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Modeling of Manganese Atom and Dimer Isolated in Solid Rare Gases: Structure, Stability, and Effect on Spin Coupling Nadezhda N. Kleshchina,† Kseniia A. Korchagina,† Dmitry S. Bezrukov,*,‡,† and Alexei A. Buchachenko*,‡ †

Department of Chemistry, M. V. Lomonosov Moscow State University, Moscow 119991, Russia Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Building 3, Moscow 143026, Russia

‡

S Supporting Information *

ABSTRACT: Structures and energies of the trapping sites of manganese atom and dimer in solid Ar, Kr, and Xe are investigated within the classical model, which balances local distortion and longrange crystal order of the host and provides a means to estimate the relative site stabilities. The model is implemented with the additive pairwise potential field based on the ab initio and best empirical interatomic potential functions. In agreement with experiment, Mn single substitution (SS) and tetrahedral vacancy (TV) occupation are identified as stable for Ar and Kr, whereas the SS site is only found for Xe. Stable trapping sites of the weakly bound Mn2 dimer are shown to be the mergers of SS and/or TV atomic sites. For Ar, (SS + SS) and (TV + TV) sites are close in energy, whereas (SS + TV) site lies higher. The (SS + SS) accommodation is identified as the only stable site in Kr and Xe at low energies. The results are compared with the resonance Raman, electron spin resonance, and absorption spectroscopy data. Reproducing the numbers of stable sites, the calculations tend to underestimate the matrix effect on the dimer vibrational frequency and spin−spin coupling constant. Nonetheless, the level of agreement is found to be informative for tentative assignments of the complex features seen in Mn2 matrix isolation spectroscopy.

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references therein) and more recent ionization studies,42−45 all the informative experiments probed the neutral dimer isolated in inert matrixes. These include ESR,10−12,14,15 magnetic circular dichroism,11 UV and visible absorption, and Raman8,9,13,16 spectroscopy. Thus, from the theoretical perspective it looks unavoidable to address matrix effects on the structure and spectra of the dimer. Interaction of two Mn(6S) atoms gives rise to 36 components of different electronic spin angular momentum quantum numbers S, MS arranged in the manifold of six distinct electronic states with S going from 0 to 5 and 2S + 1 multiplicities. All these states have Σ+ spatial symmetry and possess alternating ungerade/gerade parity: 1Σ+g , 3Σ+u , 5Σ+g , 7Σ+u , 9Σ+g , and 11Σ+u . The splitting between these states, labeled simply by S hereafter, closely follows Heisenberg spin-exchange model37,46

INTRODUCTION Chemical structures containing transition metal atoms, as clusters, complexes, organometallic frameworks, inorganic materials, and alloys, are of great interest in various respects, including catalysis and molecular magnetism, see, e.g., refs 1−4 for state of the art and recent challenges. Manganese atom is perhaps the one most frequently referred to in the context of molecular magnetism for its high electronic spin s = 5/2 (magnetic moment 5 Bohr magnetons) and extreme stability of the ground-state 3d54s2 electronic configuration.5−7 Structure and magnetic properties of the neutral MnN clusters had been the subject of numerous experimental8−19 and theoretical19−27 studies for the last four decades. While nontrivial dependence of magnetic properties of the gas-phase MnN clusters with N > 4 was successfully interpreted using density functional theory (DFT), the smallest clusters still remain puzzling for theory. Despite pioneering electron spin resonance (ESR) experiments in which the rare gas matrixes identified Mn2 dimer as a weakly bound antiferromagnetic (total spin S = 0) species,10,12,14,15 it had not been confirmed theoretically until the high-level correlated ab initio methods28−37 had revealed pitfalls in some DFT implementations. Since then, ab initio data on Mn2 dimer serves as an important benchmark for less rigorous theoretical approaches of handling more complex Mn-containing systems.38−40 The main reason preventing better alignment of theory and experiment for Mn2 is the lack of reliable gas-phase information. Except early mass spectrometric (see ref 41 and © XXXX American Chemical Society

ES = (J /2)[S(S + 1) − 2s(s + 1)]

(1)

where the spin coupling constant J is of the order of a few wavenumbers. The lowest excited states of the Mn atom resulting from the 4s → 3d, 4p electron promotion bear high spin and nonzero orbital angular momentum,47 which makes the structure of the dimer absorption spectrum very complicated.13,16 In the magnetic field applied in ESR experiment, each S state splits into Received: December 10, 2016 Revised: March 5, 2017 Published: March 6, 2017 A

DOI: 10.1021/acs.jpca.6b12444 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A 2S + 1 MS components, while hyperfine interaction with the 55 Mn nuclear spins I = 5/2 induces fine structure of the ESR band. As a result, each S ≠ 0 state manifests itself as an 11-line pattern. Three patterns were observed in solid rare gases and assigned to S = 1, 2, and 3.14 Matrix environment complicates the picture further. Raman spectroscopy revealed that Mn2 vibrational frequency varies by 15 cm−1 in Ar, Kr, and Xe and signaled the existence of at least two distinct stable trapping sites in the Ar matrix.13,16 The magnitude of these variations hints that the matrix effect is comparable with the internal electronic spin coupling, while the multiple sites introduce uncertainty in the spectroscopic assignments and temperature-dependent analysis of the spectral features. These complexities justify an attempt to model matrix isolation of the Mn2 dimer at the atomic level, the primary purpose of the present article. Reproducible observations of multiple trapping sites and, especially, reversible temperature-induced interconversion between them13 favor the theoretical interpretation of Mn2 embedded in the rare gases as a defect created in an ideal crystal rather than as a molecule dissolved in a finite cluster. Even more convincing arguments in favor of crystal model are provided by the comprehensive spectroscopic studies of a single Mn atom in solid rare gases.48−55 They revealed not only the existence of distinct trapping sites but also the crystal field splitting patterns specific for different high-symmetry placements in the crystal lattice. For this reason, we used the classical model already applied to interpret the trapping sites of atomic Yb in solid Ar.56 It compromises the local distortion of the crystal upon embedding the guest species and the long-range periodic order imposed by an ideal lattice. This model offers the possibility not only to find optimal structures of the trapping sites, but also address their stabilities by referring potential energies to a common energy zero and applying well-known thermodynamic concepts. Assessment of the model with respect to existing experimental data on manganese atom and dimer constitutes the secondary goal of this article. In the next two sections we present the model in more detail and describe the approximations used for interaction potentials. Then we apply the model to the simpler case of atomic Mn in solid rare gases to study the stable trapping sites and establish the link to experimental data. Trapping of Mn2 dimer is considered afterward, also in connection with experiments. Discussion and summary covering both atomic and diatomic cases follow.

