Grégoire David

Nadia Ben Amor
email: benamor@irsamc.ups-tlse.fr

Tao Zeng

Nicolas Suaud

Georges Trinquier

Jean-Paul Malrieu

Difficulty of the evaluation of the barrier height of an open-shell transition state between closed shell minima: the case of small C 4n rings

C 4n cyclacenes exhibit strong bond alternation in their equilibrium geometry. In the two equivalent geometries the system keeps an essentially closed-shell character. The two energy minima are separated by a transition state suppressing the bond alternation, where the wave function is strongly diradical. The paper discusses the physical factors involved in this energy difference and possible evaluations of the barrier height. The simple comparison between the restricted DFT/B3LYP for the equilibrium with the broken symmetry DFT/B3LYP of the transition state underestimates or cancels the barrier in comparison with the most reliable CASPT2 or NEVPT2 results. The minimal (2 electrons in 2 Molecular Orbitals) Complete Active Space SCF overestimates the barrier, the subsequent second-order perturbation cancels it. Due to the collective character of the spin-polarization effect, it is necessary to perform a full π CASSCF + second-order perturbation to reach a reasonable value of the barrier, but this type of treatment cannot be applied to large molecules. DFT procedures treating on an equal foot the closed-shell and open-shell geometries have been explored, such as MRSF-TDDFT and a new spin-decontamination proposal, namely DFT-dressed CI, but the results still depend on the density functional. The M06-2X without or with spin-decontamination gives the best agreement with the accurate wave function results.

I) Introduction

The present paper formulates a methodological challenge: what may be an appropriate quantum chemical computational technique for the evaluation of the isomerization barrier between two isomers of essentially closed shell (CS) character when the transition state (TS) is of open-shell character? We consider the C 4n cyclacenes as paradigmatic of this situation.

Their ground state geometry exhibits a strong bond-length alternation of the CC bonds. There exist two degenerate isomers, obtained by interchanging short and long bonds. The chemical isomerization from one isomer to the other goes through a transition state geometry where all CC bonds have practically equal bond lengths. The simplest mono-electronic descriptions, starting from the Hückel Hamiltonian, predict that for the regular (non bond-alternating) geometry a degeneracy takes place between the two highest occupied molecular orbitals (MOs), which become singly occupied MOs (SOMOs). They transform as two components of an E-type irreducible representation and cannot interact. The system is then a pure diradical, since as will be discussed later, the wave function does not have any ionic component. The geometrical distortion introduces a strong bond alternation and breaks the symmetry, split the energies of the SOMOs, the lowest-energy one becoming doubly occupied. The phenomenon is usually described as a pseudo-Jahn-Teller distortion [START_REF] Jahn | Proc. R. Soc. Lond. A[END_REF][START_REF] Pearson | [END_REF][3][4] Then along the isomerization reaction path, an initially closed-shell wave function converts into a diradical function, before returning to a closed-shell character. The wave function for the essentially closed-shell minima may a priori be constructed from a closedshell single determinantal approximation, even if correlation effects may be added later.

Oppositely the zero-order description of the transition state is intrinsically bi-determinantal.

Section II will discuss the methodological challenge raised by this qualitative difference between the minima and the transition state.

Single-determinant approaches may nevertheless be employed. Around the equilibrium geometries the optimization of the MOs will lead to a closed-shell single determinant, either in the Hartree-Fock self-consistent field (SCF) or in the Kohn-Sham Density Functional Theory (DFT) formalisms. For the transition-state geometry a spin-symmetry breaking takes place, defining different MOs for different spins, in the so-called Unrestricted or Broken Symmetry (BS) formulation [5][6][7] of the SCF or DFT methods. The wave function passes smoothly from the restricted to the BS solutions somewhere between the equilibrium and the ACCEPTED MANUSCRIPT / CLEAN COPY transition state, as occurs in stretching of the homopolar single bonds. However, the BS single determinants are not eigenfunctions of the S 2 operator, they are spin-contaminated. A spindecontamination [8][9][10][11][12][13][14][15][16] seems a minimal pre-requisite for a DFT estimation of the barrier height.

The description of open-shell low spin states is a well-known challenge in DFT and especially in the context of the evaluation of singlet-triplet gaps, leading to various DFT-based proposals to overcome the problem raised by the use of BS determinants. [17][18][19][20][21] The simplest spindecontamination using a unique single S 2 mean-value presents severe defects as shown in refs 14,15 . The best procedure will consist in rigorous projection on symmetry-adapted spaces as suggested by Scuseria from wave function theory (WFT) solutions. 12,13 A simpler solution has been recently proposed by some of us. 16 Another way is to eliminate spin-contamination of spin- 17,20,18,19,21 flip (SF) time-dependent (TD)-DFT method by the use of mixed-reference reduced density matrix (MRSF-TDDFT). 22 Moving back to wave-function based methods (WFT), one may consider that since the transition state requires to consider 2 electrons in 2 MOs (namely the two SOMOs), one should as well treat the two upper-energy electrons in the set of the Highest Occupied MO (HOMO)

and Lowest Unoccupied MO (LUMO) and perform a Complete Active Space SCF 23 with 2 active electrons in 2 active MOs (CASSCF (2,2)), which in principle treats both geometries on an equal foot. From this simple description correlation effects might be estimated using standard multireference perturbation theories.

The numerical results reported in section III show the failure of these simple approaches:

1. Comparing the restricted DFT/B3LYP [24][25][26][27] energy for the closed-shell geometry with the BS-DFT/B3LYP energy of the transition state the barrier disappears or is extremely small, which is unlikely.

2. The CASSCF (2,2) barriers are very large, while the second-order perturbation puts the non-alternating geometry well below the alternating one.

This situation compels to improve the computational tools. We first try to assess a reasonable value of the barrier from intensively correlated WFT methods. A previous work 28 has shown that the spin-polarization effect, particularly important at the transition state, presents a cooperative character, and requires a full π CASSCF followed by a second-order perturbation treatment. This treatment may be consistently applied to both geometries and provides a ACCEPTED MANUSCRIPT / CLEAN COPY reliable evaluation of the barrier. Being costly, these computations have been performed on small C 4n rings only (n=3 or 4).

For large rings, a consistent DFT based treatment would definitely be welcome. Based on an analysis of the physical factors governing the singlet-to-triplet gap in diradicals, recent works have proposed a rigorous procedure exploiting BS-DFT computations and providing spincontamination free energies 15,16 . We propose in section IV a consistent extension of this treatment for the geometries where the DFT solution is of closed-shell character. These results will be compared to those of MRSF-TDDFT and of the best WFT methods.

II)

Physics of the problem and methodological challenges.

C 2n rings are alternant hydrocarbons (free from odd-membered rings) i.e. one may give one of two colors (red and blue) to each atom in such a way that each atom of a given color is linked to atoms of the other color. In these rings the numbers of red-and blue-colored atoms are equal. The Ovchinnikov's rule 29 , which may be demonstrated from Heisenberg or from Hubbard 30,[START_REF] Malrieu | Applications of Topological Methods in Molecular Chemistry[END_REF] Hamiltonians, says that ground state is a singlet state. In the monoelectronic Hückel Hamiltonian or in any mean-field calculation a regular C 4n ring exhibits two degenerate singly-occupied Molecular Orbitals. These MOs belong to two irreducible symmetry representations. The wave function is invariant under the rotation of the two SOMOs, and according to the so-called "mirror theorem" 32 it is possible to define a set of SOMOs where one of them is localized on the blue atoms while the other is localized on the red ones. Let us call a and b these two SOMOs. The singlet wave function may be written as a linear combination of two degenerate determinants

𝛹 𝑁 1 = |𝛱 𝑘 𝑘𝑘. (𝑎𝑏 + 𝑏𝑎)|/√2
where k runs on the closed-shell MOs. The subscript N indicates that the configuration is "neutral", each SOMO bearing only one electron, the superscript refers to the spin multiplicity.

A triplet state belongs to the same space configuration, the m s =0 component of which is

𝛹 3 = |𝛱 𝑘 𝑘𝑘. (𝑎𝑏 -𝑏𝑎)|/√2.
In introduces ionic components in the wave function (and therefore reduces <S 2 >), while the spin-dependent relaxation of the core MOs k' and k'' treats partially the spin-polarization effect, to the price of increasing <S 2 >. (13) The popular Yamaguchi correction [8][9][10][11] , supposed to offer a spin decontamination of the BS-DFT solution, treats both effects as the first one and is not reliable 11 . In the C 4n series the spin polarization is the only actor, for large values of n, <S 2 > reaches values close to 2, which would explode the correction.

Notice that for C 4n rings at the transition state there is no ionic component in the wave function, t ab =0, and a and b do not relax in a' and b'. The BS-DFT evaluation of its energy simply incorporates a biased evaluation of the spin-polarization, as detailed in section V.

In the CS wave function relative to the equilibrium geometry

𝜙 𝐶𝑆 = |𝛱 𝑘 𝑘𝑘. 𝑔𝑔|
the two upper-energy electrons occupy the same gerade HOMO. An ungerade virtual MO u (the LUMO) may be associated by the mirror theorem [START_REF] Longuet-Higgins | [END_REF] , and the g/u set of MOs span a set of localized MOs a and b, similar to the previously defined SOMOs. The CS determinant imposes a half-and-half ratio of the neutral (left) and ionic (right) Valence Bond components

𝜙 𝐶𝑆 = |𝛱 𝑘 𝑘𝑘. (𝑎𝑏 + 𝑏𝑎) + (𝑎𝑎 + 𝑏𝑏)|/2,
while the second component is of higher energy. The left/right or neutral/ionic correlation is missing, while in the TS geometry the ionic component is zero and the ionic component demixing is correctly taken into account. Moreover, the CS description totally ignores the spincorrelation between the two upper-energy electrons and the core electrons. One understands that the physics incorporated in, respectively, the CS and BS treatments is not balanced, to the disadvantage of the CS geometry.

One may go beyond the single determinant evaluation in either the WFT or the DFT philosophies. In the WFT approach one should at least treat correctly the interaction of the 2 upper electrons in the HOMO/LUMO set, through a CASSCF (2,2) treatment. This treatment misses the correlation between these 2 electrons and the core, which is expected to stabilize ACCEPTED MANUSCRIPT / CLEAN COPY the TS, and the barrier should be too large. One may attempt a perturbative treatment of the correlation beyond CASSCF (2,2) according to the CAS second-order perturbation theory (CASPT2) method 33,34 or the N-electron valence state second-order perturbation theory (NEVPT2) 35,36 in its strongly contracted formulation. In CASPT2 it is recommended to introduce an empirical IPEA shift 34 to get a better balance between essentially closed-shell and essentially radical situations. Since the use of the IPEA shift is a matter of debate [37][38][39][40][41][42][43][44][45][46] calculations without and with standard IPEA (0.25 a.u.) have been performed on all systems.

NEVPT2 is parameter-free and will be considered as the reference method in all the study.

