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- 1. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Looking into the black box Computing molecular excited states with time-dependent density functional theory (TDDFT) Jiahao Chen and Shane Yost Van Voorhis group, MIT Chemistry 2010-03-12
- 2. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superﬁcial survey of two ideas: 1 Self–consistent ﬁeld methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
- 3. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superﬁcial survey of two ideas: 1 Self–consistent ﬁeld methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
- 4. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Disclaimer This is a very superﬁcial survey of two ideas: 1 Self–consistent ﬁeld methods for ground state calculations, and 2 Linear response theory for excited state calculations. There will be math and jargon. Most of it is optional. The math is not presented in the most general case. We won’t have time today to discuss calculating observables and numerical results.
- 5. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Statement of the Problem (Born–Oppenheimer) Given that we know where the atomic nuclei are, where are the electrons? ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r (1) ˆ H is the electronic, or Born–Oppenheimer Hamiltonian: N 1 ˆ H = ∑ hi + ∑ |i −j | ˆ r r (1-e + e–e repulsion) (2) i=1 ij 1 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3) ˆ 2 1 M ZI vext (r ) = − ∑ (e–nucleus interaction) (4) I =1 RI − r
- 6. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Statement of the Problem (Born–Oppenheimer) Given that we know where the atomic nuclei are, where are the electrons? ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r (1) ˆ H is the electronic, or Born–Oppenheimer Hamiltonian: N 1 ˆ H = ∑ hi + ∑ |i −j | ˆ r r (1-e + e–e repulsion) (2) i=1 ij 1 h1 = − ∇2 + vext (r ) (kinetic energy + external pot.)(3) ˆ 2 1 M ZI vext (r ) = − ∑ (e–nucleus interaction) (4) I =1 RI − r
- 7. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Practical solution of the Schrödinger equation ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r What approximations do we need to solve this practically? Approximate wavefunction Variational ansätze Orbital (one–particle) approximation Basis set approximation Approximate Hamiltonian Noninteracting particle / mean ﬁeld approximation In DFT: approximations to exchange–correlation Linear response approximation (for excited states)
- 8. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Contents 1 Basis functions 2 Hartree–Fock 3 Density functional theory 4 Time–dependent DFT
- 9. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Variational principle and ansätze We don’t know the exact solution to the many–body Schrödinger equation. But we can make an educated guess, an ansatz (pl. ansätze). Variational principle An approximate wavefunction Ψ (a, b, . . . ) always yields a higher energy than the exact solution Ψ0 , i.e. Ψ |H| Ψ Ψ0 |H| Ψ0 ≥ E0 = (5) Ψ|Ψ Ψ0 |Ψ0 This reduces a diﬀerential equation into an optimization problem for the numerical parameters a, b, . . .
- 10. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Bases in vector spaces A basis describes axes, allowing coordinates to be deﬁned. Example R3 has basis {ex , ey , ez }. Numbers (x, y , z) can be assigned as coordinates to a point with position vector v = xex + y ey + zez . A less obvious example: Example Computer monitors use RGB color space, with basis {red, green, blue}. For example, a kind of purple is described by purple = 1 red 2 + 1 blue and has coordinates 1 , 0, 1 in color space. 2 2 2
- 11. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Bases in function spaces Basis functions describe “axes” in function space. If you have a collection of mutually orthogonal, real functions φ1 (x) , . . . , φn (x) that span the entire space of functions, then f (x) = ∑ ci φi (x) (7) i whose coeﬃcients are given by projections onto the basis N f (x) φi (x) dx ´ ci = R´ 2 (8) RN φi (x) dx Then the function f has “coordinates” (c1 , c2 , . . . ) in the basis spanning the function space. Example Functions over [−π, π] have Fourier series expansions, whose basis functions are 1, sin x, cos 2x, sin 3x, cos 4x, . . .
