Structural, Optical and Magnetic Properties of Ultramarine Pigments: A

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Structural, Optical and Magnetic Properties of Ultramarine Pigments: A DFT Insight Pawel Rejmak J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09856 • Publication Date (Web): 30 Nov 2018 Downloaded from http://pubs.acs.org on December 4, 2018

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The Journal of Physical Chemistry

Structural, Optical and Magnetic Properties of Ultramarine Pigments: A DFT Insight

Paweł Rejmak

Institute of Physics Polish Academy of Sciences, Aleja Lotnikow 32/46, PL-02668 Warsaw, Poland E-mail: [email protected], [email protected] ORCID 0000-0002-0535-2107

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ABSTRACT The ultramarine pigments are among the most widely used coloring materials since the antiquity till present times. Despite many experimental studies, the characterization of ultramarines is still incomplete. In this work we reported for the first time the density functional theory results obtained for realistic periodic and large cluster models of ultramarines with blue S3- and yellow S2- chromophores. Periodic calculations provided insight into Sn- siting inside aluminosilicate cages, normally not resolved well in experimental structural data. All electron

calculations

performed

on

large

cluster

models

showed

that

the optical properties of S3- ions depend little on their orientation within cavities, unless strong distortion from free S3- ion C2v symmetry is enforced by the lattice. No magnetic coupling between S3- species occupying adjacent cages was found. Upon the present results observed differences in the averaging of electron resonance signals should be rather ascribed to different S3- dynamical effects. Though the quantitative computational treatment of S2- systems is more challenging, due to near orbital degeneracy, the qualitative results show that the electronic structure and spectroscopic properties of embedded S2- radicals are more sensitive to the environment than in the case of S3- species.


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1. INTRODUCTION Ultramarines (UM), occurring naturally as mineral lazurite, are among the most widely used blue pigments, with its application dating back to 3500 BC in Mesopotamia.1 Their intense color stems from the combination of two peculiar chemical species, namely trisulfur anion radical S3-,2,3 and the sodalite type aluminosilicate lattice. S3- radical has been identified as the blue chromophore, strongly absorbing the light at the wavelengths around 600 nm.2,3 In its ground state it has a shape of isosceles triangle (C2v group).4 As the majority of radicals, S3- easily reacts if exposed to the air, thus can be stabilized only if embedded in protecting matrices, such as microporous materials or certain liquids.5 Interestingly, recent studies showed that S3- can be stable in geological fluids, making it probably the most common sulfur form in the Earth crust.6 Sodalite

(SOD)

mineral

is

a

microporous

aluminosilicate

of

ideal

formula

[Na8(Si6Al6O24)Cl2], crystalizing in cubic system (P-43n group).7 SOD structure can be seen as a system of truncated cubooctahedral units of diameter about 7 Å, so called SOD units or β-cages,being composed of corner sharing TO4 tetrahedrons, where T = Si, Al (i.e. atoms tetrahedrally coordinated by the lattice O atoms). The void inside each cage is centered by Cl anions, whereas Na cations reside nearby the middle of hexagonal faces. Due to its flexibility SOD type lattice remains stable in the wide range of chemical composition, which can be formally derived by replacing some or all Na, Cl and/or T ions in the stoichiometric SOD. This results in the rich chemistry of SOD type solids.8,9 Depending on the chemical compositions, the members of SOD family exhibit interesting optical,10-15 magnetic,16,17 ferroelectric18,19 and mechanical20 properties, along with perspective applications in separation processes,21,22 catalysis,23-25 and as new cements.26 SOD is counted as a member of zeolite family, mainly because SOD cages are building units of many zeolites (microporous aluminosilicates). However, it differs significantly from the typical zeolites, lying on the border 3 ACS Paragon Plus Environment


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between microporous materials and condensed tectosilicates.8 Due to the small aperture of rings in sodalite units (4 and 6 Si/Al tetrahedrons), SOD essentially lacks the three dimensional system of pores, accessible for adsorbing molecules or enabling cation exchange. It rather contains disjointed cavities inside of T12O24 cages, where chemical species trapped during the synthesis may remain isolated from the environment. This feature enables permanent hosting of otherwise unstable chemical species, like S3- radical. UM are SOD type aluminosilicates with significant part of cages occupied by S3- anions. UM analogues were also synthesized by trapping S3- chromophores within certain zeolites, usually the ones containing SOD cages in their structure.27-29 The chemical composition of synthethic UM pigments and natural lazurite minerals varies somewhat, depending on the synthesis conditions and mineral locality, and usually can be summarized as Na8-x(Ca, K,…)[Al6Si6O24] (S3, S2, SO4, Cl, OH,…).1,8,30-33 UM structures have been resolved in cubic systems, with P23,30,31 P-43n,33 or I-43m32 space groups. Modulated superstructures, orthorhombic or triclinic, were also reported.31 Si:Al ratio is close to unity, which ideally would correspond to alternating occupation of T sites by Si and Al atoms, though some

29Si

nuclear magnetic

resonance results indicates the presence of Al-O-Al linkages.31,32 Such Al-O-Al connectivity, forbidden in ‘normal’ zeolites by Loewenstein rule,34 is quite common in SOD type lattices.9 The main issue with structural characterization of UM is that they are only approximately periodic solids, due to unspecified distribution of Si/Al atoms between T sites, partial exchange of Na cations by K and Ca ones and variable occupation of sodalite cages by extralattice anions (S3-,S2-, S2-, SO42-, Cl- etc.). In particular, the structural details of S3- radicals and accompanying anions inside the cages have never been fully resolved from diffraction data.29-33 This is due to the fact, that only a fraction of SOD cages is occupied by S3- anions, which may adopt several orientations within cages, therefore not giving clear diffraction pattern.