A previous work 28 has shown that spin-polarization is a cooperative effect, which means that the spin-polarization of a given shell favors the spin-polarization of the other shells. This means that the treatment of spin polarization requires to introduce products of single excitations, i.e. multiple excitations. If spin polarization plays an important role in the barrier height, one should go to high excitation levels in the π system and perform at least full valence π CASSCF calculations. They miss the non-negligible spin-polarization of the σ core electrons, which has to be evaluated at least from a second-order perturbation.

The last section first reports the results of spin-contamination free method, MRSF-TDDFT. A recent work 16 has proposed a correct use of BS-DFT calculation, treating correctly the three main contributions, namely, direct exchange, kinetic exchange and spin-polarization. It may be applied to the TS geometry. An extension of this approach to the closed-shell geometry will be explored in the last section.

III) Numerical studies

The calculations have been performed using Gaussian 46 (DFT), Molcas [START_REF] Frisch | Gaussian 09[END_REF][START_REF] Karlström | [END_REF][49] (CASSCF/CASPT2), GAMESS (MRSF-TDDFT) 50 or Orca 51 (DFT 52 , CASSCF/NEVPT2) packages. The 6-311G** basis set has been used throughout this work. The impact of enlarging the basis set has been tested on the smaller compound.

A) Choice of systems and DFT/B3LYP geometry optimizations

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We consider five C 4n rings, namely the cyclobutadiene, a centro-saturated phenalene C 13 H 7 , crown-shaped C 16 H 16 and an internally saturated C 16 ring, C 19 H 13 (Figure 1). We have added another C 12 ring, namely the carbo-cyclobutadiene, C 12 H 4 , the π conjugated system of which involves 12 π electrons. The DFT geometry optimizations have been performed using the B3LYP [24][25][26][27] functional in the (6-311G**) basis set. All the geometries are given in Table 1.

For the cyclobutadiene, the closed shell structure exhibits a strong contrast between short (1.332 Å) and very long ( . This crown-shaped geometry presents a bond-length alternation (1.449/1.360 Å).

For all these five systems, the BS treatment converges on practically equal bond lengths with a very small deviation of the CC bond lengths from the ideal value 1.40 Å.

Furthermore, despite the introduction of internal sp 3 Of course, the DFT/B3LYP optimized geometries are not necessarily the exact ones, but they are sufficiently distinct to discuss the methodological problem of the energy difference between two closed-shell geometries and the diradical transition state (TS) and, even if the DFT/B3LYP geometries deviate slightly from the exact ones, their energy difference between these geometries should be close to the barrier height.

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The HOMO/LUMO and SOMOs are pictured in figure 2. For the four first systems, the SOMOs are found similar to the bond-centered HOMO/LUMO but due to their degeneracy they may be transformed into atom-centered as directly obtained in the C 19 H 14 compound.

B) Evaluations of barrier heights

For all systems, the energy difference between the closed and open-shell geometries are evaluated by means of DFT/B3LYP, with a closed-shell description for the CS geometry and a broken-symmetry one for the open-shell geometry. The zero-point energy (ZPE) is also evaluated (Table 2). Furthermore, the energy difference is also calculated by CASSCF + perturbation treatments, using a CAS(2,2) or a full π active space. The CAS(2,2) does not necessary provide meaningful SOMOs or HOMO/LUMO pairs as discussed for instance in the case of C 16 H 16 .

1) C 4 H 4

The direct DFT/B3LYP calculations without any spin decontamination suggest a barrier of 2.2 kcal/mol. Using the same basis set, a minimal CAS(2,2) active space gives a larger barrier of 12.1 kcal/mol at the CASSCF level while the CASPT2, without IPEA, is reversed, with a value of -7.0 kcal/mol. When an IPEA shift of 0.25 a.u. is applied, the CASPT2 "barrier" remains slightly negative, -1.3 kcal/mol, while SC-NEVPT2 gives a small positive value of 0.7 kcal/mol. When all the π orbitals are included in the active space (Full π), the CASSCF(4,4) gives a barrier of 6.1 kcal/mol, to be compared with the value of 4.8 kcal/mol obtained by a previous MCSCF study with a smaller 6-31G* basis set. 4 In order to evaluate the influence of the basis sets on our results, a larger 6-311++G(3df,3pd) basis set was also used and the CASSCF barrier value is of 6.7 kcal/mol (+0.6 kcal/mol compared to the 6-311G** basis set). As expected, the effect on the B3LYP result is small (+0.2 kcal/mol). The NEVPT2 treatment enhances the barrier to 8.4 kcal/mol (8.7 kcal/mol with the largest basis set). The corresponding CASPT2 result without IPEA lowers this value to 3.3 kcal/mol (3.1 kcal/mol); but adding the IPEA shift of 0.25 a.u. the barrier returns to 6.4 kcal/mol (6.7 kcal/mol). The effect of enlarging the basis set is quite small for all these methods and the 6-311G** can be considered as sufficient to treat the larger systems. However, in view of these discrepancies we have tried to perform an enlarged CASSCF calculation, adding the valence MOs relative to the sigma CC MOs (bonding and antibonding). This enlarged CASSCF calculation enhances the barrier to 10.9 kcal/mol, the PT2

ACCEPTED MANUSCRIPT / CLEAN COPY correction lowers it to 5.0 without IPEA, or to 7.1 with IPEA=0.25 a.u. These results can be compared to the 6.4 kcal/mol given by coupled-cluster method with full inclusion of the triple excitations (CCSDT) and 6.6 kcal/mol from MR-CCSD(T) . 61 These calculations were done with a smaller split-valence (3s2p1d/1s) basis set.

The dispersion of the results shows the difficulty to perform consistent calculations on the same system in two geometries where the ground state is physically very different. They nevertheless indicate the need to introduce the empirical IPEA shift in CASPT2 calculations to avoid an over-stabilization of the diradical TS, and confirm that the barrier is about 6.4 to 8.4 kcal/mol, much larger than the crude DFT/B3LYP value.

2) Carbo-cyclobutadiene C 12 H 4

Despite the existence of non-conjugated π bonds in the σ plane of the molecule, the π electron system involves 12 electrons and behaves as an anti-aromatic ring. For B3LYP the BS minimum is 2.7 kcal/mol below the closed shell solution. Actually, one finds a BS solution even in the closed-shell preferred geometry, as illustrated in Figure 3, which follows the energies of both solutions along a linear synchronous transit (LST) between the two geometries. It is however unlikely that this molecule is a diradical. The collective character of the spin-polarization phenomenon has been demonstrated on this precise problem 28

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4) Crown-shaped C 16 H 16

There is no bond alternation according to DFT/B3LYP, the closed shell minimum is found 0.9 kcal/mol above the BS one. In that case, the CASSCF(2,2) at the open-shell geometry leads to an essentially single-determinant closed-shell wave-function. The active orbitals are either localized on a unique π bond or delocalized, as expected, on the whole π system but, in the two cases, with occupation numbers close to 2 and 0. The values reported in

6) Summary

These evaluations of the barriers are summarized in Tables 3 and4

IV) Search of reliable DFT-based barrier energies

As previously noticed, one cannot rely on methods which treat differently the equilibrium (labeled as Type IV in Ref. 22 and shown in the middle of Figure 13 of Ref. 66 ), provide spin polarization in mixing with the singlet open-shell diradical configuration.

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The collinear spin-flipping (SF) mechanism is used in MRSF-TDDFT. In the collinear SF mechanism, configurations obtained by different SF transitions are only coupled by exact exchange. 67 Therefore, MRSF-TDDFT shall be used along with a functional with a large fraction of exact exchange. We chose the BHHLYP 24 functional in this work in our MRSF-TDDFT calculations, following the developers of this method 22,63,64

B) A DFT-dressed CI method

Let us recall first our recent proposal for a consistent spin-decontaminated evaluation of the energy of diradicals 16 . The method was proposed in the context of research of Singlet-Triplet energy gaps, and was based on the analysis of the physical factors governing this energy difference in WFT methods (see 51, 53). A transposition to DFT approaches was possible and led to a decomposition of several contributions, 68 from a few set of constrained DFT computations 15,16 . By the way a perfectly spin-contamination free method was proposed. 16 Let us summarize its content for the geometries where the lowest energy single determinant is of BS character.

As for the MRSF-TDDFT one starts from the definition of a closed-shell core and of SOMOs

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Then one defines a restricted singlet configuration, which appears to be "neutral" with one electron only in the localized MOs a and b

2 / ) .( 2 / ) .( 1 1 u u g g k k b a a b k k k k N ab -  = +  =  =  .
The energy of

𝜙 𝑎𝑏 ̅ 𝑅 = |∏ 𝑘𝑘 ̅ . 𝑎𝑏 ̅ 𝑘 | ab R ab R b a R b a R b a K E H E + =   =
gives access to the direct exchange integral,

< 𝛹 𝑁 1 |𝐻|𝛹 𝑁 1 > -< 𝛷 𝑎𝑏 𝑅 |𝐻| 𝛷 𝑎𝑏 𝑅 >=< 𝑎𝑏|𝑎𝑏 > = 2𝐾 𝑎𝑏 ,
which is necessarily positive.

Since, according to the mirror theorem a and b are located on disjoint sets of atoms this integral is small. For the TS geometry the optimized SOMOs of the singlet and of the triplet are almost identical in restricted approaches. For the other geometries the g and u SOMOs of the triplet are close to the HOMO and LUMO of the CAS(2,2) of the singlet state. Beyond the direct exchange ferromagnetic correction 2K ab , two main contributions can be identified namely:

a) The mixing between 𝛹 𝑁 1 and the "ionic" configuration away from TS geometry

2 / ) .( 2 / ) .( 1 u u g g k k b b a a k k k k I +  = +  =  ,
where the two electrons are in the same SOMO and subject to a much larger repulsion. In WFT, the energy difference between the ionic and neutral configurations is called U,

1 1 1 1 N N I I H H U   -   = ,
and they interact through a hopping integral 2t

ab ab I N t H 2 1 1 =   ,
in a 2x2 valence Configuration Interaction (CI) matrix

Ψ 𝑁 1 Ψ 𝐼 1 ( 𝐾 𝑎𝑏 2𝑡 𝑎𝑏 2𝑡 𝑎𝑏 𝑈 + 𝐾 𝑎𝑏 )
ACCEPTED MANUSCRIPT / CLEAN COPY This interaction, solving the 2x2 CI matrix, results in an energy stabilization of the singlet, called "kinetic exchange"

𝛥𝐾𝐸 = 𝑈 -(𝑈 2 + 16𝑡 𝑎𝑏 2 ) 1/2 2
Of course there is no kinetic exchange contribution on the TS geometry since t ab is then zero.

