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On the Jahn-Teller Effect in Silver Complexes of Dimethyl Amino Phenyl Substituted Phthalocyanine

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Martin Breza^*

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11 September 2023

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13 September 2023

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Abstract

The structures of Ag complexes with dimethyl amino phenyl substituted phtalocyanine m[dmaphPcAg]q of various charges q and in two lowest spin states m were optimized using the B3LYP method within the D4h symmetry group and its subgroups. The most stable reaction intermediate in the supposed photoinitiation reaction is 3[dmaphPcAg]-. Group-theoretical analysis of the optimized structures and of their electron states reveals two symmetry-descent mechanisms. Stable structures of maximal symmetry of complexes 1[dmaphPcAg]+, 3[dmaphPcAg]+, 2[dmaphPcAg]0 and 4[dmaphPcAg]2- correspond to the D4 group as a consequence of the pseudo-Jahn-Teller effect within unstable D4h structure. Complexes 4[dmaphPcAg]0, 1[dmaphPcAg]-, 3[dmaphPcAg]- and 2[dmaphPcAg]2- with double degenerate electron ground states in D4h symmetry structures undergo a symmetry descent to stable structures corresponding to maximal D2 symmetry not because of a simple Jahn-Teller effect but due to hidden pseudo-Jahn-Teller effect (strong vibronic interaction between excited electron states). The reduction of the neutral photoinitiator causes symmetry descent to its anionic intermediate because of vibronic interactions that must significantly affect the polymerization reactions.

Keywords:

Subject: Chemistry and Materials Science - Theoretical Chemistry

1. Introduction

New photosensitive systems with improved efficiency for initiating free-radical and/or cationic polymerizations under light activation are developed using sophisticated chemical treatments. Very recently, Breloy et al. [1] synthesized dimethyl amino substituted phtalocyanine dmaphPcH2 and its Ag(II) complex ²[dmaphPcAg]⁰ (Figure 1). Photoexcitation of [dmaphPcAg]⁰ in the presence and absence of an iodonium salt produced acidic and radical species that initiated cationic and free-radical polymerizations, respectively. The photoinitiator properties were investigated by spectroscopic and quantum-chemical methods. The polymerization kinetics in laminate and under air was studied. In the first step, irradiation (385 and 405 nm) caused a reduction of Ag(II) to Ag(I) and nitrogen-centered radicals were formed. Subsequently, Ag nanoparticles and carbon-centered radicals developed. Intermediates [dmaphPc]^-, Ag⁰ nanoparticles, and [dmaphPcAg]^q complexes, q = -1 → + 1, are supposed within the proposed reaction mechanism. DFT calculations indicate a D₄ → D₂ symmetry descent in [dmaphPcAg]^q complexes which could be ascribed to the Jahn-Teller (JT) effect. The aim of our recent study is to shed more light on this problem and to verify the above explanation using a group theoretical analysis of the DFT optimized structures of [dmaphPcAg]^q complexes within the highest possible D_4h symmetry group and its subgroups.

2. Theoretical background

According to the Jahn-Teller theorem [2], any nonlinear configuration of atomic nuclei in a degenerate electron state is unstable. Hence, there is at least one such configuration of lower symmetry, where the above degeneracy is removed. In other words, the multidimensional representation in the high-symmetric (HS) structure is split into nondegenerate representations in the low-symmetric (LS) structure.

The JT active coordinate Q_JT describes the above mentioned symmetry descent. If the potential energy surface E = f(Q_i) of N atoms is a function of 3N - 6 independent nuclear coordinates Q_i, for the JT ‘unstable’ HS structure the following relation for any Q_JT is valid

{(\frac{\partial E}{\partial Q_{J T}})}_{H S} \neq 0

(1)

The HS → LS geometry change described by Q_JT is connected with an energy decrease which is denoted as the JT stabilization energy E_JT

E_JT = E_HS – E_LS

(2)

An analogous symmetry descent and energy decrease for pseudodegenerate electron states is known as the pseudo-Jahn-Teller (PJT) effect. It can be observed for a sufficiently strong vibronic interaction and relatively small energy difference Δ_ij between the interacting electron states Ψ_i and Ψ_j [3].

