Density functional studies of molecular polarizabilities. Part 3; ethene, buta-1,3-diene and hexa-1,3,5-triene

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1 ELECTRONIC JOURNAL OF THEORETICAL CHEMISTRY, VOL. 2, (1997) Density functional studies of molecular polarizabilities. Part 3; ethene, buta-1,3-diene and hexa-1,3,5-triene ALAN HINCHLIFFE 1 AND HUMBERTO J. SOSCÚN-M 2 1 Department of Chemistry, UMIST, Sackville Street, Manchester M60 1QD, UK 2 Departamento de Química, Fac. Exp. de Ciencias, La Universidad del Zulia, Grano de Oro Módulo No. 2, Maracaibo, Venezuela SUMMARY We report accurate ab initio studies of the dipole polarizabilities of the title molecules at the Hartree Fock and density functional levels of theory. The molecular geometry was optimized in each case at the HF/6-311G(3d,2p) level of theory, and polarizabilities were then calculated with and without the addition of additional diffuse functions. In the case of ethene, 18 different DFT formulations were used, as follows; all combinations of the GAUSSIAN94 three inbuilt exchange functionals (Slater; Xα and Becke88) with the six inbuilt correlation functionals (Vosko Wilk and Nusair = VWN; Modified VWN, Lee Yang and Parr; Perdew Local; Perdew 86 and Perdew/Wang 91). The remaining molecules were treated using the BLYP density functional formulation. We also give semi-empirical (AM1) polarizability calculations for comparison. We also report a successful attempt to correlate the mean molecular polarizability α withthe molecular volume as calculated by molecular mechanics by John Wiley & Sons, Ltd. Received 15 April 1997; Accepted 15 May 1997 Electron. J. Theor. Chem., Vol. 2, (1997) No. of Figures: 0 No. of Tables: 12 No. of References: 20 KEY WORDS polarizability; density functional; molecular mechanics; ab initio 1. INTRODUCTION The electric dipole moment p e of a molecule is a quantity of fundamental importance in structural chemistry. When a molecule is subject to an external electric field E, the molecular charge density rearranges and hence the dipole moment can change [1]. This change is described by the equation p e (E) =p e (E = 0)+α E + 1 2E β E + (1) where α is a second rank tensor property called the dipole polarizability, and β is the first of a series of dipole hyperpolarizabilities. Here p e (E = 0) is the electric dipole moment in the absence of a field, and p e (E) is the dipole moment with the field. p e (E = 0) is usually referred to as the permanent electric dipole moment. The hyperpolarizabilities are known to be small, and their effect is minimal for weak electric fields. They are however important quantities when the electric field is large. For a molecule such as benzene that has a centre of symmetry, the first hyperpolarizabilities are identically zero. Correspondence to: Alan Hinchliffe. CCC /97/ $ by John Wiley & Sons, Ltd.

2 316 A. HINCHLIFFE AND H. J. SOSCÚN-M Because the electric dipole moment changes when an external field is applied, the molecular energy W also changes according to the equation W (E) =W (E = 0) p e E (2) Molecular polarizabilities and hyperpolarizabilities can be calculated from either expression. In the early days, authors used (numerical) finite difference techniques. In recent years, direct analytical differentiation of the electric dipole moment or molecular energy with respect to the applied field has become the accepted route to these properties, provide that an analytical expression has been evaluated for the particular level of theory. In the case of frequency-dependent electric fields such as those in electromagnetic radiation, the polarizability turns out to depend on the frequency of the radiation in a complicated way and so we have to distinguish carefully between the static polarizability (the value we would get for a constant electric field) and the frequency-dependent polarizability (the value we get from experiments involving oscillating electric fields with a given frequency). It is normal to state the frequency when quoting frequency-dependent polarizabilities, and authors usually refer to frequency-dependent polarizabilities as dynamic polarizabilities. It has long been known that those organic molecules that have conjugated π-electron systems show interesting non-linear optical behaviour. Studies of the dipole polarizability and hyperpolarizabilities of such conjugated molecules are therefore timely because of their importance in the new experimental field of non-linear optics [2]. Over the years, a number of authors have reported quantum mechanical studies of the dipole polarizability (and dipole hyperpolarizability) of such molecules. In view of the size of systems such as fulvene, benzene, naphthalene and anthracene, most of the calculations to date have been done at the empirical or semi-empirical level of theory. In a later section, we will report AM1, MNDO and PM3 results; indeed some semi-empirical packages such as MOPAC have polarizability calculations built in as optional properties to be determined once the Hartree Fock (HF) wave function has been calculated. The classic paper for ab initio HF calculations on linear polyenes is that by Hurst, Dupuis and Clementi [3]. These authors used a variety of atomic orbital basis sets. In an earlier series of papers, we have reported the results of high-quality ab initio calculations of the dipole polarizabilities of a range of conjugated molecules [4]. Our calculations were mostly made with good-quality atomic orbital Gaussian basis sets and at the HF level of theory. It is usually possible to obtain respectable agreement with experiment for the dipole polarizability α provided that a careful choice of atomic orbital basis set is made. Until fairly recently, the most usual method of treating electron correlation in such large molecules was the Møller Plesset perturbation technique. Such calculations are labelled MPn where n is the order of perturbation. Most post-hf techniques have a common feature that they are extremely expensive in computer resource; MPn calculations usually involve the semi-transformation of integrals from the atomic orbital basis set to the molecular orbital basis set, and this single step can be prohibitive in disk space. In recent years, density functional techniques have received a great deal of attention in the literature. The idea is to start from the HF electronic energy expression [5] ɛ el = trace(h 1 P 1 )+ 1 2 trace(p 1J) 1 4 trace(p 1K) (3) which relates the electronic energy for a one-determinant closed shell ɛ el to the electron density matrix P 1, the matrix of one-electron integrals h 1, the coulomb matrix J and the exchange matrix K.

