Lecture 9: Advanced DFT concepts: The Exchangecorrelation functional and timedependent DFT Marie...
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 Lecture 9: Advanced DFT concepts: The Exchangecorrelation functional and timedependent DFT Marie Curie Tutorial Series: Modeling Biomolecules December 611, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New York, NY 10003
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 0 is the ground state wavefunction of a Hamiltonian H=T+W+V. Note that when =0. V=V KS and when =1, V=V ext.
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 Where f xc can be obtained by integrating over r with a Change of variables s=rr .
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 Gradient Corrections Extend the LDA form to include density gradients: Example: Becke, Phys. Rev. A (1988) Functional form chosen to have the correct asymptotic behavior:
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 Motivation for TDDFT Photoexcitation processes Atomic and nuclear scattering Dynamical response of inhomogeneous metallic systems.
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 The timedependent Hamiltonian Consider an electronic system with a Hamiltonian of the form: Where V(t) is a timedependent onebody operator. Our interest is in the solution of the timedependent Schrdinger equation: Let V be the set of timedependent potentials associated and let N be the set of densities associated with timedependent solutions of the Schrdinger equation. There exists a map G such that G : V N
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 The HohenbergKohn Theorem Since V(t) is a onebody operator: Assume the potential can be expanded in a Taylor series: Suppose there are two potentials such that Then, there exists some minimum value of k such that
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 The HohenbergKohn Theorem For timedependent systems, we need to show that both the density n(r,t) and the current density j(r,t) are different for the two different potentials, where the continuity equation is satisfied: For any operator O(t), we can show that:
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 The HohenbergKohn Theorem From equation of motion, we can show that And, in general, for the minimal value of k alluded to above: Hence, even if j and j are different initially, they will differ for times just later than t 0.
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 The HohenbergKohn Theorem For the density, since It follows that: Therefore, even if n and n are initially the same, they will differ for times just later than t 0. Hence, any observable can be written as a functional of n and a function of t.
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 Actions in quantum mechanics and DFT Consider the action integral: Schrdinger equation results requiring that the action be stationary according to: Hence, if we view A as a functional of the density,
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 HohenbergKohn and KS schemes HohenbergKohn: KohnSham formulation: Introduce a noninteracting system with effective potential V KS (r,t) that gives the same timedependent density as the interacting system. For a noninteracting system, introduce singleparticle orbitals i (r, t ) such that the density is given by KS action:
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 Timedependent KohnSham equations From : Adiabatic LDA/GGA:
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 Linear response solution for the density Strategy: Solve the Liouville equation for the density matrix to linear order. Quantum Liouville equation for the density operator (t): Timedependent density:
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 Linear response solution for the density Write the density operator as: To linear order, we have Solution:
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 Linear response solution for the density To linear order: Where the Fourier transform of the response kernel is: Hence, poles of the response kernel are the electronic excitation energies.
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 from Appel, Gross and Burke, PRL 93, 043005 (2003).
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 Lecture Summary Adiabatic connection formula provides a rigorous theory of the exchangecorrelation functional and is the starting point of many approximations. Generalization of density functional theory to timedependent systems is possible through generalization of the HohenbergKohn theorem. In linear response theory, the response kernel (or its poles) is the object of interest as it yields the excitation energies.