and Cl–H distances (A function of field strength (a.u.) for the exact potential energy surface [7] and approximate potential energy surfaces calculated at first and second order in E Potential energy surface First order Field 0.0000 0.0025 0.0055 0.0100
hRClN i 2.940 2.885 2.895
Second order hRClH i 1.447 1.544 1.688
hRClN i 2.940 2.886 2.898
Exact hRClH i
hRClN i
hRClH i
1.448 1.546 1.692
3.016 2.940 2.885 2.898
1.392 1.448 1.545 1.692
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20 Table 2 Two-dimensional anharmonic vibrational frequencies (cm1 ) for proton and heavy-atom motion in the ClH:NH3 complex as a function of field strength (a.u.) for the exact potential energy surface and approximate potential energy surfaces calculated at first and second order in Ea Vibrational assignment
0.0025 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
Potential energy surface First order
Second order
Exact
243 477 700 913 1117 1221 1685
244 478 701 915 1118 1220 1687
244 478 701 917 1127 1219 1740
19
wide range of external electric fields, both uniform and inhomogeneous. This method thus provides a simple and computationally tractable means for systematically studying the interactions of a molecule with a variety of solvents or molecular environments under a range of conditions. Acknowledgements
0.0055 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
370 673 966 1209 1458 966 1458
372 677 968 1216 1466 968 1466
372 677 967 1215 1464 967 1464
0.0100 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
387 756 1110 1454 1787 1131 1534
383 749 1100 1440 1769 1143 1547
383 749 1100 1438 1762 1137 1540
a
The notation (i,j) refers to i quanta of excitation in the heavy-atom vibration and j quanta in the proton-stretching vibration.
4. Summary The response of a molecule to an external electric field can be expanded as a power series in the field, Eq. (1). We have shown that, for external fields large enough to induce dramatic structural change in the ClH:NH3 model complex, quantitative prediction of both the structure and vibrational frequencies can be obtained using only the linear term in Eq. (1). Knowledge of how the electronic energy and dipole moment of a molecule change with molecular geometry can therefore be used to model the interaction of the molecule in a
This work has been supported by the Australian Research Council (Grant A00104447). References [1] K.A. Sharp, in: P. Schleyer (Ed.), Encyclopedia of Computational Chemistry, Wiley, Chichester, 1998, p. 571. [2] A.J. Stone, The Theory of Intermolecular Forces, Oxford University Press, Oxford, 1996. [3] M. Eckert, G. Zundel, J. Phys. Chem. 91 (1987) 5170. [4] S. Geppert, A. Rabbold, G. Zundel, M. Eckert, J. Phys. Chem. 99 (1995) 12220. [5] M. Eckert, G. Zundel, J. Phys. Chem. 92 (1988) 7016. [6] J. Yin, M.E. Green, J. Phys. Chem. A 102 (1998) 7181. [7] M.J.T. Jordan, J.E. Del Bene, J. Am. Chem. Soc. 122 (2000) 2101. [8] M. Ramos, I. Alkorta, J. Elguero, N.S. Golubev, G.S. Denisov, H. Benedict, H.-H. Limbach, J. Phys. Chem. A 101 (1997) 9791. [9] J.E. Del Bene, M.J.T. Jordan, J. Am. Chem. Soc. 122 (2000) 4794. [10] J. Bevitt, K. Chapman, D. Crittenden, M.J.T. Jordan, J.E. Del Bene, J. Phys. Chem. A 105 (2001) 3371. [11] K. Chapman, D. Crittenden, J. Bevitt, M.J.T. Jordan, J.E. Del Bene, J. Phys. Chem. A 105 (2001) 5442. [12] J.E. Del Bene, M.J.T. Jordan, S.A. Perera, R.J. Bartlett, J. Phys. Chem. A 105 (2001) 8399. [13] M.J.T. Jordan, J.S.-S. Toh, J.E. Del Bene, Chem. Phys. Lett. 346 (2001) 288. [14] J.S.-S. Toh, M.J.T. Jordan, B. Husowitz, J.E. Del Bene, J. Phys. Chem. A 105 (2001) 10906. [15] I. Rozas, I. Alkorta, J. Elguero, Chem. Phys. Lett. 275 (1997) 423. [16] J.E. Del Bene, M.J.T. Jordan, J. Chem. Phys. 108 (1998) 3205. [17] J.A. Pople, J.S. Binkley, R. Seeger, Int. J. Quantum Chem. Quantum Chem. Symp. 10 (1976) 1. [18] R. Krishnan, J.A. Pople, Int. J. Quantum Chem. 14 (1978) 91. [19] R.J. Bartlett, D.M. Silver, J. Chem. Phys. 62 (1975) 3258. [20] R.J. Bartlett, G.D. Purvis, Int. J. Quantum Chem. 14 (1978) 561. [21] T.H. Dunning Jr., J. Chem. Phys. 90 (1989) 1007.
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[22] R.A. Kendall, T.H. Dunning Jr., R.J. Harrison, J. Chem. Phys. 96 (1992) 6796. [23] D.E. Woon, T.H. Dunning Jr., J. Chem. Phys. 98 (1993) 1358. [24] J.E. Del Bene, M.J.T. Jordan, P.M.W. Gill, A.D. Buckingham, Mol. Phys. 92 (1997) 429. [25] Z. Bacic, J. Light, Annu. Rev. Phys. Chem. 40 (1989) 469.
[26] H. Wei, T. Carrington Jr., J. Chem. Phys. 97 (1992) 3029. [27] M.J. Frisch et al., GA U S S I A N 98 (Revision A.7), Gaussian Inc., Pittsburgh, PA, 1998. [28] M.G. Papadopoulos, J. Waite, A.D. Buckingham, J. Chem. Phys. 102 (1995) 371. [29] H. Larsen, J. Olsen, C. H€attig, P. Jørgensen, O. Christiansen, J. Gauss, J. Chem. Phys. 111 (1999) 1917.
The response of a molecule to an external electric field: predicting structural and spectroscopic change Meredith J.T. Jordan *, Keiran C. Thompson School of Chemistry, University of Sydney, NSW 2006, Australia Received 19 December 2002; in final form 19 December 2002
Abstract The accuracy of expanding the response of a molecule to an external electric field, E, as a power series in the field is investigated in the model hydrogen-bonded complex, ClH:NH3 . Even at field strengths large enough to cause dramatic structural change in the complex, both the structure and vibrational frequencies are quantitatively predicted using only terms linear in E. These results suggest that knowledge of the zero-field molecular potential energy and dipole moment surfaces may be sufficient to accurately model the interactions of molecules in a wide range of external electric fields. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction A common method for predicting environmental effects on molecular structure and property is to use electric fields to model long-range solvation interactions [1]. Determining how structure changes as the field (or the nature of the solvent) changes is computationally intensive and is not currently feasible for large molecules or at high levels of ab initio theory. However, the response of a molecule to an external electric field may also be expanded as a power series in the field [2]: DV ðRÞ ¼ la ðRÞEa 1=2aab ðRÞEa Eb . . . ;
ð1Þ
where DV ðRÞ is the change in energy due to the applied field E, lðRÞ is the dipole moment of the *
Corresponding author. Fax: +61-2-9351-3329. E-mail address: [email protected] (M.J.T. Jordan).
molecule at zero-field and aðRÞ its zero-field polarisability. The subscripts imply the summation convention. Note also the explicit dependence of the molecular properties on the internal coordinates, R, of the molecule. Eq. (1) can be thought of as a perturbation of the molecular energy by the field E and changes to, for example, the equilibrium geometry of the molecule or its spectroscopic characteristics may be calculated on this perturbed molecular energy surface. Thus a knowledge of the zero-field molecular potential energy surface (PES), dipole moment surface, polarisability surface, etc. can be used to model molecular behaviour as a function of external field. This approach has been used previously, at first order, in reduced dimension, to model field effects in the BrH: NH2 CH3 hydrogen-bonded complex [3], the methanesulfonic acid–dimethyl sulfoxide complex [4] and complexes of water and formic acid [5]. Molecular properties have also been explicitly
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00045-9
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
calculated as a function of external electric field for a number of hydrogen-bonded complexes, inþ cluding ½H3 CH2 N:H3 NCH3 [6], FH:NH3 [7,8], ClH:NH3 [7–9], BrH:NH3 [7,8], FH:NCH [8], ClH:NCH [8], BrH:NCH [8], ClH:NðCH3 Þ3 [10], BrH:NðCH3 Þ3 [10], ClH:pyridine [11], FHF [12], CNH:NCH [13,14], and in small dihydrogen bonded systems [15]. These studies have focussed on hydrogen-bonded systems because the high polarisability of the hydrogen bond means that large structural and spectroscopic changes can be induced by an external field. The accuracy of Eq. (1) in approximating the response of a molecule to an electric field has been examined in the ClH:NH3 hydrogen-bonded complex. In this complex, an external field can induce significant structural change. As the field increases, the nature of the hydrogen bond changes from a traditional hydrogen bond, where the Cl–H distance is only slightly elongated relative to the covalent bond length in HCl, to a protonshared hydrogen bond, where both Cl–H and H–N distances are elongated. At sufficiently large fields, a Cl :NHþ 4 ion-pair hydrogen bond [7] is formed in which the N–H distance is slightly longer than the covalent bond length in NHþ 4. These structural changes correlate with both vibrational and nuclear magnetic resonance spectroscopic parameters [7,9,11,13,14,16]. The ClH:NH3 complex has been studied in two dimensions, corresponding to the RClH and RNH bond lengths. Structural and vibrational spectroscopic characteristics, calculated using first- and second-order terms in Eq. (1), have been compared to results calculated from exact PESs, as reported in [7].
2. Methods As in [7], all calculations have been performed using second order many body Møller–Plesset perturbation theory (MP2) [17–20] and the Dunning correlation-consistent polarised valence double zeta basis set augmented with diffuse s, p and d functions on non-hydrogen atoms (aug0 ccpVDZ) [21–23]. All calculations were done by freezing the s and p electrons below the valence
15
shells in the Hartree–Fock molecular orbitals. The ClH:NH3 complex was optimised at zero-field under the constraint of C3v symmetry yielding an and an equiequilibrium NH distance of 1.02 A librium HNH angle of 107.0°. The parameters specifying the NH3 geometry were held fixed at these equilibrium values and two-dimensional potential energy, molecular dipole and molecular polarisability surfaces were generated by varying the RClH and RNH bond lengths. The zero-field PES was the same as that used in [7] and the molecular property surfaces were generated analogously, using interpolation and extrapolation techniques described previously [7,16,24]. The vibrational spectroscopic and structural properties of the ClH:NH3 complex were characterised by the anharmonic frequencies describing proton and heavy-atom motion and by the expectation values of the bond lengths, hRClH i and hRClN i. These properties were calculated as previously [7,23], treating the ClH:NH3 complex as a collinear, pseudotriatomic system. Vibrational eigenfunctions and eigenvalues were calculated using the discrete variable representation [25] of a two-dimensional basis set constructed as a direct product of onedimensional tridiagonal Morse functions [26], each optimised to the appropriate one-dimensional Schr€ odinger equation. The results quoted below used up to 250:120 primitive:contracted basis functions in each coordinate. Vibrational eigenvalues were converged to better than 0:01 cm1 , and bond length expectation values . The rewere converged to better than 0.0005 A sults obtained at first and second order in Eq. (1) were compared to exact results for ClH:NH3 computed on PESs calculated in the presence of 0.0025, 0.0055, and 0.0100 a.u. external electric fields. As in [7], electric fields were uniform and applied along the ClH:N hydrogen-bonding axis. Because of the larger vibrational basis sets used, the results quoted here are slightly more accurate than those previously reported in [7]. All calculations reported here were carried out on the computing facilities in the School of Chemistry at the University of Sydney. Ab initio data points for the molecular dipole moment and polarisability surfaces were generated using GA U S S I A N 98 [27].
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M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
3. Results and discussion When an external electric field is applied along the ClH:N direction, the field preferentially stabilises geometries in which there is charge separation along this axis. A proton-shared structure is stabilised relative to a traditional hydrogen-bonded structure and, at sufficiently high fields, proton transfer occurs. The PESs generated in this study have been calculated at field strengths of 0.0025, 0.0055, and 0.0100 a.u. for illustrative purposes. These fields have been shown to correspond, respectively, to a traditional hydrogen-bonded complex, a proton-shared hydrogen-bonded com-
plex and a hydrogen-bonded ion pair [7]. The fields chosen, therefore, encompass dramatic structural change in the complex. Approximating the response of a molecule to an electric field by Eq. (1) is appealing because a knowledge of the zero-field molecular dipole moment and molecular polarisability surfaces can be used to determine molecular properties in a range of different environments. In this case, because of the symmetry imposed on the ClH:NH3 complex and the direction of the external electric field, the only relevant components of the dipole and polarisability are lz and azz , where z is the ClH:NH3 symmetry axis. These are plotted, in Figs. 1 and 2,
Fig. 1. The absolute value of the lz component (a.u.) of the ClH:NH3 molecular dipole moment vector as a function of the RClH and RNH bond lengths (a.u.). Contours are at intervals of 0.05 a.u.
Fig. 2. The azz component (a.u.) of the ClH:NH3 molecular polarisability tensor as a function of the RClH and RNH bond lengths (a.u.). Contours are at intervals of 10 a.u.
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
respectively, as functions of the RClH and RNH bond lengths. At this point it should be noted that accurate calculation of molecular polarisabilities is difficult and requires appropriate basis sets [28] and theoretical methods that adequately account for electron correlation [29]. Indeed this is not attempted here. Instead the polarisability, aðRÞ, has been calculated at the MP2/aug0 -ccpVDZ level of theory, consistent with the zero-field PES and the dipole moment surface and consistent with the identification of Eq. (1) as a Taylor Series. That is, while the values of a are not physically accurate, they are calculated in such a way that the expression a ¼ ol=oE remains valid, ensuring that errors
17
from such a calculation could be corrected by the inclusion of higher-order terms in Eq. (1). The approximations introduced by Eq. (1) are illustrated in Figs. 3 and 4 for the highest field considered, 0.0100 au. Panel (a) in Fig. 3 is the zero-field PES and panel (b) the exact 0.0100 a.u. field surface, both from [7]. It is apparent from Fig. 3 that the nature of the PES changes significantly in the presence of the field. Fig. 4 illustrates the difference between the exact PES at 0.0100 a.u. field and approximate PESs generated at first order (panel (a)) and second order (panel (b)) in E. The exact PES is indicated by the solid line and the approximate PES the dotted line. At both first and second order in E the approximate PES
Fig. 3. ClH:NH3 potential energy surfaces at zero-field (panel (a)) and 0.0100 a.u. field (panel (b)). Contour values are at intervals of 0.001 a.u. above the global minimum on each surface.
Fig. 4. The exact ClH:NH3 potential energy surface at 0.0100 a.u. field (solid line) and approximate surfaces generated at first order (panel (a)) and second order (panel (b)) in E (dotted lines). Contour values are at intervals of 0.001 a.u. above the global minimum on each surface.
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M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20
accurately reproduces the shape of the exact PES in the chemically relevant regions of configuration space, with the second-order PES being slightly more accurate. The PESs shown in Fig. 4, however, are illustrated with respect to the global minimum on each surface. It should be noted that the minimum energy on the first-order surface is 0.0023 EH above the minimum on the exact PES and the minimum on the second-order surface is 0.000046 EH below the minimum on the exact PES. Expectation values of the Cl–N and Cl–H distances, hRClN i and hRClH i, calculated from the ground state wavefunction at the various field strengths, are reported in Table 1. It is apparent from Table 1 that even large changes in the ClH:NH3 hydrogen-bond structure can be accurately predicted using Eq. (1) to perturb the zerofield PES. At the largest field considered, 0.0100 a.u., using the first-order term in E in Eq. (1) led to in hRClN i and 0.004 A in hRClH i. errors of 0.003 A Incorporation of the second-order term in E, that is, using the molecular polarisability surface, effectively corrected these errors and led to, at worst, in hRClH i at a field strength of an error of 0.001 A 0.0100 a.u. It can also be seen from Table 1 that the hRClH i distance is more sensitive to the approximations used in Eq. (1) than the hRClN i distance. Given that the heavy-atom distance better characterises the nature of the hydrogen bond [7,9,11,14,16], these results indicate that dramatic structural change can be accurately described using only the terms linear in E in Eq. (1). Two-dimensional anharmonic frequencies calculated from approximate PESs obtained using
first- and second-order corrections in E are compared to exact results in Table 2. The frequencies have been reported as a function of vibrational state for the three different field strengths considered. The notation (i; j) represents i quanta of excitation in the heavy-atom stretching mode and j in the proton-stretching mode. It is apparent from Table 2 that errors are larger using first-order corrections and increase with the number of quanta of excitation and the vibrational frequency. It can also be seen from Table 2 that the largest errors were obtained at the largest field considered. The errors observed at field strengths of 0.0025 and 0.0055 a.u., however, were similar. The vibrational frequency correlating most strongly with hydrogen bond structure type is the proton-stretching frequency, (0,1) [7,9,11,14,16]. Although the proton-stretching fundamental occurs at significantly higher energy than the heavy-atom motion, errors associated with the approximate PESs are small, at most 6 cm1 at a field strength of 0.0100 a.u. The sensitivity of the fundamental frequencies to the procedure used to generate the PES has been previously found to be at most 7 cm1 [7]. Furthermore, errors associated with the ab initio methods used to generate the data points are likely to be at least 20 cm1 . The errors seen here in the fundamental frequencies are therefore within the intrinsic errors associated with the PESs themselves. These results indicate that changes in vibrational frequency associated with change in the hydrogen bond structure can be accurately described by the first-order term in Eq. (1).
Table 1 ) in the ClH:NH3 complex as a Expectation values hRClN i and hRClH i in the ground vibrational state of the Cl–N and Cl–H distances (A function of field strength (a.u.) for the exact potential energy surface [7] and approximate potential energy surfaces calculated at first and second order in E Potential energy surface First order Field 0.0000 0.0025 0.0055 0.0100
hRClN i 2.940 2.885 2.895
Second order hRClH i 1.447 1.544 1.688
hRClN i 2.940 2.886 2.898
Exact hRClH i
hRClN i
hRClH i
1.448 1.546 1.692
3.016 2.940 2.885 2.898
1.392 1.448 1.545 1.692
M.J.T. Jordan, K.C. Thompson / Chemical Physics Letters 370 (2003) 14–20 Table 2 Two-dimensional anharmonic vibrational frequencies (cm1 ) for proton and heavy-atom motion in the ClH:NH3 complex as a function of field strength (a.u.) for the exact potential energy surface and approximate potential energy surfaces calculated at first and second order in Ea Vibrational assignment
0.0025 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
Potential energy surface First order
Second order
Exact
243 477 700 913 1117 1221 1685
244 478 701 915 1118 1220 1687
244 478 701 917 1127 1219 1740
19
wide range of external electric fields, both uniform and inhomogeneous. This method thus provides a simple and computationally tractable means for systematically studying the interactions of a molecule with a variety of solvents or molecular environments under a range of conditions. Acknowledgements
0.0055 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
370 673 966 1209 1458 966 1458
372 677 968 1216 1466 968 1466
372 677 967 1215 1464 967 1464
0.0100 a.u. Field (1,0) (2,0) (3,0) (4,0) (5,0) (0,1) (1,1)
387 756 1110 1454 1787 1131 1534
383 749 1100 1440 1769 1143 1547
383 749 1100 1438 1762 1137 1540
a
The notation (i,j) refers to i quanta of excitation in the heavy-atom vibration and j quanta in the proton-stretching vibration.
4. Summary The response of a molecule to an external electric field can be expanded as a power series in the field, Eq. (1). We have shown that, for external fields large enough to induce dramatic structural change in the ClH:NH3 model complex, quantitative prediction of both the structure and vibrational frequencies can be obtained using only the linear term in Eq. (1). Knowledge of how the electronic energy and dipole moment of a molecule change with molecular geometry can therefore be used to model the interaction of the molecule in a
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