The (P)JT effects are usually related to the ground state Ψ₀ (and some of the excited states Ψ_j). Nevertheless, excited states can undergo to these effects as well. In some cases the corresponding vibronic interaction can be so strong that the lower surface penetrates the potential surface of the ground electron state corresponding the HS structure and becomes the lowest state in the LS structure (i.e. its ground electron state). These consequences of the strong JT and PJT vibronic interactions within excited electron states are known as the hidden JT (HJT) and hidden PJT (HPJT) effects, respectively [4].

The (P)JT potential surfaces can be described by an analytical function based on perturbation theory. However, the search for their extremal points corresponding to the ‘stable’ or ‘unstable’ (P)JT structures is too complicated for large molecular systems. In such cases, a group-theoretical treatment must be used.

The epikernel principle method [5,6] is based on the Q_JT symmetry in the HS structure. A nonzero value of the integral

< Ψ_{i} |\frac{\partial^{} \hat{H}}{\partial Q_{J T}}| Ψ_{j} > \neq 0

(3)

for the interacting electron states Ψ_i and Ψ_j and the full-symmetric Hamilton operator

\hat{H}

demands that the direct product

Γ_i* ⊗ Λ_JT ⊗ Γ_j

(4)

where Γ_i , Λ_JT, and Γ_j are representations of Ψ_i, Q_JT, and Ψ_j, respectively (the asterisk denotes a complex conjugate value), must contain a full-symmetric representation. Alternatively

Λ_JT ∈ Γ_i^* ⊗ Γ_j

(5)

In the case of the JT effect and the degenerate states Ψ_i = Ψ_j , i.e. Γ_i = Γ_j, we obtain the even stronger condition

Λ_JT ∈ [Γ_i ⊗ Γ_i]⁺

(6)

where […]⁺ denotes the symmetric direct product.

The epikernel principle states that the extrema of a JT energy surface correspond to the kernel K(G, Λ_JT) or epikernel E(G, Λ_JT) subgroups of the HS parent group G. Kernels contain the symmetry operations of G that leave the Λ_JT representation invariant. The symmetry operations of epikernels leave invariant only some components of the degenerate Λ_JT representation.

The epikernel principle was originally formulated for the systems in degenerate electron states only [5,6], but it has been successfully extended (except Eq. (6)) to pseudodegenerate electron states as well [7]. Its drawback is in the restriction to JT active coordinates that have been derived within perturbation theory for the linear Taylor expansion of the perturbation only. This method might offer incomplete results in some cases (e.g., in systems with C₅ rotations) [7].

The method of step-by-step symmetry descent [8,9] is based on consecutive splitting of a degenerate electron state within a JT symmetry descent. The corresponding multidimensional representation can be split within symmetry descent to an immediate subgroup of the parent HS group G. If the structure corresponding to this subgroup is in a nondegenerate electron state, it is JT stable and the symmetry descent stops. If the structure corresponding to an immediate subgroup of G is in a degenerate electron state described by a multidimensional representation (a JT unstable structure), the symmetry descent continues, and the whole procedure is repeated. As every group can have several immediate subgroups, various symmetry descent paths are possible and the JT stable structures can correspond to various LS symmetry groups. The only condition is its nondegenerate electron state (i.e. one-dimensional representation) obtained by splitting the degenerate electron state (i.e., multidimensional representation) of the HS structure of the parent group G.

3. Results

The highest possible structures of [dmaphPcAg]^q complexes are of the D_4h symmetry group with side phenyl groups perpendicular to the central phtalocyanine plane (Figure 2). Because of the great number of possible mutual orientations of dimethyl amino phenyl groups in less symmetric structures, we restricted our study to the stable structures of maximal symmetry group only. Another restriction is implied by the inability of standard DFT methods to optimize the atomic configurations in degenerate electron states [10]. The results of the geometry optimizations of the ^m[dmaphPcAg]^q complexes with total charges q = +1 → -2 in the two lowest spin states m are presented in Table 1. It is interesting that the [dmaphPcAg]^q complexes in the triplet spin state are more stable than these ones with the same charge in the singlet spin state. ³[dmaphPcAg]^- of D₂ symmetry is even the most stable complex under study. It must be mentioned that the use of an unrestricted ‘broken symmetry’ treatment [11] results in zero spin populations and does not reduce the energy of the systems studied. Therefore, only the restricted Kohn-Sham formalism has been used in subsequent TD-DFT calculations of our complexes in singlet ground states.

