Hyperpolarizability

Hyperpolarizability is a property of molecules that is of great importance for nonlinear optics . The induced dipole moments are no longer proportional to the electric field strength of the incident light wave.

principle

Non-centrosymmetric p -nitroaniline as an example of a molecule with great hyperpolarizability.

In a macroscopic system, the induced polarization is a function of the electrical susceptibility and the electrical field : ${\ displaystyle P}$ ${\ displaystyle \ chi}$ ${\ displaystyle E}$

{\ displaystyle P = \ chi E \ epsilon _ {0}}

In the case of a strong electric field, such as that generated by a laser , the induced polarization must be represented as a power series :

{\ displaystyle P = \ chi ^ {(1)} F + \ chi ^ {(2)} FF + \ chi ^ {(3)} FFF + \ ldots}

where and describe the nonlinear effects of the second and third order. Only if the material crystallizes in a non-centrosymmetrical space group is the second order term non-zero. ${\ displaystyle \ chi ^ {(2)}}$ ${\ displaystyle \ chi ^ {(3)}}$

If you transfer this macroscopic concept to the microscopic (molecular) level, you get a similar power series for polarizability :

{\ displaystyle p_ {i} = \ sum _ {j} \ alpha _ {ij} F_ {j} + \ sum _ {j \ leq k} \ beta _ {ijk} F_ {j} F_ {k} + \ sum _ {j \ leq k \ leq l} \ gamma _ {ijkl} F_ {j} F_ {k} F_ {l} + \ ldots}

where the variables span the molecular axis system. is the second order hyperpolarizability. It is only non-zero if the molecule is not centrosymmetric. Both the linear polarizability as well as the hyperpolarizabilities and is it is frequency-dependent tensors . ${\ displaystyle i, j, k, l}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ alpha}$ ${\ displaystyle \ beta}$ ${\ displaystyle \ gamma}$

Donor-acceptor molecules such as p -nitroaniline are a prototype because the electron density can be shifted asymmetrically here due to the symmetrical field applied. An even greater hyperpolarizability can be found in extensive systems such as those found in organic dyes. ${\ displaystyle \ pi}$

In CODATA 2018, atomic units of the first and second hyperpolarizability are defined:

{\ displaystyle \ beta _ {\ mathrm {au}} = e ^ {3} a_ {0} ^ {3} / E _ {\ mathrm {h}} ^ {2} = 3 {,} 206 \, 361 \ , 306 \, 1 (15) \ cdot 10 ^ {- 53} \, \ mathrm {C ^ {3} \, m ^ {3} / J ^ {2}}}

{\ displaystyle \ gamma _ {\ mathrm {au}} = e ^ {4} a_ {0} ^ {4} / E _ {\ mathrm {h}} ^ {3} = 6 {,} 235 \, 379 \ , 990 \, 5 (38) \ cdot 10 ^ {- 65} \, \ mathrm {C ^ {4} \, m ^ {4} / J ^ {3}}}

with a ₀ = Bohr radius , e = elementary charge of the electron and E _h = Hartree energy .

technology

The first exact determinations of the hyperpolarizability were made by A. David Buckingham with the help of the Kerr effect . Methods today are frequently used measurements of hyper-Rayleigh scattering (HRS) and the field-induced second harmonic ( English electric-field-induced second-harmonic generation , EFISH). Computer calculations based on density functional theory and the Hartree-Fock method often use the SOS approach (“ sum over states ”).

literature

Entry on hyperpolarizability (of nth order) . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.HT07053 Version: 2.3.3.
DR Kanis, MA Ratner, TJ Marks: Design and construction of molecular assemblies with large second-order optical nonlinearities. Quantum chemical aspects . In: Chemical Reviews . tape 94 , no. 1 , 1994, p. 195-242 , doi : 10.1021 / cr00025a007 (English).

Individual evidence

↑ A. Volkov, C. Gatti, Y. Abramov, P. Coppens: Evaluation of net atomic charges and atomic and molecular electrostatic moments through topological analysis of the experimental charge density . In: Acta Crystallographica Section A . tape 56 , no. 3 , 2000, pp. 252-258 , doi : 10.1107 / S0108767300001628 .
^ AD Buckingham, P. Hibbard: Polarizability and Hyperpolarizability of the Helium Atom . In: Symposia of the Faraday Society . tape 2 , 1968, p. 41-47 , doi : 10.1039 / SF9680200041 .
^ P. Kaatz, EA Donley, DP Shelton: A comparison of molecular hyperpolarizabilities from gas and liquid phase measurements . In: The Journal of Chemical Physics . tape 108 , 1998, pp. 849-856 , doi : 10.1063 / 1.475448 ( online [PDF]).
^ JP Coe, MJ Paterson: Approaching exact hyperpolarizabilities via sum-over-states Monte Carlo configuration interaction . In: The Journal of Chemical Physics . tape 141 , no. 12 , 2014, p. 124118 , doi : 10.1063 / 1.4896229 , arxiv : 1409.7276 .

[1] A. Volkov, C. Gatti, Y. Abramov, P. Coppens: Evaluation of net atomic charges and atomic and molecular electrostatic moments through topological analysis of the experimental charge density . In: Acta Crystallographica Section A . tape 56 , no. 3 , 2000, pp. 252-258 , doi : 10.1107 / S0108767300001628 .

[2] AD Buckingham, P. Hibbard: Polarizability and Hyperpolarizability of the Helium Atom . In: Symposia of the Faraday Society . tape 2 , 1968, p. 41-47 , doi : 10.1039 / SF9680200041 .

[3] P. Kaatz, EA Donley, DP Shelton: A comparison of molecular hyperpolarizabilities from gas and liquid phase measurements . In: The Journal of Chemical Physics . tape 108 , 1998, pp. 849-856 , doi : 10.1063 / 1.475448 ( online [PDF]).

[4] JP Coe, MJ Paterson: Approaching exact hyperpolarizabilities via sum-over-states Monte Carlo configuration interaction . In: The Journal of Chemical Physics . tape 141 , no. 12 , 2014, p. 124118 , doi : 10.1063 / 1.4896229 , arxiv : 1409.7276 .