Time-dependent density-functional theory University of Missouri Tutorial: Time-dependent density-functional theory Carsten A. Ullrich University of Missouri XXXVI National Meeting on Condensed Matter Physics Aguas de Lindoia, SP, Brazil May 13, 2013

Literature Time-dependent Density-Functional Theory: Concepts and Applications (Oxford University Press 2012) “A brief compendium of TDDFT” Carsten A. Ullrich and Zeng-hui Yang arXiv:1305.1388 (Brazilian Journal of Physics, Vol. 43) C.A. Ullrich homepage: http://web.missouri.edu/~ullrichc ullrichc@missouri.edu

Outline PART I: ● The many-body problem ● Review of static DFT PART II: ● Formal framework of TDDFT ● Time-dependent Kohn-Sham formalism PART III: ● TDDFT in the linear-response regime ● Calculation of excitation energies

The electronic structure problem of matter What atoms, molecules, and solids can exist, and with what properties? What are the ground-state energies E and electron densities n(r)? What are the bond lengths and angles? What are the nuclear vibrations? How much energy is needed to ionize the system, or to break a bond? R E R

The electronic-nuclear many-body problem Consider a system with Ne electrons and Nn nuclei, with nuclear mass and charge Mj and Zj , where j=1,..., Nn electronic coordinates: nuclear coordinates: All electrons and all nuclei are quantum mechanical particles, forming an interacting (Ne + Nn ) -body system. For example, consider the hydrogen molecule H2: We’ll use atomic units:

Nonrelativistic Schrödinger equation electronic Hamiltonian nuclear Hamiltonian electron-nuclear attraction : external scalar potentials acting on the electrons/nuclei.

Born-Oppenheimer approximation ► Decouple the electronic and nuclear dynamics. This is a good approximation since nuclei are much heavier than electrons. ► Treat the nuclei as classical particles at fixed positions. ► Write down electronic Hamiltonian for a given nuclear configuration (here we ignore any external fields): This is just a constant

Potential-energy surfaces (diatomic molecule) 5 4 3 2 1 molecular equilibrium 1: ground-state potential-energy surface 2,3,4,5: excited-state potential-energy surfaces

The electronic many-body problem space and spin coordinate antisymmetric N-electron wave function Even a two-electron problem (Helium) is very difficult: This is a 6-dimensional partial differential equation!

The electronic many-body problem Dirac (1929): “The fundamental laws necessary for the mathematical treatment of a large part of physics and the whole of chemistry are thus completely known, and the difficulty lies only in the fact that applications of these laws leads to equations that are too complex to be solved.” Expectation value of an observable: Energy spectrum: Probability density:

The Hartree-Fock method Solving the full many-body Schrödinger equation is impossible. Instead, try a variational approach: The wave function is assumed to be a Slater determinant:

The Hartree-Fock method This is the Hartree- Fock equation, also known as the self- consistent field (SCF) equation. Hartree potential: local (multiplicative) operator Nonlocal exchange potential, acting on the jth orbital. “Nonlocal” means that the orbital that is acted upon appears under the integral.

The Hartree-Fock method The total Hartree-Fock ground-state energy is given by “direct” energy exchange energy where the single-particle orbitals are those that come from the Hartree-Fock equation, solved self-consistently. ► Not accurate enough for most applications in chemistry ► Very bad for solids: band gap too high, lattice constants too big, cohesive energy in metals much too small

Beyond Hartree-Fock: correlation Exact ground-state energy: correlation energy Question: is the correlation energy positive, negative, or either? Answer: the correlation energy is always negative! This is because the HF energy comes from a variation under the constraint that the wave function is a single Slater determinant. An unconstrained minimization will give a lower energy, according to the Rayleigh-Ritz minimum principle. How to get correlation energy? Wave-function based approaches (configuration interaction, coupled cluster) are accurate but expensive! DFT: alternative theory, formally exact but more efficient!

One-electron example Is this always true??

The Hohenberg-Kohn Theorem Recall: Hamiltonian for N-electron system We can define the following map: Claim: this map from potentials to densities is uniquely invertible, i.e., it is a 1-1 map. In other words, it cannot happen that two different potentials produce the same ground-state density: where “different” means that the potentials differ by more than just a constant:

The Hohenberg-Kohn proof Step 1: Show that cannot happen! Proof by contradiction. Let but assume that (trivial phase factor). Then we have but subtract: Contradiction! This means that the assumption must have been wrong. We have

The Hohenberg-Kohn proof cannot happen when Step 2: Show that To prove this, we assume the contrary, namely that they both give the same density. Then we can use the Ritz variational principle, whereby The following inequality holds: (simply inter- change primed and unprimed) add: Contradiction! Therefore uniquely.