We assume that accommodation of the guest X into internal subsystem (NB)B requires removal of some n RG atoms to create a comfortable vacancy. The system to be considered can be represented as ((X, NB − n)BNA)A, with the guest molecule X embedded and n RG atoms removed. If the pairwise potential approximation is adopted, its potential energy surface (PES) is U = UX + UXB + UXA + UBB + UBA + UAA

(2)

where the first term represents internal potential energy of the guest molecule, the second and third − interactions of the guest with RG atoms belonging to internal and external subsystems, respectively, while the rest three terms accumulate RG−RG interactions within or across respective subsystems. Each term depends on the coordinates of the corresponding atoms (which makes UAA constant). Minimization of U with respect to the degrees of freedom of the guest and coordinates of all movable RG atoms gives the lowest energy structure Umin(n). To compare the structures with different n, we set the common notional zero of energy as the ideal RG crystal plus the molecule X not interacting with it. The systems ((X, NB − n)BNA)A are linked to this limit by the virtual process of moving n RG atoms out of the sphere RA. The process costs nEat energy, where Eat is the atomization energy per atom, one of the basic parameters of an ideal crystal. Thus, the quantity Emin(n) = Umin(n) − nEat

(3)

defines the energies of all structures on the same scale independent of n. Let us consider the quantity defined by eq 3 as an approximation to Gibbs free energy change upon embedding of X into infinite crystal. For (almost) crystalline systems at low temperatures the dominant source of entropy change is associated with transferring X from the free to trapped state, but it may not strongly vary with n. Removal of n atoms requires the work against constant pressure, but this factor is negligible. Despite more accurate approaches for free energy estimations are possible and will be implemented in our future works, here we explicitly take Emin(n) as the free energy of formation of the corresponding structure. As such, this quantity can also be used to assess the stability of the structure with respect to decomposition into (the set of) other structures by using convex hull concept of the variable composition phase diagrams, see, e.g., ref 57. It implies that to be stable the structure with k RG atoms removed should have energy Emin(k) lying below any line on the (Emin, n) plane connecting the other two Emin(i), i ≠ k points. As a simplest example, one can assume that the most probable decomposition process is the disproportionation conserving n, i.e.,

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MODEL The starting point of our model is a fragment of an infinite ideal face-centered cubic ( fcc) rare gas (RG) lattice confined within the sphere of the radius RA with arbitrary chosen center and consisting of N RG atoms. Concentric sphere of the smaller radius RB confines NB lattice atoms. External NA = N − NB atoms located between the spheres always keep their lattice positions fixed, while all the manipulations, guest embedding, atom removal, and geometry optimization, are performed within the internal subsystem enclosed by the smaller sphere B. The underlying idea is transparent: external fixed atoms maintain the potential field corresponding to the long-range crystal order of the host, whereas internal atoms are allowed to adapt their positions to a local perturbation nearby the sphere center. The balance between two contributions is a matter of convergence with respect to enlarging RB and RA radii. It is convenient to designate the starting system as ((NB)BNA)A, where parentheses indicate the content of internal and external subsystems.

2((X , NB − n)B NA )A ↔ ((X , NB − n − 1)B NA )A + ((X , NB − n + 1)B NA )A

Then the stability criterion simplifies to Emin(n) < [Emin(n − 1) + Emin(n + 1)]/2 requiring Emin(n) to lie below the line connecting Emin(n − 1) and Emin(n + 1) points. One may note the close correspondence of this concept to the Frost diagrams for oxidation states of an element with the number of electrons playing the role of discrete “composition” variable.58 Implementation of our model requires, first, the choice of suitable potential field represented here through addition of the pairwise atom−atom potentials (eq 2). Second, ideal crystal simulations have to be performed to find out geometric lattice parameters and Eat. Search of the minimum energy structures for B


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The Journal of Physical Chemistry A each n = 0, 1, ... follows, subjected to convergence check with respect to RB and RA model parameters. Resulting Emin as a function of n identifies the stable structures for further analysis.

Table 1. Parameters of the Pairwise Interaction Potentials Used in This Worka

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INTERACTION POTENTIALS Pairwise Potentials. Mn−Mn. To describe Mn−Mn interactions in the six S = 0−5 states, we used the ab initiobased model from our previous works.34,37 The reference potential for the spin-polarized S = 5 state was obtained by means of the coupled cluster method with single, double, and noniterative triple excitations, CCSD(T),59 as implemented in the MOLPRO program package.60 In brief, all-electron augmented correlation-consistent aug-cc-pV5Z (AV5Z) basis set optimized for use with relativistic Douglass−Kroll−Hess correction61 was employed together with the bond function (bf) set centered at the middle of atom−atom distance.62 Among the reference orbitals generated by the restricted Hartree−Fock method 1s, 2s, and 2p ones were kept as core orbitals. Counterpoise correction was included. The multireference configuration interaction method was used to calculate the energies of the S < 5 states relative to that of S = 5 state. These differences were added to reference S = 5 CCSD(T) potential to give the set of potentials described in ref 37 as the “RCCSD(T)+bf” model. Mn−RG. Interaction between Mn and RG atoms gives rise to the single state of 6Σ− symmetry. Potentials for these states for RGs from He to Xe were calculated within the CCSD(T) method in the same way as described above for the spin-polarized Mn2 dimer. All-electron AV5Z bases were used for RG from He to Kr,63−66 whereas for Xe small-core ECP28MDF pseudopotential was used with the supplementary AV5Z basis.66 Except for He with the correlated 1s2 shell, outer ns2np6 shells for RG atoms were explicitly correlated in the CCSD(T) calculations. The Mn−RG potentials are tabulated in the Supporting Information to this article. RG−RG. The RG−RG pairs were described using wellestablished interaction potentials by Aziz and co-workers, namely, Ar,67 Kr,68 and Xe.69 Comparisons. To simplify numerical implementations, all the pairwise potentials were represented in the analytical form combining short-range exponential extrapolation, cubic spline interpolation, and long-range extrapolation to dispersion interactions. Details of the fitting function and all sets of its parameters are given in the Supporting Information to this article. Optimal trapping site structures Umin compromise the contributions of different pairwise interactions to the potential energy U (eq 2). Parameterizations for Mn−Mn and Mn−RG potentials described above were obtained using the same ab initio protocol. It was proven that very similar CCSD(T) calculations with AV5Z basis sets augmented by bond functions perfectly reproduce RG−RG potentials by Aziz and co-workers.70,71 Our pairwise PES can be therefore considered as consistent at reasonably high ab initio level. The range and strength of the distinct pairwise interactions are compared in Table 1, which presents σ, the position of repulsive potential wall at zero energy (taken as the dissociation limit), equilibrium distance re, and well depth De. The only point for comparison for Mn−RG potentials is provided by Collier et al.,55 who used empirical correlation relations72 to estimate equilibrium distances and bond energies. Their values presented in Table 1 in parentheses are very close to

a

species

σ (Å)

re (Å)

De (cm−1)

