Time-dependent density functional theory : direct computation of excitation energies

Transcription

1 CECAM Tutorial Electronic excitations and spectroscopies : Theory and Codes Lyon 007 Time-dependent density functional theory : direct computation of excitation energies X. Gonze Université Catholique de Louvain, Louvain-la-neuve, Belgium gonze@pcpm.ucl.ac.be Time-Dependent Density Functional Theory 1 Plan Note : sections 1, 3, 5 and part of will be covered in V. Olevano s presentation, These sections are available here only for completeness. The actual lecture will focus on sections 4, 6, 7 and Time-Dependent Kohn-Sham equation. Runge-Gross theorem 3. Approximate XC action and TD potential 4. Implementations : time vs frequency 5. The linear regime 6. Casida s approach to excitation energies 7. Some applications of Casida s technique 8. Implementation in ABINIT References : E. Runge and E.K.U. Gross, Phys. Rev. Lett. 5, 997 (1984) M.E. Casida, in Recent Developments and Applications of Modern Density Functional Theory, edited by J.M. Seminario (Elsevier Science, Amsterdam), p. 391 (1996). G. Onida, L. Reining, and A. Rubio, Rev. Mod. Phys. 74, 601 (00) J.-Y. Raty and G. Galli, J. Electroanalytical Chemistry 584, 9 (005) Time-Dependent Density Functional Theory 1

2 Back to undergrad Quantum Mechanics (I) Starting point : the Time-Dependent Schrödinger equation i!!"!t = Ĥ" e.g. for 1 particle i!!"!t (r,t) = #! m $ " + V(r,t)" (r,t) (r,t) If the Hamiltonian does not depend on time, Ĥ =!! m " # r i + V(r 1,...,r n ) i=1,n we have the following ansatz! (r 1,...r n,t) = "(r 1,...r n ).f (t) Time-Dependent Density Functional Theory 3 Back to undergrad Quantum Mechanics (II)! (r 1,...r n,t) = "(r 1,...r n ).f (t) Now, i!!"!t = Ĥ" Identify K with the energy! i! " f 1 "t f "# $ %$ depends only on time! m (r 1,...r n ) such that Ĥ! m = E m! m i!!" = #.i!! f!t!t H ˆ! = f. H ˆ " = 1 # (Ĥ#) "# $ %$ does not depend on time = K Et #i!! (r 1,...r n,t) = "(r 1,...r n ).e General solutions %! (r 1,...r n,t) = " c m # m (r 1,...r n )exp $ ie m.t ( ' m! ) * Time-Dependent Density Functional Theory 4

3 Why are we interested by time dependence? Electrons do not stay in a stationary (eigen) state! (except GS) Spontaneously emit photon => ground state Time-dependent perturbations of a system : typically time-dependent electric field (photons) -small perturbations (linear regime for electric field) -large perturbations (LASER field) Also, some adiabatic («slow») changes, like ion-molecule collisions (or ion-surface interaction) Induced electronic transitions to excited states! Time-Dependent Density Functional Theory 5 Time-dependent Kohn-Sham equation (I) Started early 80 s : -Stott and Zaremba, Phys. Rev. A 1, 1 (1980) -Zangwill and Soven, Phys. Rev. A 1, 1561 (1980) Computed photoabsorption cross section of atoms σ(ω) Time-dependent Hartree-Fock already existed but quite inaccurate! They generalised LDA... without a formal justification Time-Dependent Density Functional Theory 6 3

