Page 84 - 71 the abc of dft_opt

21.2. PERTURBATION THEORY 167 168 CHAPTER 21. TIME-DEPENDENCE
ˆ
where δV (t) is in some sense small. Usually we will consider potentials whose time-dependence To see better what this means, we can write χ in its Lehman representation, i.e., in terms
is exp(iωt+ηt), where η → 0, and the perturbation begins at −∞, and is fully turned on by of the levels of the system:
t = 0. Under these conditions, we can ask what is the change in the system to first-order in -
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χ(xx ω) = φ 0 (x)φ 0 (x ) ... (21.29)
the perurbation? More precisely, we want to know the change in observables of the system. k!=j
Since this is a book on density functional theory, a key role will be played by the change in We define:
density in response to such a perturbation. We define the susceptibility of the system by
n jk (x) = %j|ˆn(x)k& (21.30)
1 1
δρ(xt) = dx # dt χ(x, x ; t − t ) δv(x t ) (21.22)
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the generalized transition dipole. Thus, taking the moments to construct α, we find
i.e., χ tells you the density change throughout the system induced by any external potential. 1
x jk = dx x n jk (x) (21.31)
It is also known as the density-density response function and the non-local (meaning depends
on x and x ) susceptibility. Since we are doing DFT, we will write it more neatly as are the transition dipole moments for each transition. Then
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χ[n 0 ](x, x ; t − t ) = δρ(xt)/δv(x t ) (21.23) α(ω) = - |x 0j | (21.32)
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j!=0 ω − ω 2 j0
i.e., it is a functional derivative, and depends only on the ground-state density (for a non-
degenerate system). Note that it is a function of t − t alone, since the unperturbed system To see the effect of specific transitions on the optical spectrum, we must use the well-known
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is static. Furthermore, we will speak of the retarded function, meaning χ = 0 for t < t. We formula: 1 < 1 =
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show in the rest of this section that χ contains most of what we want to know about the = P + iπ δ(ω − ω0) (21.33)
ω − ω 0 ω − ω 0
response of a system.
ie the real part is a principal value, and the imaginary part a delta-function. Thus
Next we specialize further to the specific case of the optical response of a system, i.e., to
2
a long wavelength electric field. We want to find out the time-dependent dipole moment of σ(ω) = - 2ω j0 |x j0 | δ(ω − ω 0 ) (21.34)
our system, defined by j!=0
1
d(t) = dx ρ(xt)x (21.24) The optical spectrum consists of a series of delta functions, whose intensity is determined by
the dipole matrix elements.
ie the first moment of the density. This is because the optical absorption is given by ... If we
set v(x ω) = Ex , we find Exercise 89 Dipole matrix elements
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1 1 Calculate the dipole matrix elements for optical absorption for (a) the harmonic oscillator and
α(ω) = dx x dx x χ(x, x ω) (21.25)
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(b) the particle in a box.
for the dynamic polarizability, and
We define the oscillator strength of a given transition as
σ(ω) = 2α(ω)ω/c (21.26) 2
f ji = 2ω ji |x ji | /3 (21.35)
ie the moments of the susceptibility yield the complex dynamic susceptibility, which in turn
These satisfy the TRK sum-rule,
give the optical response of the system. One can show this satisfies the well-known Thomas- -
f j jk = N (21.36)
Reich-Kuhn sum-rule, k!=j
1 2
∞ 2π
dω σ(ω) = N (21.27) Note that all oscillator strengths from the ground-state are positive, but from an excited
0 c
state, some are negative.
Thre is also the static sum-rule, which follows from the general Kramers-Kronig rule for the
polarizability: Exercise 90 Oscillator strengths
2 1 ∞ dω From above, now find oscillator strengths for both systems and show that they satisfy the
2α(ω) = 3α(0) (21.28)
π 0 ω Thomas-Reich-Kuhn sum-rule.