The Linear Response Function. - Vrije Universiteit Brussel

What about third order derivatives?3⎛ ∂ E ⎞ ⎛ ∂η⎞• ⎜ 3 ⎟ = ⎜ ⎟ : hyperhardness: known to be small⎝ ∂N⎠ ⎝ ∂N⎠vv( )( )3⎛ δ E ⎞ ⎛ δχ r,r' ⎞⎜ =δv( r) δv( r' ) δv( r'' ) ⎟ ⎜ δv r'' ⎟⎝ ⎠ ⎝ ⎠• : awkward …? Importance of mixed “2+1” derivativesNN32⎛ ∂ E ⎞ ⎛ δη⎞ ⎛ ∂ ρ r ⎞•( ) ⎛ ∂f ( r)⎞⎜ Nδv( r) ⎟= ⎜ = ⎜ ⎟ = ⎜ ⎟δv( r)⎟⎝ ⎠ ⎝ ⎠ N N ⎠2 2∂N ⎝ ∂ ⎠ ∂v ⎝vDual descriptorC. Morell, A. Grand, A. Toro-Labbé,J. Phys. Chem. A 109, 205, 2005.3⎛ δ E ⎞ ⎛ ∂χ r, r' ⎞ ⎛ δf r ⎞•Already a large number of applications: one shotrepresentation of nucleophilic and electrophilic regions( ) ( )⎟( )⎜ = ⎜ =∂Nδv ( r) δv( r' ) ⎟ ∂N ⎜ δv r' ⎟⎝ ⎠⎝ ⎠vv ⎝ ⎠NunexploredFukui kernelPolarization effects on Fukui functionMight play a role in describing change in polarizability upon ionization,electron attachment, …P. Geerlings, F. De Proft, PCCP, 10, 3028, 2008.Pag.6-1-2012 9