4 Time-dependent Kohn-Sham equation (II) $! 1 " + V ext r % n(r) = (using atomic units) LDA equations occ #! * " (r)! " (r) $! " ( r)! # ( r)dr= % "# " become... $! 1 " + V ext r,t % occ n(r,t) = #! * " (r,t)! " (r,t) $! " ( r,t)! # ( r,t)dr= % "# " ( ) + n( r' ) # dr' + V XC n r r-r' ( ) + n( r',t) Is this valid? ( ( )) ( ( )) # dr' + V XC n r,t r-r' ' ( ) * + r ( ) =, + * ( + r) ' ( ) * + r,t ( ) = i,,t * ( + r,t) Time-Dependent Density Functional Theory 7 In search of a Time-Dependent Density Functional Theory... In DFT, we have... In TD-DFT, what could we have? n(r) determines V(r), up to an overall constant, for the Ground State There is an universal functional of the density F [n] such that E[n, V] = F[n] +! n(r)v(r)dr and E[V] = min E[n,V] n n(r,t) determines V(r,t)? up to an overall constant? for...? The quantum! mechanical action t 1 A = " t dt #(t) i $! H(t) #(t) 0 $t is not extremal, but only stationary for fixed #(t 0 ) and #(t 1 ) Time-Dependent Density Functional Theory 8 4

5 The Runge-Gross theorems (I) (E. Runge and E.K.U. Gross, Phys. Rev. Lett. 5, 997 (1984)) For every single-particle potential V(r,t) which can be expanded into a Taylor series with respect to the time coordinate around t=t 0, a map G :V(r,t)! n(r,t) is defined by solving the TD Schrödinger equation with a fixed initial state!(t 0 ) =! 0 and calculating the corresponding densities n(r,t). This map can be inverted up to an additive merely TD function in the potential : V'(r,t) = V(r,t) + C(t) Note : fixed initial state ; global TD of the map (not isolated times) ; the overall additive TD function in the potential Time-Dependent Density Functional Theory 9 The Runge-Gross theorems (II) For a fixed initial state, and a given TD density,!(t 0 ) =! 0 n(r,t) The wavefunction!(t) can be determined, within a TD phase factor!(t) = e "i#(t )!'[ n](t) Note : the phase factor comes from the overall additive TD function in the potential!(t) = C(t) ; no problem of degeneracy (but the initial WF is known) Also, several maps can be defined for a same fixed initial state, corresponding to larger and larger time regions : if t 0 < t 1 < t < t 3, then n(t) within [ t 0,t 1 ] gives V(t) and!(t) within [ t 0,t 1 ] n(t) within t 0,t [ ] gives V(t) and!(t) within [ t 0,t ] etc The time-dependence is non-local, albeit causal Time-Dependent Density Functional Theory 10 5

6 The Runge-Gross theorems (III) The quantum-mechanical action, t [ ] = 1 dt A n,! 0,V " i # #t $ ˆT $ V(t) $ Ŵ! [ n ](t) t 0 is a functional of the TD density between the initial and final times (the phase factor C(t) cancels out) and of the initial wavefunction. It is stationary at the TD density corresponding to the TD potential.!a!n(r,t) = 0 The following functional is an universal functional of the TD density and the initial wavefunction t [ ] = 1 dt B n,! 0 " i # #t $ ˆT $ Ŵ! [ n ](t) t 0 Time-Dependent Density Functional Theory 11 The Runge-Gross theorems (IV) Now we introduce, in the Kohn-Sham spirit, the possibility to compare the behaviour of density functionals for electrons interacting with the physical Coulomb interaction, and for non-interaction electrons. This leads to the TD exchange-correlation action, a functional of the TD density and the initial wavefunction.! A xc [ n," 0 ] Exact framework : %! 1 " + V ext r,t ' occ n(r,t) = #! * " (r,t)! " (r,t) $! " ( r,t)! # ( r,t)dr= % "# " ( ) + n( r',t) "A and v xc [ n,! 0 ](r,t) = "n(r,t) [ ]( r,t) # dr' + V XC n,$ 0 r-r' was V XC ( ) * +, r,t ( n( r,t) ) ( ) = i - -t +, ( r,t) Time-Dependent Density Functional Theory 1 6