3 DENSITY FUNCTIONAL STUDIES OF MOLECULAR POLARIZABILITIES. PART Density functional theory seeks to write the energy expression as ɛ el = trace(h 1 P 1 )+ 1 2 trace(p 1J)+ɛ X + ɛ C (4) where ɛ X is the exchange functional and ɛ C is the correlation functional, which is of course zero for HF wave functions. In order to calculate ɛ X and ɛ C it is necessary to assume some functional form for the two potentials and then calculate the contribution to the electronic energy as an integral over the electron density (and occasionally the gradient of the electron density). These calculations are usually performed numerically and tend to consume less computer resource than traditional MPn calculations. Several different functionals are available for both ɛ X and for ɛ C. For example, GAUSSIAN94 has three inbuilt exchange functionals (Slater; Xα and Becke88) and six inbuilt correlation functionals (Vosko Wilk and Nusair = VWN; Modified VWN Lee Yang and Parr; Perdew Local; Perdew 86 and Perdew/Wang 91) [6]. The application of density functional methods to the study of molecular properties is a recent development, and there is no great pool of expertise to suggest that one formulation is better than any other for the calculation of a given property. As part of a larger study, we have investigated the density functional treatment of the polarizabilities of the title molecules, and the aim of this paper is to present our results. 2. CALCULATIONS 2.1. Ethene First of all we consider HF and MP2 geometryoptimizations (Table 1). Table 1. Comparison of the predicted and calculated geometries of ethene. The atomic unit of energy E h (the hartree) is approximately J Level of theory R CC (pm) R CH (pm) CCH angle (degrees) E (E h) HF/6-311G(3d,2p) MP2/6-311G(3d,2p) HF/ G(3d,2p) Experiment [7] The HF/6-311(3d,2p) geometry optimization gives a very reasonable molecular geometry for ethene, typical of large basis set HF calculations. The main effect of electron correlation (at the MP2 level of theory) is to increase the CC bond length by 2 pm and so give better agreement with experiment. These effects are very well known, and for our remaining geometry calculations we will simply report the ab initio result without comment. It is necessary to add further diffuse basis functions to standard basis sets in order to study molecular phenomena that involve the outer electron density accurately. These functions are needed in order to study Rydberg states, anions and polarizabilities. Most ab initio packages allow standard basis sets to be augmented with such diffuse functions, and our HF/ G(3d,2p) geometry optimization shows that their effect is essentially negligible for geometry prediction. There are a number of ab initio calculations of the polarizability of ethene in the literature, ranging from minimal basis set HF calculations to large basis set MP2 and many body ones [8]. There is good experimental data for both the static and dynamic polarizabilities. The static experimental value has

