Tuning Spectral Properties of Anthracenes through Substitution

Abstract

Heteroatom substitution offers a powerful means of tuning the spectral properties of acenes, yet a general structure–property relationship valid across different heteroacene families has been lacking. In this study, we extend a previously applied analysis based on the algebraic diagrammatic construction scheme for electronic excitations (ADC) framework to a broader family of heteroacenes, including aza-, phospha-, and halogen-substituted anthracenes. Decomposition of the ADC diagonal elements and the corresponding perturbative effects from ADC(2) → ADC­(1) → ADC(0) reveals orbital energy differences as the primary driver of excited state mixing and thus of the resulting oscillator strengths. This uncovers how substituent type and position shapes spectral intensities of the low-lying ππ* electronic transitions, namely, ¹L _s (¹L _b ) and ¹B _b . Tip substitution consistently enhances the orbital energy difference and leads to bright α-bands (¹L _s /¹L _b ), whereas side substitution produces more symmetric mixing of configurations and weak α-bands. Across halogen series, increasing electronegativity while keeping the position unchanged generally leads to higher oscillator strength of the ¹L _s (¹L _b ) state. These results provide a unified, general rule for tuning the spectral properties by substitution and/or heteroatom replacement.

Introduction

Acenes consist of linearly fused benzene rings forming an extended π-conjugated system and have the smallest energy gaps between the highest occupied molecular orbital (HOMO) and lowest occupied molecular orbital (LUMO) among polycyclic aromatic hydrocarbons (PAHs) of comparable size. , They have become a subject of increasing interest over the past few decades for their distinctive electronic, spectroscopic, and geometric properties contributing to their extensive applicability in various fields of science, especially as organic semiconductors in electronic devices. − In this regard, their high charge carrier mobilities make them especially attractive. , While unsubstituted acenes are promising candidates for p-type organic semiconductors, the introduction of nitrogen atoms in the ring also enables use as n-type semiconductors. ,,

Acenes have low-lying ππ* excited states, which are thoroughly investigated to determine the potential applicability of the system in various technological and photophysical contexts. − The first excited state of smaller acenes, except for naphthalene, is the excited state ¹L _a , which is described mainly by the molecular orbital (MO) transition HOMO → LUMO. The corresponding absorption band is called p-band and has an intermediate absorption intensity. This state is followed by the excited state ¹L _b that is described mainly by a linear combination of the transitions HOMO → LUMO+1 and HOMO–1 → LUMO. The ¹L _b state of acenes is nearly dark, and the corresponding absorption band is called α-band. The next characteristic state, described by the same orbital transitions, is ¹B _b at higher excitation energies. The corresponding absorption band is called β-band and has a high absorption intensity. The two lowest-lying states of acenes were originally labeled according to Platt’s nomenclature, which categorizes the ππ*-electronic transition based on the direction of their polarization and the number of nodes of the polarization diagram. Here, the subscripts a and b denote polarization along the short and long molecular axis, respectively. , In recent studies, it has been observed that this long-standing nomenclature no longer stands valid for heteroatom-substituted acenes and hence a more universally applicable characterization is required ideally based on physical properties of excited states. In the case of acenes, the correlation between the hole and electron of the corresponding exciton characterizing the L _a and L _b states turned out to be a relevant and conserved physical property, which can be quantified by the electron–hole correlation coefficient. − Therefore, the excited states ¹L _a and ¹L _b were labeled as ¹L _w and ¹L _s , respectively, where w and s denote weakly and strongly correlated electron–hole pairs. This nomenclature will thus be used in the following.

To retain the connection to the conventional labelsand in particular the relationship between the two states, which share the same particle–hole configurationsthe Platt label is given in parentheses throughout, i.e., ¹L _s (¹L _b ) and ¹L _w (¹L _a ). The higher-lying ¹B _b state retains its Platt label as it remains a well-defined, strongly polarized bright state for which the a/b classification is still meaningful; an analogous correlation-based analysis of this state did not reveal a clear trend across the substituted systems, so that a reclassification is not warranted.

The influence of nitrogen substitution on the character and properties of the excited states ¹L _s (¹L _b ) and ¹B _b of anthracene and (1,4,5,8)-tetraazaanthracene was explored in previous work. As stated above, the excited states of interest are linear combinations of the two transitions (or p-h basis states) HOMO → LUMO+1 and HOMO–1 → LUMO. They were therefore analyzed within a two state model with the effective Hamiltonian matrix

$$H_{\mathrm{e}\mathrm{f}\mathrm{f}}=\left(\begin{array}{ll}E& V\\ V& E+\mathrm{\Delta }E\end{array}\right)$$ 1

where E is the energy of one p-h basis state and E + ΔE is the energy of the other p-h basis state. Therefore, ΔE is the energy difference between the states and V is the coupling. Within this framework, the resulting excited states (eigenstates) are described by the positive and negative symmetric combination of the two p-h basis states for ΔE = 0. In this case, one eigenstate is bright due to a symmetric combination of the transition dipole moments, while the other is dark due to asymmetric combination of the transition dipole moment. With an increase of ΔE, the eigenstates are described by an increasingly asymmetric combination and for ΔE approaching infinity are described solely by one of the p-h basis states, respectively. For a detailed introduction, the reader is referred to the previous publication.