7 Approximate XC action and TD potential (I) A xc [ n,! 0 ] v xc [ n,! 0 ](r,t) = "A "n(r,t) Formally, the action (and TD potential) is defined for an interval of times, and, at initial time, depends on the full many-body wavefunction Approximation 1 : at initial time, we choose the ground-state, and thus, can start with the usual XC potential, a functional of the density at that initial time Time-Dependent Density Functional Theory 13 Approximate XC action and TD potential (II) A xc!a [ n] v xc [ n](r,t) =!n(r,t) There is still a causal «non-local» time-dependence, as the action and XC potential at time t depends (in principle) on the GS density, and on the TD density between initial time and present time : this is a memory effect. Approximation : suppress the memory effect! Adiabatic approximation => left with only usual spatial non-locality (leading to LDA, GGA, etc ) A xc t [ n] =! 1 a XC [ n(t) ] dt v xc [ n(t) ](r,t) =!a XC t 0!n(r) (r,t ) Time-Dependent Density Functional Theory 14 7

8 Approximate XC action and TD potential (III) Beyond adiabatic approximation : active research field Sum rules / constraints exist The adiabatic approximation fulfills many of them (compare with LDA!) Hydrodynamic formulation of TDDFT => Current DFT Add spin also... Combine corrected -1/r XC potential with simple TD behaviour Time-Dependent Density Functional Theory 15 Implementations 1) Treatment in real time domain K. Yabana and G.F. Bertsch, Phys. Rev. B 54, 4484 (1996) M. Marques et al, Comput. Physics Commun. 151, 60 (00) Example : OCTOPUS ) Treatment in frequency domain Linear and non-linear response : frequency dependent susceptibilities Example : DP Direct computation of the excitation energies Example : ABINIT Time-Dependent Density Functional Theory 16 8

9 The Nitrogen molecule under intense laser (I) from the Octopus code, and T. Burnus, M. Marques, and E.K.U. Gross.Phys. Rev. A 71, (005) LASER Excitation fields : right-hand side is >5 times bigger than left-hand side Time-Dependent Density Functional Theory 17 The Nitrogen molecule under intense laser (II) 1 Dipole moment : on the right figure, observe the non-linearities seen as higher frequency Fourier components Time-Dependent Density Functional Theory 18 9

10 The Nitrogen molecule under intense laser (III) Total energy of the system : compare the different scales Time-Dependent Density Functional Theory 19 Linear response Consider some system in its GS, and apply a small TD perturbation characterized by some frequency. Watch the density change. In the linear regime, the associated Fourier components are related by the frequency-dependent susceptibility or density-density response function n (1) (r,!) = # "(r,r';!)v (1) ext (r',!) dr' This susceptibility can be computed within TD-DFT, for a given choice of TD-XC functional Time-Dependent Density Functional Theory 0 10

11 The TD Hartree and XC kernels A change of the external potential induces a change of density, that induces a change of Hartree and XC potential. For the Hartree potential, we have : V (1) 1 H (r,!) = " r-r' n(1) (r',!) dr' Supposing that the TD XC functional is known, we have an explicit expression for the change, in term of the TD XC kernel V (1) XC (r,!) = " K XC (r,r',!)n (1) (r',!) dr' Time-Dependent Density Functional Theory 1 XC kernels : local / adiabatic approximation Combining local approximation and adiabatic approximation, the expression of the TD XC kernel is particularly simple. It is independent of the frequency (local in time), local in space, and is determined by a local XC energy density, a function of the local unperturbed density. K XC (r,r',!) = d e XC dn "( r-r' ) n (0) ( r) Note : It might be that the local XC energy density is the ground state one, but this is not mandatory. Example : LB94 / ALDA Time-Dependent Density Functional Theory 11

