Section: 10 | Polarizabilities of Atoms and Molecules |
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The *polarizability* of an atom or molecule describes the response of its electron cloud to an external field. The atomic or molecular energy shift Δ*W* due to an external electric field *E* is proportional to *E* ^{2} for external fields that are weak compared to the internal electric fields between the nucleus and electron cloud. The *electric dipole polarizability* *α* is the constant of proportionality defined by Δ*W* = –*αE* ^{2}/2. The induced electric dipole moment is *αE*. *Hyperpolarizabilities*, coefficients of higher powers of *E*, are less often required. Technically, the polarizability is a tensor quantity but for spherically symmetric charge distributions reduces to a single number. In any case, an *average polarizability* is usually adequate in calculations. Frequency-dependent or *dynamic polarizabilities* are needed for electric fields that vary in time, except for frequencies that are much lower than electron orbital frequencies, for which *static polarizabilities* suffice.

Polarizabilities for atoms and molecules in excited states are found to be larger than for ground states and may be positive or negative. Molecular polarizabilities are very slightly temperature dependent because the size of the molecule depends on its rovibrational state. Only in the case of dihydrogen (H_{2}) has this effect been studied enough to warrant consideration.

Polarizabilities are normally expressed in cgs units of cm^{3}. Ground-state polarizabilities are in the range of 10^{–24} cm^{3} = 1 Å^{3 }and hence are often given in Å^{3} units. Theorists tend to use atomic units of *a*_{0}^{3} where *a*_{0} is the Bohr radius. The conversion is *α*(cm^{3}) = 0.148184 × 10^{-24 }× *α*(*a*_{0}^{3}). Polarizabilities are only recently encountered in SI units, C m^{2}/ V = J/(V/m)^{2}. The conversion from cgs units to SI units is *α*(C m^{2}/ V) = 4π*ε*_{0} × 10^{-6} *α*(cm^{3}), where *ε*_{0} is the permittivity of free space in SI units and the factor 10^{–6} simply converts cm^{3} into m^{3}. Thus,* α*(C m^{2}/ V) = 1.11265 × 10^{-16 }× *α*_{}(cm^{3}). Measurements of excited state polarizabilities by optical methods tend to be reported using units of MHz/(V/cm)^{2}, where the energy shift, Δ*W*, is expressed in frequency units with a factor of *h* understood. The polarizability is –2 Δ*W/E*^{2}. The conversion into cgs units is *α*(cm^{3}) = 5.955214 × 10^{-16} × *α*[MHz/(V/cm)^{2}].

The polarizability appears in many formulas for low-energy processes involving the valence electrons of atoms or molecules. These formulas are given below in cgs units: the polarizability *α* is in cm^{3}; masses *m* or *μ* are in grams; energies are in ergs; and electric charges are in esu, where *e* = 4.8032 × 10^{-10} esu. The symbol *α*(*ν*) denotes a frequency (*ν*) dependent polarizability, where *α*(*ν*) reduces to the static polarizability *α* for *ν* = 0. For further information, see Bonin, K. D., and Kresin, V. V., *Electric Dipole Polarizabilities of Atoms, Molecules, and Clusters*, World Scientific, Singapore, 1997; Bonin, K. D., and Kadar-Kallen, M.A., *Int. J. Mod. Phys.* *B*, 24, 3313, 1994; Miller, T. M., and Bederson, B., *Advances in Atomic and Molecular Physics*, 13, 1, 1977; and Gould, H., and Miller, T. M., *Advances in Atomic, Molecular, and Optical Physics*, 51, 243, 2005. A helpful listing of theoretical results for atomic polarizabilities is given by P. Schwerdtfeger at http://ctcp.massey.ac.nc/dipole-polarizabilities (accessed September 2016). Details on polarizability-related interactions, especially in regard to hyperpolarizabilities and nonlinear optical phenomena, are given by Bogaard, M. P., and Orr, B. J., in *Physical Chemistry, Series Two, Vol. 2, Molecular Structure and Properties*, Buckingham, A. D., Ed., Butterworths, London, 1975, pp. 149–194. A tabulation of tensor and hyperpolarizabilities is included.

An empirical additive formula for molecular polarizabilities at 589 nm frequency has been given in Bosque, R., and Sales, J., *J. Chem. Inf. Comput. Sci.* 42, 1154, 2002:

*α*= 0.32 + 1.51#C + 0.17#H + 0.57#O + 1.05#N + 2.99#S + 2.48#P + 0.22#F + 2.16#Cl + 3.29#Br + 5.45#I

where #C denotes the number of carbon atoms in the molecule, etc. A helium-elimination additive method has been given by Kassimi, N. E.-B., and Thakkar, A. J., *Chem. Phys. Lett.* 472, 232, 2009.

