Supporting information for: Simulating Third-Order Nonlinear Optical. Properties Using Damped Cubic Response. Theory within Time-Dependent Density

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1 Supporting information for: Simulating Third-Order Nonlinear Optical Properties Using Damped Cubic Response Theory within Time-Dependent Density Functional Theory Zhongwei Hu, Jochen Autschbach, and Lasse Jensen, Department of Chemistry, The Pennsylvania State University, 104 Chemistry Building, University Park, Pennsylvania , USA Department of Chemistry, University at Buffalo, State University of New York, Buffalo, New York , USA S1

2 1 Derivation of the relationship between ε αβ and ε αβ According to the damped second-order CPKS equation, the damped second-order Lagrangian multiplier matrices with arguments (+ω 1,+ω 2 ) and ( ω 1, ω 2 ) are given as ε αβ (+ω 1,+ω 2 ) = G αβ (+ω 1,+ω 2 ) +G α (+ω 1 )U β (+ω 2 )+G β (+ω 2 )U α +ω 1 ) U α (+ω 1 )ε β (+ω 2 ) U β (+ω 2 )ε α (+ω 1 ) (1) +ε 0 U αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 )ε 0 +(ω 1 +ω 2 +iγ)u αβ (+ω 1,+ω 2 ) and ε αβ ( ω 1, ω 2 ) = G αβ ( ω 1, ω 2 ) +G α ( ω 1 )U β ( ω 2 )+G β ( ω 2 )U α ( ω 1 ) U α ( ω 1 )ε β ( ω 2 ) U β ( ω 2 )ε α ( ω 1 ) (2) +ε 0 U αβ ( ω 1, ω 2 ) U αβ ( ω 1, ω 2 )ε 0 (ω 1 +ω 2 +iγ)u αβ ( ω 1, ω 2 ) respectively. Here we note that the expression for the negative argument has already transformed +iγ to iγ by construction. Therefore, taking the adjoint of Eq. (2) leads to the following equation that maintains the sign of iγ ε αβ ( ω 1, ω 2 ) = G αβ ( ω 1, ω 2 ) +U β ( ω 2 )G α ( ω 1 )+U α ( ω 1 )G β ( ω 2 ) ε β ( ω 2 )U α ( ω 1 ) ε α ( ω 1 )U β ( ω 2 ) (3) +U αβ ( ω 1, ω 2 )ε 0 ε 0 U αβ ( ω 1, ω 2 ) (ω 1 +ω 2 +iγ)u αβ ( ω 1, ω 2 ). S2

3 Using the facts ε 0 = ε 0, U a ( ω) = U a (+ω), G a ( ω) = G a (+ω), ε a ( ω) = ε a (+ω) and G ab ( ω 1, ω 2 ) = G ab (+ω 1,+ω 2 ), and substituting the following equations ε a (+ω) = G a (+ω)+ε 0 U a (+ω) U a (+ω)ε 0 +(ω +iγ)u a (+ω) G a (+ω) = ε a (+ω) ε 0 U a (+ω)+u a (+ω)ε 0 (ω +iγ)u a (+ω) (4) into Eq. (3), one can obtain ε αβ ( ω 1, ω 2 ) = G αβ (+ω 1,+ω 2 ) +G α (+ω 1 )U β (+ω 2 )+G β (+ω 2 )U α (+ω 1 ) U α (+ω 1 )ε β (+ω 2 ) U β (+ω 2 )ε α (+ω 1 ) [ ] U α (+ω 1 )U β (+ω 2 )+U β (+ω 2 )U α (+ω 1 ) ε 0 ] +ε [U 0 α (+ω 1 )U β (+ω 2 )+U β (+ω 2 )U α (+ω 1 ) [ ] +(ω 1 +ω 2 +2iΓ) U α (+ω 1 )U β (+ω 2 )+U β (+ω 2 )U α (+ω 1 ) (5) +U αβ ( ω 1, ω 2 )ε 0 ε 0 U αβ ( ω 1, ω 2 ) (ω 1 +ω 2 +iγ)u αβ ( ω 1, ω 2 ). This can be rewritten using W αβ (+ω 1,+ω 2 ) = U α (+ω 1 )U β (+ω 2 )+U β (+ω 2 )U α (+ω 1 ) as ε αβ ( ω 1, ω 2 ) = G αβ (+ω 1,+ω 2 ) +G α (+ω 1 )U β (+ω 2 )+G β (+ω 2 )U α (+ω 1 ) U α (+ω 1 )ε β (+ω 2 ) U β (+ω 2 )ε α (+ω 1 ) W αβ (+ω 1,+ω 2 )ε 0 +ε 0 W αβ (+ω 1,+ω 2 ) +(ω 1 +ω 2 +2iΓ)W αβ (+ω 1,+ω 2 ) [ ] + W αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 ) ε 0 ] ε [W 0 αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 ) [ ] (ω 1 +ω 2 +iγ) W αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 ), (6) S3

