Theoretical studies on the photophysical properties of luminescent pincer gold(iii) arylacetylide complexes: the role of π-conjugation at the C-deprotonated [C^N^C] ligand

The facile non-radiative decay for gold(iii) complexes is due to the thermally accessible 3LLCT, but not the usually assumed 3dd excited state.


Computational Details.
In this work, the hybrid density functional, PBE0, [3] was employed for all calculations using the program package G09. [4] The 6-31G* basis set [5] is used for all atoms except Au, which is described by the Stuttgart relativistic pseudopotential and its accompanying basis set (ECP60MWB). [6] Solvent effect was also included by means of the polarizable continuum model (PCM). [7] Geometry optimizations of the singlet ground state (S 0 ) and the lowest triplet excited state (T 1 ) were respectively carried out using restricted and unrestricted density functional theory (i.e. RDFT and UDFT) formalism without symmetry constraints. Frequency calculations were performed on the optimized structures to ensure that they are minimum energy structures by the absence of imaginary frequency (i.e. NImag = 0). Stability calculations were also performed for all the optimized structures to ensure that all the wavefunctions obtained are stable.
(a) SS-PCM energy calculations. Vertical transition energies were computed using the linear response approximation for optical absorption calculations, but the state specific approach for emissions. [8] In an absorption process, the solvent is in equilibrium solvation with the ground state electron density but non-equilibrium solvation with the excited state electron density. Thus, LR-TDDFT should give reasonable estimate of the absorption energies. However, as mentioned in the main text, SS-TDDFT is more adequate for calculations involving an emission process.
Within the state-specific (SS) approach, the equilibrium solvation of the T 1 excited state at its equilibrium geometry is written out via "NonEq=write": %oldchk=cncauccphome_pbe0_t1.chk chk file of optimized T 1 that is confirmed to be stable %chk=cncauccphome_pbe0_t1ss.chk #p pbe1pbe/chkbas geom=check guess=read scrf=(solvent=dichloromethane,read) nosymm pop=full Save solvent reaction field in equilibrium with T1 density at its optimized geometry to the checkpoint file, cncauccphome_pbe0_t1ss.chk 0 3

NonEq=read
The energy that should be extracted would appear near the end of the output file: After PCM corrections, the energy is -1265.80793189 a.u.
That is, this is the S 1 energy at the T 1 optimized geometry, with non-equilibrium solvation done in an SS approach.

