J. Phys. Chem. 1993, 97, 6016-6023
6016
Atom Charge Transfer in Molecular Polarizabilities. Application of the Olson-Sundberg Model to Aliphatic and Aromatic Hydrocarbons Jon Applequist Department of Biochemistry and Biophysics, Iowa State University, Ames, Iowa 5001 1 Received: January 4, 1993
The atom monopoledipole interaction model of Olson and Sundberg is used to calculate molecular dipole polarizability tensors of aliphatic and aromatic hydrocarbons for light in the visible region. Atom monopole and dipole polarizabilities are optimized to fit experimental mean polarizabilities and anisotropies for seven alkanes and five planar aromatic hydrocarbons. For alkanes the best fit is obtained with vanishing atom monopole polarizabilities; i.e., for the model in which no atom charge transfer occurs. For the planar aromatic molecules the best fit is obtained by a model which includes both monopole and dipole atom polarizabilities for the horizontal (in-plane) components and only dipole polarizabilities in the vertical (out-of-plane) components. Only the ring carbon atoms have nonzero monopole polarizabilities in this model. The vertical polarizabilities are treated by an approximation which neglects dipole coupling among atoms in the same conjugated system in order to overcome a violation of energy conservation when dipole coupling is included. The parameters so obtained are used to calculate polarizability tensors for an additional 12 aromatic hydrocarbons, including methylbenzenes and polycyclic aromatic compounds. The calculated mean polarizabilities and anisotropies lie within the ranges of available experimental data in all cases, and the relative magnitudes of principal polarizability components agree with those assigned from experimental data. Introduction There is much interest in the theory of the response of molecules to electric fields, as this response underlies the diverse optical and dielectric properties of materials as well as many aspects of the interaction among molecules. Experimentally determined molecular dipole polarizability tensors provide a measure of the response to a uniform external field, but further information is needed to calculate the polarization in the nonuniform fields of, for example, nearby polar groups or circularly polarized light. Theories that provide information on the spatial distribution of polarization within the molecule have begun to fill this need. Such theories specify the polarizability associated with each suitably defined site in the molecule, and it is usually essential to include a measure of the relayed, or nonlocal, polarization of a given site due to the field at each other site. One of the simplest theories of this type is the atom dipole interaction model, originating in the early theories of optical rotation1-3and molecular anisotropy$and pursued more recently for practical computations by several authors.5-'5 The purpose of the present study is to employ a modification of this model, proposed and developed by Olson and Sundberg16 in 1978, in which charge transfer among atoms contributes to molecular polarization, along with atom dipole polarization. In this model, the view of the atoms in a molecule as polarizable point particles is retained, but charge transfer among atoms is permitted in response to changes in local electric potentials. Olson and Sundberg suggested that this model would be more realistic than dipole interaction models for the in-plane polarizabilities of aromatic molecules, since these involve displacement of electrons in nonlocalized ?r orbitals. Their illustrative calculations supported this view, but they did not attempt to find a best fit of the model to experimental polarizabilities. In this study, we seek atom monopole and dipole polarizability parameters that give an optimum fit of the Olson-Sundberg model to observed mean molecular dipole polarizabilities and anisotropiesof aliphatic and aromatic hydrocarbons. Thealiphatic hydrocarbons were treated in a similar way without the charge-transfer contributions in our 1972 ~ t u d y ,and ~ their inclusion here is intended to determine whether a better fit to experimental data can be obtained by introducing charge transfer. Attention is confined here to data 0022-365419312097-6016$04.00/0
at visible wavelengths and in static fields, where dispersion effects are relatively small. Much progress has been made in recent years in the calculation of molecular polarizabilities by quantum mechanical methods, which go far beyond the point-atom models in providing a detailed picture of the distortions of the charge distribution experienced by a molecule in an electric field. In particular, the studies of Stonel' and of Bader and co-workers1*J9 are relevant to the present work in that they give details of the changes in charge distribution in regions of a molecule that can reasonably be identified as atoms. Further comments on the relation of these studies to the present work are deferred to the Discussion section. The need for studies of the present type may not be obvious in view of these developments, and I offer the following points by way of justification. 