Further inspection of Table 1 shows that the studied complexes can be divided into two categories:

i): Category I contains complexes ¹[dmaphPcAg]⁺, ³[dmaphPcAg]⁺, ²[dmaphPcAg]⁰ and ⁴[dmaphPcAg]^2- with optimized structures of D_4h (unstable) and D₄ (stable) symmetry groups.
ii): Category II contains complexes ⁴[dmaphPcAg]⁰, ¹[dmaphPcAg]^-, ³[dmaphPcAg]^- and ²[dmaphPcAg]^2- where only their stable optimized structures of the D₂ symmetry group were found, while the D_4h optimized structures were absent. This can be explained by the electron configurations of the D4h complexes in Table 2. We may conclude that the category II complexes should contain partially occupied e_g molecular orbitals, which implies E_g or E_u ground electron states. Hence, they are not accessible by standard DFT methods.

3.1. Category I complexes

The optimized D_4h structures of these complexes are unstable due to several imaginary vibrations of Λ_im representations (Table 1) that coincide with the JT active coordinates. The JT stabilization energies E_JT correspond to D_4h → D₄ symmetry decrease. Stable D₄ structures with equally rotated phenyl rings (Figure 3) and without any imaginary vibration correspond to the K(D_4h,Λ_im) kernel subgroups for the coordinate of the a_1u representation. As indicated by its wavenumber ν_im, the energies of the corresponding vibrations are comparable with the highest energy ones in all category I complexes. The optimized structures of D_2d and C_4v symmetries are not stable (not presented), and hence only the D₄ ones fulfill the condition of the stable structure of the highest symmetry group.

For known ground state Γ₀ and JT coordinate Λ_im representations the Eqs. (4) and (5) enable to determine the excited state representations Γ_exc (see Table 1) interacting with the ground electron state in D_4h structures of our complexes. The comparison of the TD-DFT calculated electron states in the corresponding D_4h and D₄ structures (Table 3) shows that the energy difference E_exc between the PJT interacting states increases after symmetry descent as a consequence of their vibronic interaction. This confirms the correct assignment of the corresponding states in both groups because the standard treatment based on similarity of their oscillator strengths f is hardly usable in our cases. The ground states in the D_4h structures also correspond to those in their D₄ subgroups.

We can see (Table 3) that the PJT active excited states in the D_4h structures are relatively high. This indicates that their excitation energies are less important for possible vibronic interactions than the energies of JT active coordinates.

3.2. Category II complexes

As mentioned above, the ground state of the D_4h structures of these complexes is double degenerate (E_g or E_u representations) and only the stable D₂ structures (Figure 4) were obtained by geometry optimizations. Possible representations of the corresponding JT active coordinates can be obtained by the symmetric direct products for the HS group

[E_g ⊗ E_g]⁺ = [E_u ⊗ E_u]⁺ = A_1g ⊕ B_1g ⊕ B_2g

(6)

The full-symmetric a_1g coordinate does not change the symmetry of the structure (i.e. cannot be JT active) and the kernels

K(D_4h, b_1g) = D_2h(C₂’)

(7)

K(D_4h,b_2g) = D_2h(C₂”)

(8)

do not explain the existence of the optimized D₂ structure. Therefore, the method of the epikernel principle [5,6] cannot explain the D_4h → D₂ symmetry descent.

The method of step-by-step symmetry descent [8,9] for the structures of the D_4h symmetry group in double degenerate electron states is based on the scheme in Figure 5. Except D_2h, all immediate subgroups of the D_4h group (i. e. D₄, D_2d, C_4v and C_4h) are JT unstable because of preserved electron degeneracy. After the removal of the C₄ axis from the D₄ group we obtain its immediate subgroup D₂, which is JT stable because the degeneracy is removed here. Alternatively, the D₂ group can be obtained by symmetry descent through JT unstable D_2d group with preserved electron degeneracy. The structures of the C₄ and S₄ symmetry groups preserve electron degeneracy and therefore cannot be stable. We have not found any stable structure of D_2h, C_2v, or C_2h symmetry groups by geometry optimization. Therefore, the stable D₂ structures meet the condition of maximal groups for category II complexes.