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The properties of S3- radicals inside UM have been studied by spectroscopic techniques, mainly electron paramagnetic resonance (EPR),2,28,35-41 Raman3,35,37,40,42-44 and ultravioletvisible (UV-Vis) spectroscopy.3,28,35,36,43,45 Other techniques are also occasionally used, such as positron

annihilation,41

infrared,45

nuclear

magnetic

resonance,30,32,45

X-ray

photoelectron,44,46,47 and recently X-ray absorption spectroscopy (XAS).46,48,49 The results obtained with various techniques for UM vary in details, nevertheless some general picture can be achieved. Usually it is found that less than half cages are occupied by S3ions.30,31,36,37 UV-Vis, Raman and EPR signals from S3- species are rather similiar in various UM samples, however, in the latter case there are some differences regarding the broadening of EPR signal and recovery of g tensor components from it.2,36,37,39,41 For example Arieli et al. claimed the presence of more than one type of S3- sites, one giving only isotropic signal, and the other with rhombic g tensor structure resolved.36 S3- are regularly accompanied by S2radicals, which are yellow chromophore.35-37,40,43-48 and large admixture of the latter can make UM coloration green. Part of the sulfur present in UM is supposed to exists in the form of diamagnetic, colorless anions, such as S2- or SO42- ones.3,33,36,46-49 Even more exotic polysulfide or oxosulfur species are claimed to be present in certain UM (e.g. S4, S4-, S62-, S2O-),3,35, 43,44,46,48 notably in red or pink UM, where the nature of chromophore is still not fully clarified. As the unambiguous experimental characterization of UM is difficult, the assistance from computational approach is welcomed. So far mainly isolated S3- radicals, among other polysulfide species, have been studied computationally, including high level correlated methods.4,36,44,50-55 Arieli et al. supported their EPR studies by the density functional theory (DFT) calculations for both pristine S3- radical and (Na4S3)3+ cluster.36 With aim to model S3behavior in geological fluids, Tossell considered both single S3- and related polysulfide species at DFT and coupled cluster levels, along with DFT study on larger models, incorporating H2O molecules and Cu+ ions.50 S3- defects in NaCl and KCl have been studied using large cluster 5 ACS Paragon Plus Environment


models by Stevens et al.53 Despite these efforts, there is still a lack of computational data on polysulfide radicals embedded in aluminosilicate matrices. The atomistic simulations are invaluable tools in the characterization of zeolites and related nanoporous materials.56 With current development of computational resources the most frequently chosen computational approach to study electron structure of solids are periodic DFT calculations, usually relying on semilocal exchange correlation functionals and the approximate treatment of core electrons. However, the usage of cluster models for the specific issues of zeolites chemistry, like the description of defects, isolated catalytic centers or molecular species trapped in pores, can be still fruitful, particularly if more advanced treatment of electron correlations and/or all electron approach is applied.57-61 The aim of this work is to refine the structural properties of polysulfide radical chromophores embedded in SOD framework and help elucidating available spectroscopic data. For the first time, up to the author’s knowledge, periodic DFT calculations have been performed for the realistic models of UM lattices with blue S3- chromophore, followed by introductory study on more delicate issue of UM hosting yellow S2- chromophore. Full optimizations of structural models, namely both atom positions and unit cell relaxations, were performed. Then large clusters were cut of the optimized periodic structures and used in all electron calculations, employing hybrid functionals, to achieve insight into the electron absorption and paramagnetic resonance spectra.


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2. METHODS 2.1. Structural Models. The cubic crystal structure from ref. 32 were used as a starting point to construct the models for periodic DFT studies. In all cases Si:Al ratio was fixed to 1:1, with the alternating occupation of T sites by Si/Al assumed, i.e. obeying Loewenstein rule.34 All cations beyond aluminosilicate framework are Na ones. Hence the stoichiometry of SOD framework was always (Si6Al6O24)2-, and this repeating unit will be dropped in models’ labeling for the sake of brevity. Three types of extraframework stoichiometry were considered: (1), Na8S3Cl, (2) Na7S3 and (3) Na8(S3)2. In models Na8S3Cl and Na7S3 only half of the sodalite cages is occupied by S3- species, in line with some experimental data.30,36,37 The neutrality of the unit cell is achieved either by inserting Cl- ions (Na8S3Cl models), which are present in mineral lazurite samples,33 or by the removal of one Na+ ions (Na7S3 models). The latter Na7S3 stoichiometry is close to the value claimed by some authors to be typical for synthetic UM.1 The third class of models, with two S3- per single cubic cell, were developed to investigate the possibility of interaction between neighboring radicals, proposed upon certain EPR data.36 For each type of stoichiometry three different initial positions of S3- were considered: (i) in the middle of the cage, in 010 plane, (ii) in 110 plane, with lines connecting terminal S atoms roughly along the unit cell diagonal and (iii) in 110 plane, with lines connecting terminal S atoms being parallel to ab plane diagonal. Two models of the lowest energy of each (1)-(3) categories were further analyzed. For sake of completeness data for two the most stable (of four considered) S2- hosting models are presented, one with Na8S2Cl and one with Na7S2 extraframework stoichiometry, respectively. Large cluster models were cut from selected DFT optimized periodic models for spectroscopic studies. All atoms (Na and O) within 3.4 Å distance from S ones were included in cluster models. This distance limit corresponds to the sum of van der Waals radii of S and O atoms, and also roughly to the radius of sodalite cage. Si and Al atoms were removed and the 7 ACS Paragon Plus Environment


valencies of O atoms were saturated with H atoms, with O-H bonds oriented along former OSi/Al ones. In total 9 cluster models of stoichiometry [Na6-8(S3-/S2-)(H2O)15-20](5-7)+ were considered, with each cluster hosting single S3-/S2- chromophore (two cluster models were cut from Na8(S3)2-2 lattice and single cluster models were considered for the remaining periodic structures). More detailed information on structural models is featured in Supporting Information (see Tables S1 and S2), including geometry files attached. 2.2. Computational Methods. The DFT calculations on periodic models were performed with QUANTUM ESPRESSO code,62 using gradient Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional.63 The plane wave basis set was applied to valence electrons with energy cutoff set to 40 Ry, along with ultrasoft Vanderbilt pseudopotentials,64 downloaded from http://www.quantum-espresso.org, applied to core electrons. Brillouin zone was sampled by 2×2×2 k-points mesh. Both atomic positions and the cell parameters were optimized until the differences in the total energy and the atomic gradient norm dropped below 10-6 and 10-4 a.u., respectively. For the most relevant structures Γ phonon calculations were performed to ensure that the minimum energy was located. The DFT calculations on cluster models cut from the optimized periodic structures, involving single point energy calculations and response properties, were performed employing ORCA package.65 Ahlrich’s polarized triple zeta all electron Gaussian basis set (def2-TZVP),66 minimally augmented with diffusion functions67 for S atoms, were employed. The combined ‘resolution of identity’ and ‘chains of sphere’ type algorithms (RIJCOSX) for the fast evaluation of Coulomb and exchange integrals, respectively, were applied,68,69 with the automatic generation scheme of auxiliary basis sets.70 Hybrid Becke (3 parameter)-Lee-YangParr (B3LYP) functional71-73 was employed for EPR tensor calculations. The range separated ‘Coulomb attenuated’ version of B3LYP (CAM-B3LYP)74 was harnessed for computing of electron excitations within the time dependent DFT approach (with Tamm-Dancoff 8 ACS Paragon Plus Environment

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approximation, default in ORCA, switched off). Electron excitations were calculated employing vertical approximation, which, despite its limitations,75 is still standard approach for semi-quantitative treatment of UV-Vis spectra. Spin-orbit contributions to g tensor were calculated using mean field approximation, as implemented in ORCA.76 The results of test calculations, validating the choice of above computational approach, are presented and discussed in Supporting Information (Tables S3-5).