Mn−Mn (S = 0) Mn−Mn (S = 1) Mn−Mn (S = 5) Mn−Ar Mn−Kr Mn−Xe Ar−Ar Kr−Kr Xe−Xe Mn2−Ar (S = 5) Mn2−Kr (S = 5) Mn2−Xe (S = 5)

3.03 3.04 3.11 3.94 3.94 4.04 3.35 3.57 3.89 3.51 3.50 3.54

3.60 3.61 3.67 4.52 (4.48) 4.53 (4.54) 4.65 (4.63) 3.76 4.01 4.37 4.14 4.16 4.29

573 569 533 76 (85) 109 (115) 149 (157) 100 140 197 152 218 298

For Mn−RG, empirical values from ref 55 are given in parentheses.

the present ab initio results indicating high predictive power of the correlation relations used. The order of interaction strengths is always Mn−Mn > RG− RG > Mn−RG, but the potential range and well depth follow opposite trends along the RG sequence. The mismatch in equilibrium distances of Mn−RG and RG−RG pairs decreases from Ar to Xe, while the binding energy difference increases, as is shown pictorially in the Figure 1. The figure also marks the

Figure 1. Comparison of the Mn−Mn, RG−RG, and Mn−RG pairwise potentials referred to a common dissociation limit for RG = Ar (upper panel) and RG = Xe (lower panel). Vertical lines mark the distances between the two closest RG atoms in the ideal fcc lattice.

minimum distance between RG atoms in the ideal fcc lattice a/ √2, where the lattice parameter a is taken as optimized with the present potentials (see below). Comparison hints that single substitution of RG atom by Mn is more favorable for Xe, whereas for Ar matrix Mn may prefer larger vacancies. Mn2−RG Complexes. Within the pairwise potential approximation, Mn2−RG complexes possess T-shaped equilibrium structure and saddle point at the collinear Mn−Mn−RG arrangement. Equilibrium parameters of the effective Mn2−RG S = 5 potentials along the distance ρ between the center of Mn− Mn bond and RG atom in the T-shaped configuration are given in Table 1. In contrast to a single Mn atom, Mn2 dimer interacts with RG atom stronger than the two RG atoms do with each C


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The Journal of Physical Chemistry A other. Compatibility of the Mn2−RG and RG−RG interactions increases from Ar to Xe. However, as follows from Table 1, Mn2 itself geometrically better matches Ar lattice than the Xe one. The validity of the pairwise potential approximation for triatomic Mn2−RG systems is important for the present study. To assess it, we performed test ab initio calculations on the Mn2− Ar S = 5 complex using the same CCSD(T) approach as described above, except smaller triple-ζ AVTZ sets were adopted and bond functions were omitted. The sections of threedimensional PES on ρ for T-shaped and linear configurations and Mn−Mn separations equal to 3.5, 3.755, and 4 Å were obtained and compared to the sum of three ab initio pairwise potentials, recomputed for consistency with the same reduced basis sets. Ab initio calculations confirm the compliance of the Mn2−Ar PES topology with the pairwise potential prediction−global T-shaped minimum and shallow minimum (saddle, if the bending is taken into account) in the linear configuration. The differences between the ab initio and pairwise potentials shown in Figure 2

Table 2. Lattice Parameters a (Å) and Atomization Energies Eat (cm−1) of the Ideal Ar, Kr, and Xe fcc Latticesa Ar exptl present (2) CCSD(T) (2)79 present (2) + ZPE CCSD(T) (2) + ZPE79 CCSD(T) (2 + 3 + 4)79 CCSD(T) (2 + 3 + 4) + ZPE79

Kr

Xe

a

Eat

a

Eat

a

Eat

5.311b 5.200 5.213 5.263

646c 787 760 713

5.670d 5.553 5.556 5.579

936c 1096 1062 1039

6.132e 6.049 6.054 6.078

1328c 1524 1492 1474

5.273

697

5.591

1015

6.078

1448

5.251

709

5.573

983

6.089

1371

5.311

646

5.633

936

6.111

1328

a Numbers in parentheses in the first column indicate the orders of many-body interaction potential expansions: pairwise (2), three-body (3), and four-body (4). bRef 73. cRef 74. dRef 75. eRef 76.

ment, see Table 2. The second source of inaccuracy is the manybody interactions. Rościszewski and co-workers found that inclusion of three- and four-body interactions recovers almost all remaining difference between the experimental and ZPEcorrected pairwise potential results.79 This analysis allows us to claim that pairwise interaction model with the potentials by Aziz and co-workers provide very reasonable agreement with the ab initio-based pairwise description of the RG crystals and that the error due to omission of the many-body interactions does not exceed −1% for geometry and +8% for energy.

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ATOMIC Mn IN RG CRYSTALS In all the simulations, eq 2 with X = Mn, UMn = 0, and pairwise contributions described in the previous section were used to reproduce the PES. Lattice parameters a and Eat defined by the present model were chosen for consistency, as specified in the second line of Table 2. Also, it was checked (for Mn2@RG systems as well, see below) that the settings RB = 4a (NB ≈ 1000) and RA = 8a (N ≈ 8500 atoms) provide converged results for structure energies, geometries, and frequencies. For optimum structure search, gradient minimization of the potential energy U (eq 2) was used. Special care was taken to provide a representative sampling of the configuration space by initial conditions. As a first option, initial conditions were generated in a kind of “genealogic” way. Parent sample included the geometries with Mn atom embedded into tetrahedral and octahedral hollows of the fcc unit cell, as well as replacing one of the RG atoms, plus all these geometries with the RG atoms forming the first or the first and second Mn coordination polyhedra removed. This set of geometries gave, after minimization, the parental structures, for each of which initial conditions for descendent structures were generated by one-by-one removal of RG atoms closest to Mn, either keeping the positions of all movable atoms or displacing them randomly by up to 0.05 Å. As a second option, we used Monte Carlo-like sampling, placing Mn atom in the center of the sphere RA and removing n RG atoms closest to it one by one, each time adding random displacements to the coordinates of all movable atoms up to 0.005, 0.05, and 0.5 Å. Among all optimized structures ((Mn, NB − n)BNA)A, the one with minimum potential energy Umin(n) was selected (checking that it appears at least in two independent samples). Einstein frequencies of the Mn atom were calculated keeping all RG atoms fixed at their optimal positions, whereas the full harmonic vibrational problem was solved for all movable atoms within the internal sphere RB.