All polarizabilities in this table are experimental values except those indicated as calculated in the Note column. The experimental polarizabilities are mostly determined by measurements of a dielectric constant or refractive index and are quite precise (0.5% or better). However, one should treat many of the results with several percent of caution because of the age of the data and because some of the results refer to optical frequencies rather than static. Comments given with the references are intended to allow one to judge the degree of caution required. Interested persons should consult these references. In many cases, the reference given is to a theoretical paper in which the experimental results are quoted. These papers, noted in the references, contain valuable information on polarizability calculations and experimental data, which often includes the tensor components of the polarizability.

The table first lists atoms in order of atomic number, followed by molecules in alphabetical order by name.

Description | Formula | Remarks |

Lorentz-Lorenz relation | $\alpha \left(\nu \right)=\frac{3}{4\pi n}\left[\frac{{\eta}^{2}\left(\nu \right)-1}{{\eta}^{2}\left(\nu \right)+2}\right]$ | For a gas of atoms or nonpolar molecules; the index of refraction is η(ν) |

Refraction by polar molecules | $\alpha \left(\nu \right)+\frac{{d}^{2}}{3kT}=\frac{3}{4\pi n}\left[\frac{{\eta}^{2}\left(\nu \right)-1}{{\eta}^{2}\left(\nu \right)+2}\right]$ | The dipole moment is d, in esu·cm (= 10^{–18} D) |

Dielectric constant (dimensionless) | κ(ν) = 1+ 4πn α(ν) | From the Lorentz-Lorenz relation for the usual case of κ(ν) ≈ 1 |

Index of refraction (dimensionless) | μ(ν) = 1+ 2πnα(ν) | From η^{3}(ν) = κ(ν) |

Diamagnetic susceptibility | χ_{m} = e^{2}(a_{0}Nα)^{½} / 4m_{e}c^{2} | From the approximation that the static polarizability is given by the variational formula α = (4/9a_{0})Σ(N_{i}r_{i}^{2})^{2}; N is the number of electrons, m_{e} is the electron mass; a crude approximation is χ_{m} = (E_{i}/4m_{e}c^{2})α, where E_{i} is the ionization energy |

Long-range electron- or ion-molecule interaction energy | V(r) = -e^{2}α / 2r^{4} | The target molecule polarizability is α |

Ion mobility in a gas | κ = 13.87 / (αμ)^{½}cm^{2} / V · s | This one formula is not in cgs units. Enter α in Å^{3} or 10^{-24} cm^{3} units and the reduced mass μ of the ion-molecule pair in amu. Classical limit; pure polarization potential |

Langevin capture cross section | σ(ν_{0}) = (2πe / ν_{0})(α / μ)^{½} | The relative velocity of approach for an ion-molecule pair is ν_{0}; the target molecular polarizability is α and the reduced mass of the ion-molecule pair is μ |

Langevin reaction rate coefficient | k = 2πe(α / μ)^{½} | Collisional rate coefficient for an ion-molecule reaction |

Rate coefficient for polar molecules | k = 2π_{d}e[(α / μ)^{½} + cd(2 / μπkT)^{½}] | The dipole moment of the neutral is d in esu cm; the number c is a “locking factor” that depends on α and d, and is between 0 and 1 |

Modified effective range cross section for electron-neutral scattering | σ(k) = 4πA^{2}+32π ^{4}μe^{2}αAk / 3h^{2}+... | Here, k is the electron momentum divided by h/2π, where h is Planck’s constant; A is called the “scattering length”; the reduced mass is μ |

van der Waals constant between two systems A, B | ${C}_{6}=\frac{3}{2}\left[\frac{{\alpha}^{\text{A}}{\alpha}^{\text{B}}{E}^{\text{A}}{E}^{\text{B}}}{{E}^{\text{A}}+{E}^{\text{B}}}\right]$ | For the interaction potential term V_{6}(r)= –C_{6}r^{6}; E^{A,B} represents average dipole transition energies and α^{A,B} the respective polarizabilities of A, B |

Dipole-quadrupole constant between two systems A, B | $\begin{array}{l}{C}_{8}=\frac{15}{4}\left[\frac{{\alpha}^{\text{A}}{\alpha}_{\text{q}}{}^{\text{B}}{E}^{\text{A}}{E}_{\text{q}}{}^{\text{B}}}{{E}^{\text{A}}+{E}_{\text{q}}{}^{\text{B}}}\right]\\ +\frac{15}{4}\left[\frac{{\alpha}_{\text{q}}{}^{\text{A}}{\alpha}^{\text{B}}{E}_{\text{q}}{}^{\text{A}}{E}^{\text{B}}}{{E}_{\text{q}}{}^{\text{A}}+{E}^{\text{B}}}\right]\end{array}$ | For the interaction potential term V_{8}(r) = –C_{8}r^{8};E_{q}^{A,B} represents average quadrupole transition energies and α_{q}^{A,B} are the respective quadrupole polarizabilities of A, B |

van der Waals constant between an atom and a surface | ${\text{C}}_{3}=\frac{\alpha g{E}^{\text{A}}{E}^{\text{S}}}{8\left({E}^{\text{A}}+{E}^{\text{S}}\right)}$ | For an interaction potential V_{3}(r) = –C_{3}r^{3}; E^{A,S} are characteristic energies of the atom and surface; g = 1 for a free-electron metal and g = (ε_{∞} – 1)/(ε_{∞} + 1) for an ionic crystal |

