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

Abstract

The structures of Ag complexes with dimethyl amino phenyl substituted phthalocyanine ^m[dmaphPcAg]^q of various charges q and in the two lowest spin states m were optimized using the B3LYP method within the D_4h 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. The 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 D_4h structure. Complexes ⁴[dmaphPcAg]⁰, ¹[dmaphPcAg]⁻, ³[dmaphPcAg]⁻, and ²[dmaphPcAg]²⁻ with double degenerate electron ground states in D_4h 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 a 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.

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 phthalocyanine 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 via 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.

Figure 1.

Structure of ²[dmaphPcAg]⁰ [1].

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 abovementioned 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:

$${\left(\frac{\partial E}{\partial Q_{\mathrm{JT}}}\right)}_{\mathrm{HS}}\ne 0$$ (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].

In some cases, the existence of LS structures cannot be explained by the JT effect of the degenerate ground electron state nor by the PJT interactions of the non-degenerate ground electron state with low-lying excited states in the HS structure (e.g., because of their different spin multiplicity). (P)JT effects in these cases are “hidden” in the excited states, which can undergo vibronic interactions as well. The vibronic interaction within the degenerate excited state or between two (or more) excited states of suitable symmetries can be so strong that the lower energy 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].

(P)JT potential surfaces can be described using an analytical function based on perturbation theory. In the simplest case of single distortion coordinate Q and double electron degeneracy or two interacting electron states of different symmetries, we obtain their potential energy surface E(Q) in the form

$$E\left(Q\right)=\frac{1}{2}K{Q}^2\pm [\frac{{∆}_{\mathrm{ij}}^2}{4}+F^2{Q}^2{]}^{1/2}$$ (3)

where Δ_ij = 0 for double electron degeneracy (JT effect) or Δ_ij > 0 is the energy difference between both interacting electron states (PJT effect) in the undistorted geometry, K is the primary force constant (without vibronic coupling), and F is the vibronic coupling constant [3]. 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

$$<{\Psi }_{\mathrm{i}}\left|\frac{{\partial }^{}\widehat{H}}{\partial Q_{\mathrm{JT}}}\right|{\Psi }_{\mathrm{j}}>\ne 0$$ (4)

for the interacting electron states Ψ_i and Ψ_j and the full-symmetric Hamilton operator $\widehat{H}$ demands that the direct product

Γ_i* ⊗ Λ_JT ⊗ Γ_j (5)

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 (6)

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]⁺ (7)

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 Equation (7)) 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 vibronic interaction causes instability of the HS structure, and thus, in the sense of the JT theorem [2], some symmetry elements are removed. The probability of the symmetry elements removal decreases with the number of these elements. It implies that the symmetry decrease to the immediate subgroups with the lowest number of the removed symmetry elements are more probable than to other subgroups. The multidimensional representation corresponding to the degenerate electron state can be split within symmetry descent to an immediate subgroup of the parent HS group G (see Table S1 in Supplementary Materials). 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 to its immediate subgroups 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 through 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 phthalocyanine 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 the 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 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.

Figure 2.

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

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	EDFT [Hartree]	EJT [eV]	Λim	νim [cm−1]	K(D4h, Λim)	E(D4h, Λim)	Γexc
+1	1	D4h	1A1g	−4734.55067	-	b1u	−48	D2d(C2′)		B1u
						2eg	−47, −31	C1	C2h(C2′), C2h(C2″)	Eg
						a1u	−46	D4		A1u
						a2u	−31	C4v		A2u
						b2u	−31	D2d(C2″)		B2u
+1	1	D4	1A1	−4734.58352	0.894	-	-
+1	3	D4h	3B1u	−4734.54698	-	b1u	−18	D2d(C2′)		A1g
						eg	−18	C1	C2h(C2′), C2h(C2″)	Eu
						a1u	−18	D4		B1g
+1	3	D4	3B1	−4734.58550	1.048	-	-
0	2	D4h	2B1g	−4734.74708	-	b1u	−42	D2d(C2′)		A1u
						eg	−42	C1	C2h(C2′), C2h(C2″)	Eg
						a1u	−41	D4		B1u
						a2u	−23	C4v		B2u
						3b2u	−23(3×)	D2d(C2″)		A2u
0	2	D4	2B1	−4734.77283	0.701	-	-
0	4	D2	4B2	−4734.73113	unknown	-	-
−1	1	D2	1A	−4734.79846	unknown	-	-
−1	3	D2	3B2	−4734.83355	unknown	-	-
−2	2	D2	2B1	−4734.80520	unknown	-	-
−2	4	D4h	4B1u	−4734.77355	-	b1u	−47	D2d(C2′)		A1g
						2eg	−46, −27	C1	C2h(C2′), C2h(C2″)	Eu
						a1u	−45	D4		B1g
						a2u	−27	C4v		B2g
						b2u	−27	D2d(C2″)		A2g
−2	4	D4	4B1	−4734.80581	0.878	-	-

