Efficient Calculation of (Resonance) Raman Spectra and Excitation Profiles with Real-Time Propagation

Zurich Open Repository and Archive University of Zurich Main Library Strickhofstrasse 39 CH-8057 Zurich www.zora.uzh.ch

Year: 2018

Eﬀicient calculation of (resonance) Raman spectra and excitation profiles with real-time propagation

Mattiat, Johann ; Luber, Sandra

Abstract: We investigate approaches for the calculation of (resonance) Raman spectra in a real-time time-dependent density functional theory (RT-TDDFT) framework. Several short time approximations to the Kramers, Heisenberg, and Dirac polarizability tensor are examined with regard to the calculation of resonance Raman spectra: One relies on a Placzek type expansion of the electronic polarizability and the other one relies on the excited state gradient method. The first one is shown to be in agreement with an approach based on perturbation theory in the case of a weak ฀-pulse perturbation. The latter is newly applied in a real time propagation framework, enabled by the use of Padé approximants to the Fourier transform which allow for a suﬀicient resolution in the frequency domain. An analysis ofthe performance of Padé approximants is given. All approaches were found to be in good agreement for uracil and R-methyloxirane. Moreover it is shown how RT-TDDFT can be used to calculate Raman excitation profiles eﬀiciently.

DOI: https://doi.org/10.1063/1.5051250

Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-162379 Journal Article Published Version

Originally published at: Mattiat, Johann; Luber, Sandra (2018). Eﬀicient calculation of (resonance) Raman spectra and excitation profiles with real-time propagation. Journal of Chemical Physics, 149(17):174108. DOI: https://doi.org/10.1063/1.5051250 Efficient calculation of (resonance) Raman spectra and excitation profiles with real- time propagation

Cite as: J. Chem. Phys. 149, 174108 (2018); https://doi.org/10.1063/1.5051250 Submitted: 07 August 2018 . Accepted: 11 October 2018 . Published Online: 05 November 2018

Johann Mattiat, and Sandra Luber

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Eﬃcient calculation of (resonance) Raman spectra and excitation proﬁles with real-time propagation Johann Mattiat and Sandra Lubera) Department of Chemistry, University of Zurich, Winterthurerstrasse 190, Zurich, Switzerland (Received 7 August 2018; accepted 11 October 2018; published online 5 November 2018)

We investigate approaches for the calculation of (resonance) Raman spectra in a real-time time- dependent density functional theory (RT-TDDFT) framework. Several short time approximations to the Kramers, Heisenberg, and Dirac polarizability tensor are examined with regard to the calculation of resonance Raman spectra: One relies on a Placzek type expansion of the electronic polarizability and the other one relies on the excited state gradient method. The first one is shown to be in agreement with an approach based on perturbation theory in the case of a weak δ-pulse perturbation. The latter is newly applied in a real time propagation framework, enabled by the use of Pade´ approximants to the Fourier transform which allow for a sufficient resolution in the frequency domain. An analysis of the performance of Pade´ approximants is given. All approaches were found to be in good agreement for uracil and R-methyloxirane. Moreover it is shown how RT-TDDFT can be used to calculate Raman excitation profiles efficiently. Published by AIP Publishing. https://doi.org/10.1063/1.5051250

I. INTRODUCTION resonant case [resonance Raman spectroscopy (RRS)].20 The enhancement of the Raman signal when the excitation fre- Real time propagation (RTP) techniques for the solu- quency is near an electronic transition of the molecule opened tion of the time-dependent Schrodinger¨ equation (TDSE) in a wide research field. Among many other applications, RRS a time-dependent density functional theory (TDDFT) frame- was used to gain structural information about bio-molecules,21 work have become a viable alternative to common perturbation monitor reactions on (catalytic) surfaces,22 and to investigate theory (PT) approaches, such as Sternheimer’s1 and Casida’s2 nano-structures such as graphene23 and carbon nano-tubes.24 approach to TDDFT, for the calculation of spectroscopic Recently, RRS has provided valuable insights into mecha- properties.3,4 nisms in the field of photovoltaics,25 photo-catalysis,26 and One of the advantages of RTP techniques is that they bring artificial water splitting.27 Computational approaches for res- about the full spectrum naturally because spectra are connected onance Raman optical activity for chiral molecules have also via a Fourier transform (FT) to the characteristics of the time been presented.28–30 evolution, whereas PT methods usually cover only a limited Theoretical efforts to calculate resonance Raman spectra frequency range at a time, e.g., for the calculation of excita- can be broadly categorized into two families of approximations tion energies and oscillator strengths.5 For details regarding to the perturbation theory result of Kramers, Heisenberg,31 scaling, see, e.g., Ref. 6. This leads to an advantage in cost if and Dirac32 (commonly known as the KHD polarizability many excited states are involved, as an example for the calcula- tensor), which gives an expression for the electric-dipole– tion of spectra of dye in solution.7 Moreover, RTP approaches electric-dipole polarizability (for sake of brevity in the fol- can give the response beyond the linear response by including lowing referred to as polarizability) in the Born-Oppenheimer electro-magnetic fields non-perturbatively, which makes them (BO) approximation. computationally simpler compared to perturbative approaches The first kind of approximation was developed by to non-linear response.8,9 Albrecht33 et al. in the 1960s. In their approach, the elec- 10 Beginning with absorption spectra of C60, the appli- tric transition dipole moment is expanded into a Taylor series cation of RTP techniques to chemically relevant systems was around the equilibrium geometry. In the resonant case, the extended most notably to hyper-polarizabilities,11 electronic12 resulting terms [the first two terms of the expansion are and magnetic13 dichroism spectra, molecular conductance,14 often called A [Franck–Condon (FC)] and B (Herzberg– charge transfer,15 X-ray absorption spectroscopy,16 the inclu- Teller)34 terms] involve a computationally expensive sum over sion of relativistic effects in four-component DFT,17 time- vibrational levels.35 resolved pump probe type spectroscopy,18 and a phase cycling On the other hand, Lee, Heller et al.36–38 cast the KHD protocol for X-ray spectroscopy.19 tensor into the time-domain and justified short time approx- In this work, the focus will be on the application of real- imations (STA) to the resulting wave package dynamics. time (RT) TDDFT to Raman spectroscopy, especially in the A very useful result of them is the excited state gradient method (ESGM), which allows us to calculate relative resonance Raman intensities from the gradient of the excited state a)Electronic mail: [email protected] BO-surfaces at the ground state equilibrium geometry.

