J. Phys. Chem. 1994,98, 69036905
6903
Polarizability of Molecular Orbitals Douglas A. Smith' and Michael E. Mura Department of Chemistry, The Unviersity of Toledo, Toledo, Ohio 43606-3390 Received: March 9, 1994; In Final Form: May 27, 1994'
A finite field method to examine the polarizability and hyperpolarizabilities of individual molecular orbitals under the influence of an applied static electric field vector has been tested on two molecules, H F and C2H2. The results indicate that core u orbitals can be significantly more (hyper)polarizable than valence and ?r orbitals but may not contribute significantly to the total (hyper)polarizability. The effect of local environmental effects on valence orbitals can be qualitatively and quantitatively examined.
Introduction The concept of polarizability is important in many diverse areas of chemistry.' Integrated Raman intensities are proportional to the square of the derivative of a with respect to the corresponding normal mode. The dielectric constant, usually simplified by assuming free rotation to give the Langevin-Debye formula, and induction and dispersion energies, which are used in the Slater-Kirkwood formulation of Lennard-Jones parameters, are related to a. The refractive index depends on the frequencydependent polarizability. The polarizability is related to pK, and inductive effects? is used to calculate the electrostatic interaction en erg^,^ has been correlated with the Hansch hydrophobic parameter? and has been utilized as a component in developing theoretical linear free energy relationships (TLFER).s More recently, polarizability and hyperpolarizabilities have been of interest in calculations of nonlinear optical (NLO) properties of molecules.6 The interaction of light or electromagnetic fields with a molecule can result in a polarizationof the charge distribution. The linear response to the applied or external field is characterized by the polarizability, a,while the nonlinear responses are termed the first, second, etc., hyperpolarizabilitiesand are given the symbols 6,y, and so on. The standard definition of these quantities is as the second (for a),third (for /I) and , fourth (for y) derivatives of the energy with respect to an applied electric field evaluated at zero field strength.' The subscripts associated with the polarizability and hyperpolarizabilities (e.g., axxand yxxxx; cf. Tables 1-4) denote the component of the (hyper)polarizability tensor, and x is the long axis of the molecule. Within the finite field approach,8 the energy of a molecule interacting with an electromagnetic field can be expressed as
where p0 is the permanent dipole moment of the molecule and at/,&k, and yrjkI are the tensor elements of the linear polarizability and the first and second hyperpolarizabilities, respectively, and the Einstein convention of summed repeated indices has been used. As part of our NLO program, we have an interest in correlating a and the second hyperpolarizability, y, with molecular and electronicstructurein organic systems. We have observedvisually in AM1 semiempiricalstudies that there are significant perturbations in molecular orbitals as a result of applying an electric field vector in thecalc~lation.~ Furthermore, these perturbations are significant in many orbitals other than the highest lying and/ or T molecular orbitals. In order to confirm that these results Abstract published in Aduance ACS Abstracrs, July 1, 1994.
0022-365419412098-6903$04.50/0
were not an artifact of the finite field method implemented in MOPAC,"J the visualization routine, or due to an underlying error in our approach, we have developed a theory of molecular orbital polarizability and performed tests on two small molecule systems, H F and CzH,.
Theory and Results A particular molecular orbital, $I,may be given as a linear combination of the n atomic orbitals, cp, where the set cij is composed of the eigenvector (LCAO) coefficients for the ith molecular orbital (eq 2). We use qito denote the ith molecular orbital dipole moment integral for an applied electric field in the r direction (eq 3). Substituting for $1 in the molecular orbital moment integral of eq 2 with the summation of eq 3 gives an expression for the moment integral in terms of the eigenvector coefficients(assumedhere to be real) and atomicmoment integrals (eq 4). Since the Born-Oppenheimer approximation is used, no nuclear relaxation occurs as a function of the field vector, and the atomic moment integrals are independent of applied electric field. Thus, only the LCAO coefficients may vary with applied electric field.
(4)
The finite field method can be used to calculate the electric properties by fitting a polynomial to a set of moments q, calculated for a set of small applied electric fields. Thus, the polarizability tensor element at,of the ith molecular orbital, $1, may be calculated from the first derivative of qiwith respect to the field F6 aligned along the direction s, the first hyperpolarizability tensor element &,$ from the second derivative, the second hyperpolarizabilityy:,##from the third derivative, and so on. It can be shown that the sum over the m occupied orbitals of a molecular orbital property yields the total tensor element (eq 5).
