TDDFT in Chemistry and Biochemistry III

Transcription

1 TDDFT in Chemistry and Biochemistry III Dmitrij Rappoport Department of Chemistry and Chemical Biology Harvard University TDDFT Winter School Benasque, January 2010 Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

2 Properties of Medium-Size and Large Molecules Fluorescein Phenylthio Ag 11 Dinaphtho[2,3-b:2,3 -f] (neutral form) Complex thieno[3,2-b] thiophene Fluorescent dye Model SERS Material for excitonics substrate applications Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

3 All Roads Lead to Rome Frequency-Domain Techniques Poles of Density Response Time-Domain Techniques Linear Density Response Polarizability α(ω) Excitation Energies Ωn One-Photon Cross Sections σn(1p) Time Evolution of Electron Density Quadratic Density Response Hyperpolarizability β(ω,ω') Excitation Energies Ωn Two-Photon Cross Sections σn(2p) Line Shapes Fourier Transform Higher-Order Density Response Excitation Energies Ωn Higher-Order Cross Sections σn(np) Electronic Absorption Spectra Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

4 One- and Two-Photon Absorption Spectra of Fluorescein N. S. Makarov et al., Optics Express, 2008, 16, Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

5 Kohn Sham Orbital Response Ground state Kohn Sham determinant Φ KS (r 1, σ 1,..., r n, σ n ) = 1 n! Static (ground-state) Kohn Sham orbitals ϕ 1 (r 1, σ 1 ) ϕ 2 (r 1, σ 1 )... ϕ n (r 1, σ 1 ) ϕ 1 (r 2, σ 2 ) ϕ 2 (r 2, σ 2 )... ϕ n (r 2, σ 2 ) ϕ 1 (r n, σ n ) ϕ 2 (r n, σ n )... ϕ n (r n, σ n ) H (0),KS ϕ i (r, σ) = ε i ϕ i (r, σ) ϕ i (r, σ) i = 1,..., N occ Occupied orbitals ϕ a (r, σ) a = 1,..., N virt Virtual orbitals ϕ p (r, σ) a = 1,..., N General orbitals Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

6 Kohn Sham Orbital Response Time-dependent Kohn Sham equations i t ϕ i(t, r, σ) = H KS ϕ i (t, r, σ) External potential includes periodic perturbation H KS = v (0) ext (r) + σ dr ρ(t, r, σ ) r r + v xc [ρ](t, r, σ) c x v HF [γ](r, r, t) + λ(v (1) (ω)e iωt + v (1) ( ω)e iωt ) Coupling parameter λ defines the strength of external perturbation v (1) (e. g., electric field strength E ); c x is hybrid mixing parameter, interpolates between pure density functionals (c x = 0) and Hartree Fock (c x = 1, v xc [ρ] = 0). Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

7 Density Matrix Formulation Density matrix of Kohn Sham system γ(t, r, σ, r, σ ) = i ϕ i(t, r, σ) ϕ i (t, r, σ ) γ(t) remains idempotent for all t, γ 2 (t) = γ(t) ; Diagonal of γ(t) is time-dependent density, ρ(t, r, σ) = γ(t, r, σ, r, σ) ; Time evolution is given by one-particle Liouville equation, i t γ(t) = [HKS, γ(t)]. F. Furche, J. Chem. Phys., 2001, 114, Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

8 Density Matrix Formulation Density matrix response expansion γ(t) =γ (0) + λ(γ (1) (ω)e iωt + γ (1) ( ω)e iωt ) + λ 2 (γ (2) (ω, ω)e 2iωt + γ (2) (ω, ω) + γ (2) ( ω, ω) + γ (2) ( ω, ω)e 2iωt ) +... Response equations [H (0),KS, γ (0) ] = 0 Static KS equations [H (1),KS, γ (0) ] + [H (0),KS, γ (1) ] = ωγ (1) Linear response [H (2),KS, γ (0) ] + 2 [H (1),KS, γ (1) ] +[H (0),KS, γ (2) ] = 2ωγ (1) Quadratic response Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