0021-9606/2018/149(17)/174108/13/$30.00 149, 174108-1 Published by AIP Publishing. 174108-2 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

39 Jensen et al. also used a short-time approximation to Here dˆα denotes the electric dipole moment operator and the the wave packet dynamics in order to derive a PT-TDDFT Greek subscripts α and β denote Cartesian directions (x, y, z). expression for the electronic polarizability tensor that is Γ accounts phenomenologically for the finite life time of n applicable to the resonant case by including a finite phe- excited states. In principle, each excited state |vk i would have nomenological life time of the electronic excited states. Its a different life time Γ n , but in practice an averaged value is vk application to RRS involves a Placzek type expansion40 of taken. The KHD tensor describes a two photon process: The the electronic polarizability tensor around the equilibrium frequencies of the incoming and scattered photons, ω and ωS, f i geometry.36 are related via the energy balance ~ω = ~ω − (Ev − Ev ) S e0 e0 Besides calculations of surface-enhanced Raman spectra with ~ being Planck’s constant divided by 2π. The first term via RT-TDDFT,41,42 Thomas et al.43 have applied the calcu- dominates if the excitation frequency ω is near or on an elec- i n lation of the electronic polarizability in an RT-TDDFT frame- tronic resonance (Ev + ~ω ≈ Ev ). For the sake of brevity, the e0 ek work to RRS, also by applying a Placzek type expansion. second term is referred to as a non-resonant term (NRT) in this They compared their results to both Jensen’s polarizability paragraph. 44 method and the excited state gradient method in a PT- The Raman scattering cross section σfi,αβ for spontaneous TDDFT framework finding a good agreement of the resulting Raman scattering is related to the absolute value of the KHD RRS spectra. polarizability tensor38 as follows: In this work, Jensen’s approach to calculate the electronic = ΣKHD 2 = KHD 2 polarizability by using PT is shown to be equivalent to the RT σfi,αβ(ω) αβ (ω) ααβ (ω) . (2) approach in the limit of a weak δ-pulse perturbation. Moreover Lee, Heller et al.38, 37 cast the KHD polarizability tensor it is demonstrated how the excited state gradient method can be applied in an RTP framework, instead of using PT-TDDFT. in the time domain using the algebraic identity This is enabled by using Pade´ approximants in the context i ∞ 1 1 dt e−i(a+ib)t = (3) of the Fourier transforms (FTs) required in RTP techniques ~ 0 ~ (a + ib) which allows for shorter simulation times and, most crucially, 48 for higher resolution in the frequency domain.45 to rewrite Eq. (1) as In order to support the theoretical findings and compare i ∞ αKHD(ω) = dt hvf |he0|dˆ |eki|vni the results to previous studies, two small molecules have been αβ ~ 0 α k 0 f vnX,vi ,v chosen as model systems: uracil and R-methyloxirane. k 0 0 k 0 The paper is structured as follows: In Sec. II, the necessary e ,e n i i ω−ωv +ωv −Γ t theoretical background is given. Details about implementation n k ˆ 0 i ek e0 × hvk |he |dβ |e i|v0ie + NRT and calculations are given in Sec. III. The results are presented and discussed in Sec. IV, and a conclusion is given in (4)

Sec. V. n n with ωv = Ev /~. Considering the action of the Hamiltonian ek ek for vibrational motion belonging to the BO-surface of excited k n state |e i, Hˆ k , on the vibrational state hv |, II. THEORY e k n − ˆ ~ −iωv t A. KHD polarizability tensor n iHek t/ = n ek hvk |e hvk |e , (5) Usually computational approaches for the calculation of and introducing the Raman wave function Raman scattering cross sections rely on an expression derived i −iHˆ t/~ k 0 i | i = ek h | ˆ | i| i by Kramers, Dirac, and Heisenberg, the KHD polarizability ξek e0,β(t) e e dβ e v0 (6) tensor, which describes vibrational Raman scattering in the allows us to write Eq. (4) as BO approximation [rotational levels are accounted for by an i ∞ isotropic average (for more information see Ref. 46)]. Here αKHD(ω) = dt hξf (0)|vnihvn|ξi (t)i k αβ ~ e0ek ,α k k ek e0,β electronic states are denoted as |e i and vibrational states that 0 f vnX,vi ,v belong to the electronic BO surface k as |vni. Their energies k 0 0 k ek ,e0 vn are E k and E , respectively. i e ek v −Γ i ω+ω 0 t Then the Raman scattering cross section from an initial × e e + NRT . (7) state |vi i to a ﬁnal state |vf i, both belonging to the electronic 0 0 Now the closure over the vibrational states |vni can be carried ground state |e0i, is proportional to the KHD polarizability k KHD 47 out tensor ααβ (ω) given by i ∞ αKHD(ω) = dt hξf (0)|ξi (t)i αβ e0ek ,α ek e0,β f 0 k n n k 0 i ~ hv |he |dˆ |e i|v ihv |he |dˆ |e i|v i 0 eXk ,e0 KHD 0 α k k β 0 α (ω) = − i αβ vn vi i ω+ωv −Γ t " ~ω − (E − E ) + i~Γ 0 nX, i f ek e0 × e e + NRT . (8) vk v0,v0 ek ,e0 f 0 k n n k 0 i This time-domain expression for the KHD polarizability ten- hv |he |dˆα |e i|v ihv |he |dˆβ |e i|v i + 0 k k 0 . (1) sor is equivalent to its frequency-domain counterpart [see −~ω − (Evn − Evf ) + i~Γ # ek e0 Eq. (1)]. Computational efforts for the calculation of Raman 174108-3 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

2 spectra often require approximations to the KHD polarizabil- dσ π 4 h = (ν ˜ − ν˜q ) ity tensor, which usually involve a STA of the excited state dΩ 2 in k 8π2cν˜ ǫ0 qk dynamics in the resonant case. The time domain formulation 2 2 45|aqk | + 7γq 1 of the KHD polarizability tensor [Eq. (8)] gives an intuitive × k , (12) hcν˜ interpretation of STAs:38 Initially |ξf (0)i and |ξi (t)i 45 1 − exp(− qk ) e0ek ,α ek e0,β kBT are localized in the Franck–Condon (FC) region and their ini- whereν ˜ is the wavenumber of the incoming light,ν ˜ is the tial overlap decays rapidly, on a femto second scale. In many in qk i wavenumber of the normal coordinate qk, ǫ0 is the vacuum per- cases, recurrences of the |ξ k 0 (t)i to the FC region do not e e ,β mittivity, c is the speed of light in vacuum, aqk is the isotropic contribute signiﬁcantly to the overlap, among other reasons, derivative, due to damping and dephasing. As a consequence, it is suf- = 1 ∂ααα ﬁcient to consider the polarizability tensor only near the FC aqk , (13) 3 ∂qk ! region of the electronic ground state, also for the calculation Xα γ2 of resonance Raman spectra. and qk is the anisotropic derivative,