Within ADC, the p-h basis states correspond to the intermediate states. The distribution of oscillator strength between the excited states ¹L _s (¹L _b ) and ¹B _b could therefore be explained by the energetic splitting between the contributing intermediate states. In anthracene, the intermediate states HOMO → LUMO+1 and HOMO–1 → LUMO are almost energetically degenerate so that the resulting excited states ¹L _s (¹L _b ) and ¹B _b are described by symmetric positive and negative linear combinations of these p-h basis states. In turn, ¹L _s (¹L _b ) is essentially dark, while ¹B _b acquires all oscillator strength. For tetraazaanthracene, the intermediate states differ in energy so that this symmetry is broken and as a consequence, the oscillator strength of ¹L _s (¹L _b ) is increased and that of ¹B _b decreased compared to anthracene. By computing the excitation energies of the intermediate states at different levels of ADC, the influence of different physical contributions was analyzed. It was observed that the orbital energy difference at the zeroth order between the participating orbitals is the largest contribution to the diagonal elements. The asymmetric mixing of the intermediate states at all levels of ADC in tetraazaanthracene as opposed to anthracene could therefore be explained by the influence of nitrogen substitution on the orbital energies. Figure shows the orbitals HOMO–1, HOMO, LUMO, and LUMO+1 of anthracene and of tetraazaanthracene. It can be seen that substitution in tetraazaanthracene affects the energies of HOMO and LUMO more than those of HOMO–1 and LUMO+1 since the former do not show nodes on the substituted positions. The energies of HOMO and LUMO are therefore lowered in tetraazaanthracene, resulting in a different energy of the two participating p-h basis states. This simplified yet insightful molecular orbital picture explains the change in oscillator strength of the ¹L _s (¹L _b ) state upon nitrogen substitution. The decomposition of the ADC diagonal elements underlying this analysis was introduced and validated in previous work, and it is this established approach that the present study builds upon and applies to a broader, chemically diverse family of substituted acenes.

1.

Molecular orbitals participating in the particle–hole states HOMO → LUMO+1 and HOMO–1 → LUMO, which describe the ¹L _s (¹L _b ) and ¹B _b states of anthracene and 1,4,5,8-tetraazaanthracene based on previous work. The orbital energies are not depicted to scale.

In this work, the influence of heteroatom substitution on excited-state properties, especially the oscillator strength, of the ¹L _s (¹L _b ) and ¹B _b states of aza-, phospha-, and halogen-substituted anthracenes (shown in Figures , and ) is further explored to substantiate and, at the same time, challenge the simplified explanation for the occurrence of the α-band at the molecular orbital level. Furthermore, insight into the influence of substitution at specific positions on the oscillator strength based on the involvement of nodal planes of the respective MOs is gained. Tip positions (1,4,5,8,9,10) are located on the nodal planes of the HOMO–1 and LUMO+1 (Figure ), while side positions (2,3,6,7) are never on nodal planes. Hence, upon tip substitution with nitrogen, the energies of the MOs HOMO and LUMO are expected to be decreased, while the energy of HOMO–1 and LUMO+1 should remain mostly unchanged, as explained above. In the case of side substitution, the energy of all involved MOs should be decreased to a similar extent.

2.

Anthracene (A1) and azaanthracenes (A2-A7) investigated in this work.

3.

Phosphaanthracenes (B1–B6) investigated in this work.

5.

Halogen-substituted anthracenes (C1–C6) investigated in this work.

Excitation energies, oscillator strengths, and the contributions of the participating particle–hole states of ¹L _s (¹L _b ) and ¹B _b will be computed using ADC(2), ADC(1), and ADC(0). − Following the previously established approach, the diagonal elements of the p-h block of the ADC matrix at all employed levels of ADC will be computed to trace the effect of substitution on the energy of the p-h states of interest, with the goal to demonstrate the validity of the previously observed correlation between substitution pattern and oscillator strength for other systems and explore the targeted modulation of the absorption spectra of acene derivatives.

