Resolution of molecular polarizabilities of CH 3 – X and CH 3 – CH 2 – X derivatives into atomic terms

An additive scheme for resolving average molecular electric dipole polarizabilities into atomic contributions, based on the acceleration gauge for the electric dipole, has been applied to a series of methyl and ethyl derivatives, CH3–X and CH3–CH2–X. Extended calculations have been carried out to test the reliability of the partition method. Gross atomic isotropic contributions have been evaluated for carbon, hydrogen, and heteroatoms, showing a good degree of transferability from molecule to molecule. The theoretical values of atomic polarizabilities can be used to predict fairly accurate average polarizabilities of higher homologous molecules in the series of X-substituted alkanes.


I. INTRODUCTION
If molecules are made up of atoms, as we can concisely state relying on a widespread reductionist attitude of chemists, then a global molecular property can be thought of as a sum of corresponding atomic contributions.This idea was found empirically correct in a few cases, involving series of homologous molecules characterized by similar structure, e.g., geometrical parameters such as bond distances and bond angles, chemical and physical properties, etc.The systematic work of Pascal on magnetic susceptibilities, [1][2][3] preceding the quantum era, provides a paradigma that paved the way to many aspects of modern chemical thinking. 4 number of efforts have been reported later on to assess the reliability of a resolution of total quantities into group contributions possibly transferable from molecule to molecule.Besides its theoretical interest, the importance of this work in practical applications is evident, as it can enable researchers to estimate relevant features of big systems a priori with some degree of confidence.
6][7][8][9] Attempts have successfully been made to interpret some quantities, e.g., magnetic susceptibilities, electric polarizabilities, and so on, as a sum of atomic terms.Extension to other properties, for instance, optical rotatory power, is in principle straightforward.Accordingly, we could decide that a global molecular property can be basically resolved into contributions arising from different basins that can be expected to define atoms-inmolecules a `la Bader.