We have applied this method using the Gaussian 90 package of programs." All calculations were performed at the HF/631G** level12 utilizing applied electric field vectors of 0.000, 0 1994 American Chemical Society
6904 The Journal of Physical Chemistry, Vol. 98, No. 28, 1994
TABLE 1: HF/6-31G** Results for HF in a Static Electric Field Perpendicular to the Internuclear Axis MO symmetry ffxx Yxxxx 1 2 3 4 5
0.0008 1.0450 0.6726 0.3642 -0.1048
SG SG SG PI PI
0.0826 -17.0931 80.2486 2.6155 -18.2581
TABLE 2 HF/631C** Results for HF in a Static Electric Field Parallel to the Internuclear Axis MO symmetrv ff Brz, II
1 2 3 4 5
SG SG SG PI PI
0.0008 1.2504 2.3091 0.3496 0.3496
Letters
TABLE 3: HF/6-31G** Results for Cfi in a Static Electric Field Perpendicular to the C==C Axis and Molecular Plane MO symmetry Y ffXX
1 2 3 4
5 6 7 8
AG BlU AG BlU B2U AG B3U B3U
0.0055 0.0055 6.2653 12.2383 14.2353 -9.9376 -3.5102 0.5452
XXXX
-0.0669 -0.08 18 -109.1834 -476.9060 -2237.6748 2456.4901 741.7608 -1.5005
YZZZZ
0.0015 1.8575 -16.7954 0.3965 0.3965
-0.0201 -13.5289 137.4688 -3.5345 -3.5345
f0.001,and f0.002au. Thedegreeofconvergencein thedensity matrix necessary to obtain consistent polarizabilities and hyperpolarizabilities was examined by setting the SCF convergence parameter CONV to 6,8, 10, 12,and 14;for CONV = N the difference between the largest elements of two successive density matrices is not greater than leN. The results were system dependent and are discussed below. For the HFsystem, the optimized bond length of 0.9170A was held constant during the finite field calculations; the results for CONV = 8 are presented in Tables 1 and 2. At this level, the convergence of the molecular orbital polarizabilities, aix perpendicular to the bond axis, converge less quickly than those in parallel to the bond axis, a:,, although the variation is in the fourth decimal place. The perpendicular first hyperpolarizability components are zero by symmetry, and the parallel components are unchanged for the occupied molecular orbitals are unchanged over the range of CONV examined; for the components of y1 the parallel values vary only in the fourth significant figure while the perpendicular components can vary within an order of magnitude. While our method also gives values for the third hyperpolarizability, 6l, the values obtained are inaccurate due to insufficient precision in the calculation of the eigenvectors and are thus not reported. The most (hyper)polarizable orbital in the H F system is MO 3, the a* antibonding orbital between hydrogen and fluorine. MO 2,the u bonding orbital, is only slightly less polarizable but has a significantly larger first and second hyperpolarizability. The two highest occupied MOs are orthogonal r bonding orbitals with nodal planes along the internuclear axis and are not particularly polarizable. For the ethylene system in C1 symmetry, C2H4, the optimized carbon-carbon and carbon-hydrogen bond lengths of 1.3166 and 1.0766 A, respectively, and the C-C-H bond angles of 121.73' were held constant during the finite field calculations. Convergence to three significant figures for r:,,, required a convergence criterion of 12;moreover, the values for yizzzdiffered significantly at lower levels of convergence. For example, the sign of ~~,,, changed in going from CONV = 6 to 8. Thus, calculations for all electric field vectors were run at CONV = 14;the results for the electric field vector perpendicular to both the C = C bond and plane are reported in Table 3 while the parallel electric field vector results are shown in Table 4.13 In both cases the perpendicular first hyperpolarizability components are zero by symmetry. The most significant result of these calculations can be seen in Table4. Contrary to the generally accepted notion that valence and r orbitals are most polarizable while core orbitals are least, the two lowest energy MO's, the C-C u and u* orbitals, are an order of magnitude more polarizable than any other orbital. The second hyperpolarizability (for the field parallel to the C-C internuclear axis) is 3-4orders of magnitude larger than for any
TABLE 4 HF/6-31GS* Results for Cfi in a Static Electric Field Parallel to tbe C=C Axis MO
symmetry
ffzz
Ym.