- 12. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End The independent particle ansatz: from many to one Let’s try to solve the many–body Schrödinger equation ˆ H (r1 , . . . , N ) Ψn (r1 , . . . , N ) = En Ψn (r1 , . . . , N ) r r r As a ﬁrst guess for Ψn , assume that it is factorizable into a Hartree product of one–particle wavefunctions (orbitals) 1 Ψ (r1 , . . . , N ) = φ1 (r1 ) · · · φN (rN ) r (9) However, this wavefunction is not antisymmetric (no Pauli exclusion). The simplest fermionic ansatz is the Slater determinant2 φ1 (r1 ) · · · φ1 (rN ) 1 Ψ (r1 , . . . , N ) = √ r . . .. . . (10) . . . N! φN (r1 ) · · · φN (rN ) 1 D. R. Hartree, Proc. Cam. Phil. Soc. 24 (1928) 89; 111; 426 2 J. C. Slater, Phys. Rev. 34 (1929) 1293
- 13. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End The Hartree–Fock equation Using the ansatz that the many–body wf is a Slater determinant, the variational principle leads to3 the Hartree–Fock equation Fi φi (r ) = εi φi (r ) ˆ (11) ˆ Fi = hi + ∑ Jj − Kj (Fock operator) ˆ ˆ ˆ (12) j J is the classical Coulomb potential = Hartree potential vH . K is the exchange interaction with no classical analogue. (absent in Hartree theory) 3 A. Szabo and N. S. Ostlund, Modern Quantum Chemistry, 1982, Ch. 3
- 14. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Hartree–Fock is a mean ﬁeld theory Electrons in Hartree–Fock feel an electric ﬁeld of potential Φ = ∑N Jj − Kj . ˆ j=1 ˆ ˆ The mean ﬁeld is a self–consistent ﬁeld The electron–electron interactions are approximated by the interaction between an electron and this electric ﬁeld. This ﬁeld depends on the orbitals, yet also helps determines them. Correlation energy = the error in Hartree–Fock energy (wrt exact)
- 15. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Roothaan–Hall equation If we project the Hartree–Fock equation onto a basis {χ1 , . . . , χP }, P φi = ∑ Ciα χα (13) α=1 we get the Roothaan–Hall equation FC = ESC (14) which is a generalized eigenvalue equation. Atomic orbitals = spatially localized basis {χ1 , . . . , χP }, Molecular orbitals = eigenvectors of the Roothaan–Hall equation
- 16. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Basis sets There are many choices for the atomic basis {χ1 , . . . , χP }: Plane waves: great for periodic systems (solids), ineﬃcient for molecules (and other systems without translational symmetry) Wannier functions Slater orbitals: use hydrogenic orbitals, nice theoretical properties. In practice the Fock matrix is very diﬃcult to calculate. Not widely used today, except in ADF. Gaussian orbitals: popular for molecules. Easy to calculate. Pople basis sets: STO-3G, 3–21G, 6–31G*, 6–311+G**,... Dunning–Hays correlation–consistent basis sets: cc-pVDZ, aug-cc-pVQZ, d-aug-cc-pVTZ, ... Others, e.g. wavelets, adaptive quadrature grids...
- 17. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Post–Hartree–Fock methods Many methods to treat correlation use Hartree–Fock as a starting point: Perturbative corrections: Møller–Plesset (MP2,...),... Conﬁguration interaction: CISD, FCI,... Coupled–cluster: CCSD, CCSD(T),...
