Evaluating excited state atomic polarizabilities of chromophores

Ground and excited state atomic polarizabilities of the chromophores N-methyl-6-oxyquinolinium betaine and coumarin 153 have been evaluated via quantum mechanics.

between two coordinate sets v (6-31G(d)) and w (Sadlej) to describe the induced changes, Table S1. Table S1 furthermore lists the three largest deviations in bond length and angle for all four systems. No significant difference could be found. The corresponding coordinates are given in the following. Table S1: RMSD of MQ, as well as three largest deviations in bond lengths, ∆b, and angles, ∆a, between geometry optimizations using a 6-31G(d), or Sadlej's polarizable pVTZ basis set.

Definition of bond charges and the resulting dipole moment
The net charges on atomic sites i of a neutral molecule can be distributed among the bonds connected to the site so that where q b(ij) are the directed contributions to the charge q i . Since where the first four rows arise from Eqn (2) and (3), and the last from the constraint that bond charges within a ring add up to 0. Reformulating Eqn (4) and using the respective This set of equations can be solved, yielding for our minimal working example Figure S2: Bond charges of an artificial molecule as also depicted in Fig. S2. The dipole (charge transfer contribution) with respect to the atomic site is then where R i is the nuclear position, and R b(ij) the bond critical point, or here simply Summing up all contributions leads to µ c = i µ ic = 0 0.6 . This result can also be obtained via the traditional definition of the dipole moment µ c = i q i R i = 0 0.6 . Calculating µ ic at different external fields and numerically differentiating finally yields α ic .

S7
3 Romberg differentiation procedure The Romberg differentiation procedure is based on the recursive formula with k=1,2 ..., p the number of Romberg iteration and P the current differential. Before the first iteration (for p=0), the differential is evaluated directly for multiple values of k.
We applied an electric field of magnitude 0.0008, 0.0016, 0.0032, 0.0064 and 0.0128 au in each of the positive and negative x, y and z directions and obtained the resultant energy (ie 33 separate calculations) For example, in G09 to apply an electric field in the x direction of 0.0008 au include the keywords density=current and field=x+8. Then for a second derivative with k=1 and h=0.0004 au This result is iteratively improved by usage of the recursive formula, Eqn (13).

Python script
A python script was developed to read the output of GDMA (multipole rank 1) and transform the atomic dipoles and charges to atomic and molecular polarizabilities. The script is reported at the end of this section and can also be downloaded from http://www.mdy.univie.ac.at/python-stuff/atomic polarizabilities.py. It requires python3, as well as the packages numpy, re and sys. To use the script, six output files are needed, where an electric field of the same strength was applied in the positive and negative x, y and z direction respectively. Furthermore, the script requires an input file, where the connectivity of the atoms, as well as optional rings can be specified. For the minimal working example from Section 2, the input file reads as: bond 1 2 bond 2 3 bond 3 4 bond 4 2 ring 2 3 4 There is no order of indices that needs to be taken care of ('bond 1 2' is equivalent to 'bond 2 1'). The atom numbering must be the same as in GDMA (which is the same atom numbering as in Gaussian09). Then, execute $ python3 atomic_polarizabilities.py

S8
The program asks where the six output files and the connectivity input files are located. Per default, 'x.out' is the GDMA output for field in positive x direction, 'mx.out' for a field in negative x direction, and so on, all located in the subdirectory 'out'. The connectivity file is per default named as 'connec.inp'. Other directories and names can be specified. The program now parses through the files, and lists the coordinates, charges and dipole moments of each atom (atom names and order as in GDMA output). Then, the field strength needs to be specified. For example, when ±0.0008 au were applied (Gaussian option 'field=x+8'), enter 0.0008. Furthermore the script asks whether atomic polarizabilities should be given in [au]   ") ) answer = input (" Do you want to change the default units ( au .) of the output to angstrom^3? yes / no ") if answer ==" yes ": unit = 0 .5 29 17 7 24 9* * 3 # transform au^3 into Â 3 else : if answer ==" no ": unit =1 else : print ( In implicit solvent models the solute is placed into a cavity, surrounded by an continuous dielectric medium mimicking the solvent. Depending on the cavity radius, the charges on the cavity surface move further away from the solute, thus weakening the interaction between solute and solvent. A infinitely large cavity corresponds thus to a vacuum calculation. The dipole moments and polarizabilities of the solute vary largely with the scaling factor determining the difference between van der Waals radii and the actual cavity volume, Table S2. 9.7 9.3 7.9 6.9 α S0 30.8 29.8 24.9 21.9 α S1 31.7 29.9 23.7 20.4 The default scaling factor, 1.1, was used for all calculations with PCM solvents, as well as 1.0 for SMD solvents. S14 6 Atomic polarizabilities of coumarin 153 The atomic polarizability of C153 from charge transfer and polarization is reported in Table S3.