Supporting Information. Evaluation of spin-orbit couplings with. linear-response TDDFT, TDA, and TD-DFTB

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1 Supporting Information Evaluation of spin-orbit couplings with linear-response TDDFT, TDA, an TD-DFTB Xing Gao, Shuming Bai, Daniele Fazzi, Thomas Niehaus, Mario Barbatti, Walter Thiel Table of Contents SI-1. BLOCK MATRIX TRANSFORMATION FOR GAUSSIAN SI-. STO FITTING... 3 SI-3. BMT FOR ATOMIC INTEGRALS IN THE INTERFACE TO DFTB SI-4. ENERGY OF EXCITED STATES... 5 SI-5. CARTESIAN COORDINATES

2 SI-1. Block Matrix Transformation for Gaussian 09 In this section, we iscuss the block matrix transformation (BMT) for atomic integrals in the interface to Gaussian 09. In Gaussian 09, the output orer for orbitals is XX, YY, ZZ, XY, XZ, YZ; for f orbitals, it is XXX, YYY, ZZZ, XYY, XXY, XXZ, XZZ, YZZ, YYZ, XYZ. In MolSOC, the orer for orbitals is XX, XY, XZ, YY, YZ, ZZ; for f orbitals, it is XXX, XXY, XXZ, XYY, XYZ, XZZ, YYY, YYZ, YZZ, ZZZ. The transformation matrix for orbitals is T T xx T xx yy xy zz xz T T =,6 6,,6 6, C C xy yy xz yz yz zz while for f orbitals, it is T xyz 3 z T 3 3 x x T y x y z x z xy xy x y xyz T T = f,, C C f,. x z xz y xz yz y z y z yz For s an p orbitals, these matrices are ientity matrices. I Suppose we have a matrix H for atomic integrals obtaine from MolSOC, which inclues 1 s, 3 p, an 6 orbitals. This matrix can be blocke as Hss,1 1 Hsp,1 3 H s,1 6 I H = H ps,3 1 H pp,3 3 H p,3 6. Hs,6 1 Hp,6 3 H,6 6 II Applying the transformation matrix efine above, the atomic integral matrix H use for the Gaussian 09 interface can be reconstructe as

3 1s,1 1 1s,1 1 II I H = 1p,3 3 H 1p,3 3. T C,6 6 C,6 6 SI-. STO Fitting Some fits of Slater type orbitals (STOs) with Gaussian type orbitals (GTOs) are shown in Figure 1S. Fitting errors are usually on the orer of -5 ~ -7 epening on the number of noes in the wave function. l (r) O_p_STO O_p_GTO fitting error =5.71E-007 a l (r) fitting error =1.58E-005 b Zn_4s_STO Zn_4s_GTO l (r) r (a.u.) c S_3s_STO S_3s_GTO fitting error =1.67E r (a.u.) l (r) r (a.u.) S_3_STO S_3_GTO fitting error =6.65E-007 r (a.u.) Figure 1S. Raial part of the atomic wave function versus istance for orbitals: a. O p, b. Zn 4s, c. S 3s,. S 3 SI-3. BMT for atomic integrals in the interface to DFTB+ In DFTB+, the angular-epenent part of the atomic orbitals is of tesseral harmonic form S1. Therefore, the transformation matrix between spherical l, m ( { = 0, = 1, =, = 3}, m { l, l 1,..., l 1, l} l s p f { 1,,..., l 1} n = + ) can be written as, = + ) an Cartesian GTOs n, l ( = C, s s,0 s,1 1 1, s where ( c ) C, 1 s,1 1 s c s =, 4π 3

4 p, 1 1, px p,0 = C p,3 3 1, p, y p,1 1, pz 0 c p 0 3 where C p, c p, c = p c p 0 0 4π, 1, xx,, xy, 1 3, xz,0 = C,5 6, 4, yy, + 1 5, yz, + 6, zz 0 c c 0 where C,5 6 c 0 0 c 0 c, 0 0 c c 0 0 c 0 0 f, 3 f, f, 1 f,0 f,7 f, + 1 f, + f, + 3 f, x f, x y f, x z f, xy f, xyz = C, f, xz 3 f, y f, y z f, yz 3 f, z c = , c π = 4 π, 4