4 318 A. HINCHLIFFE AND H. J. SOSCÚN-M been obtained from a mixture of refractive index and Rayleigh scattering dispersion measurements. The dynamic values have been obtained using electromagnetic radiation of wavelength nm in a number of different experiments (rotational Raman spectroscopy, refractive index studies and Rayleigh scattering). In our studies of the (static) polarizabilities, we have chosen to use those standard basis sets available in a typical ab initio package such as GAUSSIAN94. Other authors have taken a different approach and have sought to develop systematically basis sets of highly polarized functions specially designed to provide accurate estimates of these properties [8]. The input to all our polarizability calculations is a molecular geometry optimized at the HF/6-311(3d,2p) level of theory. Ideally one would optimize the molecular geometry at the relevant level of theory before calculating the polarizability; if one s goal in such calculations were the prediction of force constants then this step would be essential because force constants are calculated as the second derivatives of the energy with respect to the corresponding normal coordinate at the potential minimum. Dipole polarizabilities are usually calculated as derivatives of the dipole (or energy) expression given above with respect to the applied electric field, and the procedure we have used is therefore acceptable (and much less costly). Of particular importance for the calculation of dipole polarizability are the extra diffuse functions on each centre, and these are indicated by the notation ++. The largest basis set G(3d,2p) therefore involved 138 primitive Gaussians contracted to 104 basis functions. For conjugated molecules in their principal axis system, it is conventional to write the three principal values of the polarizability tensor as α LL, α MM and α NN. The subscripts L, N and M are understood to refer to the long axis (which is in the molecular plane and contains most of the atoms), the normal axis which is normal to the molecular plane and the remaining medium axis. Quantities of interest to experimentalists are the mean polarizability α and the anisotropy α, defined in terms of these principal values as α = 1 3 (α LL + α MM + α NN ) α = ( 1 2 ((α LL α MM ) 2 +(α LL α NN ) 2 +(α MM α NN ) 2 )) 1/2 (5) Our HF and MP2 polarizability calculations for ethene are shown in Table 2. Table 2. HF and MP2 polarizability calculations for ethene at the HF/6-311(3d,2p) geometry. Table values are atomic units (a.u.). The atomic unit of polarizability is e 2 a 2 0 E 1 h and the following conversion factors are appropriate; 1 a.u. of polarizability = C 2 m 2 J 1 and this is equivalent to cm 3 Level of theory α NN α MM α LL α α HF/6-311G(3d,2p) MP2/6-311G(3d,2p) HF/ G(3d,2p) MP2/ G(3d,2p) Experiment [9] Experiment [10] Previous work (e.g. [3]) shows that small basis set calculations give a very poor value for the component of α perpendicular to the molecular plane (the NN component). For example, a HF/6-31G* calculation gives just a.u. Large basis sets correct this difficulty, and our HF calculations of α NN

5 DENSITY FUNCTIONAL STUDIES OF MOLECULAR POLARIZABILITIES. PART are in very reasonable agreement with experiment. The effect of including electron correlation at the MP2 level for a given basis set is to increase α MM by some 3 4%, to decrease α LL by about 5% and to leave the perpendicular component roughly unchanged. In order to investigate the density functional model, we took all 18 combinations of the (three) exchange and (six) correlation functionals given as standard options in GAUSSIAN 94, and calculated the polarizability using the G(3d,2p) basis set, using the standard geometry found from a HF/6-311G(3d,2p) optimization. The DFT acronyms are summarized in Table 3 below. Table 3. Acronyms used in density functional calculations, together with primary references Acronym Description Ref. Exchange functionals S Slater s local spin density exchange functional, equal to 2/3 ρ 4/3 11 where ρ is the electron density at each point in space XA Variation on S above with a factor of α replacing the 2/3 11 B Becke s 1988 exchange functional which corrects S above for the 12 gradient of ρ at points in space Correlation functionals VWN Vosko, Wilk and Nusair uniform electron gas functional 13 VWN5 Refinement of VWN 13 LYP The correlation functional of Lee, Yang and Parr which 14 includes local and non-local terms. PL The local electron density functional of Perdew 15 P86 The electron density gradient corrections to Perdew PL above 16 PW91 Perdew and Wang s 1991 gradient-corrected correlation 17 functional A typical calculation is referenced by the exchange functional abbreviation plus that for the correlation functional. Thus, the acronym BLYP means a Becke 88 exchange functional combined with the Lee Yang Parrcorrelationfunctional (Table 4). The Xα exchange functional gives the smallest values of the mean polarizability and the polarizability anisotropy, but there seems little to choose between the other combinations of exchange/correlation functionals. For the remaining density functionals we have therefore chosen to standardize on the BLYP combination Buta-1,3-diene We considered both the cis and trans isomers, and first optimized the geometry at the HF/6-311G(3d,2p) level of theory. The Cartesian coordinates are given in Tables 5 and 6 for reference, and in either case the molecule lies in the x y plane. Those extra, diffuse functions denoted ++ in GAUSSIAN94 were then added to the basis set, and the polarizability was calculated at the HF/ G(3d,2p) and BLYP/ G(3d,2p) levels of theory. The results are shown in the Tables 7 and 8. The only experimental value in the literature is the mean polarizability for the trans isomer, and both our calculations are in reasonable agreement with this value.