During the symmetry descent from D_4h to D₂ group the double degenerate representation is split into the nondegenerate B₂ and B₃ ones

E_g or E_u (D_4h) → E (D₄ or D_2d) → B₂ ⊕ B₃ (D₂)

(9)

None of these representations corresponds to the calculated ground electron state of the stable D₂ structures. (Table 4). This discrepancy can be explained by the above-mentioned HPJT effect and the ground state of the D₂ structure corresponds to the lower PJT interacting excited state in its supergroup. In our case, the situation is complicated by the fact that D₂ is not an immediate subgroup of the D_4h group. Therefore, three possible two-step symmetry descent paths are possible:

i) D 4_{h} \overset{a_{1 u}}{\to} D_{4} \overset{b_{1} {o r b}_{2}}{\to} D_{2}

(10)

ii) D 4_{h} \overset{b_{1 g} {o r b}_{2 g}}{\to} D 2_{h} \overset{a_{u}}{\to} D_{2}

(11)

iii) D 4_{h} \overset{b_{1 u} {o r b}_{2 u}}{\to} D 2_{d} \overset{b_{1}}{\to} D_{2}

(12)

Based on group-subgroup relations we can assign the electron state representations of the D₂ symmetry group to the corresponding D₄, D_2h or D_2d ones despite there can be several alternatives. The same holds for the D_4h to D₄, D_2h, or D_2d symmetry descent. For known representations of ground states and JT active coordinates, it might be possible to determine the HPJT excited state representations according to Eq. (4). For the two-step symmetry descent there are too many possibilities where almost all excited-state representations (except two-dimensional) might be involved. Therefore, we shall not deal with this problem.

4. Methods

We have performed geometry optimization of the complexes ^m[dmaphPcAg]^q with charges q = +1 → -2 in the two lowest spin states (singlet to quartet) with spin multiplicities m within the D_4h symmetry and its subgroups. In agreement with our previous study [1], B3LYP hybrid functional [12], GD3 dispersion correction [13], cc-pVDZ-PP pseudopotential and basis set for Ag [14] and cc-pVDZ basis sets for remaining atoms [15] were used. The optimized structures were tested on the number of imaginary vibrations. Excited states (up to 50) were calculated for every optimized structure using time-dependent DFT (TD-DFT) treatment [16,17] analogously to our previous studies [1,18,19]. All calculations were performed with Gaussian16 (Revision B.01) software [20]. MOLDRAW (https://www.moldraw.software.informer.com, accessed on 9 September 2019) software [21] was used for visualization and geometry modification purposes. Finally, a group-theoretical analysis of the obtained results was carried out using the methods of the epikernel principle [5,6,7] and of step-by-step symmetry descent [8,9].

5. Conclusions

To shed more light on the photopolymerization action of the Ag(II) complex with dimethylamino phenyl-substituted phtalocyanine ²[dmaphPcAg]⁰, we have performed a quantum-chemical model study of possible reaction intermediates [dmaphPcAg]^q with charges q = +1 → -2 with the aim to explain the possible role of the JT effect. Group-theoretical analysis of the obtained results shows that the complexes under study are of two categories.

Stable structures of maximal symmetry of ¹[dmaphPcAg]⁺, ³[dmaphPcAg]⁺, ²[dmaphPcAg]⁰ and ⁴[dmaphPcAg]^2- complexes (category I) correspond to the D₄ group as a consequence of the PJT effect within the unstable D_4h structure. On the other hand, complexes ⁴[dmaphPcAg]⁰, ¹[dmaphPcAg]^-, ³[dmaphPcAg]^- and ²[dmaphPcAg]^2- (category II) with double degenerate electron ground states in (JT unstable) D_4h symmetry structures undergo a symmetry descent to stable structures corresponding to maximal D₂ symmetry not because of a simple JT effect, but due to HPJT effect. The most stable reaction intermediate in the supposed photoinitiation reaction [1] is surprisingly ³[dmaphPcAg]^- (a singlet electron state was expected). Therefore, the reduction of ²[dmaphPcAg]⁰ photoinitiator (D₄ symmetry) to ³[dmaphPcAg]^- intermediate (D₂ symmetry) must be significantly affected by vibronic interactions (PJT and HPJT effects). Their influence on reaction rates and thermodynamics must be investigated in solutions.