1. Non-resonant case and Placzek type expansion 1 ∂ααβ ∂ααβ ∂α ∂αββ γ2 = 3 − αα . (14) qk 2 " ∂q ! ∂q ! ∂q ! ∂q !# In the case of non-resonant Raman spectroscopy (NRS), Xαβ k k k k the KHD tensor can be reduced to the expression derived by The derivatives are performed numerically using the scheme Placzek,40 by replacing the vibronic energies with their elec- vn given in Refs. 51 and 52. tronic counterparts at the equilibrium geometry q , E ≈ E k , 0 ek e There has been a long standing effort to enable the use vi E ≈ E 0 , setting ω = ω , and omitting Γ, which allows us e0 e S of Placzek’s expansion also in the resonant case: Warshel to perform the closure over the vibrational states. This leads and Dauber53 suggested a way to evaluate Albrecht’s A and to the following expression for the non-resonant electronic B terms directly in the resonant case by introducing a Tay- el, NRS 36 polarizability αα,β : lor expansion of the vibronic transition dipole moments. Lee searched for appropriate approximations of the KHD tensor by he0|dˆ |ekihek |dˆ |e0i 36 el, NRS = α β using the time domain description. In the non-resonant case, ααβ (ω, q) − " ~ω − (E k − E 0 ) Placzek’s expansion reproduces Albrecht’s A and B terms Xk , 0 e e e e exactly.36 0 k k 0 he |dˆα |e ihe |dˆβ |e i + ~ . (9) 2. Jensen’s polarizability method − ω − (Eek − Ee0 ) # In order to use a Placzek type treatment of vibration also These assumptions are valid for a clear separation of the energy in the resonant case, Jensen et al.39 derived a semi-classical scales of the electronic transitions, the exciting photon energy, (SC) expression for the electronic polarizability tensor fol- and the vibrational levels.36,46 lowing Lee’s original SC treatment of the KHD tensor.36 Here The electronic polarizability αel (ω, q) depends paramet- αβ the propagation in Eq. (8) is cast into Wigner phase space rically on the nuclear coordinates q. Then the Raman scattering and the propagator is approximated to first order in t by cross section is proportional to its classical counterpart. A restriction to the Franck–Condon Σadiabatic 2 = f el i 2 region leads to the semi-classical Raman scattering cross sec- (ω, q) hv |ααβ(ω, q)|v i . (10) 2 fi,αβ 0 0 tion σSC (ω) = ΣSC (ω) [|ξi i are the Raman wave fi,αβ fi,αβ e0ek ,β Placzek expanded the electronic polarizability into a Taylor functions given in Eq. (6), with the time-dependence made 40 series around the equilibrium geometry q0, leading to explicit] adiabatic el f i ∞ Σ (ω, q) = α (ω, q) = hv v i SC i f −iE /~t i i(E /~+ω)t−~Γt fi,αβ αβ q q0 0 0 Σ (ω) = dt hξ |e ek |ξ ie 0 fi,αβ ~ e0ek ,α ek e0,β ∂αel (ω, q) 0 Xek αβ f i + hv |qk |v i + ... , ∞ = 0 0 i f iE /~t i ∂qk q q0 ek Xk − dt hξ 0 k |e |ξ k 0 i ~ e e ,α e e ,β 0 Xk (11) e ~ ~Γ × ei(−E0/ +ωS)t− t. (15) where qk are mass-weighted normal coordinates for normal mode k. In the harmonic approximation, the terms of the Transferring to the frequency domain and assuming ω ≈ ωS Taylor expansion can be assigned to different sorts of spec- give for the adiabatic polarizability troscopy49,50 and the usual selection rules apply: The 0th order ΣSC, adiabatic(ω, q) = hvi |αel, SC(ω, q)|vf i, (16) term represents Rayleigh scattering and the 1st order term fi,αβ 0 αβ 0 fundamental Raman scattering. he0|dˆ |ekihek |dˆ |e0i The differential Raman scattering cross section per solid el, SC = α β Ω ααβ (ω, q) angle for Stokes scattering can then be calculated according " −~ω + (E k − E 0 ) − i~Γ eXk ,e0 e e to the following formula for a specific normal coordinate qk, 0 ˆ k k ˆ 0 a scattering angle of 90◦, and linearly polarized light with the he |dα |e ihe |dβ |e i + . (17) 46 ~ ~Γ electric field vector perpendicular to the scatter plane: ω + (Eek − Ee0 ) + i # 174108-4 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

