Section: 10 | Polarizabilities of Atoms and Molecules |

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POLARIZABILITIES OF ATOMS AND MOLECULES

Thomas M. Miller

The polarizability of an atom or molecule describes the response of its electron cloud to an external field. The atomic or molecular energy shift ΔW 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 ΔW = –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 (H₂) has this effect been studied enough to warrant consideration.

Polarizabilities are normally expressed in cgs units of cm³. Ground-state polarizabilities are in the range of 10^–24 cm³ = 1 Å³and hence are often given in Å³ units. Theorists tend to use atomic units of a₀³ where a₀ is the Bohr radius. The conversion is α(cm³) = 0.148184 × 10^-24× α(a₀³). Polarizabilities are only recently encountered in SI units, C m²/ V = J/(V/m)². The conversion from cgs units to SI units is α(C m²/ V) = 4πε₀ × 10^-6 α(cm³), where ε₀ is the permittivity of free space in SI units and the factor 10^–6 simply converts cm³ into m³. Thus, α(C m²/ V) = 1.11265 × 10^-16× α(cm³). Measurements of excited state polarizabilities by optical methods tend to be reported using units of MHz/(V/cm)², where the energy shift, ΔW, is expressed in frequency units with a factor of h understood. The polarizability is –2 ΔW/E². The conversion into cgs units is α(cm³) = 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 cm³; 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 results for atomic polarizabilities is given by P. Schwerdtfeger at <http://ctcp.massey.ac.nc/Tablepol.pdf> (accessed October 2021). 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.

Column definitions for the table are as follows.

Column heading	Definition
Name	Name of atom or molecule; atoms are listed in order of atomic number, followed by molecules in alphabetical order by name
Mol. form.	Molecular formula of atom or molecule
α	Polarizability, in units 10^-24 cm³
Ref.	Reference; separate reference lists are given for atoms and molecules
Notes	Additional information about data
Z	Atomic number (for atoms only)

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Formulas Involving Polarizability*

Description	Formula	Remarks
Lorentz-Lorenz relation	$α (ν) = \frac{3}{4 π n} [\frac{η^{2} (ν) - 1}{η^{2} (ν) + 2}]$	For a gas of atoms or nonpolar molecules; the index of refraction is η(ν)
Refraction by polar molecules	$α (ν) + \frac{d^{2}}{3 k T} = \frac{3}{4 π n} [\frac{η^{2} (ν) - 1}{η^{2} (ν) + 2}]$	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 η³(ν) = κ(ν)
Diamagnetic susceptibility	χ_m = e²(a₀Nα)^½ / 4m_ec²	From the approximation that the static polarizability is given by the variational formula α = (4/9a₀)Σ(N_ir_i²)²; N is the number of electrons, m_e is the electron mass; a crude approximation is χ_m = (E_i/4m_ec²)α, where E_i is the ionization energy
Long-range electron- or ion-molecule interaction energy	V(r) = -e²α / 2r⁴	The target molecule polarizability is α
Ion mobility in a gas	κ = 13.87 / (αμ)^½cm² / V · s	This one formula is not in cgs units. Enter α in Å³ or 10^-24 cm³ units and the reduced mass μ of the ion-molecule pair in amu. Classical limit; pure polarization potential
Langevin capture cross section	σ(ν₀) = (2πe / ν₀)(α / μ)^½	The relative velocity of approach for an ion-molecule pair is ν₀; 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_d = 2πe[(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) = 4πA² +32π⁴μe²αAk / 3h² +...	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	$C_{6} = \frac{3}{2} [\frac{α^{A} α^{B} E^{A} E^{B}}{E^{A} + E^{B}}]$	For the interaction potential term V₆(r)= –C₆r⁶; 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	$\begin{array}{l} C_{8} = \frac{15}{4} [\frac{α^{A} α_{q}^{B} E^{A} E_{q}^{B}}{E^{A} + E_{q}^{B}}] \\ + \frac{15}{4} [\frac{α_{q}^{A} α^{B} E_{q}^{A} E^{B}}{E_{q}^{A} + E^{B}}] \end{array}$	For the interaction potential term V₈(r) = –C₈r⁸;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	$C_{3} = \frac{α g E^{A} E^{S}}{8 (E^{A} + E^{S})}$	For an interaction potential V₃(r) = –C₃r³; 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	$α (v) = \frac{e^{2} h^{2}}{4 π^{2} m_{e}} Σ \frac{f_{k}}{E_{k}^{2} - {(h v)}^{2}}$	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	$α (v) = \frac{α E_{r}^{2}}{E_{r}^{2} - {(h ν)}^{2}}$	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	$\begin{array}{l} α (v) = \frac{8 π}{9 c^{4}} {(2 π ν)}^{4} \\ \times [3 α^{2} (ν) + 2 γ^{2} (ν) / 3] \end{array}$	The photon frequency is ν; the polarizability anisotropy (the difference between polarizabilities parallel and perpendicular to the molecular axis) is γ(ν)
Verdet constant	$V (ν) = \frac{ν n}{2 m_{e} c^{2}} [\frac{d α (ν)}{d ν}]$	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.

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Polarizability of Atoms and Molecules

Name	Synonym	Mol. form.	Formula	CAS Reg. No.	α/10^-24 cm³	Ref.	Notes	Z
Continued on next page...
*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

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