3. RESULTS AND DISCUSSION 3.1. Structural and Vibrational Properties. Selected optimized structures of S3- UM are shown at Figure 1. Cell parameters of DFT models are gathered in Supporting Information (Table S6). For comparison and testing purpose, stoichiometric SOD lattice were relaxed at DFT/PBE level. In that case the computed cell parameters perfectly match experimental ones.77 The direct comparison of computed and experimental unit cell parameters of UM should be done with cautiousness, given the limitations of both diffraction experiment and DFT models stressed above, nevertheless the agreement in cell volumes is very good. In all cases optimized unit cells deviates somewhat from ideal cubic settings. Both experiments and DFT predict UM unit cell to be expanded with respect to stoichiometric SOD. Si/Al-O bond lengths predicted for SOD and UM are virtually the same (about 1.75 and 1.64 Å, respectively), hence the lattice expansion stems from the mutual rotation of TO4 polyhedra. The latter is indicated by an increase in mean Si-Al-O angle (137° in SOD vs. 140-148° in UM), which correlates with lattice expansion in SOD type solids.9,78 Note that crystal structures, resolved within cubic symmetry, have unusually short mean T-O bond lengths and rather high T-O-T angles,30,32 whereas DFT values are more reasonable. The structural data for S3- radical are summarized in Table 1. In all UM models S-S bond lengths is nearly equal 2.0 Å, whereas S-S-S angle lies in the range of 108-114°. These values are very close to the experimental ones by Weller, who, upon combined XAS/neutron diffraction efforts claimed values of 2.05 Å and 110° for S-S bond length and S-S-S angle, respectively.8 DFT predicted bond length and angle for gas phase S3- are slightly larger than for entrapped radicals. The shortest distance between the nearest S3- radicals (either periodic images or the second radical in Na8(S3)2 models) in neighboring SOD cages is at least 6 Å. Terminal S atoms are coordinated by 2-4 Na atoms, usually with 1-3 of them in the distance range of 2.8-3.0 Å (see Supporting Information, Table S7). Middle S atoms are more loosely 10 ACS Paragon Plus Environment

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bound to 1-3 Na+ ions, with Na-S distance well above 2.9 Å. In total, S3- species have 3-5 Na+ ions in close vicinity (2.8-3.0 Å), and another 1-2 Na+ ions further (3.1-3.3 Å). Each terminal S atoms has 1-2 lattice O atoms in the vicinity of 3.1-3.3 Å. S3- species may adopt several orientation within SOD cage: e.g. (i) in Na8ClS3-1 both terminal S atoms points toward Na+ ion in the middle of 6T ring (Figure 1a), (ii) in Na7S3-1 one terminal atom is directed toward the empty 6T ring and the second toward the small void of 4T ring (Figure 1b), (iii) in Na8(S3)2-1 model, one terminal S atom points toward the Na+ ion in 6T ring, and the other one toward 4T ring (Figure 1c). Note that in the case of Na8(S3)2-1 both S3- chromphores in unit cell are symmetrically equivalent (the converged structure can be refined as a primitive cell of orthorhombic Cc group), whereas in Na8(S3)2-2 they are different.

Figure 1 DFT optimized periodic structures for selected S3- ultramarine models: (a) Na8S3Cl1, (b) Na7S3-1, and (c) Na8(S3)2-1. S-Na and S-O distances shorter than 3.4 Å depicted with sticks and dotted lines, respectively. Element coloring: S – yellow, Na – purple, O – red, Cl – green and Al/Si -dark/light blue. 11 ACS Paragon Plus Environment



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Table 1 Structural data for S3- in ultramarine models: distances (R, in Å), angles (θ, in °), S3planes Miller indices (hkl, rounded to the nearest small integers) and coordination number of S atoms by Na and O atoms (CNS-Na/O, within 3.4 Å radius). S1 and S3 are terminal atoms, S2 is the middle one. RS1-S2, RS2-S3, R(S1-S3)inter1 θS1-S2-S3 S3 hkl CNS1-Na, CNS3-Na, CNS2-Na RS1-S3 CNS1-O2 CNS3-O2 2 (S3 )gas 2.012, 2.012, 116.33 3.419 Na8S3Cl-1 1.996, 1.997, 7.086 110.05 (3 -3 2) 4, 2 4, 2 3 3.272 Na8S3Cl-2 1.996, 1.996 7.077 110.44 (1 1 0) 4, 3 4, 3 4 3.278 Na7S3-1 1.987, 2.019, 6.405 111.20 (0 1 1) 2, 2 4, 2 2 3.305 Na7S3-2 1.996, 1.997, 7.196 108.17 (1 -1 10) 4, 2 4, 2 1 3.234 Na8(S3)2-13 1.989, 2.004, 6.025 112.76 (4 9 -1) 3, 2 3, 1 1 3.325 (9 4 -1) Na8(S3)2-2 1.996, 1.996, 113.69 (1 5 0) 3, 1 3, 1 2 3.342 1.997, 1.997, 6.074 113.43 (1 6 0) 3, 1 3, 1 2 3.338 (6.074) 1The nearest inter-anion distance between S - occupying neighboring cages (S1 and S3 3 atoms). 2PBE/ma-def2-TZVP data computed with ORCA. 3Both S3- radicals in this model symmetrically equivalent (the optimized structure can be resolved as belonging to Cc space group). Model

Vibrational frequencies for gas phase and embedded S3- in selected periodic models are summarized in Table 2, with graphical representation of relevant modes postponed to Supporting Information (Figures S1-2). Computed harmonic DFT frequencies are close to the experimental values obtained from Raman measurements for ultramarines.1 The good agreement between DFT harmonic and experimental goes in line with data by Tossell, who claimed the anharmonic effect for S3- vibrations to be small.50 Gas phase and embedded S3- are predicted to vibrate with similar frequencies, the latter being slightly blue shifted due to shortening S-S bonds within SOD cage. 13 ACS Paragon Plus Environment