It can be used in a Placzek type expansion to calculate NRS and hBˆ (t)i = Tr ρˆ(t)Bˆ . (21) RRS spectra. The only difference to the non-resonant expres- f g The assumption in linear response theory is that this expec- sion in Eq. (9) is the inclusion of a finite life time Γ of the tation value changes linearly with f (t). If perturbations at electronic states. different times are independent of each other, this assumption 3. Excited state gradient method can be cast into the following formula: t By taking a purely classical propagation scheme for the ′ ′ ′ hBˆ (t)i − hBˆ i = dt ΦBA(t − t )f (t ), (22) short time wave packet dynamics in Eq. (8) and assuming a −∞ harmonic ground state BO surface, Heller et al. derived the fol- where ΦBA is called the (linear) response function (of lowing formula for the relative Raman intensities of different Bˆ Aˆ 38 operator with respect to a perturbation operator ) and vibrational modes: hBˆ i = hBˆ (t → −∞)i being the expectation value of the unper- k 2 I ω ′ V turbed system. The perturbation is assumed to be turned on = k k ′ ′ ′ , (18) adiabatically and ΦBA(t − t ) = 0 for t > t due to causality. Ik ω V ′ ! k k In the frequency domain, the response is expressed by the where Laplace transform χBA(ω) of the response function, = ∂Eex ∞ Vk = , (19) = Φ i(ω+iǫ)t ∂q q q0 χBA(ω) lim dt BA(t)e . (23) k ǫ→0+ 0 with the mass-weighted normal coordinate q corresponding k The generalized Fourier transform of the expectation value can to the normal mode frequency ω and E the excitation energy k ex then be written as of the electronic transition. This became known as the excited ∞ state gradient method (ESGM) and should only be used on and hBˆ (ω)i = lim dt hBˆ (t)i − hBˆ i eiωte−ǫt. (24) 38 ǫ→0+ near resonances (i.e., in the Condon approximation). In this −∞ paper, we will apply this method in an RT-TDDFT framework Using the convolution theorem for FTs, one obtains the as shown in Sec. III. Equations (18) and (19) are applicable following relation for the generalized susceptibility χAB(ω): in that form if only a single electronic state is contributing to ˆ = the resonance and should only be used for well separated (i.e., hB(ω)i χBA(ω)f (ω). (25) 39,54 usually low-lying) electronic transitions. 1. Linear response in a real-time framework and real-time polarizability method B. Linear response theory In the case of a perturbation by an electric field in the In this paragraph, the real-time propagation and the per- dipole approximation and a measurement of the electric dipole turbative approach to obtain the electronic polarizability are moment Aˆ = dˆ, Bˆ = dˆ , the generalized susceptibility χdd(ω) formulated in the framework of response theory. This presen- is called polarizability tensor ααβ and thus given by tation of linear response theory is closely following Jensen’s ˆ book, and we refer for more details to Ref. 55. Two general hdα(ω)i ααβ(ω) = , (26) expressions for the linear response, one in an RTP frame- Eβ(ω) work and the other in a PT framework will be derived and where E (ω) denotes the FT of the electric field in direction β applied to the specific case of a light–matter electric-dipole– β and hdˆ (ω)i the FT of the electric dipole moment in direction electric-dipole interaction. The discussion will be restricted α α to electronic responses, which will later be used for the cal- ∞ culation of Raman spectra according to Placzek’s expansion iωt −ǫt hdˆα(ω)i = lim dt hdˆα(t)i − d0,α e e . (27) ǫ→0+ [see Eq. (11)]. In order to simplify notation, electronic states −∞ are denoted as |ki and their respective energies as Ek. Hˆ0 is d0,α denotes the static electric dipole moment of the unper- the unperturbed, time-independent Hamiltonian of the (elec- turbed system. In real-time propagation, the expectation value tronic) system in the BO approximation, giving rise to a of the time-dependent electric dipole moment is simply ˆ = complete set of eigenstates H0|mi Em |mi and the den- expressed in the Schrodinger¨ picture using the electronic wave = 2 sity operatorρ ˆ0 m pm |mihm| with pm = |cm| denoting functions Ψ(t) of the system at time t, the probability amplitudeP of finding a general superposition ˆ = Ψ ˆ Ψ |Ψi = mcm|mi in eigen-state |mi. Hˆ 1 is a time-dependent hdα(t)i h (t)|dα | (t)i. (28) perturbation Hamiltonian of the general form, P Equations (26)–(28) are our working equations for obtain- ing the frequency dependent electronic polarizability tensor Hˆ1(t) = −Afˆ (t), (20) ααβ(ω) in an RT-TDDFT framework. A brief summary of e.g., a perturbation due to an electric field in the dipole approx- RTP in a time-dependent Kohn–Sham framework is given in imation with Aˆ = dˆ and f (t) = E(t) describing the strength and Appendix B. time dependence of the field. If only the linear response of a system under study is The density matrixρ ˆ(t) of the total Hamiltonian Hˆ (t) required, it can be calculated in RTP methods by applying an = Hˆ0 +Hˆ1(t) then becomes time-dependent and the expectation initial weak δ-pulse, Eα(t) = καδ(t), yielding the perturbation value of an operator Bˆ at time t can be written as Hamiltonian 174108-5 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

Hˆ 1(t) = −καdˆαδ(t), (29) using the algebraic identity given in Eq. (3). Setting ǫ to a finite small value Γ, this expression is the same as the one derived where κα denotes the field strength in the α-direction. This method is referred to as the RT polarizability method in the by Jensen from a SC approximation of the KHD polarizability remainder of this work, and the resulting electronic polariz- tensor [see Eq. (17)]. In the regime of validity of perturbation ability tensor is called αel, RT(ω). The derivatives required for theory, thus the RT polarizability method and the PT polariz- αβ ability method are expected to yield very similar results. This the Raman scattering cross section [see Eq. (11)] are performed 43 numerically. The great advantage of employing an RTP tech- was indeed observed by Thomas et al. nique is that the whole frequency-dependent polarizability is In fact, for a weak δ-pulse perturbation [see Eq. (29)], the obtained in one simulation run via the FT in Eq. (27) and allows RT polarizability method can be shown to correspond to the us to calculate NRS and RRS at once. PT polarizability method as follows: The linear response for- The absorption strength function S(ω) is related to the mula (A6) can be used directly to calculate the time-dependent el, RT expectation value of the electric dipole moment to first order, imaginary part of the electronic polarizability ααβ (ω) as follows: under the assumption that κβ is sufficiently small 4πω el, RT t S(ω) = Tr Im(α (ω)) . (30) ˆ ˆ ′ ′ ′ 3c αβ hdα(t)i − hdαi = dt Kαβ(t − t )κβ δ(t ). (33) ( ) −∞ 2. Linear response in a perturbation theory framework Inserting Eq. (31), starting from the ground state, and integrat- Instead of explicitly calculating the evolution of per- ing out the δ distribution give turbed time-dependent wave functions, the linear response i −i(Ek −E0)t/~ can be expressed in terms of unperturbed wave functions and hdˆα(t)i − hdˆαi = κβ h0|dˆα |kihk|dˆβ |0ie ~ " the perturbation Hamiltonian by using time-dependent PT. A Xk,0 derivation of relevant formulae is sketched in Appendix A. −i(E0−Ek )t/~ − h0|dˆα |kihk|dˆβ |0ie For the specific case of a perturbation by an electric field # in the dipole approximation and a measurement of the electric 2 (Ek − E0)t dipole moment Aˆ = dˆ, Bˆ = dˆ , the linear response function to = κβ sin h0|dˆα |kihk|dˆβ |0i. ~ ~ ! first order is Xk,0