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 D_4h 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 through standard DFT methods.

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	…(b2g)2(eu)4(a2g)2(b1g)0(eg)0(b1u)0…	1A1g
+1	3	α: …(b1g)1(a2g)1(eu)2(b2g)1(eg)0(b1u)0… β: …(eu)2(b2g)1(a1u)0(eg)0(b1g)0(b1u)0…	3B1u
0	2	α:…(eu)2(b2g)1(b1g)1(a1u)1(eg)0(b1u)0… β:…(eu)2(b2g)1(a1u)1(eg)0(b1g)0(b1u)0…	2B1g
0	4	unknown	4Eg or 4Eu
−1	1	unknown	1Eg or 1Eu
−1	3	unknown	3Eg or 3Eu
−2	2	unknown	2Eg or 2Eu
−2	4	α:…(eg)2(b2g)1(a1u)1(b1g)1(eg)2(b2u)0… β:…(eg)2(a2u)1(a1u)1(eg)0(a2u)0(b2u)0…	4B1u

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.

Figure 3.

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

For known ground state Γ₀ and JT coordinate Λ_im representations, the Equations (4) and (5) can be used 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.

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	Γexc	Eexc [eV]	f	G	Γ0	Γexc	Eexc [eV]	f
+1	1	D4h	1A1g	11A2g	0.156	0.000	D4	1A1	11B1	0.167	0.000
				11Eu	0.158	0.002			11B2	0.255	0.000
				11B2g	0.159	0.000			11E	0.259	0.000
				11B1g	0.349	0.000			11A2	0.267	0.000
				21Eu	0.353	0.081			21E	0.631	0.014
				11B1u	0.377	0.000			11A1	0.642	0.000
				11A1g	0.487	0.000			21B1	0.858	0.000
				11Eg	1.010	0.000			31E	1.187	0.050
				11A1u	1.010	0.000			21A1	1.192	0.001
				11B1u	1.010	0.000			21A2	1.200	0.000
+1	3	D4h	3B1u	13B2g	0.011	0.000	D4	3B1	13E	0.188	0.007
				13A2g	0.015	0.001			13B2	0.190	0.000
				13Eu	0.016	0.001			13A2	0.195	0.000
				13A1g	0.235	0.000			13A1	0.600	0.000
				23Eu	0.235	0.014			23E	0.617	0.298
				13B2u	0.239	0.000			13B1	0.779	0.000
				23Eu	1.255	0.000			33E	1.244	0.001
				33Eu	1.327	0.003			43E	1.391	0.033
				13Eg	1.493	0.008			23B1	1.400	0.000
0	2	D4h	2B1g	12Eu	1.195	0.000	D4	2B1	12E	1.157	0.000
				12Eg	1.306	0.000			22E	1.305	0.000
				12B1u	2.014	0.000			32E	1.901	0.624
				22Eu	2.050	0.710			12B2	1.942	0.000
				22Eg	2.140	0.000			12A2	1.944	0.000
				12B2u	2.140	0.000			42E	1.956	0.004
				12A2u	2.140	0.000			22A2	1.969	0.000
				32Eg	2.163	0.000			12A1	2.055	0.000
				22B2u	2.164	0.000			32A2	2.056	0.000
				22A2u	2.164	0.000			12B1	2.059	0.001
−2	4	D4h	4B1u	14Eg	0.957	0.000	D4	4B1	14E	0.888	0.191
				14A2u	0.976	0.000			14B1	1.356	0.000
				14B2u	0.978	0.000			24E	1.360	0.010
				24Eg	1.025	0.000			14A1	1.360	0.000
				14A1u	1.042	0.000			34E	1.396	0.010
				14B1u	1.048	0.000			14A2	1.415	0.000
				34Eg	1.118	0.110			14B2	1.417	0.000
				14A1g	1.495	0.000			44E	1.436	0.012
				14B1g	1.495	0.000			24B2	1.472	0.001
				14Eu	1.496	0.000			24A2	1.481	0.000
				24A1u	1.501	0.000			34B1	1.523	0.000