The Physical Contributions to the Intermediate State Energies

The algebraic diagrammatic construction scheme for the polarization propagator (ADC) is a family of ab initio methods, used to compute excitation energies and excited-state properties. ADC originates from Green’s function theory, where its expressions were constructed using a diagrammatic approach, but it can also be formulated within the intermediate state representation (ISR), which has become the more widely used framework. ,,−

ADC schemes are Hermitian eigenvalue problems

$$\mathbf{M}\mathbf{Y}=\mathbf{Y}\mathbf{\Omega }$$ 2

where M is the ADC secular matrix, which gives access to the diagonal matrix Ω consisting of the excitation energies ω _n and the excitation vectors Y _n . ,,, The ADC secular matrix has a block-like structure (Figure S1 in the Supporting Information) and is represented on the basis of correlated intermediate states with singly-(p-h), doubly-(2p-2h), and so forth excited configurations. The p-h block is always treated at the same order of perturbation theory as the Møller–Plesset ground state used to construct the intermediate states. These present correlated excited configurations rather than Slater determinants so that for ADC(2) they contain electron–hole interaction and correlation. The diagonal elements of the ADC matrix before diagonalization correspond to the energies of the individual intermediate states. Due to the hierarchical nature of the ADC­(n) schemes, it is thus possible to relate the energy of the ISR states to specific physical properties and interactions. At zeroth-order (ADC(0)), the diagonal elements of the ADC matrix are just the energy differences between the orbital energies ϵ of the two molecular orbitals involved in the particle–hole excitation

$$D_{ia}^{\mathrm{A}\mathrm{D}\mathrm{C}(0)}=({ϵ}_a-{ϵ}_i)$$ 3

The labels i,j,k,.. denote occupied orbitals, while a,b,c,.. are used for virtual orbitals. The orbital energies ϵ entering eq are the canonical Hartree–Fock orbital energies; the zeroth-order diagonal element, and hence the zeroth-order contribution Δ⁰ to the term differential analyzed below is therefore directly available from the Hartree–Fock reference without any post-Hartree–Fock treatment. At first-order (ADC(1)), the diagonal elements become

$$D_{ia}^{\mathrm{A}\mathrm{D}\mathrm{C}(1)}=({ϵ}_a-{ϵ}_i)-⟨ia\parallel ia⟩$$ 4

The antisymmetrized integral ⟨ia∥ia⟩ has a clear physical meaning as it corresponds to the electron–hole Coulombic and exchange interactions (eq ). Since the Coulombic attraction dominates, this usually leads to an energy decrease of the ISR states at first order. At the ADC(2) level, the diagonal elements read as

$$\begin{array}{ll}{D}_{ia}^{(2)}=& ({ϵ}_a-{ϵ}_i)-⟨ia\parallel ia⟩\\ & +\frac{1}{2}\sum _{ckl}\frac{⟨ac|kl⟩⟨kl|ac⟩}{{ϵ}_a+{ϵ}_c-{ϵ}_k-{ϵ}_l}\\ & +\frac{1}{2}\sum _{cdk}\frac{⟨cd|ik⟩⟨ik|cd⟩}{{ϵ}_c+{ϵ}_d-{ϵ}_i-{ϵ}_k}\\ & -\sum _{ck}\frac{⟨ac|ik⟩⟨ik|ac⟩}{{ϵ}_a+{ϵ}_c-{ϵ}_i-{ϵ}_k}\end{array}$$ 5

The first two additional terms describe the correlation due to individual electron and hole orbitals, respectively, while the third term describes the electron–hole pair correlation. In order to include also the correlation contributions provided by the doubles block, the diagonal matrix elements of the singles block of the folded ADC(2) matrix are analyzed, which are given by

$$\begin{array}{ll}{D}_{ia}^{\mathrm{A}\mathrm{D}\mathrm{C}(2)}=& ({ϵ}_a-{ϵ}_i)-⟨ia\parallel ia⟩+\frac{1}{2}\sum _{ckl}\frac{⟨ac\parallel kl⟩⟨kl\parallel ac⟩}{{ϵ}_a+{ϵ}_c-{ϵ}_k-{ϵ}_l}\\ & +\frac{1}{2}\sum _{cdk}\frac{⟨cd\parallel ik⟩⟨ik\parallel cd⟩}{{ϵ}_c+{ϵ}_d-{ϵ}_i-{ϵ}_k}-\sum _{ck}\frac{⟨ac\parallel ik⟩⟨ik\parallel ac⟩}{{ϵ}_a+{ϵ}_c-{ϵ}_i-{ϵ}_k}\\ & +\frac{1}{2}\sum _{bc}\frac{⟨ia\parallel bc⟩⟨ia\parallel bc⟩}{{ϵ}_i+{ϵ}_i-{ϵ}_b-{ϵ}_c-\omega }-2\sum _{jb}\frac{⟨ij\parallel ib⟩⟨ja\parallel ab⟩}{{ϵ}_i+{ϵ}_j-{ϵ}_a-{ϵ}_b-\omega }\\ & -\frac{1}{2}\sum _{jbc}\frac{⟨ja\parallel bc⟩⟨ja\parallel bc⟩}{{ϵ}_i+{ϵ}_j-{ϵ}_b-{ϵ}_c-\omega }+\frac{1}{2}\sum _{jk}\frac{⟨jk\parallel ia⟩⟨jk\parallel ia⟩}{{ϵ}_j+{ϵ}_k-{ϵ}_a-{ϵ}_a-\omega }\\ & -\frac{1}{2}\sum _{jkb}\frac{⟨jk\parallel ib⟩⟨jk\parallel ib⟩}{{ϵ}_i+{ϵ}_j-{ϵ}_a-{ϵ}_b-\omega }\end{array}$$ 6

where the new terms are dependent on the excitation energy of the corresponding excited state.