i ˆ ~ ˆ ~ (34) K (t) = Tr ρˆ eiH0t/ dˆ e−iH0t/ , dˆ dα dβ ~ 0 α β Taking the FT according to Eq. (27), identifying ǫ with Γ i iEmt/~ −iEk t/~ and dividing by Eβ(ω) = κβ [Eq. (26)] one arrives at the = pm e hm|dˆα |kie hk|dˆβ |mi ~ " Xm,k same expression for the electronic polarizability as the direct calculation of the susceptibility in Eq. (32). iEk t/~ −iEmt/~ − hm|dˆα |kie hk|dˆβ |mie # C. Pade´ approximants and RT excited state i gradient method −i(Ek −Em)t/~ = pm hm|dˆα |kihk|dˆβ |mie ~ " Instead of using PT-TDDFT for the excited state gradient Xm,k approximation, as is routinely done,43,54 the latter can also be −i(Em−Ek )t/~ − hm|dˆα |kihk|dˆβ |mie . (31) performed in an RT-TDDFT framework: In order to do so, a # very fine frequency resolution of the Fourier transform [see Eq. (27)] is required, which is usually not achieved by the fast Starting from the electronic ground state |mi = |0i, i.e., p0 = 1, and taking the Laplace transform [Eq. (23)], one arrives to Fourier transform (FFT) algorithm due to computational limits (the resolution is ∝ 1 ; i.e., at a time step of ∆t = 0.1 a.u., first order in the perturbation at the following expression ∆tNsteps for the frequency-dependent electric-dipole–electric-dipole a resolution of 0.0001 eV would require roughly 30 000 000 el, PT steps). susceptibility, which we call ααβ (ω): As pointed out by Bruner et al.,45 the application of Pade´ ∞ 56,57 αel, PT(ω) = lim dt K ei(ω+iǫ)t approximants to the Fourier transform may be used in αβ + dd ǫ→0 0 order to achieve sufficient resolution and decrease the required ∞ i simulation time. = lim dt h0|dˆ |kihk|dˆ |0i + α β The idea is to write the discrete Fourier transform of the ǫ→0 0 ~ " Xk,0 dipole moment d(t) as a polynomial in z = e−iω∆t as ~ ~ ~ × ei( ω+i ǫ−Ek +E0)t/ M M = −iωtk = k i(~ω+i~ǫ−E0+Ek )t/~ d(ω) d(tk)e ckz ≡ d(z), (35) − h0|dˆα |kihk|dˆβ |0ie # Xk=0 Xk=0 h0|dˆ |kihk|dˆ |0i where ck = d(tk) is the electric dipole moment at time step = α β − lim tk = k∆t. Then it can be approximated by the Pade´ approxima- ǫ→0+ " ~ω − (Ek − E0) + i~ǫ Xk,0 tion N k h0|dˆα |kihk|dˆβ |0i = akz − , (32) d(z) = k 0 . (36) ~ − ~ PN k ω + (Ek E0) + i ǫ # k=0 bkz P 174108-6 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

The coefficients ak and bk are found by solving the set The time step in the simulations was chosen to be 0.1 a.u. of linear equations given by equating the terms in Eq. (36) for the GPW method and 0.05 a.u. for the GAPW method. in powers of z. With these at hand, the Fourier transform can The smaller time step for the GAPW method is necessary in be extrapolated for an arbitrary resolution in the frequency order to get a sufficient sampling rate in the RTP because the domain. This in turn allows us to determine the peak positions last resolved feature in the full absorption spectrum appears in the absorption spectrum (excited state energies) up to the at higher frequencies compared to using GPW with pseudo- desired precision and thus, by taking a numerical derivative potentials. The RTP runs were run for a total amount of 5000 with respect to normal coordinates, the gradients of the excited a.u. (≈120 fs) simulation time in both cases. state surfaces at the ground state geometry are required for the All RT-spectra shown in this work are calculated using application of the excited state gradient method [see Eq. (19)]. Pade´ approximants to the Fourier transform, if not mentioned otherwise. An assessment of the convergence of the Pade´ approximants is given in Sec. IV. III. COMPUTATIONAL DETAILS The phenomenological damping factor Γ was set to A. RT polarizability method 0.1 eV ≈ 0.0037 a.u. in accordance with the choice in the literature,28,39,54 if not mentioned otherwise. The RT-TDDFT runs were performed using the pack- In order to obtain the full absorption (Raman) spectrum, age CP2K.58,59 In this implementation, the numerical solu- ∗ ∗ a total of 3 (3 2 Nq, with Nq being the number of normal tion of the time-dependent Kohn–Sham (TDKS) [Eq. (B1)] modes of interest) runs are necessary, if the differentiation of proceeds as follows: The electron density (and therefore the the polarizability tensor along mass-weighted normal coordi- KS-Hamiltonian) is extrapolated in time by the always sta- nates is carried out numerically using a standard three point ble predictor corrector method and then converged self- differentiation scheme.69 consistently. The enforced time reversible symmetry (ETRS) The actual protocol used to obtain Raman intensities in and the exponential mid-point (EM) propagator are imple- the RTP framework can be summarized as follows: mented to propagate the electronic density, among others.7 For sufficiently small time steps, no significant difference 1. A geometry optimization with the chosen basis set and between those two propagators was detectable. For pseudo- functional was performed. potential basis sets, ETRS was used and for all electron basis 2. A normal mode analysis on the optimized geometry is sets EM because the EM propagator appeared to be more carried out in order to obtain the normal mode frequencies stable for the Gaussian and augmented plane wave (GAPW) and coordinates. method. 3. Two geometries, one displaced along the positive and one Gaussian type Goedecker–Teter–Hutter (GTH) pseudo- along the negative normal mode coordinates, were gen- potential basis sets60 and all electron basis sets from Ahlrichs61 erated for every normal mode of interest using a step size were used in combination with CP2K’s Gaussian and plane of 0.005 bohr. Several tests were performed to confirm wave (GPW) and GAPW methods,62,63 respectively. Three that this is a reasonable choice. functionals were investigated: The Perdew-Burke-Ernzerhof 4. RTP runs were performed for each of the displaced (PBE) functional,64 its hybrid version PBE0,65,66 and the BP86 geometries, and the electric dipole moment was recorded functional.67,68 in time. The initial δ-pulse perturbation corresponds to

FIG. 1. Comparison between absorption spectra of R-methyloxirane calculated with the FFT algorithm and Pade´ approximants. In contrast to the FFT, which is only deﬁned at certain points, the Pade´ approximants extrapolate the FT to an arbitrary resolution. The RTP step size was set to 0.10 a.u. The PBE exchange– correlation functional and the TZV2P-GTH basis set were used. (a) Absorption spectrum calculated using FFT for two geometries slightly displaced along a normal coordinate, one positively (yellow, triangles pointing downwards), one negatively (green, triangles pointing upwards). (b) Absorption spectrum calculated using Pade´ approximants for two geometries slightly displaced along a normal coordinate, one positively (yellow), one negatively (green). 174108-7 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