4 where the fact U αβ ( ω 1, ω 2 ) = W αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 ) is also adopted. Eq. (6) can be further simplified as ε αβ ( ω 1, ω 2 ) = G αβ (+ω 1,+ω 2 ) +G α (+ω 1 )U β (+ω 2 )+G β (+ω 2 )U α (+ω 1 ) U α (+ω 1 )ε b (+ω 2 ) U β (+ω 2 )ε α (+ω 1 ) +ε 0 U αβ (+ω 1,+ω 2 ) U αβ (+ω 1,+ω 2 )ε 0 (7) +(ω 1 +ω 2 +iγ)u αβ (+ω 1,+ω 2 ) +iγw αβ (+ω 1,+ω 2 ), which is equivalent to ε αβ ( ω 1, ω 2 ) = ε αβ (+ω 1,+ω 2 )+iγw αβ (+ω 1,+ω 2 ) (8) by substituting Eq. (1) into Eq. (7). S4

5 2 Derivation of the reduced IDRI second hyperpolarizability The full expression of IDRI can be obtained by substituting ω 1 = ω, ω 2 = ω, and ω 3 = ω into the general SOS equation for the second hyperpolarizability. To obtain the reduced form of it (γ TPA ) when ω is far from any one-photon resonances, we start by eliminating all pure one-photon terms in the full expression. This leads to a total of eight two-photon terms, where the dominant ones are given as 1 3 m 0 { n 0 p 0 0 µ α m m µ δ n n µ γ p p µ β 0 (ω m0 ω iγ)(ω n0 2ω iγ)(ω p0 ω iγ) 0 µ α m m µ + δ n n µ β p p µ γ 0 (ω m0 ω iγ)(ω n0 2ω iγ)(ω p0 ω iγ) 0 µ δ m m µ + α n n µ γ p p µ β 0 (ω m0 ω +iγ)(ω n0 2ω iγ)(ω p0 ω iγ) } 0 µ δ m m µ + α n n µ β p p µ γ 0, (ω m0 ω +iγ)(ω n0 2ω iγ)(ω p0 ω iγ) (9) equivalent to 1 3 n { S αδ 0n ( ω +iγ,ω iγ ) S βγ 0n ω n0 2ω iγ The expression above can be further rewritten as ( ω +iγ,ω +iγ ) }. (10) 1 3 n { S αδ 0n + ( ) ( ) ω iγ,ω iγ S βγ 0n ω +iγ,ω +iγ ω n0 2ω iγ [ S αδ 0n ( ω +iγ,ω iγ ) S αδ 0n ( ω iγ,ω iγ ) ] S βγ 0n ω n0 2ω iγ ( ω +iγ,ω +iγ ) }, (11) where the first part corresponds to the reduced IDRI second hyperpolarizability (γ TPA ), and the second part that represents a two-photon interference term, equivalent to 1 3 n { m [ 0 µ α m m µ δ n ω m0 (ω +iγ) 0 µα m m µ δ n ω m0 (ω iγ) S5 ] Sβγ 0n ( ω +iγ,ω +iγ ) ω n0 2ω iγ }, (12)