A BS calculation of the determinant

𝛷 𝑎 ′ 𝑏 ′ ̅̅̅̅̅ 𝑓𝑐 = |𝛱 𝑘 𝑘𝑘. 𝑎 ′ 𝑏 ′ ̅̅̅̅ |
relaxing the magnetic orbitals in the field of the frozen core (fc) gives access to the values of the t and U parameter 11,16 through the knowledge of its energy lowering and of <S 2 >

ab fc U b a fc U b a R b a K S E E U 2 ) 1 ( ) ( 2 , ' ' 2 , ' ' - - - = , 2 / 1 , ' ' 2 , ' ' ) 1 ( ) ( fc U b a fc U b a R b a ab S E E t - - = .
However, extracting t and U through the relaxation of the magnetic MOs is no longer valid when they lead to a closed-shell solution, and one requires to use a more general approach.

This may be done by using the two non-optimized lowest closed-shell determinants, This effect stabilizes the neutral component of the wave function. One may consider that one shifts the energy of the neutral component of the wave function by 𝐸 1 𝑆𝑃 [START_REF] Pearson | [END_REF] , and increases the

|∏𝑘𝑘 ̅ . 𝑔𝑔̅ | and |∏ 𝑘𝑘 ̅ . 𝑢𝑢 ̅|, ) ) .( ( ) ) .( ( 4 u u k k E g g k k E t k k  -  = ab k k k K gu k k E u u k k E g g k k E U +  -  +  = ) .( ( ) ) .( (
effective value of U to U' SP E U U 1 ' - = .
The final energy of the singlet state is given by defining a

𝐸 𝑆 = 𝐸 𝑎𝑏 𝑅 + 2𝐾 𝑎𝑏 + 1 2 (𝑈′ -(𝑈′ 2 + 16𝑡 𝑎𝑏 2 ) 1/2 ) + 𝐸 1 𝑆𝑃 .

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This procedure represents an extension of our previously proposed spin-decontaminated technique, exploiting BS solutions, to situations where spin-symmetry breaking does not occur between the HOMO and the LUMO to produce non-orthogonal SOMOs a' and b'. The method:

-starts from a restricted description of the lowest triplet state, which defines a pair of active MOs, -constructs a 2x2 CI matrix between a neutral and an ionic configuration from mean values of closed-shell determinants, -dresses this matrix under the spin-polarization effects obtained from constrained BS calculations,

-the final energy being the eigenvalue of the dressed CI matrix,

( E ab ̅ + K ab + 1 E SP 2 t ab 2 t ab U + K ab )
Notice that diagonalizing the dressed CI matrix by the spin polarization effects improves the weighting of the neutral component. We propose to call this procedure "DFT-dressed CI" (DFT-dCI).

C) Results

The overall results regarding the barriers are reported in Table 5 for the five members of the series, using different methods and different exchange correlation potentials. They should be compared with our best WFT estimates from large CASSCF + PT2 calculations.

The impact of the exchange correlation potential is crucial. According to Jacquemin et. al 70 , BHHLYP and M06-2X deliver reasonably balanced description of the single and double bond lengths. If one simply relies on the energy difference between the CS single determinant for the bond-alternating geometry and the BS determinant for the "transition state", BHHLYP 24,25,71 functional increases the defect of B3LYP since the bond alternation is destroyed in all systems, including C 4 H 4 . M06-2X 72 functional restores barriers, with reasonable values of their heights at this crude level of description. On the basis of this study, M06-2X functional ACCEPTED MANUSCRIPT / CLEAN COPY may be considered as the best functional for the evaluation of the barrier heights in such systems.

The MRSF-TDDFT single point calculations using the CS and BS structures indicate barriers for all five species. This is in qualitative agreement with the CASPT2-IPEA-0.25 reference results.

For the smallest C 4 H 4 system, the error is as small as 0.8 kcal/mol. For the larger systems, MRSF-TDDFT tend to overestimate the barriers by about 5 kcal/mol. Other than the typical ~3 kcal/mol error in energy calculations (see above for the ST gap estimation), another possible factor is that the "active space" with two electrons in two orbitals of the MRSF-TDDFT formalism is relatively smaller for those large systems. All the 𝜋 orbitals and electrons are not treated on the same footing, and this is especially so when the 𝜋 space becomes larger. When the spin polarization within the 𝜋 space is large, it is necessary to include the whole space in the zeroth-order description of the system, so that the cooperative spin polarization is treated in a self-consistent manner. The spin polarization is more significant at the TS structures.

Therefore, the inability of treating spin polarization through one-electron spin-flipping from doubly occupied orbitals to virtual orbitals becomes more pronounced, resulting in an overestimated energy. For comparison, the spin polarization is less significant at the CS structure and therefore, this inability does not overestimate the energy as much.

Consequently, the barrier is overestimated.

The electronic states of the MRSF-TDDFT calculations are consistent with expectation. At the CS geometries, the states are dominated by the |𝛱 𝑘 𝑘𝑘. 𝑔𝑔| closed-shell configurations, with the 0.98 amplitude for all five species. At the BS geometries, the states are dominated by the 𝛹 𝑁 1 open-shell configuration, with the amplitudes ranging from 0.97 to 0.98.

The DFT-dCI technique systematically improves the results with respect to the crude DFT values, stabilizing more the bond-alternating geometry than the TS, but does not provide sufficient barriers for B3LYP [24][25][26][27] , nor for BHHLYP 24,25,71 . M06-2X 72 overestimates the barriers at the crude DFT level but DFT-dCI now diminishes them to a good agreement with the best WFT values. The DFT-dCI always works in the right direction. For intermediate structures connecting the diradical TS and closed shell minima, the mixed character of the wave function is frequently problematic. 73 This problem has been considered in the carbo-cyclobutadiene

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C 12 H 4 . The DFT/MO6-2X-dCI presents satisfactory behavior as shown in Figure 4, the potential energy curve being in qualitative agreement with the Full π CASPT2 and NEVPT2 results.

Our DFT-dCI procedure enables us to identify the origin of the divergent values for different functionals. Table 6 summarizes the values of the t and U parameters which govern the neutral/ionic mixture and the kinetic exchange amplitude. The question of the dependence to the density functional approximation used, and especially the amount of exact exchange, has been intensively investigated in the context of magnetic couplings. The answer directly relies on the physics of the t and U parameters. [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88] The hopping integral is weakly sensitive to the functional, while the effective bi-electronic repulsion U, although nearly identical in B3LYP and BHHLYP, is dramatically reduced by the M06-2X potential. This reduction increases the amplitude of the kinetic exchange, acting on the bond-alternating region, and increases the barrier. We think that the decomposition and consistent combination of the three contributions: direct exchange, kinetic exchange and spin-polarization, are both numerically effective and physically enlightening.

V) Conclusion

The problem of the evaluation of the barrier height in C 4n rings appears to be difficult.

DFT/B3LYP approaches relying on energy minimizations of the CS determinant for the equilibrium geometry and of the BS open-shell determinant for the transition state underestimate or even suppress the barrier. The reasons of this unbalance have been analyzed

and discussed, the TS is subject to a strong spin-polarization, the counterpart of which (a spincorrelation effect) is not considered in the CS treatment. Moreover, the CS function imposes equal weights to the neutral and ionic components of the outer-valence wave function.

The use of a minimal active space with 2 active electrons in either the HOMO and LUMO or in the two SOMOs seems in principle to solve this unbalance, but numerical explorations show that the so-calculated barriers are far too large, and that the subsequent second-order perturbation reverses the energetic order. This dramatic impact of the post-CAS energy is due to the large spin-polarization correction. Going to a full valence π CASSF treatment provides more reasonable values of the barrier, but it is necessary to perform an additional second-ACCEPTED MANUSCRIPT / CLEAN COPY order correction. In CASPT2 treatments, even with this large CAS, the IPEA shift is required to restore a barrier.

We have finally exploited the MRSF-TDDFT method and formulated an extension of the recently proposed spin-decontaminated DFT method 16 for the evaluation of the singlet-state energy via a consistent treatment of kinetic and spin-polarization effects. The current proposition can be extended to situations where the singlet state is no longer subject to Symmetry Breaking. Of low computational cost, it may be applied to large systems. Indeed, here-addressed problem is not limited to the rather exotic problem of the isomerization barrier of C 4n rings. It concerns many isomerization problems between equilibrium geometries of essentially closed shell character, passing through a diradical transition state. Many isomerizations proceeding through the rotations around double bonds are of this type, but other reactions involve charge transfer transition state 73,89,90 . Cyclization reactions should also be considered.

In some sense the here-addressed question is linked to a more general problem, namely the dual vision of the molecular electronic population, either as composed of local on-bond electron pairs, or as of on delocalized energy-governed MOs. In the case of C 4n rings, at the transition-state geometry the on-bond localization 91,92 leads to two degenerate solutions, none of them being a satisfactory starting point for the building of the wave function. It is noteworthy that the symmetry-breaking taking place around this geometry calls the attention on a third representation of the electronic assembly as governed by effective spin antiferromagnetic interactions 93 . This multiplicity of viewpoints and entrance doors is not a defect but a richness of Quantum Chemistry. ** The two lines refer to butyne and butatriene edges respectively (see labels in Figure 1)

ACCEPTED MANUSCRIPT / CLEAN COPY ACCEPTED MANUSCRIPT / CLEAN COPY

  principle the minimization of the BS-DFT determinant ACCEPTED MANUSCRIPT / CLEAN COPY 𝛷 𝑈,𝑎′𝑏′ = |𝛱 𝑘 𝑘′𝑘". 𝑎′𝑏′| incorporates two distinct effects. The mixing of the orthogonal SOMOs a and b in a' and b'

  carbons the conjugated rings are almost planar in C 13 H 7 and C 19 H 13 . The Hessian presents a single negative root, the imaginary frequency being very low, and the associated change of geometry corresponds to the creation of bond-length alternation.