In the following, the difference between the ADC­(n) diagonal elements of two different particle–hole intermediate basis states (D _ia – D _jb ) at the ADC­(n) level will be expressed as

$$\begin{array}{ll}\mathrm{\Delta }{D}_{ia,jb}^{\mathrm{A}\mathrm{D}\mathrm{C}(n)}& ={\mathrm{\Delta }}_{ia,jb}^{(0)}+{\mathrm{\Delta }}_{ia,jb}^{(1)}\\ & +{\mathrm{\Delta }}_{ia,jb}^{(2)}+...{\mathrm{\Delta }}_{ia,jb}^{(n)}\end{array}$$ 7

where Δ _ia,jb gives their difference at zeroth-order ((ϵ _a – ϵ _i ) – (ϵ _b – ϵ _j )), Δ _ia,jb gives the difference of the first-order terms (⟨ia∥ia⟩ – ⟨jb∥jb⟩), etc. Therefore, Δ⁽⁰⁾ corresponds to the difference in electron and hole molecular orbital energies, Δ⁽¹⁾ gives the difference in electron–hole interaction, i.e., Coulombic attraction and exchange repulsion, and Δ⁽²⁾ represents the difference in electron–hole correlation effects.

Throughout this work, the two particle–hole intermediate states that are compared are those describing the ¹L _s (¹L _b ) and ¹B _b states, namely, (i, a) = (HOMO −1, LUMO) and (j, b) = (HOMO, LUMO+1). For this fixed pair, we abbreviate the order-resolved energy differences as Δ⁽⁰⁾≡Δ _ia,jb , Δ⁽¹⁾≡Δ _ia,jb , and Δ⁽²⁾≡Δ _ia,jb , and their sum as Δ^tot = Δ⁽⁰⁾ + Δ⁽¹⁾ + Δ⁽²⁾.

Computational Methodology

Geometry optimizations followed by frequency calculations were performed using the Q-Chem 6.0 program package at the RI-MP2/def2-TZVP level of theory. Excitation energies and oscillator strengths were computed using ADC(2) with the basis set def2-SVPD both using adcc 0.15.17 with PYSCF as the backend for the SCF procedure as well as Q-Chem 6.0. The diagonal elements of the p-h block of the ADC matrix were computed at ADC(0), ADC(1), and ADC(2) using adcc 0.15.17. The python script employed to compute the diagonal elements of the p-h block of the folded ADC(2) matrix in adcc is given in the Supporting Information S6. Molecular structures were drawn using ChemDraw 21.0.0, and molecular orbitals were visualized using Avogadro 1.97.0 and IQmol 2.8.0.

Results and Discussion

Azaanthracenes

In this section, the spectral properties of the excited states ¹L _s (¹L _b ) and ¹B _b of azaanthracenes shown in Figure are analyzed. Table shows the excitation energy ω, oscillator strength f, transition dipole moment μ _T , and the mixing of the intermediate p-h states given by their respective expansion coefficient in the corresponding excitation vector of ¹L _s (¹L _b ) and ¹B _b as ϕ _H–1 /ϕ _H at the ADC(2) level. The normalized value is given as ∥ϕ _H–1 /ϕ _H ∥. The relative contributions of the p-h intermediate states and the oscillator strengths of the excited states ¹L _s (¹L _b ) and ¹B _b show the same correlation as observed in previous work, a more asymmetric contribution of the p-h states leads to an increased oscillator strength of ¹L _s (¹L _b ). Indeed, the difference in the energies of the involved MOs is influenced differently by substitution at tip or side positions. Table shows the second- Δ⁽²⁾, first- Δ⁽¹⁾, and zeroth-order energy differences Δ⁽⁰⁾ (see eq ) as well as the sum of all three differences Δ^tot.

1. Excitation Energy ω [eV], Oscillator Strength f, Transition Dipole Moment μ _T [a.u.], and Contributions of the Single-Excitation Basis States ϕ _H–1 /ϕ _H to the ¹L _s (¹L _b ) and ¹B _b States at ADC(2)/def2-SVPD Level of Theory for Molecules in Subset A .