6 320 A. HINCHLIFFE AND H. J. SOSCÚN-M Table 4. Density functional polarizability calculations using the G(3d,2p) standard basis set at the HF/6-311G(3d,2p) geometry. Table values are a.u. Level of theory α NN α MM α LL α α = X/ G(3d,2p) where X = SVWN SVWN SLYP SPL SP SPW XAVWN XAVWN XALYP XAPL XAP XAPW BVWN BVWN BLYP BALP BAP BPW Table 5. Cartesian coordinates for trans buta-1,3-diene, calculated at the HF/6-311G(3d,2p) level of theory Coordinates (Å) Atom x y z C C C C H H H H H H

7 DENSITY FUNCTIONAL STUDIES OF MOLECULAR POLARIZABILITIES. PART Table 6. Cartesian coordinates for cis buta-1,3-diene, calculated at the HF/6-311G(3d,2p) level of theory Coordinates (Å) Atom x y z C C C C H H H H H H Table 7. trans Buta-1,3-diene dipole polarizability calculations using the G(3d,2p) standard basis set at the HF/6-311G(3d,2p) geometry. Table values are a.u. trans α NN α MM α LL α α HF/ G(3d,2p) BLYP/ G(3d,2p) Experimental [18] Table 8. cis Buta-1,3-diene dipole polarizability calculations using the G(3d,2p) standard basis set at the HF/6-311G(3d,2p) geometry. Table values are a.u. cis α NN α MM α LL α α HF/ G(3d,2p) BLYP/ G(3d,2p)

8 322 A. HINCHLIFFE AND H. J. SOSCÚN-M Table 9. Cartesian coordinates for hexa-1,3,5-triene, calculated at the HF/6-311G(3d,2p) level of theory Coordinates (Å) Atom x y z C C C C C C H H H H H H H H Hexatriene We simply considered the linear configuration. A geometry optimization was performed at the HF/6-311G(3d,2p) level of theory, and the resulting Cartesian coordinates are shown in Table 9. Polarizabilities were then calculated using this geometry at the HF/ G(3d,2p) and BLYP/ G(3d,2p) levels of theory. The results are shown in Table 10. Table 10. Hexa-1,3,5-triene dipole polarizability calculations using the G(3d,2p) standard basis set at the HF/6-311G(3d,2p) geometry. Table values are a.u. trans α NN α MM α LL α α HF/ G(3d,2p) BLYP/ G(3d,2p) SEMI-EMPIRICAL CALCULATIONS Ab initio polarizability calculations are expensive even for such simple conjugated molecules, and both empirical and semi-empirical models offer attractive routes to these quantities. Table 11 shows the result of an AM1 investigation. In each case, the molecular geometry was optimized before the dipole polarizability calculation, and we used MOPAC [19] to do the calculations. Table 11. AM1 polarizability calculations. Table values are a.u. Molecule α NN α MM α LL α α ethene trans buta-1,3-diene cis buta-1,3-diene hexa-1,3,5-triene