II. BASINS OF ATOMIC OPERATOR AS ''ATOMS-IN-MOLECULES''
Whereas a ''Bader atom-in-molecule'' is geometrically and topologically defined by asymptotic lines of the gradient field, we could alternatively suppose that an ''atom'' within a given molecule is a region of space which defines the actual domain of a certain ''atomic'' operator.Such a domain is not necessarily closed and is not uniquely defined, depending on the form of the operator itself.At any rate, the reliability of partitioning methods based on this definition can only be established via empirical criteria and numerical tests.Some examples may help us understand its evident limits and its possible merits.
Static electric dipole polarizability is customarily defined within the dipole length representation (R,R) via the equation for a molecule in the singlet electronic reference state ͉⌿ a (0) ͘ϵ͉a͘.Such an expression, obtained allowing for the Goeppert-Mayer form of the interaction Hamiltonian 17,18 within Rayleigh-Schro ¨dinger perturbation theory, is not suitable to provide a rationalization of a global molecular property in terms of atomic contributions.However, a different definition of electric dipole polarizability can be arrived at a͒ Member of Carrera del Investigador del CONICET.
b͒ Electronic mail: lazzeret@unimo.itallowing for the interaction Hamiltonian in the dipole acceleration picture, or ͑which is fully equivalent͒ using offdiagonal Ehrenfest relations. 18Thus we replace matrix elements of the position operator in Eq. ͑1͒ by those of the operator for the force of N nuclei on n electrons, to define electric polarizability in mixed length-acceleration formalism, The nice feature of the dipole acceleration picture is that the F n␣ N operator can readily be written as a sum over nuclei, i.e., The operators for the force of nucleus I on n electrons, where r i and R I denote electronic and nuclear coordinates, possess the peculiar ''atomic'' form that we have in mind: due to its overall dependence on ͉rϪR I ͉ Ϫ2 , each operator actually samples the charge distributions in a domain in the environment of the I-nucleus.Accordingly, it leads to a precise definition of gross ''atomic polarizabilities,'' via Eq.͑3͒, Total average molecular polarizability can now be resolved as Such a resolution is physically sound, as it is based on the rigorous definition of the total force, Eq. ͑4͒.It was also found reliable in some numerical tests on the electric polarizabilities of the alkane series, where it provides values of hydrogen and carbon atomic polarizabilities that are transferable from molecule to molecule with good accuracy. 13It should be noticed that the (R,R) polarizability, Eq. ͑1͒, is a tensor characterized by symmetry in the exchange of ␣Ϫ␤ indices, and can therefore be reduced to the principal axis system in any case.Tensors like those defined via Eqs.͑3͒ and ͑6͒ are not in general symmetric, unless certain hypervirial theorems 19 are identically fulfilled.If necessary, symmetrized properties could be defined in numerical calculations, e.g., At any rate, as we are essentially interested in the diagonal components and trace of the polarizability tensor, this aspect will not be further investigated here.
We could consequently attempt to yield a definition of the Ith atom in a molecule as that region which essentially coincides with the domain weighed by the F n I operator.At first sight, such a definition may seem quite loose for evident reasons.The main objection is that it depends on the type of operator, that for the atomic force in the present case, which implies that other operators, e.g., the atomic torque 20,18,14 K n I ϭR I ϫF n I , might actually gauge slightly different domains of the electronic charge.In any event, both show the same dependence on ͉rϪR I ͉ Ϫ2 .As a piece of evidence assessing the practicality of the definition of atomic basin proposed above, we observe that the definition of Thomas-Reiche-Kuhn ͑TRK͒ atomic populations provided by an analogous partition of oscillator strengths in mixed lengthacceleration picture via Eq.͑5͒ seems quite reliable on empirical grounds.In fact, the TRK populations are to a good extent transferable within a set of homologous molecules, and are related to measurable quantities, i.e., infrared intensities. 21,15,16esolutions of molecular properties allowing for similar mathematical procedures may also serve to exclude some possibilities, e.g., that of resolving the total electric dipole moment as a sum of bond moment vectors.As a matter of fact, the rotational constraint for nuclear electric shieldings clearly shows that the molecular property which can actually be written as a sum of atomic terms is instead a dipole moment tensor. 18,21

III. DISCUSSION OF RESULTS
Attempting to extend the investigations of Ref. 13 on alkanes, a larger series of molecules has been considered in the present study.Two high-quality Gaussian basis sets, referred to as I and II, were used to obtain numerical estimates for gross atomic polarizabilities in methyl and ethyl derivatives, CH 3 -X and CH 3 -CH 2 -X, with XϭF, OH, NH 2 , within the coupled Hartree-Fock ͑HF͒ level of accuracy.They are constructed according to the contraction schemes (14s14p5d/5s5 p)→͓6s6 p2d/2s2 p͔ and (13s8p4d/8s2p) →͓8s6 p4d/6s2 p͔, respectively, where functions on heavy atoms and hydrogens are separated by a slash as usual.The detailed description of these basis sets, as well as the molecular geometries adopted in the calculations, can be found in Ref. 16, where Thomas-Reiche-Kuhn atomic populations have been evaluated via nuclear electric shieldings.
The conclusions arrived at by inspection of the results reported in Tables I-VI are as follows.