The presented study shows how group-theoretical treatment can be profitable by solving chemical problems for large molecules. The method of epikernel principle [5,6,7] is restricted to JT active coordinates based on perturbation theory and thus grants incomplete results but can also be used for the PJT effect. The method of step-by-step symmetry descent [8,9] based on splitting degenerate electron states during symmetry descent is more universal but not suitable for the PJT effect. We have shown the usefulness of the combination of both methods. Further theoretical studies in this field are welcome.

Author Contributions

Investigation, writing—original draft preparation, writing—review and editing, M.B. All authors have read and agreed to the published version of the manuscript.

Funding

The work has been supported by the Slovak Research and Development Agency (no. APVV-19-0087), by the Slovak Scientific Grant Agency VEGA (no. 1/0139/20), and by Ministry of Education, Science, Research and Sport of the Slovak Republic within the scheme “Excellent research teams”.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data is contained within the article.

Acknowledgments

The authors thank the HPC center at the Slovak University of Technology in Bratislava, which is a part of the Slovak Infrastructure of High Performance Computing (SIVVP project ITMS 26230120002 funded by the European Region Development Funds) for the computational time and resources made available.

Conflicts of Interest

The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Sample Availability

Not applicable.

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Figure 1. Structure of ²[dmaphPcAg]⁰ [1].

Figure 2. Optimized D_4h structure of ²[dmaphPcAg]⁰ (C – black, N – blue, H – white, Ag – grey).

Figure 3. Optimized D₄ structure of ²[dmaphPcAg]⁰ (see Fig. 2 for atom notation).

Figure 4. Optimized D₂ structure of ¹[dmaphPcAg]^- (see Figure 2 for atom notation).

Figure 5. Possible symmetry descent paths of D_4h structures in double degenerate electron states [9]. The top and bottom lines of individual rectangles denote symmetry groups and the corresponding irreducible representations, respectively.

Table 1. Charge q, spin multiplicity m, symmetry group G, representation of the ground electron state Γ₀, DFT energy E_DFT, JT stabilization energy E_JT, representations Λ_im and wavenumbers ν_im of imaginary vibrations, kernel K(D_4h,Λ_im) and epikernel K(D_4h,Λ_im) subgroups of D_4h, and representations of relevant PJT excited states Γ_exc of ^m[dmaphPc]^q complexes under study (the preserved symmetry elements in the kernel and epikernel subgroups are in parentheses). The most stable structure is shown in bold.

q	m	G	Γ₀	E_DFT [Hartree]	E_JT [eV]	Λ_im	ν_im [cm^-1]	K(D_4h,Λ_im)	E(D_4h,Λ_im)	Γ_exc
+1	1	D_4h	¹A_1g	-4734.55067	-	b_1u	-48	D_2d(C₂’)		B_1u
						2e_g	-47, -31	C₁	C_2h(C₂’), C_2h(C₂”)	E_g
						a_1u	-46	D₄		A_1u
						a_2u	-31	C_4v		A_2u
						b_2u	-31	D_2d(C₂”)		B_2u
+1	1	D₄	¹A₁	-4734.58352	0.894	-	-
+1	3	D_4h	³B_1u	-4734.54698	-	b_1u	-18	D_2d(C₂’)		A_1g
						e_g	-18	C₁	C_2h(C₂’), C_2h(C₂”)	E_u
						a_1u	-18	D₄		B_1g
+1	3	D₄	³B₁	-4734.58550	1.048	-	-
0	2	D_4h	²B_1g	-4734.74708	-	b_1u	-42	D_2d(C₂’)		A_1u
						e_g	-42	C₁	C_2h(C₂’), C_2h(C₂”)	E_g
						a_1u	-41	D₄		B_1u
						a_2u	-23	C_4v		B_2u
						3b_2u	-23(3×)	D_2d(C₂”)		A_2u
0	2	D₄	²B₁	-4734.77283	0.701	-	-
0	4	D₂	⁴B₂	-4734.73113	?	-	-
-1	1	D₂	¹A	-4734.79846	?	-	-
-1	3	D₂	³B₂	-4734.83355	?	-	-
-2	2	D₂	²B₁	-4734.80520	?	-	-
-2	4	D_4h	⁴B_1u	-4734.77355	-	b_1u	-47	D_2d(C₂’)		A_1g
						2e_g	-46,-27	C₁	C_2h(C₂’), C_2h(C₂”)	E_u
						a_1u	-45	D₄		B_1g
						a_2u	-27	C_4v		B_2g
						b_2u	-27	D_2d(C₂”)		A_2g
-2	4	D₄	⁴B₁	-4734.80581	0.878	-	-