Figure 4

 4 Figure4reports the evolution between the bond-alternating geometry and the transition state for full π CASSCF, CASPT2 (without and with IPEA shift) and strongly or partially contracted NEVPT2 methods. One cannot rely neither on the CASSCF nor the CASPT2 results without IPEA. One can notice the remarkable agreement between the partially and strongly contracted NEVPT2 versions. The consistent results of the NEVPT2 and the CASPT2 with IPEA methods suggest that the bond length alternation at the minimum is less pronounced than the result obtained by DFT/B3LYP. It is not so surprising that such large correlation effects on the energy difference between two extreme geometries also impact the optimum bondlength alternation.62

  . The most reliable estimates are those provided by the valence π CASSCF + NEVPT2 or CASPT2 with the large value of the empirical IPEA parameter (0.25 a.u.). Without the perturbative correction the barrier is too large. The minimal CASSCF(2,2) gives exceedingly large values and the secondorder perturbation eliminates the barrier, with a deep diradical minimum. Comparing the DFT/B3LYP energies of the closed shell determinant for the minimum to that of the BS solution for the TS similarly destroys the barrier or propose a severely underestimate of its value. But the large CASSCF+ PT2 cannot be applied to large rings and an improvement of DFT approaches would be welcome.

  geometry from a closed-shell description and the transition-state geometry from an open-ACCEPTED MANUSCRIPT / CLEAN COPY shell approach, since the physics incorporated in these treatments are different. One needs a uniform treatment. We consider two proposals satisfying this requirement, namely the Mixed-Reference Spin-Flip Time-Dependent DFT and a new proposal, DFT-dressed CI. But of course one cannot forget the impact of the choice of the exchange-correlation potential, which will be discussed in the last part of this section.A) Mixed-Reference Spin-Flip Time-Dependent Density Functional TheoryThe details of MRSF-TDDFT are presented in Refs.22,[63][64][65] . Only a brief overview of this method is given here. MRSF-TDDFT was devised to solve the spin contamination problem of conventional the SF-TDDFT method. After a regular restricted open-shell Kohn-Sham DFT calculation with a setting of triplet spin multiplicity, a |𝛱 𝑘 𝑘𝑘. 𝑎𝑏| type Kohn-Sham determinant is obtained as the reference triplet state with m S = +1. M S is the magnetic quantum number of the total spin. 𝑎𝑏 here represent the two generic SOMOs with 𝛼 spin and can be replaced by 𝑔𝑢. Then a mixed-reference reduced density matrix is constructed, equally averaging the density matrix of this reference state and that of its m S = -1 analogue, |𝛱 𝑘 𝑘𝑘. 𝑎𝑏|. The Tamm-Dancoff approximation is then applied to such a mixed-reference reduced density matrix, resulting in decoupled linear-response equations of singlet and triplet states. There is hence no spin contamination in the resultant singlet and triplet (with m S = 0) states obtained from one-electron excitations and de-excitations along with spin-flipping. The 𝑎 → 𝑎 ̅ and 𝑏 → 𝑏 ̅ spin-flippings in the |𝛱 𝑘 𝑘𝑘. 𝑎𝑏| reference and the spin-reversed counterparts in the |𝛱 𝑘 𝑘𝑘. 𝑎𝑏| reference result in the singlet open-shell diradical configuration, and also the corresponding triplet configuration with m S = 0. The spin-reversed counterparts will not be mentioned below but they are implicitly included. The 𝑎 → 𝑏 ̅ and 𝑏 → 𝑎 ̅ spin-flippings and electron transfers result in the |𝛱 𝑘 𝑘𝑘. 𝑎𝑎| and |𝛱 𝑘 𝑘𝑘. 𝑏𝑏| ionic configurations, which can interact with the singlet open-shell diradical configuration through the transfer integral t ab . The spin-flipping and excitation of one electron from a doubly occupied orbital h to a virtual orbital p results in the |𝛱 𝑘≠h 𝑘𝑘. 𝑝𝑎𝑏ℎ| and |𝛱 𝑘≠h 𝑘𝑘. 𝑝𝑎𝑏ℎ| type configurations, where 𝑝 and ℎ stand for particle and hole orbitals. These configurations

from the energy minimization of a restricted open-shell triplet determinant

  localized SOMOs a and b generate symmetry-adapted MOs g and u, or HOMO and LUMO, by a 45° rotation,

1 𝑆𝑃 ≅ 3 (

 13 results in a set of parameters consistent with those obtained by the orbital relaxation. 69 b) The spin polarization of the core electrons by the exchange field of the unpaired electrons acting on the neutral component of the wave function is partly treated by relaxing ACCEPTED MANUSCRIPT / CLEAN COPY the core MOs in the differential exchange field of the unpaired electrons in the BS-DFT treatment, minimizing the energy of 𝛷 𝑈𝑎𝑏 = |𝛱 𝑘 𝑘′𝑘". 𝑎𝑏| of energy 𝐸 𝑎𝑏 . This function incorporates the effect of the singly-excited determinants 𝛷 ℎ ̅ →𝑝̅ ,𝑎𝑏 = |𝛱 𝑘≠h 𝑘𝑘 ℎ𝑝̅ 𝑎𝑏| and 𝛷 ℎ→𝑝,𝑎𝑏 = |𝛱 𝑘≠h 𝑘𝑘 𝑝ℎ ̅ 𝑎𝑏| but is not an S 2 eigenfunction. To satisfy this property one must include the spin-flip determinants 𝛷 ℎ→𝑝̅ 𝑆𝐹 = |𝛱 𝑘≠h 𝑘𝑘 ℎ ̅ 𝑝̅ 𝑎𝑏| 𝛷 ℎ ̅ →𝑝 𝑆𝐹 = |𝛱 𝑘≠h 𝑘𝑘 ℎ𝑝 𝑎 ̅𝑏 ̅ | which are doubly excited. Their effect may be called a spin-correlation effect between the unpaired electrons and the core electrons. It is proportional to that of the singly excited determinants, and it is possible to evaluate the energy brought by the spin-polarization and spin-correlation as 𝐸 𝐸 𝑈,𝑎𝑏 -𝐸 𝑎𝑏 ).

Figure 2 .

 2 Figure 2. HOMO/LUMO and SOMOs of the five C 4n rings obtained at the full π CASSCF level.MOs CS geometryMOs BS geometry

Figure 3 .

 3 Figure 3. Evolution of the DFT/B3LYP energy as a function of the geometry of the carbomerized cyclobutadiene C 12 H 4 for the BS (ms=0) and the closed shell singlet. LST stands for linear synchronous transit.

Figure 4 .

 4 Figure 4. Evolution of the full π CAS(12,12) based energies as a function of the geometry of the carbo-cyclobutadiene C 12 H 4 .The zero of energy has been taken at the BS geometry. LST stands for linear synchronous transit.

  1.579 Å ) bonds while a D 4h symmetry is kept for the BS structure with an intermediate value (1.440 Å) for the CC bond length . The carbo-cyclobutadiene, the less typical member of the series, exhibits two triple bonds and two cumulenic systems with four different CC bond length for the CS minimum (D 2h ) and only two, keeping the D 4h

symmetry, for the BS transition state. The closed-shell treatment always provides clear bond length alternation and marked deviations from planarity in C 13 H 10 and C 19 H 13 . In these systems the double bonds keep an in-plane geometry while the single bonds are twisted. The CC double bond lengths are between 1.351 and 1.352 Å for the C 13 H 10 and between 1.354 and 1.361 Å for the larger C 19 H 14 system. The long lengths are typical of sp 2 -sp 2 single bonds: 1.454/1.455 Å and 1.443/1.452 Å for C 13 H 10 and C 19 H 14 respectively. The considered C 16 H 16 system is not the most stable isomer but is a true minimum and the most regular one. For various structural studies of 16-annulenes, see for instance refs. 53- 60

Table 3

 3 IPEA produces reasonable value of 4.4 kcal/mol. This value is practically identical to that obtained from NEVPT2 (4.6 Kcal/mol).The internally graphanized C 19 H 14 nano-flake, which exhibits a peripheral conjugated C 16 ring, has also been studied through an all π valence CASSCF calculation. The DFT/B3LYP calculations converge to two well-characterized minima. The two singly-occupied MOs are essentially localized on "red" and "blue" sites as in CASSCF (see Fig.2). The closed-shell solution is lower by 1.0 kcal/mol only. The convergence of the CASSCF(2,2) is difficult and predicts a huge barrier of 40.7 kcal/mol. The corresponding NEVPT2 reverses the result (-8.1 kcal/mol). The full valence π CASSCF(16,16) calculation still predicts a large barrier (12.0 kcal/mol), which is dramatically cancelled to -0.7 kcal/mol by the CASPT2 treatment without IPEA. The recommended IPEA value restores a barrier to 6.0 kcal/mol. The NEVPT2 result was not obtained.

	are then

previous systems, over-stabilizes the non-alternating geometry. The CASPT2 with 0.25 a.u.

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Table 1 :

 1 Calculated B3LYP/6-311G** CC bond lengths (Å) at equilibrium geometries for closed-shell and open-shell (broken-symmetry) solutions. Mean bond lengths are given, with standard deviations in parenthesis when required. All species are transition states, except for C 12 H 4 and C 16 H 16 , which are true minima.

	Systems	Closed shell geometry	Broken symmetry geometry *
		Short CC-bond (Å)	Long CC-bond (Å)	CC-bond (Å)
	C 4 H 4	1.332	1.579	1.444
	C 12 H 4 **	1.211 (a) 1.261 (a')	1.429 (b) 1.331 (b')	1.234 1.380
	C 13 H 10	1.352 (0.001)	1.455 (0.001)	1.401 (0.000)
	C 16 H 16	1.360	1.449	1.403
	C 19 H 14	1.358 (0.003)	1.447 (0.003)	1.401 (0.003)

*

Table 2 .

 2 DFT/B3LYP calculations: <S 2 > and imaginary frequency (i) of the BS solution, barrier and corrected ZPE barrier between the two equivalent bond-alternating forms. For C 12 H 4 , the BS solution is a real minimum while the C 16 H 16 BS is a local minimum.

				E (kcal/mol)
	system	<S 2 > i (cm -1 ) Barrier ZPE
	C 4 H 4	1.05	405	2.2	-0.7
	C 12 H 4	1.12	-	(-2.9) -5.0
	C 13 H 10	1.12	250	0.5	-2.4
	C 16 H 16	1.20	-	(-0.9)	-3.5
	C 19 H 14	1.18	365	1.0	-2.1

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Table 3 .

 3 Energy difference (kcal/mol) between open-shell and closed-shell minima obtained by means of single determinant (CS or BS) DFT/B3LYP and CASSCF+PT2 calculations with minimal CAS(2,2) active spaces. The main weights of the wave functions of the two states are also given.

		DFT		CASSCF(2,2)	CASPT2	CASPT2	NEVPT2
		B3LYP			Ipea 0 a.u.	Ipea 0.25 a.u.
	C 4 H 4	2.2	ΔE	12.1	-7.0	-1.3	0.7
			Weight (CS geo) 0.96 |20>
				0.04 |02>
			Weight (BS geo) 0.50 |20>
				0.50 |02>
	C 12 H 4 -2.7 ΔE	24.7	-17.6	-4.1	-20.6
			Weight (CS geo) 0.99 |20>
				0.01 |02>
			Weight (BS geo) 0.50 |20>
				0.50 |02>
	C 13 H 10 0.5	ΔE	25.2	-10.9	1.2	-10.9
			Weight (CS geo) 0.99 |20>
				0.01 |02>
			Weight (BS geo) 0.51 |20>
				0.49 |02>
	C 16 H 16 -0.9 ΔE	25.7	(11.3)	(11.5)	(4.8)
			Weight (CS geo) 0.997 |20>
				0.003 |02>
			Weight (BS geo) 0.98 |20>
				0.02 |02>
	C 19 H 14 1.0	ΔE	40.7	X	X	-8.1
			Weight (CS geo) 0.97 |20>
				0.03 |02>
			Weight (BS geo) 0.50 |20>
				0.50 |02>

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Table 4 .

 4 . Energy difference (kcal/mol) between open-shell and closed-shell minima obtained by means of CASSCF+PT2 calculations with full π active spaces. The main weights of the wave functions of the two states are also given, expressed on the HOMO/LUMO or SOMOs.