12 The independent-particle susceptibility The TD external, Hartree, and XC changes of potential combine to give a total change of Kohn-Sham potential seen by independent electrons : V (1) KS (r,!) = V (1) ext (r,!) + V (1) H (r,!) + V (1) XC (r,!) The change of density due to this potential can be computed from the independent-particle susceptibility! 0 (r,r';") n (1) (r,!) = # " 0 (r,r';!)v (1) KS (r',!) dr' Adler and Wiser have given an explicit expression for! 0 (r,r';") in terms of the one-particle occupied and unoccupied eigenfunctions, and eigenenergies ( ) % n# ((( # n m ( ) $ " $ i'! 0 (r,r';") = f m# $ f n# * * (r)% m# (r)% m# m# $ n# (r')% n# (r') Time-Dependent Density Functional Theory 3 The Dyson equation for the susceptibility (I) One combines these equations or their inverse (assuming they can be inverted) : V (1) KS (r,!) = V (1) ext (r,!) + V (1) H (r,!) + V (1) XC (r,!) V (1) KS (r',!) = $ " #1 0 (r,r';!) n (1) (r,!)dr' V (1) ext (r',!) = $ " #1 (r,r';!)n (1) (r,!) dr' V (1) 1 H (r,!) = " r-r' n(1) (r',!) dr' (1) (r,!) = " K XC (r,r',!)n (1) (r',!) dr' V XC They are valid for all perturbations, thus we must have:! "1 0 (r,r';#)=! "1 (r,r';#) K XC (r,r',#) r-r'! "1 (r,r';#) =! 0 "1 (r,r';#)" 1 r-r' " K XC (r,r',#) Time-Dependent Density Functional Theory 4 1

13 The Dyson equation for the susceptibility (II) This is an exact formula for the inverse of the susceptibility! "1 (r,r';#) =! 0 "1 (r,r';#)" 1 r-r' " K XC (r,r',#) Exact XC kernel, to be approximated ( ) % n# ((( # n m ( ) $ " $ i'! 0 (r,r';") = f m# $ f n# * * (r)% m# (r)% m# m# $ n# The latter is an exact expressionfor the independent-particle susceptibility, if the exact KS eigenenergies and eigenfunctions (occ/unocc) are known. In practice these must be approximated. (r')% n# (r') Time-Dependent Density Functional Theory 5 Photoabsorption cross section For a finite system, the polarizability tensor is given by! ij (") = # r i $(r,r';") % r j drdr' and the photoabsorption cross section corresponds to its imaginary part, with suitable coefficients! ij (") = 4#" $m % ij (" + i + ) c Independently of the TDDFT formulation of the susceptibility, it is known (in the many-body theory), that the systems can absorb energy at the frequencies corresponding to transitions to excited states, and that the polarizability can be formulated as a sum over states of the many-body system. This is the basis of the Casida technique to find excitation energies. Time-Dependent Density Functional Theory 6 13

14 Casida s approach to excitation energies (I) For a finite system, the polarizability must be given by f I,ij! ij (") = $ I " I # " where the I index labels different excited states,! I and f I,ij are the corresponding excitation energy and oscillator strength. After some non-trivial mathematics (see later), for adiabatic XC kernels, Casida (1995) deduced that the square of the excitation energies were all eigenvalues of a specific matrix equation! F I = " I F I while the oscillator strengths could be deduced from the eigenvectors of the same equation. F I Time-Dependent Density Functional Theory 7 Casida s approach to excitation energies (II)! This matrix is defined in the space of electron-hole excitations, namely, combinations of one occupied and one unoccupied wavefunction, for one spin channel. (Note that we now explicitely takes into account the spin, unlike in the previous slides) One line (or one row) of the matrix will be characterized by a composite index, i.e. klσ (k unoccupied and l occupied states for one spin channel), associated to KS energy change! kl" = # k" $ # l" and spatial electron-hole wavefunction product! k" (r)! l" (r) Note : for a finite system, all wavefunctions can be taken real. n (1) (r!,") = $ # 0 (r!,r'!';")v (1) KS (r'!',") dr' occ unocc ( 1! 0 (r",r'" ';#) = $% "" ' ' ' n" (r) m" (r) m" (r') n" (r') # mn" $ # ) * # mn" + #, - n m occ unocc ( = $% "" ' ' ' n" (r) m" (r) m" (r') n" (r') ) * n m # mn" # mn" + $ #, - Time-Dependent Density Functional Theory 8 14