Table 2. Charge q, spin multiplicity m, electron configuration, and ground electron state representation Γ₀ of studied ^m[dmaphPc]^q complexes of D_4h, symmetry group.

q	m	Electron configuration	Γ₀
+1	1	…(b_2g)²(e_u)⁴(a_2g)²(b_1g)⁰(e_g)⁰(b_1u)⁰…	¹A_1g
+1	3	α: …(b_1g)¹(a_2g)¹(e_u)²(b_2g)¹(e_g)⁰(b_1u)⁰…$$$β: …(e_u)²(b_2g)¹(a_1u)⁰(e_g)⁰(b_1g)⁰(b_1u)⁰…	³B_1u
0	2	α:…(e_u)²(b_2g)¹(b_1g)¹(a_1u)¹(e_g)⁰(b_1u)⁰…$$$β:…(e_u)²(b_2g)¹(a_1u)¹(e_g)⁰(b_1g)⁰(b_1u)⁰…	²B_1g
0	4	?	⁴E_g or ⁴E_u
-1	1	?	¹E_g or ¹E_u
-1	3	?	³E_g or ³E_u
-2	2	?	²E_g or ²E_u
-2	4	α:…(e_g)²(b_2g)¹(a_1u)¹(b_1g)¹(e_g)²(b_2u)⁰…$$$β:…(e_g)²(a_2u)¹(a_1u)¹(e_g)⁰(a_2u)⁰(b_2u)⁰…	⁴B_1u

Table 3. Charge q, spin multiplicity m, symmetry group G, ground electron state representation Γ₀, representations Γ_exc, excitation energies E_exc and oscillator strengths f of the low excited electron states of the studied ^m[dmaphPc]^q complexes in D_4h and D₄ symmetry groups. The excited states that interact with the ground states are in bold.

q	m	G	Γ₀	Γ_exc	E_exc [eV]	f	G	Γ₀	Γ_exc	E_exc [eV]	f
+1	1	D_4h	¹A_1g	1¹A_2g	0.156	0.000	D₄	¹A₁	1¹B₁	0.167	0.000
				1¹E_u	0.158	0.002			1¹B₂	0.255	0.000
				1¹B_2g	0.159	0.000			1¹E	0.259	0.000
				1¹B_1g	0.349	0.000			1¹A₂	0.267	0.000
				2¹E_u	0.353	0.081			2¹E	0.631	0.014
				1¹B_1u	0.377	0.000			1¹A₁	0.642	0.000
				1¹A_1g	0.487	0.000			2¹B₁	0.858	0.000
				1¹E_g	1.010	0.000			3¹E	1.187	0.050
				1¹A_1u	1.010	0.000			2¹A₁	1.192	0.001
				1¹B_1u	1.010	0.000			2¹A₂	1.200	0.000
+1	3	D_4h	³B_1u	1³B_2g	0.011	0.000	D₄	³B₁	1³E	0.188	0.007
				1³A_2g	0.015	0.001			1³B₂	0.190	0.000
				1³E_u	0.016	0.001			1³A₂	0.195	0.000
				1³A_1g	0.235	0.000			1³A₁	0.600	0.000
				2³E_u	0.235	0.014			2³E	0.617	0.298
				1³B_2u	0.239	0.000			1³B₁	0.779	0.000
				2³E_u	1.255	0.000			3³E	1.244	0.001
				3³E_u	1.327	0.003			4³E	1.391	0.033
				1³E_g	1.493	0.008			2³B₁	1.400	0.000
0	2	D_4h	²B_1g	1²E_u	1.195	0.000	D₄	²B₁	1²E	1.157	0.000
				1²E_g	1.306	0.000			2²E	1.305	0.000
				1²B_1u	2.014	0.000			3²E	1.901	0.624
				2²E_u	2.050	0.710			1²B₂	1.942	0.000
				2²E_g	2.140	0.000			1²A₂	1.944	0.000
				1²B_2u	2.140	0.000			4²E	1.956	0.004
				1²A_2u	2.140	0.000			2²A₂	1.969	0.000
				3²E_g	2.163	0.000			1²A₁	2.055	0.000
				2²B_2u	2.164	0.000			3²A₂	2.056	0.000
				2²A_2u	2.164	0.000			1²B₁	2.059	0.001
-2	4	D_4h	⁴B_1u	1⁴E_g	0.957	0.000	D₄	⁴B₁	1⁴E	0.888	0.191
				1⁴A_2u	0.976	0.000			1⁴B₁	1.356	0.000
				1⁴B_2u	0.978	0.000			2⁴E	1.360	0.010
				2⁴E_g	1.025	0.000			1⁴A₁	1.360	0.000
				1⁴A_1u	1.042	0.000			3⁴E	1.396	0.010
				1⁴B_1u	1.048	0.000			1⁴A₂	1.415	0.000
				3⁴E_g	1.118	0.110			1⁴B₂	1.417	0.000
				1⁴A_1g	1.495	0.000			4⁴E	1.436	0.012
				1⁴B_1g	1.495	0.000			2⁴B₂	1.472	0.001
				1⁴E_u	1.496	0.000			2⁴A₂	1.481	0.000
				2⁴A_1u	1.501	0.000			3⁴B₁	1.523	0.000