9 Density Matrix Formulation Coupling due to Hartree and xc interactions H (1),KS = σ Hartree kernel ( ) dr f H (r, r ) + f xc (ω, r, σ, r, σ ) ρ (1) (r, σ ) c x v HF [γ (1) ] + v (1) f H (r, r ) = xc Kernel in adiabatic approximation 1 r r ; f xc (ω, r, σ, r, σ ) = f xc (0, r, σ, r, σ ) = Non-local (Hartree Fock) exchange v HF [γ](r, r ) = γ(r, r ) r r. δ 2 E xc δρ(r) δρ(r ) ; Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

10 Density Matrix Formulation Expansion of linear response in static KS orbitals γ (1) (ω, r, σ, r, σ) = ia (X ω ia ϕ a (r, σ)ϕ i (r, σ) + Y ω iaσϕ i (r, σ)ϕ a (r, σ)), equivalent to expansion of time-dependent KS orbitals ϕ i (t, r, σ) = ϕ i (r, σ) + λ a (X ω ia e iωt + Y ω ia e iωt ) ϕ a (r, σ). Hilbert space notation X ω ia, Y ω ia X ω, Y ω L = L occ L virt L occ L virt L occ, L virt Hilbert spaces of occupied and virtual KS orbitals (L L one-particle Liouville space) Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

11 Linear Response Equation of motion for linear response jb (A + B) iajb(x + Y ) ω jb ω(x Y )ω ia = (P + Q) ia jb (A B) iajb(x Y ) ω jb ω(x + Y )ω ia = (P Q) ia Orbital rotation Hessians (A + B) iajb = (ε a ε i )δ ij δ ab + 2fiajb H xc + 2fiajb c x[fjaib H + f abij H ] (A B) iajb = (ε a ε i )δ ij δ ab + c x [fjaib H f abij H ] f iajb = d 3 r d 3 r ϕ i (r, σ)ϕ a (r, σ)f (r, σ, r, σ )ϕ j (r, σ )ϕ b (r, σ ) Right-hand side contains the external perturbation (P + Q) ia = 2 v (1) ia ; (P Q) ia = 0 Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

12 Linear Response Operator notation Define operators Λ and on L, ( ) A B Λ = ; = B A Linear response equation takes the form ( ) (Λ ω ) X, Y = P, Q Frequency-dependent polarizability α mn (ω) External perturbation is external electric field (dipole approximation) P n, Q n = µ n, µ n ; α mn (ω) = µ m, µ m X ω,n, Y ω,n µ m ia = d 3 r ϕ i (r, σ) r m ϕ a (r, σ), m, n {x, y, z} Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

13 Linear Response Frequency-dependent polarizability α(ω) of fullerene C α, a.u Wavelength, nm Polarizability α(ω) of C 60 diverges at λ = 350 nm (3.54 ev) PBE0/SVPs basis set Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

14 Higher-Order Response Expansion of nth-order response in static KS orbitals γ (n) (ω, ω,...,r, σ, r, σ) = ia (X (n) ia ϕ a(r, σ)ϕ i (r, σ) + Y (n) iaσ ϕ i(r, σ)ϕ a (r, σ) + K (n) ij ϕ i (r, σ)ϕ j (r, σ) + K (n) ab ϕ a(r, σ)ϕ b (r, σ)) Occ-occ and virt-virt blocks K (n) ij, K (n) ab are products of lower-order response by virtue of idempotency constraint; Occ-virt and virt-occ blocks are determined from nth-order equation of motion, (Λ nω ) X (n), Y (n) = P (n), Q (n) ; Right-hand sides P (n), Q (n) contain only lower-order response. Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

15 Higher-Order Response Frequency-dependent hyperpolarizability β lmn (ω, ω) β lmn (ω, ω) = 2 ij µ l ijk (2),mn ij + 2 ab µ l ab K (2),mn ab + µ l, µ l X (2),mn, Y (2),mn = 2 tr(µ l K (2),mn ) µ l, µ l (Λ 2ω ) 1 P (2),mn, Q (2),mn = 2 tr(µ l K (2),mn ) + X 2ω,l, Y 2ω,l P (2),mn, Q (2),mn, l, m, n {x, y, z} nth-order response determines (quasi-)energy to order 2n + 1 (Wigner s rule); occ-virt and virt-occ blocks of quadratic response X (2), Y (2) required only for second hyperpolarizabilities δ klmn (ω, ω, ω, ω) and higher-order properties; Computation of kth-order (quasi-)energy requires solution of k 1 equations of motion (here for ω and 2ω). Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