9 DENSITY FUNCTIONAL STUDIES OF MOLECULAR POLARIZABILITIES. PART Table 12. Correlation of the molecular mechanics molecular volumes calculated using the DTMM package with the BLYP mean molecular polarizabilities Molecule DTMM volume (Å 3 ) BLYP α (a.u.) ethene cis buta-1,3-diene trans buta-1.3-diene trans hexa-1,3,5-triene The main problem with all such semi-empirical calculations is that they give very small values for the component of α perpendicular to the molecular plane. This behaviour is well documented and well understood [5]. The in-plane values are in reasonable agreement with the ab initio values, and could be easily scaled for predictive purposes Molecular volumes It is an old idea that the polarizability of a molecule is directly related to its volume. A simple classical argument runs as follows; consider a nucleus of charge q surrounded by a spherical distribution of (negative) electron density of radius a. An applied electric field of magnitude E displaces the nucleus by distance d (relative to the electron distribution). Balancing the forces on the nucleus, which are qe from the applied electric field and q 2 d/4πɛ 0 a 3 from Gauss flux theorem [1], and noting that the induced electric dipole moment is qd gives an expression for the dipole polarizability (on rearrangement) as α = 4πɛ 0 a 3. Now a 3 is the volume of the atomic sphere and so this simple argument gives a relationship between the dipole polarizability α and the atomic volume a 3. It is conventional to carry this relationship to molecules with volume V, where we relate the molecular volume V to the mean polarizability α. Because of this simple relationship, people often speak about polarizability volume, and indeed a repeat derivation in cgs gives a direct relationship (without the 4πɛ 0 ) between polarizability and volume. Molecular volumes can be modelled at the semi-empirical and ab initio levels of theory. A rough and ready estimate can be obtained from molecular mechanics calculations, and Table 12 shows the results given by the DeskTop Molecular Modeller (DTMM3) package [20]. What we did was to optimize the molecular geometry, in each case according to the standard DTMM3 algorithm, and then calculate the molecular volume. This gives a good straight line fit with a correlation coefficient of r 2 = 96%, suggesting that the mean value of the polarizability tensor can be calculated very simply from the molecular volume as calculated by standard molecular mechanics. REFERENCES 1. A. Hinchliffe and R. W. Munn, Molecular Electromagnetism, John Wiley and Sons, Chichester (1985). 2. J. H. Day, Chem. Rev., 53, 167 (1953). 3. G. J. B. Hurst, M. Dupuis and E. Clementi, J. Chem. Phys., 89, 385 (1988). 4. A. Hinchliffe and H. J. Soscún-M, J. Mol. Struct., 300, 1 (1993). 5. A. Hinchliffe, Modelling Molecular Structures, John Wiley & Sons, Chichester (1996). 6. Gaussian 94, Revision B.2, M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill, B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A. Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L. Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox, J. S. Binkley, D. J. Defrees, J. Baker, J. J. P. Stewart, M. Head-Gordon, C. Gonzalez and J. A. Pople, Gaussian, Inc., Pittsburgh PA, USA (1995).

10 324 A. HINCHLIFFE AND H. J. SOSCÚN-M 7. M. D. Garmony, V. W. Laurie, R. L. Kuczkowsky, R. H. Schwendeman, D. A. Ramsay, F. J. Loras, W. J. Lafferty and A. A. G. Maki, J. Phys. Chem. Ref. Data, 8, 619 (1979). 8. J. J. Perez and A. J. Sadlej, J. Mol. Struct. (THEOCHEM), 371, 31 (1996) and references therein. 9. G. W. Hills and W. J. Jones, J. Chem. Soc., Faraday Trans. 2, 71, 812 (1975). 10. M. P. Bogaard, A. D. Buckingham and G. L. D. Ritchie, Chem. Phys. Lett., 74, 3008 (1978). 11. P. Hohenberg and W. Kohn, Phys. Rev., 136, B864 (1964); W. Kohn and L. J. Sham, Phys. Rev., A140, 1133 (1965). 12. A. D. Becke, Phys. Rev., A38, 3098 (1988). 13. S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys., 58, 1200 (1980); C. Lee, W. Yang and R. G. Parr, Phys. Rev., B37, 785 (1988); B. Miehlich, A. Savin, H. Stoll and H. Preuss, Chem. Phys. Lett., 157, 200 (1989). 14. C. Lee, W. Yang and R. G. Parr, Phys. Rev., B37, 785 (1988). 15. J. P. Perdew and A. Zunger, Phys. Rev., B23, 5048 (1981). 16. J. P. Perdew, Phys. Rev., B33, 8822 (1986). 17. J. P. Perdew and Y. Wang, Phys. Rev., B45, (1992). 18. R. C. Weast, M. J. Astle and W. B. Beyer (eds) Handbook of Chemistry and Physics, 66th edn, CRC, Boca Raton (1985). 19. J. J. P. Stewart, MOPAC Manual, Fujitsu Limited, Tokyo, Japan (1993). 20. J. C. Crabbe, J. R. Appleyard and C. R. Lay, DTMM3, Desktop Molecular Modeller 3.0, Oxford University Press, Oxford (1994).