Figure 2. Difference between the ab initio (Uabi) and pairwise (Upair) potential energy surfaces of the Mn2−Ar complex. Solid and dashed lines correspond to the cuts in T-shaped and linear geometries, respectively; numbers indicate Mn−Mn distance (Å). Solid and dashed arrows point to the locations of the potential minima along the T-shaped and linear cuts, respectively.

indicate that the latter work reasonably well at least close to equilibrium of the Mn2−Ar complex. If one takes 15 cm−1 (10% of interaction energy) as a still acceptable error, pairwise approximation can be used safely if all separations between Mn and surrounding Ar atoms exceed 80% of the Mn−Ar equilibrium distance. RG Crystals. Description of the RG fcc crystal provides the basic parameters of our model and allows us to assess it more quantitatively. In particular, we are interested in two fundamental quantities, lattice parameter a and crystal atomization energy Eat. Minimization of the lattice energy with respect to a gave the values listed in Table 2 in comparison with experimental data.73−76 Our calculations underestimate the lattice parameters by 2% and overestimate atomization energies by 22, 17, and 15% for Ar, Kr, and Xe crystals, respectively. Detailed ab initio studies77,78 and, especially, that by Rościszewski and co-workers who used CCSD(T) method,79 make it possible to disentangle two sources of inaccuracy. The first one is the quantum correction to zero-point vibrational energy (ZPE). According to ref 79, it improves a from 1% (Ar) to 0.3% (Xe) and Eat from 10% (Ar) to 3% (Xe). Applying simple ZPE correction within Einstein theory,79 we achieved very similar degree of improveD


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The Journal of Physical Chemistry A

also emphasizes the first coordination polyhedra, cuboctahedron for SS, and truncated tetrahedron for TV structures. The local point symmetry groups are Oh and Td, respectively. While the symmetries of the stable structures are the same as can be expected from the purely geometric considerations,80 embedding of Mn causes significant local distortions of the host. Table 3 compares the geometries of the optimized stable

Figure 3 shows the energies of optimal structures Emin(n) (eq 3) found for Mn in Ar, Kr, and Xe crystals. One may note that

Table 3. Distances between Mn Atom (or Equivalent Position in the Ideal fcc lattice) and ni RG Atoms Forming First Three Coordination Polyhedra i in the SS and TV Trapping Sites (or Ideal Lattice), in Å shell i

Figure 3. Energies of the minimum structures Emin(n) for Mn atom embedded in Ar, Kr, and Xe crystals.

only Kr and Xe n = 1 structures fall in energy below the zero line representing free atom not interacting with the crystal. However, even these unlikely correspond to absolute energy minimum of the system. For a given RG the Mn−RG interaction is weaker than RG−RG one, so the absolute minimum should correspond to Mn adsorbed on the crystal surface.80 The convex hull concept implies that the structures with n = 1 and n = 4 are stable for all hosts. By construction, the structures keep the local symmetry of the crystal and can easily be interpreted in terms of lattice vacancies. At n = 1, Mn atom replaces one RG atom forming single substitution (SS) trapping site. The structure at n = 4 is formed by removing one vertex atom of the unit fcc cell and three neighboring ones from the face centers. Mn occupies the resulting tetrahedral vacancy (TV). SS and TV structures are schematically shown in the Figure 4, which

ni

Ar fcc

1 2 3

12 6 24

3.68 5.20 6.37

1 2 3

12 12 4

4.31 5.67 6.76

Mn@Ar

Kr fcc

Mn@Kr

SS, Single Substitution 3.88 3.93 4.04 5.19 5.55 5.55 6.43 6.80 6.83 TV, Tetrahedral Vacancy 4.31 4.60 4.56 5.63 6.05 6.01 6.75 7.21 7.19

Xe fcc

Mn@Xe

4.28 6.05 7.41

4.32 6.05 7.42

5.02 6.59 7.86

4.95 6.55 7.82

trapping sites with those of an ideal fcc lattice in terms of the distances between the Mn (or its equivalent position in an ideal lattice) and RG atoms forming the first three coordination polyhedra i = 1, 2, 3. Substitution of RG atom by Mn always leads to some swelling of the first coordination shell, in part compensated by slight contraction of the second. The degree of these lattice distortions decreases from 5% for Ar to less than 1% for Xe favoring single substitution for heavier rare gases. It features decreasing compression of the Mn−RG distances: the shortest Mn−RG distance corresponds to repulsive wall of the pairwise potential for Ar, almost matches σ for Kr, and falls in the attractive well region for Xe (see Table 1). Placement of Mn atom in the tetrahedral vacancy in Ar keeps the remaining lattice atoms almost (within 0.04 Å) intact still gaining attraction from the Mn−Ar interactions, which contract much less than at single substitution. In Kr and Xe, tetrahedral vacancy is too big; in contrast to SS substitution, the nearest coordination shells all shrink down, but the resulting Mn−RG distance is still longer than it would be in the gas phase. Atomic coordinates of the Mn@RG structures found here are presented in the Supporting Information to this article. These simple steric arguments explain the relative energies of the stable structures. For Ar, SS and TV accommodations are almost equally appropriate, with the latter lying higher in energy by 100 cm−1 only. The gap increases for Kr to 1460 cm−1 and further to 3200 cm−1 for Xe. Some of the early experimental works on Mn atom isolated in RG matrixes11,48,49 attributed the complexity of spectral features to the multiple trapping sites. Recent comprehensive absorption, excitation, and emission spectroscopic studies by McCaffrey and co-workers51−55 provided evidence that Mn occupies two stable sites in Ar and Kr (red and blue, according to the shift of absorption bands). One stable site was found for Xe, and it was aligned with blue ones in lighter RGs. Matching the literature Mg−RG potentials51 (later on, the empirical Mn−RG potentials mentioned above55) with the ideal RG lattices, McCaffrey and co-workers attributed blue and red sites to SS and TV occupations, respectively. More explicit evidence have been obtained in the recent analysis of excitation and emission spectra of D-symmetry electronic states.54,55 Observation of the narrow

Figure 4. Structures of the SS (a) and TV (b) Mn trapping sites with the polyhedra formed by the first coordination shell. Mn atom is represented by large sphere, RG atoms by small spheres. For TV site, removed RG atoms, which form tetrahedral hollow, are shown by empty circles. The reference unit fcc cell is represented by the cube. E