6 was also eliminated as being found contributing insignificantly to the two-photon poles when ω is far from any one-photon resonances. S6

7 3 Comparison between the SZ and STO-3G basis sets for the response properties of LiH at the TDDFT level of theory In Table S1, we represent the static α, β and γ values calculated at the TDDFT level of theory. All values in Table S1 are given in a.u. and only non-duplicate ones are shown based on Kleinman symmetry. Note that all static values were obtained without the lifetime, i.e., Γ = 0 a.u.. Table S1: Simulated static α, β and γ for LiH at the TDDFT level ADF Response Dalton Response Dalton SOS Property α xx (0;0) α zz (0;0) β zxx (0;0,0) β zzz (0;0,0) γ xxxx (0;0,0,0) γ xxyy (0;0,0,0) γ xxzz (0;0,0,0) γ zzzz (0;0,0,0) Within the TDDFT framework, all the properties calculated using response theory through ADF are in good agreement with those calculated using both response theory and the SOS approach through Dalton. This is an indication that, though identical minimal basis sets are not available, adopting SZ basis set for ADF while STO-3G basis set for Dalton is a reasonable approach for comparison purpose. Despite the same basis set (STO-3G) is ensured, discrepancies between those Dalton response and Dalton SOS results can still be seen for NLO properties at the TDDFT level of theory. S7

8 4 Comparison between response theory and the SOS approach for the response properties of LiH at the FCI level of theory In Table S2, we represent the static α, β and γ values calculated at the FCI level of theory. Including the damping factor, in Table S3 we also represent the resonant α and β values calculated at the FCI level of theory. All values in Tables S2 and S3 are given in a.u. and only non-duplicate ones are shown based on Kleinman symmetry. Note that all static values were obtained without the lifetime, i.e., Γ = 0 a.u., whereas the resonant α and β values were obtained using Γ = a.u at frequency equal to and a.u., respectively. The resonant γ values are not provided here because the FCI response theory approach to damped γ is currently not available in Dalton. Table S2: Simulated static α, β and γ for LiH at the FCI level Dalton Response Dalton SOS Property α xx (0;0) α zz (0;0) β zxx (0;0,0) β zzz (0;0,0) γ xxxx (0;0,0,0) γ xxyy (0;0,0,0) γ xxzz (0;0,0,0) γ zzzz (0;0,0,0) Table S3: Simulated resonant α and β for LiH at the FCI level Dalton Response Dalton SOS Property α xx ( ω;ω) α zz ( ω;ω) β xxz ( 2ω;ω,ω) β zxx ( 2ω;ω,ω) β zzz ( 2ω;ω,ω) S8

9 Within the FCI framework, all the properties calculated using response theory are nearly identical to those calculated using the SOS approach. This is expected for such an exact wavefunction method when the same basis (STO-3G) is ensured. S9

10 5 Comparison between response theory and the SOS approachfor thenlo processesoflih at thetddft level of theory EOPE: βeope( ω;ω,0) ( 10 4 a.u.) one photon pole βeope( ω;ω,0) ( 10 4 a.u.) one photon pole 6 6 (a) TDDFT Response (b) TDDFT SOS Figure S1: Simulated EOPE spectra for LiH using (a) response theory and (b) the SOS approach at the TDDFT level of theory. The vertical - - line indicates the one photon resonance due to the excited state. OR: S10

11 βor(0;ω, ω) ( 10 4 a.u.) one photon pole βor(0;ω, ω) ( 10 4 a.u.) one photon pole 6 6 (a) TDDFT Response (b) TDDFT SOS FigureS2: SimulatedORspectraforLiHusing(a)responsetheoryand(b)theSOSapproach at the TDDFT level of theory. The vertical - - line indicates the one photon resonance due to the excited state. EFIOR: γefior(0;ω, ω,0) ( 10 7 a.u.) one photon pole γefior(0;ω, ω,0) ( 10 7 a.u.) one photon pole (a) TDDFT Response (b) TDDFT SOS Figure S3: Simulated EFIOR spectra for LiH using (a) response theory and (b) the SOS approach at the TDDFT level of theory. The vertical - - line indicates the one photon resonance due to the excited state. EFISHG: S11