- 18. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Kohn–Sham equations In 1965, Kohn and Sham proposed a noninteracting system for use with density functional theory.4 This turns out to look like Hartree–Fock theory with a correction for electronic correlation. The resulting one–particle equation is written as: 1 2 Hs φi (r ) = − ∇s + vs (r ) φi (r ) = εi φi (r ) (15) 2 where − 1 ∇2 is the noninteracting part of the kinetic energy and vs 2 s is the Kohn–Sham potential: vs (r ) = vext (r ) + vH (r ) + vxc (r ) (16) The exchange–correlation potential vxc has three contributions: exchange, correlation, and a correction to the kinetic energy. 4 W. Kohn and L. J. Sham, Phys. Rev. 140, 1965, A1133
- 19. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Kohn–Sham is a density functional theory The Kohn–Sham equation is equivalent to ﬁnding the density ρ that minimizes the energy functional ˆ E [ρ] = vext (r ) ρ (r ) d + Ts [ρ] + VH [ρ] + Exc [ρ] r (17) R3 1 N ˆ Ts [ρ] = − ∑ φi (r ) ∇2 φi (r ) d r (18) 2 i=1 R3 1 ρ (r ) ρ (r ) ˆ VH [ρ] = + d d r r (19) 2 R6 |r − r | where the density is constructed from N ρ (r ) = ∑ |φi ( )|2 = ∑ r Ci µ Ciν χµ (r ) χν (r ) (20) i=1 µνi
- 20. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Density functionals The form of the exchange–correlation functional Exc is unknown. There are many, many approximations to it: “Jacob’s Ladder” of approximations to Exc [ρ]:5 1 Exc [ρ] LDA X α, LDA,... 2 Exc [ρ, ∇ρ] GGA BLYP, PBE, PW91... 3 Exc [ρ, ∇ρ, τ] meta–GGA VSXC, TPSS, ... 4 Exc [ρ, ∇ρ, τ, {φ }o ] hybrids 6 ,... B3LYP, PBE0,... 5 Exc [ρ, ∇ρ, τ, {φ }] fully nonlocal - LDA, local density approximation; GGA, generalized gradient approximation; τ, kinetic energy density; {φ }o , occupied orbitals; {φ }, all orbitals. Green function expansions for Exc : GW, Görling–Levy PT,... Approximations to kinetic energy: Thomas–Fermi, von Wiezsäcker,... 5 M. Casida, http://bit.ly/casidadft 6 Hybrid functionals mix in Hartree–Fock exchange.
- 21. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Rules of thumb7 for the accuracy of some ab initio methods Property HF DFT MP2 CCSD(T) IPs, EAs ±0.5 ±0.2 ±0.2 ±0.05 eV Bond lengths –1% ±1 ±1 ±0.5 pm Vib. freqs. +10% +3% +3% ±5 cm−1 Barrier heights +30—50% –25% +10% ±2 kcal/mol Bond energies –50% ±3 ±10 ±1 kcal/mol 7 T. van Voorhis, MIT 5.74 course notes, 2010.
- 22. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Polarizabilities as linear response Polarizability: a dipole’s response to an electric ﬁeld The isotropic ground state polarizability can be calculated exactly from time–dependent perturbation theorya ∂ ˆ r f0→n α (ω) = ¯ =∑ 2 (21) ∂ ε (ω) n ω0→n − ω 2 where ω0→n is excitation frequency (energy) to state n and f0→n are oscillator strengths. This is a sum over states (SOS) formula. a Kramers, H. A.; Heisenberg, W. (1925) Z. Phys. 31: 681–708. Dirac, P. A. M. (1927) Proc. Roy. Soc. Lond. A 114: 243–265; 710–728. Linear response is a quest for resonances Resonant frequencies of the system’s reaction to small perturbations says something about its excitation spectrum.
- 23. Introduction Basis functions Hartree–Fock Density functional theory Time–dependent DFT End Resonances in density response Main idea Using the density response δ ρ to a time–dependent perturbing potential δ v , we can back out what the excited state characters are. Consider the generalized susceptibility χ (t) as deﬁned by ˆ ˆ δ ρ (r , t) = χ , , t − t δ v , t dt d r r r r (22) R R3 A SOS formula for the Fourier–transformed susceptibility χ can be ˜ derived, and furthermore the conditions for resonance can be expressed as an eigenvalue equation that takes the form8 A B X I 0 X =ω (23) B ∗ A∗ Y 0 −I Y X is the hole–particle (occupied–virtual) density change Y is the particle–hole (virtual–occupied) density change 8 M. Casida, in Recent Advances in Density Functional Methods I, p. 155
- 24. End of Part I

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