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The Journal of Physical Chemistry A bands associated with b4D, a4D, and a6D transitions in the blue site and b4D transition in the red site made it possible to resolve phonon structure and identify multiple zero-phonon lines assigned to the crystal field splitting induced by the host. The patterns found agree with the Td symmetry for the red sites and Oh for the blue sites. Though the lack of Mn−RG potentials for excited 6P state prevented us from calculating absorption band shifts, as we did before for Yb@Ar,56 agreement between simulations and experiment in the number of sites for each host and their assignment is convincing. The SS structure is found here to be present in all RGs, as was observed for the blue site, while the TV structure may not be detectable in Xe due to high energy and marginal stability, as confirmed experimentally for the red sites. The site attribution is confirmed by the crystal field splitting patterns. It is also instructive to note that, analyzing the phonon band structure, Collier et al. used the line shape function81,82 that contains, as the parameter, the effective phonon frequency ωph assumed to be identical for the ground and excited states.54,55 They found ωph equal to 32 and 12 cm−1 for Ar SS and TV sites, 25 and 10 cm−1 for Kr SS and TV sites, and 19 cm−1 for Xe SS site. Our vibrational analysis confirms that Mn frequencies in all stable optimized structures differ substantially. Taking for simplicity the Einstein frequencies, 55 and 30 cm−1 for Ar SS and TV sites, 52 and 24 cm−1 for Kr SS and TV sites, and 48 cm−1 for Xe SS site, one can see that their variation is in good correlation with that of the measured broadening parameters. Thus, phonon broadening can indirectly hint the structure of the trapping sites. One disagreement with experimental data should be mentioned. Taking absorption intensity as the measure of site population, McCaffrey and co-workers51 concluded that the blue SS site dominates for Kr, whereas Ar favors the red TV occupation. In contrast, our simulations place Ar TV site 100 cm−1 higher in energy than the SS one. First, absorption intensity may reflect not only the population of the given site but also its structure through the variation of transition dipole moment. Second, as pointed out in the previous section, our model overestimates atomization energy (Table 2). If we take Eat as it was measured, the Ar TV site goes down by 300 cm−1 with respect to SS one to become the ground. Kr and Xe sites preserve their order, but the gap between SS and TV structures is reduced to 980 and 2610 cm−1, respectively. Third, better approximations for structure free energies are to be invoked to verify the relative stabilities of the trapping sites obtained here.

generation of the descent structures, with and without random bias, were also made as described for Mn atom. In addition, five Monte Carlo samples were generated for each ((Mn2, NB − n)BNA)A system. Three samples used one-by-one removal of n RG neighbors of the dimer accompanied by random displacements of all remaining atoms up to 0.005, 0.05, and 0.5 Å, respectively. In two samples, n RG neighbors were removed at once with random displacements up to 0.05 and 0.5 Å, respectively. The number of starting structures in each sample increased from 5 to 10 and then to 20, with no new lowest energy structures found at the last step. In contrast to atomic trapping case, where both genealogic and Monte Carlo samplings gave identical results, the former failed to locate the lowest energy structures with n = 0, 1, 4, 5, and 9 for trapped dimer even with rather strong random bias. It hints that intuitive sampling through the high-symmetry configurations is not enough for thermodynamic stability analysis of the molecular trapping sites. The energy diagrams featuring Emin(n) are shown in Figure 5. Note that zero energy now corresponds to an infinite crystal plus

Figure 5. Energies of the minimum structures Emin(n) for Mn2 dimer embedded in Ar, Kr, and Xe crystals.

isolated Mn2 molecule. For Ar, two stable low-energy structures with two and six lattice atoms removed are evident. In what follows, we will label them as S2 and S6, respectively. The latter has the lowest energy, while the former lies 100 cm−1 above. Noteworthy, both structures have positive energy indicating that Mn2 dimer, like Mn atom, favors free (or adsorbed) state rather than trapping in crystal. Same structures are present in Kr, but now S2 goes down in energy to become the ground one, whereas S6 lies 1690 cm−1 above. Convex hull for Xe shows S2 as the single stable low energy structure. Dimer favors the trapping in both Kr and Xe crystals. An interesting case is presented by n = 5 point in the Ar diagram. The corresponding S5 structure resulting from removal of five Ar atoms, which lies 265 cm−1 above the ground S6 structure, is generally unstable, but withstands disproportionation process lying below the line connecting n = 4 and n = 6 points. Figure 6 schematically shows S2, S5, and S6 structures emphasizing the first coordination polyhedra formed by RG atoms closest to one or another Mn atom and indicating RG atoms removed. From the first glance, the structures may seem weird, but in fact they can be straightforwardly interpreted as the mergers of atomic trapping sites described in the previous section. Indeed, S2 structure can be viewed as the combination of two single substitutions within a unit fcc cell, one at vertex, another at the nearest face center. If one can imagine the conjugation of two cubooctahedra (one as plotted in the Figure

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Mn2 DIMER IN RARE GASES Energy and Structure of the Trapping Sites. The PES of the dimer can be represented as the generalized eq 2 U = UXY + UXB + UYB + UXA + UYA + UBB + UBA + UAA (4)

with X, Y = Mn and the same choice of all parameters as for the atomic simulations. Note that this equation describes the dimer in the particular spin state S depending on the choice of the UMnMn potential. Search of the minimum energy structures was performed with the ground-state S = 0 Mn−Mn potential. Genealogic sampling was performed as described for atomic case, with the difference that for each atomic parent configuration the dimer at equilibrium was oriented along three directions (001), (011), and (111) of the fcc lattice with adjustment of its position to maintain the symmetry. Update of the parent structure set and F


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Table 4. Distances between Each Mn Atom and Closest RG Atoms in the Atomic SS and TV Sites and Mn2 S2, S6, and S5 Structures (Å)a RG

Mn, SS

Mn, TV

Mn2, S2

Mn2, S6

Mn2, S5

Ar

3.88 (12)

4.31 (12)

3.90 (4) 3.94 (8) 3.95 (4) 3.96 (2)

4.05 (4) 4.22 (8) 4.41 (4)

3.71 (1) 3.84 (2) 3.94 (2) 4.08 (2) 4.26 (1) 4.29 (2)

Kr

4.04 (12)

4.56 (12)

4.30 (4) 4.46 (8) 4.62 (4)

Xe

4.32 (12)

4.95 (12)

4.05 (4) 4.06 (4) 4.13 (8) 4.18 (2) 4.28 (4) 4.32 (4) 4.45 (8) 4.54 (2)

a

Number in parentheses indicates the number of RG atoms located at a given distance.

embedding of the second Mn atom leads to slight swelling of the SS atomic site and remarkable contraction of the TV one. Atomic coordinates of the Mn2@RG structures discussed above are presented in the Supporting Information to this article. The relative stability of the dimer structures can also be understood with the reference to that of atomic ones. In Ar, SS site is slightly more stable than TV one, so S5 (SS + TV) lies higher in energy than S2 (SS + SS), while S6 (TV + TV) lies lower as gaining additional stabilization (−2Eat) by two lattice atoms remaining in their places. In Kr, atomic TV site has significantly higher energy than SS; S2 is therefore more stable, S5 disappears, and extra stabilization helps S6 to survive, but at rather high energy. In Xe, atomic TV site goes even higher in energy and becomes practically inaccessible. It leaves the combination of two SS sites in the S2 structure as the only option for Mn2 accommodation. We will return to energetic issues in the discussion below. We also note that all the Mn−Ar distances found exceed 80% of the gas-phase values indicating that three-body Mn2−Ar interactions unlikely introduce significant error. Dimer Vibrational Frequencies and Spin Coupling. All stable structures found for Ar, Kr, and Xe hosts exhibit noticeable distortion of the Mn−Mn distance r(Mn2) with respect to the gas phase. It should be accompanied by the changes in the vibrational frequency and spin coupling detectable by Raman, electron spin resonance, and electronic absorption spectroscopy. In this subsection, we will attest the properties of Mn2 dimer trapped in the stable sites and discuss the correlations with experimental data. In addition to simulations described above for the ground S = 0 spin state of Mn2 dimer, we studied S = 1 and S = 5 states by simply changing the Mn−Mn interaction potential in eq 4. Energy differences between the ground singlet S = 0 state and excited state S were estimated as vertical ΔEv(S), at the geometry optimized for the ground state, and as adiabatic ΔEa(S), the differences in energy between the structures optimized with S = 0 and S Mn−Mn interaction potentials. It was shown that the splittings of the S = 0 − 5 manifold in the isolated Mn2 dimer closely follow the Heisenberg model (eq 1) with some minor contribution of the term quadratic in S(S + 1).37 Heisenberg model implies the Landé interval rule for energy differences

