Excited State Structural Analysis: TDDFT and Related Models

Abstract

We review and further develop the excited state structural analysis (ESSA) which was proposed many years ago [Luzanov AV (1980) Russ Chem Rev 49: 1033] for semiempirical models of $\pi {\pi^{\ast}}$-transitions and which was extended quite recently to the time-dependent density functional theory. Herein we discuss ESSA with some new features (generalized bond orders, similarity measures etc.) and provide additional applications of the ESSA to various topics of spectrochemistry and photochemistry. The illustrations focus primarily on the visualization of electronic transitions by portraying the excitation localization on atoms and molecular fragments and by detailing excited state structure using specialized charge transfer numbers. An extension of ESSA to general-type wave functions is briefly considered.

Appendix

We consider the conventional exciton-like model [79, 80], slightly modified here for triplet states of a finite-size system (assembly) of m weakly interacting identical subunits (fragments) ${\{ {A_k}\}_{{1 \leqslant k \leqslant m}}}$. Let ${\Xi_k}$ be the the kth basis vector which corresponds to the local excitation of the kth site of the assembly, viz. ${\Xi_k} = \left| {A_1^0 \ldots A_{{k - 1}}^0\,A_k^{*}\,A_{{k - 1}}^0 \ldots A_m^0} \right\rangle$. Then in a tight-binding approximation we can approximate the exciton-like Hamiltonian matrix $\Lambda = \left\| {{\Lambda_{{kl}}}} \right\|$, as follows

$$ {\Lambda_{{kl}}} = ({\lambda_0} - {\upsilon_k}\,{V_{*}}){\delta_{{kl}}} + \{ {K_{*}}{\mbox{ if }}{A_k}{\mbox{ and }}{A_l}{\mbox{ are neighbors, 0 otherwise\} }} $$

where ${\lambda_0}$ is an excitation energy of the isolated fragment; the positive $\,{V_{*}}$ and ${K_{*}}$ are effective local level shift and exchange parameters, respectively. The factor ${\upsilon_k}$ in term ${\upsilon_k}{V_{*}}$ is equal to a number of neighbours of site ${A_k}$, which takes into account the local environment effects.

Eigenvalues, ${\lambda_k}$, of $\Lambda$ are excitation energies of the whole system. For instance, in a linear trimer we have

$$ \Lambda = \left\| \begin{array}{ccc} {\lambda_0} - {V_{*}}& {K_{*}} & 0 \hfill \\ {K_{*}} & \ \ {\lambda_0} - 2{V_{*}}& {K_{*}} \\ 0&{K_{*}}&{\lambda_0} - {V_{*}}\\ \end{array} \right\| $$

with simply computed excitation energies. They take the form

$$\begin{array}{ll} {\lambda_1} &= {\lambda_0} - {V_{*}}(\sqrt {{1 + 2{q_{*}}^2}} + 3)\,/2,\,{\lambda_2} = {\lambda_0} - {J_{*}},\\ {\lambda_3} &= {\lambda_0} + {V_{*}}(\sqrt {{1 + 2{q_{*}}^2}} - 3)\,/2 \end{array}$$

where the dimensionless parameter ${q_{*}} = 2{K_{*}}\,/{V_{*}}$ is used. The corresponding (non-normalized) eigenvectors are of the form

$$ \left\| \begin{gathered} - {q_{*}} \hfill \\ \,\,\sqrt {{1 + 2{q_{*}}^2}} + 1 \hfill \\ - {q_{*}} \hfill \\ \end{gathered} \right\|\;\left\| \begin{gathered} \,\,\,\,1 \hfill \\ \,\,\,0 \hfill \\ - 1\, \hfill \\ \end{gathered} \right\|,\;\left\| \begin{gathered} \,\,\,\,\,\,\,{q_{*}} \hfill \\ \,\,\sqrt {{1 + 2{q_{*}}^2}} - 1 \hfill \\ \,\,\,\,\,\,\,{q_{*}} \hfill \\ \end{gathered} \right\| $$

From them we obtain the distribution localization indices $L_A^{*}\,$ by merely squaring the corresponding “coordinates” in the above eigenvectors. The results become more clear for very weakly coupled fragments (chromophores) when ${q_{*}} \ll 1$, and we have

$$\begin{array}{ll} {\lambda_1} &= {\lambda_0} - {J_{*}} - 2{K_{*}}^2/{V_{*}},\;\{ L_A^{*}\,\} = \{ 1/2 - {({K_{*}}/{V_{*}})^2},\,\,\,2\,{({K_{*}}/{V_{*}})^2},\\ &\qquad\qquad\qquad\ 1/2 - {({K_{*}}/{V_{*}})^2}\}; \end{array}$$

$$ {\lambda_2} = {\lambda_0} - {V_{*}},\;\{ L_A^{*}\} = \{ 1/2,\,\,\,0,\,\,\,1/2\}; $$

$$ {\lambda_3} = {\lambda_0} - 2{J_{*}} + 2{K_{*}}^2/{V_{*}},\;\{ L_A^{*}\} = \{ {({K_{*}}/{V_{*}})^2},\,\,\,1 - 2\,{({K_{*}}/{V_{*}})^2},\,\,{({K_{*}}/{V_{*}})^2}\}. $$

We see that the lowest excitation is predominantly localized on the internal fragment A whereas the rest are shared between the two terminal fragments of the trimer. The qualitatively similar picture is appropriate in the case of moderate values of ${q_{*}}$ as well.