				CAS Full π
			CASSCF	CASPT2	CASPT2	SC-NEVPT2
				Ipea 0 a.u.	Ipea 0.25 a.u.
	C 4 H 4	ΔE	6.1	3.3	6.4	8.4
		Weight (CS geo) 0.88 |20>
			0.04 |02>
		Weight (BS geo) 0.46 |20>
			0.46 |02>
	C 12 H 4	ΔE	6.1	-1.9	2.6	2.3
		Weight (CS geo) 0.75 |20>
			0.03 |02>
		Weight (BS geo) 0.36 |20>
			0.36 |02>
	C 13 H 10	ΔE	9.4	0.0	5.1	4.4
		Weight (CS geo) 0.74 |20>
			0.02 |02>
		Weight (BS geo) 0.35 |20>
			0.34 |02>
	C 16 H 16	ΔE	10.3	-1.5	4.4	4.6
		Weight (CS geo) 0. |20>
			0.01 |02>
		Weight (BS geo) 0.27 |20>
			0.27 |02>
	C 19 H 14	ΔE	11.8	-0.2	6.0
		Weight (CS geo) 0.66 |20>
			0.01 |02>
		Weight (BS geo) 0.51 |11>
			0.04 |20>
			0.04 |02>

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Table 5 .

 5 Energy difference (kcal/mol) between open-shell (OS) and closed-shell (CS) minima obtained by means of single determinant (CS or OS) DFT or DFT-dressed CI (DFT-dCI) using different functionals and MRSF-TDDFT/BHHLYP. C 4 H 4 C 12 H 4 C 13 H 10 C 16 H 16 C 19 H 14

	CASPT2-IPEA 0.25 (table 4) 6.4	2.6	5.1	4.4	6.0
	DFT/B3LYP	2.2 -2.7	0.5	-0.9	1.1
	DFT-dCI B3LYP	5.6 -0.9	1.2	0.0	1.0
	DFT/M06-2X	6.7	2.5	6.3	5.5	7.6
	DFT-dCI M06-2X	8.6	1.1	4.5	2.9	4.3
	DFT / BHHLYP	-5.3 -9.7	-5.9 -10.1 93.4 *
	DFT-dCI / BHHLYP	1.5 -4.3	-2.3	-4.0	-3.7
	MRSF-TDDFT/BHHLYP	5.6	7.2	9.7	9.4	11.0

* <S 2 > = 1.57

Table 6 .

 6 Hopping integral t and on-site repulsion U (in a.u.) from various functionals in the case of carbomerized cyclobutadiene. Figure 1. Structures of the five C 4n rings: cyclobutadiene C 4 H 4 , carbo-cyclobutadiene C 12 H 4 , centro-saturated phenalene C 13 H 10 , crown-shaped C 16 H 16 and internally saturated C 16 ring

	B3LYP	M06-2X	BHHLYP
	t -0.0171831 -0.020175 -0.020851
	U 0.0230019 0.0158654 0.0243505

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Acknowledgments: G.D. benefits of research funding from the European Union's Horizon 2020 research and program under the Marie Skłodowska-Curie grant agreement No 899546. Mésocentre of Aix-Marseille Université is acknowledged for allocated HPC resources. T.Z. thanks York University for the start-up grant (481333) and the Natural Sciences and Engineering Research Council (NSERC) of Canada for funding (RGPIN-2016-06276), and thanks Compute Canada for computational resources.

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Micheàl Moloney

Clément Cabaud

Nicolas Massoni

Shelly Stafford

Yurii K Gun'ko

Munuswamy Venkatesan

CEA, DEN, Agnès Grandjean
email: agnes.grandjean@cea.fr

Mícheál P Moloney

Searching for the nano effect in Cu-HCF (II) particles to improve Cs sorption efficiency: Highlighting the use of intrinsic magnetism

Introduction

Transition metal hexacyanoferrates (HCF's) have attracted great interest over the last several decades from the nuclear sector. This interest is due to this materials ability to selectively extract trace amounts of radioactive Cs from waste water at varying pH's, and also in the presence of competitive ions at high concentration. [START_REF] Draouil | Copper hexacyanoferrate functionalized single-walled carbon nanotubes for selective cesium extraction[END_REF][START_REF] Grandjean | Effect of the chemical nature of different transition metal ferrocyanides to entrap Cs[END_REF][START_REF] Ding | Remediation of radiocesium-contaminated liquid waste, soil, and ash: a mini review since the Fukushima Daiichi Nuclear Power Plant accident[END_REF][START_REF] Lee | Improved adsorption properties of granulated copper hexacyanoferrate with multi-scale porous networks[END_REF][START_REF] Bokor | Development and validation of an automated unit for the extraction of radiocaesium from seawater[END_REF][START_REF] Causse | Facile one-pot synthesis of copper hexacyanoferrate nanoparticle functionalised silica monoliths for the selective entrapment of Cs-137[END_REF] Prepared through a simple aqueous co-precipitation process, HCF's low toxicity combined with their aforementioned ability to quickly and selectively remove low concentrations of Cs from effluents which contain high concentrations of other M +1 , and M +2 , ions has made HCF's highly valuable materials. Here we look at copper (II) hexacyanoferrate (II) or simply Cu-HCF(II).

While Cu-HCF could be classified as a). an inorganic polymer, or b). a bimettic supramolecular complex, they are also in fact cube shaped nanoparticles with a generally reported size range of below 50 nm. [START_REF] Soek | Anisotropic behavior of layer-by-layer films using highly disordered copper hexacyanoferrate(II) nanoparticles[END_REF][START_REF] Catala | Nanoparticles of Prussian blue analogs and related coordination polymers: From information storage to biomedical applications[END_REF] However, problematically, a high degree of interparticle aggregation exists in these materials. [START_REF] Causse | Facile one-pot synthesis of copper hexacyanoferrate nanoparticle functionalised silica monoliths for the selective entrapment of Cs-137[END_REF] Particle aggregation leads to a to a loss of exposed surface area and therefore can affect the Cs sorption kinetics. While nanoparticle aggregation is generally avoided through the use of surface bound stabilising molecules, this approach is (generally speaking) not used in the nuclear industry. The reasons for this are two-fold; firstly,the presence of such a molecular layer may inhibit Cs sorption, and secondly, the industry prefers that waste forms avoid the use of organic species due to the radiolysis effect. Instead HCF nanoscale is normally tuned by encasing, or growing, the HCF particles inside template materials. These templates can range from soft polymers to hard inorganic materials such as silica microspheres, or highly ordered silica monoliths. [START_REF] Causse | Facile one-pot synthesis of copper hexacyanoferrate nanoparticle functionalised silica monoliths for the selective entrapment of Cs-137[END_REF][START_REF] Catala | Nanoparticles of Prussian blue analogs and related coordination polymers: From information storage to biomedical applications[END_REF][START_REF] Loos-Neskovic | A copper hexacyanoferrate/polymer/silica composite as selective sorbent for the decontamination of radioactive caesium[END_REF][START_REF] Sommer-Marquez | Reinforced silica monoliths functionalised with metal hexacyanoferrates for cesium decontamination: a combination of a onepot procedure and skeleton calcination[END_REF][START_REF] Turgis | An Original "Click and Bind" Approach for Immobilizing Copper Hexacyanoferrate Nanoparticles on Mesoporous Silica[END_REF] By employing template chemistry the particles are separated and dispersed throughout a porous template material thereby allowing for greater access to their surfaces. However, for financial reasons industrial setups are slow to adapt to new technologies and in fact aggregated HCFs are still used today to extract Cs from waste pools using physical aggregation to disperse the HCF in the effluent, and solid/liquid separation techniques to remove it once Cs has been extracted. [START_REF] Flouret | Design of an intensified coprecipitation reactor for the treatment of liquid radioactive wastes[END_REF] To our knowledge, there is a lack of an easy way to disaggregate HCF present as a suspension in aqueous solution.

In this paper, we propose to compare highly monodisperse Cu-HCF nanoparticles which were synthesised within the pores of a silica template to a "bulk" Cu-HCF suspensions, which has been disaggregated by the by the assistance of a magnetic field. [START_REF] Stuyven | Magnetic field assisted nanoparticle dispersion[END_REF] In the first part of this publication, we observe the spectroscopic properties (FT-IR, UV-Vis, XRD, and Elemental Analysis) of templated Cu-HCF nanoparticles and compared them to those of the highly aggregated Cu-HCF particle systems. The idea here is to determine if it is possible to elucidate any physical or chemical differences between the nano and bulk systems. If found, any electronic or structural differences, could then be discussed in relation to these materials ability to sorb Cs. In the second part, we propose a simple way to improve the extraction kinetics of bulk material without having to resort to template chemistry. The use of an external magnetic field to orientate or align magnetic particles is well recorded in the literature but it normally involves superparamagnetic particles. [START_REF] Tanase | Magnetic Alignment of Fluorescent Nanowires[END_REF][START_REF] Corr | Linear Assemblies of Magnetic Nanoparticles as MRI Contrast Agents[END_REF][START_REF] Dai | Magnetically-responsive self assembled composites[END_REF] By simply applying a magnetic field to the aggregated particles it is possible to disaggregate them, [START_REF] Stuyven | Magnetic field assisted nanoparticle dispersion[END_REF] we then access their intrinsic nano-property of high surface area to mass ratio. This nano effect improves the rapid Cs sorption kinetics of these materials, [START_REF] Grandjean | Effect of the chemical nature of different transition metal ferrocyanides to entrap Cs[END_REF][START_REF] Causse | Facile one-pot synthesis of copper hexacyanoferrate nanoparticle functionalised silica monoliths for the selective entrapment of Cs-137[END_REF]17] when compared with a true bulk material. In fact, recently a number of groups have begun to examine combining functionalisable magnetic nanoparticles and hexacyanoferrates to create two in one particles cable of magnetically [17] separating Cs from waste waters. [START_REF] Mobtaker | Magnetic CuHCNPAN nano composite as an efficient adsorbent for strontium uptake[END_REF][START_REF] Yang | Sodium-copper hexacyanoferrate-functionalized magnetic nanoclusters for the highly efficient magnetic removal of radioactive caesium from seawater[END_REF][START_REF] Hwang | Highly efficient removal of radioactive cesium by sodium-copper hexacyanoferrate-modified magnetic nanoparticles[END_REF][START_REF] Mobtaker | Cesium removal from nuclear waste using a magnetical CuHCNPAN nano composite[END_REF][START_REF] Zhang | Sorption behavior of cesium from aqueous solution on magnetic hexacyanoferrate materials[END_REF] However, this functionalisation appears not necessary as Copper hexacyanoferrate (II) appears to be intrinsically magnetic (despite both metals being in the +2 oxidation state). We therefore report here for the first time that Cu-HCF can be affected by an external magnetic field without the need for further modification.

Experimental

Chemical products

All chemical products (Cu(NO3)2, K4[Fe(CN)6], CsNO3, silica gel, 3-aminopropyltriethoxysilane (APTES), solvent (acetone, ethanol) were purchased from Sigma-Aldrich. Potable mineral water that contains competitive cation was purchased from Evian®, and its mineral composition is given in supplementary information (SI).

