2.2.1 Density Functional Theory and Kohn-Sham scheme
Density functional theory is the most popular method for many-electron system, atoms, molecules, and solid s. In 1964, Hohenberg and Sham develop the basic theorem of density functional theory[26]. We give simple concept about the theorem.
First theorem, we can represent the energy as a functional of the electron density
for given potential. We don’t need to use wave functionΨ(r1, r2, . . . , rN), which has 3N variables in N-electron system, but use electron densityρ(r), which has only 3 variables in N-electron system. That avoid the main computational difficulty. Second theorem, The energy functional is minimized by the ground state density. We can find the ground state potential by variantional principle. In Hohenberg-Kohn theorem, if we know the exact form of the universe functional, we can find the ground state by minimizing the functional. But we don’t know the exact functional and systematic way to find such functional. In 1965, Kohn and Sham develop systematic way to approximate the functional and find the ground state density.
Kohn and Sham develop another method[27] for the functional, by using non-interacting system as auxiliary system. Therefore, the ground-state wave function is the single slater-determinant. Ts[ρ] is the non-interacting kinetic energy functional, the J [ρ] is the Hartree en-ergy functional and the Exc[ρ] is called exchange-correlation energy functional. The exchange energy is from the Pauli-expulsion and the correlation is from the ap-proximation of single slater-determinant. The complexity of system is inside the exchange-correlation functional. And finally we take the density derivative of the energy functional, we get the Schrödinger-like equation
HˆKSϕiσ(r) = [−1
2∇2+ veff,σ(r)]ϕiσ(r) = εiσϕiσ(r), i = 1, 2, . . . , Nσ,
(2.36)
where vef f,σ is the effective KS potential and σ is the spin index. The effective potential is
vef f,σ = vext(r) + δJ [ρ]
δρσ(r) + δExc[ρα, ρβ]
δρσ(r) (2.37)
where vxc,σ(r) is the exchange-correlation potential vxc,σ(r) = δExc[ρα, ρβ]
δρσ(r) (2.38)
The KS equations are solved self-consistently. One guesses the initial density at first and then solves the KS equation to get the new density from new orbitals until the convergence.
2.2.2 Optimized Effective Potential method and Krieger-Li-Iafrate approximation
The self-interaction is from the classical Coulomb repulsion. The effect of self-interaction should be cancelled by exchange-correlation functional, but for most of exchange-correlation functionals, the self-interaction correction is not consider. One of the most important error is the incorrect long-range tail of Kohn-Sham potential, which will affect the ionization energy. Therefore, the self-interaction correction is crucial for excited states.
Perdew and Zunger proposed the self-interaction correction(SIC)[28] by giving the approximate exchange-correlation energy functional Exc[ρα, ρβ],
ExcSIC[ρα, ρβ] = Exc[ρα, ρβ]−X where ρiσ is the one-electron density of the ith KS spin orbital.
However, the SIC energy functional is explicit orbital-dependence, so for each electron orbital they have different potentials. That cause each orbital to be nonorthogonal and be complicated.
Another approach is the optimized effective potential method[29, 30] .In this approach, one solves the set of one-electron equations, similar to the KS equations in Eq. 2.36.
HˆOEPϕiσ(r) = [−1
2∇2+ vOEPσ (r)]ϕiσ(r) = εiσϕiσ(r), i = 1, 2, . . . , Nσ
(2.40)
The optimized effective potential vσOEP(r) is obtained by the orbitals{ϕiσ} in Eq. 2.40 which minimized the energy functional E[ϕiα, ϕjβ],
δExc[{ϕjσ}]
δvOEPσ (r) = 0 (2.41)
Eq. 2.41 can be written as, using chain rule for functional derivative, X Eq. 2.42 leads to an integral equation that is complicated. Krieger, Li and Iafrate[31, 32, 33] make an approximate procedure to simplify the original OEP in-tegral equations into the set of linear equations. Although the KLI procedure can’t reach the exact exchange functional, it reduces the computational difficulty and the its result is pretty close to OEP method.
2.2.3 KLI-SIC method
The OEP method and KLI approximation uses the exchange part of the density functional contains a Hartree-Fock-like nonlocal functional.