The Hohenberg-Kohn Theorem (1964) 1:1 Therefore, uniquely determines The Hamiltonian formally becomes a functional of the density: and all wave functions become density functionals as well. Every physical observable is a functional of n0 The energy is a density functional: Minimum principle:

The Kohn-Sham formalism (1965) The HK theorem can be proved for any type of particle-particle interaction—in particular, it holds for noninteracting systems, too! Therefore, there exists a unique noninteracting system that reproduces a given ground-state density. This is the Kohn-Sham system. We can write the total ground-state energy as follows: kinetic energy functional for interacting systems kinetic energy functional for noninteracting systems This defines the xc energy functional.

The Kohn-Sham equation The Kohn-Sham many-body wave function is a single Slater determinant, whose single-particle orbitals follow from a self-consistent equation: where is the exact ground-state density. Exchange-correlation (xc) potential: The total ground-state energy can be written as

The Kohn-Sham equation: spin-DFT In practice, almost all Kohn-Sham calculations are done with spin-dependent single-particle orbitals, even if the system is closed-shell and nonmagnetic:

Exact properties (I) The Kohn-Sham Slater determinant is not meant to reproduce the full interacting many-body wave function: The Kohn-Sham energy eigenvalues do not have a rigorous physical meaning, except the highest occupied ones: ionization energy of the N-particle system electron affinity of the N-particle system Eigenvalue differences between occupied and empty levels, cannot be interpreted as excitation energies of the many-body system.

Exact properties (II) The asymptotic behavior of the KS potential for neutral systems is very important. For an atom with nuclear charge +N, we have for If an electron is “far away” in the “outer regions” of the system, it should see the Coulomb potential of the remaining positive ion. Therefore, for

Exact properties (III) The exact Kohn-Sham formalism must be free of self-interaction. This implies that for a 1-electron system the Hartree and xc potential cancel out exactly. We have, for The self-Hartree energy is fully compensated by the exchange energy, (evaluated with the exact KS orbitals) The self-correlation energy vanishes by itself.

Exchange-correlation functionals The exact xc energy functional is unknown and has to be approximated in practice. There exist many approximations! K. Burke, J. Chem. Phys. 136, 150901 (2012)

The local-density approximation (LDA) The xc energy of an inhomogeneous system is where exc[n] is the xc energy density. LDA: at each point r, replace the exact xc energy density with that of a uniform, homogeneous electron gas whose density has the same value as n(r).

The homogeneous electron gas The xc energy per unit volume of a uniform electron gas only depends on the uniform density n. It can be separated into exchange and correlation. The exchange energy can be calculated exactly from Hartree-Fock. The HF solutions are plane waves, and the total ground-state energy is kinetic energy density exchange energy density

The LDA exchange potential The LDA exchange potential is The LDA correlation energy and correlation potential have more complicated expressions (from Quantum Monte Carlo data).

Performance of the LDA ● Atomic and molecular ground-state energies within 1-5% ● Molecular equilibrium distances and geometries within ~3% ● Fermi surfaces of metals: within a few percent ● Vibrational frequencies and phonon energies within a few percent ● Lattice constants of solids within ~2% Czonka et al., PRB 79, 155107 (2009)

Shortcomings of the LDA ► The LDA is not self-interaction free. As a consequence, the xc potential goes to zero exponentially fast (not as -1/r): and the KS energy eigenvalues are too low in magnitude. ► LDA does not produce any stable negative ions. ► LDA underestimates the band gap in solids ► Dissociation of heteronuclear molecules produces ions with fractional charges. Overestimates atomization energies. ► LDA in general not accurate enough for many chemical applications. typically 30-50% too small

The Jacob’s Ladder of functionals Heaven: chemical accuracy LDA GGA Meta-GGA Hyper-GGA hybrids RPA double hybrids Unoccupied orbitals 1 2 3 4 5 Earth: the Hartree world

C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988) Generalized Gradient Approximations (GGA) There exists hundreds of GGA functionals. The most famous are the B88 exchange functional and the LYP correlation functional, A.D. Becke, Phys. Rev. A 38, 3098 (1988) C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988) and the PBE functional, J.P. Perdew, K. Burke, and M. Ernzerhof, PRL 79, 3865 (1996) where

Hybrid functionals Hybrid functionals mix in a fraction of exact exchange: where a ~ 0.25. The most famous hybrid is B3LYP: where

Mean absolute errors for large molecular test sets V.N.Staroverov, G.E.Scuseria, J. Tao, and J.P. Perdew, JCP 119, 12129 (2003) A good introduction to ground-state DFT: K. Capelle, Brazilian Journal of Physics 36, 1318 (2006).