15 Casida s approach to excitation energies (III) The change of density can be represented by a sum over electron-hole pairs : n (1) occ unocc (r!,") = $ $ c nm! # n! (r)# m! (r) n m with coefficients obtained from the Adler-Wiser susceptibility expression c nm! = " # % $ mn! ( ' $ mn! " $ ) * + m! (r')+ n! (r')v KS (1) (r'!',$)dr' The change of Kohn-Sham potential comes from the change of external potential, combined with the induced (self-consistent) change of potential : V (1) KS (r'!',") = V (1) (1) ext (r'!',") + V HXC (r'!',") = V (1) ext (r'!',") + # $ 1 r'-r'' + K XC % (r'!',r''!'';") ' ( ) n(1) (r''!'')dr'' = V (1) occ unocc $ 1 ext (r'!',") + * * * # c nm!! n m r'-r'' + K ' XC(r'!',r''!,") % ( ) + n!(r'')+ m! (r'')dr'' Time-Dependent Density Functional Theory 9 Casida s approach to excitation energies (IV) Now, all the quantities defined in term of spatial functions, can be replaced by coefficients within the electron-hole pair basis. So,we introduce : (1) V ext, nm! = " # m! (r')# n! (r')v ext (1) (r'!',$)dr' (") = # $ n! (r)$ m! (r) % 1 r-r' + K XC ' (r!,r'!',") ( ) * $ n'!' (r')$ m'!' (r')drdr' and we find for the coefficients of the density : HXC K mn!,m'n'!' $ c nm! = " % ' $ " # ( ) V ext, % # occ unocc mn! (1) HXC nm! + * * # mn!!' n' m' (! mn" #! % (1) )c nm" = #! mn" V ext, nm" + occ unocc HXC $ $ ' "' n' m' occ unocc HXC m' * K mn!,m'n'!' $ K mn",m'n'"' (#)c n'm'!' The latter equation is characteristic of a forced oscillator. The right-hand side member is the driving force. Resonances will have non-zero solutions, even without a driving force. Time-Dependent Density Functional Theory 30 ' ( ) (!)c n'm'"'!!! ((# mn" $ # )% mm' % nn' % ""' + # mn" K mn",m'n'"' (#))c n'm'"' = $# mn" V ext, "' n' ( ) * (1) nm" 15

16 Casida s approach to excitation energies (V) Resonances are found by solving : occ unocc!!! ((# mn" $ # )% mm' % nn' % ""' + # mn" K HXC mn",m'n'"' (#))c n'm'"' = 0 "' n' m' occ unocc!!! (# mn" $ mm' $ nn' $ ""' + # mn" K HXC mn",m'n'"' (#))c n'm'"' = # c nm" "' n' m' The latter equation is an eigenvalue equation. Note that if the HXC kernel contribution is omitted, the eigenvalues are simply the Kohn-Sham electron-hole excitation energies. Two problems are introduced by the HXC kernel term, however : (1) It depends on the frequency () Combined with the multiplication by the frequency, it is not hermitian The second problem is easily solved by incorporating the square root of the electron-hole pair excitation into the eigenvector : occ unocc!!! # mn" $ mm' $ nn' $ ""' + # mn" K mn",m'n'"' (#) # m'n'"' "' n' m' ( ) HXC c n'm'"' # m'n'"' = # c nm" # mn" Time-Dependent Density Functional Theory 31 Casida s approach to excitation energies (V) Concerning the first problem, we simply remark that for an adiabatic approximation to the true TD functional, there is no dependence of the HXC kernel on the frequency. Thus : occ unocc!!! # mn" $ mm' $ nn' $ ""' + # mn" K mn",m'n'"' "' n' m' ( HXC # m'n'"' ) c n'm'"' # m'n'"' = # c nm" # mn" That is,! F I = " I F I with! ( ) = # mn",m'n'"' mn" HXC $ mm' $ nn' $ ""' + # mn" K mn",m'n'"' Such matrix will be diagonalized, yielding all the eigenvalues and corresponding eigenvectors ( ) mn!! I F I # m'n'"' just in one diagonalization. The coefficients of the density change in the electron-hole pair basis can be reconstructed, and allow to find the oscillator strength for each excitation. Time-Dependent Density Functional Theory 3 16