12 γefishg( 2ω;ω,ω,0) ( 10 6 a.u.) one photonpole two photonpole γefishg( 2ω;ω,ω,0) ( 10 6 a.u.) one photonpole two photonpole (a) TDDFT Response (b) TDDFT SOS Figure S4: Simulated EFISHG spectra for LiH using (a) response theory and (b) the SOS approach at the TDDFT level of theory. The vertical - - and -. lines indicate the oneand two-photon resonances due to the excited states, respectively. THG: For thepair ofbothβ OR andγ OKE spectra, we notethat the zero imaginary part is due to the orientational averaging rather than all zero tensor components. For the pair of γ THG spectra, the magnitude and shape of the second three-photon poles, as well as the first two-photon poles, slightly differ from each other. This is because the band extrema distance (the one within 1.5 to 2.0 ev) in Figure S5(b) is smaller than that in Figure S5(a), which causes a more pronounced doubly resonant effect that contributed by the overlap between two- and three-photon resonances. S12

13 γthg( 3ω;ω,ω,ω) ( 10 6 a.u.) one photon pole two photon pole three photon pole (a) TDDFT Response one photon pole two photon pole three photon pole (b) TDDFT SOS Figure S5: Simulated THG spectra for LiH using (a) response theory and (b) the SOS approach at the TDDFT level of theory. The vertical - -, -., and.. lines indicate the one-, two- and three-photon resonances due to the excited states, respectively. S13

14 6 Derivation of the spectral representation for the first hyperpolarizability We start with the first order transformation matrix (U) that can be obtained by solving the follow set of linear equations Uα (ω) U α ( ω) = A B B A ω V α V α, (13) wherev α isthedipolematrixinthemobasisandthespectral representationoftheresponse matrix reads ω B A X n n + ω n n ω Y N Y n 1 = 1 ω n +ω Y n X n Y n X n. (14) Then, the elementary form of the transition dipole moment can be written as µ α 0n,ia = V α ia (X n,ia +Y n,ia ) (15) and µ α n0,ia = V α ia(x n,ia +Y n,ia), (16) where i,j,... occupied (occ) orbitals, a,b,... virtual (virt) orbitals, and the Cartesian component is represented as α, β, γ... to avoid duplicate indices. This leads to the elementary form for the spectral representation of U as U α ia(+ω) = n X n,ia µ α n0,ia ω n ω + Y n,ia µα 0n,ia ω n +ω (17) S14

15 and U α ia ( ω) = n Y n,ia µ α n0,ia ω n ω + X n,ia µα 0n,ia ω n +ω. (18) Recall that the trace (Tr) operation in the 2n+1 expression for β can be rewritten as the sums over both occ and all (occ + virt) orbitals, S1 i.e., occ all Tr[nU α G β U γ ] = n UikG α β kl Uγ li (19) i kl and occ all Tr[nU α U γ G β ] = n Uik α Uγ kl Gβ li. (20) i kl Notice that Ref. S1 uses Tr[nU α U γ ε β ] in Eq. (20) but it is valid to replace ε with G here due to the zero diagonal block of U. Using the elementary form of U matrices and the first-order KS matrix G α ia (±ω) = Hα ia + j,b K HXC ia,jb Uα jb (±ω)+khxc ia,bj Uα bj ( ω), (21) Equations (19) and (20) can be rewritten as occ all n Uik( ω α σ )G β kl (+ω 1)U γ li (+ω occ all { 2) = n i [ m j,b kl X m,ik µ α m0,ik + Y ][ m,ik µα 0m,ik H β kl ω m +ω σ ω m ω + σ j,b i kl K HXC kl,jb ( Y Kkl,bj HXC n,jb µ β n0,jb + X n,jb µβ )][ 0n,jb ω n n ω 1 ω n +ω 1 p ( n X p,li µ γ p0,li X n,jb µ β n0,jb + Y n,jb µβ ) 0n,jb + ω n ω 1 ω n +ω 1 ω p ω 2 + Y p,li µγ 0p,li ω p +ω 2 ]} (22) S15