Analytical techniques

XRD data were recorded using a D8 Bruker Advance diffractometer (Cu Kα radiation). FT-IR and UV-Vis measurements were carried out using a Nicolet iS50 and Shimadzu UV-2600 respectively. TEM of the bulk particles was performed on a JEOL 2100 Lab TEM operating at 200 kV while the silica grain particles were examined using a Zeiss Leo EM910 operating at 120 kV. Chemical compositions were measured by inductively coupled mass spectrometry for Cu, Fe, K, Si after dissolution. Magnetisation measurements were carried using a SQUID magnetometer at both 300 K and 4 K, and thermal scans of the magnetisation in a 1 T field of the particles were examined were from 4 -300 K.

Stirrer/agitation

Several forms of mixing were used in the sorption study. In all cases bar one (mechanical stirring) the same vessel was used; 100 ml glass bottle. For mechanical stirring with a propeller stirrer a 250 ml beaker was used, as seen in Figure 3 of the SI.

The magnets used in this study were PTFE encased 12.4 g oval shaped magnetic stirrers purchased from VWR (VWRI422-0405). The dimensions of which were 300 mm x 16 mm. The stirring speed was set to 700 rpms. To mimic the mechanical effect of the magnetic stir bar without magnetic field, we use a plastic surrogate with a weigh and a size comparable to the one of the magnetic bar used. Rotary agitation was carried out using a Heidolph Reax 2, purchased from Heidolph-Instruments. The instrument was set to speed 6. Mechanical stirring was carried out with a IKA labotechnik RW 20 DZM with power set to 65%.

Bulk Cu-HCF (II)

Copper (II) Hexacyanoferrate (II) (here on referred to as Cu-HCF, or bulk Cu-HCF) was prepared by pouring 45 ml of an aqueous 1 x 10 -1 M Cu(NO3)2.3H2O solution directly into 30 ml of a mixed aqueous solution of 1 x 10 -1 M K4[Fe(CN)6].3H2O (K-HCF) and 1 x 10 -1 M KNO3. Upon addition of the blue Cu(NO3)2.3H2O solution to the light yellow K-HCF/KNO3 solution a dark burgundy coloured precipitate was immediately formed. The contents of the reaction vessel were then allowed to mix for an hour by rotary mixing. The resulting solid was then collected by centrifugation and washed several times with Millipore water. As the washing steps accumulate the HCF-Cu becomes more and more stable in water and therefore equal volumes of acetone were needed as a non-solvent to help crash-out the particles. Once the washing steps are complete (4 washes) the Cu-HCF was dried under rotary evaporation at 60 0 C. The resulting dark solid was then examined using UV-Vis, FT-IR, XRD and TEM. Elemental analysis showed the structure formula to be K0.8Cu1.58Fe(CN)6.4.8H2O

Cu-HCF (II) nano templated into Silica Grains

Commercially available silica gel having a 15 nm mesoporosity was used as silica support. The gel was firstly functionalized with 3-aminopropyl-triethoxysilane (APTES). This was accomplished by dispersing the silica gel in an absolute ethanol solution and adding APTES. The resulting solution was the stirred at 60°C for 24hours. The -(CH2)3-NH2 functionalised silica gel was then washed several times with water and then acetone, in order to remove any unreacted APTES. The functionalised gel was then immersed in a 1 x 10 -2 M copper nitrate solution and placed under rotary mixing for 2h. After washing with water, the now Cu functionalised silica gel was added to a 1 x 10 -2 M K4[Fe(CN)6].3H2O/KNO3 solution and allowed to stir for 2h. The as formed Cu-HCF@SiO2 gel was then washed with water. This two-step Cu-HCF impregnation process was repeated six times in total allowing for the growth of Cu-HCF on the silica surface(s). The resulting Cu-HCF@SiO2 grains were finally dried under air at 40°C for 48h. Elemental analysis performed as described in Michel et al. [START_REF] Michel | Ion exchange and structural properties of a new cyanoferrate mesoporous silica material for Cs removal from natural saline waters[END_REF] showed the structure formula to be K1,36Cu1,46Fe(CN)6.45SiO2. This material is therefore 11.8% Cu-HCF and 88.2% SiO2 by mass.

Sorption experiments

Simple Cs sorption experiments were carried in Evian © mineral water (composition in SI). This was done by first re-dispersing the Cu-HCF (or Cu-HCF@SiO2) in Evian® water. All batches contained 1g of Cu-HCF per 99 ml of mineral water and each batch was allowed to re-disperse overnight (see Table 1). Once particle dispersion was completed, the dispersions were then transferred to a magnetic stirring plate and a magnetic stirring bar was added and activated seconds before Cs solution addition. This Cs sorption reaction was initiated by addition of 1mL of a fixed concentration CsNO3. This resulted in a solution 70 mg.L -1 of Cs + ions at the beginning of the sorption reaction for silica grains and batches 1 to 3 and 700 mg.L -1 for batch 4. Aliquots of the resulting solution were taken at fixed intervals and filtered (the filters were 200 nm for the Cu-HCF and 450 nm for the Cu-HCF@SiO2) to remove the Cu-HCF (or Cu-HCF@SiO2) particles stopping the sorption process. The Cs concentration of each aliquot was then measured using a Perkin Elmer Analyst 400 atomic absorption spectrometer. This allowed us to monitor the Cs concentration against time.

The Cs sorption properties of hard templated Cu-HCF@SiO2 particles were first compared to non-magnetically dispersed Cu-HCF particles. As described above the particles were dispersed in water overnight before Cs addition. Particle dispersion was performed using rotary agitation only (no magnet). This non-magnetically dispersed Cu-HCF sample is referred to as batch 1, where dispersion were done in presence of surrogate of stir bar (see below). The Cu-HCF@SiO2 sample will henceforth be referred to as Gr or grain sample.[ Note : To aid contact time between the active sites in the nano Cu-HCF@SiO2 particles and the Cs in solution the silica grain sample was firstly crushed down to ≤ 20 µm]. Once the particles were allowed to disperse for 19 hours the Cs sorption reaction was initiated as described above.

Table 1 : Cs sorption data for bulk (B1-B4) and ground nano-templated silica grain (Gr) samples. Data has been normalised to account for actual Cu-HCF content as given by elemental analysis. Permanent Magnet and magnetic stirrer refers only to Cu-HCF dispersion

* Normalised with respect to % Cu-HCF in sample ** the % of sorption capacity refers Q(t)/Qmax at equilibrium.

In order to study if a magnetic disaggregation effect as seen by Stuyven et al. was possible to induce, batches 2 and 3 were prepared. [START_REF] Stuyven | Magnetic field assisted nanoparticle dispersion[END_REF] Batches 2 and 3 were identical to batch 1 except for the following details (see Table 1). While batches 1 and 2 were in fact simultaneously dispersed using the same rotary mixer a magnetic stirrer (permanent magnet) was placed in batch 2 Batch 1 also contained a non-magnetic plastic object of similar size and shape to the stirring bar to mimic the magnetic stirrer (surrogate). Batch 3 was dispersed by conventional magnetic stirring with no rotary mixing. After a 19 hours period of dispersion, the sorption reaction began as described above.

Finally the Cs sorption capacity of bulk Cu-HCF was determined by repeating the batch 3 experiment and increasing the initial Cs concentration by a factor of ten. This experiment is referred to as batch 4.

The sorption capacity was calculated by the classical formula

= -

Where [Cs]init and [Cs]final are respectively the initial and final concentration of Cs in the solution (mg.L -1 ); V is the volume of solution used (L) and m the mass of solid used (g). In case of silica grain, this mass was taken equal to the mass of Cu-HCF inserted into the silica.

In each we report also the ratio (%) of Cs uptake by the solid at t (s or min): this ratio is equal to Q(t)/Qmax (%) where Qmax corresponds to the capacity at equilibrium for each batches.

Results and Discussion

Comparison between bulk and nano-templated Cu-HCF

XRD

The XRD patterns show that the Cu-HCF in both the unfunctionalised particles and the Cu-HCF@SiO2 sample possessed a face centred cubic (FCC) structure with the space group Fm-3m. This corresponded to the PDF04-010-1963 card, i.e. K2CuFe(CN)6 (see Figure 1). The amorphous state of silica is also confirmed. The backgrounds of the XRD patterns of Figure 1 were removed for clarity. The space group and lattice parameters are in good agreement with the literature. [START_REF] Ojwang | Structure Characterization and Properties of K-Containing Copper Hexacyanoferrate[END_REF][START_REF] Liao | The role of potassium ions in iron hexacyanoferrate as a cathode material for hybrid ion batteries[END_REF] The lattice parameter of the bulk as-prepared sample was found to be 10.0039±0.0040 Å while the silica templated nano was quite similar at 10.0007±0.0042 Å. The smaller size of the templated HCF unit cell (-0.0032 Å) is probably due to its relatively greater K + content, (see elemental analysis) as increased K + content has been reported to slightly reduce the lattice parameter. [START_REF] Liao | The role of potassium ions in iron hexacyanoferrate as a cathode material for hybrid ion batteries[END_REF] However, despite this difference the near identical XRD patterns demonstrate that there is little structural difference between the materials formed by the two synthetic approaches (Figure 1).

FT-IR

FT-IR was carried out using a Nicolet iS50 infrared spectrometer. Looking firstly at the aggregated material, the v(CN) stretch of the Cu-HCF is seen at 2,089 cm -1 , (Figure 2-inset top centre). Cu-HCF (II) generally possesses a v(CN) stretch at around 2,095 cm -1 , while the v(CN) stretch for Cu-HCF(III) (where the iron is in the Fe (III) oxidation state) is typically located around 2,177 cm -1 . [START_REF] Ng | Structure and magnetic properties of copper(II) hexacyanoferrate(III) compound[END_REF][START_REF] Ghosh | Infrared spectra of the Prussian blue analogs[END_REF] Therefore, we can safely assume that Cu-HCF (II) is the dominant form here, (Figure 2). The v(Fe-C) stretch is seen at 594 cm -1 , (Figure 2-inset top right), this position again corresponds to the dominant presence of Fe (II) . The δ(HOH) signal is seen at 1,614 cm -1 , with a shoulder at 1626 cm -1 indicating multiple water environments (Figure 2inset bottom). The multiple structured v(OH) signals which are seen above 3,000 cm -1 confirm this assertion. This indicates the presence of several coordinated water sites. The boarder hump-like peak that they are buried in is indicative of zeolitic water, (Figure 2-inset top left). This is all in good agreement with the literature.