the multiplication of a phase factor70 of eiκα rα (in atomic Jensen’s PT-TDDFT polarizability method. At last, the calcu- units) and was applied to the ground state DFT wave lation of a full Raman excitation profile of R-methyloxirane is function. The field strength parameter in CP2K was set presented. to 0.001 a.u., resulting in an effective field strength of 2.2166 × 10−4 a.u. for uracil and 2.7708 × 10−4 a.u. for A. Convergence of the Pade´ approximation R-methyloxirane. All simulations were carried out using In order to assess the performance of the Pade´ approxi- non-periodic boundary conditions. mants, the absorption spectrum of uracil was calculated using 5. The resulting polarizabilities and its numerical deriva- the standard FFT algorithm with maximal signal length and tives were evaluated according to Eqs. (12)–(14). compared to Pade´ approximants taking different lengths of the Geometry optimizations, normal mode analysis, and RTP signal into account for determining the coefficients in Eq. (35). runs were carried out using CP2K; a python suite was cre- To measure the convergence of the Pade´ approximants, the root ated for displacing the geometries, parsing the output of mean square of the differences between the FFT and the Pade´ CP2K, and evaluating the electric dipole signals. The python approximants spectra was calculated at the points where the code for performing the Pade´ approximants was adapted from FFT is defined and normalized by the number of these points. https://github.com/jjgoings/pade. This procedure was carried out for different time steps and the number of steps: 0.02 a.u. (500 000 steps), 0.05 a.u. (300 000 B. RT excited state gradient method steps), and 0.10 a.u. (150 000 steps). The first 15 eV of the spectrum were included in the analysis. The results are shown For the real-time excited state gradient method (RT- in Fig. 2. As soon as a certain amount of the signals is taken into ESGM), the same RTP runs as for the RT polarizability method account for the Pade´ approximants, the difference between the are used and the same choice of parameters applies. Here the FFT and the Pade´ approximants spectrum reaches a plateau, derivatives of the excited state energies with respect to normal indicating that the respective absorption spectra are converged. coordinates [see Eqs. (18) and (19)] are performed numerically Not surprisingly, this threshold depends more on the propaga- as follows: As mentioned in Sec. II, Pade´ approximants allow tion time than on the amount of steps, at least for the time for a sufficient resolution in the frequency domain and there- steps of 0.10 a.u. and 0.05 a.u. For a well converged absorp- fore to obtain the excitation energies (peak positions) in the tion spectrum, up to an excitation energy of 15 eV,a simulation absorption spectrum to the desired precision by using a simple time of ∼8000 a.u. (≈200 fs) is necessary for an RTP step of minimax search such as the golden section search algorithm.71 0.10 a.u. The gradient of a specific normal mode is then obtained by However, for the calculation of Raman spectra the excita- the finite difference scheme for the geometry displaced posi- tion frequency was tuned to one of the first excitations visible tively and the geometry displaced negatively along the normal in the absorption spectrum, which are usually below 10 eV. In coordinate. that case, an insignificant error of ∼1% was found for the result- The shift of absorption peaks along a normal mode is ing Raman intensities between using either ∼50 fs or ∼100 fs illustrated in Fig. 1 for using a conventional FFT on the one of the signal for the calculation of the Pade´ approximants for hand and Pade´ approximants on the other hand. an RTP time step of 0.05 a.u. Additionally, the precision for the calculation of excitation C. PT-TDDFT Raman energies with Pade´ approximants was evaluated as follows: One excitation energy was calculated for a time step of 0.10 In order to compare the RT results with traditional PT- a.u. in the RTP and a maximal signal length of 50 000 steps. TDDFT, the program Turbomole72 was used for the calculation of NRS spectra. This implementation is based on analytical derivatives of a polarizability Lagrangian.73 It is restricted to non-resonant and near-resonance Raman spectra.74 The TZVP basis set and the PBE and BP86 functionals were used.

IV. RESULTS AND DISCUSSION The results are discussed in Secs. IV A–IV E: First the convergence of Pade´ approximants to the FFT algorithm is evaluated. Then absorption spectra calculated with RT-TDDFT and PT-TDDFT, respectively, are discussed for uracil and R- methyloxirane as a validation of the RTP approach. Then NRS spectra obtained by the analytical gradient (PT-TDDFT) method in Turbomole are compared with the NRS spectra from RT-TDDFT in CP2K for the same molecules. Resonance Raman (RR) spectra obtained with the RT FIG. 2. Convergence of Pade´ approximants toward the FFT spectrum for polarizability method and the RT-ESGM are compared for dif- different time steps in the RTP. On the y-axis, the root mean square between ferent functionals and basis sets and discussed in relation to the FFT and the Pade´ approximants spectra is given. 174108-8 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

Then the difference to this ﬁnal value at full signal length was calculated while taking different lengths of the signal into account for determining the coefﬁcients in Eq. (35). The result is shown in Fig. 3. The resulting excitation energy is well converged after ∼3000 a.u. of simulation time with a time step of 0.10 a.u.

B. Absorption spectra Absorption spectra of uracil and R-methyloxirane were calculated using RT-TDDFT in CP2K and PT-TDDFT in Turbomole. For uracil, the BP86 exchange–correlation functional was used with Ahlrichs’ TZVP basis set for both methods in order to be consistent with the work of Jensen39 et al. on uracil. The resulting spectra are shown in Fig. 4. The RT-TDDFTspectrum is calculated according to Eq. (30) and is naturally broadened Γ FIG. 4. Absorption spectrum of uracil calculated with PT-TDDFT (Turbo- due to the damping factor , whereas the PT-TDDFT spec- mole, solid red) and RT-TDDFT (CP2K, dashed black). The BP86 exchange– trum is given in terms of excitation energies and corresponding correlation functional was used with Ahlrichs’ TZVP basis set. The damping oscillator strengths. Both methods agree perfectly as expected factor Γ was set to 0.01 eV in order to have more pronounced peaks in the RT from the discussion in Sec. II since they are basically given by absorption spectrum. the same response in the case of a weak δ-pulse perturbation for the calculation of the RT-polarizability. This is consistent out for the PBE exchange–correlation functional with the 5 with the literature. Also note that the RT-TDDFT spectrum TZV2P-GTH and aug-QZV2P-GTH basis sets, respectively, extends itself to give the full spectrum from one simulation run, and for the hybrid exchange–correlation functional PBE0 with while the PT-TDDFT spectrum is limited to a certain energy the TZV2P-GTH basis set. The resulting NR spectra at an range. excitation wavelength of 633 nm are shown in Fig. 7. In Fig. 5, the RT-TDDFTand PT-TDDFTabsorption spec- The normal mode frequencies differ slightly between tra for R-methyloxirane are shown. Here the PBE functional different exchange–correlation functionals and basis sets, was used with Ahlrichs’ TZVP basis set. Again, both methods which is especially pronounced in the case of the hybrid agree well. exchange–correlation functional PBE0, where they are shifted up to ∼50 cm−1 to higher wavenumbers. The relative and C. Non-resonant Raman spectra absolute intensities agree well across different exchange– correlation functionals and basis sets. The RT NR spectra in Non-resonance Raman (NR) spectra of R-methyloxirane Fig. 7 also compare very well to the PT-TDDFT NR spec- were calculated with the RT polarizability method in CP2K trum in Fig. 6, showing only a small difference in absolute and PT-TDDFT in Turbomole. The PT-TDDFT NR spectrum intensities. is shown in Fig. 6 for the PBE exchange–correlation functional with Ahlrichs’ TZVP basis set. RTP runs were carried

FIG. 5. Absorption spectrum of R-methyloxirane calculated with PT-TDDFT (Turbomole, solid red) and RT-TDDFT (CP2K, dashed black). The PBE FIG. 3. Difference between the excitation energy predicted by the Pade´ exchange–correlation functional was used with Ahlrichs’ TZVP basis set. approximants using the total signal and less for an RTP time step of 0.10 The damping factor Γ was set to 0.01 eV in order to have more pronounced a.u. (for details see text). peaks in the RT absorption spectrum. 174108-9 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

FIG. 6. Non-resonant Raman scattering cross section of R-methyloxirane for the TZVP basis set and PBE functional calculated with PT-TDDFT FIG. 8. Polarizability of uracil calculated with the RT polarizability method (Turbomole) at an excitation wavelength of 633 nm. using the BP86 exchange–correlation functional with Ahlrichs’ TZVP basis set. The real part is plotted in black, the imaginary part in green.

D. Resonance Raman spectra a maximum at ∼263 nm (4.71 eV) consistent with the ﬁrst excitation in the RT and PT absorption spectra (see Fig. 4). In this section, three methods are compared for the calcu- For the calculation of the RR spectrum, an excitation wave- lation of resonance Raman (RR) spectra: The RT polarizability length of 266 nm was chosen as well as a damping factor of method described in Sec. II B 1, the RT excited state gradi- 0.004 a.u. and the BP86 exchange–correlation functional with ent method described in Sec. II C, and the PT polarizability Ahlrichs’ TZVP basis set, in order to be consistent with the method described in Sec. II A 2. The RT calculations were work of Jensen et al.39 The resulting RR spectrum is shown in carried out by the authors, for the PT results it is referred 39 Fig. 9. to the work of Jensen et al. on uracil. The presentation is The polarizability and consequently the RR spectrum split into two parts: First the RT polarizability method and agree perfectly with the result of Jensen et al. (Ref. 39, Fig. the PT polarizability method are compared for uracil, then the 4). Slight differences may be accounted for, among others, RT polarizability method is compared to the RT excited state by the use of Gaussian type basis sets for the RT polarizabil- gradient method for R-methyloxirane. ity method in contrast to the Slater type basis sets used by Jensen et al. for the PT polarizability method. This agreement 1. Uracil—RTP and PT polarizability method between the two methods is expected because they are essen- The polarizability of uracil calculated with the RT polar- tially in accordance in the regime where perturbation theory izability method is shown in Fig. 8. The imaginary part shows is valid, as shown in Sec. II. Thomas et al. observed a good agreement between the two methods for ortho-nitrophenol43 as well.

FIG. 7. Non-resonant Raman scattering cross section of R-methyloxirane calculated with the RT polarizability method at an excitation energy of 633 nm. FIG. 9. RR scattering cross section of uracil calculated with the RT polariz- The phenomenological damping factor Γ was set to 0.1 eV ≈ 0.0037 a.u. ability method using the BP86 exchange–correlation functional with Ahlrichs’ The Raman peaks were broadened by a Lorentzian with a full width at half TZVP basis set. The phenomenological damping factor Γ was set to 0.004 a.u. maximum (FWHM) of 10 cm−1. The Raman peaks were broadened by a Lorentzian with a FWHM of 10 cm−1. 174108-10 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

FIG. 12. Excitation proﬁle of R-methyloxirane using PBE exchange– correlation functional and the TZV2P-GTH basis set. The resonance Raman FIG. 10. RR scattering cross section of R-methyloxirane calculated with the effect is clearly visible. The Raman peaks were broadened by a Lorentzian RT polarizability method (upper) and the RT excited state gradient method with a FWHM of 10 cm−1. (lower). The PBE exchange–correlation functional was used with the TZV2P- GTH basis set. The peaks in the RT polarizability spectrum have been broad- −1 ened by a Lorentzian with a FWHM of 10 cm . Selected normal modes are of the intensity compared to the non-resonant case (see Fig. 7) shown in the upper panel. of a factor of 105–106, which is consistent with experimental results.75 The relative RR intensities given by the RT excited state 2. R-methyloxirane—RT polarizability and RT excited gradient method match perfectly with the relative intensities state gradient method predicted by the RT polarizability method. This is expected In this section, the RT excited state gradient method, since both methods involve STAsto the excited state dynamics. which is enabled by using Pade´ approximants, is compared This correspondence was also observed by Thomas et al.43 to the RT polarizability method for R-methyloxirane. In the and Kane and Jensen54 who applied the excited state gradient following, the TZV2P-GTH basis set is used in combination method in a PT-TDDFT framework. with the PBE exchange–correlation functional and its hybrid The RR spectra computed with the PBE0 hybrid version PBE0. The excitation frequency for the calculation of exchange–correlation functional with the TZV2P-GTH basis RR was set to the ﬁrst excitation visible in the absorption spec- set are shown in Fig. 11. They are comparable to the ones trum, at 7.09 eV in the case of the PBE and at 7.93 eV in the calculated with the PBE functional (see Fig. 10), but the rel- case of the PBE0. ative intensities and the features between ∼1000 cm−1 and The resulting RR spectra together with pictures of ∼1200 cm−1 show considerable differences. As noticed in the selected normal modes are shown in Fig. 10 for the PBE NRS case, the normal mode frequencies are shifted to higher exchange–correlation functional. There is a strong increase wavenumbers.

E. Excitation proﬁle of R-methyloxirane The key advantage of the RT polarizability method is that the NR and RR spectra are calculated from one and the same set of simulations and the excitation frequency can be tuned to an arbitrary value by virtue of the Pade´ approximants. Thus the whole Raman excitation proﬁle can be obtained in one run. This is illustrated for R-methyloxirane in Fig. 12.

V. CONCLUSION We have presented efﬁcient methods to calculate (resonance) Raman spectra via real-time propagation. Besides the automatic evaluation of entire excitation proﬁles, several short time approximations to the KHD tensor for the calculation of FIG. 11. RR scattering cross section of R-methyloxirane calculated with Raman spectra were examined: The RT polarizability method, the RT polarizability method (upper) and the RT excited state gradient method (lower). The PBE0 exchange–correlation functional was used with where RTP of the time-dependent Kohn–Sham equations is the TZV2P-GTH basis set. The peaks in the RT polarizability spectrum have used to calculate the electronic polarizability, and the excited been broadened by a Lorentzian with a FWHM of 10 cm−1. state gradient method from Heller et al.38 174108-11 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018)

Using the unified language of (linear) response theory and change the description to the interaction picture since here the calculations, the RT polarizability method and the PT polar- unperturbed wave functions are used naturally as basis and the izability method from Jensen et al.39 were shown to be in time evolution is shifted to the operators. Again we closely excellent agreement in the case of a weak δ-pulse perturbation follow Ref. 55 for the presentation of the theory. as earlier results from Thomas et al.43 suggested. This has been The equation of motion (e.o.m.) in the interaction picture demonstrated for absorption and resonance Raman spectra of can be written as uracil. d i ˆ ~ ˆ ~ ρˆ (t) = − eiH0t/ Hˆ (t),ρ ˆ(t) e−iH0t/ , (A1) The great advantage of the RT method, that the whole exci- dt I ~ 1 tation frequency range is obtained in just one set of simulations, f g where is illustrated by a Raman excitation profile of R-methyloxirane. ˆ ~ ˆ ~ ρˆ (t) = eiH0t/ ρˆ(t)e−iH0t/ (A2) With the help of Pade´ approximants to the FT required in an I RTP framework, the excitation frequency can be tuned to an is the time-dependent density operator in the interaction pic- arbitrary value and the transition from non-resonant and res- ture andρ ˆ(t) = ρˆ0 +ρ ˆ1(t) is the time-dependent density oper- onance Raman spectra can be monitored in fine detail (see ator split into the unperturbed partρ ˆ0 and a time-dependent Fig. 12). partρ ˆ1(t). The convergence of Pade´ approximants dependent on the Since Hˆ 1(t) is linear in f (t) [see Eq. (20)], one can replace simulation time was evaluated for absorption spectra, and a ρˆ(t) byρ ˆ0 in linear response; i.e., only terms linear in the simulation time of at most ∼5000 fs was found to be nec- perturbation strength are considered in the e.o.m., essary for the calculation of Raman spectra at an RTP time d i ˆ ~ ˆ ~ ρˆ ≈ − eiH0t/ Hˆ ,ρ ˆ e−iH0t/ step of 0.10 a.u. Taking further advantage of Pade´ approxi- dt I ~ 1 0 mants, the excited state gradient method has been extended f g = i ˆ to an RTP framework: Pade´ approximants allow us to achieve ~ A0(t),ρ ˆ0 f (t), (A3) the necessary resolution in the frequency domain to perform f g with the derivatives of excited state energies in the Condon approx- ˆ ~ ˆ ~ Aˆ t = eiH0t/ Aeˆ −iH0t/ imation numerically, by determining the peak positions (exci- 0( ) , (A4) tation energies) in the RT absorption spectra for the displaced which corresponds to evolution of the perturbation operator in geometries. The formula found by Heller et al.38 can then be the Heisenberg picture of the unperturbed system. Using Eqs. applied directly. Subsequently this new RT excited state gra- (A2) and (A3), the time evolution of the density operator to dient method was compared to the RT-polarizability method first order in the perturbation can be expressed as for RRS spectra of R-methyloxirane and found to be in good t agreement. i ρˆ(t) = ρˆ + dt′ A (t′ − t),ρ ˆ f (t′), (A5) Additionally the NR spectra calculated with the RT polar- 0 ~ 0 0 izability method were compared with the NR spectra calcu- −∞ lated in a PT-TDDFT framework for R-methyloxirane, and a allowing us to write the expectation value in Eq. (22) as very good agreement was found. hBˆ (t)i − hBˆ i = Tr (ρ ˆ(t) − ρˆ )Bˆ In summary, RTP is a promising approach to efficiently 0 ( t ) calculate Raman spectra for off- and on resonance cases also i ′ ′ ′ = dt Bˆ , Aˆ 0(t − t) f (t ). (A6) for overlapping excited states. A newly created toolbox allows ~ 0 −∞ Df gE now automatic evaluation of whole excitation profiles as well The braket hi0 is the same as in Eq. (21), but referring to the as the application of the excited state gradient method via equilibrium density ρ and not ρ(t). Identifying the response ´ 0 RTP and the use of Pade approximants. These approaches will function from Eq. (22) with be very valuable for spectroscopic investigation of functional systems, in particular for light-driven catalysis such as water ′ i ′ ′ ΦBA(t − t ) = θ(t − t ) Bˆ 0(t), Aˆ 0(t ) (A7) ~ 0 spitting. Df gE gives the so-called Kubo formula (θ is the Heaviside function). Bˆ 0(t) is operator Bˆ propagated in time according to Eq. (A4). ACKNOWLEDGMENTS A general response function can now be written as (omitting This work has been supported by the University of Zurich, the index 0) the University Research Priority Program “Solar Light to i − ′ = ˆ ˆ ′ Chemical Energy Conversion” (LightChEC), and the Swiss KBA(t t ) ~ B(t), A(t ) National Foundation (Grant No. PP00P2 170667). Our calcu- Df gE = i ˆ ˆ ′ lations have been supported by the Swiss National Supercom- ~Tr ρˆ0 B(t), A(t ) . (A8) puting Center, Account Nos. s745 and s788. ( f g) APPENDIX B: REAL TIME PROPAGATION APPENDIX A: TIME-DEPENDENT OF THE TIME-DEPENDENT KOHN–SHAM EQUATIONS PERTURBATION THEORY Here the RTP in a TDDFT framework which was used in For the derivation of a time-dependent perturbation theory this work is briefly sketched. Analogous to ground state DFT,76 expression for the susceptibility [Eq. (23)], it is convenient to the Kohn–Sham approach77 combined with the Runge-Gross 174108-12 J. Mattiat and S. Luber J. Chem. Phys. 149, 174108 (2018) theorem78 allows us to treat the time-dependent many-body The errors induced by the RTP scheme itself are twofold:4 Schrodinger¨ equation (TDSE)79 in terms of one electron func- First, the propagator (i.e., the Hamiltonian) depends on time tions |ψi(r, t)i (orbitals), which constitute the solution for a and the electron density has to be interpolated for ∆t. It has non-interacting reference system, a Slater determinant,80 that been suggested to do this self consistently,87 but there is an reproduces the real electron density. The TDSE is then repre- ongoing debate whether the additional computational cost is sented by the following form, known as the time-dependent worth the gained precision of the calculation.88 Second, for the Kohn–Sham (TDKS) equations: reasonably sized system the matrix exponentials necessary for the propagation have to be approximated as well.89,90 Usu- ∂ i i i |ψ (r, t)i = Hˆ KS|ψ (r, t)i ally this is done by series expansions, subspace algorithms, or ∂t splitting techniques. ∆ r i = − + Vˆ eff (r, t) |ψ (r, t)i (B1) " 2 # 1R. M. Sternheimer, Phys. Rev. 96, 951 (1954). with the effective potential 2M. E. Casida, Recent Advances in Computational Chemistry (World Scientific, 1995), Vol. 1, pp. 155–192. 3 ′ ρ(r, t) J. J. Goings, P. J. Lestrange, and X. Li, Wiley Interdiscip. Rev.: Comput. Vˆ (r, t) = Vˆ (r, t) + dr + Vˆ (r, t) (B2) 8 eff ext | − ′| xc Mol. Sci. , e1341 (2017). r r 4M. R. Provorse and C. M. Isborn, Int. J. 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