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4.7 Polarizability Effect

Polarizability values are calculated from additive increments typical of an atom in a specific hybridization state. When values for the effect of polarizability on charge stabilization are calculated these increments are considered only to an extent that reflects the number of bonds between the atom considered and the charge or reaction center.

It is important to note that these two publications give the procedure that was originally developed by modification of a method for the estimation of the sum of atomic static polarizability () as given by K. J. Miller and J. A. Savchik (J. Am. Chem. Soc. 101, 7206 (1979)). The presently used procedure is no longer the one referenced in the above two publications but is a modification of a method for the estimation of the sum of atomic static polarizability as published by Y. K. Kang and M. S. Jhon (Theor. Chim. Acta 61, 41 (1982)). The results of the new method are for the most part similar to those of the originally published one. However, the new approach removes a deficiency of the old method that atoms with many electrons (Cl, S) far away from the reaction center unduly lowered the values for effective polarizability.
Mean molecular polarizability (MMP), mol, can be estimated from additive contributions, i, of atoms as shown by Kang and Jhon (see above). The values of those atomic increments are different depending on the hybridization state of the atoms.

The stabilization of charge in a molecule due to polarizability rapidly decreases with distance from the charge center. This decrease is described by a simple damping model, which uses the contribution of each atom, i, in a molecule only to an extent that accounts for the number of bonds, nij, between the charge center, j, and the atom, i.

It should be noted that the value of the effective polarizability, jd, is thus a property of the atom being considered as the charge or reaction center. This is illustrated in the next figure for the nitrogen as the center of protonation.

Figure 10. Evaluation of the effective polarizability of N as center of protonation

The value of effective polarizability, jd, must be seen as a quantitative measure of the stabilization energy resulting from the effect of polarizability.
The bond polarizability, b, is calculated as the mean value of the polarizabilities of the two atoms of a bond.

Values calculated

mol(POLARIZA): mean molecular polarizability

jd(APOLARIZ(A)): effective atom polarizability

b(BPOLARIZ(AB)): bond polarizability, which is the mean value of the polarizabililies of the atoms A and B

Example
The site dependence of the values for the effective polarizability jd can be seen with the example of N,N-dimethyl ethylenediamine. The value at the tertiary nitrogen atom, 1d, is with 6.93 Å3 much larger than the one at the primary nitrogen atom with, 2d, 4.73 Å3, reflecting the increased tendency of the former atom to accept a positive charge on protonation.

Figure 11. Site dependence of d

Results
Only values for the mean molecular polarization are given as they can directly be compared with experimental data.

Molecule

PETRA

Kang, Jhon

water

1.43

1.44

diethyl ether

8.78

8.78

diethyl ketone

9.96

9.96

acetic acid

5.11

5.12

acetamide

5.93

5.92

triethylamine

13.27

13.27

Table 15. Mean molecular polarizabilities [Å3]

Y. K. Kang, M. S. Jhon, Theor. Chim. Acta 61, 41 (1982)

Scope and Limitations
Presently, the following atoms and hybridization states have been parameterized:

Parameters of aromatic and fused aromatic carbon atoms are not included in the program.

Applications
The use of mean molecular polarizability can be found in textbooks on physical chemistry and will not be further discussed here.
Rather, we want to demonstrate the usefulness of the damping model and the values of effective polarizability calculated with it. Effective polarizability is a parameter that has found many uses in investigations of data on chemical reactivity. However, to get a clear picture of the significance of effective polarizability values we were searching for physical data that were solely dependent on such values. We found such data in combinations of ESCA and Auger spectroscopy.
Through ESCA experiments shifts in the core electron binding energies of chlorine atoms in different molecules may be determined. Auger spectroscopy deals with an additional ionization (KLL spectra) and can be used to determine shifts in Auger kinetic energy.

The shifts in core-ionization energy and in the Auger kinetic energy make it possible to determine relaxation energies that were shown to be directly related to the polarizability of the ligands of the ionized atom (E. J. Aitken, M. K. Banl, K. D. Bomben, J. K. Gimzewski, G. S. Nolan, T. D. Thomas, J. Am. Chem. Soc. 102, 4873 (1980)). We could show that these relaxation energies correlate directly with the effective polarizability values on chlorine calculated for 13 different organochlorine compounds (r = 0.958; std. dev. 0.10 eV) (ref. 2).
If values on mean molecular polarizability, , are used, the correlation is much worse (r = 0.885; std. dev. 0.21 eV), showing the superior modelling power of the damping procedure.
Similarly good results were obtained for measurements of such ESCA/Auger relaxation energies in organo germanium compounds (ref. 2).
The largest benefits of the values of effective polarizability have been obtained in studies of data on chemical reactivity. A series of data on gas phase reactions has been investigated as these reactions show the reactivity of individual molecules uncorrupted by solvent effects. Furthermore, data of high quality have become available through high pressure mass spectrometry and ion cyclotron resonance measurements.
Protonation of amines in the gas phase is exothermic, the energy released is called proton affinity (PA).
Polarizability is thought to be a primary source for stabilization of such ions produced in the gas phase.

In fact, it can be shown that the proton affinity of unsubstituted alkyl amines can be reproduced by an equation having effective polarizability as the only parameter (ref. 2).

Figure 12. Proton affinity of alkyl amines

The data set consisted of 34 primary, secondary, and tertiary amines of different skeletal type.
If substituents are introduced into the alkylamines, the inductive effect of these heteroatoms has to be accounted for. This was accomplished by introducing, in addition to the values of effective polarizability, an electronegativitiy parameter, s. The resulting two parameter equation reproduced well the proton affinity of 80 unsubstituted and substituted alkylamines (ref. 1).
(2) PA(kJ/mol) = 1435.5 - 116.3s + 12.5Nd
Such multiparameter equations could be derived for a series of fundamental polar gas phase reactions. The next table gives a summary of these investigations indicating which parameters were found to be necessary for a quantitative description of the data. Effective polarizability values were needed in all cases underlining the importance of this effect, at least in the gas phase.

Reaction

parameters

comment

 

d

hyp.

ref.

 

x

   

1

unsubst. only

x

x

 

2

80 subst. amines

x

x

 

3

ether,alcohols

x

x

 

3

thiols, thioethers

x

x

 

4

acidity of alcohols,

x

x

x

4

 

x

x

x

4

aldehydes,

x

x

x

4

ketones

Table 16. Parameters used in linear equations of type (2) for calculating data on gas phase reactions

d = effective polarizability
= electronegativity parameter for inductive effect
hyp.: parameter for hyperconjugation

  1. Quantification of Effective Polarisability. Applications to Studies of X-Ray Photoelectron Spectroscopy and Alkylamine Protonation
    J. Gasteiger, M. G. Hutchings,
    J. Chem. Soc. Perkin
    2, 1984, 559-564
  2. Empirical Models of Substituent Polarisability and their Application to Stabilisation Effects in Positively Charged Species
    J. Gasteiger, M. G. Hutchings,
    Tetrahedron Lett.
    24, 2537-2540 (1983)
  3. Quantitative Models of Gas-Phase Proton Transfer Reactions Involving Alcohols, Ethers, and their Thio Analogs. Correlation Analyses Based on Residual Electronegativity and Effective Polarizability
    J. Gasteiger, M. G. Hutchings
    J. Amer. Chem. Soc.
    106, 6489-6495 (1984)

A Quantitative Description of Fundamental Polar Reaction Types. Proton and Hydride Transfer Reactions Connecting Alcohols and Carbonyl Compounds in the Gas Phase
M. G. Hutchings, J. Gasteiger
J. Chem. Soc. Perkin
2, 1986, 447-454

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