Deciphering excited state properties utilizing algebraic diagrammatic construction schemes of decreasing order

Excited state properties are difficult to trace back to the common molecular orbital picture when the excited state wavefunction is a linear combination of two or more Slater determinants. Here, a theoretical methodology is introduced based on the algebraic diagrammatic construction scheme for the polarization propagator (ADC(n)) that allows to make this connection and to eventually derive structure–function relationships. The usefulness of this approach is demonstrated by an analysis of the transition dipole moments of the low‐lying 1B3u and 2B3u states of anthracene and (1,4,5,8)‐tetraazaanthracene.


| INTRODUCTION
Optical properties and molecular photochemistry is determined by the electronic structure of the excited electronic states involved. Hence, analysis of their electronic structure is a prerequisite for understanding molecular photochemistry in general. [1][2][3][4] The most straight forward analysis consists of simply inspecting the molecular orbitals involved in an electronic excitation. 5,6 However, this quickly falls short when multiple molecular orbital pairs are needed to describe the transition into an excited electronic state. Alternatively, the difference between ground (P 0 ) and excited state (P I ) one-electron density matrices that is, the difference density matrix Δ 0I , can be analyzed. Diagonalization and splitting of its eigenvalues into electron detachment (negative sign) and attachment (positive sign) contributions offers an even more transparent picture of the electronic structure of the excited state I. 7,8 Alternatively, one can obtain information by analyzing the one-particle transition density matrix γ 0I : γ 0I r h , r e ð Þ= ð ÁÁÁ ð Ψ 0 r h ,r 2 , …, r N ð Þ Ψ I r e , r 2 ,…, r N ð Þ dr 2 …dr N : Here, r e and r h describe the coordinates of the excited electron and hole, respectively. 9,10 By performing a singular value decomposition of the transition density matrix, one obtains the so-called natural-transition orbitals which in contrast to the simple molecular orbitals allow a more concise visualization of the electronic transition. [11][12][13] Moreover, the transition density can be identified as an exciton wavefunction and used to calculate physical properties of the exciton as expectation values, such as electron-hole distance, exciton size and correlation coefficient. [14][15][16][17][18] Although the density-and exciton wavefunction descriptors yield valuable physical insight into excitation character and properties of excited electronic states, the connection to the canonical molecular orbital picture, the current natural language of chemists, gets lost. This happens immediately when already two singly excited determinants ϕ constitute the excited state wavefunctions, for example, with ϕ L + 1 H describing a single electron transition from the HOMO to the LUMO+1 and ϕ L H− 1 from the HOMO-1 to the LUMO. This is by no means unusual and this pattern (Equation 4) typically occurs in polyenes and related aromatic compounds. [19][20][21] Going to densities and integration over the individual MO contributions eventually limits the interpretability and usefulness of quantum chemical results for synthesis in particular, since its rules are practically all based on the molecular orbital picture.
To return to a molecular orbital interpretation once the correlated excited state wavefunction has been calculated, the molecular factors determining the expansion coefficients c (Equation 4) as well as the individual properties of the determinants need to be known. This can be achieved by exploiting the systematic structure of ADC schemes for the polarization propagator with decreasing order of perturbation theory, that is, by analyzing the contributions of individual excited states at different orders. The methodological procedure will be outlined in the following and its usefulness demonstrated by explaining the differences in the absorption spectra of anthracene and (1,4,5,8)tetraazaanthracene.

| PRELIMINARY THEORETICAL CONSIDERATIONS
Considering the simplest case consisting of the linear combination of two singly excited determinants according to Equation (4) allows one to construct a two-state model Hamiltonian to analyze the factors governing the expansion coefficients c. Within this two-state model the effective Hamiltonian matrix is given as with E being the energy of one p-h basis state and (E + ΔE) being the energy of the second basis state, ΔE is thus the energy difference between the two p-h basis states and V the coupling between them.
The resulting eigenvalues after diagonalization are the energies of the excited states given as with their energetic splitting being The corresponding non-normalized eigenvectors are given as Since the majority of energetically low-lying excited states consist of the linear combinations of singly excited configurations, we will restrict ourselves to the p-h,p-h block of the ADC secular matrix and its diagonal elements which correspond to the energies E of the p-h basis states in Equation (5). In zeroth-order perturbation theory the diagonal element, that is, the energy of the p-h intermediate basis state is simply given as the difference between the orbital energies ε of the two molecular orbitals involved in the particle-hole excitation The (1) reads The antisymmetrized integral hiakiai which is now subtracted from The first two terms describing the correlation effects due to the individual electron and hole orbital, respectively, whereas the third term describes electron-hole pair correlation, reminiscent of the second-order Møller-Plesset expression for the ground state.
Therefore, in the case of ADC (2) Expanding up to third-order perturbation theory, yields additional 29 terms for the diagonal elements of the singles block of the ADC (3) matrix, 28 whose explicit expressions can be found in the literature. 29,30 Calculating the ADC secular matrix using different orders of pertur- yields the first-order correction Δ  and thus the properties of the computed excited state, proper connection to the molecular orbital picture can finally be made by successively lowering the ADC order, as will be demonstrated below.

