2.2.1 An introduction to time-dependent density functional theory Although the HK theorems lay the foundation for a density representation of the fully inter-acting many-particle system, they do not establish a direct relation between the GS density and truly dynamic or excited-state properties. The investigation of such properties using density functionals is based on time-dependent density functional theory. In TDDFT, the Runge-Gross theorem [RG84] is the complement of the HK theorems of GS DFT. Given an initial state and particle-particle interaction, it establishes a one-to-one correspondence between the time-dependent (TD) density n(r, t) and the TD external potential vext(r, t) up to a purely TD function c(t). With vext(r, t) and the initial state Ψ(t0), also the TD wave-function is determined uniquely up to a TD phase via solution of the TD Schrödinger equation. As expectation values of any operator are not sensitive to the phase of the wave function, in principle, each observable is a functional of n(r, t) and the initial state. The Runge-Gross proof has be rened by van Leeuwen [vL99] who covers the non-interacting v -representability question of TD densities by a construction procedure of the external potential of the alternative reference system [vL99, MUN+06]. Moreover, the initial-state dependence of the density representation has been discussed in Refs. [MB01] and [MBW02].
To derive a calculation scheme for dynamic properties, Runge and Gross [RG84] suggested a variational principle that rests upon on an action functional. However, Refs. [vL98] and [vL01] demonstrate that TDDFT based on the Runge-Gross action leads to contradictions in the symmetry and causality requirements of one of the most important ingredients of TDDFT linear response theory, namely the xc kernel
fxc(r, t;r0, t0) = δ2Axc
δn(r, t)δn(r0, t0), (2.10) where Axc is the xc part of the Runge-Gross action [RG84]. Van Leeuwen [vL98, vL01]
solved this problem by introducing a new action functional that is based on the time contour method due to Keldysh (for more details, see Refs. [vL98, vL01, MUN+06, Mun07]). The thus obtained variational principle yields a set of time-dependent Kohn-Sham equations
i∂
∂tϕjσ(r, t) =hKS,σ(r, t)ϕjσ(r, t), (2.11) where the TD KS Hamiltonian reads
hKS,σ(r, t) =−∇2
2 +vH(r, t) +vxc,σ(r, t) +vext(r, t). (2.12) The TD xc potentialvxc,σ(r, t)follows from the functional derivative
vxc,σ(r, t) = δAxc
δn(r, τ)
n=nσ(r,t)
(2.13) of the xc part of the new action functional with respect to the densityn(r, τ)where the time variableτ is the Keldysh pseudo time, but the functional derivative is taken at the physical TD densityn(r, t)[vL98, vL01]. By the basic theorems of TDDFT,vxc,σ(r, t)is a functional
2.2. TIME-DEPENDENT DENSITY FUNCTIONAL THEORY 7 of the TD density and its entire history, the initial interacting wave function, and the initial state of the KS system. Finally, based on xc approximations, the TD density follows from theNσ occupied orbitals per spin channel of the TD KS system according to
n(r, t) = X
σ=↑,↓
Nσ
X
j=1
|ϕjσ(r, t)|2. (2.14) Practical calculations of TDDFT rely either on the linear response formalism or on real-time propagation of the KS system. Both approaches are introduced in the following.
2.2.2 TDDFT linear response formalism
Today, most TDDFT excitation energy investigations are based on the linear response for-malism. The linear response of the GS density to small perturbations δvext(r0, t0) of the external potential reads
δn(r, t) = Z Z
χ[nGS](r,r0, t−t0)δvext(r0, t0) d3r0dt0, (2.15) whereχ[nGS](r,r0, t−t0) is the linear response function of the interacting system. For the sake of clarity, I use a spin-independent notation here and in the next section. Based on the fundamental theorems of TDDFT, the density response may be expressed in terms of the linear response of the KS system due to changes of the KS potential [PGG96]. Hence, a relation between χ[nGS](r,r0, t−t0) and the linear response function of the KS system χKS
exists [PGG96, MUN+06]. In frequency space, the interacting linear response function is χ(r,r0, ω) =χKS(r,r0, ω)
+ Z Z
χKS(r,r1, ω) 1
|r1−r2|+fxc(r1,r2, ω)
χ(r2,r0, ω) d3r1d3r2, (2.16) whereχKS(r,r0, ω) is the frequency-dependent linear response of the KS system reading
χKS(r,r0, ω) = 2 lim
η→0+
X
i,a
ξia(r)ξia∗(r0)
ω−ωia+ iη − ξia∗(r)ξia(r0) ω+ωia+ iη
. (2.17)
It depends on the eigenvalue dierencesωia =εa−εi between all possible combinations of occupied KS orbitals i and unoccupied KS orbitals aand on the orbital products ξia(r) = ϕ∗i(r)ϕa(r) of the corresponding GS KS orbitals [MUN+06]. It has poles at frequenciesωia. The KS response contribution to Eq. (2.16) includes only those eects that are encoded in the single-particle GS KS system, whereas the Hartree-exchange-correlation (Hxc) kernel
fHxc(r,r0, ω) = 1
|r−r0|+fxc(r,r0, ω) (2.18) needs to cover all many-particle eects beyond that.