17 Spin-singlet and spin-triplet excitations (I) In case of a spin-unpolarized ground state, the matrix to be diagonalized has important symmetry properties, so that its eigenvectors will be either symmetric or anti-symmetric with respect to spin flip. n! (r) = n " (r) # i! (r) = # i" (r) $ i! = $ i" ( ) = # mn",m'n'"' mn" Indeed! $ mm' $ nn' $ ""' + # mn" K mn",m'n'"' # m'n'"' HXC K mn!,m'n'!' (") = # $ % 1 n!(r)$ m! (r) r-r' + K ( XC(r!,r'!',") ' ) * $ n'!'(r')$ m'!' (r')drdr' $! ""! "# ' So such that! "" =! ##! "# =! #" %! #"! ) ## ( We already knew! "" = (! "" ) t! "# = (! #" ) t (1) F!I = F "I with (#!! + #!" )F!I = $ I F!I () F!I = "F #I with ( $!! " $!# )F!I = % I F!I HXC Time-Dependent Density Functional Theory 33 Spin-singlet and spin-triplet excitations (II) What is the meaning of the symmetric spin-flip eigenvector? occ unocc n (1) (r!,") = $ $ c nm! # n! (r)# m! (r) whereas F nm! = c nm! n m " mn! F!I = F "I # for a given electron-hole pair, both spin channels interfere constructively, TD total charge change, no TD magnetization change! ( â m" â n" + â m# â n# )$ 0 Spin-singlet excited state F!I = "F #I $ for a given electron-hole pair, both spin channels interfere destructively, no TD total charge change, TD magnetization change! ( â m" â n" # â m$ â n$ )% 0 Spin-triplet excited state Time-Dependent Density Functional Theory 34 17

18 Spin-singlet and spin-triplet excitations (III) Why is this technique able to detect spin-triplet states?!ˆ S! = s(s +1)"! Ŝ z! = m s "! s = 1! m s = "1,0,1 s = 0! m s = 0 Triplet Singlet With superposition of electron-hole excitations for the same spin channel, one only sees m s = 0. Still, with such excitations both triplet and singlet states can be reached. But the applied external potential was NOT able to change the s value... Yes, so the oscillator strengths for the triplet excitations will be zero. If one applies a spin-flipping external potential (not done in the previous slides) the! matrix is the same. But in this case, the oscillator strength does not vanish for the triplet excitations. Actually ABINIT gives non-zero oscillator strengths for triplet excitations, corresponding to matrix elements where the spin has been flipped by the perturbation. Time-Dependent Density Functional Theory 35 Casida s approach : additional remarks (1) Knowing the excitation energies allows one to compute Born-Oppenheimer hypersurfaces for excited states, as E I { R} = E GS R { } +! I { R} () Casida s formalism allows to identify well-separated excitation eigenenergies. OK for finite systems, not for bulk, periodic chains or slabs. More fundamentally, kernels based on LDA/GGA are unable to modify band edges within TDDFT, they are only able to shift oscillator strengths. Time-Dependent Density Functional Theory 36 18

19 Nitrogen molecule : tight-binding analysis Diagram of molecular orbitals σ* u π* gx π* gy p z p x p y σ π π π ux π uy px p y p z π π σ σ g Atom Diatomic molecule Atom However, due to repulsion with the lower lying states, generated by the s orbitals, the σ g orbital is pushed above the orbitals, and the σ* u is also repelled. π u Time-Dependent Density Functional Theory 37 Nitrogen molecule : Kohn-Sham energies (from the tutorial : non-converged values) σ g π* gx π* gy 0.40 ev ev! g! u + " u 9.91 ev p 8.46 ev σ g π uy π ux ev ev 9.36 ev! g ev s σ* u σ g ev ev! u KS energies lowest KS transitions Time-Dependent Density Functional Theory 38 19

