Time-Dependent Electron Localization Function - Fachbereich ...

2.3 Derivation of the time-dependent ELF
n σ (r, t) := n σ (r, r, t). (2.10)
Eq. (2.10) gives the probability of finding a particle with spin σ at r and is normalized
to the particle number N σ . For the ELF we need the spin-diagonal (σ 1 = σ 2 )
of the reduced two-particle density matrix
D σ1 σ 2
(r 1 , r 2 , t) = N(N − 1) ∑ ∫ ∫
d 3 r 3 · · · d 3 r N |Ψ(r 1 σ 1 , r 2 σ 2 , . . . , r N σ N , t)| 2 ,
σ 3···σ N
(2.11)
which describes the probability of finding an electron with spin σ 1 at r 1 and another
electron at r 2 with spin σ 2 . For σ 1 = σ 2 , it is known as the same-spin pair probability.
Central for the electron localization function is the so-called conditional pair
probability. It is defined analogously to the static case as
P σ (r, r ′ , t) := D σσ(r, r ′ , t)
n σ (r, t)
(2.12)
and gives the probability of finding an electron with spin σ at r ′ at time t knowing
with certainty that another electron with the same spin is at r at that time.
2.3.1 Spherical average and Taylor expansion of P σ (r, r + s, t)
Since we are only interested in the probability of finding an electron in the vicinity
of r, we substitute r ′ by r + s and do a spherical average of the Taylor expansion
in s for small s, s := |s|.
Expanding the wave function in terms of s (in the second argument) gives
Ψ(rσ, (r + s)σ, . . . , r N σ N , t) = Ψ(rσ, (r + s)σ, . . . , r N σ N , t) ∣ s=0
∣ 3∑ ∂Ψ ∣∣∣si
+ s i + O(s 2 ). (2.13)
∂s i =0
The first term surely vanishes since two electrons with the same spin cannot be at
the same location (Pauli exclusion principle). Thus we get
|Ψ| 2 ∝
3∑
i,k=1
i=1
s i s k c ik + O(s 3 ), (2.14)
where c ik contains the factors and the s-independent derivative of Ψ and Ψ ∗ . s can
be written in spherical coordinates as s = (s 1 , s 2 , s 3 ) T = s (cos φ s sin θ s , sin φ s sin θ s , cos θ s ) T .
Doing the spherical average of Eq. (2.14) leads to
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