1 2 3 4 5 6 7 8
AG BlU AG BlU B2U AG B3U B3U
211.0728 -2 1 1.0648 11.2235 13.1961 31.8087 -1 8.0729 -23.4798 17.5790
-4,160,259 4,160,260 797.2625 -409.338 1 -2342.2230 -154.2055 2585.1124 -1 602.6741
other orbital, including the rorbital, MO 8. For small molecules with several identical nuclei the core orbitals are typically almost degenerate; this degeneracy is removed upon perturbation by the electric field. Thus, while the (hyper)polarizabilityof each orbital is large, the contribution of these orbitals to the total is almost negligible (ca. 0.01au for the contribution of the core A, and BI, orbitals to a,,) relative to the valence contributions, as the values are virtually equal in magnitude but of opposite sign. These results may not be in conflict with accepted dogma if one views orbital polarizability and hyperpolarizabilities as measures of the change in the shape and/or volume of the MO relative to the unperturbed state rather than an absolute measure of deformation of the orbital. Two recent reports have appeared which espouse philosophically similar evaluations of polarizability. Zimmerman14has coupled Hiickel and ab initio molecular orbital theories to examine the bond-bond and atom-bond polarizability in several organic reactions and has concluded that such an analysis can be used as an independent method to assess reactivity. Pearlman15 has developed a sum-over-states approach to the molecular polarizability based on semiempirical SCF calculations which can be partitioned into atom-centered or orbital-centered polarizability tensors and which are affected by conformation and local environment in ways not readily obtainable from atomic polarizabilities. In conclusion, we have carried out initial studies of molecular orbital polarizability and hyperpolarizability as a function of an applied static electric field vector and have found that, contrary to general belief, u orbitals in the core can be significantly more (hyper)polarizable. In addition, the (hyper)polarizability of valence orbitals due to an external perturbation such as an electric field, whether applied or due to local environmental effects such as solvent, receptor, or reactant, can now be assessed. The utility of this data in areas such as quantitative measures of orbital hardness, prediction of reactivity and interactions in frontier orbital theory,16 and ligand-receptor binding can certainly be envisioned. Further studies on basis set effects, the relationship of orbital symmetry to (hyper)polarizability, and correlation effects are planned.
Acknowledgment. We thank Cray Research Inc. for a grant, the Ohio Supercomputer Center for generous allocations of time, and Drs. Dudis, Guo, and Huang for helpful discussions.
Letters
References and Notes (1) Le Fevre, R. J. W. Rev. Pure Appl. Chem. 1970,20,67.
(2) Hehre, W. J.; Pople, J. A. Tetrahedron Lrtt. 1970, 2959. (3) Oh,J. J.; Hillig, K. W.; Kuczlowski, R. L.; Bohn, R. K. J . Phys. Chem. 1990,94,4453. (4) Lewis, D. F. V. J . Comput. Chem. 1989,10,145.
( 5 ) See, for example: Famini, G. R.; Marquez, B. C.; Wilson, L. Y. J . Chem. Soc., Perkin Trans. 2 1993,773. (6) Nonlinear Optical Properties of Organic and Polymeric Materials; Williams, D. J., Ed.; ACS Symposium Series 233; American Chemical Society: Washington, DC, 1985. (7) Buckingham, A. D. Adu. Chem. Phys. 1967,12,107. (8) For the introduction of finite field methods, see: Cohen, H. D.; Roothaan, C. C. J. J. Chem. Phys. 1965,43,S34. For an excellent discussion of the use of finite field methods in the calculation of hvucmlarizabilities bv ab initio SCF methods, see: Chopra, P.;Carlacci, L.,i(iig, H. F.; Prasad: P. N. J . Phys. Chem. 1989,93,7120. (9) Smith, D. A,; Dudis, D. S.; Yeates, A. T.Manuscript in preparation. See also: Smith, D. A. US. Air Force Technical Report 151-7035. Development of Molecular Orbital Visualization Methods for tfe Study of Nonlinkar Optical Properties, 1990.
The Journal of Physical Chemistry, Vol. 98, No. 28, 1994 6905 (IO) Kurtz, H. A.;Stewart,J. J.P.; Dieter,K. M.J. Comput. Chem. 1990, 11, 82. (1 1) Gaussian 90,Revision H: Frisch, M. J.; Head-Gordon, M.; TNC~S, G. W.; Foresman, J.B.;Schlegel,H. B.;Raghavachari,K.;Robb,M.;Binkley, J. S.; Gonzalez, C.; Defrm, D. J.; Fox, D. J.; Whitesides, R. A,; Secger, R.; Melius, C. F.; Baker, J.; Martin, R. L.; Kahn, L. R.; Stewart, J. J. P.; Topiol, S.;Pople, J. A. Gaussian Inc., Pittsburgh, PA, 1990. (12) Hehre, W. J.; Radom, L.; Schleyer, P. v. R.; Pople, J. A. Ab Initio Molecular Orbital Theory; John Wiley and Sons: New York, 1986. (13) The molecular orbital polarizabilities and hyperpolarizabilities for an electric field vector applied perpendicular to the C = C bond but in the plane are uninteresting and not reported here. (14) Zimmerman, H. E.;Weinhold, F. J . Am. Chem. Soc. 1994, 116,
1579. (15) Smith, K. M.;Pearlman, R. S.Calculated Polarizabilitiesfrom SemiEmpirical SCF Calculations, 207th ACS National Meeting, San Diego, CA, Abstract COMP 151, 1994. (16) Huang, X. L.; Dannenberg, J. J.; Duran, M.; Bertran, J. J. Am. Chem.Soc. 1993,lIS,4024. Huang, X.L.;Dannenberg, J. J. Ibid 1993,lIS. 6017.
