DFT Reactivity Descriptors and Catalysis - Vrije Universiteit Brussel

Softness and hardness: from global to local level
2
E
η ⎛ ∂ ⎞
2
⎝ ∂N
⎠v
⎛ ∂ µ ⎞
= ⎜ ⎟ = ⎜ ⎟
⎝ ∂N
⎠
v
η S =
1
( r)
⎛ ∂N
⎞ ∂
⎛ ∂ρ
⎞
S = ⎜ ⎟ = ρ ( r)
d r = ⎜ ⎟ d r = s( r)
d r
∂µ ∂µ ∫ ∫
∂µ
∫
⎝ ⎠v
⎝ ⎠v
s( r ) = local softness
Chain rule ( )
( ) ∂ρ
( )
∂ρ
r ⎛ r ⎞ ⎛ ∂N
⎞
s r = = ⎜ ⎟ ⎜ ⎟ = Sf r
∂µ ⎝ ∂N
⎠ ∂µ
v ⎝ ⎠v
( )
⎛
δµ
⎞
• Local Hardness η ( r)
= ∫η ( r) s ( r) d r = 1 but η
⎜
!!
δρ ( r) ⎟
( r)
≠ 1
s ( r)
⎝ ⎠ v
∫
η
( r) ( )
f r d r
= η
Pag.
30-9-2008 16