Excitonic Configuration Interaction: Going Beyond the Frenkel Exciton Model

We present the excitonic configuration interaction (ECI) method—a fragment-based analogue of the CI method for electronic structure calculations of multichromophoric systems. It can also be viewed as a generalization of the exciton approach, with the following properties: (i) It constructs the effective Hamiltonian exclusively from monomer calculations. (ii) It employs the strong orthogonality assumption and is exact within McWeeny’s group function theory, thus requiring only one-electron density matrices of the monomer states. (iii) It is agnostic of the monomer electronic structure method, allowing us to use/combine different methods. (iv) It includes an embedding point charge scheme (called excitonic Hartree–Fock, EHF) to improve the accuracy of the monomer states, but such that the effective full-system Hamiltonian is not explicitly dependent on the embedding. (v) It is systematically improvable, by expanding the set of monomer states and by including configurations where two or more monomers are excited (defining the ECIS, ECISD, etc., methods). The performance of ECI is assessed by computing the absorption spectrum of two exemplary multichromophoric systems, using CIS as the monomer electronic structure method. The accuracy of ECI significantly depends on the chosen embedding charges and the ECI expansion. The most accurate assessed combinations—ECIS or ECISD with EHF embedding—yield spectra that agree qualitatively and quantitatively with full-system direct calculations, with deviations of the excitation energies below 0.1 eV. We also show that ECISD based on CIS monomer calculations can predict states where two monomers are excited simultaneously (e.g., triplet–triplet double-local excitations) that are inaccessible in a full-system CIS calculation.

In this section, we summarize the full ECI Hamiltonian for the readers.The entire ECI Hamiltonian is given by Equation ( 20

+ adjoint. (S1)
We note that this ECI Hamiltonian looks the same regardless of the employed embedding, except for the modifications in Equation ( 18).In the case of EHF embedding, some terms will tend towards zero, e.g., ĤGS−LE due to the excitonic version of Brillouin's theorem. With It contains not only the sum of site state energies, but also an interaction term for each pair of fragments.
The ĤLE−LE term is conveniently divided according to the number of the differing site states (noted in the superscript) as: The Ĥ(0) LE−LE gives rise to the diagonal matrix elements of all LE products, Ĥ(1) LE−LE forms the coupling between two LEs on the same fragment, while Ĥ(2) LE−LE gives the couplings between LEs on two different sites.
The DLE-DLE term is also conveniently separated in the same way as The Ĥ(0) On the contrary, the ĤLE−DLE term covers the coupling between ESDs differing in one or two site states and hence can be separated again as: For comparison, the Frenkel Hamiltonian can be written in this notation as which is equivalent to equation ( 11) in the main text.The fields are gray-white shaded to ease the distinguishing between different site states.The matrices of the density-matrix coefficients that are read directly from Gaussian's fchk file are coloured blue, the ones that are calculated by the Equation (51) are green, while those that are calculated from the CI and MO coefficients of the site states (read from Gaussian's rwf and fchk file, respectively) are coloured red.The S is the AO-overlap matrix.The density-matrix coefficients for the pairs of states in the lower triangle of the figure are obtained by transposing the matrices of the density-matrix coefficients from the upper triangle.While transposing, the spin labels also swap positions, e.g.D αβ (T 1 2 , S 1 ) = D βα (S 1 , T 1 2 ) T .Note that the figure can trivially be generalized from S 1 , S 2 , T 1 , and T 2 to any pair of excited singlet/triplet site states.

Figure S1 :
Figure S1: Molecular orbitals of guanine on FC geometry of G 4 needed to descripe guanine site states in both G 4 and MgG 2+4 systems with all three embedding schemes.

Figure S2 :
FigureS2: Schematic of how the density-matrix coefficients for different (pairs of) site states are obtained.For each row-column combination, a rhomb in the middle of the square represents the total density ( D).The four corners then correspond to D αα (upper left), D αβ (upper right), D βα (lower left) and D ββ (lower right).The fields are gray-white shaded to ease the distinguishing between different site states.The matrices of the density-matrix coefficients that are read directly from Gaussian's fchk file are coloured blue, the ones that are calculated by the Equation (51) are green, while those that are calculated from the CI and MO coefficients of the site states (read from Gaussian's rwf and fchk file, respectively) are coloured red.The S is the AO-overlap matrix.The density-matrix coefficients for the pairs of states in the lower triangle of the figure are obtained by transposing the matrices of the density-matrix coefficients from the upper triangle.While transposing, the spin labels also swap positions, e.g.D αβ (T 1 2 , S 1 ) = D βα (S 1 , T 1 2 ) T .Note that the figure can trivially be generalized from S 1 , S 2 , T 1 , and T 2 to any pair of excited singlet/triplet site states.

Table S1 :
Excitation energies of guanine site states and oscillator strengths of the singlet site states on the Frank-Condon geometries of G 4 and MgG 2+ 4 systems with three different embedding schemes.

Table S4 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECIS method without embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S5 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECISD method without embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S6 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECIS method with ESP-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S7 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECISD method with ESP-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S8 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECIS method with EHF-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S9 :
ECI wave functions of the first 9 singlet states of G 4 , calculated with ECISD method with EHF-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S10 :
Excitation energies (eV), oscillator strengths, guanine-centered (GC) and guanine-to-guanine charge-transfer (GGCT) percentage of the character of the first 28 excited singlet states of G 4 on symmetric geometry.

Table S11 :
Excitation energies (and oscillator strengths from the ground state for singlets) of the first eight singlet and first eight triplet excited states of MgG 2+ 4 , calculated with the direct full-system method, the FEM, ECIS+FDA and ECISD+FDA, where in the FEM and ECI calculations the "real" FDA was employed, i.e., not even the +2 charge of Mg 2+ was present in the embedding charges.

Table S12 :
ECI wave functions of the first 9 singlet states of MgG 2+ 4 , calculated with FEM.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S13 :
ECI wave functions of the first 9 singlet states of MgG 2+ 4 , calculated with ECIS method without embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S14 :
ECI wave functions of the first 9 singlet states of MgG 2+ 4 , calculated with ECISD method without embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S15 :
ECI wave functions of the first 9 singlet states of MgG 2+ 4 , calculated with ECIS method with ESP-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S16 :
ECI wave functions of the first 9 singlet states of MgG 2+ 4 , calculated with ECISD method with ESP-embedding.Only the ECSFs having the squared value of the ECI coefficient in any of considered states greater than 0.005 are shown.

Table S19 :
Excitation energies (eV), oscillator strengths, guanine-centered (GC), guanine-toguanine charge-transfer (GGCT) and guanine-to-metal charge-transfer (GMCT) percentage of the character of the first 28 excited singlet states of MgG 2+ 4 on symmetric geometry.The MC and MGCT percentages in all states are found to be negligible.

Table S20 :
Excitation energies (eV), oscillator strengths, guanine-centered (GC), guanine-toguanine charge-transfer (GGCT) and guanine-to-metal charge-transfer (GMCT) percentage of the character of the first 28 excited singlet states of MgG 2+ 4 on non-symmetric geometry R 1 .The MC and MGCT percentages in all states are found to be negligible.S8 Errors of the ECI calculations from the Section 5.5

Table S21 :
MADs (in eV) of excitation energies of the first 8 singlet excited states of G 4 and MgG 2+ 4 for different ECI+EHF calculations and for different t Q threshold for the EHF convergence, corresponding to Figure8in the manuscript.