20 Nitrogen molecule : TDDFT (from the tutorial : non-converged values, ecut=5, nband 1) Kohn-Sham TDDFT triplet singlet! u ev 9.91 ev ev! u (ionized) 9.08 ev (ionized)! u! u! g! g 9.91 ev 9.36 ev 8.46 ev ev 8.15 ev 9.91 ev 9.17 ev 9.86 ev 7.84 ev 9.45 ev Time-Dependent Density Functional Theory 39 Nitrogen molecule : TDDFT (better values, ecut=45, nband 30) Kohn-Sham TDDFT triplet singlet Exp. triplet singlet! u 10.9 ev ev ev ev! u 9.78 ev (ionized) 9.78 ev 8.90 ev 10.3 ev 7.87 ev 9.78 ev! u! u 9.67 ev 8.88 ev 10.7 ev 8.04 ev 9.9 ev! g 9.5 ev 9.3 ev 9.89 ev! g 8.39 ev 7.75 ev 9.36 ev 7.75 ev 9.31 ev exp. taken from JCP 110, 785 (1999) Time-Dependent Density Functional Theory 40 0

21 Optical properties of nanodiamonds (I) J.-Y. Raty and G. Galli, J. Electroanalytical Chemistry 584, 9 (005) Nanometer size diamond : interstellar dust, carbonaceous residues of explosion, constituent of diamond-like films. Generation : spherical truncation of bulk, followed by hydrogen passivation. Fig. 1. Absorption spectra of fully hydrogenated nanodiamonds up to 1 nm in diameter. At least 600 excitations have been included in the linear response treatment to ensure the convergence of the spectra in a range of a minimum of ev above the onset of the absorption. The spectra are truncated at high energy (dashes). The vertical dashed line corresponds to the absorption gap of bulk diamond with the same level of accuracy and converged with respect to k-points. Time-Dependent Density Functional Theory 41 Optical properties of nanodiamonds (II) J.-Y. Raty and G. Galli, J. Electroanalytical Chemistry 584, 9 (005) Fig.. Fully hydrogenated 66 carbon atom cluster, 0.8 nm diameter, (left) and (1 0 0) reconstructed cluster (right). The (1 0 0) reconstruction occurs spontaneously upon removal of pairs of hydrogen atoms, such as those circled on the left. The HOMO and LUMO are represented by the light- and dark isosurfaces, respectively, and are drawn at 33% of their maximal value. Fig. 3. Absorption spectra of unreconstructed and (1 0 0) reconstructed C9 clusters (0.6 nm in diameter). The arrow indicates the new absorption peak induced by the surface reconstruction. Time-Dependent Density Functional Theory 4 1

22 Implementation of Casida s formalism in ABINIT (I) Restricted to finite systems : - big supercell - nkpt=1 Restricted to spin-unpolarized ground-state + no spinor (in ABINITv5.3 : ability to treat spin-polarized) - nsppol=1 - nspinor=1 First dataset Ground State SCF (with occupied bands) Second dataset iscf=-1 Non-SCF calculation (with many bands) +TDDFT Time-Dependent Density Functional Theory 43 Implementation of Casida s formalism in ABINIT (II) If you need the dynamical polarisability, you need to give the supercell a «box center», so that there is vacuum at the discontinuity of the r potential. The spectrum is to be produced by yourself, on the basis of excitation energies and oscillator strengths (utility : ~abinit/extras/post_processing/dynamic_pol.f) Parallelism present (although documentation is lacking). Tutorial available In order to get acceptable ionization energy, the GS potential must be better than LDA/GGA. Use van Leeuwen-Baerends (ixc=13) Time-Dependent Density Functional Theory 44

23 Discussion TDDFT formalism well-established Approximations : adiabatic OK ; beyond is still an active research area In the Casida formalism, need good starting XC potential and XC kernel Powerful for finite systems Accuracy : might be as good as 0.1 ev, but as bad as 1 ev. With usual kernels, not useful for bulk solids, surfaces and chains. Time-Dependent Density Functional Theory 45 3