Finding the excitation energies of the interacting systems in linear response amounts to nding the poles of χ(r,r0, ω). Casida [Cas95, Cas96] developed a matrix-equation for-mulation for practical implementations of this strategy. In this approach, solution of the eigenvalue problem [Cas96, MUN+06]
X
i0,a0
Ria,i0a0Fi0a0 = Ω2Fia, (2.19)
where
Ria,i0a0 =ω2iaδii0δaa0 + 4√ωiaωi0a0Kia,i0a0 (2.20) and
Kia,i0a0(ω) = Z Z
ξia∗(r)fHxc(r,r0, ω)ξi0a0(r0) d3rd3r0, (2.21) yields the excitation energies Ω of the interacting system. The corresponding oscillator strength is encoded in the eigenvectors [Cas96]. This procedure is well-known as the Casida approach of linear response TDDFT. It is most frequently used in today's TDDFT based applications and implemented in most quantum chemistry codes.
An approximate approach to ndingΩfrom TDDFT linear response provides means for more insight into the inuence of the individual contributions of Eq. (2.16), i.e., contribu-tions from the GS KS system and Hartree-exchange-correlation eects. It is based on the single-pole approximation [PGG96]: Assuming that the true excitation Ω˜ is dominated by one transition from a single occupied orbital j to a single unoccupied orbital b, all other contributions to the total response can be neglected. Thus, one obtains [PGG96, AGB03]
Ω˜ ≈ωjb+ 2 Re{Kjb,jb(ωjb)}. (2.22) Equation (2.22) is used in Sec. 4.1 to explain the CT problem of TDDFT.
2.2.3 Real-time propagation TDDFT
In this thesis, I mainly used the real-time (RT) propagation approach to TDDFT which does not require explicit linear response theory, but is directly based on the TD KS equations of Sec. 2.2.1. The central idea of this method is to compute the time evolution of the density from RT propagation of the TD KS system
ϕj(r, t) =U(t, t0)ϕj(r, t0) =T exp
−i Z t
t0
hKS(r, t0) dt0
ϕj(r, t) (2.23) via application of the propagator U(t, t0). All other observables need to be obtained from n(r, t). This is a crucial point because in some situations it is dicult to extract information that is easily available in the Casida approach from the time evolution of the density, e.g., see Chap. 4. In such cases, new investigation ideas need to be developed as for instance the transition density analysis tool of Sec. 2.3. Yet, some important observables are explicit functionals of the density, in particular the TD dipole moment
d(t) = Z
rn(r, t) d3r. (2.24)
Excitation energies emerge as peaks in the spectrum of the TD density after some initial excitation [ZS80, CRS97]. Most importantly, optical excitations that are sensitive to the dipole moment cause peaks at frequencies ω in the Fourier transformation d(ω) of the TD dipole moment that correspond to optical excitation energies. To compute the latter, the system is typically excited by an initial moment boost and the resulting TD dipole moment is used as basic observable [YB96, YB99a, YB99b, MCBR03, CAO+06, MK07, Mun07, Mun09].
Here, the boost is applied initially by multiplyingexp ir·pboost
~
to the GS KS orbitals. This introduces an excitation energy Eexcit via the momentum |pboost| = p
2mEexcit/N. The
2.3. THE TRANSITION DENSITY ANALYSIS TOOL 9