| COMPUTATIONAL DETAILS
Geometry optimizations as well as frequency calculations were conducted with MP2 using the resolution-of-identity approximation (RI) and the def2-TZVP basis set 31 within the Psi4 software package in version 1.4. 32 Subsequent ADC calculations in conjunction with the def2-SVPD basis set were performed using adcc in version 0. 15.1. 33 together with PySCF 34 for the SCF procedure. Molecular structures were drawn using MarvinSketch. 35

| CASE STUDY: EXCITED STATES OF HETEROACENES
Acenes and polycyclic aromatic hydrocarbons in general possess two low-lying excited ππ * states which are observed in the low-energy region of experimental UV/VIS absorption spectra. These states have been termed previously according to Platt as 1 L a and 1 L b states. 19 Recently, an alternative nomenclature was proposed for the lowlying excited ππ * states based on the physical properties of the corresponding exciton. 36 The 1 L b state was shown to consistently exhibit a more strongly correlated electron-hole pair than the 1 L a state and was thus labeled as the 1 L s (strongly correlated) state next to the 1 L w (weakly correlated) state. This nomenclature was highlighted to be more strictly applicable independent of substitution patterns and the point group symmetry of nitrogen substituted acenes.
In the absorption spectrum of anthracene, for example, the 1 L w (1B 2u ) state is visible at 3.67 eV, whereas the 1 L s (1B 3u ) state has practically no oscillator strength at an excitation energy of 3.89 eV at the theoretical level of ADC(2) (Figure 2). However, when acenes are chemically modified by substitution or, as is the case here, by heteroatom introduction, the intensity of the 1 L s state increases. 37,38 Focusing only on the 1 L s (1B 3u ) state, this effect is here demon-  (17), the diagonal element difference found within the ADC (2) matrix before diagonalization (0.01 eV, see Table 2) results from the sum of the three contributions (Table 3 Last, removing also electron-hole interaction by going to ADC (0) and thus only accounting for the difference in the energy gap between the respective virtual (electron) and occupied (hole) molecular orbitals leads to a diagonal element difference of 0.05 eV ( Table 2).
It is thus recognized that the differences in electron-hole interaction (Δ (1) ) and correlation (Δ (2) ) are in magnitude never significantly different from the difference in energy gap of the involved molecular orbitals (Δ (0) ). Consequently, the mixing ratio stays almost constant with 0.5 throughout different orders of ADC schemes (Figure 3).  (Table 1).
Since the two eigenstates 1B 3u and 2B 3u are the difference and the sum of the two basis functions ϕ L H− 1 and ϕ L + 1 H , it is clear that 1B 3u exhibits no transition dipole moment while 2B 3u has a larger transition dipole moment. Eventually, we were thus able to translate the excited state properties of the 1B 3u and 2B 3u states computed at the ADC (2) level. This also provides a suggestion for a recipe how to modulate the excited state properties of the 1B 3u and 2B 3u states by chemical modifications: the energies of the contributing orbitals need to change differently to module Δ (0) which appears to be the dominating term influencing the composition of the excited state wavefunctions, as will be demonstrated with the next example.
In (1,4,5,8)-tetraazaanthracene the 1B 3u ( 1 L s ) state exhibits a significantly enhanced transition dipole moment of 1.24 a.u. (osc. strength: 0.15) compared to the unsubstituted anthracene (see Table 1). The enhanced transition dipole moment goes in hand with a considerable degree of de-mixing of both p-h basis states as seen by the coefficients for the excited states at the ADC(2) level: T A B L E 1 Transition dipole momentμ of the p-h basis states ϕ (ADC(0)/def2-SVPD) as well as the excitation energy, transition dipole moment and contributions (squared expansion coefficients with the signs of the coefficients preserved) of the p-h basis states ϕ L H −1 = ϕ L + 1 H of the 1B 3u ( 1 L s ) and 2B 3u excited states of anthracene (1) and (1,4,5,8)-tetraazaanthracene (2) on the ADC(2)/def2-SVPD level of theory   Finally, it is important to realize that this kind of mixing of two intermediate state basis functions is not unique to the two molecules investigated here but that it is typically observed in π-extended systems. This is related to momentum conservation, as it has already been analyzed in the context of plasmonic states in molecules. 39,40