16 and occ all n Uik α ( ω σ)u γ kl (+ω 1)G β li (+ω occ all { 2) = n i [ m ( n kl X m,ik µ α m0,ik + Y ][ m,ik µα 0m,ik ω m +ω σ ω m ω σ X n,jb µ β n0,jb + Y n,jb µβ ) 0n,jb + ω n ω 2 ω n +ω 2 j,b p i kl X p,kl µ γ p0,kl ω p ω 1 + Y p,kl µγ 0p,kl ω p +ω 1 ][ H β li + j,b K HXC li,jb ( Y Kli,bj HXC n,jb µ β n0,jb + X n,jb µβ )]} 0n,jb, ω n n ω 2 ω n +ω 2 (23) respectively. We focus on the ALDA in response theory by extracting the terms containing thecouplingmatrixk HXC fromthe2n+1expressionofβ, inwhichtheone-photondominant ones are found as n p occ virt all exc. no. ij b kl mnp 0 [ Y m,ik µ α 0m,ik ω m ω σ K HXC kl,jb Xn,jbµ β n0,jb ω n ω 1 Xp,liµ γ p0,li ω p ω 2 Y m,ik µα 0m,ik ω m ω σ Xp,klµ γ p0,kl ω p ω 1 K HXC li,jb Xn,jbµ β n0,jb ω n ω 2 + Y m,ik µα 0m,ik ω m ω σ K HXC kl,bj Yn,jbµ β n0,jb ω n ω 1 Xp,liµ γ p0,li ω p ω 2 Y m,ik µα 0m,ik ω m ω σ Xp,klµ γ p0,kl ω p ω 1 K HXC li,bj Yn,jbµ β n0,jb ω n ω 2 ], (24) For the expression above, X n and Y n are the spectral representation of the response vectors obtained from the linear response equations, and p represents a summation over corresponding terms obtained by permuting ( 2ω,α), (+ω,β), and (+ω,γ). If one considers the S16

17 SHG case and takes the finite lifetime of the excited state into account, Eq. (24) becomes n p occ virt ij all b kl exc. no. mnp 0 [ Y m,ik µ α 0m,ik KHXC kl,jb X n,jb µ β n0,jb X p,liµ γ p0,li (ω m 2ω iγ)(ω n ω iγ)(ω p ω iγ) Y m,ik µα 0m,ik X p,klµ γ p0,kl KHXC li,jb X n,jb µ β n0,jb (ω m 2ω iγ)(ω p ω iγ)(ω n ω iγ) + Y m,ik µα 0m,ik KHXC kl,bj Y n,jb µ β n0,jb X p,liµ γ p0,li (ω m 2ω iγ)(ω n ω iγ)(ω p ω iγ) Y m,ik µα 0m,ik X p,klµ γ p0,kl KHXC li,bj Y n,jb µ β n0,jb (ω m 2ω iγ)(ω p ω iγ)(ω n ω iγ) ]. (25) Notice that, in addition to the part p Uα ( ω σ )[G β (+ω 1 ),U γ (+ω 2 )] being addressed above, the 2n+1 expression of β has another part that contains the second-order xc kernel, i.e., Tr[g ALDA xc D α ( ω σ )D β (+ω 1 )D γ (+ω 2 )]. (26) This term is also adiabatically approximated and potentially has a significant contribution to β in the vicinity of one-photon poles. However, as shown in Figure S6, we find it barely affects the shape or intensity of the one-photon structure in the SHG spectrum, and thus was not included within the derivation above. References (S1) Karna, S. P.; Dupuis, M. J. Comput. Chem. 1991, 12, S17

18 βshg( 2ω;ω,ω) ( 10 4 a.u.) No g xc No g xc Figure S6: Simulated SHG spectrum for LiH using response theory at the TDDFT level of theory. The dashed lines indicate the real and imaginary SHG values that were calculated by setting Eq. (26) equal to zero, respectively S18