# Comparison of Model Systems (M+)n·[CrX63−] and M3CrX6 + 18MX Based on Quantum-Chemical Calculations (X: F, Cl)

Authors: Vyacheslav Kremenetsky; Sergey Kuznetsov
Journal: Journal of Chemistry
 (2016)

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## Abstract

On the basis of quantum-chemical calculations the most stable particle compositions are estimated in such model systems as (M+)n·[CrCl6] and M3CrCl6 + 18MCl (M = Na, K, and Cs). In all systems these particles are positively charged. For systems (M+)n·[CrCl6], (M+)n·[CrF6], M3CrF6 + 18MCl, M3CrF6 + 18MF, and M3CrCl6 + 18MCl (M = Na, K, and Cs) a number of energy parameters characterizing the state of the system before and after electron transfer are calculated. The results indicate the possibility of electron transfer from the cathode to the melt system, which is in the initial state. However, this possibility cannot be realized in systems where LUMOs (lowest unoccupied molecular orbitals) have purely ligand character. In this case, the preliminary deformation of a cationic shell of electroactive species is required; it transforms the initial system to the transition state. However, in all considered systems the search of the transition state should be carried close to the initial state Pi. This greatly simplifies a problem and transforms it from a purely theoretical sphere to the field of practical tasks that do not require exceptional cost of computer time.


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## Body

## 1. Introduction

Previously [1–4], quantum-chemical calculations of such model systems as (M+)n·[NbF7], (M+)n·[NbCl6], and (M+)n·[CrCl6] (M = Na, K, Cs; n=1÷nlim⁡) were carried out. These systems contain a complex anion with a cation outer-sphere (OS) shell. It was found that compositions with an intermediate number of OS cations m<nlim⁡, where nlim⁡ is the limiting number of OS cations bound to a given complex, are thermodynamically most stable. Such systems (designating them as the typeI system) are often used for modeling of the charge transfer process in melts [5–9]. The activation energy of electron transfer is estimated here based on Marcus theory [10]. These systems are suitable for express-evaluation of the charge transfer parameters, because it does not require large expenditures of computer time. However, it is clear that such simple systems can have only limited application. In particular, the effect of the anion composition of the electrolyte cannot be incorporated in the typeI systems.To verify the main findings for model systemsI quantum-chemical computations of some parameters have been carried out in extended model systems of typeII M3CrX6 + 18MX (M = Na, K, and Cs; X: F, Cl).Some results for the CrF6-containing systemsI andII were given previously [11–13]; for CrCl6-containing systemsI andII calculations were made in the nonrelativistic basis sets for M: Na and K in the main [2, 4, 14, 15]. Here we provide data for the systems (M+)n·[CrCl6] and M3CrCl6 + 18MCl in a quasi-relativistic ECP basis for M: Na, K, and Cs and an additional analysis of all these systems.


## 2. Computational Methods

The quantum-chemical calculations were performed with the Firefly program package [16], partially based on the codes of the GAMESS (US) program [17], by the density functional theory methods (DFT/UHF). The spin-polarized version of the Kohn-Sham equations and the B3LYP hybrid functional were used. All calculations are made with the quasi-relativistic ECP basis set of Stuttgart/Dresden groups [18–20], more exactly, with Stuttgart RSC 1997 ECP for Cr, K, and Cs and Stuttgart RLC ECP for F, Cl, and Na. In addition Na-systems in some cases used CRENBL ECP basis set [20]. The energy values correspond to true minima of the potential energy surface (the imaginary frequencies are absent). In all complexes chromium is in a high-spin state: s=3/2 for Cr(III) and s=2 for Cr(II). Spin contamination can be ignored: for the Cr(III) and Cr(II) particles, the S2 values were 3.76–3.82 and 6.01–6.03, respectively (3.75 and 6.00 in the ideal case).