For gas phase S3- three vibrational modes are possible, for embedded ones more modes with notable S3- contribution are present, due to coupling with lattice motions. In Na8(S3)2-1 each mode is found doubly degenerate, due to ‘in-phase’ and ‘out-of-phase’ movements of two S3radicals in single unit cell. Whereas in Na8S3Cl-1 and Na8(S3)2-1 bands around 540 and 580 cm-1 correspond to symmetric and antisymmetric S3- stretching, somewhat different character is found in Na7S3-1: the lower mode has mainly character of longer S-S bond stretching (i.e. with S more tightly bound to Na ions), while the higher one is mostly due to stretching of shorter S-S bond. Apart from the bending and stretching S3- modes several low frequency lattice vibrations (below 200 cm-1) are found to contain partial S3- rotation or translation within SOD cage.

Table 2 The experimental and harmonic DFT frequencies (, in cm-1) for S3- modes, computed with ORCA (gas phase models, PBE and B3LYP functionals) and Quantum Espresso (periodic models, PBE functional). bending

symmetric-

antisymm.-

-stretching

-stretching

Exp.1

240

540

580

S3- (PBE)

217

514

550

S3-(B3LYP)

224

520

558

Na8S3Cl-1

233, 264

543

576

Na7S3-1

271

521, 530

572-597

Na8(S3)2-1

264, 267

539

565, 586

1Ref.

1.


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3.2. Electronic and Optical Properties. Electronic structure of gas phase S3- radical computed here at DFT level is consistent with previously reported results.4,50 The orbitals relevant for further discussion are shown at Figure 2 and their importance for optical spectra is presented in Table 3. Singly occupied molecular orbital (SOMO), for S3- in C2v point group, belongs to 4b1 representation. The only electron excitation of high intensity in optical range is due to electron transfer from 2a2 SOMO-1 to 4b1 SOMO. Other electron transitions in the range of 400-715 nm, even if not formally forbidden (in electric dipole approximation) are predicted by TD-DFT to be very weak in comparison to 2a2 → 4b1 transition. TD-DFT results agree nicely with available UV-Vis data for gas phase S3- ions (bands in the range 500-750 nm, with maximum intensity around 620 nm)79 and previous high-level correlated methods calculations (623 nm).4 In S3- ion negative charge is localized mainly at terminal S atoms, whereas spin is more uniformly distributed throughout S3- ion, with small excess on the central S atom, see Table 4.

Table 3 Optical/UV Electron excitations calculated at CAM-B3LYP level (at B3LYP geometry) for isolated S3- radical: wavelengths (λ, in nm), relative intensities (I/Imax, expressed as a fraction of the highest intensity) and dominant orbital contribution to a given transition, the maximum intensity transition depicted with bold. λ

I/Imax

Excitation type

S3- (C2v group, SOMO: 4b1) 715

0.1%

11a1 → 4b1 (99%)

715

forbidden

8b2 → 4b1 (99%)

641

100%

2a2 → 4b1 (95%)

419

3.4%

4b1 → 12a1 (74%)

344

0.3%

8b2 → 12a1 (60%) 15 ACS Paragon Plus Environment


Figure 2 DFT calculated molecular orbitals for gas phase S3- radical (C2v symmetry) around singly occupied molecular orbital (SOMO), which are the most relevant for its optical and magnetic properties. The blue coloration of S3- chromophore is mainly due to excitation from 2a2 SOMO-1 orbital to 4b1 SOMO.

Electronic structure of S3- embedded in SOD lattice are quite similar to these of free radical., see Figure 3. Singly occupied crystal orbital lies in the gap of SOD lattice bands, being spatially confined to S3- radical and closely resembling 4b1 SOMO in free S3- ion. Top doubly occupied state in UM valence band are also strongly localized at S3- radical, looking like the corresponding S3- molecular orbitals, with only small lattice O 2p contributions. Consequently, 16 ACS Paragon Plus Environment

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in further discussion these predominantly S3- localized states in periodic and cluster models will be labeled according to their representations in C2v symmetry S3- free ion.

Table 4 Atomic Löwdin charges (Q) and spin polarization (Nα-β)calculated for S3- periodic models at DFT/PBE level (if not stated otherwise). S1 and S3 are terminal atoms, S2 is the middle one. S1 S3 Q -0.43 -0.43 Nα-β 0.32 0.32 NaS3 Q -0.39 -0.39 Nα-β 0.29 0.29 Na8S3Cl-1 Q -0.29 -0.29 Nα-β 0.29 0.30 Na8S3Cl-2 Q -0.29 -0.29 Nα-β 0.29 0.29 Na7S3-1 Q -0.21 -0.37 Nα-β 0.37 0.21 Na7S3-2 Q -0.31 -0.31 Nα-β 0.27 0.27 Na8(S3)2-1 Q -0.29 -0.33 Nα-β 0.31 0.28 Na8(S3)2-2 Q -0.32 -0.32 Nα-β 0.28 0.28 1Calculated at PBE level with ORCA. S3-(gas)1

S2 -0.15 0.37 0.31 0.41 0.26 0.37 0.26 0.39 0.26 0.37 0.31 0.41 0.28 0.37 0.29 0.39

Charge and spin distribution are similar as in free S3- moiety, with negative charge located at terminal S atoms (about -0.3 and 0.3 for each terminal and middle S atom, respectively) and spin showing reverse localization (0.3 and 0.4 spin at each terminal and middle S atom, respectively). Only in Na7S3-1model, with strongly asymmetric S3- geometry, the charge and spin distribution is slightly different: terminal S atom with the shortest bond to middle S one accommodates half of negative charge of the other terminal S atom (about -0.2 and -0.4, respectively), with spin population equal to this of central S atom (both about 0.4).



Figure 3 Electronic structure of S3- ultramarine (Na8S3Cl-1 model): (a) total and atom projected density of states for periodic model, (b) orbital density (Γ point) of the states close to the Fermi level (set to 0), (c) corresponding molecular orbitals in cluster model. It can be seen that both in periodic and crystal models top valence states are predominantly located at S3- chromophore and closely resemble 2a2 and 4b1 (SOMO) orbitals for the isolated S3- ion (viz. Figure 2).