Figure 6. Structures of the S2 (a), S5 (b), and S6 (c) trapping sites of the Mn2 dimer with the polyhedra formed by the first coordination shell. Mn atom is represented by large sphere, RG atoms by small spheres. Removed RG atoms, which form tetrahedral hollows, are shown by empty circles. The reference unit fcc cell is represented by the cube.

4, another the same, but shifted 0.5a, 0.5a in the right vertical plane), the result is exactly the same polyhedron as depicted in Figure 6 (we tried to visualize this merging on the Graphical Abstract to this article). Upon the combination of two SS atomic vacancies, their local Oh point symmetry group is lowered to D2h. S6 structure can be understood in the same way, as the merger of two TV atomic structures. Creation of two tetrahedral vacancies at neighboring vertices of the unit cell necessitates the removal of six RG atoms (not eight, as these tetrahedra share two RG atoms at the common edge) and provides the space for two Mn atoms. In turn, conjugation of two truncated tetrahedra forming the first atomic coordination shells (Figure 4) gives the shell with the local D2h symmetry depicted in Figure 6 (Graphical Abstract). Finally, S5 structure is the merger of SS and TV atomic sites when substitution and vacancy formation occur at neighboring vertices. Its local point group is Cs with only one symmetry plane containing the dimer. This low-symmetry structure was found only with strongly biased Monte Carlo sampling. The distances to the closest RG atoms are quantified in the Table 4. Despite the perfect cubooctahedral or tetrahedral symmetries are destroyed, the good correspondence exists between the shell in S2 structure, more compact “half-shell” in S5 structure and atomic SS site, on the one hand, and between the shell in the S6 structure, less compact “half-shell” in the S5 structure and atomic TV site, on the other. In average, G


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The Journal of Physical Chemistry A Table 5. Parameters of the Mn2 in the Gas Phase and in the Low-Energy RG Structuresas parameter

gas

Ar

Ar

Ar

Kr

Kr

Xe

state

phase

S2

S5

S6

S2

S6

S2

r(Mn2,S = 0) r(Mn2,S = 1) r(Mn2,S = 5) ωE S = 0

3.60 3.61 3.67 43.6

3.38 3.38 3.43 78.0

ωE S = 1 ωE S = 5 ωH S = 0 ωav S = 0 −Jv(1) −Jv(5) −Ja(1) −Ja(5)

43.4 41.2 43.6 43.6 3.8 2.9 3.8 2.6

77.9 75.9 84.8(57%) 80.1 6.7 5.5 6.7 5.1

3.49 3.50 3.55 59.5(73%) 53.4(27%) 59.5 59.8 57.7(3%) 58.3 5.0 3.9 4.9 3.6

3.42 3.42 3.48 67.8

3.48 3.48 3.52 66.4

3.50 3.51 3.56 58.4

3.68 3.68 3.73 48.3

67.6 67.4 73.4(12%) 68.6 6.0 4.9 6.0 4.5

66.5 67.7 69.4(85%) 68.3 5.2 4.1 5.2 3.8

58.4 58.9 59.2(94%) 59.1 4.9 3.9 4.8 3.5

48.2 49.1 52.4(65%) 49.4 3.1 2.4 3.1 2.2

a Distances in Å, frequencies and spin couplings in cm−1. For harmonic frequencies and for splitted Einstein frequencies the percentage contributions of Mn−Mn stretching into the normal mode are given in parentheses.

ES − ES − 1 = −JS

along S2, S6, and S5 sequence of the structures. But again, very significant matrix effect, from 40 to 100% with respect to the gas phase, is worthy of mentioning. Trapping Mn2 in Kr also compresses the dimer, but to a lesser extent, whereas the single S2 vacancy in Xe is too large and forces dimer to stretch. While it still shifts vibrational frequency to the blue by ca. 10%, −J parameter goes below its gas-phase value. Therefore, for equivalent structures S2 and S6 both frequencies and spin-coupling strength gradually decreases from Ar to Xe. As was mentioned in Introduction, Mn2 dimer in RG matrixes was studied by means of ESR,10−12,14,15 magnetic circular dichroism,11 UV and visible absorption, and resonance Raman8,9,13,16 spectroscopies. The Raman spectra provide vibrational frequencies of the dimer directly comparable with the present results. The resonance Raman measurements gave the Mn2 vibrational frequencies 76 cm−1 for Kr13 and 68 cm−1 for Xe.16 Simulations unambiguously assign them to the most stable S2 structure, which is characterized by the Einstein frequencies of 66 and 48 cm−1, respectively (see Table 5). Comparison confirms decreasing frequency shift from Kr and Xe and its “blue” sign, though our model tends to underestimate the shift and gives worse result for heavier Xe matrix. We are not aware of any experimental evidence for multiple trapping sites in Kr and Xe. In Ar, the situation is much more complex. Bier et al.13 identified three vibrational progressions at resonance Raman frequency 15516 cm−1 based on ωe = 59, 68, and 71 cm−1 and the forth one appearing only at 14840 cm−1 excitation with ωe = 84 cm−1. The first three were interpreted by studying the intensity dependence on temperature and laser irradiation. It was proven that the low-frequency progressions 59 and 68 cm−1 belong to distinct stable trapping sites, which undergo reversible interconversion upon heating and irreversible upon irradiation. The progression at 71 cm−1, which does not depend on temperature, was attributed to Mn2 excited to the second spin state S = 1. Following this interpretation, two stable sites found here, S2 and S6, could be related to those observed. This assignment suggests to link the measured frequencies 68 and 59 cm−1 to calculated 78 and 68 cm−1, i.e., overestimation of the experiment in contrast to Kr and Xe cases. To gain better insight on the frequency variations, we correlate them in Figure 7 with the static dipole polarizability of RG atoms as the measure of interaction

