Resolution of alkane molecular polarizabilities into atomic terms

Two additive schemes for resolving average molecular electric dipole polarizabilities into atomic contributions, based on the acceleration gauge for the electric dipole, are outlined. Extended calculations have been carried out for a few terms of the alkane series to test the reliability of the partition method. Gross atomic isotropic contributions evaluated for carbon, αAvC≈5.7 a.u., and hydrogen, αAvH≈2.7 a.u., are actually transferable from molecule to molecule, and can be used to predict fairly accurate average polarizabilities of higher homologous molecules in the alkane series.


I. INTRODUCTION
The idea that molecular magnetic susceptibilities can be rationalized in terms of atomic contributions transferable from one molecule to another is due to Pascal. 1 A wide series of experimental results proves the reliability of Pascal's hypothesis for a given set of structurally and chemically related homologous molecules. 2,3ttempts have also been made to define a resolution of electric dipole polarizabity into atomic terms.Denbigh 4 and Vogel 5 succeeded in determining some sets of transferable contributions.
Alternative partitions have been proposed, adopting simple quantum mechanical methods based on molecular orbital localization procedures. 6,7Although atomic and bond contributions to average electric dipole polarizabilities can be reasonably defined in this way, the theoretical values depend on the molecular orbital ͑MO͒ localization scheme adopted in the calculation.Moreover, due to the fact that complete orbital localization is not actually feasible, small discrepancies are to be expected between total theoretical polarizability and corresponding sums of atomic and bond contributions. 6,7everal molecular orbital approaches for the multicenter distribution of molecular polarization have been proposed, see a recent article by Nakagawa 8 for extended bibliography.A general formalism has been established by Stone. 9Bader and co-workers 10 used the theory-of-atoms-in-molecules to demonstrate the additivity of group polarizabilities.
Quite different partitioning schemes have been made available to define atomic and bond contributions to the total average dipole polarizabilities in molecules, based on rigorous definitions of the quantum mechanical operators suitable to estimate these quantities. 11,12Related theoretical procedures and algorithms have been implemented within the SYSMO suite of computer programs, 13 but so far they have not been applied to rationalize molecular polarizabilities in terms of transferable contributions.
The present article sets out to investigate the reliability of methods discussed in Refs.11 and 12.A wide numerical test has been performed, in order to document the transferability of ͑i͒ gross atomic, ͑ii͒ net atomic, and bond polarizabilities.Section II contains a brief outline of the theoretical methods employed in the calculation.Numerical results are presented in Sec.III for a few molecules belonging to the alkane series.

II. THEORETICAL PROCEDURE
Static electric dipole polarizability of a molecule in its electronic reference state ͑singlet ͉⌿ a (0) ͘ϵ͉a͒͘ is defined, within the length gauge, in the notation of Ref. 12. Alternative definitions can be obtained using dipole velocity and/or dipole acceleration gauges.From off-diagonal hypervirial relationships, it is easily shown that the electric polarizability can be written in a number of different ways, e.g., in the velocity formalism, in the mixed length-acceleration formalism, where the operators P ␣ ͑canonical momentum͒ and F n␣ N ͑total force of N nuclei acting on n electrons͒ replace the position operator R ␣ .

␣ ␣␤
Since the hypervirial theorem is only satisfied if the electronic wave functions are exact eigenstates of a model Hamiltonian, 14 relationship ͑2͒ implies that the alternative definitions of Eqs.͑1͒, and ͑3͒-͑5͒ of a given tensor component provide the same numerical value in a restricted number of ideal cases, e.g., within the ''exact'' Hartree-Fock method, or the multiconfiguration self-consistent field ͑SCF͒ scheme allowing for a complete basis set.
In actual calculations employing the algebraic approximation, the off-diagonal hypervirial conditions ͑2͒ are only approximtely fulfilled.Consequently, the numerical estimates of electric dipole polarizability provided by length, velocity, and accelerations gauges are, usually, not identical.
At any rate, whenever corresponding values furnished by different formalisms are sufficiently close one to another, it can be reasonably concluded that the basis set is fairly complete ͑at least as far as the problem of accurate representation of the set of R ␣ , P ␣ , and F n␣ N operators is concerned͒, and the degree of accuracy of theoretical polarizabilities can therefore be assessed.
Other auxiliary theoretical quantities useful to test the quality of a given calculation of electric polarizabilities are the Thomas-Reiche-Khun ͑TRK͒ sum rules in different gauges, 12 e.g., The interesting property of the dipole acceleration gauge is due to the fact that the operator F n␣ N for the total force of N nuclei on n electrons can obviously be written as a sum over nuclei, i.e., It is worth noticing that the atomic contributions in Eq. ͑10͒ closely resemble the analytic expression for the dipole electric shielding 12 of nucleus I, where the operator for the electric field of n electrons on nucleus I is related to the force terms in Eq. ͑11͒, It can be reasonably argued that transferability of atomic and bond polarizabilities defined via Eqs.͑10͒ and ͑11͒ is actually expected in a limited number of cases, i.e., an additive scheme should probably work only within a restricted set of molecules characterized by similar structural and chemical properties.
As in the case of magnetic susceptibilities, 2 strain effects, delocalized electrons, functional groups inducing charge polarization, etc., can cause large perturbations, and would imply the need for ''constitutive corrections,'' ''exaltations contributions,'' etc.
To simplify the problem, in this work we limited ourselves to investigate the electric dipole polarizability of some lower homologous terms of the alkane series, namely methane, ethane, propane, and butane, which meet the aforementioned requirements.

