Page 90 - 71 the abc of dft_opt

180 CHAPTER 23. LINEAR RESPONSE
where all objects are functionals of the ground-state density. This is called a Dyson-like
equation, because it has the same mathematical form as the Dyson equation relating the
one-particle Green’s function to its free counterpart, with a kernel that is the one-body
potential. However, it is an oddity of DFT that it has this form, since χ is a two-body
Chapter 23 response function.
This equation contains the key to electronic excitations via TDDFT. This is because when
ω matches a true transition frequency of the system, the response function χ blows up, i.e.,
Linear response has a pole as a function of ω. Thus χ S has a set of such poles, at the single-particle excitations
of the KS system:
φ (r) φ a (r) φ (r ) φ i (r )
∗
∗
#
#
- i a
#
χ S (rr ω) = 2 − c.c.(ω → −ω) (23.5)
#
Note that the difference between n(x) and n (x) is non-vanishing already in first order of ω − (& i − & a ) + i0 +
iocc,aunocc
v(x) − v (x), ensuring the invertibility of the linear response operators of section ??.
#
In the absence of Hartree-exchange-correlation effects, χ = χ S , and so the allowed transi-
tions are exactly those of the ground-state KS potential. But the presence of the kernel in
23.1 Dyson-like response equation and the kernel Eq. (23.4) shifts the transitions away from the KS values to the true values. Moreover,
the strengths of the poles can be simply related to optical absorption intensities (oscillator
When the perturbing field is weak, as in normal spectroscopic experiments, perturbation
strengths) and so these also are affected by the kernel.
theory applies. Then, instead of needing knowledge of v XC for densities that are changing In linear response, this implies that the exchange-correlation kernel has the form:
significantly with time, we only need know this potential in the vicinity of the initial state, gs
which we take to be a non-degenerate ground-state. Writing n(x) = n(r) + δn(x), we have f adia (rr , t − t ) = δv [n 0 ](r) | n 0 (r)=n(rt) δ(t − t ). (23.6)
XC
#
#
#
XC
1 1 δn 0 (r)
#
# #
#
v XC [n + δn](x) = v XC [n](r) + dt # d r f XC [n](rr , t − t ) δn(r t ), (23.1) adia
3 #
When Fourier-transformed, this implies f is frequency-independent.
XC
where f XC is called the exchange-correlation kernel, evaluated on the ground-state density:
23.2 Casida’s equations
# #
#
#
f XC [n](rr , t − t ) = δv XC (rt)/δn(r t ). (23.2)
While still a more complex beast than the ground-state exchange-correlation potential, the The various methods for extracting excitations have been described in the overview. Here we
exchange-correlation kernel is much more manageable than the full time-dependent exchange- focus on the results. Casida showed that, finding the poles of χ is equivalent to solving the
correlation potential, because it is a functional of the ground-state density alone. To un- eigenvalue problem:
˜
-
derstand why f XC is important for linear response, we define the point-wise susceptibility Ω qq !(ω)v q ! = Ωv q , (23.7)
χ[n 0 ](rr , t − t ) as the response of the ground state to a small change in the external poten- q !
#
#
tial: where q is a double index, representing a transition from occupied KS orbital i to unoccupied
1 1 2 ∗
#
#
3 #
# #
δn(x) = dt # d r χ[n 0 ](rr , t − t ) δv ext (r t ), (23.3) KS orbital a, ω q = & a − & i , Ω = ω , and Φ q (r) = φ (r)φ a (r). The matrix is
i
˜
%
#
#
i.e., if you make a small change in the external potential at point r and time t , χ tells you Ω qq !(ω) = δ qq !Ω q + 2 ω q ω %q|f HXC (ω)|q &, (23.8)
#
#
q
how the density will change at point r and later time t. Now, the ground-state Kohn-Sham where
1 1
3
system has its own analog of χ, which we denote by χ s , which says how the non-interacting %q|f HXC (ω)|q & = d r d r Φ (r)f HXC (r, r , ω)Φ q !(r ). (23.9)
∗
#
#
#
3 #
q
KS electrons would respond to δv S (r t ), which is quite different from the interacting case.
# #
In this equation, f HXC is the Hartree-exchange-correlation kernel, 1/|r − r | + f XC (r, r , ω).
#
#
But both must yield the same density response, so using the definition of the KS potential, A selfconsistent solution of Eq. (23.7) yields the excitation energies ω and the oscillator
and of the exchange-correlation kernel, we find the key equation of TDDFT linear response:
strengths can be obtained from the eigenvectors [?]. In the special case of an adiabatic
 
1 3 1 3 1 approximation, these equations are a straightforward matrix equation. Algorithms exist for
#
χ(rr ω) = χ S (rr ω) + d r 1 d r 2 χ S (rr 1 ω)  + f XC (r 1 r 2 ω) χ(r 2 r ω), (23.4)
#
#

|r 1 − r 2 | extracting just the lowest transitions.
179