͑i͒
Basis sets I and II provide two alternative sets of gross atomic polarizabilities.The internal consistency and homogeneity of data from basis set I is guaranteed by the accurate values of the TRK sum rule within the mixed length-acceleration and lengthvelocity gauges. 15,16Analogous features of consistency and homogeneity characterize theoretical atomic polarizabilities from basis set II, whereby TRK sum rules of near-Hartree-Fock quality are obtained ͑almost the same values for all formalisms͒.
We recall that the closeness or otherwise of the results arrived at in different pictures gives a measure of ba-sis set quality. 19Thus basis set I is suitable to represent the force operator, and basis set II is flexible enough to guarantee accurate predictions of a number of second-order properties in different ''pictures.''͑ii͒ Atomic, group, and total polarizabilities from basis set II are larger than corresponding values from basis set I. The former are expected to be more accurate owing to a well-known analysis of ''quadratic'' secondorder properties presented by Moccia, 22 which can be empirically formulated by the statement ''the larger, the better.''At any rate, basis set I, owing to its reduced size, can be adopted in calculations on bigger molecules attempting to build up a larger set of transferable atomic polarizabilities which include, for in-  Whereas theoretical atomic polarizabilities of Y slightly increase for both basis sets on passing from CH 3 -X to CH 3 -CH 2 -X, the opposite trend observed for ␣ Av N from basis set II is not easy to explain.It might depend on a number of reasons, e.g., geometrical parameters.In any event, the numerical difference, roughly 0.05 a.u., is quite small.͑iv͒ In the CH 3 -CH 2 -X series, for Y ϭF, O, N, C, ␣ Av C of the methyl group from basis set I vary within the interval 5.35↔5.44a.u.The corresponding values from basis set II range within 5.91↔5.99a.u.We can see  that the effect of the perturbation introduced in the molecule by the Y heteroatom is strongly attenuated, or virtually lost, after two bonds.Similar trends are observed for the atomic polarizabilities of hydrogens belonging to the methyl group of CH 3 -CH 2 -X.Therefore, the group polarizability of CH 3 is fairly transferable in this series.The largest deviations from full transferability were found for Y ϭC of a methyl group of propane ͑see Table VI of Ref. 13͒.

͑v͒
The atomic polarizabilities of methylic carbon in CH 3 and methylenic carbon in CH 3 -CH 2 -X are clearly correlated with the electronegativity 23 of X ͑see Fig. 1͒.Therefore, allowing for ͑iv͒ it can be argued that X produces essentially a nearest-neighbor effect in the CH 3 -CH 2 -X series.
stance, second-row substituents as -PH 2 , -SH, and -Cl.In such a case, accuracy and consistency of atomic polarizabilities should be sought within that set.͑iii͒ Let us denote by Y the heavy atom entering the substituent group X in the series CH 3 -X and CH 3 -CH 2 -X, then Y ϭF, O, N, C for XϭF, OH, NH 2 , CH 3 .The ͑average͒ gross atomic polarizabilities of a given Y atom, characterized by electronegativity higher than that of carbon, estimated via each basis set, are virtually transferable from CH 3 -X to CH 3 -CH 2 -X compounds.Tiny differences between ␣ Av Y from basis set I in ethyl and methyl derivatives can be entirely explained in terms of the inductive effect of the aliphatic chain always acting upon X in the same direction.In fact, for Y ϭF, O, N, C ͑see also Tables V and VI of Ref. 13͒ the values 3.49↔3.53,5.14↔5.23,6.61↔6.65,and 5.13↔5.17a.u.were obtained for ␣ Av Y respectively in CH 3 -X and CH 3 -CH 2 -X.The corresponding estimates from basis set II are comparatively more accurate, and can be interpreted as near-HF gross atomic polarizabilities.From the tables in this work, and Tables V and VI of Ref. 13, ␣ Av F ϭ3.69↔3.71,␣ Av O ϭ5.36↔5.44,␣ Av N ϭ7.49↔7.44,␣ Av C ϭ5.68↔5.72 a.u., respectively for CH 3 -X↔CH 3 -CH 2 -X.

FIG. 1 .
FIG. 1.Average atomic polarizability, ␣ Av C , of methyl carbon in CH 3 -X and C 2 H 5 -X molecules as a function of Pauling electronegativity of the heavy atom Y.

TABLE I .
Atomic contributions to electric polarizability ͑a.u.͒ for the CH 3 NH 2 molecule.

TABLE III .
Atomic contributions to electric polarizability ͑a.u.͒ for the CH 3 F molecule.

TABLE V .
Atomic contributions to electric polarizability ͑a.u.͒ for the C 2 H 5 OH molecule.

TABLE VI .
Atomic contributions to electric polarizability ͑a.u.͒ for the C 2 H 5 F molecule.