III. RESULTS AND DISCUSSION
The random-phase approximation ͑RPA͒ to the electronic polarization propagator 15 has been adopted in the calculation.
Zero-order molecular orbitals are expanded over atomic gaussian functions.Three different basis sets have been employed.The first one, hereafter referred to as I, is an ad hoc basis set, developed by Sadlej to evaluate near Hartree-Fock electric dipole polarizabilities within the length gauge. 16Basis set II, also developed by Sadlej et al. 17,18 to yield accurate representation of the force operator in Eq. ͑9͒, has been successfully used to predict near Hartree-Fock estimates of nuclear electric shieldings 19,20 and infrared ͑IR͒ intensities. 21,22Owing to the similarity between relations ͑10͒ and ͑12͒, both involving transition matrix elements within the mixed length-force gauge, as underlined above, we expect basis II to be suitable also in the prediction of gross atomic polarizabilities.The reasons of our choice of basis sets are therefore evident: as the theoretical determinations of ␣ Av (R,R) via basis set I are expected to be of near Hartree-Fock quality, a resolution of ␣ Av (R,F) evaluated by basis set II into transferable atomic contributions of Eq. ͑10͒ should furnish a reliable partition of total electric polarizabilities of alkanes, provided that the identity holds.In other words, even if relationship ͑14͒ would be exactly fulfilled only in a complete basis set calculation, it should be met with sufficient accuracy throughout the calculations, when the left-hand side ͑lhs͒ ͓right-hand side ͑rhs͔͒ has been obtained via ad hoc basis set I ͑II͒, proving a posteriori the near Hartree-Fock ͑HF͒ quality of theoretical polarizabilities.
On the other hand both basis sets I and II are presumably unsuitable to give accurate values of ␣ Av (F,F) , the average polarizability in the full acceleration gauge.As a matter of fact, one should apply Sadlej's ''polarization'' recipe 17,18 one more time to account for the second force operator appearing in definitions of Eqs.͑5͒ and ͑8͒, in order to develop doubly polarized basis sets for carbon and hydrogen, which lies beyond the aims of the present article.In any event, attempts have been made to evaluate ␣ Av IJ via relationship ͑11͒ by means of basis sets I, II, and III.The extended basis set III is similar to the one previously employed for a large number of molecular properties and related sum rules. 13,23Its (s/p) substratum has been taken from van Duijneveldt tables, 24 the exponents of 3d functions on carbon are 1.61, 0.43, 0.15, and 0.062.The exponents for 2p functions on hydrogen are 4.02, 0.952, and 0.294.Owing to its size and flexibility, the corresponding theoretical estimates of several response properties are expected to be close to the Hartree-Fock limit.
The molecular geometries of ethane, propane, and butane molecules used in the present study have been opti-mized by means of the GAMESS program 25 using the 6-31G** basis set. 26The geometry of methane is the same as in previous articles. 13he results of our calculations are reported in Tables I to XI. Basis sets are described in Table I, where SCF energies are also given.Incidentally, it can be observed that total energies from basis sets I and II are poor in comparison with those obtained via basis sets of the same ͑or smaller͒ size using energy-optimized exponents: it is worth recalling that basis sets I and II are especially meant for electric polarizabilities in different gauges.In fact, the nice features of basis set II can be judged from inspection of Table II, reporting TRK sum rules in the mixed length-acceleration gauge for methane.The results obtained for the other alkane molecules have not been shown for the sake of space.In any event, for all the molecules studied here the (R,F) values are virtually equal to the number n of electrons ͑a little bit larger in every case͒.It is also evident that the same basis set is usually unsuitable to predict accurate TRK sum rules in other formalisms, which, on the other hand, are nicely satisfied by basis set III.In addition, basis set I, for all of the molecular systems investigated here, provides insufficiently accurate TRK constraints also within the full length gauge, although corresponding estimates for ␣ Av (R,R) in Tables III-VII are close to the HF limit, 16 as it can be verified by a comparison with corresponding values from basis set III for methane, ethane in eclipsed and staggered conformations, propane, and butane.
As a matter of fact, the diagonal components of the polarizability tensor in the length gauge, ␣ xx (R,R) , etc., are ''quadratic'' properties, which tend to the HF limit from below ͑see, in any event, an article by Moccia͒. 27Therefore, in any calculation adopting the algebraic approximation, it can be usually expected that the criterion ''the larger, the better'' holds for these theoretical quantities. 13,28,29Accordingly, owing to the special features of Sadlej's basis set, 16 one can reasonably argue that, in many cases, the theoretical ␣ Av (R,R) values arrived at in this study furnish accurate lower bounds to the corresponding HF values.