	1L s (1L b )					1B b
	ω [eV]	f	μ T [a.u.]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥	ω [eV]	f	μ T [a.u.]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥
A1	3.89	0.00	0.06	–0.44/–0.42	0.51/0.49	5.42	2.43	4.28	–0.42/0.44	0.49/0.51
A2	3.87	0.15	1.24	0.63/0.23	0.73/0.27	5.47	1.62	3.48	0.20/–0.58	0.26/0.74
A3	3.86	0.05	0.73	0.55/0.30	0.65/0.35	5.16	1.31	3.22	0.23/–0.42	0.35/0.65
A4	3.88	0.14	1.20	–0.63/0.22	0.74/0.26	5.48	1.86	3.72	–0.21/–0.61	0.26/0.74
A5	3.84	0.26	1.65	–0.70/–0.14	0.83/0.17	5.37	0.81	2.49	–0.47/0.09	0.16/0.84
A6	3.95	0.002	0.15	0.37/–0.45	0.45/0.55	5.62	1.59	3.39	–0.16/–0.23	0.41/0.59
A7	3.91	0.02	0.39	–0.49/–0.37	0.57/0.43	5.35	1.72	3.62	–0.30/0.40	0.43/0.57

^aFor A7, the involved particle–hole states are HOMO–1 to LUMO and HOMO to LUMO+2 (Figure S5 in Supporting Information).

2. Zeroth Δ⁽⁰⁾, First Δ⁽¹⁾, and Second Order Δ⁽²⁾ and Total Δ^tot Diagonal Element Energy Differences of the Two Particle–Hole States ϕ _H–1 and ϕ _H of Molecular Systems in Subset A .

	energy differences [eV]
derivative	Δ(0)	Δ(1)	Δ(2)	Δtot
A1	0.04	–0.08	–0.04	–0.08
A2	–1.41	0.28	0.19	–0.94
A3	–0.65	0.08	0.05	–0.52
A4	–1.45	–0.05	0.46	–1.04
A5	–2.13	0.06	0.63	–1.44
A6	0.30	–0.14	–0.11	0.05
A7	–0.17	0.09	–0.20	–0.28

Further comparing A3 and A4, it is observed that molecular symmetry has an influence on the oscillator strength. Looking at Tables and , substitution on the central ring leads to stronger asymmetry in p-h state mixing. The total energy difference (Δ^tot) of A4 (Δ^tot = −1.04 eV) is twice as large as that of A3 (Δ^tot = −0.52 eV), which leads to a further enhancement of the oscillator strength of the ¹L _s (¹L _b ) state. Detailed tables of ADC diagonal elements and perturbation contributions are provided in the Supporting Information (Tables S1 and S2). Systems A2 and 1,4,9,10-tetraazaanthracene (A5) are tip-substituted derivatives, with a large zeroth-order difference, resulting in an asymmetric mixing and increased oscillator strength. The increased zeroth-order energy difference can be explained by the larger number of introduced nitrogen atoms. From Table , it can also be seen that A5 has a larger total energy difference (Δ^tot = −1.44 eV) than A2 (Δ^tot = −0.94 eV). In short, due to the increased zeroth-order difference and resulting higher second-order difference, nitrogen substitutions in the central ring, at positions 9 and 10 result in a ¹L _s (¹L _b ) state with a larger oscillator strength.

The systems 1,4,5,8-tetraazaanthracene A2, 1,4-diazaanthracene A3, and 9,10-diazaanthracene A4 are substituted with nitrogen at tip positions and show an asymmetric mixing of the p-h states compared with anthracene A1 and accordingly also a higher oscillator strength for ¹L _s (¹L _b ). Substitution at side positions in 2,7-diazaanthracene A6, however, leads to a similar mixing of p-h states to those of A1 and a similar oscillator strength of the ¹L _s (¹L _b ) state in the absorption spectrum.

Analyzing the properties of side-substituted molecule A6, it can be noted that the oscillator strength of the ¹L _s (¹L _b ) state is almost negligible, similar to A1. It also has a smaller orbital energy difference and total energy difference (Δ^tot = 0.05 eV), resulting in a symmetric mixing of p-h states. In the case of 1-azaanthracene (A7), similar trends are observed as for the tip derivatives but to a weaker extent due to the presence of only one single nitrogen atom. Inspecting the individual values of energy differences in Table indicates that Δ⁽⁰⁾ remains the largest contribution in most cases where substitutions are made at tip positions. One can also see that the total energy difference remains negative for the tip-substituted systems, which means that the energy of the ϕ _H state is larger than the one of ϕ _H–1 . In contrast, side-substituted systems display a positive difference, reflecting the reversed contribution.

Phosphaanthracenes

The same analysis is performed for the phosphaanthracenes shown in Figure , to challenge the generalizability of the observed trend and to investigate how replacing nitrogen with phosphorus influences the excited-state properties. Table shows the spectral properties calculated at the ADC(2) level of the excited states ¹L _s (¹L _b ) and ¹B _b . The excitation energies of the involved p-h states and the perturbation theoretical contributions from the ADC matrix are presented in Tables S3 and S4 in the Supporting Information, respectively. In 1-phosphaanthracene (B1), the zeroth-order difference is relatively small, the first-order interaction effects are negligible, and a reduced second-order difference results in a total energy difference of Δ^tot = 0.13 eV (Table ), ultimately leading to a practically dark ¹L _s (¹L _b ) state. With the introduction of a methyl group in 1-methyl-1-phosphaanthracene (B6), the absolute value of the total energy difference is significantly reduced (Δ^tot = −0.06 eV), leading to a very symmetric mixing (0.42/0.41) and thereby an even darker ¹L _s (¹L _b ) state.