(5)

which approximates −J(1) = ΔE(1) or −J(5) = ΔE(5)/15 using both vertical and adiabatic quantities. Harmonic vibrational analysis was performed either for all movable atoms within the inner sphere RB or for two Mn atoms keeping all RG atoms fixed. The former calculations give the normal modes with the maximum contribution form the Mn− Mn stretching displacement and the corresponding harmonic frequencies ωH assigned to Mn2 vibration. Using the Mn−Mn stretching contribution as a weighting factor, we also computed ωav, frequency of Mn−Mn vibration averaged over all modes. From the calculations with fixed RG atoms, Einstein frequencies ωE were determined for S = 0, 1, and 5 using the same criterion. Table 5 provides the summary of all related theoretical results for the trapped dimer in comparison with the gas-phase data on Mn2 dimer obtained using the same interaction potentials and the same computational approaches. For a given structure, frequency estimates normally follow the ωH > ωav > ωE order, except for Ar S5 one with the reverse relation. The difference between ωH, ωav, and ωE is roughly proportional to the frequency shift with respect to the gas phase. In what follows we will mainly consider the simplest Einstein estimates commenting on others when required. In all three low-energy structures found in Ar, Mn−Mn bond is shorter than in the gas phase and degree of contraction decreases from S2 to S6 and further to S5 structure. This contraction manifests itself in the shift of the vibrational frequency, which increases from 50% to almost 100% depending on the tightness of the structure. S2 and S6 structures give single Einstein frequency, well separated by symmetry of the normal displacement with respect to D2h symmetry plane orthogonal to Mn−Mn axis that coincides with D∞h plane for bare Mn2. In the S5 structure of Cs symmetry, stretching of Mn−Mn bond is in resonance with Mn2 center-of-mass translation and two Einstein frequencies appear, as indicated in Table 5 together with percentage contributions of Mn−Mn stretching. When RG atoms are allowed to move, the dimer vibration further spreads over the phonon modes with the maximum contribution per mode 3% only. Spin-coupling parameter −J is a strongly decreasing function of the distance between the spin-bearing atoms. No wonder that it shows exactly the same trend as does the vibrational frequency H


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allowing the authors to deduce the single average estimate −J = 9 ± 3 cm−1. Earlier work gave −J = 8 ± 4 cm−1 for Kr and Xe only.10 Cheeseman and co-workers14,15 added the measurement in cyclopropane matrix to estimate the average equilibrium −J value as 8.5 cm−1. Magnetic circular dichroism spectroscopy11 estimated −J as 10.3 ± 0.6 cm−1 for Ar, but this result was questioned by Kirkwood et al.16 In the latter work, temperature variations of the rich structure of UV−visible absorption were investigated. Depending on assignment and temperature interval analyzed, different −J values were deduced, for instance, 10.8, 11.9, and 10.4 cm−1 for S = 2 in Ar, Kr, and Xe and 7.6, 10.2 cm−1 for S = 0 in Kr and Xe, respectively, with the average value 10 ± 2 cm−1.16 All the experimentally deduced parameters are higher than calculated (see Table 5) and do not discern the trend predicted by the matrix compression, i.e., remarkable decrease from Ar to Xe. As far as most of the experimental estimates are indirect and averaged, this likely reflects a deficiency of the parametrization for the gas-phase Mn2 spin state splitting rather than systematic problems in more subtle Mn2@RG simulations.

Figure 7. Vibrational frequencies of the Mn2 dimer in rare gases plotted with respect to static dipole polarizability of RG. Crosses: Einstein frequencies calculated for indicated structure, S2, S5, or S6. Squares: Experimental resonance Raman data for Ar (15516 cm−1 resonance), Kr, and Xe.13,16 Star: Experimental resonance Raman data for Ar (14840 cm−1 resonance).13

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DISCUSSION As advertised in the Introduction, advantage of the present model is the thermodynamic grounds allowing us to assess the stability of the trapping sites in the rare gas crystals. Using the common reference energy, the model permits one to compare the potential energies for different placements of the guest inside the lattice, including cavities, direct substitutions, and vacancies formed by missing lattice atoms. Such comparison can be supplemented by the analysis of phase stability for the systems of variable composition. In this way, the model is capable to identify a few well-defined structures that correspond to the stable trapping sites and can be used in subsequent in-depth spectroscopic or dynamical simulations. Ability of the model to predict multiple stable trapping sites is confirmed by the present study, which reproduces the numbers of sites observed for Mn in Ar, Kr, and Xe and Mn2 in Xe and predicts the second highenergy stable site for Mn2 in Kr. The case of Mn2@Ar is more controversial, but prediction of the third stable trapping site, in addition to two firmly identified, provides the tentative assignment of the measured Raman frequencies no less plausible than the existing one. Symmetry conservation maintained by the external crystal potential field represents another advantage of the model. Each trapping site is characterized, rigorously, by the point symmetry group related to local symmetry group of the unit lattice cell. It makes it possible to analyze the splitting of degenerate electronic or vibrational states of the guest and to assess their spectral manifestations, very much in line with the works by McCaffrey and co-workers on atomic Mn,54,55 but supplementing such analysis with the realistic estimates of the host−guest interactions. We consider present results as convincing demonstration for these advantages, admitting that further work is needed to invoke better approximations to free energy of the system, simply identified here with its potential energy (which to some extent excused by overall small entropy contribution for an ideal crystal). To the best of our knowledge, Mn2 is the first weakly bound metal dimer whose trapping sites in RG crystals are analyzed in detail. The striking result of this analysis is that the crystal tends to accommodate the dimer “per atom”, by merging two stable low-energy atomic sites within a single unit fcc cell. In particular, S5 structure stable in Ar has low symmetry and cannot be derived from high-symmetry placements of Mn2 dimer in an ideal lattice. Yet, the dimer maintains its molecular identity. With respect to

strength, in a way widely accepted for atomic guests.83,84 Frequencies calculated for S2 site almost linearly decrease from Ar to Xe, and so do the measured ones from Kr to Xe. The correlation (if trusted) leaves the single choice for S2 assignment in Ar: 84 cm−1. The fact that the corresponding progression was observed only for one resonance Raman band13 could be explained by the difference of site structures in one or another excited state. The frequency of S6 structure perfectly matches the experimental 68 cm−1, while “semistable” S5 structure can pretend to explain the experimental 59 cm−1 frequency. Moreover, S5 and S6 sites may well exhibit rapid interconversion that requires migration of one Mn atom from the position inside tetrahedral hollow to its empty vertex (see Figure 6). However, such a hypothetical assignment implies the energy order S6 > S5, in contrast to the calculations, S6 < S5. Intensity of the progression observed at 71 cm−1 and assigned to excited S = 1 state of the dimer did not vary with temperature.13 This may happen if the gain in thermal population is compensated by the site interconversion loss. Thus, this S = 1 excitation should be linked with the S = 0 state in the more stable site, i.e., that which gives 59 cm−1 progression. It is unlikely that S = 1 excitation of the dimer increases the frequency by as much as 12 cm−1, in contrast to a small decrease expected for the gas phase and found in the simulations, see Table 5. It should be also noted that features related to excited S = 1 state of Mn2 were not detected by the resonance Raman spectroscopy in other hosts13,16 and, unlike S = 2 and S = 3 ones, are hardly seen in the ESR spectra.12 If the above rationale based on polarizability plot (Figure 7) is wrong, then one can assign 71 cm−1 progression to S2 trapping site. In any case, we believe that existence of the third trapping site provides an alternative explanation of the origins of multiple resonance Raman progressions. Calculated harmonic frequencies shown in Table 5 provide better agreement with the measurements for Kr and Xe than the Einstein ones and do not contradict suggested assignments for Ar. Despite spin-coupling parameter −J had been estimated in several experimental works, it is difficult to assess quantitatively the results and trends found in our simulations. The ESR measurements, the most relevant in this respect, gave only eight bands related to the S = 1−3 states for all three RG matrixes12 I