It is surprising to observe that also basis II is capable of predicting quite precise ␣ ␣␤ (R,R) values ͑usually slightly smaller than corresponding ones evaluated via basis set I͒, although it has been specifically designed for nuclear electric shieldings. 17,18At any rate, the theoretical predictions yielded by basis sets I and II fulfill the condition ͑14͒ quite satisfactorily.
Accordingly, it can be expected that the quantities calculated from relationship ͑10͒ via basis set II provide reliable numerical values for the ''gross atomic contributions'' defined in this way.The degree of transferability of these atomic polarizabilities from molecule to molecule within the alkane series can be assessed by inspecting Tables III-VII.Average carbon contributions ␣ Av C seem to depend very slightly on molecular conformation, compare, in Tables IV and V, ␣ Av C Ϸ5.2 a.u. and ␣ Av C Ϸ5.1 a.u.from basis set II, respectively, for eclipsed and staggered conformers.The results for hydrogen are fairly close one another, basis set II yields the value ␣ Av H Ϸ2.5 a.u.Very similar values have been obtained, via basis set II, for the other alkane molecules examined here.Therefore their transferability from molecule to molecule is excellent, as it can be observed in Tables III-VII.The estimates arrived at via basis set III are slightly higher, i.e., ␣ Av C Ϸ5.7 a.u. and ␣ Av H Ϸ2.7 a.u.Three main conclusions emerge from these findings: ͑i͒ the electric dipole polarizability of alkanes can actually be rationalized in terms of gross atomic contributions; ͑ii͒ the values ␣ Av C Ϸ5.7 a.u. and ␣ Av H Ϸ2.7 a.u.for methyl groups, and ␣ Av C Ϸ5.9 a.u. and ␣ Av H Ϸ2.8 a.u.for methylene groups, can be used to predict average electric polarizabilities of higher homologous terms in the alkane series; ͑iii͒ basis set II provides good approximations to near Hartree-Fock values from basis set III: the discrepancies are about 10%.From the data obtained for propane one can suggest that the values 13.7 and 11.5 a.u., respectively, for the CH 3 Ϫ and CH 2 Ϫ groups, can be used to obtain average polarizabilities of alkane molecules of near HF quality.
Owing to its reduced size, basis set II is useful to obtain reliable gross atomic polarizabilities in larger molecular systems.
The gross isotropic atom polarizabilities arrived at via the theoretical approaches described in the present article are numerically different from those reported by Nakagawa, see Table II of Ref. 8, ␣ Av C Ϸ4.27 a.u. and ␣ Av H Ϸ3.16 a.u.In addition, correlation effects and vibrational contributions were not taken into account in our study.Accordingly, it can probably be argued that different calculation methods only prove the reliability of additive schemes for electric dipole polarizability, as they seem to sample different features and domains of a given molecular wave-function.In other terms, the definitions of ''atomic'' contributions to molecular properties are apparently not univocal, they rather depend on the basic assumptions of the localization procedure.The ''pair polarizabilities'' evaluated in this study via Eq.͑11͒ are reported in Tables VIII-XI.Basis set I, which is obviously unsuitable to represent the force operator, see for instance Table XI, provides very poor results.Accordingly, only theoretical values from basis sets II and III are given in Tables VIII-X.The former is not flexible enough to guarantee accurate estimates of properties in the (F,F) formalism, compare the corresponding TRK sum rules in Table II and polarizabilities in Tables III-VII, as it is especially designed for (R,F) properties only. 17,18he overall performance of basis set III, as far as the (F,F) gauge is concerned, seems to be much better, even if TRK sum rules and polarizabilities are constantly overestimated.For instance, in the case of propane, the value ␣ Av (F,F) Ϸ39.5 a.u., probably lies beyond the Hartree-Fock limit, close to ␣ Av (R,R) Ϸ38.4 a.u., predicted via basis sets I and III.
The partial failure of basis set III to provide more accurate representation of electron distribution around pairs of nuclei is due to the intrinsic inadequacies of gaussian functions.
Accordingly, the attempts made here to evaluate ''net atomic'' and ''bond polarizabilities'' give only rough estimates.In any event, the degree of transferability of ␣

TABLE I .
Specification of basis sets and SCF energies ͑in Hartree͒.

TABLE VIII .
Pair polarizabilities ␣ Av IJ of methane molecule from basis sets II and III ͑a.u.͒.

TABLE VII .
Atomic contributions to electric polarizability ͑a.u.͒ of butane.a

TABLE IX .
Pair polarizabilities ␣ Av IJ of ethane molecule from basis sets II and III ͑a.u.͒.

TABLE X .
Pair polarizabilities ␣ Av IJ of propane molecule from basis sets II and III ͑a.u.͒.

TABLE XI .
Pair polarizabilities ␣ Av IJ of butane molecule from basis sets I and II ͑a.u.͒.