5 where, C f,7 0 3c f c f c f 0 c f c f 3 0 4c 3 0 f 0 0 3c f c f 4 0 c f 4, c f c f 3 0 4c f c f c f 0 0 c f c f c f 1 =, c f =, c f 3 =, c f 4 =. 3π 16π 3π 16π Now, we can apply the MBT following the same strategy as in Section S1, I Suppose we have a matrix H14 14 for atomic integrals obtaine from MolSOC, which inclues 1 s, 3 p an f orbitals. This matrix can be blocke as Hss,1 1 Hsp,1 3 H sf,1 I H14 14 = H ps,3 1 H pp,3 3 H pf,3. H fs, 1 H fp, 3 H ff, II Applying the transformation matrix efine above, the atomic integral matrix H11 11 use for the DFTB+ interface can be reconstructe as C s,1 1 C s,1 1 II I H11 11 = C p,3 3 H14 14 C p,3 3. T C f,7 C f, 7 SI-4. Energy of Excite States Table S1. Basis set effects on state energies for tthy at TDDFT/B3LYP level. E (ev) (TD-B3LYP) cc-pvdz aug-cc-pvdz cc-pvtz S 0 1 gs T 1 3 ππ* T 3 nπ* nπ* S 1 Table S. Density functional effects on state energies for tthy at TDDFT/TZVP level. B3LYP TDDFT B3LYP TDA PBE0 TDDFT E (ev) (TZVP) CAM-B3LYP ωb97xd TDDFT TDDFT ωb97xd TDA M06X TDDFT PBE TDDFT TD-DFTB 5

6 S T T S Table S3. Vertical excitation energies an ominant excitations for formalehye an acetone at TDDFT/TZVP/(CAM-)B3LYP an TD-DFTB levels. B3LYP CAM-B3LYP TD-DFTB E (ev) ominant excitation E (ev) ominant excitation E (ev) ominant excitation formalehye S n H π * 4.00 n H π * L(0.99) 4.50 n H π * T n H π * 3.6 n H π * L(0.99) 4.50 n H π * T acetone S n H π * L(0.99) 4.5 n H π * L(0.98) 4.7 n H π * T n H π * L(0.99) 3.88 n H π * L(0.98) 4.71 n H π * T 5.73 L(0.94) 5.69 L(0.91) 6.79 Table S4. State energies an ominant excitations for thymine (Thy), 4-thiothymine (4tThy), an,4-thiothymine (,4tThy) from TD(A)DFT/TZVP (B3LYP, ωb97xd, an PBE), TD-DFTB, an CASPT. 1 (n π*) S 1 or S 3 (π π*) T 3 1 (n π*) T E (ev) ominant excitation E (ev) ominant excitation E (ev) ominant excitation Thy TD-B3LYP 3.93 n H-1 π * L(0.96) n H-1 π * L(0.93) TD-ωB97XD 4.44 n H-1 π * L(0.90) n H-1 π * L(0.84) TD-PBE 3.1 n H-1 π * n H-1 π * TDA-B3LYP 3.94 n H-1 π * L(0.97).16 L(0.98) 3.63 n H-1 π * L(0.94) TD-DFTB.93 n H-1 π * L(0.87).64 L(0.94).93 n H-1 π * L(0.94) CASPT 4.0 n H- π * L(0.58) n H-1 π * L(0.15).56 L(0.76) 4.17 n H- π * L(0.41) n H-1 π * L(0.0) 4tThy TD-B3LYP.51 n H π * L(0.99).11 L(0.98).0 n H π * TD-ωB97XD.79 n H π * L(0.96).08 L(0.95).49 n H π * TD-PBE.4 n H π * 1.95 n H π *. TDA-B3LYP.5 n H π * L(0.99). n H π * L(0.98).9 L(0.98) TD-DFTB 1.91 n H π * 1.91 n H π *. CASPT.7 n H π * L(0.89).67 L(0.8).70 n H π * L(0.81),4tThy TD-B3LYP.48 n H π * L(0.86) n H- π * L(0.13).09 π H-3 π * L(0.89).17 n H π * L(0.78) n H- π * L(0.19) TD-ωB97XD.76 n H π * L(0.7) n H- π * L(0.4).05 π H-3 π * L(0.75) L(0.5).45 n H π * L(0.69) n H- π * L(0.6) TD-PBE.13 n H π * L(0.95) 1.91 n H π * L(0.97).0 π H-3 π * L(0.84) TDA-B3LYP.49 n H π * L(0.86) n H- π * L(0.13).19 n H π * L(0.78) n H- π * L(0.19).8 π H-3 π * L(0.85) L(0.13) TD-DFTB 1.83 n H π * 1.83 n H π *.0(T 3 ) π H-3 π * L(0.89) CASPT.64 n H π * L(0.49) n H- π * L(0.37).60 L(0.81).6 n H π * L(0.46) n H- π * L(0.36) 6