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 through geometry optimizations. Possible representations of the corresponding JT active coordinates can be obtained via the symmetric direct products for the HS group:

[E_g ⊗ E_g]⁺ = [E_u ⊗ E_u]⁺ = A_1g ⊕ B_1g ⊕ B_2g (8)

Figure 4.

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

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′) (9)

K(D_4h, b_2g) = D_2h(C₂″) (10)

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 via symmetry descent through the 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 through geometry optimization. Therefore, the stable D₂ structures meet the condition of maximal groups for category II complexes.

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.

During the symmetry descent from D_4h to D₂, 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) (11)

None of these representations corresponds to the calculated ground electron state of the stable D₂ structures (Table 4). This discrepancy can be explained using 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, there are three possible two-step symmetry descent paths:

$${\mathrm{D}}_{4\mathrm{h}} \stackrel{a_{1u}}{\to } {\mathrm{D}}_4 \stackrel{b_1 {or b}_2}{\to } {\mathrm{D}}_2$$ (12)

$${\mathrm{D}}_{4\mathrm{h}} \stackrel{b_{1g}{ or b}_{2g}}{\to } {\mathrm{D}}_{2\mathrm{h}} \stackrel{a_u}{\to } {\mathrm{D}}_2$$ (13)

$${\mathrm{D}}_{4\mathrm{h}} \stackrel{b_{1u}{ or b}_{2u}}{\to } {\mathrm{D}}_{2\mathrm{d}} \stackrel{b_1}{\to } {\mathrm{D}}_2$$ (14)

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 through splitting the two-dimensional representations of the degenerate ground state in D_4h structures are in bold.

q	m	Γ0	Γexc	Eexc [eV]	f	q	m	Γ0	Γexc	Eexc [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

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 the 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 Equation (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 (Release 2.0, 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].

The B3LYP functional is most frequently used in quantum-chemical studies and produces relatively reliable excitation energies. The basis sets used are restricted by our technical capabilities.

5. Conclusions

To shed more light on the photopolymerization action of the Ag(II) complex with dimethylamino phenyl-substituted phthalocyanine 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.

The 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 the ²[dmaphPcAg]⁰ photoinitiator (D₄ symmetry) to the ³[dmaphPcAg]⁻ intermediate (D₂ symmetry) must be significantly affected by vibronic interactions (PJT and HPJT effects), primarily the reaction barrier height and reaction equilibrium. Moreover, the reactivity of ³[dmaphPcAg]⁻ is supported by the non-equal spin density at nitrogen atoms of the D₂ structure (doubled at one pair of these atoms; see Table S2 in Supplementary Materials). Nevertheless, their exact influence on reaction rates and thermodynamics must be investigated in solutions.

The presented study shows how group-theoretical treatment can be profitable through 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 it 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.

Supplementary Materials

The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/molecules28207019/s1, Table S1. Relations between the irreducible representations of the parent group D_4h and of its D₄ and D₂ subgroups. Table S2. Charge q, spin multiplicity m, symmetry group G, Mulliken atomic charges and spin populations of Ag and pyrrole N atoms in ^m[dmaphPc]^q complexes under study. Tables S3–S14. Atom coordinates of ^m[dmaphPc]^q in D_4h, D₄ or D₂ symmetry (in Å).

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

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.

Funding Statement

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”.