1. There is a need for a theory that can be easily scaled up to molecules of arbitrary size and complexity. The point-atom models have proven to be easily within the capacity of a moderatesized computer for molecules with several hundred atoms,20 and the main limitation on molecular size at present is the number of unknown structural variables that would have to be explored. Corresponding calculations by quantum mechanical methods are still limited to much smaller molecules. 2. The extension of a study on small molecules, such as the present one, to larger molecules requires knowledge of atomic parameters that are transferable within a series of related molecules. The optimization method used to determine the transferable parameters gives significantresults only if the number of parameters is small relative to the number of target data. Therefore, we use a simple model for a molecule in order to limit the parameters to those needed to predict the main features of the phenomena of interest. 3. One of the desirable features of a model where information on the spatial distribution of polarization is required is that the atoms or other subunits of a molecule interact with each other in such a way as to influence the response of each other to an external electric field. This interaction should be expressible in terms of an explicit function of the molecular geometry and the atomic parameters to permit extension to molecules of arbitrary structure. In the point-atom models the well-defined multiple 0 1993 American Chemical Society
Atom Charge Transfer in Molecular Polarizabilities field tensors sewe this purpose. Stone's17 calculations show how the interaction among atoms can be expressed in numerical form by quantum mechanical methods, but this approach requires a complete self-consistent-field treatment of the wave function for each structure. 4. Finally, I would emphasize that the point-atom model is not simply a mathematical approximation for real molecules but is a well-defined system whose behavior can be calculated to a high degree of accuracy on the basis of known physical laws. Where the behavior of a real molecule is similar to that of the model, an insight is gained into the originsof the behavior. Where discrepancies are found, it can be concluded that the molecule is not like the model. This introduction is intended not to minimize the importance of studies such as those of Stone and of Bader et al., but to show that each study serves a different purpose. In fact, part of the motivation for the present study comes from the finding of these authors that, in diatomic molecules, the dipole polarizability parallel to the bond axis has a charge-transfer contribution amounting to 15-90% of the total axial polarizability.17J* The magnitude of this contribution depends on the manner in which the bounding surfaces of the atoms are defined, but the results indicate that electron mobility among atoms should be considered in a study such as the present one.
Theory The theory used here is equivalent to that of Olson and Sundberg.'6 A matrix organization corresponding to the polytensor formalism of a more general multipole interaction model21 is adopted here, with modifications to incorporate DeVoe's22 linear oscillator treatment of dipole polarizabilities. This formulation is convenient for the treatment of anisotropic atoms required here and lends itself to future applications to absorption spectra. The model consists of an assembly of N particles, which will be referred to here as atoms, though they may be any suitable molecular units or subunits. Atom i, located at ri, has a monopole polarizability ui such that the charge qi induced by the electric potential c#+ at atom i is qi = -uf#Jj (1) Following DeVoe, we place on atom i a set of pi linear dipole oscillators oriented along unit vectors ui, (s = 1, ...,pi),each with polarizability ai,, such that the dipole moment induced in oscillator is by an electric field E,at atom i is
where Eis = E,-ui,. Both ai and aismay be arbitrary complex functions of the frequency of the external field, though we do not consider that frequency dependence here. If the potential and field due to external sources at atom i are 40i and Eoi, then the potentials as modified by the presence of charges and dipoles on other atoms are
The Journal of Physical Chemistry, Vol. 97, No. 22, 1993 6017 of 9 are related to those of the Cartesian multipole field tensors21 TF"'as follows. For i # j ,
(5)
T ~= TF').uj, ~ ~ ,= r.r3rjj.ujt, t = 1, ...,pi /I
(6)
Tisjo= T$").u, = -r.:3rji.uis, s = 1, ...,pi JI
(7)
TiSji= u ~ , ~ T $ * = ~ )rj;3~is-~jr ~ u ~ , - 3rj;5u,,~r,irji~uj,, s = 1, ...,pi; t = 1, ...,pi(8) where rii = rrrb Also, for i = j , Ti,j, = 0. To complete the matrix formulation of the problem, we define the polarizability matrix Pi of atom i as
Pi = (q
...
ail
D
CYip,)
(9)
where the superscript D denotes a diagonal matrix of the elements shown. Pi is a version of the polarizability polytensor*I adapted to the DeVoe formalism. (The vanishing of off-diagonalelements in the first row and first column of Piimplies that induced charges are independent of field and induced dipoles are independent of potential.) We then define the matrix
P = (P1
...