Moving to the templated (silica grain) sample, we see that the v(CN) stretch has increased in frequency to 2,103 cm -1 , (Figure 2-inset top centre). The v(Fe-C) stretch has also increased in frequency to 594 cm -1 , (Figure 2-inset top right). Despite these slight increases in frequency Cu-HCF (II) is also the dominant form in the templated sample. The δ(HOH) signal is seen at 1,624 cm -1 , and is broad and tapering which suggests zeolitic water is more prevalence here, (Figure 2-inset bottom). This is confirmed by the v(OH) signals which now consist of only a single coordinated water peak at 3,625 cm -1 and a board zeolitic peak at 3,412 cm -1 , (Figure 2-inset top left). As elemental analysis showed that the ratio of copper to iron was the same for the bulk and templated samples, that means that they have the same number of vacant Fe(CN)6 4-sites and therefore hydrated copper. This therefore indicates that the "other" coordinated water peaks in the bulk sample correspond to surface bound water; the surface coppers of the templated sample being bound to APTES's -NH2 groups. Looking closer (Table 2) we see that the v(CN) stretch of the nano-templated sample not only increases in the frequency by 15 cm -1 , but that its full width half max (FWHM), decreases by 25 cm -1 , when compared to the bulk. Taking the decrease in the FWHM first, the FWHM of the v(CN) can be used to measure any change in the local chemical environment inside the HCF cage. An increase in the chemical uniformity sees a decrease in FWHM, and vice versa. This effect was discussed in detail by Lejeune et al. where it was pointed out that the larger the guest ion the easier it is for it to "seen" by the greatest number of cyano bridging ligands. [START_REF] Lejeune | Application of the infrared spectroscopy to the structural study of Prussian blue analogues[END_REF] The much narrower v(CN) stretch of the templated sample is therefore probably due to its higher K content as seen in elemental analysis. It should be noted that the controlled growth of the particles on to the silica surface may also contribute to this sharper v(CN) stretch. As for the v(CN)'s relatively higher frequency, this suggests an enhanced interaction between the metals and CN ligands in the templated samples when compared to that of the bulk. Avila et al. saw similar high v(CN) for Cu-HCF and pointed out that Cu 2+ has the ability to fill its 3d shell by receiving electrons from the CN's 5σ orbital, thereby enhancing the bond strength and consequently causing a relative increase in the frequency of the v(CN) stretch when compared with other HCF analogues. [START_REF] Avila | Porous framework of T2[Fe(CN)6]•xH2O with T=Co, Ni, Cu, Zn, and H2 storage[END_REF] However, if this were the explanation here, why is this phenomenon not seen in the bulk samples? A likely explanation for the relatively higher frequency of the nano v(CN) stretch is electron donation from the APTES amine groups to their adjacent coppers, strengthening the Cu-NC bonds which in turn effects the electronically sensitive cyano ligands. This would explain why the v(Fe-C) stretches of both the templated and bulk samples occupy identical frequencies but the v(CN)s do not.

Solid UV-Vis

The as-prepared bulk sample showed a broad peak at 476 nm and a narrower one at 319 nm, (Figure 3 and Table 3). A shoulder at 269 nm was also present. The signals at 319 nm and 269 nm can be assigned to Fe (II) (CN)6, as they can also be seen in K4Fe (II) (CN)6 starting material (SI). [START_REF] Gray | The Electronic Structures of Octahedral Metal Complexes. I. Metal Hexacarbonyls and Hexacyanides[END_REF] The wider peak at 476 nm is assigned to Cu d-d transitions. [START_REF] Babu | Facile immobilization of potassium-copper hexacyanoferrate nanoparticles using a room-temperature ionic liquid as an ionic binder and its application towards BHA determination[END_REF][START_REF] Tokarev | Synthesis and study of Prussian blue type nanoparticles in an alginate matrix[END_REF] The nano templated sample showed the Cu d-d peak at 483 nm and the ligand field (LF) bands of Fe (II) (CN)6 at 314 nm and 269 nm. The lower energy of the copper peak is probably due to the passivating effect of the APTES surface ligands while the higher energy of the Fe co-ordination sphere is probably due to the template effect as the Fe(CN)6 units are forced to adjust to the coppers spacing in the silica surface. However, despite these variances in peak positions, again little difference is seen between the nano templated and as prepared bulk samples. This again indicates that the silica template has not altered the electronic properties of the Cu-HCF with respect to the aggregated sample.

v(CN) B v(CN) G v(Fe-C) B v(Fe-C) G Frequency (cm

TEM

TEM showed the bulk particles were heavily aggregated with a 20 nm ± 5 nm, (Figure 4). Templated particles inserted into silica grains were evaluated to be close to 15 ± 5 nm, (Figure 4)

Cs Extraction Kinetics and Capacity

Comparison bulk and nano-templated

As shown by spectroscopic analysis, it was realised structurally there was very little difference between the templated and freely grown samples. In fact the only technique which showed a stark difference between the nano templated and bulk samples was TEM, (Figure 4), which showed, (as expected), that the unfunctionalised sample was heavily aggregated in contrast to the silica grains Cu-HCF@SiO2 which contained individual pore based Cu-HCF particles.

The Cs extraction properties of the aggregated Cu-HCF (batch 1) were compared to those of the silica grain templated Cu-HCF@SiO2 particles (batch Gr). This was done to determine if the increased monodispersity (and therefore increased surface availability) provided by nano-templating would increase the Cs sorption efficiency of Cu-HCF. When noting Cs sorption data one must remember that mg/g normally refers to mg of Cs sorbed per gram of Cu-HCF. Elemental analysis of the Cu-HCF@SiO2 by ICP-MS (after dissolution of the sample) showed the chemical formula to be K1,36Cu1,46Fe(CN)6.45SiO2 with the silica making up 88.2% by mass of the sample. Consequently, Cu-HCF makes up only around 11.8 % m/m of the Cu-HCF@SiO2 sample, (Batch Gr). Therfore, unless otherwise stated all Cs sorption results for Cu-HCF@SiO2 have been normalised to account of this. Figure 5 shows the results of the kinetic experiment for the Cu-HCF (II) @ SiO2 sample and batches 1 and 2. The Cs extraction kinetics of the silica templated particles are close to those of batch 1. The silica grain sample exchanges 52% of the total Cs added within 1 minute, while batch 1 exchange 45% at this time (see Table 1). Both sample (silica grain and aggregated) has reached near equilibrium at 10 minutes. However, Figure 5 also shows by applying a magnetic field to the particles (batch 2) and disaggregating them it is possible to enhance the Cs kinetics.

Use of intrinsic magnetism to disaggregate Cu-HCF bulk suspension

It was observed over the course of this study that the batch 1 rotary dispersed Cu-HCF particles separate and disperse far more uniformly if a magnetic stirrer is included. By casual observation it appeared that after 3 hours a black homogenous dispersion was acquired. In fact, once rotary aggregation was halted after 19 hours the magnet containing (batch 2) sample did not immediately flocculate, unlike batch 1, (Figure 6). After this was observed attempts were made to disperse Cu-HCF particles, again using a Heidolph Reax 2, but this time including an object of similar composition, shape and size to a magnetic stirrer to determine if it was the magnetic nature of the stirrer, or simply the presence of a "grinding assistant", which aided dispersion Figure 6). However, the presence of a non-magnetic surrogate had no effect. Mechanical stirring was also tried but again did not produce a homogenous dispersion (SI). The only other technique which yielded similar results was when the particles were dispersed by classic magnetic stirring (batch 3). The Cs extracting abilities of magnetically dispersed Cu-HCF (batches 2 and 3) were then examined and compared non-magnetically dispersed (batch 1). Magnetic dispersion is characterised as either active, (under magnetic stirring (batch 3)), or passive (physically agitated in the presence of a permanent magnet (batch 2)). For the nonmagnetically dispersed sample a plastic Eppendorf (filled with water) was added to mimic the stirring bar's size and shape (batch 1).

Figure 7 shows the resulting sorption trace in a form of the diffusing model first proposed by Morris and Weber. [START_REF] Weber | Kinetics of adsorption on carbon from solution[END_REF] In it milligrams of Cs sorbed per gram of Cu-HCF are plotted against the square root of time (minutes). This model allows for the determination of the diffusive limiting step of a sorption by a solid. That is the diffusion of ions in solution, followed by their diffusion in the transition layer between the liquid and solid (called boundary layer), and then through the solid (intra-particle diffusion). The initial measurement was taken 20 seconds after contact with the dispersed Cu-HCF. The final measurement was taken after an hour, by which time all three batches (1-3) had almost completely sorbed the total added Cs.

As seen in Table 1, batch 1 (no magnet and physical stirring) showed 45 % Cs sorption after one minute (close to sorption half-life). However, while initial sorption was quite fast, it began to rapidly slow with 18 % of the total sorbed Cs still remaining after 5 minutes (Figure 7). Near total sorption was completed between 10 and 30 minutes. These results plus the shape of the sorption curve would indicate a 2 or even 3 step process, as describe elsewhere. [START_REF] Zhu | Efficient removal of Pb2+ by Tb-MOFs: identifying the adsorption mechanism through experimental and theoretical investigations[END_REF] We believe that step 1 is probably the rapid removal of Cs from the solution by exchanging it with the K ions present on, and close to, the aggregated HCF's surface. This initial exchange is accomplished in the first minute. This is then followed by a slower ion diffusion process as the sorbed Cs is exchanged with K held deeper in the HCF cage structure (intraparticle diffusion). Once this internal exchange (with K now once again close to the surface) is complete the removal of Cs from solution begins again. This process is continued until all Cs has been removed from the surrounding solution. The addition of a magnet has a drastic effect on the sorption kinetics. Batch 2 (again dispersed by rotary agitation, but now in the presence of a permanent magnet) displayed an enhanced Cs sorption kinetic. The sorption half-life was calculated at 6 seconds, while 84 % of the total Cs has been sorbed within the first 20 seconds and 98 % within the first minute. Batch 2 seems also to exhibit a 2 step process, with over double the time required to sorb the remaining 50% of the solutions Cs, however, as this batches sorption half-life is over an order of magnitude greater than batch 1, (6 vs 74 seconds) almost total sorption is complete within two minutes. Finally, the dispersion which had been magnetically stirred overnight (batch 3) had sorbed 89 % of the total Cs + after 20 seconds, this increased to 98% by the end of the first minute. Its half-life was recorded as 3 seconds. This is twice as fast as batch 2.

After 30 minutes all three batches contained around 0.5 ppm of Cs indicating that the equilibrium pointis not affected by the dispersion method. As is clear the slow Cs/K diffusion step is present in batch 2. However, thanks to its uniform dispersion, and therefore better access to each particles individual surface, ultra-fast sorption of 99 % of the Cs content is achieved within 60 seconds after contact with the sorbent. While it would be easy to assume the improved dispersion of batches 2 and 3 are simply a result of the difference between magnetic stirring and rotary mixing, the lack of a magnetic stirring plate in batch 2 demonstrates that it is in fact the presence of an external permanent magnetic field within the container aided by rotary agitation that allows the Cu-HCF particles to disaggregate. Once disaggregated the Cu-HCF particles arrange themselves more homogenously (throughout the magnetic field) allowing the Cs + ions a more unobstructed access to their surfaces. The presence of an alternating field in batch 3 further improves this effect.

Capacity Kinetics

Once normalised for % mass of Cu-HCF, the Cs capacity of the Cu-HCF@SiO2 was found to be around 215 mg/g. Therefore to determine capacity of the bulk sample the batch 3 experiment was repeated using a 700 ppm Cs solution (from now on referred to as batch 4).