Relationship between α(ν) and oscillator strengths | $\alpha \left(\text{v}\right)=\frac{{e}^{2}{h}^{2}}{4{\pi}^{2}{m}_{\text{e}}}\Sigma \frac{{f}_{k}}{{E}_{k}^{2}-{\left(hv\right)}^{2}}$ | Here, f_{k} is the oscillator strength from the ground state to an excited state k, with excitation energy E_{k}. This formula is often used to estimate static polarizabilities (ν = 0) |

Dynamic polarizability | $\alpha \left(\text{v}\right)=\frac{\alpha {E}_{\text{r}}{}^{2}}{{E}_{\text{r}}{}^{2}-{\left(h\nu \right)}^{2}}$ | Approximate variation of the frequency-dependent polarizability α(ν) from ν = 0 up to the first dipole-allowed electronic transition, of energy E_{r}; the static dipole polarizability is α(0); infrared contributions ignored |

Rayleigh scattering cross section | $\begin{array}{l}\alpha \left(\text{v}\right)=\frac{8\pi}{9{c}^{4}}{\left(2\pi \nu \right)}^{4}\\ \times \left[3{\alpha}^{2}\left(\nu \right)+2{\gamma}^{2}\left(\nu \right)/3\right]\end{array}$ | The photon frequency is ν; the polarizability anisotropy (the difference between polarizabilities parallel and perpendicular to the molecular axis) is γ(ν) |

Verdet constant | $V\left(\nu \right)=\frac{\nu n}{2{m}_{\text{e}}{c}^{2}}\left[\frac{\text{d}\alpha \left(\nu \right)}{\text{d}\nu}\right]$ | Defined from θ = V(ν)B, where θ is the angle of rotation of linearly polarized light through a medium of number density n, per unit length, for a longitudinal magnetic field strength B (Faraday effect) |

- * The gas number density,
*n*, in the list of formulas is usually taken to be that of 1 atm at 0 ºC in reporting experimental data.

Name | Synonym | Mol. form. | Formula | CAS Reg. No. | α/10^{-24} cm^{3} | Ref. | Note | Z |

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Atoms | ||||||||

Hydrogen (atomic) | H | H | 12385-13-6 | 0.666793 | 1 | calculated ("exact") | 1 | |

Helium | He | He | 7440-59-7 | 0.2050522 | 2 | calculated ("exact") | 2 | |

Helium | He | He | 7440-59-7 | 0.2050519 | 3 | dielectric constant (±0.009%) | 2 | |

Lithium | Li | Li | 7439-93-2 | 24.33 | 4 | beam (±0.66%) | 3 | |

Beryllium | Glucinium | Be | Be | 7440-41-7 | 5.60 | 42 | calculated (±1.2%) | 4 |

Boron | B | B | 7440-42-8 | 3.051 | 31 | calculated (±0.07%) | 5 | |

Carbon | C | C | 7440-44-0 | 1.67 | 5 | calculated (±2%) | 6 | |

Nitrogen (atomic) | N | N | 17778-88-0 | 1.13 | 2 | index of refraction (±5.3%) | 7 | |

Nitrogen (atomic) | N | N | 17778-88-0 | 1.1 | 40 | calculated (<2.5%) | ||

Oxygen (atomic) | O | O | 17778-80-2 | 0.77 | 2 | index of refraction (±7.8%) | 8 | |

Fluorine (atomic) | F | F | 14762-94-8 | 0.557 | 1 | calculated (±2%) | 9 | |

Neon | Ne | Ne | 7440-01-9 | 0.39432 | 6 | dielectric constant (±0.003) | 10 | |

Sodium | Natrium | Na | Na | 7440-23-5 | 24.11 | 7 | interferom (±0.50%) | 11 |

Sodium | Natrium | Na | Na | 7440-23-5 | 24.11 | 8 | interferom (±0.83%) | 11 |

Magnesium | Mg | Mg | 7439-95-4 | 8.8 | 36 | beam (±26%) | 12 | |

Magnesium | Mg | Mg | 7439-95-4 | 10.5 | 25 | calculated (<2.9%) | 12 | |

Aluminum | Al | Al | 7429-90-5 | 6.8 | 11 | beam (±4.4%) | 13 | |

Aluminum | Al | Al | 7429-90-5 | 8.22 | 31 | calculated (±0.1%) | ||

Silicon | Si | Si | 7440-21-3 | 5.53 | 5 | calculated (±2%) | 14 |