3. Excitation Energy ω [eV], Oscillator Strength f, Transition Dipole Moment μ̂ [a.u.], and Contributions of the Single-Excitation Basis States ϕ _H–1 /ϕ _H of the ¹L _s (¹L _b ) and ¹B _b States at ADC(2)/def2-SVPD Level of Theory for Molecules in Subset B .

	1L s (1L b )					1B b
	ω [eV]	F	μ T [a.u]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥	ω [eV]	f	μ T [a.u]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥
B1	3.77	0.007	0.27	0.38/0.46	0.45/0.55	5.14	1.86	3.86	–0.49/0.34	0.59/0.41
B2	3.37	0.13	1.25	–0.21/0.62	0.25/0.75	4.64	1.48	3.60	–0.62/0.27	0.70/0.30
B3	3.62	0.03	0.54	0.31/–0.51	0.38/0.62	4.83	1.41	3.46	0.52/–0.28	0.65/0.35
B4	3.39	0.06	0.84	0.29/–0.56	0.34/0.66	4.68	1.36	3.44	0.56/0.27	0.67/0.33
B5	3.11	0.20	1.61	0.14/0.70	0.17/0.83	4.68	0.65	2.39	0.28/–0.07	0.80/0.20
B6	3.68	4.87 × 10–4	0.54	–0.42/0.41	0.51/0.49	4.73	0.93	2.83	–0.27/–0.21	0.56/0.44

^aFor B2 and B5, the involved particle–hole states are HOMO–2 → LUMO and HOMO → LUMO+2 (Figure S6 in Supporting Information).

4. Zeroth Δ⁽⁰⁾, First Δ⁽¹⁾, and Second Order Δ⁽²⁾ and Total Δ^tot Diagonal Element Energy Differences of the Two Particle–Hole States ϕ _H–1 and ϕ _H of Molecular Systems in Subset B .

	energy difference [eV]
derivative	Δ(0)	Δ(1)	Δ(2)	Δtot
B1	0.26	0.00	–0.13	0.13
B2	1.05	–0.13	–0.20	0.72
B3	0.40	0.12	–0.20	0.32
B4	0.63	–0.14	–0.07	0.42
B5	–1.49	0.19	2.38	1.08
B6	0.08	0.21	–0.35	–0.06

Comparing 1,4-diphosphaanthracene (B3) and 9,10-diphosphaanthracene (B4), the pattern remains consistent with the one previously identified for the nitrogen-substituted derivatives, where the zeroth-order energy difference is larger for the 9,10-derivative, while the first-order difference remains almost similar. However, the second-order difference is notably higher for B3 than B4, indicating that B3 exhibits a stronger electron–hole correlation. The total energy difference for B4 (Δ^tot = 0.42 eV) is slightly larger than for B3 (Δ^tot = 0.32 eV), resulting in a larger oscillator strength of the ¹L _s (¹L _b ) state in B4.

Finally, to discuss 1,4,9,10-tetraphosphaanthracene (B5)² and 1,4,5,8-tetraphosphaanthracene (B2), and to discuss the influence of symmetry on the excited states, it can be seen that the total energy difference is larger for B5 (Δ^tot = 1.08 eV) than for B2 (Δ^tot = 0.72 eV), with second-order correlation effects being significantly enhanced. This increased second-order energy difference, along with higher first- and zeroth-order energy differences (Table ), leads to a brighter ¹L _s (¹L _b ) state in B5 than B2.

In contrast to the observations for azaanthracenes, Table shows that the total energy difference values are positive across all phosphaanthracene derivatives, suggesting that the energy of the ϕ _H–1 intermediate basis state is larger than that of ϕ _H . The exception occurs for B6, where methyl substitution alters this trend.

As shown in Figure , unsubstituted anthracene exhibits degenerate MO energy differences for the MOs involved in the intermediate basis states ϕ _H–1 and ϕ _H . The introduction of nitrogen atoms into the ring at tip positions leads to a lowering of the orbital energies of HOMO and LUMO due to the higher electronegativity of nitrogen compared to carbon. This results in a reduction in the excitation energy of ϕ _H–1 and an increase in that of ϕ _H , which results in a negative zeroth-order energy difference. In contrast, the introduction of a phosphorus atom, which is less electronegative than carbon, leads to an increase in the HOMO and LUMO energies. As a result, the energy of the intermediate state ϕ _H–1 increases, while that of ϕ _H decreases, resulting in a positive zeroth-order energy difference.

4.

Schematic representation of the effect of nitrogen and phosphorus substitution on the energies of MOs involved in the intermediate basis states ϕ _H–1 and ϕ _H of anthracene.