PN)D
(10) Both T and P are of order M = N p1 + ... + p ~and , the row and column indices correspond to the atom monopoles and dipoles in the same sequence in both matrices. The relay polytensor matrix for the system without constraint on the total charge is21
+
+
= ( I PT)-'P (11) where 3 is the identity matrix of order M. Introduction of the constraint of constant total charge16J produces the correct relay polytensor matrix B O
where U ' is the column vector whose M elements are U,,,= 1 if s = 0 and U,, = 0 if s # 0, and the superscript T denotes the matrix transpose. (Equation 12 is obtained by expansion of eq B12, ref 21.) The molecular uniform-field dipole polarizability tensor, including contributions from induced atom monopoles and dipoles, is a = XBXT
(13)
where
x = (XI
XN)
and
Xi = (ri uil N
Pi
where the primes indicate that the sums omit the terms j = i. Equations 1-4 constitute a set of linear equations in the induced charges and dipole moments, and their solution describes the response of the system to an external field. The interaction coefficients in eqs 3 and 4 comprise a matrix I whose rows are indexed by the pair is and the columns by the pair j t ( i ,j = 1, ..., N, s = 0, ...,pi; t = 0, ...,p,). The elements
...
ujp)
(15)
where the Cartesian vectors are arrayed as columns so that Xi is a 3 X (pi + 1) matrix, and X is a 3 X M matrix. (Equation 13 is derived in the same manner as eq 43 of ref 21, adapted to the DeVoe formalism and reduced to matrix form.) The primary computational problem is that of obtaining a via eq 13 from a given set of ri, ui,, q,and ai,. The ut, for a given atom are taken to be a set of three orthogonal vectors in the cases studied here. For an isotropic atom these vectors are arbitrarily chosen, and only a single value of aisis required. We give particular attention to the mean molecular polarizability (Y and the anisotropy (61defined by
Applequist
6018 The Journal of Physical Chemistry, Vol. 97,No. 22, 1993
TABLE I: Optimized Atom Mono le and Dipole Polarizabilitiesfor ~uranesat d ' i m A -0.427f0.432
B
c Finally, as a check on the validity of parameters for any system, it is useful to note that B must be positive semidefinite to satisfy energy conservation. (It has previously been stated that B must be positive definite;' I,21 however, if any atom polarizability parameter is zero, B may have one or more zero eigenvalues. Conservation of energy is violated only if the matrix has negative eigenvalues.) In the present study, the eigenvalues of B were routinely calculated to confirm or refute the validity of any set of parameters. As a point of reference it may be noted that, for a diatomic molecule with polarizabilities u, a for each atom and internuclear distance r, B becomes indefinite when u > r or a > r3/2. Roughly similar limitations occur in polyatomic molecules.
Computational Methods Calculations were carried out on a Hitachi Data Systems 9180 computer and a Digital Equipment 2100 workstation, using Fortran programs and double precision arithmetic. The major matrix manipulations were carried out using subroutines from the IMSL,23 LINPACK,24and NAG2Slibraries. Molecular Structures. Structures of alkanes were calculated as described in our previous study.s For planar aromatic molecules, preliminary calculations were carried out using regular These hexagonal rings with C-C = 1.40 A and C-H = 1.083 structures were refined for all of the calculations reported here by optimizing the atom coordinates in the following way. C-C bond lengths were targeted to the values predicted by Pauling's2' equation, d = 1.504-0.3128(n - 1)/(0.84n + 0.16), where d is the bond length and n is the bond number predicted from the Kekul6 structures. Bond angles were simultaneously targeted to the value 120°, with weighting factors that favored on accurate fit of the bond lengths. C-H lengths were kept at the value 1.083 A found for benzene.26 The optimized atom polarizability parameters differed by