Table 4. Charge q, spin multiplicity m, representations of the ground electron state Γ₀ and of the excited states Γ_exc, excitation energies E_exc and oscillator strengths f of low excited electron states of stable ^m[dmaphPc]^q complexes under study in the D₂ symmetry group. The irreducible representations obtained by splitting the two-dimensional representations of the degenerate ground state in D_4h structures are in bold.

q	m	Γ₀	Γ_exc	E_exc [eV]	f	q	m	Γ₀	Γ_exc	E_exc [eV]	f
0	4	⁴B₂	1⁴B₁	0.349	0.000	-1	1	¹A	1¹B₂	0.523	0.000
			1⁴B₃	0.814	0.012				1¹B₁	0.539	0.000
			2⁴B₁	0.829	0.000				1¹B₃	0.815	0.000
			1⁴B₂	0.927	0.006				2¹B₂	1.337	0.023
			3⁴B₁	0.948	0.000				2¹B₃	1.478	0.028
			2⁴B₃	1.253	0.000				3¹B₂	1.551	0.005
			2⁴B₂	1.254	0.213				4¹B₂	1.793	0.809
			1⁴A	1.291	0.000				3¹B₃	1.965	0.018
			3⁴B₃	1.356	0.164				1¹A	2.056	0.000
			2⁴A	1.423	0.000				4¹B₃	2.097	0.011
-1	3	³B₂	1³B₁	0.272	0.000	-2	2	²B₁	1²B₁	0.048	0.000
			1³B₃	1.256	0.000				2²B₁	0.377	0.000
			1³B₂	1.350	0.021				1²B₂	0.900	0.007
			2³B₃	1.356	0.014				2²B₂	1.095	0.283
			2³B₂	1.796	0.792				1²B₃	1.112	0.306
			3³B₃	1.835	0.007				3²B₂	1.168	0.000
			4³B₃	1.987	0.012				1²A	1.343	0.000
			1³A	1.993	0.000				4²B₂	1.348	0.002
			2³B₁	2.000	0.000				2²A	1.362	0.000
			2³A	2.092	0.000				2²B₃	1.367	0.002

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On the Jahn-Teller Effect in Silver Complexes of Dimethyl Amino Phenyl Substituted Phthalocyanine

Abstract

1. Introduction

2. Theoretical background

3. Results

3.1. Category I complexes

3.2. Category II complexes

4. Methods

5. Conclusions

Author Contributions

Funding

Institutional Review Board Statement

Informed Consent Statement

Data Availability Statement

Acknowledgments

Conflicts of Interest

Sample Availability

References

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