The cluster TD-DFT results for electron excitations are summarized in Table 5. The only electron excitations of high intensity are these with strong contribution of 2a2 → 4b1 transfer. 18 ACS Paragon Plus Environment

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In all cases but one, these computed excitations fall in range of 580-645 nm, in an excellent agreement with experimental wavelengths of maximum absorption (λmax) in optical spectra, namely 59536-62037 nm. Electron excitations for embedded S3- models usually cannot be factorized as electron transfer between a single pair of orbitals, as in the case of free S3- (Table 2). Apparently, the lowering of S3- symmetry, due to embedding in nonhomogeneous electric field of SOD cavity and/or geometrical distortion from ideal C2v symmetry, leads to more pronounced interactions between the states which symmetry has been disturbed by embedding. The discussed transitions in S3- UM models have rather strong 8b2 → 4b1 and/or 11a1 → 4b1 component. In a few cases, in addition to the main peak a satellite transition is predicted, having predominantly 11a1 → 4b1 nature (strongly blue shifted in comparison to free S3-), with some 2a2 → 4b1 share. Somewhat different is the case of (Na7S3)-1 model, with the most intense transition exhibiting significant blue shift. In this model, S3- is rather strongly deformed, with S-S bond differing by 0.03 Å. Consequently, the most intense electronic transitions originates not from pure 2a2 state, but the one having intrinsically strong 3b1 orbital admixture, namely with a π bonding between S 3p states along the shortest S-S bond (see Supporting Information, Figure S3). Interestingly, at least two experimental studies on UM reported the presence of band at 540 nm, apart from λmax around 600 nm, with no conclusive assignment.29,35 Upon this computational study such signal can be tentatively assigned to the transition in assymetrically distorted S3- species, like in (Na7S3)-1 model. Note, however, that also some tetrasulfur species may absorb light at wavelengths around 530 nm.51 All remaining TD-DFT predicted excitations in S3- UM models falls in UV range (usually well below 370 nm) and have very weak intensities. Therefore, it can be safely assumed that none optical transitions observed in UM below 500 nm can be ascribed to S3- species.



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In summary, top valence states in embedded S3- are well separated both spatially and energetically from lattice ones, hence their molecular-like character is largely retained and the maximum intensity optical transition (SOMO-1 → SOMO for free radical) in most cases falls in similar region as for the isolated S3- radical, with the exception of the most asymmetrically distorted radical in Na7S3-1 model.

Table 5 Selected optical electron excitations calculated at CAM-B3LYP level for cluster models (cut from PBE periodic models): wavelengths (λ, in nm), relative intensities (I/Imax, expressed as a fraction of the strongest intensity) and dominant orbital contribution (orbital naming as for C2v symmetry of gas phase S3-). Only the excitations with I/Imax larger than 5% presented, maximum intensity transition distinguished with bold. λ

I/Imax

Excitation type λ(nm) I/Imax Excitation type Na8ClS3-1 Na8(S3)2-1 645 100% 2a2 → 4b1 (88%) 638 11% 11a1 → 4b1 (81%) 8b2 → 4b1 (8%) 2a2 → 4b1 (15%) Na8ClS3-2 607 100% 2a2 → 4b1 (46%) 628 100% 2a2 → 4b1 (88%) 8b2 → 4b1 (36%) 11a1 → 4b1 (8%) 11a1 → 4b1 (15%) 623 10% 11a1 → 4b1 (90%) Na8(S3)2-2 1 2a2 → 4b1 (8%) 655/665 14%/11% 11a1 → 4b1 (64%/66%) Na7S3-1 2a2 → 4b1 (34%) 1 537 100% 2a2/3b1→ 4b1 (95%) 578/582 100% 2a2 → 4b1 (62%) Na7S3-2 11a1 → 4b1 (34%) 602 100% 2a2 → 4b1 (66%) 8b2 → 4b1 (31%) 1Data for two types of cluster models, each for inequivalent S - chromphore in Na (S ) -2 3 8 3 2 model.

3.3. Magnetic Properties. The principal elements of g tensor computed for S3- models at B3LYP level are shown in Table 6. The data predicted for gas phase and embedded S3- radicals are virtually identical, namely gxx = 2.002 (perpendicular to S3- plane), gyy = 2.056±0.003 (in 20 ACS Paragon Plus Environment

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S3- plane, perpendicular to C2 axis) and gzz = 2.040±0.002 (along C2 axis), with isotropic g being 2.032±0.001 (note that for S3- ion principal g tensor axis coincides with C2v group coordinates). The insensitivity of S3- g tensor components to the environment is understood, in view of the fact that non-degenerate spin polarized orbital is essentially localized at S3- ion, and well separated energetically from the other states. This makes spin-orbit contributions to g tensor rather small and nearly constant in both isolated and embedded S3- species. Predicted values are in excellent agreement with experimental data2,28,35-41 and previously reported computational results for S3- radical (isolated or in small cluster models).35,50 While EPR data reports essentially identical value of giso for S3- species in UM (giso = 2.028-2.032), the shape of spectra varies between experiments and the assignment of g components not always is possible. Usually, at room temperature only the averaged signal is observed, while three component structure may appear at lowered temperatures (4-77 K).2,36 However, UM samples with S3- species giving isotropic signal up to 4-8 K temperature were reported,36,39 as well as the one with rhombic structure of g tensor visible at room temperature.41 The observed averaging of EPR signal is explained either as a result of S3- rotation inside SOD cages,2,37 or spin coupling between radicals occupying neighboring cages.36 Both these possibilities will be discussed further. We started with investigation of magnetic coupling between the nearest neighboring S3- ions. For that purpose Na7S3-1 and Na8(S3)2-1 models were selected: the former model has the shortest intermolecular S-S distance among the models with single S3- per unit cell (i.e. distance between S1 atom of a given S3- species and S3 atom of its periodic image), the latter has the shortest S-S distance among models with two S3- ions per unit cell (and the shortest in general). To study the magnetic coupling between radicals in Na7S3-1 model, its unit cell were doubled along the direction of the shortest inter-radical S-S distance (i.e. along a direction). For periodic models with two S3- centers per (super)cell single point 21 ACS Paragon Plus Environment


energies were calculated for ferromagnetic (spin number S=1) and antiferromagnetic solution (net spin polarization set to 0), see Figure 4. For each considered model the unit cell energy was identical for both magnetic solutions (up to 10-3 kJ/mol), thus indicating no spin coupling between S3- paramagnetic sites in neighboring SOD cages. This results is somewhat expected, given discussed previously strong confinement of spin polarized states, their negligible overlapping with lattice states and the spatial separation of S3- ions (over 6 Å).

Table 6 g tensor calculated at B3LYP level for isolated S3- and for cluster models (cut from PBE periodic models): isotriopic value (giso) and main axis components (gxx/yy/zz). See Figure 5 for axis orientation. giso

gxx

gyy

gzz

S3-

2.033

2.002

2.038

2.059

Na8S3Cl-1

2.033

2.002

2.042

2.054

Na8S3Cl-2

2.032

2.002

2.041

2.053

Na7S3-1

2.033

2.002

2.038

2.058

Na7S3-2

2.032

2.002

2.041

2.053

Na8(S3)2-1

2.033

2.002

2.040

2.057

Na8(S3)2-2

2.031

2.002

2.039

2.053


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Figure 4 The spin density for the antiferromagnetic solution of the Na7S3-1 model, with cell doubled along a direction corresponding to the nearest intermolecular S-S distance (6.405 Å). Spin up and down densities depicted with blue and green, respectively. The energy of antiferromagnetic and ferromagnetic solutions are the same, which indicates no magnetic coupling between the radicals occupying adjacent cages.

The issue of dynamic averaging EPR spectra is somewhat more challenging computationally, as it would demand expensive ab initio molecular dynamics long time (up to nanosecond) runs, to study the problem in fully systematic manner. Nevertheless, qualitative understanding of S3mobility

within

SOD

cages

can

be

achieved

from

static

calculations.

The inspection of Table 6 shows that the largest deviation from giso occurs for gxx and gzz components, thus one could expect that the most important motion for the averaging of EPR signal would be 90º flipping along y direction (i.e. C2 axis), exchanging x and z axis, and hence the corresponding g tensor components. To estimate the energetic of this S3- flipping, for three periodic models (Na8S3Cl-1, Na7S3-1 and Na8(S3)2-1) S3- radical was rotated along its C2 axis by 45 and 90º (only one of two S3- species in case of Na8(S3)2-1 model), then the remaining atomic positions within the unit cell were relaxed with S3- ion kept fixed. The computed energy differences between fully optimized models and the ones with S3- rotated are shown at Figure 5. The largest flipping energy is found for Na8S3Cl-1 model, mounting up to 15 kJ/mol for 90º 23 ACS Paragon Plus Environment


rotated structure, where relatively large rearrangement of Na+ ions was needed to accommodate rotated S3- species. Much smaller energy, about 5 kJ/mol, were found for two other S3- UM models.. Although the above numbers should be taken with care (given limited DFT accuracy and that no thermal contribution to free energy at finite temperature are considered here), they show clearly that S3- motion can be either hindered or allowed, depending on the local Na arrangement, which should make impact on the averaging of EPR signal.

Figure 5 Estimation of S3- 90º flipping energy along y direction for selected periodic models. The principal g tensor axis orientation shown in the inset. Additionally flipping energy calculated for Na7S2 model is shown (angle between S2- main axis in optimized structure and upon arbitrary rotation of S2- radical is given).

To conclude this part it should be stressed that DFT calculations indicates that no magnetic coupling is expected between S3- radicals in neighboring SOD cages. One cannot exclude the possibility of spin coupling between S3- radicals and some nearer paramagnetic centers (e.g. Fe3+ ions partially substituting T sites) but this is beyond the scope of present study. The plausible mechanisms for reported differences in shape of EPR spectra, isotropic or not, is the presence of various S3- sites, where local environment (Na+ coordination) allows or not for fast movement of S3- radicals. 24 ACS Paragon Plus Environment

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3.4. Comments on S2- Ultramarines. The case of S2- UM appears to be harder task for DFT modeling, due to strong static correlation in S2- radical. Within C∞v group S2- shell formed by 3p orbitals has the occupation σ(3pz)2π(3px,3py)4π*(3px,3py)3σ*(3pz)0, hence doubly degenerated π*(3px,3py)3 should be populated by three electrons (i.e. equally weighted contribution (π*x)2(π*y)1 and (π*x)1(π*y)2 to the wavefunction), see Figure 6. The latter effect cannot be properly captured by single determinant methods, as the standard implementation of Kohn-Sham approach to DFT. On the other hand, the interaction of S2- with low symmetry environment may lifts π*(3p) orbital degeneracy and diminish multireference character, making DFT treatment at least qualitatively plausible. Even then, the quantitative DFT performance may be not fully satisfactory and strongly functional dependent (see Supporting Information). On the other hand, the successful description of S2- defect in alkali halides at DFT level was reported.80 Bearing in mind possible DFT limitations, we decided to follow with brief qualitative discussion of DFT results on S2- UM. Structural parameters for S2- models are gathered in Table 7, see also Figure 7. Similarly to S3- UM, the unit cell is somewhat expanded with respect to SOD lattice, and deviates slightly from cubic settings. S-S bonds slightly shortened (up to 0.01 Å) upon embedding in SOD cages, hence the small blue-shift in S2- stretching vibration (about 20 cm-1) with respect to DFT value for gas phase radical. Quite different S2- coordination was found for both models: (i) in Na8ClS2 model (Na4S2)3+ complex of approximately C2v symmetry, with two tightly bound Na+ ions bridging both S atoms, (ii) in Na7S2 model asymmetric coordination is found with more elongated Na-S bonds. The different Na-S binding is followed by the distinct charge and spin distributions, being asymmetric for Na7S2 model. The shortest distances between S2- occupying different SOD cages is always well above 7Å. The inspection of crystal orbitals shows that πg* states are separated energetically from the other ones and spatially confined to S2- radical, see Figure 8. Filled πu states are shifted down, overlapping in energy range with lattice O 2p states. 25 ACS Paragon Plus Environment


This may lead to more prominent mixing of πu states with lattice ones, though this effect depends on the functional (being smaller for more localized orbitals predicted with hybrid functionals), compare Figures 6 and 8.

Figure 6 DFT/CAM-B3LYP frontier orbitals for gas phase S2- radical and cluster model cut from periodic Na7S2 model. The incompletely filled π*(3p) shell makes the system multireference one and hence problematic for the accurate description at DFT level.


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Table 7 The periodic DFT/PBE data for S2- UM models: lattice constants (a, b, c, in Å), angles (α, β, γ, in °), cell volume (Vcell, in Å3), interatomic distances (R, , in Å, mean values for Al/SiO), mean Al-O-Si angles (θ, in °), S atoms coordination numbers (CN), DFT harmonic S2stretching freuqencies (, in cm-1), Loewdin charges (Q) and spin population (Nα-β). Data for gas phase S2- radical, computed with ORCA, added for comparison. Wavelengths of the most intense optical/UV transition (λ, in nm) are computed for cluster models in ORCA with CAMB3LYP functional. Na8S2Cl Na7S2 a, b, c 9.025, 8.973, 9.164, 9.118, 8.982 9.122 α, β, γ 90.09, 89.84, 89.79, 90.22, 89.85 90.24 Vcell 727.31 762.2 RAl-O/RSi-O 1.750/1.639 1.746/1.636 θAl-O-Si 139.48 145.75 RS1-S2 2.016 (7.663)1 2.010 (7.160)1 RS1-Na 2.736, 2.837, 2.904, 2.946, 2.936 3.142, 3.144 CNS1-Na 3 4 RS2-Na 2.743, 2.840, 2.874, 2.875 2.940 CNS2-Na 3 2 579 579 S1-S2 Q/Nα-β S1 -0.25/0.49 -0.29/0.40 Q/Nα-β S2 -0.25/0.49 -0.20/0.61 λ 289 383 S2- (gas) RS1-S2 2.023 (B3LYP), 2.020 (PBE) 562 (B3LYP), 564 (PBE) S1-S2 λ 415 S2 (Exp.) 5802 S1-S2 λ 4672 (gas)/3933 (in UM) 1The nearest interanion distance between S - occupying neighboring cages. 2Ref. 79. 3Ref. 3. 2



Figure 7 DFT/PBE optimized models for S2- ultramarine models (a) Na8ClS3 and (b) Na7S2. Element coloring as in Figure 1.


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Figure 8 Electronic structure of S2- ultramarines: total and atom projected density of states for (a) Na8ClS2 and (b) Na7S2 periodic models (Fermi level set to 0), (c) orbital densities (Γ point) of the crystal states with S2- πu and half-filled πg* character, (d) corresponding molecular orbitals in Na8ClS2 cluster model. Note that both crystal and molecular S2- πu orbitals have visible admixture from O 2p lattice states (unlike in Na7S2 model, viz. Figure 6).



The accurate prediction of spectroscopic properties at single determinant Kohn-Sham approach to DFT may be more problematic, due to reasons described above. The most intense optical excitations in S2- radical stems from πu → πg* transition. This transition in TD-DFT calculations is (i) too much blue-shifted well with respect to experimental data (i.e. 467 nm)79 for S2- in gas phase, (ii) matches perfectly data for S2- UM within Na7S2 model (this agreement may be accidental, though) and (iii) is blue-shifted toward UV region (below 300 nm) for Na8ClS2 model. The DFT fails to deliver reliable values of g tensor in S2- UM (see Supporting Information). Unlike in the case of S3- species, the g tensor values strongly depends on the applied functional. This issue can be ascribed to the DFT problems with proper handling of near degeneracy within πg* shell and the magnitude of arising spin-orbit coupling contribution to g tensor, evaluated in perturbative manner.76 The performance of latter approach should depend on πg* splitting, which vary between functionals (generally larger for hybrid functionals). Nevertheless, it is understood that near degeneracy of incompletely filled πg*states is the source of strong spinorbit coupling contribution to gzz , hence much larger g tensor anisotropy for S2- should be expected than for S3- species. As the splitting between S2- π* states is mediated by their interaction with local environment, EPR signal of S2- species should be much more environmental dependent than in the case of S3- ones. Indeed, EPR experiments on S2- defects in alkali halides reported g components strongly varying across these systems.81,82 The calculated flipping energy for S2- is nearly zero (see Figure 5), therefore this small radical is expected to rotate freely inside SOD cages, which should give averaged signal. Above conclusions are in qualitative agreement with the recent EPR study on S2- UM, where strongly broaden (likely due to fast transverse relaxation) signal was ascribed to S2-, with its anisotropic components resolved only at low temperature.35 Periodic DFT calculations performed for doubled cells with two S2- radicals shows that there is no magnetic coupling between S2- species 30 ACS Paragon Plus Environment

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in neighboring SOD cages (i.e. the energies of corresponding ferro- and antiferromagnetic solutions are the same, as in S3- UM case). At this point it is not possible to conclusively state, whether the issues with the reproduction of spectropscopic data for S2- UM, as well as the large discrepancy between data computed for both S2- UM models, stem from the DFT limitation or rather from the imperfection of applied models. Certainly, beyond DFT calculations, involving multireference correlated methods of quantum chemistry, are still needed for the accurate treatment of embedded S2- species and the assessment of DFT performance for such systems. Nevertheless, it is sound to say, that S2optical and magnetic properties should be more fragile to the environment than it is in the case of S3-radical, as they depend on the lifting πg* degeneracy and, in the case of UV-Vis spectra, perhaps also on πu interaction with lattice states. Possibly more than one optical band and/or EPR signal may arise from S2- species in UM, depending on their siting in SOD cages.



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4. CONCLUSIONS For the first time the systematic DFT study on S3- and S2- ultramarine pigments was performed, consisting of periodic structure optimization and all electron calculations of spectroscopic properties for large cluster models. Several types of S3- siting in SOD cages, not resolved well in available crystal structure, were identified. Different orientations of S3- ions within SOD cages are shown to have little effect on their electronic structure and spectroscopic properties,

due

to

rather

weak

interaction

between

S3-

and

lattice

states.

The most intense optical transitions (corresponding to 2a2 → 4b1 for S3- in ideal C2v symmetry) are predicted to be in range of 575-645 nm, in excellent agreement with experiments (595-625 nm). Only in the case when SOD lattice distorts S3- from C2v symmetry, prominent blue-shift is observed, which may explained 540 nm satellite band, reported in some experiments. Calculated g tensor principal components are nearly the same for all S3- models and match perfectly EPR measured values. Present study excludes the possibility of magnetic coupling between S3- centers in neighboring SOD cages, hence this factor should not affect the shape of measured EPR spectra. The differences in observed averaging of S3- EPR signal should be rather addressed to different dynamics of S3- ions in various sites. Though the quantitative predictions for S2- ultramarines at DFT level are less reliable due to near orbital degeneracy, one can see that the embedded S2- ions are more sensitive to the local environment than S3ones, predominantly due to various split of degenerated π* shell upon the interaction with lattice Na+ ions. S2- g tensor is expected to have strongly anisotropic structure, though the nearly free rotation within SOD cages should average it at ambient temperature.


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SUPPORTING INFORMATION DESCRIPTION Supporting Information Available: the overview of all structural models of S3- and S2- UM, test

calculations for validation of computational method (functional, basis set, cluster size), the detailed structural data for S3- UM models, atomic displacements for S3- vibrational modes in selected periodic models, figure of SOMO for Na7S3-1 model, the DFT g tensor for S2- UM models. This material is available free of charge via the Internet at http://pubs.acs.org.

ACKNOWLEDGEMENTS Prof. Ewa Broclawik from Jerzy Haber Institute of Catalysis and Surface Chemistry PAS and Prof. Piotr Pietrzyk from Faculty of Chemistry, Jagiellonian University are kindly acknowledged for their valuable comments. This work was supported in part by PL-Grid infrastructure.



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(17) Srdanov, V. I.; Stucky, G. D.; Lippmaa, F.; Engelhardt, G. Evidence for an Antiferromagnetic Transition in a Zeolite-Supported Cubic Lattice of F Centers. Phys. Rev. Lett. 1998, 80, 2449-2452. (18) Maeda, Y; Wakamatsu, T; Konishi, A; Moriwake, H; Moriyoshi, C; Kuroiwa, Y; Tanabe, K; Terasaki, I; Taniguchi, H Improper Ferroelectricity in Stuffed Aluminate Sodalites for Pyroelectric Energy Harvesting. Phys. Rev. Appl. 2017, 7, 034012. (19) Setter, N.; Mendoza-Alvarez. M. E.; Depmeier, W.; Schmid, H. Aluminate Sodalite Sr8[Al12O24](CrO4)2 – A New Ferroelectric Material. Ferroel. 1984, 56, 49-52. (20) Yang, Y; Jiang, XX; Gong, PF; Molokeev, MS; Li, XD; Li, YC; Wu, X 4; Wu, YC; Lin, ZS High Mechanical Strength in Zn4B6O13 with an Unique Sodalite-Cage Structure. RSC Adv. 2017, 7, 2038-2043. (21) Nabavi, M. S.; Mohammadi, T.; Kazemimoghadam, M. Hydrothermal Synthesis of Hydroxy Sodalite Zeolite Membrane: Separation of H2/CH4. Ceram. Int. 2014, 40, 5889-5896. (22) Khajavi. S.; Sartipi, S.; Gascon, J.; Jansen, J. C.; Kapteijn, F. Thermostability of Hydroxy Sodalite in View of Membrane Applications. Micropor. Mesopor. Mater. 2010, 132, 510-517. (23) Sibi, M. G.; Rai, A., Anand, M.; Farooqui, S. A.; Sinha, A. K. Improved Hydrogenation Function of Pt@SOD Incorporated inside Sulfided NiMo Hydrocracking Catalyst. Catal. Sci. Technol. 2016, 6, 1850. (24) Hiyoshi, N. Nanocrystalline Sodalite: Preparation and Application to Epoxidation of 2-cyclohexen-1-one with Hydrogen Peroxide. Appl. Catal A 2012, 419-420, 164-169.


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Figure 1 DFT optimized periodic structures for selected S3- ultramarine models: (a) Na8S3Cl-1, (b) Na7S3-1, and (c) Na8(S3)2-1. S-Na and S-O distances shorter than 3.4 Å depicted with sticks and dotted lines, respectively. Element coloring: S – yellow, Na – purple, O – red, Cl – green and Al/Si -dark/light blue. 76x211mm (300 x 300 DPI)

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Figure 2 DFT calculated molecular orbitals for gas phase S3- radical (C2v symmetry) around singly occupied molecular orbital (SOMO), which are the most relevant for its optical and magnetic properties. The blue coloration of S3- chromophore is mainly due to excitation from 2a2 SOMO-1 orbital to 4b1 SOMO. 81x127mm (300 x 300 DPI)


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Figure 3 Electronic structure of S3- ultramarine (Na8S3Cl-1 model): (a) total and atom projected density of states for periodic model, (b) orbital density (Γ point) of the states close to the Fermi level (set to 0), (c) corresponding molecular orbitals in cluster model. It can be seen that both in periodic and crystal models top valence states are predominantly located at S3- chromophore and closely resemble 2a2 and 4b1 (SOMO) orbitals for the isolated S3- ion (viz. Figure 2). 82x133mm (300 x 300 DPI)



Figure 4 The spin density for the antiferromagnetic solution of the Na7S3-1 model, with cell doubled along a direction corresponding to the nearest intermolecular S-S distance (6.405 Å). Spin up and down densities depicted with blue and green, respectively. The energy of antiferromagnetic and ferromagnetic solutions are the same, which indicates no magnetic coupling between the radicals occupying adjacent cages. 79x50mm (300 x 300 DPI)


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Figure 5 Estimation of S3- 90º flipping energy along y direction for selected periodic models. The principal g tensor axis orientation shown in the inset. Additionally flipping energy calculated for Na7S2 model is shown (angle between S2- main axis in optimized structure and upon arbitrary rotation of S2- radical is given). 67x50mm (300 x 300 DPI)



Figure 6 DFT/CAM-B3LYP frontier orbitals for gas phase S2- radical and cluster model cut from periodic Na7S2 model. The incompletely filled π*(3p) shell makes the system multireference one and hence problematic for the accurate description at DFT level. 82x105mm (300 x 300 DPI)


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Figure 7 DFT/PBE optimized models for S2- ultramarine models (a) Na8ClS3 and (b) Na7S2. Element coloring as in Figure 1. 76x138mm (300 x 300 DPI)



Figure 8 Electronic structure of S2- ultramarines: total and atom projected density of states for (a) Na8ClS2 and (b) Na7S2 periodic models (Fermi level set to 0), (c) orbital densities (Γ point) of the crystal states with

S2- πu and half-filled πg* character, (d) corresponding molecular orbitals in Na8ClS2 cluster model. Note that both crystal and molecular S2- πu orbitals have visible admixture from O 2p lattice states (unlike in Na7S2 model, viz. Figure 6). 176x114mm (300 x 300 DPI)


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Structural, Optical and Magnetic Properties of Ultramarine Pigments: A

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