7 Table S5. Vertical excitation energies an ominant excitations for psoralens from TDDFT/TZVP/B3LYP, ωb97xd, TD-DFTB. B3LYP ωb97xd TD-DFTB E ominant E ominant E ominant (ev) excitations (ev) excitations (ev) excitations psoralenoo S L(0.9) 4.30 L(0.89) 3.09 n H-1 π * S 4.40 L(0.85) 4.78 L(0.78) 3.41 L(0.99) L+1(0.17) S n H- π * L(0.95) 4.96 n H- π * L(0.80) 4.04 π H- π * L(0.99) T 1.85 L(0.6).85 L(0.64).96 L(0.6) L+1(0.14) L(0.13) T 3.04 L(0.6) 3.3 L(0.61) 3.09 n H-1 π * L+1(0.17) L+1(0.1) L(0.13) T L+1(0.65) 4.03 L+1(0.44) 3.34 π H- π * L(0.17) L(0.19) L(0.14) T n H- π * L(0.90) 4.44 L+1(0.9) 3.99 L+1(1.00) π H-4 π * L(0.15) L+(0.14) T L+1(0.61) 4.54 n H- π * L(0.75) 4.19 n H-1 π * L+1 n H- π * L+(0.16) (1.00) psoralenos S L(0.95) 4.04 L(0.90).79 n H π * S 3.91 n H- π * L(0.95) 4.1 n H- π * L(0.8) 3.05 S L(0.75) 4.57 L(0.70) 3.63 π H- π * L(0.88) L+1(0.19) L+1(0.3) (S 4 ) T 1.76 L(0.48).75 L(0.65).73 L(0.4) L+1(0.14) L(0.) T.90 L(0.50) 3.15 L(0.6).79 n H π * L(0.4) L+1(0.1) L+1(0.18) T L+1(0.65) 3.64 L+1(0.4) 3.17 π H- π * L(0.19) L(0.19) L(0.18) T n H- π * L(0.91) n H- π * L(0.78) L+1 (1.00) T L+1(0.77) 4.34 L+1(0.43) 3.54 n H π * L+1(1.00) psoralenso S L(0.96) 4. L(0.66).96 n H-1 π * L(0.0) S 4.11 L(0.8) 4.5 L(0.6) 3.13 L+1(0.13) L(0.1) S n H- π * L(0.95) 4.93 n H- π * L(0.80) 3.57 π H- π * L(0.93) T 1.77 L(0.66).76 L(0.59).84 L+1(0.1) L+1(0.15) L(0.) T.99 L(0.66) 3.1 L(0.38).96 n H-1 π * L(0.0) L+1(0.7) L(0.) T L+1(0.70) 3.7 L+1(0.43) 3.05 π H- π * L(0.18) L(0.37) T L+1(0.89) 4.37 L+1(0.76) 3.41 L+1(1.00) T n H- π * L(0.90) 4.51 n H- π * L(0.75) 3.61 n H-1 π * L+1 (T 6 ) (T 6 ) (1.00) 7

8 Table S6. Vertical excitation energies an ominant excitations for BODIPY at TDDFT/TZVP/(CAM-)B3LYP level. B3LYP CAM-B3LYP E (ev) ominant excitations E (ev) ominant excitations BODIPY S 1.58 L(0.99).69 L(0.98) T T.70 L(0.87).83 L(0.54) L+1(0.17) SI-5. Cartesian coorinates Cartesian coorinates in Angstrom, for all molecules investigate in this work. formalehye C O H H acetone C O C H H H C H H H thiothymine N C N C C C S O C H H

9 H H H H thymine C C C C H O O C H H H N H N H thiothymine N C N C C C O S C H H H H H H ,4-thiothymine N C N C

10 C C S S C H H H H H H psoralenoo O C O C C C C C C C C O C C H H H H H H psoralenos O C C C C C C C C

11 C O C C H H H H H H S psoralenso O C O C C C C C C C C C C H H H H H H S BODIPY B F N H C F N C H

12 N C C C C C H C H C H C H C C C H C C C H C H H H H Reference [S1] Görrler-Walran, C.; Binnemans, K. Rationalization of Crystal-Fiel Parametrization, In: Hanbook on the Physics an Chemistry of Rare Earths Vol. 3, es. Gschneiner, K. A. an Eyring, L. North Hollan,

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