16 Implementation Using Molecular Basis Sets LCAO expansion of KS orbitals ϕ p (r, σ) = µ C µpσχ µ (r) Response equations are transformed into finite-dimensional linear equations, amenable to matrix algebra techniques; Atomic orbitals are atom-centered, mostly contracted Cartesian Gaussian-type orbitals (CGTOs), χ µ (r) = d c d x i y j z k e ζ d (r R µ) 2 ; Atomic orbitals form a local, non-orthogonal basis set; Integrals over CGTOs can be evaluated analytically using Gaussian product theorem and recursion w.r.t. l = i + j + k; Basis set parameters (c d, ζ d ) tabulated, several hierarchies of basis sets of increasing size (Pople, Dunning, Ahlrichs, etc.) available. Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

17 Strategies for Solving Matrix Equations Direct methods Dimension of matrix equation is N occ N virt, (A + B) iajb = (ε a ε i )δ ij δ ab + 2(ia jb) + 2f xc iajb c x[(ja ib) + (ab ij)] (A B) iajb = (ε a ε i )δ ij δ ab + c x [(ja ib) (ab ij)] Electron-repulsion integrals in Mulliken notation, (ia jb) = fiajb H = d 3 r d 3 r 1 ϕ i (r, σ)ϕ a (r, σ) r r ϕ j(r, σ )ϕ b (r, σ ) Computational cost of direct solution scales as N 6 ; Requires computation and simultaneous storage of all matrix elements (A + B) iajb, (A B) iajb, scales as N 5 ; Feasible for small systems only. Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

18 Strategies for Solving Matrix Equations Iterative methods Solution of linear equation on iteratively expanded subspace S L; Matrix elements (A + B) iajb, (A B) iajb need not to be stored; Time-determining step is computation of matrix-vector products jb (A ± B) iajb(x ± Y ) jb, very similar to ground-state DFT calculations; n iter = iterations are sufficient for most systems; Integral pre-screening reduces computational cost to N 2 n iter ; Resolution-of-the-identity (RI-J) techniques reduce computational cost by factor for non-hybrid functionals. Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

19 Strategies for Solving Matrix Equations Choose subspace S L Compute projection Λ S Λ of Λ on S Increase subspace S = S U δx, δy Solve projected linear equation (Λ S ω ) X S,Y S = P S,Q S no yes Converged Check convergence δx, δy < thr Compute gradient δx, δy =(Λ ω ) X S,Y S + P, Q Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

20 Electronic Polarizabilities of Molecules Accuracy of TDDFT for static polarizabilities Isotropic polarizabilities α iso, a.u. LDA PBE PBE0 CCSD Exp. N H 2 O CO C 2 H PH d-aug-cc-pvtz basis set C. Van Caillie, R. D. Amos, Chem. Phys. Lett., 2000, 328, 446. Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

21 Electronic Polarizabilities of Molecules Basis set dependence of electronic polarizabilities Exact HCN PBE0 Basis set error % α, a.u MP2 HF 12 HF/SVP 2-5 % Methodical error SVP TZVPP QZVP Molecular polarizabilities converge slowly with basis set size; Large diffuse basis (low-exponent) sets are often required; Reduced numerical stability with diffuse basis sets (ill-conditioned). Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22

22 Electronic Polarizabilities of Molecules Longitudinal polarizabilities of polymethyleneimines α, a.u H C H N H n 6-31G(d) basis set n PBE PBE0 CAM-B3LYP MP2 HF 35 Semi-local and global hybrid functionals overestimate polarizabilities of π-conjugated polymers; long-range corrections reduce errors. D. Jacquemin et al., J. Chem. Phys , 126, Dmitrij Rappoport (Harvard U.) TDDFT in Chemistry III TDDFT Winter School / 22