## 3. Results and Discussion

### 3.1. Composition of the Most Stable Particles

In systemsII calculation of the interaction energy of the second coordination sphere fragments (M+)n with both the complex and the external environment of a given fragment can be made. Figures 1(a) and 1(b) show the plots of the energy of cationic OS shell formation Eos in systemsI (M+)n·[CrCl6]3− (Figure 1(a)) andII M3CrCl6 + 18MCl (Figure 1(b)) calculated by (1)Eos=Ep-Efc-nEM,where Ep, Efc, and EM are the energies of systemsI(or an equivalent fragment of systemsII), free [CrCl6]3− complex, and free M+ cation, respectively. The energy value Eos reflects the total effect of the interaction of OS cations M+ with the complex [CrCl6]3− and with each other.Figure 1
Energy of formation of the complex cationic OS shellEos (kJ/mol) versus the number of OS cations (n) in systemsI(M+)n⋅[CrCl6]3− (a) and systemsII M3CrCl6 + 18MCl (b) for (1) Na, (2) K, and (3) Cs.
(a) (b)Dependence of this type always has a minimum at some intermediatenmin value. The existence of the minimum is mainly caused by an increase in the repulsion between OS cations as their number increases. The composition of the system at point nmin is the most stable. In the case of systemsInmin=4 for all M (Figure 1(a)). The maximal number of OS cations retained by the chromium complex in systemsIis 6.Figure1(b) shows the energy of the OS shell formation Eos for analogous fragments of systemsII (M+)n·[CrCl6]3−. Note, in systemsII, the OS cations are ranked by increasing the (Cr-M) distances (r); that is, with increase of the number n on Figure 1(b) the distance r(Cr-M(n)) increases, too. The second coordination sphere of chromium in these systems contains 8-9 cations; however the Eos minimum corresponds to nmin equal to 4 (Na, K) or 5 (Cs) only. Figure 2 shows the examples of model structures for systemsIandII.Figure 2
Optimized structures of (a) 4Na+⋅[CrCl6]3− and (b) Na3CrCl6 + 18NaCl.
(a) (b)Thus, no significant displacement of the minimum was observed during the transition from systemsIto systemsII. This allows considering data based on results for systemsIas a valid initial estimate.Unlike the systemsI, for the systemsII the interaction energy of the second coordination sphere (M)n with the complex andthe rest of the system can be calculated directly. If the interaction energy of this shell with the complex exceeds the energy of its interaction with the outer environment, we can state that there is a dynamic equilibrium in the system responsible for the existence of rather stable complex species of definite composition [11]. Relevant data are presented below.Figure3 shows the Eint(f) energy of interaction of the (M+)n·[CrCl6]3− fragment with the outer environment (the rest of the systemsII) as a function of the OS cations number. The Eintf values were calculated from the following equation:(2)Eintf=Es-Ef-Er.Figure 3
Interaction energyEintf (kJ/mol) of the (M+)n⋅[CrCl6]3− species in systemsII M3CrCl6 + 18MCl versus the number of OS cations for (1) Na, (2) K, and (3) Cs.Here the symbolsEs, Ef, and Er denote energies of the entire model system, its fragment (Mn)·[CrCl6]3−, and the rest of systemsII, respectively.The maximal value of this energy is observed atnmax=3(4). The compositions closest in energy to the composition with the minimum energy are given in parentheses. Thus, the composition of the fragment (M+)n·[CrCl6]3− at which its interaction with the environment is minimal (n=3) is close to the most stable compositions mentioned in Figure 3 (n=4, 5).Finally, let us consider the function ΔEversus n in Figure 4. It was obtained in the following way. First, two types of the interaction energy of the cation OS shell (M+)n were calculated: (i) the energy of interaction with the complex Eint(Mn·com) and (ii) the energy of interaction with the rest of systemsIIEintMn·rest. The ΔE value in Figure 4 is equal to the difference:(3)ΔE=EintMn·rest-EintMn·com.Figure 4
The ΔE value (kJ/mol) in systems M3CrCl6 + 18MCl versus the number of OS cations for (1) Na, (2) K, and (3) Cs. Explanation in the text.If we denote such fragments as [CrCl6], (Mn), and the rest of system by characters A, B, and C, respectively, then the first energy Eint(Mn·com) corresponds to equation A+B=AB. The second energy, that is, EintMn·rest, characterizes the equation B+C=BC, while the energy ΔE corresponds to equilibrium (AB)+C=A+(BC).As follows from Figure4, in the range n=1–6, the energy of OS shell interaction (M+)n· with the complex is larger than with the external rest of system in all model systems. Maximum of this function corresponds to the value nmax=3(4).Thus, the most stable particles in systemsIIare (M+)n·[CrCl6]3− with n=4 (Na, K) or 5 (Cs), as follows from Figure 1(b). According to Figure 4, such fragments can be considered as a single particle relatively weakly bound to the surroundings; they have a positive charge for all M.


### 3.2. Comparison of the System Energies before and after Electron Transfer

Let us introduce the following notation.Pi is initial state of the system (before electron transfer); Pf is final state of the system (after electron transfer); Pfi is state of the system with the geometric structure Pi, but after the electron transfer (ET); Pif is state of the system with the geometric structure Pf, but before ET; E(P) is the energy of system in state P; ΔE1=EPf-E(Pi); ΔE2=EPfi-E(Pi); ΔE3=EPf-E(Pif). Electron transfer here means the transfer of one electron in the cathode process to the complex of chromium: [CrX6]3− + ē → [CrX6]4− (X – F, Cl).The energy values ΔE are given in the Table 1 for systemsI (M+)n·[CrF6]3− and (M+)n·[CrCl6]3− (n=2–4) and systemsII M3CrF6 + 18MCl, M3CrF6 + 18MF, and M3CrCl6 + 18MCl (M – Na, K, Cs).Table 1
The energy parameters (kJ/mol) of the systems before and after electron transfer. Explanation in text.
S (M+)n⋅[CrF6]3− M Na K Cs n 2 3 4 2 3 4 2 3 4 Δ E 1 291 14 −313 190 −31 −261 172 −51 −220 Δ E 2 293 17 −288 191 −29 −249 173 −33 −220 Δ E 3 289 10 −538 188 −34 −640 171 −28 −220  S (M+)n⋅[CrCl6]3− M Na K Cs n 2 3 4 2 3 4 2 3 4 Δ E 1 −186 −380 −569 −133 −389 −503 −152 −351 −457 Δ E 2 153 −151 −462 157 −87 −375 133 −49 −303 Δ E 3 −257 −516 −698 −177 −489 −681 −172 −484 −628  S M3CrF6 + 18MCl M3CrF6 + 18MF M3CrCl6 + 18MCl M Na K Cs Na K Cs Na K Cs Δ E 1 −265 −150 −111 −156 −67 −47 −332 −368 −354 Δ E 2 −43 −47 −38 −34 −34 −23 −223 −187 −122 Δ E 3 −428 −475 −419 −428 −91 −47 −543 −532 −495Here we should pay attention to the following. In the range ofn=2–4, the energy values ΔE2 and ΔE3change the sign in the system (M+)n·[CrF6]3−. Therefore, in this range of the n value mechanism of ET will depend on the composition of the electroactive particle. In particular for Na-system at n=3 there is intermediate structure state P∗ in which the intersection of the potential energy surfaces of states, before and after ET, occurs. This fact is the basis for estimation of the ET activation energy according to the Marcus theory [10]. If composition of the electroactive species in the Na-system is close to the one with n equal 4, then ET may occur directly to the complex in the initial state Pi (in case when there are no other restrictions). These considerations are also applicable to K-, Cs-systems.In the system (M+)n·[CrCl6]3− only functions ΔE2 change the sign. However, this is quite enough to use for states with n=3 and 4, the conclusions obtained above in relation to the initial state Pi. Low stability of the state Pif in comparison with the Pf state is due to the strong influence of the Jahn-Teller effect on the structure of the [CrCl6]4− complex after ET. For this reason, the state Pif is strongly destabilized and the ΔE3 energy is less than zero for all values of n.In the extended systemsII the ΔE values are less than zero for all M. In this case, formally electron transfer can occur at the initial state Pi. Some comments on this conclusion are made in the next section.


## 3.1. Composition of the Most Stable Particles

In systemsII calculation of the interaction energy of the second coordination sphere fragments (M+)n with both the complex and the external environment of a given fragment can be made. Figures 1(a) and 1(b) show the plots of the energy of cationic OS shell formation Eos in systemsI (M+)n·[CrCl6]3− (Figure 1(a)) andII M3CrCl6 + 18MCl (Figure 1(b)) calculated by (1)Eos=Ep-Efc-nEM,where Ep, Efc, and EM are the energies of systemsI(or an equivalent fragment of systemsII), free [CrCl6]3− complex, and free M+ cation, respectively. The energy value Eos reflects the total effect of the interaction of OS cations M+ with the complex [CrCl6]3− and with each other.Figure 1
Energy of formation of the complex cationic OS shellEos (kJ/mol) versus the number of OS cations (n) in systemsI(M+)n⋅[CrCl6]3− (a) and systemsII M3CrCl6 + 18MCl (b) for (1) Na, (2) K, and (3) Cs.
(a) (b)Dependence of this type always has a minimum at some intermediatenmin value. The existence of the minimum is mainly caused by an increase in the repulsion between OS cations as their number increases. The composition of the system at point nmin is the most stable. In the case of systemsInmin=4 for all M (Figure 1(a)). The maximal number of OS cations retained by the chromium complex in systemsIis 6.Figure1(b) shows the energy of the OS shell formation Eos for analogous fragments of systemsII (M+)n·[CrCl6]3−. Note, in systemsII, the OS cations are ranked by increasing the (Cr-M) distances (r); that is, with increase of the number n on Figure 1(b) the distance r(Cr-M(n)) increases, too. The second coordination sphere of chromium in these systems contains 8-9 cations; however the Eos minimum corresponds to nmin equal to 4 (Na, K) or 5 (Cs) only. Figure 2 shows the examples of model structures for systemsIandII.Figure 2
Optimized structures of (a) 4Na+⋅[CrCl6]3− and (b) Na3CrCl6 + 18NaCl.
(a) (b)Thus, no significant displacement of the minimum was observed during the transition from systemsIto systemsII. This allows considering data based on results for systemsIas a valid initial estimate.Unlike the systemsI, for the systemsII the interaction energy of the second coordination sphere (M)n with the complex andthe rest of the system can be calculated directly. If the interaction energy of this shell with the complex exceeds the energy of its interaction with the outer environment, we can state that there is a dynamic equilibrium in the system responsible for the existence of rather stable complex species of definite composition [11]. Relevant data are presented below.Figure3 shows the Eint(f) energy of interaction of the (M+)n·[CrCl6]3− fragment with the outer environment (the rest of the systemsII) as a function of the OS cations number. The Eintf values were calculated from the following equation:(2)Eintf=Es-Ef-Er.Figure 3
Interaction energyEintf (kJ/mol) of the (M+)n⋅[CrCl6]3− species in systemsII M3CrCl6 + 18MCl versus the number of OS cations for (1) Na, (2) K, and (3) Cs.Here the symbolsEs, Ef, and Er denote energies of the entire model system, its fragment (Mn)·[CrCl6]3−, and the rest of systemsII, respectively.The maximal value of this energy is observed atnmax=3(4). The compositions closest in energy to the composition with the minimum energy are given in parentheses. Thus, the composition of the fragment (M+)n·[CrCl6]3− at which its interaction with the environment is minimal (n=3) is close to the most stable compositions mentioned in Figure 3 (n=4, 5).Finally, let us consider the function ΔEversus n in Figure 4. It was obtained in the following way. First, two types of the interaction energy of the cation OS shell (M+)n were calculated: (i) the energy of interaction with the complex Eint(Mn·com) and (ii) the energy of interaction with the rest of systemsIIEintMn·rest. The ΔE value in Figure 4 is equal to the difference:(3)ΔE=EintMn·rest-EintMn·com.Figure 4
The ΔE value (kJ/mol) in systems M3CrCl6 + 18MCl versus the number of OS cations for (1) Na, (2) K, and (3) Cs. Explanation in the text.If we denote such fragments as [CrCl6], (Mn), and the rest of system by characters A, B, and C, respectively, then the first energy Eint(Mn·com) corresponds to equation A+B=AB. The second energy, that is, EintMn·rest, characterizes the equation B+C=BC, while the energy ΔE corresponds to equilibrium (AB)+C=A+(BC).As follows from Figure4, in the range n=1–6, the energy of OS shell interaction (M+)n· with the complex is larger than with the external rest of system in all model systems. Maximum of this function corresponds to the value nmax=3(4).Thus, the most stable particles in systemsIIare (M+)n·[CrCl6]3− with n=4 (Na, K) or 5 (Cs), as follows from Figure 1(b). According to Figure 4, such fragments can be considered as a single particle relatively weakly bound to the surroundings; they have a positive charge for all M.


## 3.2. Comparison of the System Energies before and after Electron Transfer

Let us introduce the following notation.Pi is initial state of the system (before electron transfer); Pf is final state of the system (after electron transfer); Pfi is state of the system with the geometric structure Pi, but after the electron transfer (ET); Pif is state of the system with the geometric structure Pf, but before ET; E(P) is the energy of system in state P; ΔE1=EPf-E(Pi); ΔE2=EPfi-E(Pi); ΔE3=EPf-E(Pif). Electron transfer here means the transfer of one electron in the cathode process to the complex of chromium: [CrX6]3− + ē → [CrX6]4− (X – F, Cl).The energy values ΔE are given in the Table 1 for systemsI (M+)n·[CrF6]3− and (M+)n·[CrCl6]3− (n=2–4) and systemsII M3CrF6 + 18MCl, M3CrF6 + 18MF, and M3CrCl6 + 18MCl (M – Na, K, Cs).Table 1
The energy parameters (kJ/mol) of the systems before and after electron transfer. Explanation in text.
S (M+)n⋅[CrF6]3− M Na K Cs n 2 3 4 2 3 4 2 3 4 Δ E 1 291 14 −313 190 −31 −261 172 −51 −220 Δ E 2 293 17 −288 191 −29 −249 173 −33 −220 Δ E 3 289 10 −538 188 −34 −640 171 −28 −220  S (M+)n⋅[CrCl6]3− M Na K Cs n 2 3 4 2 3 4 2 3 4 Δ E 1 −186 −380 −569 −133 −389 −503 −152 −351 −457 Δ E 2 153 −151 −462 157 −87 −375 133 −49 −303 Δ E 3 −257 −516 −698 −177 −489 −681 −172 −484 −628  S M3CrF6 + 18MCl M3CrF6 + 18MF M3CrCl6 + 18MCl M Na K Cs Na K Cs Na K Cs Δ E 1 −265 −150 −111 −156 −67 −47 −332 −368 −354 Δ E 2 −43 −47 −38 −34 −34 −23 −223 −187 −122 Δ E 3 −428 −475 −419 −428 −91 −47 −543 −532 −495Here we should pay attention to the following. In the range ofn=2–4, the energy values ΔE2 and ΔE3change the sign in the system (M+)n·[CrF6]3−. Therefore, in this range of the n value mechanism of ET will depend on the composition of the electroactive particle. In particular for Na-system at n=3 there is intermediate structure state P∗ in which the intersection of the potential energy surfaces of states, before and after ET, occurs. This fact is the basis for estimation of the ET activation energy according to the Marcus theory [10]. If composition of the electroactive species in the Na-system is close to the one with n equal 4, then ET may occur directly to the complex in the initial state Pi (in case when there are no other restrictions). These considerations are also applicable to K-, Cs-systems.In the system (M+)n·[CrCl6]3− only functions ΔE2 change the sign. However, this is quite enough to use for states with n=3 and 4, the conclusions obtained above in relation to the initial state Pi. Low stability of the state Pif in comparison with the Pf state is due to the strong influence of the Jahn-Teller effect on the structure of the [CrCl6]4− complex after ET. For this reason, the state Pif is strongly destabilized and the ΔE3 energy is less than zero for all values of n.In the extended systemsII the ΔE values are less than zero for all M. In this case, formally electron transfer can occur at the initial state Pi. Some comments on this conclusion are made in the next section.


## 4. Conclusions

Results of this work indicate the possibility of ET directly to the initial particlePi, rather than through a transition state P∗. This finding needs to be clarified.Earlier [21] analysis of the nature of frontier molecular orbitals was held in systems M3CrX6 + 18MCl (X: F, Cl; M: Na, K, and Cs) and it was shown that ET through the OS cations is only possible in system Cs3CrF6 + 18CsCl. In other systems, the nature of the LUMO (lowest unoccupied molecular orbital) requires preliminary adjustment of the OS shell to provide direct contact ligands of the first coordination sphere with the electrode surface. That is, in these systems, ET has to take place in a transition state P∗.In work [21] it was also suggested that for the system Cs3CrF6 + 18CsCl charge transfer takes place in a state with geometric structure which is close to the structure of the initial system. In this work we obtain confirmation of this assumption.In the Cs3CrF6 + 18CsCl system ET occurs with abnormally high rate and apparently is limited by diffusion. The approach of the active particles to the electrode surface requires the overcoming of the activation barrier, which depends on the particle charge. Data related to the composition and the possible charge of such particles can be obtained from data such as in Figures 1, 3, and 4. An example of use of such approach is contained in the work [22] where the abnormal rate of ET in the system Na2NbCl6 + 18MCl in comparison with systems M2NbCl6 + 18MCl (M – K, Cs) has been explained by distinction of charge of the electroactive species.Furthermore, according to calculations, in some cases a higher rate of ET can occur for particles with less stability [13].Thus, analysis of the mechanism of electrochemical ET in melts requires taking into account a number of factors. Nevertheless, the main conclusion of this work remains true: the search of the transition state in the considered systems should be carried close to the initial statePi. This greatly simplifies a problem and transforms it from a purely theoretical sphere to the field of practical tasks that do not require exceptional cost of computer time.



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