NonEq=read
And the energy of the ground state from a non-equilibrium solvation in solution is: SCF Done: E(RPBE1PBE) = -1265.92332980 A.U. after 13 cycles (b) Radiative decay rate calculations. The spin wavefunctions of the triplet sub-states T 1 α are expressed along the three Cartesian coordinates, x, y, and z as: x T i and the singlet spin wavefunction as: Phosphorescence, being a spin-forbidden process, borrows its emission intensity by first-order perturbative interactions with the singlet excited states via spin-orbit coupling (SOC). Therefore, the singlet excited state energies should also be evaluated at non-equilibrium solvation with the emitting triplet excited state electronic density. Therefore, in principle, the SS approach is more appropriate than the LR approach for calculating the energy difference between the singlet and triplet excited states in eq.(9) of the main text. If the singlet excited state energies are computed within the LR-TDDFT, the singlet excited state energies are obtained either through nonequilibrium solvation with ground state electronic density or equilibrium solvation with the singlet excited state of interest (by specifying the "root" in the LR-TDDFT calculation). In either case, the energies obtained are not the solvent response to the emitting triplet excited state electronic density.
To calculate the radiative decay rate constant using the SS-TDDFT results, the metal coefficients (c jk ), the CI coefficients (a j ), and the transition dipole moments (M x , M y , M z ) of each singlet excited state (S m ) of interest, could be extracted from the output files of the SS-TDDFT calculations at the T 1 optimized geometry at the last step of the iterative procedure for each S m excited state considered. For the singlet-triplet energy gap, , it should be emphasized that this is not the energy difference for the transitions , S m  S 0 and T 1  S 0 , from TDDFT calculations (i.e., the section where the CI coefficients are extracted), but those with PCM corrections added, i.e., the energies mentioned in the previous section ("After PCM corrections…").
In addition, the singlet excited state energies may shift, depending on the electron density of the emitting T 1 state. This is particularly important if there is a large difference in dipole moments between the emitting T 1 and S m excited states in a polar medium. This means that one has to do a SS-TDDFT calculation for each singlet excited state S m . For example, if one wants to include the first ten singlet excited states in estimating the radiative decay rate constant through eq. (9), then one has to do a SS-TDDFT calculation for each singlet excited state, i.e. a total of ten SS-TDDFT jobs (with root = 1, 2, …, 10 in the "NonEq=read" step). This could be quite a lengthy task; hence, for simplicity, only the closest-lying singlet excited state(s) that could have effective SOC with the T 1 excited state were included in estimating k r as these states would dominate in the calculation of radiative decay rate constant (further details for the specific complexes would be given in a later section).
The SOC matrix elements are listed in Table S4. To calculate the Huang-Rhys factor, S j , and to simulate the emission spectrum for , a Franck-Condon calculation is done using the keyword "freq=fc". One could 〈̃〉 request the printing of the "shift vector" (which relates to ) for the computation of S j . For a ∆ more detailed theoretical background for the Franck-Condon calculation implemented in the G09 program, please consult the references cited in G09 and its document titled "Vibrationallyresolved electronic spectra in Gaussian 09".
Following our previous work, [9] we have grouped the normal modes into 5 sets: For the non-radiative decay rate calculations, k B T is assumed to be ~200 cm 1 .
For 3 IL  S 0 transition, the single-mode expression (eq.(11) of the main text) is used; however, for 3 LLCT  S 0 transition, as both aromatic CC/CN stretching and CC stretching normal modes could act as effective accepting modes, the two-mode expression is used instead: [10] ( Here, S C and n C are respectively the Huang-Rhys factor and change in the vibrational quantum number of the CC stretching normal mode (ħ C = ħ CC ~ 22002300 cm 1 ) and I nM is the modified Bessel function of order n M .         1 and 2, the lowestlying 1 LLCT excited state is S 1 and k r is calculated for m = 1 for these two complexes using eq.(9) of the main text. On the other hand, for complexes 3-exo and 3-endo, the closest-lying 1 LLCT excited state is S 2 ; hence, for these two complexes, both S 1 and S 2 are included in the k r calculations.
Similarly, for the 3 LLCT excited state of complexes 1 and 3-endo, the singlet excited state that could have appreciable SOC is 1 IL/ 1 MLCT. For complex 1, S 3 is of the character 1 IL/ 1 MLCT from the NonEq SS-TDDFT calculation, and it is found to be mainly of "H4"  L transition ("H4" in the NEQ SS-TDDFT calculation of the S 1 excited state; this orbital becomes H1 in the NonEq SS-TDDFT calculation of the S 3 excited state). But S 4 , though also a 1 LLCT and is derived from a "H2"  L transition ("H2" in the NEQ SS-TDDFT calculation of the S 1 excited state; this orbital becomes H1 in the NonEq SS-TDDFT calculation of the S 4 excited state), could also has significant SOC with the 3 LLCT excited state of complex 1 because this H-2 has the Au(d) orbital orthogonal to the HOMO, see Figure S8. Therefore, in estimating the k r of 3 LLCT  S 0 of complex 1 using eq.(9) of the main text, both m = 3 and 4 are considered to be the main states that contribute to the radiative decay rate.
MO100 (H4) MO102 (H2) MO104 (HOMO) Figure S8. MO surfaces relevant to the SOC calculation of the 3 LLCT excited state of complex 1. The relative MO orders are those obtained from a NEQ SS-TDDFT calculation of the S 1 excited state.
Likewise, for the 3 LLCT excited state of complex 3-endo, both the 1 IL/ 1 MLCT and 1 LLCT singlet excited states were considered to be the two major S m excited states contributing to the radiative decay rates. Therefore, NonEq SS-TDDFT calculations were also performed for the S 3 (mainly of 1 IL/ 1 MLCT character and is derived from H1  L and H5  L transitions) and S 4 (mainly of 1 LLCT character and is derived from "H3"  L transition; "H3" in the NEQ SS-TDDFT calculation of the S 1   cm 1 ) > 1 (vide infra), eq.(S1) would be used: (S1) where  includes contributions from both solvent modes and the low-frequency normal modes ( lf1  200 cm 1 ), i.e., Table S10. Average normal mode  A (cm 1 ) and the corresponding Huang-Rhys factor S A and reorganization energy  A (cm 1 ) of 3 IL  S 0 transition of complex 1. Is PBE0 calculation reliable in predicting the relative energies of the 3 LLCT and 3 IL excited states?
Global hybrid functionals are known to fail in zero-overlap charge transfer transition. [11] However, it has also been shown that TD-PBE0 can also provide accurate descriptions with partial charge transfer character, without resorting to range-separated hybrid density functionals. [12] To confirm that our present calculation results using the PBE0 functional are valid, we have also done SS-TDDFT calculation using the same basis set but a long-range corrected density functional, CAM-B3LYP [13] for complex 1. It was found that with the CAM-B3LYP functional at the PBE0 optimized triplet excited state geometries, the 3 LLCT excited state is only ~0.04 eV above the 3 IL excited state using the unrestricted formalism (UDFT); when SS-TDDFT was employed, the 3 LLCT excited state was found to be ~0.05 eV below the 3 IL excited state. That is, even with a long-range corrected functional, the TD-CAMB3LYP results also showed that the 3 LLCT excited state is slightly lower-lying than the 3 IL excited state. Therefore, the 3 LLCT excited state is a thermally accessible excited state that contributes to the fast non-radiative decay rate of complex 1. Table S14. Optimized S 0 structures of complex 1.