Halogen-Substitued Anthracenes

Subsequently, the study is extended to the halogen-substituted anthracene derivatives displayed in Figure , to explore whether the relation between substitution pattern and oscillator strength of ¹L _s (¹L _b ) is also valid for substitution of hydrogen on the ring, and to investigate the influence of increasing electronegativity.

All of the studied molecules, except 2,3,6,7-tetrafluoranthracene (C2), are substituted at tip positions. The 1,4,5,8,9,10-hexafluoranthracene (C1) has fluorine atoms at all of the tip positions. In the case of 9,10-difluoranthracene (C3), 9,10-dichloranthracene (C4), and 9,10-dibromanthracene (C6), the influence of electronegativity on the oscillator strength is investigated. The comparison of C4 and 1,4-dichloranthracene (C5) allows for the investigation of the influence of symmetry. Similar to the aza- and phosphaanthracene derivatives, we focus on the ¹L _s (¹L _b ) and ¹B _b states. Table depicts the excited-state properties, and Tables S5 and S6 in the Supporting Information show the excitation energies of the p-h states and the perturbation theoretical contributions, respectively. Table shows the energy differences at each order of ADC and the corresponding total energy differences.

5. Excitation Energy ω [eV], Oscillator Strength f, Transition Dipole Moment μ̂ [a.u.], and Contributions of the Single-Excitation Basis States ϕ _H–1 /ϕ _H to the ¹L _s (¹L _b ) and ¹B _b States at ADC(2)/def2-SVPD Level of Theory for Molecules in Subset C .

derivative	1L s (1L b )					1B b
	ω [eV]	f	μ T [a.u]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥	ω [eV]	f	μ T [a.u]	ϕ H–1 /ϕ H	∥ϕ H–1 /ϕ H ∥
C1	3.53	0.21	1.56	–0.16/–0.70	0.19/0.81	5.50	2.02	3.49	–0.69/0.16	0.81/0.19
C2	3.89	0.028	0.54	–0.52/–0.34	0.60/0.40	5.43	2.422	4.27	0.33/0.52	0.39/0.61
C3	3.79	0.036	0.63	0.31/0.55	0.36/0.64	5.36	2.207	4.10	–0.55/0.30	0.65/0.35
C4	3.76	0.002	0.14	0.38/–0.47	0.45/0.55	5.23	1.99	3.64	0.47/0.38	0.55/0.45
C5	3.83	0.0005	0.07	–0.40/–0.45	0.48/0.52	5.24	1.85	3.79	0.40/–0.35	0.53/0.47
C6	3.76	0.0006	0.08	–0.39/0.47	0.45/0.55	5.18	1.83	3.80	–0.46/–0.38	0.55/0.45

6. Zeroth Δ⁽⁰⁾, First Δ⁽¹⁾, and Second Order Δ⁽²⁾ and Total Δ^tot Diagonal Element Energy Differences of the Two Particle–Hole States ϕ _H–1 and ϕ _H of Molecular Systems in Subset C .

	energy differences [eV]
derivative	Δ(0)	Δ(1)	Δ(2)	Δtot
C1	1.43	–0.22	0.16	1.37
C2	–0.34	0.10	–0.16	–0.40
C3	0.47	–0.03	–0.01	0.43
C4	0.10	–0.06	0.09	0.13
C5	0.14	0.21	–0.01	0.34
C6	0.03	–0.08	0.14	0.09

In line with the previous observations, C1, which has fluorine substitution at each tip position, shows a significantly larger oscillator strength of the ¹L _s (¹L _b ) state than the other molecules in the subset, while C2, which has only side fluorine substitution, shows a low oscillator strength. The absolute total energy difference is also significantly higher for C1 (Δ^tot = 1.37 eV) than C2 (Δ^tot = −0.40 eV). In C3, C4, and C6, with the decrease in the electronegativity of the substituted halogen, a decrease in the total energy difference and oscillator strength (Table ) is noticed. This can also be observed in the simulated absorption spectra of the molecules in Figure and in Table . The absorption intensities of both ¹L _s (¹L _b ) and ¹B _b increase with increasing electronegativity. C5 has a more symmetric mixing of the p-h states and a reduced oscillator strength compared to C4, which shows that molecular symmetry has the same influence as observed for azaanthracenes. However, the zeroth-order energy difference does not align with the oscillator strength showing that Δ⁽⁰⁾ is not the only determining factor.

6.

Simulated absorption spectra of C3, C4, and C6 obtained by convolution of excitation energies (ADC(2)/def2-SVPD) using a Gaussian broadening function with a standard deviation of 0.1 eV.

To sum it up, for halogen-substituted anthracenes, the overall energy difference exhibits the opposite trend to that observed for azaanthracenes. Substitution at the tip positions yields positive total energy differences, indicating that the energy of the ϕ _H–1 intermediate basis state is larger than that of the ϕ _H intermediate state. In contrast, substitution at the side position (C2) produces negative total energy differences, reflecting the opposite effect. Furthermore, the oscillator strength increases with the electronegativity of the substituent, as illustrated in Figure , and this relationship holds consistent for both excited states of interest.

