2.2. Density Functional Theory
2.2.2. Time-Dependent Density Functional Theory
In principle, the ground state density also includes the information about the excited states of a system. This can be derived from the logical chain in Equation (2.37). The ground state density determines the Hamiltonian through the external potential that in turn also comprises all its eigenstates. However, there has been no efficient method to extract the excitation energies from static DFT until now. Time-dependent density functional theory (TD-DFT) is the dynamical counterpart of stationary DFT, which allows to evaluate time-dependent properties of a system, including the calculation of excitation spectra in an efficient manner.[68]
The Runge-Gross (RG) theorem can be regarded as the dynamical analogue to the ex-istence theorem of Hohenberg and Kohn in static DFT.[69] The former states that when starting from the same initial state Ψ(t0), two different time-dependent potentials al-ways cause different time-dependent densities. Therefore, an one-to-one correspondence between these two quantities is given:
ρ(r, t)↔vext(r, t) . (2.50) The RG theorem only holds for time-dependent potentials that are Taylor-expandable around the initial time. This limitation is mostly satisfied in practical applications, especially for the calculation of excitation spectra, which deals with potentials arising from monochromatic light.
The van Leeuwen theorem provides the formal foundation to apply the KS formalism in TD-DFT.[70] This theorem states that the same time-dependent density of a system in an initial state Ψ(t0) with the electron-electron interactionVˆee and external potential vext can be completely reproduced by another system in an initial state Ψ′(t0) with a different electron-electron interaction Vˆee′ and a unique external potential vext′ . Thus, a fictive system with absent electron-electron interaction (KS system) is appropriate to describe the exact time-dependent density.
In the following, the system is assumed to be in its electronic ground state before
Density Functional Theory 23
a time-dependent potential is switched on at a certain time t0. The KS orbitals then satisfy the time-dependent Schrödinger equation
[
−∇2
2 +vext(ni)[ρ](r, t) ]
ϕi(r, t) =i∂
∂tϕi(r, t) , (2.51) where the effective potential is related to the real system by
vext(ni)[ρ](r, t) =vext[ρ](r, t) +vCoul[ρ](r, t) +vxc[ρ](r, t). (2.52) Here, vCoul(r, t) denotes the classical Coulomb potential, which only depends on the instantaneous density. vxc(r, t) stands for the time-dependent xc potential and is the only unknown in this formalism. An ubiquitous approximation is to evaluate the time-dependent xc potential at the instantaneous densities within the ground state xc poten-tial
vxcA(r, t)≈ v0xc[ρ0](r)
ρ0(r)→ρ(r,t) , (2.53)
and thus, ignores memory effects during the time propagation. The concept is called adiabatic approximation and used throughout this section.
For the calculation of excitation spectra, Equation (2.51) is rarely solved explicitly.
The essential task is to consider the response of the system to a weak perturbation.
Therefore, linear response theory is applied,[68] as already introduced in Section 2.1.2.3.
In the context of TD-DFT, the formalism results in the frequency-dependent density response function for the KS system:
ρ1(r, ω) =
∫
d3r′χni(r,r′, ω)vext(ni)(1) (r′, ω) , (2.54) with χni(r,r′, ω)being the KS density-density response function
χni(r,r′, ω) = lim
η→0+
∑
i,j
(nj−ni)ϕi(r)ϕ∗j(r)ϕ∗i(r′)ϕj(r′)
ω−ωij +iη . (2.55) Here, ni and nj stand for the occupation numbers of the KS orbitals i and j in the ground state, respectively and ωij denotes the difference between their corresponding orbital energies. The summation runs over all occupied and virtual KS orbitals. The
first-order effective perturbation is given by In Equation (2.56) the xc kernel fxcA(r′,r′′) was introduced in the adiabatic approxima-tion, which is the functional derivative of the xc potential with respect to the density
fxcA(r′,r′′) = δv0xc[ρ0](r′)
δρ0(r′′) = δ2Exc[ρ0]
δρ0(r′)δρ0(r′′) . (2.57) Electronic excitations can be regarded as characteristic eigenmodes of an interacting system, which are induced by an external perturbation. Nevertheless, those eigenmodes exist already in the absence of such a perturbation (vext[ρ](r, t) = 0) and there are non-trivial solutions of Equation (2.54) at the exact excitation energiesΩn. Casida has recast this problem,[71] resulting in the expression
( A B
The column vectors X and Y contain the coefficients to describe the transition. The elements of the matrices A and B are given by
Aiaσ,jbτ =δijδabδστωiaσ +Kiaσ,jbτ , (2.59) where the usual notation for occupied (i, j) and virtual (a, b) orbitals is used. σ and τ denote the orbital spin index.
The Casida equations result in pairs of excitation energies with positive and negative values corresponding to excitation and deexcitation processes, respectively. These two types of transitions are decoupled within the Tamm-Dancoff approximation (TDA) by setting the B matrix to zero.[72] In this case, the Casida formalism results in a simple
Molecular Mechanics 25
eigenvalue problem of the form
AX = ΩX . (2.62)
The Casida equations are closely related to the time-dependent framework of HF theory, also called random-phase approximation (RPA).[48]The latter is obtained by interchang-ing the xc kernel with the exact exchange operator. If the TDA approximation is ad-ditionally applied, then the CIS equations are obtained. However, the major difference constitutes the application of already correlated KS orbitals from DFT ground state calculations instead of uncorrelated HF orbitals. The calculated excitation energies are dominated by the diagonal elements featuring the MO energy differences, which has a significant effect on the accuracy. TD-DFT benchmark calculations for valence excited states of organic molecules have shown an accuracy of about 0.3 eV for global hybrid functionals such as B3LYP.[73;74] This is a large improvement compared to RPA or CIS excitation energies, for which the errors often exceed 1 eV.[48]