Article


of the experimental frequencies still remains for Kr and Xe. Though some inaccuracies of the ab initio gas-phase Mn2 potential used here cannot be ruled out, other source of errors, like omission of the zero-point effect for the host, many-body interactions (not assessed here for Mn2 in Kr and Xe), phonon structure of the Raman bands, etc., may well contribute to this divergence and cannot be disentangled without extra theoretical efforts. In addition, one should be cautious with high expectations applying an ideal crystal model to experimental situations, which do not directly address the perfectness of the matrix. In contrast, persistent underestimation of the spin−spin coupling constant −J likely reflects the deficiency of the present gas-phase parametrization based on the multireference ab initio calculations.37

the gas phase, the bond of the dimer trapped dimer in Ar and Kr gets shorter and spin−spin interaction stronger, whereas the stretching frequency increases in all hosts. Moreover, our model provides an evidence that the binding energy of two Mn atoms also increases upon trapping. For instance, the energy gain of Mn2 formation in the S2 site from two Mn atoms in the SS sites, Mn(SS) + Mn(SS) → Mn2(S2) can be estimated as De(Mn2@RG) = Emin(Mn2@RG; S2) − 2Emin(Mn@ RG; SS) + De(Mn2)

where the first two terms are the trapping energies of the dimer and the atom depicted in Figures 5 and 3, respectively, while the third one, binding energy of Mn2 in the gas phase (570 cm−1), accounts for the difference in two zero-energy levels. Effective binding energy amounts to 1250 cm−1 for Ar, 960 cm−1 for Kr, and 610 cm−1 for Xe, all well in excess of the gas-phase value. The largest value for Ar is in line with the comment “that Ar was the most difficult of the solid rare gas hosts to achieve atomic isolation” for Mn due to dimerization.51 Stabilization of Mn2 dimer in S6 or S5 sites formed by Mn(TV) + Mn(TV) → Mn2(S6) or Mn(SS) + Mn(TV) → Mn2(S5) fusion in Ar, corrected to two or one Eat for Ar atoms remaining in the lattice, is equal to 1570 or 1200 cm−1, respectively. These estimates not only reveal the strengthening of Mn−Mn bond upon trapping but also indicate the potential use of our model for thermodynamics of slow processes associated with matrix restructuring and relaxation. Accommodation of the Mn2 dimer in RG crystals can be compared with that of the Na2 dimer with stronger bonding: De ≈ 5900 cm−1, re = 3.08 Å, ωe = 159 cm−1.85 Though no consistent thermodynamic stability analysis was performed for sodium dimer, Gervais and co-workers86 considered five of its symmetric placements (a)−(e) in Ar lattice to compare their potential energies and the matrix shifts of the X−A and X−B electronic transitions. They found the lowest energy and the most realistic shifts for (a) site, which is equivalent to S2 Mn2 site in our notations. Noteworthy, other four structures suggested for Na2 were included in our genealogic samples as parent ones and none of them appeared as the lowest energy structure at corresponding n. Our S6 and S5 structures were not considered in ref 86. This limited comparison hints that S2 accommodation is preferred by diatomics whose bond length is much shorter than RG−RG lattice distance (Na2@Ar, Mn2@Xe). More detailed study of various trapped dimers using, e.g., thermodynamic stability analysis, suggested here, is of interest for understanding generic trapping site structures and their sensitivity to intra- and intermolecular potentials. Theoretical results are compared with the Mn2 vibrational frequencies measured by resonance Raman spectroscopy. If predicted frequency shifts in Ar reasonably agree with the measured ones enabling us to hypothesize on assignments, significant underestimation is found for Kr and Xe. To check its origin, test calculations were made with the UMnMn pairwise potential scaled by a factor λ and stretched by an increment Δ added to equilibrium distance. Larger interaction strength and longer equilibrium distance both push the frequency up, but not to an extent enough to reproduce the experiment. For instance, at λ = 1.5 and Δ = 0.15 Å (50 and 4% variation of the original ab initio parameters), Einstein vibrational frequencies amount to 86, 77, and 61 cm−1 for S2 sites in Ar, Kr, and Xe hosts, respectively. Providing that our tentative assignment of the measured 84 cm−1 progression to S2 site in Ar is correct, 8 and 10% underestimation

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SUMMARY AND CONCLUSIONS The classical model compromising the local distortion of the RG matrix by embedded guest with the long-range ideal crystal order was proposed for studying the matrix isolation species. The model enables one to find the structures of the trapping sites, rank them in energy, and assess their thermodynamic stability. The principal advantage of the model is the possibility to identify a few symmetrical structures for subsequent in-depth spectroscopic or dynamical simulations. The model was applied to the trapping sites of Mn atom and its dimer using the consistent ab initio-based pairwise potential energy surface. Good agreement with experimental findings for atom was achieved. All stable trapping sites found for weakly bound Mn2 dimer were interpreted as the mergers of atomic trapping sites. Only one of them, S2, corresponds to an evident symmetric placement of the diatom in the fcc lattice. Extensive sampling of the configuration space was in need to identify the other two, S5 and S6. It was found that trapping in Ar and Kr hosts compresses Mn2 dimer causing blue frequency shift and increase of the spin− spin coupling constant with respect to the gas phase. Slight elongation of the Mn−Mn bond in Xe leads to weaker spin coupling, but still higher frequency. These findings, together with the presence of multiple trapping sites in Ar, are in line with experimental observations. At the quantitative level, however, the calculations underestimate the frequency shifts in Kr and Xe and predict too weak spin−spin coupling for all RG hosts.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.6b12444. Tabulated ab initio Mn−RG interaction potentials and analytical fits to all pairwise potentials used in this work (PDF) Geometries of the Mn@RG stable site structures and relevant fragments of ideal RG lattices in XYZ format (PDF) Geometries of the Mn2@RG stable site structures and relevant fragments of ideal RG lattices in XYZ format (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. J


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The Journal of Physical Chemistry A ORCID

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Alexei A. Buchachenko: 0000-0003-0701-5531 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank Dr. John G. McCaffrey for inspiring comments and suggestions and Dr. Andrei V. Scherbinin for discussions.

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