From this experiment the capacity of the bulk sample was determined to be 266 mg/g, (after one day). This high concentration experiment also allowed us to compare the Cs sorption kinetics at capacity of disaggregated bulk against the silica templated particles. Cs (mg.g -1 ) t 1/2 (min 0.5 ) Batch 4 showed a sorption half-life of 14 seconds, and within the first minute of contact with the Cs solution it had sorbed 96% of the total sorbed Cs, (Figure 8 and Table 1). [This is in agreement with batch 3 and shows the reproducibility of these experiments in both high and low concentration Cs solutions]. Compared to Cu-HCF@SiO2 batch (Gr) which showed a Cs sorption half-life of 58 seconds, and 52 % Cs sorbed in one minute, we see that batch 4 is far more efficient. Cs measurements at equilibrium showed that batch 4 had sorbed 266 mg of Cs per gram of Cu-HCF. For the silica samples, once normalised the grains showed a Cs sorption capacity of 215 mg/g. These results were unusual as the templated sample should have shown a much quicker sorption kinetic than the bulk sample, unless of course the bulk sample was as disperse as the templated one. The capacity of batch 4 is also higher than the Cu-HCF@SiO2 Gr batch. This difference however falls within human/experimental error. This result clearly shows the benefit of the dispersion of bulk samples assisted by a magnetic field.

SQUID

Magnetisation measurements were carried out on the bulk Cu-HCF (II) particles (Figure 9) and confirmed that the particles are paramagnetic. Powder samples (bulk Cu-HCF samples) were mounted in gel caps, which have a temperature -independent diamagnetic susceptibility of -5.9 10 -9 m 3 kg -1 . The magnetisation curve at 300 K and 4 K is shown in Figure 9, after correction for the diamagnetism of the sample holder. The curve is linear and paramagnetic at 300 K, with no trace of ferromagnetic behaviour. The paramagnetic contribution at 4 K can be well fitted by Brillouin function [Fe (II) and J = 4):

M(x) = NJµBgJ coth( ) -coth( ) (1)
where x = gJJµBHe/kBT is the ratio of the Zeeman energy of the magnetic moment in the effective field He to the thermal energy, the effective field He = H + Hm, where the Hm is the molecular field proportional to magnetic moment M, gJ is the g-factor, J the angular momentum number, N the number of spins, kB the Boltzmann constant. The paramagnetic moment per iron atom obtained from fitting the Brillouin function to the magnetisation at 5K is 4.8 µB. Figure 10 shows the magnetisation data of aggregated particles, as a function of temperature in an applied magnetic field of 100 mT. Furthermore the plot of the inverse susceptibility (Figure 10 b), there is no clear magnetic transition and the samples exhibit paramagnetic behaviour. The susceptibility is fitted to the equation χ(T) = χ0 + χCW(T) between 4 K and 100 K. Here χ0 is the temperature independent term and χCW(T)=C/(T-ΘCW) is the Curie-Weiss term with Curie constant C = NApeff 2 μB 2 /(3kB) and Curie-Weiss temperature ΘCW (NA is Avogadro number, peff is the effective Bohr magneton number, μB is the Bohr magneton, and kB is the Boltzmann constant). The effective magnetic moment peff, obtained from the Curie law to the magnetisation versus temperature data, is 4.7 μB. These values agree but are slightly lower than what would be expected if all the iron in the sample were present as high spin Fe 2+ ions (typically 5.4 μB). Although Mössbauer spectroscopy would be needed to elucidate the exact spin state of the Fe it was not carried out as SQUID was determined to be sufficient to show that Cu-HCF particles are indeed magnetic. Finally, it should be noted that there is a large paramagnetic contribution arising from Cu(II) ions, as illustrated by the non-saturated paramagnetic behaviour in magnetization curves, both in field and temperature dependent magnetization curves. [START_REF] Cheng | Paramagnetic and diamagnetic binuclear copper(II) complexes with different bidentate exogenous bridges: synthesis, spectroscopy and magnetic properties[END_REF][START_REF] Okubo | Ion-Induced Transformation of Magnetism in a Bimetallic CuFe Prussian Blue Analogue[END_REF] As Corr et al. have previously shown it is possible to disaggregate magnetic particles and align individual super paramagnetic particles in an applied field. [START_REF] Corr | Linear Assemblies of Magnetic Nanoparticles as MRI Contrast Agents[END_REF] We believe that this goes some way to explaining how the presence of paramagnetism aides in the disaggregation of the Cu-HCF particles. However going further Stuyven et al pointed out that the stability of a suspension of charged particles can be seen as being in dynamic equilibrium between van der Waals (attractive) forces and electrostatic repulsion. [START_REF] Stuyven | Magnetic field assisted nanoparticle dispersion[END_REF] By placing the particles under rotary mixing the number of aggregate collisions, as well as the hydrodynamic sheer stress, will be increased causing the aggregates to disassemble. Eventually reaching equilibrium the frequency of aggregate growth and break up will define the mean aggregate size. The addition of an external magnetic field will complicate this. The addition of accompanying Lorentz forces will affect the van der Waals/electrostatic balance as the Lorentz forces act on opposite directions of opposite sides of rotating charged particles separating them. This will also enhance the effect of sheer stress as they move from the inside to the outside of the moving container (and back again). This due to the difference in local velocity gradients along the container walls.

Conclusion

Cu-HCF nanoparticles were prepared by two routes, the classic co-precipitation method, and grown step by step inside silica grains. XRD, FT-IR and UV-Vis measurements demonstrated that the aggregated Cu-HCF and the more monodisperse Cu-HCF@SiO2 particles are chemically near identical. Comparison of the Cs sorption kinetics show that by using a magnetic field to aid with the dispersal of the aggregated particles the time taken to sorb 50% of the total Cs was decreased from over a minute to 6 seconds, with total sorption time being almost complete after 60 seconds instead of 10 minutes. By comparing the aggregated HCF and ground silica sample we see that the silica templated particles are the more efficient of the two, sorbing 53% of the total Cs after 1 minute compared to the aggregated samples 44%. However, once the aggregated particles are magnetically dispersed the sample efficiency increases to 96-98% of total Cs sorbed after 1 minute. This implies we have dispersed the Cu-HCF particles within a magnetic template equaling the monodispersity of the silica template.

Figure 1 :

 1 Figure 1: XRD of bulk (black) and silica grain (blue). The pattern backgrounds were removed to highlight the Cu-HCF reflections. Both samples possess a face centred cubic structure. The asterisks indicate the presence of impurities or secondary phases.

Figure 2 :

 2 Figure 2 : FT-IR of as prepared templated Cu-HCF@silica-grains (red), and bulk Cu-HCF (black)). Close up of the v(OH) (top left inset 3,500-4,000 cm -1 ), cyano v(CN) stretch (centre top inset -1,950-2,200 cm -1 ) and M-L stretches (top right inset -590 cm-1 ). The δ(HOH)close up is seen just below the main spectra.

Figure 3 :

 3 Figure3: Solid UV-Vis of as prepared bulk Cu-HCF (green) and Cu-HCF@silica-grain (blue).

Figure 4 :

 4 Figure 4: TEM of Cu-HCF Bulk particles (left) and templated (right). Average particles size is 20 nm ± 5 nm and 15 nm ± 5 nm respectively

Figure 6 :

 6 Figure6: Photograph of as-prepared Cu-HCF nanoparticles. Particles were prepared, dried, weighed, and then redispersed in water. Each bottle contains 1 g of Cu-HCF in 100 ml of commercially available mineral water, (composition in SI). The image was taken immediately after both samples were removed from the same rotary agitator. They had previously spent 4 days undergoing rotary mixing. . This far longer mixing time was used to remove any ambiguity. The bottle on the right had a magnetic stirring bar placed in it before rotary mixing began (batch 2). These particles were fully dispersed after 3 hours. The one on the left (batch 1) acted as a control contained a non-magnetic plastic object of similar size and shape to the stirring bar.

Figure 7 :

 7 Figure 7 : Cs sorption spectrum of aggregated batch 1 (no magnet, physical stirring-red circles), batch 2 (with magnet but physical agitated-black squares) and batch 3 (with magnet and magnetically stirred-blue triangles). Data is displayed as mg Cs sorbed per gram of Cu-HCF against the square root of time.

Figure 8 :

 8 Figure 8 : Cs capacity sorption spectrum of Cu-HCF batch 4 (red circles). The black squares show Cs sorption onto crushed Cu-HCF @ silica grains. Data is displayed as Cs/Cmax of Cs against the square root of time. Silica grain sorption data has been normalised according to amount of HCF particles per gram of crushed grain (11%).

Figure 9 :

 9 Figure 9: Magnetisation data of Cu-HCF aggregated particles at 300 K and 4 K

Figure 10 :

 10 Figure 10 : Above, temperature dependence of magnetisation of aggregated particles as a function of T. Below, temperature dependence of magnetic susceptibility (χm) of aggregated particles as a function of 1/T.

Cs init (mg.L - 1 ) Permanent magnet Magnetic stirrer Half Life (sec) Capacity (mg/g) [%]**

					At 20	At 1 min Maximum
					seconds
	Gr 35			58	16[8]	112[52]	215*
	B1 70			74	15[21]	33[45]	N/a
	B2 70	X		6	61[84]	71[98]	N/a
	B3 70	X	X	3	67[89]	74[98]	N/a
	B4 700	X	X	14	143[54]	225[96]

Table 2 :

 2

v(CN) positons and full width half max (FWHM) values for aggregated (B) and grain (G) samples. K= as prepared. Cs = Caesium exchanged. All values are in cm -1 .

Table 3 :

 3

		1.8
		1.6
		1.4
	A	1.2
	b
	s	1
		0.8
		0.6
		220	320	420		520	620	720	820
						Wavelength (nm)
		Origin		State	Type bulk (nm) Templated (nm)
		Fe (II) (CN)6 1 A1g 1 T2g	LF	269	269
		Fe (II) (CN)6 1 A1g 1 T1g	LF	319	314
		Cu(NC)6	Eg-T2g	d-d	476	483

UV-Vis data for bulk (b) and nano templated (g) samples. All values are in nm, (LF= Ligand Field) .

  1/2 (min 0.5 ) Figure

	Batch 1
	Silica Gr
	Batch 2

5: Cs sorption spectrum of batch 1 (no magnet, physical stirring) (blue circle), batch 2 (magnet, physical stirring) (green square) and Gr (silica grain) (black squares). Data is displayed as % Cs sorbed (Cs/Cmax where Cs is concentration of Cs at time t and Cmax is total Cs exchanged) against the square root of time.

Acknowledgements

Elemental Analysis was carries out by the Laboratoire de metallographie et d'analysis chemique, DEN/MAR/SA2I/DIR, CEA Marcoule, France.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Research was conducted in part by the Center for Hierarchical Waste Form Materials (CHWM), an Energy Frontier Research Center (EFRC) supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award DE-SC0016574. We also thank the EDDEM-CEA project for funding this work.

Data Availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations