Exploiting the quasi-invariance of atomic displacements

Spectrochimica Acta Part A 79 (2011) 2017–2019

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Exploiting the quasi-invariance of atomic displacements John Tomkinson ∗ , Stewart F. Parker ISIS Facility, Science and Technology Facilities Council, The Rutherford Appleton Laboratory, The Harwell Campus, OX 11 0 QX, UK

a r t i c l e

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Article history: Received 17 January 2011 Accepted 27 April 2011

a b s t r a c t The quasi-invariance of the magnitude of atomic displacements is briefly introduced and demonstrated for the case of indole. The usefulness of this property for the assignment of neutron vibrational spectra is underlined. © 2011 Published by Elsevier B.V.

Keywords: Eigenvectors Neutron scattering Scaling factors

The solution of the simultaneous equations of motion lies at the heart of vibrational spectroscopy: (F − E)K = 0

(1)

where E is the unit matrix and the  are eigenvalues of the force constant matrix, F, with K the column vector containing the mass weighted eigenvectors, k, of each mode, . The elements of k are normalised, where for atom, l, of mass, ml (of the n atoms in the molecule): v u √m l l v k = {v u ˜ l } for v u ˜ l =  (2)   v u )2 ) (m ( j j j=1,n The eigenvalues of Eq. (1) are available through solutions to its secular determinant but the eigenvectors are often ignored. Except, that is, when visualising vibrational modes; for this eigenvectors are essential since they are the patterns of atomic displacement of a molecule in its given modes. Pattern is an appropriate description, since only relative atomic displacements are determined. The mean squared displacement, u2 , in a mode at an energy transfer, ω (cm−1 ), is given in conventional units (Å2 ), by [1]: vu ˜2 l

 16.9  ωv

= v u2l ;

and u = |u|

(3)

Here, we focus on an unusual aspect of molecular vibrational eigenvectors; the quasi-invariance of the magnitudes of the atomic ul . The  ˜ ul values are quasi-invariant in that, for a displacements,  ˜ given molecular geometry, they vary little for any reasonable forcefield. Further, there is also little variation between the  ˜ ul values

∗ Corresponding author. Tel.: +44 01235 44 6686. E-mail address: [email protected] (J. Tomkinson). 1386-1425/$ – see front matter © 2011 Published by Elsevier B.V. doi:10.1016/j.saa.2011.04.066

obtained for the same force-field when applied to slightly different molecular geometries. It might be thought that this is simply a reflection of the mathematical property that eigenvectors are stationary [2], which has been exploited in the solving of Eq. (1) by iterating on approximate eigenvectors [3], or saving calculational effort in an era when computers were uncommon [4]. However, the quasi-invariance that we outline here stems not from a mathematical property of eigenvectors and eigenvalues but rather from the fundamental correctness of Bjerrum’s hypothesis that valence forces govern molecular vibrations [5]. The molecular shape determines where the major forces act and, hence, the overall pattern of displacements in an eigenvector. Other interactions refine the directions of atomic displacement but, for a given structure, have a modest impact on the magnitude of the displacement. This is most obvious in the case of highly symmetric molecules where the matrix F is reduced to a block diagonal form and off-diagonal blocks are zero. But even where there is no symmetry the form of the eigenvectors is restrained by the molecular shape [6]. In Fig. 1 we compare two calculated sets of ˜ for the vibrations of benin-plane atomic displacement vectors, u, zene. (These were calculated under the most severe (D6h ), and least severe (Cs ), symmetry constraint using a high, and a low, quality force-field respectively, see below.) The displacement directions are clearly different in the two cases but both sets cover the same u values. range of ˜ Most vibrational spectroscopy assignments appeal to ab initio methods to calculate the harmonic vibrations of the molecule under study. Then, by comparing experimental and computed frequencies, one may determine which transitions are associated with particular modes of vibration. The more accurate the calculated frequencies are, the easier it is to relate them to the observed data and much effort is expended in choosing appropriate techniques, functionals and basis sets. We shall show that, irrespective of the quality

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J. Tomkinson, S.F. Parker / Spectrochimica Acta Part A 79 (2011) 2017–2019

Fig. 1. Showing the distribution of the normalised eigenvectors of the in-plane vibrations of benzene for two slightly different structural models calculated with different force-fields, see text, colours distinguish the two sets. Each point gives the x, y coordinates of the head of an atomic displacement vector with (0,0) as their origin. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

of the calculation, the u values display only modest changes, which is an important finding for neutron vibrational spectroscopy. Molecular vibrational spectroscopy with neutrons uses the well established Inelastic Neutron Scattering (INS) technique that is broadly available today [7]. In INS, neutrons, instead of the photons of optical spectroscopy, are used to excite the molecular modes and the spectral transition strengths are straightforwardly related to the u values. In a simplified form; the contribution to the INS intensity, S(Q,ω␯ )• l , arising from an atom with crosssection,  l , is given by [7]: •

S(Q, ωv )l = Q 2 v u2l exp(Q 2

 v

v u2 ) l

(4)

where the momentum transfer, Q, Å−1 , is determined by the neutron spectrometer used. The GAUSSIAN03 program [8] was used to calculate the eigenvalues and eigenvectors of several molecules, some with optimised geometries and some not, using different methods; simple mechanics UFF, H-F, DFT, B3LYP, B3PW91 and different basis sets, from STO-3G to 6-31G++ (d,p). Here, we report results for a typical planar organic molecule of low symmetry, indole (Cs ). We can conveniently demonstrate the quasi-invariance of the calculated u diagrammatically, using ACLIMAX [9]. This free-ware program takes GAUSSIAN output files and produces the related S(Q,ω␯ )• l , using more sophisticated variants of Eq. (4). The results emphasise the motions of hydrogen, since its displacements are greater and its crosssection bigger than for other atoms. However, before we can successfully compare the S(Q,ω␯ )• l values of the different calculational methods we must overcome the problem of poorly calculated eigenvalues. Each calculational method associates a given eigenvector with a specific ω␯ , these eigenvalues are strongly dependant on the method used and, so, differ widely. However, as we see from Eq. (3), the values of  u2 l are related to ω␯ and, to compare S(Q,ω␯ )• l from different calculations correctly, they must be calculated at the same ω␯ . Fortunately, we can use ViPA [10] to identify those eigenvectors of different calculational methods that are closely related in displacement pattern. Then, with this information, we bring the different ω␯ of related eigenvectors to a common value, which is then used in the calculation of S(Q,ω␯ )• l .

Fig. 2. Showing how the S(Q,ω␯ )• l values obtained for four different ‘poor quality calculations’ ( (black), #1; (blue), #2; (green), #3; and (red), #4) correlate with those obtained for a ‘high quality calculation’: see main text for details. The inset diagram shows the valence bond description of the molecular structure of indole. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)

In Fig. 2 we correlate indole’s S(Q,ω␯ )• l values for a ‘good quality calculation’ (structure optimised under DFT/B3LYP/6-31G and its dynamics calculated at the same level), with other ‘low quality calculations’. Of the low-quality calculations; the first, used the good-quality structure with the dynamics calculated under the STO3G basis, which gave a correlation factor, Rf , of 0.93. The second, used a non-optimised structure [11] with dynamics calculated under the mechanics/UFF method, Rf , ∼0.88. The third, used the non-optimised structure [11] with dynamics calculated under DFT/B3LYP/6-31G, Rf , ∼0.85. The fourth, used the structure optimised under HF/STO3G with dynamics calculated at the same level, Rf , ∼0.76. (The average, for all four groups of data, gave Rf , ∼0.84.) The relatively good showing of the non-optimised structure, #2 above, compared to an optimised structure, #4 above, probably relates to the optimised source structure used in its generation [11]. Higher symmetry molecules, such as benzene, give better correlations between the results from non-optimised/UFF and fully optimised/DFT/B3LYP/6-31G, Rf , ∼0.92. However, as minor differences develop between the structures used to represent the same molecule, so the degree of correlation amongst their eigenvectors falls; a 20◦ displacement of the nitrogen atom from the indole molecular plane finds Rf , ∼0.3. Globular molecules, like adamantane (Td ), have relatively modest coefficients, Rf , ∼0.74 and are more sensitive to structural differences. Eventually, the correlations falter badly if the calculational methods obtain quite different optimised structures for the molecule under consideration and in these circumstances the u values are no longer quasi-invariant. These results become less surprising when we recall that the eigenvectors of a short, one-dimensional, linear chain of n atoms are fixed and given by [12]: vu ˜ 2l =



sin(vl/(n + 1)) 2



(5)

(mj (vj/(n + 1))) ) j=1,n

Here the eigenvectors are constrained such that they lie along the chain and the molecular centre of mass does not move, these requirements fix the eigenvectors. When we move to twodimensional, planar, systems the displacements are no longer fixed but they are restricted. It is this restriction that reduces the range u2 l values that can occur for similar structures. Unsurprisof  ˜

J. Tomkinson, S.F. Parker / Spectrochimica Acta Part A 79 (2011) 2017–2019

ingly, three-dimensional, globular, molecules are least restricted and here, as we see above, slight differences in the geometric u2 l description of the molecule lead to wider variations in the  ˜ values. It must not be imagined that, based on the quasi-invariance of u, we need only use the very simplest calculational methods to enable us to assign INS spectra. Poor quality calculations still produce poor eigenvalues and the u values will never be so well calculated, or measured, that they can be used alone to circumvent this difficulty. However, even with a good quality calculation there will still be a need to scale individual eigenvalues to match the observed INS spectrum but, to date, the scaling factors have been modest—generally staying within the limits suggested by the mathematical stationary properties of eigenvalues and eigenvectors. We can now see that this limitation is overly restrictive: definitely, in the case of linear molecules; almost certainly, in the case of planar molecules; and probably, in the case of globular molecules. We may conclude that calculated eigenvalues can be scaled to any desired extent, without concern that the resulting S(Q,ω␯ )• l values will be very wrong. References [1] S.J. Cyvin, Molecular Vibrations and Mean Square Amplitudes, Elsevier, Amsterdam, 1968.

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[2] K.F. Riley, M.P. Hobson, S.J. Bence, Mathematical Methods for Physics and Engineering, Cambridge University Press, 2006, p. 8.17.1. [3] E. Bright-Wilson, J.C. Decius, P.C. Cross, Molecular Vibrations, Dover Publications Inc., New York, 1980, pp. 9.6–9.7. [4] P. Gans, Vibrating Molecules, Chapman and Hall Ltd., London, 1971, p. 50. [5] G. Hertzberg, Molecular Spectra and Molecular Structure, vol. II, Van Nostrand, New York, 1945, p. 168. [6] M. Lu, J. Ma, Biophys. J. 89 (2005) 2395–2401. [7] P.C.H. Mitchell, S.F. Parker, A.J. Ramirez-Cuesta, J. Tomkinson, Vibrational Spectroscopy with Neutrons, World Scientific Press, Singapore, 2005. [8] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery Jr., R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, J.L. Andres, C. Gonzalez, M. Head-Gordon, E.S. Replogle, J.A. Pople, GAUSSIAN03, Revision B.05, Gaussian, Inc., Wallingford, 1998. [9] A.J. Ramirez-Cuesta, Comput. Phys. Commun. 157 (2004) 226–238. [10] A.K. Grafton, R.A. Wheeler, Comput. Phys. Commun. 113 (1998) 78– 84. [11] The input structure was that of iso-indene, C2v , from Gauss-View with one carbon replaced by a nitrogen atom. The delocalised bonding description, as given by Gauss-View, was then changed into the valence-bond pattern of single and double bonds, see Fig. 1. [12] For example: N.W. Ashcroft and N.D. Mermin ‘Solid State Physics’ Saunders College Publishing, Fort Worth, 1976.