Exexact[{ϕjσ}] = −1
Even though Eq. 2.43 provides more accurate exchange potential, it’s computa-tionally more expensive than the traditional DFT functional with only local func-tional. Therefore,we present the extension of KLI procedures to the SIC term[34, 35]
in Eq. 2.39. This new KLI-SIC procedure can speed up the static DFT calcula-tion and time dependent DFT calculacalcula-tion. This KLI-SIC procedure make the self-interaction-free effective potential orbital-independent. In other word, this avoid the problems with respect to nonorthogonal spin-orbitals. And the KLI-SIC procedure give the optimized effective potential with the correct long-range behavior(−1/r)in Fig. 2.6 and surprisingly improvement of ionization energy and excited states.
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Figure 2.6: The effective potential rVef f with the LSDA and LSDA-KLI-SIC in neon and argon(left to right).
Define the total energy functional with SIC to be ESICOEP[{ϕiα, ϕjβ}] = EOEP[{ϕiα, ϕjβ}] −X
σ
X
i
{J[ρiσ] + Exc[ρiσ, 0]} (2.44)
where EOEP[{ϕiα, ϕjβ}] is normal energy functional in Eq. 2.35.Following the OEP-KLI procedure, one finds that
vSIC,σOEP (r) = vext(r) +
and
viSIC,σ =< ϕiσ|vSIC,σ(r)|ϕiσ > (2.48) viσ =< ϕiσ|viσ(r)|ϕiσ > (2.49) The value of the vSIC,σ is unknown, but we can solve it through linear equations
NXσ−1 The highest occupied orbital dominates the potential at the long-range. We choose vi=NSIC,σσ = vNσ to make sure the potential has correct asymptotic behavior.
2.2.4 TD-KLI-SIC method
We extend the KLI-SIC method into the time dependent system. The basic theorem of time dependent density functional theory is from Runge and Gross[36], mainly the similar structure of HK theorem and KS scheme. We’ll give the main theorems of TDDFT.More detail proofs and discussions can be found in.
First theorem, there is a one-to-one correspondence between time dependent den-sity and time dependent potential for any fixed initial states. In general, potential, hamiltonian and wave function is the functional of the time dependent density. This is the basic existence theorem of TDDFT.
Second theorem, we define the action A of the many-body system as the functional of many-body wave function,
If the variation of the action is We do integration by parts in the second term:
δA[ψ] = leads to the time dependent Schrödinger equation (i∂t∂ − ˆH(t))φ(t) = 0. With the first theorem, the wave function is the functional of density, so we define
A[ρ] = Z t2
t1
dt < ϕ[ρ](t)|i∂
∂t− ˆH(t)|ϕ[ρ](t) > (2.56) And we rewrite the Eq. 2.56 as
A[ρ] = A0[ρ]− The action A0 is the universal functional from kinetic and electron-electron interac-tion term. The time dependent density can be solved from the variainterac-tional principle.
δA[ρ]
δρ(r, t) = 0 (2.58)
Eq. 2.58 leads to the set of one-electron time dependent Schrödinger-like equation(TD Kohn-Sham equation).
(−1
2∇2+ vef f,σ(r, t))ϕiσ(r, t) = i∂
∂tϕiσ(r, t) (2.59) where the effective potential is
vef f(r, t) = v(r, t) +
Z ρ(r0, t)
|r − r0|dr0+ δAxc[ρ]
δρ(r, t) (2.60)
The last term is the time dependent exchange-correlation potential.
The OEP method is also available in the time dependent system[37].
Axc[ϕ]
vef f,σOEP[(r, t)] (2.61)
We present the TD-KLI-SIC method with adiabatic approximation[38]. The action is defined as
ASICxc = Z t2
t1
dtExcSIC[ρ0]|ρ0−→ρ(r,t) (2.62) The adiabatic approximation means that the system is only dependent on instant time. The derivative of the action ASICxc leads to the time dependent potential,
vxc,σSIC(r, t) = δExc[ρα, ρβ]
We solve the vSIC,σ(r, t) with the same procedures as before.
Finally, we solve the TDKS Eq. 2.60 with TDGPS method. We define ˆH0 as Hˆ0 =−1
2∇2− Z
r + vσ,ef f(r, 0) (2.66)
And
V (r, t) = v[ρ]ˆ σ,ef f(r, t)− vσ,ef f(r, 0) (2.67) Therefore, we have to solve the static Kohn-Sham equation by the self-consistency.
And we follow the TDGPS method that we discuss before to propagate the wave function.