General Considerations

To further quantify the influence of substitution on the total energy differences, the standard deviation of the second-, first-, and zeroth-order energy differences of the p-h states ϕ _H–1 and ϕ _H of the different subsets was computed and can be seen in Figure . In most cases, the zeroth-order energy difference has a larger standard deviation than the first- and second-order energy differences. It can be concluded that the orbital energy difference Δ⁽⁰⁾ is influenced stronger by chemical substitution than the difference in electron–hole interaction Δ⁽¹⁾ and the difference in electron–hole correlation Δ⁽²⁾.

7.

A bar graph representing the standard deviation of the absolute values of the individual contributions to the intermediate state energies at zeroth (0), first (1), and second order (2) of perturbation theory in green, red, and blue, respectively, for all investigated systems.

Furthermore, Figure shows the oscillator strength of the ¹L _s (¹L _b ) state of all investigated molecules in relation to the zeroth-order energy difference Δ⁽⁰⁾, where a clear correlation between increasing orbital energy difference and increasing oscillator strength can be observed. This shows that the previously observed connection between substitution, orbital energy difference, and oscillator strength is valid for aza-, phospha-, and halogen-substituted anthracenes. Individual plots of each subsets A, B, and C can be seen in the Supporting Information in Figures S2–S4, respectively.

8.

Scatter plot of oscillator strength f of the ¹L _s (¹L _b ) state against the zeroth-order energy difference Δ⁽⁰⁾ of all investigated systems.

Conclusion

The analysis of the spectral properties of the ¹L _s (¹L _b ) and ¹B _b states of aza-, phospha-, and halogen-substituted anthracenes reveals distinct trends in the absorption intensity of the ¹L _s (¹L _b ) state in the UV/Vis absorption spectrum. In azaanthracenes, substitution at different positions modulates the ¹L _s (¹L _b ) absorption intensity, where two distinct classes of substitution sitestip and sidewere identified. Substitutions at tip positions significantly affect the oscillator strength of the ¹L _s (¹L _b ) state compared to anthracene, whereas side substitutions show negligible influence. Extending the investigation to phospha- and halogen-substituted anthracenes demonstrates that the same underlying mechanism governs the intensity of the ¹L _s (¹L _b ) state across chemically diverse substitutions. In all cases, the oscillator strength is controlled by the degree of asymmetry in the mixing of the dominant particle–hole configurations, which is primarily driven by the zeroth-order orbital energy difference Δ⁽⁰⁾. Higher order electron–hole correlation contributions modulate this effect, particularly for heavier heteroatoms, but remain secondary to the zeroth-order energy differences.

The investigation of halogen-substituted anthracenes provides further insight into the role of electronegativity in governing the excitation energies of the involved p-h states. Comparison of derivatives with different halogen substituents at identical positions (C3, C4, and C6) shows that with increasing electronegativity from bromine to fluorine, the magnitude of the zeroth-order energy difference Δ⁽⁰⁾ increases due to a larger molecular orbital energy gap difference. This enhancement promotes greater decoupling of the p-h states and consequently leads to an increase in the oscillator strength of the ¹L _s (¹L _b ) state. In contrast, the second-order energy difference exhibits the opposite trend, decreasing with increasing oscillator strength, indicating a reduced contribution from electron–hole correlation effects.

Consequently, for aza-, phospha-, and halogen-substituted anthracene derivatives, clear and transferable strategies can be derived for manipulating the p-h state energy differences and, in turn, the oscillator strength of the ¹L _s (¹L _b ) state through the control of the relevant molecular orbital energy differences. This structure–property relationship can be readily extended to other conjugated systems. Notably, the relative intensities of low-lying electronic absorption bands can be anticipated without explicit calculations or measurements by considering a small set of guiding principles: (i) substitution at tip positions enhances the p-h energy difference due to asymmetric perturbation of the participating molecular orbitals, and (ii) substitution at the 9,10-positions leads to a comparatively larger p-h energy difference.

Beyond the oscillator strength studied here, the same analysis can be applied to properties governed by the higher-order contributions. A natural example is the singlet–triplet gap, which carries no zeroth-order (orbital-energy) contribution and is determined entirely by the electron–hole exchange and correlation terms Δ⁽¹⁾ and Δ⁽²⁾ , a complementary regime we intend to explore in future work.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsphyschemau.6c00105.

• Diagonal elements of the single-excitation intermediate states at each order of ADC (ADC(0), ADC(1), ADC(2)) and the corresponding energy differences at each order for the aza- (A), phospha- (B), and halogen-substituted anthracene (C) series; description of the ADC(2)-s matrix; scatter plots of the ¹L _s (¹L _b ) oscillator strength against the zeroth-order energy difference Δ⁰ for each subset; Python script for the folded ADC(2) matrix calculation; and molecular orbitals involved in the participating p-h states for selected molecules (PDF)

The authors declare no competing financial interest.